Multiscale and probabilistic modelling of micro ... · Multiscale and probabilistic modelling of micro electromechanical systems PROEFSCHRIFT ter verkrijging van de graad van doctor

Multiscale and probabilistic modelling

of micro electromechanical systems



PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema,

voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 12 oktober 2009 om 15.00 uur

door

Clemens Vitus VERHOOSEL

ingenieur luchtvaart en ruimtevaart

geboren te Diessen.

Dit proefschrift is goedgekeurd door de promotoren:

Prof. dr. ir. R. de Borst

Prof. dr. ir. M.A. Gutiérrez

Samenstelling promotiecommissie:

Rector Magnificus Voorzitter

Prof. dr. ir. R. de Borst Technische Universiteit Eindhoven, promotor

Prof. dr. ir. M. A. Gutiérrez Technische Universiteit Delft, promotor

Prof. Dr.-Ing. E. Ramm University of Stuttgart

Prof. dr. S. Krenk Technical University of Denmark

Prof. dr. ir. H. Askes University of Sheffield

Prof. dr. V. Deshpande Technische Universiteit Eindhoven

Prof. dr. ir. D. J. Rixen Technische Universiteit Delft

Keywords:

Multiscale modelling, cohesive zone modelling, partition of unity method,

stochastic finite element methods, micro electromechanical systems

Acknowledgement:

This research is performed within MicroNed, part of the BSIK research pro-

gram of the Dutch government.

Copyright © 2009 by Clemens V. Verhoosel

Printed in The Netherlands by Ipskamp Drukkers

ISBN 978-90-79488-75-9

Preface

Minitiaturisation of components is a trend observed in many in-

dustries, and micro systems technology is recognised as a niche mar-

ket. In order to strengthen its role as a player in this market, in 2004 the

Dutch government initiated the MicroNed program. In this program, uni-

versities, research institutes and companies collaborate to develop a solid

knowledge infrastructure for micro systems technology.

After having performed my Master’s on stochastic fluid-structure interac-

tions at the Engineering Mechanics group at the Faculty of Aerospace En-

gineering in 2005, I was enthused to continue conducting research. The

MicroNed project “Stochastic Analysis of Micro Electromechanical Systems”

offered me the possibility to continue working on stochastic finite element

methods, while having to master the field of numerical fracture mechanics.

Over the last four years, the broadness and versatility of the field of en-

gineering mechanics has become obvious to me. I have been lucky enough

to be able to cover quite a few aspects of the field, and am looking forward

to discover many more in the years to come. I am very grateful to my pro-

motors, René de Borst and Miguel Gutiérrez, who always encouraged me to

explore new research topics. Their inspiring enthusiasm for mechanics, and

science in its broadest sense, has been a source of motivation.

The work that you will find in this dissertation would not have been pos-

sible without the help of quite a few people. The many discussions with Joris

Remmers and Doo Bo Chung really helped me making a smooth start with

my Ph.D. project. The number of discussions with Joris has grown exponen-

tially over the past couple of years and have always been very constructive.

In particular, I am grateful that he offered me the possibility to work with his

partition of unity code. I also would like to thank Erik-Jan Lingen and Gertjan

van Zwieten for resolving many programming issues and for assisting me in

improving my programming skills. Joost van Bennekom is acknowledged for

providing the experimentally obtained images presented in Chapter 6. For the

electrostatic pull-in problem studied in Chapter 8, my thanks go to Stephan

Hannot for the useful discussions.

My sincere thanks also go to Thomas Hille, Marcela Cid Alfaro and Wij-

nand Hoitinga for the many discussions and nice time we had while sharing

an office. I would also like to express my gratitude to Thomas Scholcz, Timo

van Opstal and Edwin Schimmel for the discussions we had regarding their

Master’s projects. Carla Roovers and Harold Thung are acknowledged for the

excellent support they provided. Finally my gratitude also goes to all my col-

leagues for the many fruitful discussions and the good working atmosphere.

Clemens Verhoosel

Delft, September 2009

Voor Simone

Contents

1 Introduction 1

1.1 Miniaturisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Micro electromechanical systems . . . . . . . . . . . . . . . . . . . . 2

1.3 Computational challenges for micro electromechanical systems 3

1.4 Scope and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Notational issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Partition of unity-based fracture modelling of piezoelectric ceramics 9

2.1 Fundamental assumptions and limitations . . . . . . . . . . . . . . 10

2.2 The partition of unity method for electromechanical systems . . 12

2.3 Constitutive behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Algorithmic aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Inter- and transgranular fracture in piezoelectric polycrystals 31

3.1 Microscale finite element model . . . . . . . . . . . . . . . . . . . . 32

3.2 Constitutive behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . 34



4 Multiscale modelling of fracture in piezoelectric microsystems 61

4.1 Multiscale constitutive modelling . . . . . . . . . . . . . . . . . . . 63

4.2 Finite element formulation . . . . . . . . . . . . . . . . . . . . . . . . 75



5 Dissipation-based arc-length control for the simulation of failure 89

5.1 Path-following in quasi-static solid mechanics problems . . . . . 90

5.2 Energy release rate path-following constraint . . . . . . . . . . . . 91

ix

Contents



6 Characterisation of microstructural randomness 115

6.1 Characterisation of the microstructural geometry . . . . . . . . . 116

6.2 Local representation of the microstructure . . . . . . . . . . . . . 131

6.3 Homogenisation of the random fields of material properties . . 134

6.4 Parametrisation of the random fields of material properties . . . 143

7 Partition of unity-based stochastic fracture modelling 145

7.1 Stochastic finite elements for ultimate load computations . . . . 146

7.2 Sensitivities computation . . . . . . . . . . . . . . . . . . . . . . . . 152


8 Stochastic analysis of the electrostatic pull-in instability 167

8.1 Deterministic pull-in problem . . . . . . . . . . . . . . . . . . . . . . 168

8.2 Sensitivities computation . . . . . . . . . . . . . . . . . . . . . . . . 176


9 Conclusions and recommendations 185

A Path-following constraints for prescribed displacement problems 191

B Discretisation of the electrostatic pull-in problem 193

Bibliography 195

Summary 205

Samenvatting 209

Curriculum Vitæ 213

x

Chapter 1

Introduction

Miniaturised components have changed our way of living, which is

most evidently illustrated by the development of the microprocessor.

The first electronic computers† used vacuum tubes as switches. As a con-

sequence, they occupied complete rooms and were only available for a very

small community. The development of the transistor and its incorporation

in integrated circuits has downscaled the size of the switches by several or-

ders of magnitude. This miniaturisation has lead to microprocessors used

in desktop computers, laptops, phones and many other customer electronic

devices. The enormous impact of all these devices on everyday life is beyond

doubt.

1.1 Miniaturisation

The microprocessor is probably the most prominent example of a minia-

turised component, but is certainly not the only one. Many microscopic elec-

tric systems are nowadays commercially available. Examples of such com-

ponents are ink jet print heads, high-frequency switches and accelerometers.

Although many miniaturised components are electric, miniaturisation is also

used for non-electric devices. Typical examples of such devices are the micro

reactor and micro truster.

These are only a few examples of devices where downscaling has taken

place. Miniaturisation of components has been a global trend in industry

over the past decades and will likely continue at an even stronger pace. Where

state-of-the-art technologies nowadays commonly carry the label "micro", the

†The British Colossos (1944) and American ENIAC (1945) are nowadays recognised as the

first two electronic computers.

1

Introduction

next step towards "nano" is already made. This continuing trend of miniatur-

isation is undeniable, but where does it come from?

An important driving force for downscaling is the reduction in cost price

per functional unit that can be achieved. Where the first electronic computer

in the United States cost half a million dollars in 1946 (this is without in-

flation correction!), personal computers are now as cheap as a few hundred

dollars. The reason for this enormous cost reduction is partly due to the

miniaturisation of the components. Only little material is required to make

the small devices and also the amount of material required for packaging is

very limited. The reduction in price per functional unit and emergence of an

enormous range of novel customer electronic devices has lead to mass pro-

duction. These, often waver-based, production processes further reduce the

cost per unit.

Another driving force for miniaturisation is a more physical one. For ex-

ample, while downscaling a specimen, volume related physical effects (e.g.

gravity) decrease with an order of three, while surface related effects (e.g.

pressure) only decrease with an order of two. As a result of these different

scaling factors, the behaviour of small scale devices differs from that of large

scale devices (Wautelet, 2001). A device might be inefficient (or not work at

all) on the macroscale, but can be efficient on the microscale.

Although miniaturisation has many advantages, some difficulties are asso-

ciated with it as well. Primarily, the controllability of production processes of

microscale components is considerably more difficult than that of traditional

macroscale production processes. This leads to devices with relatively many

(and relatively large) imperfections, which can have a significant impact on

the performance and reliability of such devices.

1.2 Micro electromechanical systems

This thesis focuses on micro electromechanical systems (MEMS). This is a

class of micro systems where interaction between mechanical fields (displace-

ments, strains, stresses, etc.) and electric fields (electric potential, electric

field, electric flux density) is used to give a system functional properties. A

subdivision of MEMS can further be made by distinguishing systems using

electromechanical materials and devices where the materials are not elec-

tromechanically coupled, but where the coupling is achieved by the design of

the system.

2

Computational challenges for micro electromechanical systems

Nozzle

50 μm

Piezoelectric component

Diaphragm

Ink supply

Electrodes

Figure 1.1 Schematic representation of a miniaturised printer head (left)and microscopic image of the piezoelectric components used for the actua-tion of the device (right).

A typical example of the first class of MEMS is the miniaturised ink jet

printer head, which is schematically shown in Figure 1.1. A piezoelectric film,

i.e. a material with electromechanical coupling, is used to deflect a membrane

and push ink out of a reservoir and project it onto a piece of paper. The de-

flection of the membrane is achieved by application of a voltage over the

attached electrodes. The motivation for size reduction of these components

is the potential reduction in price by using waver-based manufacturing. The

printing quality of the device is likely to be enhanced (up to 2400 dpi) since

smaller droplet volumes can be achieved in combination with a higher noz-

zle density. Moreover, the performance of the device is improved by the

allowance of a higher operating frequency.

A typical example of a device belonging to the second class of MEMS is

the electrostatic bridge, as schematically shown in Figure 1.2. In this device,

an electric potential difference is applied over the gap. Upon increasing the

voltage, the charges on both walls increase, consequently also increasing the

electrostatic forces. At a certain voltage, known as the pull-in voltage, the

upper beam hits the bottom electrode. Such a system can for example be

used as a (high-frequency) switching device or microscopic actuator.

1.3 Computational challenges for micro electromechanical systems

Numerical models have aided in the design of almost all complex structures.

The prediction of both the performance and reliability of these structures

has led to more efficient and robust designs. Computational modelling of

3

Introduction

Micro bridge

10 μm

Rigid electrode

Figure 1.2 Schematic representation (left) and microscopic image (right) ofa capacitive micro electromchanical component.

MEMS poses several additional challenges. The most important of which are

discussed in the subsequent sections.

1.3.1 Multiphysics modelling

MEMS are by definition components designed to exploit the coupling between

electric and mechanical fields. In order to study the behaviour of such com-

ponents, incorporation of both these fields in a computational model is in-

evitable. As a consequence, generally a strongly coupled multiphysics prob-

lem needs to be considered.

In this thesis two kinds of electromechanical coupling are considered. In

the first case, the fields are coupled as a consequence of the constitutive

behaviour of a piezoelectric material. The main challenge in this kind of

problem is the design of novel constitutive models to mimic experimentally

observed phenomena. In particular, the description of crack nucleation and

propagation in a piezoelectric medium is a relatively unexplored topic of in-

terest. The second type of coupling is caused by electrostatic effects. In

that case, generally multiple electric and mechanical subdomains are cou-

pled, leading to an electrostatic problem with a moving boundary. The devel-

opment of efficient and robust computational models for such free-boundary

problems is an ongoing research topic.

Although the models in this thesis only incorporate mechanical and elec-

tric fields, it should be emphasised that consideration of additional physical

fields might be required for appropriate description of certain phenomena.

A magnetic field is probably the most obvious additional physical field that

4

Computational challenges for micro electromechanical systems

can be involved, but also thermal fields or polarisation fields† can be of cru-

cial importance to model certain aspects of MEMS. Moreover, especially in the

case of the electrostatic type of MEMS, it might be necessary to model the

medium in which the component is submerged.

1.3.2 Multiscale modelling

In traditional structures, the characteristic dimensions of the device are typ-

ically orders of magnitude larger than these of the microstructure. With the

downscaling of devices this separation has largely vanished and hence a more

direct influence of the microstructure is experienced by miniaturised compo-

nents.

From a computational point of view this means that the influence of the

microstructure must be incorporated in the numerical model. On one hand

the usage of analytical constitutive laws to represent the complex microstruc-

ture often leads to inaccurate results. On the other hand, full-resolution mod-

elling of the microstructure is often impractical due to the computational ef-

fort involved. Incorporation of the microstructure in numerical simulations,

while keeping the computational effort limited, is a topic which has gained a

lot of attention over the past few decades.

The development of efficient models to capture multiscale effects is one of

the main concerns in this thesis. It should, however, be emphasised that the

real challenge in multiscale analyses lies in the identification of the dominant

physical phenomena of interest on each of the scales. Experiments across the

different length scales play a crucial role in this identification process.

1.3.3 Modelling of microscale randomness

Closely related to the previous computational challenge is that of the influ-

ence of microscopic imperfections. Although these imperfections will affect

the performance of a macroscale device, the random character of these imper-

fections is filtered out due to the length scale difference between the device

and the imperfections. In other words, when performing measurements on

†The polarisation describes the alignment of electric dipoles and is as a consequence ameasure of the degree of piezoelectricity (Jaffe et. al, 1971). Description of the polarisation

by means of a field is for example useful when examining domain switching (Zhang and Bhat-tacharya, 2005), i.e. the reorientation of the polarisation direction in regions of uniformly

oriented electric dipoles.

5

Introduction

a number of macroscale devices with microscale imperfections, the results

will practically coincide. However, in the case of MEMS this changes. Since

the separation of the length scales between the device and imperfections is

considerably smaller, the randomness of the imperfections is reflected in the

performance of the device. When performing experiments with such devices,

a significant spread in results will be seen as a consequence of small scale

imperfections.

The necessity of the incorporation of randomness in a computational

model in the case of MEMS is considerably larger than in the case of mac-

roscale devices. On the one hand, this requires the characterisation of the

microscale randomness. Experimental observations need to be translated

into random fields in order to incorporate the randomness in computational

models. On the other hand, the computations themselves need to be capable

of dealing with random input data.

1.4 Scope and outline

Numerical prediction of the reliability of micro electromechanical compo-

nents is the main topic of interest in this thesis. Obviously, this research

field cannot be covered in a single thesis. Despite that, the various methods

introduced should provide insight in the most important aspects that need

to be taken into account when using computational models to gain insight in

the reliability of micro electromechanical systems.

This thesis is comprised of nine chapters. In Chapter 2, the partition of

unity method is applied to model fracture in piezoelectric ceramics. Numer-

ical simulations on macroscale specimens are performed to demonstrate the

applicability of the method on that length scale. In Chapter 3, an interface

elements-based cohesive zone model is introduced to model piezoelectric

fracture in microscale polycrystals. In Chapter 4, a constitutive multiscale

model is introduced that couples the two models discussed in Chapters 2

and 3. This multiscale framework is used to efficiently model fracture in

micro electromechanical components, i.e. components with a length scale in

between the two length scales considered before. The arc-length method used

for the simulations across all the scales is then discussed in Chapter 5. Chap-

ter 6 focuses on the characterisation of imperfections at the microscale and

discusses a homogenisation framework to derive expressions for the random

fields for the bulk and cohesive properties. Using these properties, stochastic

6

Notational issues

finite element simulations are performed to gain insight in the reliability of

miniaturised components in Chapter 7. In Chapter 8, a reliability analysis is

performed on a different type of electromechanical problem, the electrostatic

pull-in problem. Finally in Chapter 9, conclusions are drawn and recommen-

dations are made.

1.5 Notational issues

In this thesis two types of notation are used. In the case that a continuum

formulation is considered, index notation is employed. In this notation, ten-

sors are printed in regular font with roman indices. The order of the tensor

is determined by the number of indices. Unless otherwise specified, Einstein

summation† is assumed over repeated indices. In the case that a finite el-

ement formulation is considered, matrix-vector notation is used in order to

stay as close as possible to the actual implementation. In this notation, bold

symbols are used to indicate vectors and matrices. Voigt notation‡ is used to

represent higher-order tensors in matrix-vector form.

†Let ai and bi be two first-order tensors with i = 1,2. The dot product of these tensors is

then written using Einstein notation as aibi, which should be interpreted as a1b1 + a2b2.‡Let Aij be a symmetric second-order tensor with i = 1,2. The Voigt form of this tensor is

then a vector given by A = (A11, A22, A12). Occasionally, extra weighing factors are applied to

the off-diagonal terms, e.g. A = (A11, A22,2A12), which is then indicated in the text.

7

Chapter 2

Partition of unity-based fracturemodelling of piezoelectric ceramics

Over the past decades numerical simulation of fracture in piezoelec-

tric ceramics has primarily been based on linear elastic fracture me-

chanics models (Pak, 1992; Suo et. al, 1992; Sosa, 1992). An overview of these

methods can be found in (Qin, 2001). The use of either impermeable or per-

meable boundary conditions has been studied extensively. In the case of an

impermeable crack, charge free boundaries are used, whereas in the case of

a permeable crack, continuity requirements for the electric field and electric

flux density are employed. The permeable crack assumption was demon-

strated to be the most appropriate (Shindo et. al, 1997; Gao and Fan, 1999).

The definition of a failure criterion that correctly mimics the influence of an

electric field has also been addressed frequently. The fracture criterion pro-

posed by Park and Sun (1995) has been demonstrated to be in good agreement

with experimental observations. Improvements to this fracture criterion by

incorporation of nonlinear effects have been suggested (Gao et. al, 1997; Ful-

ton and Gao, 1997) as well as models for simulating fatigue in piezoelec-

tric ceramics (Arias et. al, 2006). Recently, the partition of unity concept has

been employed for the enrichment of crack tip fields in piezoelectric ceramics

(Béchet et. al, 2009).

The above-mentioned studies have primarily focussed on the study of

fracture in relatively large specimens, i.e. specimens with dimensions in the

order of centimetres. In that case, the size of the process zone, i.e. the zone

in which gradual degradation of the material takes place, is negligible com-

9

Partition of unity-based fracture modelling of piezoelectric ceramics

pared to the size of the specimen. Linear elastic fracture mechanics† is in

these situations a very useful tool for modelling fracture, since the material

in the vicinity of the crack tip can be assumed to behave linearly. When the

size of the specimen is downscaled, as is the case for MEMS, the process

zone remains of the same order of magnitude, whereas the specimen size

can decrease with one or more orders of magnitude. The process zone is

then no longer negligible compared to the specimen size. As a consequence

the assumption of linear material behaviour in the vicinity of a crack tip can

no longer be made, which restricts the applicability of linear elastic fracture

mechanics. A cohesive zone approach, which incorporates material nonlin-

earities in the region around a crack tip, is then more appropriate to mimic

the fracture process.

In this work, a partition of unity-based cohesive zone formulation is used

to model fracture in piezoelectric ceramics (Verhoosel et. al, 2009b). Elec-

tromechanical constitutive laws are used to describe the constitutive behav-

iour of the specimens. Although this method is primarily useful at small

length scales, it can be applied at larger length scales as well. This comes

at the cost of increased computational effort, since small meshes need to be

considered to appropriately discretise the small process zone. In this chapter,

the macroscopic experiments as discussed in Park and Sun (1995) are consid-

ered as a benchmark for the proposed model. Application of the model to

fracture in miniaturised components is the topic of interest of Chapter 4.

2.1 Fundamental assumptions and limitations

The problems studied in this work are treated within the context of classical

mechanics, with the most important assumptions being that the specimen

sizes are considerably larger than the atomic length scales and velocities are

considerably smaller than the speed of light. Under these conditions, the

behaviour of an electromechanical continuum can generally be described by

means of three fields: a displacement field, an electric field and a magnetic

field. On the one hand these equations obey Newton’s second law. On the

other hand, these fields are governed by the Maxwell equations.

This work is restricted to the static analysis of electromechanical prob-

lems, thereby assuming all rate-dependent terms in both Newton’s second

†In the context of this dissertation, linear elastic fracture mechanics refers to fracture

analyses based on a linear description of both the mechanical and electric fields.

10

Fundamental assumptions and limitations

law and Maxwell equations to be negligible. In addition it is assumed that

the studied materials are not magnetic, which (in combination with the as-

sumption of negligible rate terms) makes the magnetic field independent of

the other two fields. Except for some of the problems considered in Chap-

ter 5 and the problem studied in Chapter 8, displacements and displacement

gradients are assumed to be small. This allows for the formulation of the

coupled equilibrium equations on an undeformed domain.

As already outlined in the preamble, a cohesive zone model is adopted

to describe the behaviour of a crack in a piezoelectric continuum. Such an

approach assumes that the accumulated damage in the microstructure can

be lumped in a zero area crack surface on the macroscale. In other words,

from a macroscopic point of view the crack is not smeared out over a finite

volume. In Peerlings et. al (1996) it is shown that insight in this localisation

phenomenon under quasi-static loading can be obtained by studying the phe-

nomenon under dynamic loading. In the dynamic case, the occurrence of a

zero volume localisation zone is closely related to the question whether the

medium is dispersive† as outlined by Sluys and de Borst (1994). In contrast to

an elastic medium, a piezoelectric medium with linear constitutive behaviour

is shown to be dispersive (Auld, 1973). However, due to the large separation

of the acoustic and electromagnetic wave speeds, the inherent capability of

such media to regularise the formulation is negligible. This means that an

internal length scale exists, but that it is too small to solve the observed spu-

rious mesh dependencies (de Borst, 2004) in finite element discretisations

using practical meshes. From a macroscopic viewpoint it is therefore reason-

able to assume the damage to localise in a (zero volume) surface. Moreover,

it is emphasised that the use of continuum damage models for piezoelectric

materials requires the introduction of an artificial internal length scale using

e.g. a gradient enhanced description (Peerlings et. al, 2002). The results as

reported in Yang et. al (2003) and Yang et. al (2005) might suffer from spuri-

ous mesh dependencies as a consequence of a missing (or unclear) means of

regularisation.

†A medium is called dispersive when the velocity of the waves travelling through it de-

pends on the wave number. In non-dispersive media, the shape of these waves is not altered.

11


Ω+

Ω−

ΓΦΓu

Γt Γq

Γ

x1

x2

n

Γd

Figure 2.1 Schematic representation of an electromechanical domain Ω witha crack Γd.

2.2 The partition of unity method for electromechanical systems

The partition of unity approach for modelling cohesive fracture (Wells and

Sluys, 2001; Moës and Belytschko, 2002) is commonly used to simulate crack

growth in arbitrary directions in a solid material. Application of the parti-

tion of unity method to crack propagation problems in which multi-physical

phenomena are incorporated has recently been studied in e.g. Réthoré et. al

(2008) and Kraaijeveld et. al (2009), where the influence of a fluid on crack

propagation is considered.

The partition of unity-based cohesive zone formulation is derived for a

two-dimensional piezoelectric body, Ω ⊂ R2, subject to mechanical and elec-

tric boundary conditions as schematically shown in Figure 2.1. A crack, rep-

resented by the internal boundary Γd, splits the body in two parts, Ω+ and

Ω− (satisfying Ω = Ω+ ∪ Ω−). The discussion is here restricted to a single

crack, but can be extended to the case of multiple cracks (Remmers et. al,

2008a). The formulation is extendible to the three-dimensional case (Moës et.

al, 2002), but the implementation of such an extension is cumbersome.

2.2.1 Kinematical formulation

The state of the body in Figure 2.1 is determined by a displacement field,

ui (with i = 1,2), and electric potential field, Φ. A linear description of the

kinematics of the body is employed, hence assuming small displacements

and displacement gradients. Upon formation of a crack, Γd, discrete jumps

in both the displacement field and electric potential field occur. These jumps

12

The partition of unity method for electromechanical systems

are caused by the decreased stiffness and permittivity of the fractured ma-

terial. Appropriate description of these jumps requires the formulation of a

discontinuous basis for both fields, given by

ui = ui +HΓdui;

Φ = Φ +HΓdΦ,(2.1)

with HΓd the Heaviside function, defined as

HΓd(x) ={

1 ∀x ∈ Ω+

0 ∀x ∈ Ω− . (2.2)

Both fields in equation (2.1) are decomposed in a continuous part (denoted

by �) and a discontinuous part (denoted by �). The displacement jump �ui�

and potential jump �Φ� over the crack are given by

�ui�(x) = ui(x) ∀x ∈ Γd;

�Φ�(x) = Φ(x) ∀x ∈ Γd.(2.3)

Under the assumption of linear kinematics, the infinitesimal strain tensor (or

Cauchy strain tensor) is regarded as an appropriate measure for the deforma-

tion of the bulk material. This infinitesimal strain and corresponding electric

field (i.e. also defined under the assumption of small displacements) are given

by

εij = 1

2

(∂ui∂xj

+ ∂uj∂xi

);

Ei = − ∂Φ∂xi

.

(2.4)

In this thesis, the finite element method is used for the discretisation of

both the electric and mechanical field. It was demonstrated by Babuška and

Melenk (1997) that a discontinuous field f (x) can be discretised using con-

tinuous finite element shape functions φi(x) in combination with multiple

enhanced basis functions γj(x) according to

f (x) = φi(x)[ai + γj(x)aij

], (2.5)

with ai and aij being the nodal degrees of freedom. Discretisation of the

fields in equation (2.1), which requires only one enhanced basis functionHΓd,

yieldsu(x) = N(x)a +HΓdN(x)a;

Φ(x) = M(x)b+HΓdM(x)b,(2.6)

13


where the displacement field and electric potential field are described in

terms of nodal displacement vectors, a and a, and nodal potential vectors,

b and b. The matrices N(x) and M(x) are arrays containing the shape func-

tion. Similarly as the approximation of the displacement field and electric

potential field, the strain field and electric field (2.4) are expressed in terms

of the nodal vectors as

ε(x) = B(x)a +HΓdB(x)a;

E(x) = C(x)b+HΓdC(x)b,(2.7)

where Voigt notation is used to reduce the order of the rank two strain tensor

to yield the engineering strain ε = (ε11, ε22,2ε12). Note that in the definition

of the engineering strain, the shear component is multiplied by a factor of

two in order to let it be the work conjugate of the Cauchy stress, as will be

demonstrated in the next section. In equation (2.7), the matrices B(x) and

C(x) contain the gradients of the finite element shape functions.

2.2.2 Electromechanical equilibrium equations and boundary conditions

In the absence of body forces and charges, the displacement field and elec-

tric potential field as given in equation (2.1) are governed by the quasi-static

equilibrium equations∂σij

∂xj= 0;

∂Di∂xi

= 0,

(2.8)

with σij and Di denoting the Cauchy stress and electric flux density†, respec-

tively. Note that the repeated indices imply summation over that index. In

order to be solved, these equilibrium equations are supplemented with the

boundary conditions

ti = ti ∀x ∈ Γt; ui = ui ∀x ∈ Γu;

q = q ∀x ∈ Γq; Φ = Φ ∀x ∈ ΓΦ,(2.9)

with ti = σijnj and q = −Dini being the traction and surface charge density,

respectively. The weak form of both partial differential equations (2.8) is then

†In literature, the “electric flux density” is often called the “electric displacement”. In thisdissertation, the term “electric flux density” is preferred, in order to avoid confusion with the

mechanical displacements.

14

The partition of unity method for electromechanical systems

obtained as ∫Ω

σijδεijdΩ +∫Γd

tiδuidΓ =∫Γt

tiδuidΓt ;∫Ω

DiδEidΩ +∫Γd

qδΦdΓ =∫Γq

qδΦdΓq,(2.10)

with δ� being arbitrary admissible perturbations of the same form as the

corresponding fields (2.1) and satisfying the essential boundary conditions

on Γu and ΓΦ.

Following the derivation by Wells and Sluys (2001), discrete equilibrium

equations are obtained using the finite element discretisation presented in

equation (2.6) and (2.7) as

fint = fext; fint = 0;

gint = gext; gint = 0,(2.11)

with the mechanical and electrical internal force vectors defined as

fint(a, a, b, b) =∫Ω

BTσdΩ;

fint(a, a, b, b) =∫Ω+

BTσdΩ +∫Γd

NTt dΓd;

gint(a, a, b, b) =∫Ω

CTD dΩ;

gint(a, a, b, b) =∫Ω+

CTD dΩ +∫Γd

MTqdΓd.

(2.12)

In this expression, σ is the Voigt form of the Cauchy stress, which is the work

conjugate of the engineering strain, i.e. σijδεij = σTδε. Note that the traction

t and surface charge density q on the discontinuity boundary Γd are provided

by means of constitutive laws, i.e. they are related to the jumps in the dis-

placement field and electric potential field over the discontinuity boundary.

Upon satisfaction of the equilibrium conditions (2.10), the described traction

and surface charge density are equal to the projections on the discontinuity

plane of the Cauchy stress and electric flux density .

Under the assumption that no boundary conditions are imposed at the

intersection of the discontinuity with the boundary (at Γ ∩ Γd), the external

mechanical and electric force are given by

fext =∫Γt

NTt dΓt ;

gext =∫Γq

MTqdΓq.(2.13)

15


The traction and charge boundaries can overlap (in general Γt∩Γq ≠ �), but the

traction and displacement boundaries cannot overlap (Γt ∩ Γu = �). The same

holds for the charge and potential boundaries (Γq ∩ ΓΦ = �). The constitutive

laws required for the closure of the system of nonlinear equations (2.11) are

provided in Section 2.3.

2.3 Constitutive behaviour

Evaluation of the integrals in equation (2.12) requires the Cauchy stress, elec-

tric flux density, mechanical traction and surface charge density to be known.

These quantities are related to the mechanical displacement field and electric

potential field (2.1) by means of electromechanical constitutive laws. More-

over, a failure criterion needs to be supplemented to govern the evolution of

the crack.

2.3.1 Bulk constitutive behaviour

The mechanical Cauchy stress and electric flux density are related to the engi-

neering strain and electric field using linear piezoelectricity (Jaffe et. al, 1971).

Using Voigt notation, this can be written as(σ

D

)=[

H −eT

e λ

](ε

E

). (2.14)

In this expression H is the Hookean matrix for a material under plane strain

and λ is the permittivity tensor. The actual electromechanical coupling is a

consequence of the piezoelectric matrix e. The specific shape of e depends

on the type of piezoelectric material and is discussed in Section 2.5.

It should be noted that, considering the experiments in Section 2.5, the

assumption of either plane strain or plane stress conditions is precarious.

Although the depth of the specimens is of the same order of magnitude as the

other dimensions, use of a plane stress assumption can be problematic due to

the presence of the supports (or hinges) and electrodes. Also the influence of

the electric field on the in-depth strain state is uncertain. In this dissertation

it is assumed that the in-depth direction of the electric field is equal to zero.

In line with that assumption, a plane strain state is used. Reality is that

an accurate description of the experiments considered requires the use of

a three-dimensional model (in which also the electrodes and supports are

16

Constitutive behaviour

incorporated). The implementation of such a model is, however, considered

beyond the scope of this work. It is important though to realise that the effect

of either a plane stress or plane strain assumption on the results presented in

this work is limited. In the plane stress case, the bulk material will respond

somewhat more flexible, but the influence on the computed fracture loads

will be limited.

2.3.2 Cohesive behaviour

The employed electromechanical cohesive model can be regarded as an ex-

tension of an existing mechanical cohesive law using relations based on a

parallel plate capacitor. Since this cohesive law is an extension of the co-

hesive law used for grain boundary failure on the microscale, the complete

derivation of this law is postponed until Section 3.2.3. The relation between

the traction and surface charge density on a crack and the crack opening and

potential jump is provided by(t

q

)=[

I

−eintH−1int

]tm − Ecp,n

(eT

int

λint

)+(

12λintE

2cp,nn

0

), (2.15)

with Hint, eint and λint being the interfacial elastic, piezoelectric and dielectric

tangents, respectively. Damage is taken into account in these tangents by a

scalar damage parameter. Furthermore, Ecp,n is the electric field inside the

capacitor and n is the normal vector of the crack plane (which is a line in

the two-dimensional case). In the remainder of this work, the subscripts �nand �s are used to indicate the normal and shear component of a vector,

respectively. In equation (2.15), the mechanical traction tm (the subscript �m

is used to indicate that the mechanical traction is concerned) is based on

the cohesive law proposed by Wells and Sluys (2001). The normal and shear

traction are given by

tm,n = t0,n exp

(−tult

Gcκ

);

tm,s = exp (hsκ)(t0,s + ksus),(2.16)

and are shown in Figure 2.2. In this mechanical law κ is a history param-

eter defined as the maximum achieved value of the normal opening un up

to the current time instance. The loading condition is checked using the

Kuhn-Tucker conditions. The cohesive law unloads with the secant stiffness

17


Unloading

Loading

tult

Gc un

t m,n

t 0,n

543210

1

0.8

0.6

0.4

0.2

0

κ = −h−1s

κ = 0

t0,s + ksust m,s

420-2-4

4

2

0

-2

-4

Figure 2.2 Traction-opening relations in pure normal (left) and pure shear(right) directions according to the relations given in equation (2.16).

corresponding to the history parameter. The parameters t0,n and t0,s are the

normal and shear traction in the undamaged state (κ = 0) with zero opening

(u = 0), respectively. Furthermore Gc is the mechanical fracture toughness

and tult a prescribed ultimate traction. The parameter hs governs the degra-

dation of the shear stiffness ks and is here directly related to the damage in

normal direction by taking hs = −tult/Gc . Finally, note that the cohesive law

as used in Wells and Sluys (2001) has been adapted in order to ensure the

traction continuity condition tm(0) = t0,m in the undamaged state.

2.3.3 Failure criterion

In Park and Sun (1995) a linear elastic fracture mechanics approach is used

to predict the ultimate load of a piezoelectric specimen under combined me-

chanical and electric loading. The crack-closure method (Jih and Sun, 1990) is

used to determine the maximum fracture energy. It is observed that a failure

criterion based on the mechanical fracture toughness yields the best repre-

sentation of the experimentally observed dependence of the maximum load

on the electric field strength.

In the partition of unity-based cohesive zone formulation used here, a

failure criterion based on the local stress state of the system is used. Both

the position of the crack and the angle at which the crack propagates (or

nucleates) are determined on the basis of a local stress representation. The

fracture criterion is here based on the mechanical stress, which is defined as

σm = Hε = H[I+H−1eTλ−1e

]−1H−1

(σ+ eTλ−1D

). (2.17)

18


Following the argumentation in Park and Sun (1995), this mechanical stress

is assumed to be a more appropriate failure measure than the total stress.

Reason for this is that piezoelectric materials, as a consequence of the electric

contributions to the stress, can experience a zero stress state while being

deformed. As a consequence, using the total stress as a failure measure can

indicate that a specimen is significantly deformed, but does not experience

any cracking. This counterintuitive behaviour is not experienced when using

the mechanical stress as a failure indicator, since it is related directly to the

strain. The reason for not using the strain directly as a failure measure is

that, in contrast to the mechanical stress, it cannot be related directly to

the (mechanical) fracture strength, which is a material parameter suitable

for experimental determination. Also note that the mechanical stress is only

used as a failure indicator. Mechanical equilibrium remains based on the total

stress. As a consequence, the model can predict failure under pure electric

loading. A comparison of the failure criterion based on the mechanical stress

with a criterion based on the total stress is presented in Section 2.5.

Using the mechanical stress (2.17), the failure criterion is constructed as

failure ={

true max (Σm) > σult

false otherwise, (2.18)

in which Σm is a second-order tensor containing the principal stresses of σm

and σult is an ultimate value for the maximum mechanical principal stress.

Generally, the ultimate stress σult coincides with the fracture strength tult,

which is a parameter of the cohesive zone models. When the criterion (2.18)

is satisfied, a crack propagates (or nucleated) perpendicular to the direction

of the maximum mechanical principal stress.

For stability reasons, the direction of propagation of a crack is commonly

based on a smoothed stress measure around a crack tip (Jirásek, 1998). The

instance of propagation should, however, be based on a non-smoothed crack

tip stress, in order to avoid delayed crack growth. In this dissertation, both

effects are achieved by determining the stress and electric flux density at the

crack tip using (Remmers et. al, 2009)

σ = VRΣ0V−1R ;

D = ‖D0‖‖DR‖DR,

(2.19)

where VR contains the normalised eigenvectors of the smoothed stress σR,

Σ0 is a diagonal matrix containing the eigenvalues of the approximated local

19


stress σ0 and D0 is an approximation of the local electric flux density. The

local estimates are based on their values in the integration point closest to

the crack tip. The smoothed stress and electric flux density are determined

using

σR(x) =

∫y∈Ω

σ(y)exp(−‖x−y‖2

l2R

)dΩ

∫y∈Ω

exp(−‖x−y‖2

l2R

)dΩ

;

DR(x) =

∫y∈Ω

D(y)exp(−‖x−y‖2

l2R

)dΩ

∫y∈Ω

exp(−‖x−y‖2

l2R

)dΩ

,

(2.20)

with lR being typically three times the element length in the process zone.

Comparison of numerical and experimental results, as discussed in detail

in the next section, demonstrates that a propagation criterion based on (2.17)

overestimates the influence of the electric field on the propagation instance.

A likely cause of this discrepancy is that around a crack tip as well as around

an initial notch, the magnitude of the piezoelectric tensor is overestimated.

In reality, the high stresses involved in manufacturing the initial notch locally

cause domain switching (Huber et. al, 1999). When domain switching occurs,

the polarisation direction of domains that exist within the grains is altered.

As a consequence, the effective piezoelectric effect experienced is smaller

than expected from computations assuming an undamaged (i.e. not affected

by domain switching) piezoelectric tensor.

In this work, this diminished influence of the electric field is accounted

for by the introduction of a third measure for the electric flux density, Dρ,

corresponding to the averaging length lρ upon which the propagation electric

flux density is based. As a consequence, equation (2.17) is then written as

σm = H[I+H−1eTλ−1e

]−1H−1

(σ+

∥∥Dρ∥∥‖DR‖eTλ−1D

). (2.21)

A reduction in electric field dependence is then accomplished by taking the

smoothing length for the electric flux density lρ considerably larger than that

used for the stresses, lR. This is the case since the stress peak around a crack

tip will be smoothened out more using a larger smoothing radius. The ratio∥∥Dρ∥∥ /‖DR‖ will then become smaller, consequently decreasing the influence

of the electric flux density D on the mechanical stress σm.

20

Algorithmic aspects

2.4 Algorithmic aspects

Implementation of the partition of unity formulation outlined in the previ-

ous sections requires careful treatment of many algorithmic details. A good

overview of these algorithmic issues is given in the theses of Wells (2001) and

Remmers (2006). Some of these issues, specifically interesting for this work,

are briefly discussed here.

In general, the system of equilibrium equations (2.11) is solved incremen-

tally by stepwise adjustment of the external force vector. Within each of these

load steps, the nonlinear equations are solved iteratively using a Newton-

Raphson procedure. From this point of view, the partition of unity model is

considered to be implicit, since equilibrium is satisfied after each converged

load step. A detailed discussion on the tracing of the equilibrium path is the

topic of interest of Chapter 5.

From the expressions for the internal force vectors (2.12) it is observed

that these depend on the crack path Γd. Since one of the primary goals of the

partition of unity method is to model crack nucleation and propagation, this

discontinuity surface depends on the state of the system and therefore varies

in time. To reduce the complexity of the model, this variation of the crack

path is not directly accounted for in the Newton-Raphson iterations that solve

for the material nonlinearities. The evolution of the discontinuity is consid-

ered after each converged load step. If a failure criterion (see Section 2.3.3)

is satisfied, a crack is either nucleated (if the failure criterion is violated in a

bulk point) or propagated (if violated in a crack tip). In the current work, the

crack is always extended by the length of the bulk elements and the crack tips

are constructed by constraining the enhanced degrees of freedom at the tip

of the discontinuity (Wells and Sluys, 2001), but alternative formulations in

which the crack tip is allowed to fall within a bulk element exist (Belytschko

and Black, 1999).

The fact that the propagation algorithm is decoupled from the nonlinear

iterative procedure to solve for the material nonlinearities makes that the

complete formulation is not fully implicit. To ensure that the determined

points on the equilibrium path are satisfying the system of equations (2.11),

a step is only considered as converged if the crack path is stable (i.e. it does

not nucleate or propagate). If after a converged Newton-Raphson iteration

the crack path in changed, the load step is repeated with the new discontinu-

ity. In general, this corrective step improves the approximation of the exact

21


Elastic constants [GPa]

H11 113.0

H22 139.0

H33 25.6

H12 74.3

Dielectric constants [ NV2 ]

λ11 617.8λ0

λ22 677.6λ0

Piezoelectric constants [ NVmm]

e11 13.84 · 10−3

e12 −6.98 · 10−3

e23 13.44 · 10−3

Table 2.1 Parameters used for the numerical simulation of PZT specimens(Park and Sun, 1995). The permittivity of vacuum is λ0 = 8.8542 · 10−12 N

V2 .The poling direction corresponds with index 1.

solution (Remmers et. al, 2009). It does, however, only partially compensate

for the explicitness of the propagation algorithm. Therefore, careful selec-

tion of both the spatial and temporal discretisation parameters is required.

Most importantly, the load steps should be chosen such that the crack is only

allowed to extend a single bulk element per load step.

2.5 Numerical simulations

The finite element formulation presented in the previous section is tested us-

ing benchmark experiments. Following the approach in Park and Sun (1995),

experimental results for a compact tension specimen are used to fit unknown

material and geometry parameters. A three-point bending test with varying

initial crack position is then considered to test the proposed method. The nu-

merical results for that case are compared with the experimentally obtained

results presented in Park and Sun (1995).

The specimens considered are composed of lead zirconate titanate (PZT)

with chemical formula Pb (ZrxTi1−x)O3. Experiments show that the magni-

tude of the piezoelectric effect in PZT significantly depends on the stoichio-

metric ratio of zirconate and titanate (Jaffe et. al, 1954). A significant piezo-

electric effect is observed in the case that x ≈ 0.5 when the PZT is in the

morphologic phase boundary (MPB). On the scale of the crystal lattice, the

piezoelectric effect observed in PZT is caused by the off-centred zirconium

or titanium atom. In order to obtain a piezoelectric bulk specimen, a strong

22

Numerical simulations

0.23ε

4.9

5

F

F

V

3.20

0.46

4.60 6.90 14.0

Electrode

9.5

54.6

0

0.23

Figure 2.3 Schematic representation (with all units in millimetres) and finiteelement mesh of a compact tension specimen with an applied electric field.The specimen has a thickness of 5.1 mm and is poled in the vertical direction(i.e. the x1-direction).

electric field is applied to a specimen in order to align the directions of the

off-centred atoms. This process is referred to as the poling process and the

direction in which the electric field is applied is called the poling direction.

The bulk parameters for this material, poled in the x1-direction, are given in

Table 2.1. The matrices required for the evaluation of the bulk constitutive

behaviour (2.14) are then constructed as

H =

⎡⎢⎣ H11 H12 0

H12 H22 0

0 0 H33

⎤⎥⎦ ; e =

[e11 e12 0

0 0 e23

]; λ =

[λ11 0

0 λ22

]. (2.22)

Note that the piezoelectric tensor is described by only three parameters,

which is in accordance with its class 6mm† symmetry (Jaffe et. al, 1971).

The fracture strength and fracture toughness are taken from literature as

σult = tult = 80 MPa (Xiang et. al, 2003) and Gc = 2.34 · 10−3 N/mm (Park and

Sun, 1995), respectively. The shear stiffness ks is taken as 5 · 106 MPa/mm

and the averaging length lR is taken as 2.5μm.

2.5.1 Compact tension specimen

The considered compact tension specimen with a thickness of 5.1 mm and a

0.46 mm thick initial crack is shown in Figure 2.3. The specimen is poled in

†Hermann-Mauguin notation is used to describe the symmetries in the piezoelectric tensor

(Sands, 1993).

23


the vertical direction and is immersed in a tub filled with silicon oil with per-

mittivity λ∞ = 2.5λ0. A direct current power supply is attached to electrodes

on the top and bottom of the specimen. Mechanical loading is performed by

two steel hinges that are moved apart by application of a force F . The modu-

lus of elasticity and Poisson’s ratio of these hinges are taken as 200 GPa and

0.3, respectively.

As mentioned in the introduction, the brittleness of the material causes

the process zone to be small compared to the specimen size (Verhoosel and

Gutiérrez, 2009a). In fact, the process zone is even observed to be small com-

pared to the size of the initial notch. As a consequence, the ultimate load is

significantly influenced by the shape of the initial notch. To vary the shape of

the crack tip, the eccentricity ε of the ellipse shown in Figure 2.3 is modified.

To correctly predict the experimentally measured fracture load, the notch ec-

centricity is taken as ε = 4.5. For the same reason, the averaging length lρ is

taken as 150μm, which is of the same order of magnitude as the notch width.

The model is discretised using 3249 nodes and 6083 linear triangular el-

ements (Figure 2.3), with 3 degrees of freedom per node, yielding a system

with 9747 degrees of freedom. The mesh is significantly refined near the

crack tip in order to have an appropriate discretisation of the process zone.

Prior to crack nucleation, the forces F are stepwise increased by 5.1 N (i.e. 1 N

per millimetre thickness). After nucleation, energy release rate control (as

discussed in Chapter 5) is employed with a maximum dissipation increment

of 1 · 10−9 J.

As a consequence of the brittleness of the material, visualisation of the

response by a force-displacement (separation of the two hinges) curve is not

meaningful due to the severe snapback that occurs. A better visualisation

of the response of the structure is obtained by plotting the force versus the

amount of energy dissipated (Figure 2.4). Note that using the selected values

for the notch eccentricity and electric flux density smoothing parameter, a

fracture load of 92.8 N is found in the absence of an external electric field

(with V = 0 V), which is in agreement with experimental results.

The influence of the discretisation is studied by mesh refinement. The re-

sult is predominantly influenced by the mesh size around the crack tip, which

is parametrised by the mesh length, le†. The previously discussed mesh with

le = 2.5μm was refined to le = 1.5μm. The ultimate load obtained for this

†The characteristic mesh length used for the mesh generation corresponds to the average

element edge length.

24


×10−9

Dissipated energy [J]

F[N

]

76543210

100

80

60

40

20

0

Figure 2.4 Force versus dissipated energy curve for the compact tensionspecimen with no external electric field applied (V = 0 V). The responsecomputed on the original mesh with le = 2.5μm (solid line) is comparedwith that computed on a refined mesh with le = 1.5μm (dashed line).

refined mesh equals 93.4 N, which differs from the coarser mesh result by

less than one percent. The corresponding response is shown in Figure 2.4.

In Figure 2.5 the influence of the externally applied potential difference

on the fracture load is shown. As can be seen, the numerical results closely

resemble the experimentally obtained results for all measurement voltages

except for the one at 5 kV/cm. The highly nonlinear behaviour for that mea-

surement point, which is also observed for even higher electric field strengths

(Park and Sun, 1995) is not captured by the model. Correct representation of

this behaviour would require incorporation of additional nonlinear phenom-

ena in the model, such as domain switching.

The electric potential field as obtained by the finite element simulations

is shown for two settings of the applied electric voltage in Figure 2.6. The

most important observation regarding these fields is that the piezoelectri-

cally induced field (at 0 V applied voltage and ultimate mechanical load) is

significantly smaller than the field caused by the externally applied voltage

of 5 kV. Zooming in on the tip of the notch, as shown in Figure 2.7, reveals

some characteristics of the partition of unity-based fracture mechanics ap-

proach. As can be seen in the stress contour, numerical results confirm that

the cohesive zone length is indeed considerably smaller than the notch size.

One of the fundamental characteristics of the proposed electromechanical

25


Applied electric field [kV/cm]

Fra

ctu

relo

ad

[N]

6420-2-4-6

160

140

120

100

80

60

40

Figure 2.5 Dependence of fracture load on the strength of the applied elec-tric field for the compact tension specimen as obtained experimentally (�),numerically without smoothing of the electric flux density (◦), numericallywith smoothed electric flux density (+) and numerically using a failure cri-terion based on the total stress (•).

cohesive law is demonstrated in the electric potential field as shown in in Fig-

ure 2.7. Upon opening of the crack, a jump in potential is observed. The way

in which this phenomenon is incorporated in the cohesive law is discussed in

detail in Section 3.2.2.

As already mentioned in Section 2.3.3, the averaging length lρ is neces-

sary to correctly predict the dependence on the external electric field. This is

illustrated in Figure 2.5 by means of a numerical simulation without the addi-

tional smoothing of the electric flux density . The notch eccentricity, which is

then the only remaining parameter to be tuned is taken as ε = 2.65. As can be

seen, this simulation significantly overpredicts the influence of the external

electric field. For completeness, in Figure 2.5 also the fracture load depen-

dence on the external electric field is plotted while using a fracture criterion

based on the total Cauchy stress σ (in contrast to the mechanical stress σm).

A notch eccentricity of ε = 2.73 is used in this case. As can be observed,

this fracture criterion significantly underestimates the effect of the external

electric field.

26


×103

-320-192 -64 64 192 320

Electric potential [V]

0 1 2 3 4 5

Electric potential [V]

Figure 2.6 Contour plots showing the electric potential over the compacttension specimen at the ultimate loading condition for the case of an exter-nally applied voltage of 0 V (left) and 5 kV (right).

×103

-10 15 40 65 11590Cauchy stress σ11 [MPa] Electric potential [V]

2.552.49 2.51 2.532.472.45

Figure 2.7 Contour plots showing the σ11-stress component (left) and elec-tric potential (right) at the notch of the compact tension specimen for thecase of an externally applied voltage of 5 kV, after the crack has propagatedinto the bulk material.

27


0.46

2.0

8.55

2.04.0

5.0

19.1

V

F1.0

Figure 2.8 Schematic representation (with all units in millimetres) of a three-point bending specimen with an applied electric field. The specimen has athickness of 5.1 mm and is poled in the horizontal direction.

2.5.2 Three-point bending specimen

The three-point bending test as shown in Figure 2.8 is considered as a second

application of the proposed model. Specimens with a 4 mm long initial crack

at an offset of 0 mm, 2 mm and 4 mm are studied. The initial cracks have been

made in the same way as that of the compact tension specimen. Therefore it

is assumed that the geometry of the initial crack tip is the same. The speci-

mens are poled in the horizontal direction. Besides the notch eccentricity ε,

the smoothing parameter lρ found to be appropriate for the compact tension

specimen is used for the simulation of the three-point bending experiment.

For all specimens the ultimate load is computed for various electric field

strengths. The results of the finite element simulations are compared with

experimental data in Figure 2.9. For negative and zero electric fields, the

finite element method is capable of determining the experimentally obtained

results with approximately 10 percent accuracy. For positive electric fields it

correctly predicts the downward trend of the maximum load, but the accuracy

is limited.

The most important difference between the experiments and simulations

is the dependence on the electric field. This dependence is significantly over-

estimated by the finite element result. As already mentioned in the previous

section, this dependence is dictated by the averaging length lρ. Optimisation

of the values for the notch eccentricity and averaging length to better fit the

electric field dependence is possible, but is beyond the scope of this work.

28



Fra

ctu

relo

ad

[N]

129630-3-6

160

140

120

100

80

60


Fra

ctu

relo

ad

[N]

129630-3-6

180

160

140

120

100

80

60


Fra

ctu

relo

ad

[N]

129630-3-6

220

200

180

160

140

120

100

Figure 2.9 Dependence of fracture load on the strength of the applied elec-tric field for the three-point bending specimen with a centred crack (top-left), 2 mm off-centred crack (top-right) and 4 mm off-centred crack (bottom)as obtained experimentally (�) and numerically (+).

29

Chapter 3

Inter- and transgranular fracture inpiezoelectric polycrystals

In the previous chapter, numerical modelling of failure in macroscopic

specimens has been considered. The term “macroscopic” is used to indi-

cated that the microstructure is not directly incorporated in the models. In

fact, the generally complex fracture behaviour of the microscale is introduced

by means of analytical constitutive relations. In the case of large specimens

such an approach can often be adopted since the microstructural influence

on the macroscale properties of interest is generally negligible. When con-

sidering specimens of considerably smaller dimensions, the microstructural

influence can no longer be incorporated by means of analytical constitutive

laws. In that case, the fracture process needs to be studied at the microscopic

length scale.

Here quasi-static fracture of a piezoelectric polycrystal is studied using a

finite element model. The fracture process is described by a cohesive zone

model, capable of modelling crack nucleation and propagation. The influence

of an electric field as well as some material nonlinearities are incorporated

in the model. Upon the evolution of damage, the effective permittivity of

the cohesive zone is diminished. In this chapter, the proposed finite element

model is used to gain more insight in the microscale phenomena that lead to

macroscale fracture. In the next chapter, the developed model is applied in a

multiscale framework for the simulation of failure in micro electromechanical

systems.

31

Inter- and transgranular fracture in piezoelectric polycrystals

ΩΓgb

Γtr

x2

x1

Figure 3.1 Schematic representation of the considered polycrystalline mi-crostructure.

3.1 Microscale finite element model

Consider the schematic representation of a polycrystal shown in Figure 3.1,

consisting of convex grains Ω separated by grain boundaries Γgb with both in-

tergranular cracks (along Γgb) and transgranular cracks Γtr. Such polycrystals

can effectively be generated using Voronoi tessellations (Okabe et. al, 1992).

Some details on this geometrical procedure that can mimic the phenomenon

of isotropic grain growth are discussed in Section 6.2. In contrast to the

partition of unity-based finite element model discussed in the previous chap-

ter, predefined interface elements are used to model inter- and transgran-

ular fracture in piezoelectric polycrystals. This choice is motivated by the

relatively simple implementation for this approach (compared to a partition

of unity-based formulation with multiple, possibly branching and merging,

cracks) and is further commented upon in Section 3.3.

3.1.1 Kinematical formulation

As on the macroscale, the state of the microscale polycrystal is described by

a displacement field and an electric potential field. As in the case of the parti-

tion of unity method, additional degrees of freedom are added to the system

to allow for a discrete jump in these fields over an interface. Since in the

interface elements formulation the discontinuities need to coincide with the

element edges, this enhancement of the fields is effectuated by introducing

additional nodes (and corresponding degrees of freedom) at the interfaces,

as schematically shown in Figure 3.2.

Using this formulation, the displacement field and electric potential field

32

Microscale finite element model

n

s�u�, �Φ�

Γd

t, q

t, q

Figure 3.2 Schematic representation of the interface element formulation.The original nodes (◦) are duplicated (•) in order to model a jump in the fieldquantities over an element edge.

are described using C−1-continuous (i.e. allowing for discontinuities over the

interfaces) finite elements as

u(x) = N(x)a;

Φ(x) = M(x)b,(3.1)

with a and b being the nodal displacement vector and nodal electric potential

vector, respectively. Note that since the nodes on the interface are dupli-

cated, also the corresponding degrees of freedom a and b are duplicated.

These nodal vectors are mapped onto the piecewise continuous fields by the

arrays N(x) and M(x). Similarly, the matrices B(x) and C(x) map the nodal

quantities onto the strain and electric field as

ε(x) = B(x)a;

E(x) = C(x)b.(3.2)

The kinematical description is completed by expressing the jumps in the dis-

placement field and electric potential field in terms of the nodal quantities

as�u�(x) = P(x)a;

�Φ�(x) = Q(x)b,(3.3)

where the arrays P(x) and Q(x) contain the C0-continuous shape functions

defined on the interface elements.

33


3.1.2 Electromechanical equilibrium equations and boundary conditions

The microscopic fields defined in the previous section satisfy the equilibrium

equations (2.8) and are supplemented with the boundary conditions (2.9). Fol-

lowing the derivation in Verhoosel and Gutiérrez (2009a), the discrete equi-

librium equations are obtained as

fint = fext;

gint = gext,(3.4)

in which the mechanical and electric internal force vector are given by

fint(a,b) =∫Ω

BTσdΩ +∫Γd

PTt dΓd;

gint(a,b) =∫Ω

CTD dΩ +∫Γd

qQT dΓd,

(3.5)

and the external force vectors by equation (2.13). In the considered descrip-

tion, the grain boundaries are considered as physical interfaces and are there-

fore always modelled with interface elements. For that reason, the grain

boundaries are always part of the discontinuity (Γgb ⊆ Γd). In the case that

transgranular cracks Γtr appear, the discontinuity Γd is considered as the

union of the grain boundaries and transgranular cracks (Γd = Γgb ∪ Γtr).

3.2 Constitutive behaviour

In order to solve the electromechanical equilibrium equations (3.4), the con-

stitutive behaviour of the material needs to be described. Three constitu-

tive laws are required for complete description of the polycrystal. First the

stresses and electric flux densities need to be related to the strains and elec-

tric fields in the bulk material. Second, the traction and surface charge den-

sity on a grain boundary need to be described in terms of the opening of a

grain boundary and the jump in electric potential over it. A similar law needs

to be derived for the cracks in the bulk material. Finally, a failure criterion

governing the evolution of transgranular cracks needs to be provided.

3.2.1 Bulk constitutive behaviour

The constitutive behaviour of the crystalline bulk material is described us-

ing linear piezoelectricity as discussed in Section 2.3. As on the macroscale,

34


the microscopic polycrystal is considered to be made of the commonly used

piezoelectric ceramic PZT. Since using a two-dimensional model presumes the

grains to extend in the in-depth direction (leading to planar cracks), a plane

strain condition is adopted. These simplifying assumptions significantly re-

duce the involved computational effort of the microscale model, while many

interesting phenomena are preserved in the model. Since the primary goal

of this chapter is to gain insight in the microstructural behaviour of piezo-

electric polycrystals, these assumptions are considered acceptable. Although

considered beyond the scope of this work, it should be mentioned that an

improved material description is of special interest on the small scales con-

sidered. The use of a three-dimensional model is also inevitable when more

accurate results are demanded.

3.2.2 Initially elastic cohesive behaviour

Microscopic studies by Tan and Shang (2002) of the material in the grain

boundaries show that the molecular composition of the material inside the

grain boundaries differs significantly from the material inside the grains†.Since a significant piezoelectric effect is only expected in the case that PZT

is near the morphologic phase boundary (as explained in Section 2.5), it is

assumed that the material in the grain boundary itself is not piezoelectric. It

is therefore assumed that a purely mechanical cohesive law can be employed

to relate the traction to the opening of a crack. This law is then enhanced to

include electrical effects, yielding a combined electromechanical cohesive law

of the form

t = t(�u�, �Φ�

) = tm

(�u�)+ te

(�u�, �Φ�

);

q = q (�u�, �Φ�) = qe

(�u�, �Φ�

).

(3.6)

The absence of a mechanical contribution to the surface charge density is

caused by the fact that the material in the grain boundary is assumed not to

be piezoelectric. Nonetheless, the above relations are fully coupled by means

of the surface charges and electrostatic forces. The mechanical and electrical

contributions to equation (3.6) are discussed in the following paragraphs.

†Compared to the material inside the grains, the grain boundary material contains largeamounts of lead, silicon and aluminium, whereas titanium and zirconium are practically ab-

sent.

35


Mechanical contributions In the present work, two initially elastic mechan-

ical traction-opening laws are considered. For the numerical simulations in

this chapter, the commonly used Xu-Needleman law is used. An alternative

law for which the slope of the initial tangent can be adjusted independent

of the fracture toughness and fracture strength is also proposed and is em-

ployed in the multiscale simulations presented in the next chapter.

The traction-opening law proposed by Xu and Needleman (1993) is based

on the definition of a mechanical potential function. Assuming the fracture

toughness in normal (mode I) and shear (mode II) to be equal to Gc and as-

suming the opening in the normal direction after complete shear failure to

be equal to zero in the case of zero normal traction, gives this mechanical

potential function as

φm

(�u�) = Gc

[1−

(1+ �un�

δn

)exp

(−�un�

δn

)exp

(−�us�

2

δ2s

)], (3.7)

where �un� = �u� · n and �us� = �u� · s are the normal and shear components

of the opening, respectively. Note that, due to the linearised kinematics, the

normal and shear directions are the same on both sides of a crack. The

parameters δn and δs are the characteristic length parameters that are related

to the ultimate traction tult (the same in the normal and in the shear direction)

and fracture toughness by δn = Gc/(tulte) and δs = Gc/(tult

√12e) with e =

exp (1). The mechanical traction components are obtained by differentiation

of equation (3.7) with respect to the corresponding opening components to

yield

tm,n = ∂φm

∂�un�= Gc

δn

�un�

δnexp

(−�un�

δn

)exp

(−�us�

2

δ2s

);

tm,s = ∂φm

∂�us�= 2Gc

δs

�us�

δs

(1+ �un�

δn

)exp

(−�un�

δn

)exp

(−�us�

2

δ2s

).

(3.8)

The relations for pure mode I and pure mode II opening are illustrated in

Figure 3.3. Secant unloading is assumed and the loading condition is checked

on the basis of the Kuhn-Tucker conditions with history parameter κ equal to

the magnitude of the opening (κ =∥∥�u�

∥∥). Finally an additional penetration

stiffness kp is added in the normal direction in the case that a negative crack

opening in the normal direction is present.

A downside of this Xu-Needleman law is that the initial stiffness of the

interface is fully determined by the fracture toughness and fracture strength.

36


Compression

Unloading

Loading

�un�δn

δn G c·t

m,n

86.44.83.21.60

0.36

0.24

0.12

0

-0.12Unloading

Loading

�us�δs

δs

2G c·t

m,s

2.41.20-1.2-2.4

0.4

0.2

0

-0.2

-0.4

Figure 3.3 Traction-opening relations in pure normal (left) and pure shear(right) directions according to the relations given in equation (3.8).

For that reason, an alternative law is introduced. This law is based on an

effective traction-opening relation (Ortiz and Pandolfi, 1999), such that the

mechanical traction components in normal and shear direction are given by

tm,n = �un�

ΔTm;

tm,s = �us�

β2ΔTm.

(3.9)

In these expressions, Δ is the effective opening, which is written in terms of

the opening components as

Δ =√〈〈�un�〉〉2 + �us�2

β2, (3.10)

in which 〈〈�〉〉 represents the Macaulay brackets† and β is introduced as the

mode-mixity factor. Note that the effective relations (3.9) and (3.10) are cho-

sen such that the power delivered by the effective traction and opening is

equal to the combined power of both traction and opening components,

i.e. Tm · Δ = tm · ˙�u�. The effective mechanical traction Tm as introduced

in equation (3.9) is related to this effective opening by means of the relations

Tm =

⎧⎪⎨⎪⎩kinitΔ Δ < Δult

Tult exp

(− Tult(Δ−Δult)

Gc− 12kinitΔ

2ult

)Δ ≥ Δult

, (3.11)

†The Macaulay brackets are defined as 〈〈�〉〉 = 12 (�+ |�|), with |�| being the absolute value

of �.

37


Δ

Δult

Tm

Tu

lt

876543210

0.8

0.4

0

-0.4

Figure 3.4 Exponentially decaying effective traction-opening relation corre-sponding to equation (3.11) with Gc = TultΔult + 1

2kinitΔ2ult.

n

s

−q

�Φ�

q

n

Δcp

te,n

te,n

−n

−qD = Dcp,nn

D = 0

Figure 3.5 Parallel plate capacitor used for the derivation of electromechan-ical cohesive laws.

and is shown in Figure 3.4. As in the case of the Xu-Needleman law, an addi-

tional penetration stiffness kp is added in the case that the effective opening

is negative.

Using either one of these mechanical cohesive laws, the mechanical trac-

tion contribution to the electromechanical cohesive law as defined in equa-

tion (3.6) is given by

tm = tm,nn+ tm,ss, (3.12)

with the traction components given by either equation (3.8) or equation (3.9).

Electrical contributions In order to enhance the mechanical traction-open-

ing laws introduced in the previous paragraph with electrical properties, the

interface is considered as a parallel plate capacitor (Figure 3.5). For such a

capacitor it is assumed that:

38


◦ The plates are infinitely long, such that the effect of fringing fields can

be neglected.

◦ The plates are not tilted with respect to each other, such that the electric

field is aligned with the normal direction (n).

Since, strictly speaking, both assumptions are violated by the grain bound-

aries considered, the parallel plate capacitor assumption needs to be consid-

ered carefully. The appropriateness of this assumption for the configurations

considered in this chapter is demonstrated in Section 3.4.

Assuming the grain boundaries to behave like a parallel plate capacitor,

the electric field in the normal direction is given by

Ecp,n = − �Φ�

Δcp, (3.13)

where the subscript �cp is used to indicate that a variable is related to the

capacitor. The opening of the capacitor Δcp is further elaborated as

Δcp = �un�+ dgb, (3.14)

where dgb is the thickness of the capacitor at zero normal opening. This

thickness is required in order to avoid the singularity arising when the nor-

mal jump is equal to zero while a potential jump is present. Physically the

thickness dgb is interpreted as the grain boundary thickness, which for the

material considered is typically of the order of 10 nm. It should be empha-

sised that an interface element has no thickness and that the thickness dgb is

a material parameter.

The permittivity of the medium in between the capacitor plates should

represent the permittivity of the grain boundary. Initially, the permittivity of

the grain boundary material λgb has a value depending on its constituents.

Upon opening, cracks will emerge in the grain boundaries, decreasing the ef-

fective permittivity. Once the crack has fully opened, the permittivity of the

grain boundary has attained the value of vacuum λ∞ (or of the medium that

filled the crack). This deterioration of the material is incorporated in the con-

stitutive behaviour by introduction of a scalar damage parameter, ω. Since

the Xu-Needleman traction-opening law does not provide a natural expression

for this damage parameter, it is expressed in terms of the secant stiffness of

the mechanical cohesive laws using

ω = 1− knk0, (3.15)

39


where kn is the secant stiffness in the normal direction and k0 is the secant

normal stiffness in the undamaged state. The electric flux density in the

normal direction is found as

Dcp,n =[(1−ω)λgb +ωλ∞

]Ecp,n, (3.16)

which upon substitution of (3.13) gives the surface charge density as

q = −Dcp,n =[(1−ω)λgb +ωλ∞

]�Φ�

Δcp. (3.17)

Note that the minus sign in the relation between the electric flux density in the

capacitor and the surface charge density on the crack surface follows from

the direction of the normal vector in Figure 3.5. The signs in this figure are

consistent with the weak form of the electrostatic differential equation (2.10)

introduced in the previous chapter. Application of Gauss’ theorem, as illus-

trated in Figure 3.5, shows that in the definition of the surface charge density

on the capacitor electrodes, it is assumed that the electric flux density is equal

to zero outside the capacitor. From a physical point of view, it is assumed

that the electric flux density in the material surrounding a crack is consid-

erably smaller than that in the crack. The electric potential differs from the

one side of the crack to the other, but is assumed to be relatively constant on

each side.

From equation (3.17) it is observed that the scalar damage parameter re-

duces the effective permittivity from that of the undamaged grain boundary

λgb in the undamaged state (ω = 0) to that of the surrounding medium λ∞in the fully damaged state (ω = 1). The positive and negative charges on

the plates of the capacitor are attracting each other, causing an additional

electrostatic traction contribution

te,n = q2

2[(1−ω)λgb +ωλ∞

] = 1

2

[(1−ω)λgb +ωλ∞

]E2

cp,n. (3.18)

Since no electrical traction contribution in the shear direction is assumed, the

electrical traction vector as used in equation (3.6) can be written as

te = te,nn = 1

2

[(1−ω)λgb +ωλ∞

]( �Φ�

Δcp

)2

n, (3.19)

where (3.13) is used to substitute the electric field.

40


3.2.3 Initially rigid cohesive behaviour

The cohesive law used for the bulk interfaces was already introduced in Sec-

tion 2.3.2. Its derivation is discussed now since it is a logical extension of the

initially elastic interface discussed in the previous section. The cohesive law

for the interfaces in the bulk material differs from that for the grain bound-

ary on two major points. First the cohesive law is initially rigid, which means

that it has an infinite stiffness before opening and hence a non-zero traction

at zero opening. After failure initiation, a constitutive law with finite stiffness

is employed. The algorithmic treatment of the initially rigid interfaces is elab-

orated in Section 3.3. Secondly, the interface resides in the piezoelectric bulk

material and hence a piezoelectric effect is to be expected. The combined

electromechanical cohesive laws are given by

t = t(�u�, �Φ�

) = tp

(�u�, �Φ�

) + te

(�u�, �Φ�

);

q = q (�u�, �Φ�) = qp

(�u�, �Φ�

).

(3.20)

In these expressions, the subscript �p is used for the piezoelectric contribu-

tions to the electromechanical traction-opening law. The mechanical traction

tm and electric surface charge density qe are incorporated in these piezoelec-

tric contributions. Comparison with equation (3.6) shows that the piezoelec-

tric terms can be written as

tp = tm + tadd;

qp = qe + qadd,(3.21)

in which the subscript �add is used to indicate the additional terms. Since the

electrostatic contributions in equation (3.20) were discussed in the previous

section, the discussion is here restricted to the piezoelectric contributions.

Piezoelectric contributions In order to derive the additional contributions

to the traction and surface charge density coming from the piezoelectric ef-

fect in the bulk material, it is postulated that the material inside the capacitor

is deformed as a consequence of an applied traction tp and that the electric

field inside the capacitor is caused by the applied surface charge density qp.

Assuming the material inside the capacitor to be in equilibrium (both me-

chanically and electrically) this traction and surface charge density can be

written astp = NTσcp;

qp = −nTDcp,(3.22)

41


with σcp and Dcp being the Cauchy stress and electric flux density of the ma-

terial inside the capacitor. In equation (3.22), N is a matrix that projects the

Voigt form of the stress tensor on the traction†. Following equation (2.14), the

stress and electric flux density in the capacitor are related to the engineering

strain and electric field by

σcp = Hcpεcp − eTcpEcp;

Dcp = ecpεcp + λcpEcp,(3.23)

withHcp = (1−ω)H;

ecp = (1−ω)e;

λcp = (1−ω)[λ− λ∞I]+ λ∞I,

(3.24)

being the elastic tangent, piezoelectric matrix and dielectric matrix of the

material inside the capacitor. Deterioration of the material is incorporated

in the model by means of the scalar damage parameter ω. Consistently with

the employed mechanical cohesive law presented in equation (2.16) (Wells

and Sluys, 2001) this parameter is taken as ω = 1 − exp (−tultκ/Gc), while

assuming that once fully opened the permittivity in the crack is dropped to

the permittivity of the surrounding medium λ∞.

Substitution of equation (3.23) in equation (3.22) then yields the piezo-

electric traction and surface charge density as

tp = NTHcpεcp − NTeTcpEcp;

qp = −nTecpεcp − nTλcpEcp.(3.25)

In this expression, the purely mechanical traction as already described in

Section 2.3.2 and purely electric surface charge density as described in Sec-

tion 3.2.2 are identified as

tm = NTHcpεcp;

qe = −nTλcpEcp,(3.26)

such that the piezoelectric traction and surface charge density can be rewrit-

ten astp = tm −NTeT

cpEcp︸︷︷︸tadd

;

qp = −nTecpεcp︸︷︷︸qadd

+ qe.(3.27)

†The projection ti = σijnj of the second-order stress tensor is written as t = NTσ in Voigt

notation.

42


in which the additional terms in equation (3.21) are recognised. In the special

case that no piezoelectric coupling is present, the piezoelectric traction and

surface charge density resemble the results obtained in the previous section.

The additional terms in equation (3.27) resulting from a non-zero piezoelec-

tric matrix can be elaborated by assuming the strain and electric field in the

capacitor to be written in terms of the opening and potential jump as

εcp = 1

ΔcpN�u�;

Ecp = − 1

Δcpn�Φ�.

(3.28)

These expressions are chosen such that the energy conditions

σcp · εcp = 1

Δcptp · �u�;

Dcp · Ecp = 1

Δcpqp�Φ�,

(3.29)

are satisfied. Note that these conditions are stronger than the requirement

of total (i.e. electromechanical) power conservation. This means that in ad-

dition to energy preservation in the coupling between the traction and stress

(and surface charge density and electric flux density), also energy exchange

between the mechanical and electric fields is prohibited. Remind, however,

that coupling in the constitutive behaviour remains present since σcp and Dcp

are coupled via equation (3.23).

Combination of equations (3.26) and (3.28) then yields

εcp = N[NTHcpN

]−1tm;

Ecp = −n[nTλcpn

]−1qe,

(3.30)

which can be substituted in equation (3.27) to obtain

tp = tm + eint [λint]−1 qe;

qp = −eint [Hint]−1

tm + qe,(3.31)

with the elastic, piezoelectric and dielectric tangents of the interface (indi-

cated by �int) given byHint = NTHcpN;

eint = nTecpN;

λint = nTλcpn,

(3.32)

43


and the mechanical traction tm and electric surface charge density qe given

by the relations derived in Section 2.3.2 and Section 3.2.2, respectively.

Note from equation (3.31) that the presence of piezoelectric material in

the capacitor introduces an additional dependence of the total traction on

the electric field. In the proposed model, this additional contribution rep-

resents the experimentally observed dependence of the fracture strength on

the electric field (Park and Sun, 1995).

3.2.4 Failure criterion

As will be discussed in Section 3.3, transgranular fracture is modelled by

activation of specific bulk interface elements. This is done by checking a

mode I fracture criterion on the basis of the mechanical traction as

tm,n ≥ tult, (3.33)

where tult is the prescribed ultimate mechanical traction.

Since a piezoelectric material is considered, the traction is composed of

two parts: a mechanical part and an electrical part. Since the mechanical trac-

tion is required to evaluate the fracture criterion (3.33), this part of the trac-

tion needs to be computed from the given total traction and surface charge

density. In order to do this, the electromechanical cohesive law as presented

in equation (3.20) is elaborated as(t

q

)=[

I

−eintH−1int

]tm − Ecp,n

(eT

int

λint

)+(

12λintE

2cp,nn

0

). (3.34)

Given the total traction t and surface charge density q, the only unknown

quantities in this equation are the components of the mechanical traction tm

and the normal component of electric field in the capacitor Ecp,n. The non-

linear system of equations (3.34) can therefore be solved for these unknowns

by using a Newton-Raphson procedure. As a starting point for this iterative

method, the solution of the linear system which is obtained by neglecting the

electrostatic traction contribution is used.


Various algorithmic aspects need to be clarified in order to robustly imple-

ment the microscale formulation. The most important aspect is that of the

44

Algorithmic aspects

use of interface elements to model transgranular cracks. The discussion re-

garding the algorithmic aspects concerning the iterative solution procedure

and corrective step as treated in the previous chapter also applies to the mi-

croscale finite element simulations discussed here.

In the case that interface elements are used to model transgranular cracks,

the crack path is only allowed to run over the element edges, thereby making

the crack trajectory dependent on the discretisation. Besides that, a spuri-

ous mesh dependency is observed in the case that initially elastic traction-

opening relations are used for the interfaces in the bulk material (de Borst,

2004). This spurious mesh dependency is caused by the fact that additional

flexibility is added to a body by initially elastic interfaces with finite stiffness.

Since the interface length increases upon mesh refinement, so does the flexi-

bility of the body. The practical implications of this fundamental problem can

be limited for relatively coarse meshes by selecting a high value for the initial

stiffness. Using a too high value for this stiffness will, however, make the

system ill-conditioned. An additional disadvantage of using initially elastic

interfaces for transgranular cracks is that every node is duplicated, making

the method inefficient from a computational effort point of view.

Since the problems mentioned above are not present in the partition of

unity (PUM) formulation discussed in the previous chapter, it could be justly

remarked that the PUM description should also be used on the microscale.

However, from an implementation point of view, the interface elements-based

approach is more attractive. Besides that, in quasi-static analyses† the fun-

damental issue regarding the method can be circumvented and the computa-

tional effort of the method can be made comparable to that of the partition

of unity approach.

The afore-mentioned issues of interface elements for modelling transgran-

ular cracks can be overcome by only making these interfaces active after a

nucleation criterion is satisfied. Prior to the nucleation of a transgranular

crack, the bulk interfaces are made inactive by constraining the duplicated

nodes to the original node (Camacho and Ortiz, 1996; Pandolfi et. al, 1999).

In that case, the transgranular interfaces behave initially infinitely stiff and

the ill-conditioning problem does not occur since the concerned degrees of

†In dynamic crack propagation problems, the partition of unity method performs betterthan interface elements-based methods. In Falk et. al (2001) it is shown that initially rigid

interfaces prohibit crack branching when a regular (triangular) mesh is used. In Remmers et.

al (2008b) it is demonstrated that the partition of unity method is capable of modelling this

phenomenon.

45


II.II

III.I

IIA

Figure 3.6 Schematic representation of the open (dotted) and closed (solid)connectors through which the constraints are applied.

freedom are constrained. Moreover, since the constrained degrees of freedom

do not appear in the matrix which is solved for, the computational effort of

the formulation is comparable to that in the partition of unity formulation.

However, it should be mentioned that the construction of a constrained ma-

trix should be done carefully in order not to destroy the bandwidth of the

original matrix.

The interface constraints are implemented by means of connectors as

schematically illustrated in Figure 3.6. These connectors can be considered

as rigid bars (carrying both normal and transversal loads) connecting the du-

plicated nodes. As shown in Figure 3.6, these connectors can either form a

closed loop or they can form a non-branching sequence of connected bars

(here referred to as an open connector). In the case that a connector is open,

the internal force vector in each rod can be retrieved directly from the in-

ternal force vectors assembled according to equation (3.5). In the case of a

closed connector, this is not possible since an arbitrary internal force contri-

bution can be added to each of the nodes without harming the equilibrium

condition. From an algorithmic point of view this is not a problem, since the

failure criterion is only evaluated for open connectors.

A bulk interface is activated once the nucleation criterion (3.33) is violated.

This activation is carried out by destruction of some of the bars of the con-

cerned connectors. Consider for example that at some point in a computation

the force in rod A of connector I is given by [fTA,g

TA]. Since a nodal integration

scheme (integration points are located at the same position as the nodes in

the undeformed state) is used for the interface elements, the traction in the

46


adjacent interface integration point is determined using(fAgA

)=wA

(tAqA

), (3.35)

where wA is the weight of the nodal integration point at the position of A

and tA and qA are the traction and surface charge density in that point. If

the obtained traction violates criterion (3.33), rod A is removed. As is sche-

matically illustrated in Figure 3.6, this generally splits an open connector (I)

into two other open connectors (I.I and I.II) and makes a closed connector

(II) open. Note that despite breaking connector II, the opening of the crack

tip remains constrained until the open connector (II) is broken again and the

crack propagates into the next element.

From a robustness point of view, it is important that traction continu-

ity (Papoulia et. al, 2003) is satisfied upon activation of an interface. This

means that the traction and surface charge density at zero opening (and zero

potential jump) should be chosen such that the forces resulting from the co-

hesive interface are equal to the reaction forces. This continuity condition

is satisfied by adjustment of the parameters in the mechanical cohesive laws

discussed in Section 2.3.2.


The proposed piezoelectric finite element model is tested using 40× 40μm2

polycrystals with various average grain sizes. Each polycrystal is subjected

to periodic boundary conditions for the displacements and electric potential,

i.e.uright = uII + uleft; utop = uIV + ubottom;

Φright = ΦII︸︷︷︸=V

+Φleft; Φtop = ΦIV + Φbottom, (3.36)

and is loaded mechanically in the horizontal direction by a force F in the

horizontal direction (Figure 3.7). Note that, as a consequence of the periodic

boundary conditions, this force F represents the resultant force of a traction

tright over the right edge. Therefore the combination of this discrete force

and the periodic boundary conditions can be interpreted as a distributed load

over the edges. The shape of these distributed loads is unknown in advance

and generally changes during a simulation. The resultant force and charge on

the top and bottom edge are equal to zero as a consequence of the applied

47


x2

V

tright = −tleftqright = −qleft

tleftqleft

tbottomqbottom

qtop = −qbottom

ttop = −tbottom

F = R l0 trightdy

V = 1l

R l0(Φright −Φleft)dy

I II

IIIIV

x1

l=

40μ

m

l = 40μm

FΦleft

Φtop = ΦIV + Φbottom

utop = uIV + ubottom

ubottomΦbottom

uright = uII + uleftΦright = ΦII + Φleft

uleft

Figure 3.7 Geometrically periodic polycrystal of 40×40μm2 (left) subjectedto periodic displacement and electric potential boundary conditions (right),resulting in anti-periodic traction and surface charge distributions. A volt-age meter measures the average jump in electric potential from the left tothe right edge.

loading conditions. Upon loading, a potential difference between the left and

right edge will appear due to the piezoelectric effect of the considered ma-

terial. The average potential jump V is measured using a voltage meter as

indicated in Figure 3.7.

The commonly used piezoelectric ceramic PZT-4 is used to define the

material parameters. It is assumed that the grains are perfectly poled in

the horizontal direction. The bulk material parameters for PZT-4 are used

as in Park and Sun (1995) which are assembled in Table 2.1, with that dif-

ference that the elastic tangent is assumed to be isotropic with modulus

of elasticity Y = 82.3 GPa and Poisson’s ratio ν = 0.36. In Park and Sun

(1995) the fracture toughness of a bulk PZT-4 specimen is determined to be

Gc = 2.34 ·10−3 N/mm. Here this value for the fracture toughness is used for

both the grain boundary constitutive law and bulk constitutive relation. The

ultimate mechanical traction tult for PZT-4 is found as 80 MPa (Xiang et. al,

2003). For the ultimate traction, no clear distinct values for trans- and inter-

granular fracture were found. For that reason, the value of 80 MPa is used

for both cohesive laws. The grain boundary thickness is approximated to be

48


10 nm (Tan and Shang, 2002). Note that the grain boundaries are modelled us-

ing zero thickness interface elements and that this grain boundary thickness

is merely a parameter used for the constitutive model. The grain boundary

permittivity λgb is assumed to be equal to λ11. The shear stiffness ks used for

the transgranular cohesive law is taken as 7.4·106 MPa/mm, which equals the

initial shear stiffness of the Xu-Needleman law (Xu and Needleman, 1993).

3.4.1 Accurateness of the parallel plate capacitor approximation

As outlined in Section 3.2.2, it is assumed that the electrical constitutive be-

haviour of the grain boundaries can be based on a parallel plate capacitor.

Since the sides of a grain boundary are in general neither parallel, nor in-

finitely long, the appropriateness of this assumption is debatable. In the

absence of experimental results, the accuracy of the parallel plate capacitor

assumption is verified numerically. The effects of tilting of the plates, plate

misalignment and fringing fields are examined on the basis of a micro me-

chanical model for the grain boundary.

In the presented numerical model, grain boundaries are considered as in-

finitely thin line elements. This assumption is made since the thickness of

the grain boundaries is small compared to their lengths. Alternatively, the

grain boundaries could have been modelled as very slender 2D (in a 2D sim-

ulation) material domains. This approach would require fewer assumptions

and effects like grain boundary tilting and grain boundary shearing would

be incorporated in the simulation. However, from a numerical point of view

this approach is impractical. Discretisation of the grain boundaries would

introduce an enormous amount of additional degrees of freedom, leading to

a significant increase in computation time. Alternative numerical techniques,

such as e.g. the boundary element method (BEM), might significantly reduce

the involved computational effort. The performance of such techniques has,

however, not been assessed in this work.

Although micro mechanical modelling (using finite elements) of all grain

boundaries throughout complete simulations is impractical, it can be em-

ployed to verify the assumption of the parallel plate capacitor. An inter-

granular crack (as simulated by the interface element model) is considered to

derive boundary conditions for a micro mechanical finite element model that

solves the electrostatic equation in the grain boundaries (see Figure 3.8). The

considered crack is chosen such that the total charge on a crack side is at its

maximum, since in that case the largest absolute error in the internal force

49


vector (3.5) is to be expected.

Considering a typical polycrystal, the crack for which the largest abso-

lute error is expected is shown in Figure 3.9. Although this is an intergran-

ular crack, no fundamentally different results for transgranular cracks are

expected when considering the appropriateness of the parallel plate capaci-

tor. The magnitude of the opening, the mode-mixity of the opening†, the tilt

angle of the crack sides and the potential jump across the crack are shown in

Figure 3.10. The maximum opening is equal to 16.3 nm, the maximum mode-

mixity is 6.41 and a maximum tilt angle of 2.93 degrees is observed. The max-

imum jump in potential is equal to 3.9 V. From Figure 3.10 is also observed

that parts of the grain boundary are already significantly damaged. The micro

mechanical model for the grain boundary is discretised using 336454 linear

triangular elements, leading to a system of 168233 degrees of freedom.

The surface charge density resulting from the grain boundary finite ele-

ment simulation is shown in Figure 3.11. The surface charge density on the

left side of the crack q− is of opposite sign of the surface charge density at

the right side q+. As can be seen, the mismatch in surface charge density

magnitude over the crack is small as a consequence of the slenderness of the

crack. Furthermore it is observed that the parallel plate approximation ac-

curately fits the results from the micro mechanical finite element simulation.

The largest relative errors are observed in the regions where the mode-mixity

is relatively large. In these regions, however, the absolute error remains small.

It can therefore be concluded that, for the problem considered, the parallel

plate capacitor assumption is appropriate for the derivation of the electrical

contributions to the cohesive laws.

3.4.2 Fracture in piezoelectric polycrystals

The proposed numerical model is employed to model the fracture process of

the six polycrystals shown in Figure 3.12. The considered grain sizes corre-

spond with typical sizes observed in literature by e.g. Kim et. al (1990) and

are assembled in Table 3.1. The average grain size is determined using the

mean intercept length (Kim et. al, 1990).

The polycrystals are discretised using approximately 6000 three node tri-

angular elements with a Gaussian integration scheme. The grain boundaries

†The mode-mixity is here defined as the ratio between the shear opening and normal open-ing of a crack, and should not be confused with the material parameter β used for the defini-

tion of the effective opening in equation (3.10).

50


of ω

Linear interpolationof u, Φ and ω

Bilinear interpolation

dgb

Discretised grain boundaryInterface element

On grain boundary sideInside grain boundary

Figure 3.8 The state of the interface element (left) is used to derive bound-ary conditions for a finite element model for the grain boundary (right). Notethat the physical thickness of the grain boundary dgb is incorporated in theconstruction of the deformed grain boundary domain. In this domain, theelectrostatic equation is solved.

22 33 44

14.94 - 14.97 18.3 - 22.1 15.51 - 51.52

A B C

s

V0 5511

Figure 3.9 An intergranular crack resulting from the proposed model is ex-amined using a micro mechanical model for the fractured grain boundaries.The details show the electric potential in the crack at: A) the maximummode-mixity; B) the maximum opening; C) the maximum tilt angle.

51


s [μm]

∥ ∥ �u�∥ ∥ [n

m]

403020100

20

16

12

8

4

0

s [μm]

∣ ∣ �u s�/

�un�∣ ∣

403020100

7.5

6

4.5

3

1.5

0

s [μm]

θ[d

eg]

403020100

4

3.2

2.4

1.6

0.8

0

s [μm]

∥ ∥ �Φ�∥ ∥ [

V]

403020100

4.5

3.5

2.5

1.5

0.5

-0.5

s [μm]

ω

403020100

1

0.8

0.6

0.4

0.2

0

Figure 3.10 Boundary conditions used for the micro mechanical simulationof the considered crack. The crack coordinate s is defined in Figure 3.9.Furthermore, ‖�u�‖ is the magnitude of the crack opening,

∣∣�us�/�un�∣∣ is

the mode-mixity, θ is the tilt angle, �Φ� is the potential jump and ω is thedamage parameter.

52


Grain boundary FEM right edge

Grain boundary FEM left edge

Parallel plate capacitor

s [μm]

q− ,−q

+[C/m]

403020100

0.25

0.2

0.15

0.1

0.05

0

-0.05

Figure 3.11 Surface charge density along the crack sides resulting fromthe micro mechanical simulation (lines), compared with the results from theparallel plate capacitor assumption (markers).

are discretised using four node interface elements with a nodal integration

scheme. The number of grain boundary interfaces depends on the total

length of grain boundaries. In between the bulk elements approximately 8000

four node bulk interfaces with a nodal integration scheme are present. In

the undamaged state this discretisation leads to systems of approximately

50000 degrees of freedom. The equilibrium path is traced using an ultimate

dissipation increment Δτ of 5 · 10−7 mJ, which is significantly smaller than

the amount of energy dissipated in an individual interface element since

the characteristic element edge size le is equal to 7.5 · 10−4 mm. The co-

hesive zone length can be approximated (for a mechanical problem) using

various methods (Turon et. al, 2007). Hillerborg’s model for example yields

YGc/t2ult = 30μm. Although this result might be affected by the electrome-

chanical coupling, the characteristic element length is sufficiently small to

assume that the cohesive zones are discretised appropriately. This setting

typically requires 200 to 300 load steps†.The response of a polycrystal is measured in terms of average stresses and

†A simulation takes about half an hour on a 2 GHz Intel Core Duo processor with 2 GB

SDRAM.

53


A.I B.I C.I

A.II B.II C.II

Figure 3.12 Polycrystals used for numerical simulations.

ngr lgr [μm] IG [%] TG [%] 〈σ11〉ult [MPa] Vult [V]

A.I 29 8.5 93 7 54.9 42.0

A.II 29 9.2 91 9 63.4 49.1

B.I 8 16.8 78 22 71.7 55.1

B.II 8 17.1 81 19 53.2 41.0

C.I 4 21.1 70 30 78.9 61.1

C.II 4 22.7 68 32 70.9 54.7

Table 3.1 Parameters used for the generation of the considered polycrystalsand results of the finite element simulations performed on these polycrys-talline structures. Here, IG and TG respectively indicate inter- and trans-granular fracture.

54


strains. The average horizontal strain 〈ε11〉 is defined as the average horizon-

tal displacement over the right edge divided by the width of the specimen.

The average horizontal stress 〈σ11〉 is defined as the sum of the reaction

forces over the right edge divided by the length of that edge. The responses

of the considered polycrystals to the applied loading are shown in Figure 3.13.

Initially a linear relation between the average stress and average strain is

observed for all polycrystals. Also the potential difference as measure by the

voltmeter increases linearly. The ultimate loads and potential differences for

the six polycrystals are collected in Table 3.1. As can be seen, the ultimate

load (and potential difference) varies significantly for polycrystals with ap-

proximately the same average grain size (e.g. B.I and B.II). This variation is a

consequence of the differing grain shapes. The variation depends on the total

number of grains in the domain considered. If a domain of e.g. 100×100μm2

would be considered, the ultimate values would have a significantly smaller

spread. For the 40×40μm2 polycrystal considered here, significant variations

are encountered. The results, however, show some important characteristics

of the proposed model.

After the ultimate load is reached, the polycrystals start to fracture. The

wiggles observed in the fracturing part of the response curves are caused by

the fact that not all grain boundaries are failing at the same moment. At

the points where a crack propagates from one grain boundary into another,

a strong curvature in the response curves is observed. It is in these regions

where the use of an appropriate path-following constraint is required. The

σ11 stress field and electric potential field for the cracked polycrystals A.II

and C.II is shown in Figure 3.14. A potential jump over both inter- and trans-

granular cracks is noticed. The contribution of the electrostatic forces (3.18)

is observed to be less than 1 · 10−3 MPa and is therefore negligible compared

to the mechanical traction contribution. Comparison of the crack patterns

for polycrystal A.II and CII (Figure 3.14) with the crack patterns for A.I and CI

(Figure 3.15) shows the dependency of the crack pattern on the grain shapes.

An important aspect of the proposed numerical model is its capability

to correctly mimic the transition from mainly intergranular fracture for rel-

atively small grains to transgranular fracture in the case of relatively large

grains. This transition is experimentally observed for PZT in Kim et. al (1990),

the results of which are collected in Table 3.2. The results of the numerical

simulations are also incorporated in this table. As can be seen the relative

amount of transgranular fracture increases upon increasing the average grain

55


Average grain size [μm ] Intergranular [%] Transgranular [%]

9.4± 0.7 10.5 89.5

11.8± 0.2 20.0 ± 3.0 80.0± 3.0

14.5± 1.2 37.0 ± 7.0 63.0± 7.0

16.0± 0.8 70.0 ± 4.0 30.0± 4.0

17.8± 1.7 92.5 ± 2.5 7.5± 2.5

Table 3.2 Grain size dependence of the fracture mode of PZT as observedin experiments (from literature).

size. Although the model mimics this behaviour correctly from a qualitative

point of view, quantitatively there is a significant difference between the re-

sults from the numerical simulations and these reported in Kim et. al (1990)

and Table 3.1. These differences can on the one hand be explained by the fact

that the parameters used for the numerical simulations are obtained from dif-

ferent sources. On the other hand, assumptions like a plane strain condition

and isotropy of the elasticity tensor are also likely to be sources of error.

In order to investigate the influence of the piezoelectric coupling, the sim-

ulations for polycrystal A.I and C.I have been done with the piezoelectric co-

efficients taken equal to zero. The responses are shown in Figure 3.16. From

the responses it is observed that, according to the numerical model, the max-

imum load is hardly affected by the piezoelectric coupling. From Figure 3.16

it is also observed that the (initial) slope of the force-displacement curves,

i.e. the effective stiffness of the polycrystals, is increased by the piezoelectric

effect. The simulations also show that the amount of transgranular fracture

increases significantly for both polycrystals (from 7%to 20% for A.I and from

30% to 65% for C.I). This implies that the piezoelectric coupling hinders trans-

granular fracture, which is in agreement with the experimental observations

by Park and Sun (1995).

56


A.II

A.I

×10−3

〈ε11〉 [-]

〈σ11〉[

MPa]

21.510.50

80

60

40

20

0

A.II

A.I

×10−3

〈ε11〉 [-]

V[V]

21.510.50

80

60

40

20

0

B.II

B.I

×10−3

〈ε11〉 [-]

〈σ11〉[M

Pa]

21.510.50

80

60

40

20

0

B.II

B.I

×10−3

〈ε11〉 [-]

V[V]

21.510.50

80

60

40

20

0

C.II

C.I

×10−3

〈ε11〉 [-]

〈σ11〉[M

Pa]

21.510.50

80

60

40

20

0

C.II

C.I

×10−3

〈ε11〉 [-]

V[V]

21.510.50

80

60

40

20

0

Figure 3.13 Average horizontal stress 〈σ11〉 (left) and average potentialjump V (right) versus average strain 〈ε11〉 for the performed numerical sim-ulations.

57


0 21 42 10563 84

MPa

0 13 26 39 52 65

V

Figure 3.14 Contour plots showing the horizontal stress (left) and elec-tric potential (right) for the simulations of polycrystals A.II (top) at 〈σ11〉 =15.7 MPa (F = [0.63,0]N) and C.II (bottom) at 〈σ11〉 = 19.1 MPa (F =[0.76,0]N). The displacements shown in the plots are amplified by a fac-tor of 10.

58


0 13 26 39 52 65

V

0 13 26 39 52 65

V

Figure 3.15 Contour plots showing the electric potential for the simulationof polycrystal A.I (left) at 〈σ11〉 = 4.95 MPa (F = [0.20,0]N) and C.I (right) at〈σ11〉 = 32.7 MPa (F = [1.31,0]N). The displacements shown in the plots areamplified by a factor of 10.

Without piezoelectricity

With piezoelectricity

×10−3

〈ε11〉 [-]

〈σ11〉[M

Pa]

21.510.50

80

60

40

20

0 ×10−3

〈ε11〉 [-]

〈σ11〉[M

Pa]

21.510.50

80

60

40

20

0

Figure 3.16 Average horizontal stress 〈σ11〉 versus average strain 〈ε11〉 forthe simulations of polycrystal A.I (left) and C.I (right), with (solid) and with-out (dashed) piezoelectric effect.

59

Chapter 4

Multiscale modelling of fracture inpiezoelectric microsystems

In chapter 2, the partition of unity method was applied to electrome-

chanical fracture problems in specimens with dimensions in the order of

centimetres. Since the characteristic dimensions of the granular microstruc-

ture as discussed in Chapter 3 are in the range of micrometers, the influence

of the microstructure was not taken into account directly. Rather, the mi-

crostructural influence was introduced by means of phenomenological con-

stitutive models. In other words, the macroscale behaviour (involving length

scales corresponding to the specimen dimensions) was considered indepen-

dently of the microscale behaviour (involving length scales corresponding to

the polycrystalline microstructure).

Upon miniaturisation of components, as is done in the case of micro elec-

tromechanical systems (MEMS), the influence of the microstructure on their

performance increases. Therefore, appropriate prediction of the performance

of piezoelectric MEMS requires numerical models that correctly incorporate

microscale effects that are generally omitted in macroscale analyses as car-

ried out in Chapter 2. Conceptually, the most straight-forward approach to

achieve this is to model the microstructure on the complete device. Such full-

resolution modelling of the microscale is, however, often impractical from a

computational effort point of view. For the type of applications considered

here, the two length scales involved cannot be considered independently. The

analysis in which both scales are incorporated is referred to as a multiscale

analysis. Such a multiscale analysis is fundamentally different from an anal-

yses of multiple scales (as considered in Chapter 2 and 3) since the results of

both scales are directly coupled in a single analysis.

61

Multiscale modelling of fracture in piezoelectric microsystems

In a multiscale analysis, information is exchanged between length scales

by means of homogenisation techniques. The development of such tech-

niques dates back to the works of Voigt (1889) and Reuss (1929) in which

rules of mixtures are applied to achieve effective properties of multiphase

materials. Many homogenisation methods in which the solution of a micro-

structural problem is determined to obtain effective material properties were

proposed, of which especially the pioneering work of Eshelby (1957) is worth

mentioning. In more recent years, these types of closed-form homogenisation

techniques have been used to determine effective material properties and ho-

mogenised constitutive laws using numerical models to compute the solution

of the microscale problem. This type of numerical homogenisation methods

are generally referred to as unit-cell methods and have proven their worth

in many fields. However, a significant disadvantage of the unit-cell method

is that assumptions regarding the form of macroscale constitutive behaviour

are required, making the method less appropriate for many nonlinear prob-

lems.

Computational homogenisation (Suquet, 1985) is nowadays recognised as

a technique suitable for overcoming the problems with the unit-cell meth-

ods. The main feature of computational homogenisation is to describe the

macroscopic constitutive behaviour using finite element models (or any other

discretisation technique) on the microscale, rather than to use analytic con-

stitutive laws. This approach implies that an additional finite element model

is solved in all macroscopic integration points. Although computationally

expensive (despite the fact that use can be made of parallel computing),

computational homogenisation has been demonstrated to be capable of ef-

fectively capturing nonlinear microscale behaviour in macroscale analyses

(Feyel, 1999; Kouznetsova et. al, 2002). The focus of the works on compu-

tational homogenisation has primarily been on the homogenisation of bulk

constitutive relations. Recently, computational homogenisation techniques

have also been applied to determine constitutive relations for predefined in-

terfaces (Matouš et. al, 2008; Hirschberger et. al, 2008) as well as for cohesive

cracks (Belytschko et. al, 2008).

In this chapter, an alternative computational homogenisation framework

is proposed that can be used to derive homogenised traction-opening rela-

tions for cracks inside a bulk material. These interfaces, modelled using the

partition of unity method, can run in arbitrary directions and their position

is changing throughout the computation. A numerical homogenisation pro-

62

Multiscale constitutive modelling

cedure is employed for the bulk constitutive behaviour. The coefficients for

the presumed linear material behaviour are obtained only once, prior to sim-

ulation. As a consequence, microscopic finite element models are only used

for the homogenisation of the cohesive behaviour of the material.

The proposed model can be considered as a FEn model as introduced by

Feyel (1999). In the case that a microscale finite element model is solved for

every macroscopic integration point, one speaks of a FE2 model†. In the pro-

posed model, a microscale FE model is only solved for a relatively small num-

ber of macroscale integration points. In the considered two-dimensional set-

ting, the number of multiscale integration points scales by the square root of

the total number of integration points, making the model a FE32 model‡. In

three dimensions, it should be considered as a FE53 model. It should be em-

phasised that this discussion is not a pedantic naming issue, but is of crucial

importance for the performance of the multiscale framework.

4.1 Multiscale constitutive modelling

The goal of this section is to derive homogenisation laws for the constitutive

behaviour of a piezoelectric continuum, such that the complex behaviour of

the microscale is appropriately described on the macroscale. The considered

microstructure is a piezoelectric polycrystal, which upon loading fails inter-

granularly. It is emphasised that the proposed framework is not restricted to

this specific micro model and can be applied to other types of microstructural

geometry and microscale nonlinear behaviour. In order to clearly distinguish

the two scales, the superscripts �M and �m are used to indicate if a quantity

belongs to the macroscale or microscale, respectively.

For a proper description of a multiscale constitutive model it is of crucial

importance to realise that the failure description is fundamentally different

on the two scales. This difference is schematically shown in Figure 4.1. On

the macroscale, a cohesive segment is only inserted upon satisfaction of a

failure criterion. It is assumed that the material behaves linearly until crack

†Let the macroscale and microscale be described by nM and nm ∼ nM degrees of freedom,

with nM, nm � 1. In a FE2 analysis, the number of micro models scales linearly with thenumber of macroscale degrees of freedom. The total number of degrees of freedom then

scales as n ∼ nM ×nm ∼ (nM)2, hence the name FE2.‡Since micro models are only used for the evaluation of line integrals, their number scales

with the square root of the number of macroscale degrees of freedom. Hence the total number

of degrees of freedom scales as n ∼ √nM ×nm ∼ (nM)32 .

63


Microscale

F,uF,u F,uF,u

F

u

F

u

Macroscale

Figure 4.1 Schematic representation of failure processes at the micro- andmacroscale.

nucleation and hence energy is only dissipated after insertion of a cohesive

segment. In contrast, the interfaces on the microscale are present through-

out the complete simulation since they represent the grain boundaries. Upon

loading, the microscale model gradually loses strength due to damage accu-

mulation in these grain boundary interfaces. As a consequence, an obvious

definition for the instance of failure is not available on the microscale.

The pre-failure nonlinear behaviour of the microscale cannot be repre-

sented by the macroscale formulation. Therefore, it is assumed that the

macroscale constitutive behaviour prior to the insertion of a cohesive seg-

ment can be described on the basis of homogenised microscale properties

in the undeformed state. This assumption is advantageous from a computa-

tional effort point of view, since these homogenised properties are obtained

in a computationally relatively cheap pre-processing step. The appropriate-

ness of this assumption fully depends on the relative importance of the grain

boundary opening prior to macroscale crack insertion. This assumption will

further be commented upon in the numerical simulations section.

4.1.1 Homogenisation of bulk constitutive behaviour

The bulk constitutive behaviour is described by the microscale model sche-

matically shown in Figure 4.2. Note that for notational convenience a single

grain boundary Γgb is considered that splits the microstructure in two parts.

The derivation presented here, however, remains fully valid for the multiple

grain boundary case. The microscale displacement and electric potential field

64


are decomposed as

um

i =⟨εm

ij

⟩Ωmx

m

j + um

i +HΓgbum

i︸︷︷︸umi

;

Φm = −⟨Em

i

⟩Ωmx

m

i + Φm +HΓgbΦ

m︸︷︷︸Φm

,(4.1)

with Hgb being defined similarly as in equation (2.2). The boundary condi-

tions for the micro model are then given in terms of the microscopic fluctu-

ation fields �. On the four control nodes (I to IV), both the mechanical dis-

placement fluctuations and electric potential fluctuations are equal to zero.

Moreover, the fluctuation fields are periodic from the left to the right edge

and from the bottom to the top edge (as indicated by the dotted lines in Fig-

ure 4.2). From these boundary conditions, it follows that the homogenised

infinitesimal strain tensor⟨εm

ij

⟩Ωm and homogenised electric field

⟨Em

i

⟩Ωm are

defined as the volume average of the corresponding microscopic fields

⟨εm

ij

⟩Ωm = 1

wh

∫Ωmεm

ij dΩm;

⟨Em

i

⟩Ωm = 1

wh

∫ΩmEm

i dΩm,

(4.2)

in which w and h are the width and height of the micro model, respectively.

This is demonstrated by differentiation of equation (4.1) to obtain

εm

ij =⟨εm

ij

⟩Ωm + 1

2

(∂um

i

∂xm

j

+∂um

j

∂xm

i

)+

+HΓgb

1

2

(∂um

i

∂xm

j

+∂um

j

∂xm

i

)+ δΓgb

1

2

(um

i nm

j + um

j nm

i

);

Em

i =⟨Em

i

⟩Ωm − ∂Φ

m

∂xm

i

−HΓgb

∂Φm

∂xm

i

− δΓgbΦmnm

i ,

(4.3)

with δgb being the Dirac-delta function corresponding to the Heaviside Hgb.

Substitution of these expressions in equation (4.2) then yields

1

wh

∫Ωmεm

ijdΩm = ⟨εij⟩Ωm + 1

2wh

∫Γm

(um

i nm

j + um

j nm

i

)dΓ

m;

1

wh

∫ΩmEm

i dΩm = ⟨Ei⟩Ωm − 1

wh

∫Γm

Φmnm

i dΓm.

(4.4)

65


Since both boundary integrals will disappear as a consequence of the periodic

boundary conditions, it follows that volume averages indeed represent the

homogenised kinematical quantities⟨

�⟩Ωm.

The definition of the homogenised stress and electric flux density then

follows by consideration of the Hill-Mandel energy condition (Hill, 1963) for

both the mechanical and electric contributions

⟨σm

ij

⟩Ωmδ

⟨εm

ij

⟩Ωm = 1

wh

∫Ωmσm

ijδεm

ij dΩm;

⟨Dm

i

⟩Ωmδ

⟨Em

i

⟩Ωm = 1

wh

∫ΩmDm

i δEm

i dΩm.

(4.5)

Using microscale equilibrium these expressions are rewritten as

⟨σm

ij

⟩Ωmδ

⟨εm

ij

⟩Ωm = 1

wh

∫Γmtm

i δum

i dΓm;

⟨Dm

i

⟩Ωmδ

⟨Em

i

⟩Ωm = 1

wh

∫ΓmqmδΦm dΓ

m,

(4.6)

which can be rewritten by using the decompositions in equation (4.1) as

⟨σm

ij

⟩Ωmδ

⟨εm

ij

⟩Ωm = 1

wh

∫Γmtm

i xm

j dΓmδ⟨εij⟩Ωm;

⟨Dm

i

⟩Ωmδ

⟨Em

i

⟩Ωm = − 1

wh

∫Γmqmxm

i dΓmδ⟨Ei⟩Ωm ,

(4.7)

where use is made of the anti-periodicity of the traction and surface charge

density corresponding to the periodic boundary conditions for the displace-

ments and electric potential. From this expression, the macroscopic stress

and electric flux density follow as

⟨σm

ij

⟩Ωm = 1

wh

∫Γmtm

i xm

j dΓm = 1

wh

∫Ωmσm

ij dΩm;

⟨Dm

i

⟩Ωm = − 1

wh

∫Γmqmxm

i dΓm = 1

wh

∫ΩmDm

i dΩm.

(4.8)

Hence, the homogenised kinetic properties are the volume averages of the

microscopic quantities, when use is made of the boundary conditions speci-

fied. Furthermore note that the homogenised stress tensor is symmetric by

virtue of the equilibrium of moments.

For given values for the homogenised strain⟨εm

ij

⟩Ωm and electric field⟨

Em

i

⟩Ωm, equation (4.1) can be used to determine the corresponding microscale

66


boundary conditions. Equation (4.8) can then be used to determine the micro-

scopic stresses and electric flux densities. The (possible) nonlinear behaviour

of the microscale prior to macroscopic crack nucleation then generally leads

to nonlinear constitutive behaviour. This is obviously not desirable from a

computational effort point of view, since a microscopic finite element sim-

ulation is then required for each evaluation of the stresses and electric flux

densities. Since in the present work the nonlinearities of the bulk material

are assumed to be of minor importance compared to the nonlinearities com-

ing from the fracture process, an additional assumption is made by assuming

linear bulk behaviour based on the homogenised material tangent in the un-

deformed state (denoted by �|0), resulting in

σM

ij =∂⟨σm

ij

⟩Ωm

∂⟨εm

kl

⟩Ωm

∣∣∣∣∣0

εM

kl +∂⟨σm

ij

⟩Ωm

∂⟨Em

k

⟩Ωm

∣∣∣∣∣0

EM

k ;

DM

i =∂⟨Dmi

⟩Ωm

∂⟨εm

kl

⟩Ωm

∣∣∣∣∣0

εM

kl +∂⟨Dmi

⟩Ωm

∂⟨Em

k

⟩Ωm

∣∣∣∣∣0

EM

k.

(4.9)

The tangents are computed by multiple evaluations of the stresses and elec-

tric flux densities using the microscale stiffness matrix in the undeformed

state. Since these operations are performed in a pre-processing step, their

influence on the overall computational cost is negligible. During the macro-

scopic finite element simulation, the bulk stresses and electric flux densities

are computed by the evaluation of equation (4.9).

4.1.2 Homogenisation of cohesive behaviour

For the homogenisation of the bulk behaviour, the volume averages of the

fields describing the kinematics are used as homogenised kinematical quan-

tities. Based on this choice the corresponding averaging schemes for the ki-

netic quantities were derived by exploiting the Hill-Mandel energy condition.

In the case of homogenisation of the cohesive behaviour, a similar approach

is troublesome, since an obvious choice for a crack opening averaging scheme

is missing. However, since it is possible to find an obvious expression for the

homogenised traction and surface charge density, the original procedure is

inverted here. Given the averaging scheme for the kinetics, the correspond-

ing homogenisation rules for the kinematics are derived using the Hill-Mandel

energy condition.

67


ΓmD

I

III

xm1

xm2

Γgb

IV

w

h

I II

IIIIV

xm1

xm2

Γgb

II htM, hqM

vM, γM

ΓmA

ΓmB

ΓmC

Figure 4.2 Microscale boundary value problem used for the homogenisa-tion of the bulk (left) and cohesive (right) constitutive behaviour. The micro-scopic domain Ωm is bounded by the external boundary Γm = Γ

mA∪Γ

mB ∪Γ

mC ∪Γ

mD .

To derive a homogenisation scheme for the macroscopic crack opening

and macroscopic potential jump, the microscale model shown in Figure 4.2 is

considered. The macroscopic traction and surface charge density are defined

as the projection of the homogenised stress and electric flux density on the

fracture plane, yielding

tM

i =⟨σm

ij

⟩Ωmn

M

j ;

qM = −⟨Dm

i

⟩Ωmn

M

i ,(4.10)

with nM

i being the normal to the macroscopic crack, which coincides with the

xm1 -direction on the microscale. The (generally anisotropic) microscale mate-

rial tangents are rotated in accordance with the direction of the macroscopic

cohesive segment. In the present work, the microscale geometry is not ad-

justed to the direction of the crack, hence assuming isotropy in the geometry

of the microstructure. For problems where this assumption is violated, e.g. if

the microstructure is columnar, the rotation of the geometry also needs to be

taken into account. Substitution of the earlier derived expressions (4.8) for⟨σm

ij

⟩Ωm and

⟨Dm

i

⟩Ωm in equation (4.10) then yields

tM

i =1

h

∫Γ

mB

tm

i dΓmB ;

qM = 1

h

∫Γ

mB

qm dΓmB .

(4.11)

68


For the case considered here the Hill-Mandel energy condition reads

σM

ijδεM

ij +1

wtM

i δuM

i =1

wh

∫Γmtm

i δum

i dΓm;

DM

i δEM

i +1

wqMδΦM = 1

wh

∫ΓmqmδΦm dΓ

m.

(4.12)

As can be seen, this expression is composed of a part concerning the internal

work of the bulk material and a part representing the work performed by the

cohesive surface. Using the boundary conditions, the boundary integrals in

equation (4.12) are rewritten as

1

wh

∫Γmtmi δu

mi dΓ

m = 1

wtMi δv

Mi ;

1

wh

∫ΓmqmδΦm dΓ

m = 1

wqMδγM,

(4.13)

with vM

i and γM being the displacement and potential of the bottom right

control node (node II), respectively. Substitution of equation (4.13) in equa-

tion (4.12) then yields

tM

i δvM

i = wσM

ijδεM

ij + tM

i δuM

i ;

qMδγM = wDM

i δEM

i + qMδΦM.(4.14)

Note that for the boundary conditions used, the macroscopic stress and elec-

tric flux density as given in (4.8) can be written as

σM

ij =δj1

h

∫Γ

mB

tm

i dΓmB +

δj2

w

∫Γ

mC

tm

i dΓmC = Δijkt

M

k ;

DM

i = −δi1h

∫Γ

mB

qm dΓmB −

δi2w

∫Γ

mC

qm dΓmC = −δi1qM,

(4.15)

in which δij is the Kronecker delta and Δijk = δi1δj1δk1+(δi1δj2+δi2δj1)δk2

is a third-order tensor. Moreover, using

δεM

ij =∂εM

ij

∂σMkl

δσM

kl +∂εM

ij

∂DMk

δDM

k ;

δEM

i =∂EM

i

∂σM

kl

δσM

kl +∂EM

i

∂DM

k

δDM

k,

(4.16)

69


it follows that

tM

i δvM

i = wtM

i

[Δjki

∂εM

jk

∂σMpn

Δpnl

]︸︷︷︸

Cil

δtM

l −wtM

i

[Δjki

∂εM

jk

∂DM1

]︸︷︷︸

−Ci3

δqM + tM

i δuM

i ;

qMδγM = −wqM

[∂EM

1

∂σM

jk

Δjkl

]︸︷︷︸

−C3l

δtM

l +wqM∂EM

1

∂DM1︸︷︷︸

C33

δqM + qMδΦM,

(4.17)

in which the 3×3 second-order tensor C is identified as the projection of the

compliance tensor on the discontinuity plane. Since equation (4.17) should

hold for any value of the macroscopic traction and surface charge density, it

follows thatδvM

i =wCilδtM

l +wCi3δqM + δuM

i ;

δγM =wC3lδtM

l +wC33δqM + δΦM.

(4.18)

This equation is then rewritten as

δvM

i = wC0ilδt

M

l +wC0i3δq

M + δuM

i + δuM

i ;

δγM = wC03lδt

M

l +wC033δq

M + δΦM + δΦM,(4.19)

with C0 being the tensor C evaluate in the undeformed state and

δuM

i =w[Cil − C0

il

]δtM

l +w[Ci3 − C0

i3

]δqM;

δΦM =w [C3l − C03l

]δtM

l +w[C33 − C0

33

]δqM.

(4.20)

Here it is assumed that after macroscopic crack nucleation, i.e. the part where

the homogenisation scheme is used, the nonlinearity in δvM

i and δγM is fully

covered by the macroscopic jump. Hence it is assumed that δuM

i and δΦM are

zero. Making this assumption allows to rewrite (4.19) as

vM

i = wC0ilt

M

l +wC0i3q

M + uM

i + uM

i ;

γM = wC03lt

M

l +wC033q

M + ΦM + Φ

M.(4.21)

The material derivatives in this expression can directly be related to those

used for the bulk constitutive behaviour in equation (4.9). Moreover, uMi and

ΦM are computed at the moment of nucleation, i.e. when uM

i = ΦM = 0. Since

both vM

i and γM are functions of the macroscopic traction tM

i and macroscopic

surface charge density qM, the equations (4.21) are merely a set of nonlinear

70


tM

uM

tM

uM uM

tM

Homogenised traction-opening relation for a micro model in point A

Crack geoemtry and corresponding traction distribution

tMinit

A

tMulttM

ult

tMinit

A A

tMult

tMult

tMinit

Figure 4.3 Schematic representation of the crack tip propagation proce-dure. The homogenised traction-opening law at point A is shown at threeinstances (top). The corresponding crack geometries and traction distribu-tions are shown in the bottom figure.

equations. Hence, for a given macroscopic crack opening uM

i and macroscopic

potential jump ΦM, the corresponding traction and surface charge density can

be computed. The equations (4.21) thus serve as constitutive laws for the

interfaces.

4.1.3 Determination of propagation instance and direction

The instance and direction of propagation of a crack is determined on the ba-

sis of the smoothed Cauchy stress as computed by equation (2.20). Obviously,

the microstructural influence is captured by the fact that the bulk stresses are

computed by means of the homogenised constitutive relations (4.9).

The maximum principal stress is used to determine the propagation in-

stance as well as the propagation direction. Since the cohesive law is not

explicitly known in the framework considered here, it is not trivial to relate

the ultimate strength of the constitutive law (tMult) to the propagation traction

71


vM

xm

IV III

III

Γgb

FM = htM

w

h

Figure 4.4 Simplified mechanical microstructure.

(tMinit). Therefore, a crack is propagated before the homogenised maximum

strength of a micro model is reached, as illustrated in the leftmost picture in

Figure 4.3. This method of crack propagation implies that after the extension

of a crack, the load bearing capacity will initially further increase, as shown in

the middle illustration in Figure 4.3. As a consequence, the cohesive zone will

slightly run behind the tip of the discontinuity. After the ultimate strength

of a micro model is reached, the traction will gradually decrease, as shown in

the rightmost picture in Figure 4.3.

As already mentioned in the previous subsection, upon insertion of a co-

hesive segment the micro model is solved for the traction corresponding to

the local stress state. In order to increase the robustness of the method, this

external load is increased in a number of steps (typically 10). Once the exter-

nal load matches the local stress corresponding to the propagation instance,

the compatibility quantities ui and Φ are obtained.

4.1.4 Illustration of the homogenisation procedure

In Section 4.4, the homogenisation framework is applied in the partition of

unity-based finite element framework discussed in Chapter 2. In order to

clarify the homogenisation procedure, the scheme is first applied to the sim-

plified microstructure shown in Figure 4.4. Moreover, the focus in this sub-

section is on the homogenisation of the traction-opening law. Therefore only

the homogenisation procedure for the mechanical constitutive behaviour is

considered here.

72


w ×h [μm2] ∂FM

∂vM

∣∣∣0

[GPa] ∂σM

∂εM

∣∣∣0

[GPa] vM(tMult) [mm] vM [mm]

1 10× 10 88.1 88.1 13.7 · 10−6 2.33 · 10−6

2 20× 10 44.0 88.1 27.4 · 10−6 4.65 · 10−6

3 10× 5 44.0 88.1 13.7 · 10−6 2.33 · 10−6

Table 4.1 Homogenised properties for various sizes of the simplified mi-crostructure.

×10−6

vM [mm]

FM

[N]

35302520151050

1

0.8

0.6

0.4

0.2

0

321

×10−6

uM [mm]

tM[M

Pa]

35302520151050

100

80

60

40

20

0

Figure 4.5 Force-displacement curves (left) and homogenised traction-opening laws (right) for the simplified microstructure with three differentmicrostructure sizes as given in Table 4.1.

Let the microscopic constitutive behaviour be given by

σm = Hmεm;

tm = (tm

ulte)2um

Gmc

exp

(−t

m

ulteum

Gmc

),

(4.22)

with um being the displacement jump over the grain boundary Γgb. Using

these constitutive relations the response of the representative volume ele-

ment can be determined. The result for Hm = 100 GPa, tmult = 100 MPa and

Gmc = 0.01 N/mm is shown in Figure 4.5 for the three different microstructure

sizes as indicated in Table 4.1. As can be seen, the responses for the three

cases differ significantly.

Following from equation (4.2) and (4.8) the homogenised strain and stress

73


prior to macroscopic crack opening are obtained by

⟨εm⟩Ωm = v

M

w;

⟨σm⟩Ωm = F

M

h,

(4.23)

from which the tangent is obtained as

∂⟨σm⟩Ωm

∂⟨εm⟩Ωm

= wh

∂FM

∂vM. (4.24)

In Table 4.1 the initial values of both the force-displacement curve slope and

averaged tangent are shown. As can be seen, the initial slope depends on

both the width and height of the microstructure, but the tangent does not.

For the considered case, the macroscopic stress is approximated by

σM = ∂⟨σm⟩Ωm

∂⟨εm⟩Ωm

∣∣∣∣∣0

εM. (4.25)

More interesting is the homogenisation procedure for the cohesive law.

For the simple case considered, equation (4.21) reads

vM =w ∂⟨εm⟩Ωm

∂⟨σm⟩Ωm

∣∣∣∣∣0

tM + uM + uM. (4.26)

Assuming the macroscopic crack to be inserted at the maximum traction, tMult,

yields

uM = vM(tM

ult)−w∂⟨εm⟩Ωm

∂⟨σm⟩Ωm

∣∣∣∣∣0

tM

ult. (4.27)

For a given value of the opening, uM, equation (4.26) can be solved for tM. The

resulting traction-opening laws are shown in Figure 4.5. The traction-opening

relations for microstructure 1 and 3 coincide. Microstructure 2 dissipates

less energy, which is a consequence of the fact that more energy is dissipated

before crack nucleation. This is a consequence of the larger displacement at

zero opening, vM. As will be demonstrated in the Section 4.4, this difference

in dissipation is negligible for the numerical simulations considered.

74

Finite element formulation

4.2 Finite element formulation

The multiscale constitutive framework presented in the previous section does

not make presumptions on the employed discretisation technique used for

the microscale model. As already mentioned in the preamble of this chapter,

the finite element model introduced in Chapter 3 is used on the microscale.

The application of the multiscale framework presented above to this FEn case

is discussed next.

4.2.1 Bulk homogenisation

The homogenised strain and electric field as defined in equation (4.2) are

expressed in terms of the nodal degrees of freedom am and bm as introduced

in Section 3.1.1 by

⟨εm

ij

⟩Ωm = 1

wh

∫Ωmεm

ij dΩm = 1

2wh

∫Γm

(um

i nm

j +um

j nm

i

)dΓ

m;

= 1

2w

(am

II,iδj1 + am

II,jδi1

)+ 1

2h

(am

IV,iδj2 + am

IV,jδi2

);

⟨Em

i

⟩Ωm = 1

wh

∫ΩmEm

i dΩm = − 1

wh

∫Γm

Φmni dΓ

m;

= − 1

wbm

IIδi1 −1

hbm

IVδi2,

(4.28)

where use is made of Gauss’ theorem and the periodic boundary conditions

discussed in Section 4.1.1. Equation (4.28) demonstrates that the macro-

scopic strain and electric field can be applied to the micro model by pre-

scribing the degrees of freedom in node II and node IV in combination with

the constraint on node I and periodic constraints on the edges of the micro

model. Furthermore it should be noted that on the basis of the presumed

expansion of the microscopic fields (4.1), the nodal displacements aII,2 and

aIV,1 are equal, which reflects the symmetry of the infinitesimal strain tensor.

The homogenised stress and electric flux density defined in equation (4.8)

can be written in terms of the nodal force vectors fmint and gm

int as defined in

75


equation (3.5) as

⟨σm

ij

⟩Ωm = 1

wh

∫Ωmσm

ij dΩm = δ1j

h

∫Γ

mB

tm

i dΓmB +

δ2j

w

∫Γ

mC

tm

i dΓmC ;

= δ1j

h

∑k∈Γ

mB

f m

intk,i +δ2j

w

∑k∈Γ

mC

f m

intk,i;

⟨Dm

i

⟩Ωm = 1

wh

∫ΩmDm

i dΩm = −δ1i

h

∫Γ

mB

qm dΓmB −

δ2i

w

∫Γ

mC

qm dΓmC ;

= −δ1i

h

∑k∈Γ

mB

gm

intk −δ2i

w

∑k∈Γ

mC

gm

intk.

(4.29)

Hence, the homogenised stress and electric flux density are determined on

the basis of only the values of the internal force vectors in the nodes on the

right (B) and top (C) boundary.

The consistent tangents required for the homogenised law (4.9) are ob-

tained by using the equations (4.28) and (4.29). The external forces required

for the evaluation of the stress and electric flux density are related to the

displacements of the control points using condensation of the microscopic

stiffness matrix (Kouznetsova et. al, 2001).

4.2.2 Cohesive homogenisation

The homogenised cohesive law (4.21) is written in terms of the discretised

microscale quantities by expressing the displacement and electric potential

of node II in terms of the nodal degrees of freedom vector as(vM

γM

)= AT

(am

bm

). (4.30)

Using this expression, equation (4.21) can be written in matrix-vector format

as

AT

(am

bm

)=wC0

(tM

qM

)+(

uM

ΦM

)+(

uM

ΦM

). (4.31)

This system of 3 equations (4.31) complements the microscale equilibrium

equations (fmint

gmint

)= hA

(tM

qM

), (4.32)

76

Algorithmic aspects

in which the external force vector is written in terms of the homogenised

traction and surface charge density (as presented in Figure 4.2). Note that the

equations (4.31) and (4.32) merely form an augmented system of n+ 3 equa-

tions and unknowns, which is solved using a Newton-Raphson procedure. In

order to preserve the sparsity of the stiffness matrix of the micro model, the

augmented system is solved using the Woodbury formula (Golub and Loan,

1996). Application of this formula requires the solution of four conventional

microscale systems of equations. Since the LU-decomposition can be reused

these solutions are obtained without much additional computational effort.

Once the homogenised traction and surface charge density are computed

by solving equation (4.31), the material derivatives also need to be supplied

to the macroscale in order to construct a consistent tangent on that scale.

This material derivative is obtained by differentiation of the equations (4.31)

and (4.32) to the macroscopic jumps to obtain

AT ∂ (am, bm)

∂(uM, ΦM

) = wC0 ∂(tM, qM

)∂(uM, ΦM

) + I;

Km ∂ (am, bm)

∂(uM, ΦM

) = hA∂(tM, qM

)∂(uM, ΦM

) , (4.33)

with Km being the microscopic consistent tangent. Combination of these re-

sults then yields the macroscopic material tangent

∂(tM, qM

)∂(uM, ΦM

) = [hAT [Km]−1A−wC0

]−1. (4.34)

Since the term [Km]−1 A is also required for the solution of equation (4.31),

this material tangent can be computed with negligible added computational

effort.


In principle, the constitutive multiscale model presented above can be re-

garded as a rather complex constitutive model. Some of the algorithmic as-

pects, especially concerning the storage of micro model data, are elaborated

here on the basis of the finite element program shown in Figure 4.6. Various

other algorithmic aspects of computational homogenisation are discussed in

theses of Kouznetsova (2002) and Gitman (2006). In the flow diagram, the

macro- and microscale finite element models are illustrated in the left and

77


right (dark grey) blocks, respectively. Note that for convenience, the correc-

tive step which is performed in the case of a violation is not shown in the flow

diagram on the microscale. In Figure 4.6 it is clearly observed that the two

models are coupled by means of a homogenisation layer (light grey block). It

is also recognised that microscale information is only introduced via the con-

stitutive behaviour, which is evaluated in the macroscopic assembly of the

force vectors and corresponding tangents.

The bulk constitutive behaviour on the macroscale is evaluated on the

basis of the homogenised bulk tangents as discussed in Section 4.1.1. As ob-

served in the flow diagram, the tangents required for the evaluation of the

stress and electric flux density are obtained during the initialisation phase of

the microscale model. Since in this case the flow of information is only going

in the direction from the microscale to the macroscale, the bulk homogeni-

sation procedure is a typical case of numerical homogenisation, rather than

computational homogenisation. The influence of the bulk homogenisation

procedure on the computational effort of the complete multiscale framework,

in terms of both memory usage and computation time, is limited.

The implementation of the cohesive homogenisation scheme as discussed

in Section 4.1.2 is significantly more complex. From the flow diagram it is

obvious that information is flowing in both directions, making this averaging

procedure a true computational homogenisation technique. In order to eval-

uate the macroscopic traction and surface charge density, the micro model

is solved using the constraint equation (4.31). It is important to realise that

this procedure is repeated for every macroscale assembly. The traction and

surface charge density obtained from the homogenisation procedure are only

satisfying macroscopic equilibrium in the case that the macroscale finite el-

ement model has converged. In the case that the macroscale model has not

converged, but requires an additional Newton-Raphson iteration, the state

and history of the micro models are reset to their previous macroscale equi-

librium settings. These settings are only updated upon convergence of the

macroscale model.

Since a micro model is solved for each macroscopic integration point on

the cohesive surface for each macroscopic Newton-Raphson procedure, the

procedure is very expensive from a computational effort point of view. A sig-

nificant reduction in computation time can be achieved by using parallel com-

puting, which is most conveniently done by creating multiple micro models

to be evaluated simultaneously for different macroscopic integration points.

78


Since more micro models are created, the memory usage of the framework

will also increase significantly. In the case that an anisotropic microstructure

is considered, the definition of micro models for each individual macroscopic

interface integration point is required. In that case, parallel computing is an

even more attractive option since the disadvantage of added memory usage

is no longer relevant.


The three-point bending test with 4 mm off-centred notch as studied in Sec-

tion 2.5 and schematically shown in Figure 2.8 is considered with all dimen-

sions scaled down by a factor of 20. Hence, the studied specimen has a width

of 955μm and a height of 450μm. The specimen is poled in the horizontal

direction and a potential difference of V = 100 V is applied.

On the microscale, a polycrystalline microstructure is considered with 1

grain/μm2. The microscale finite element model discussed in Chapter 3 is

used to model the fracture of the microstructure. In order to have the mi-

croscale simulation as robust as possible, the polycrystal is assumed to fail

intergranularly, which appears to be a reasonable assumption considering the

employed grain size.

The constitutive behaviour of the grains is described using the linear

piezoelectric relations (2.14) with the parameters in Table 2.1. Note that

the polarisation direction on the microscale depends on the direction of the

macroscopic crack (since the microscopic coordinate system is not aligned

with the macroscopic coordinate system). The cohesive behaviour is based

on the initially elastic mechanical cohesive law (3.9) with initial stiffness 1 ·109 MPa/mm is used. The fracture strength and fracture toughness are taken

as 40 MPa and 1·10−2 N/mm, respectively. The mode-mixity parameter β and

penetration stiffness kp are assumed to be equal to 2 and 1 · 1014 MPa/mm,

respectively. The grain boundary thickness is approximated to be 10 nm

(Tan and Shang, 2002). Remind that this grain boundary thickness is merely

a material parameter and that the grain boundaries are modelled using zero

thickness interface elements. The grain boundary permittivity is assumed to

be equal to the permittivity of the bulk material in the direction of polarisa-

tion.

For the computations presented here the only nonlinearity considered on

the microscale is the failure process of the grain boundaries. It should, how-

79


and meshParameters

Macroscale Microscale

Bulk

Cohesive

Output

Y Y

N N

N

Y

Homogenisation

Evaluatebulk constitutive

behaviour

Evaluatecohesivebehaviour

Homogenisecohesivebehaviour

Initialise

Assembleforce vectorsand tangents

Updatestate vector

Conver-gence?

Parametersand mesh

Initialise

Assembleforce vectorsand tangents

Updatestate vector

Conver-gence?

Violation?

Nucleate orpropagate

crack

(Re)setloading, stateand history

Commithistory

parameters

Homogenisebulk

tangents

(Re)setloading

and state

Figure 4.6 Flow diagram of the developed multiscale finite element program.

80


ever, be emphasised that the multiscale framework presented here does not

put restrictions on the microstructural complexity. Hence, phenomena like

domain switching, transgranular fracture and many others can be incorpo-

rated in the same framework. Consideration of such complex microstructural

behaviour is beyond the scope this work. In order to focus on the proposed

homogenisation scheme, the microstructural model is intentionally kept rel-

atively simple.

4.4.1 Determination of the representative volume element size

In multiscale analyses, determination of an appropriate size for the micro-

structural model is one of the most important issues. Generally it can be

stated that the size should be chosen such that:

◦ The microstructural domain is large enough such that the homogenised

properties become independent of microstructural variations.

◦ The microstructural domain is small enough such that separation of

scales is guaranteed. That is, over the size of the microscale domain, the

homogenised kinetic and kinematic properties are practically constant.

Upon satisfaction of both conditions we speak of a representative volume

element (RVE). Determination of the RVE size is here done by a priori con-

sideration of the first condition. Since the second condition depends on the

macroscale solution, in particular on the size of the process zone, it can only

be checked a posteriori.

The same representative volume element is here used for the bulk and

cohesive homogenisation schemes. It is important to note that using the ho-

mogenisation scheme proposed in this dissertation, a homogenised traction-

opening relation can be determined that is independent of the micro model

size. This was already illustrated by the one dimensional example discussed

in Section 4.1.4. In contrast to the case in which homogenisation is used to

derive a stress-strain relation, as discussed by Gitman et. al (2007), a rep-

resentative volume element can be determined for the homogenisation of

a softening traction-opening law. Obviously, the proposed homogenisation

scheme cannot be employed in a macroscale continuum damage formulation,

since such a formulation requires the provision of a stress-strain law instead

of a traction-opening law.

81


×10−5

Microstructure size (w = h) [μm]

∂<ε

m 11>

Ωm

∂<σ

m 11>

Ωm

∣ ∣ ∣ 0[M

Pa−1]

97531

1.05

1.02

0.99

0.96

0.93

0.9

×108

Microstructure size (w = h) [μm]∂<E

m 1>

Ωm

∂<D

m 1>

Ωm

∣ ∣ ∣ 0[V

2/N]

97531

1.022

1.02

1.018

1.016

1.014

1.012


tM 1

∣ ∣ ult[M

Pa]

97531

36

34

32

30

28

26

×10−6


qM∣ ∣ u

lt[N/(V

mm)]

97531

-2.2

-2.25

-2.3

-2.35

-2.4

-2.45

Figure 4.7 Dependence of homogenised properties on the dimensions of thesquare polycrystalline microstructure. The lines indicate the mean values aswell as the (plus and minus) half times the standard deviation values.

In this work square polycrystals of 2μm, 4μm, 6μm and 8μm are con-

sidered. For each size, 20 realisations are used. In order to study the ap-

propriateness of the polycrystals for homogenisation, two components of

the homogenised initial tangent are studied. Moreover, the ultimate traction

in normal direction and ultimate surface charge density are considered for

the case that the polycrystals are loaded in horizontal direction with an ap-

plied electric field of 1 kV/mm. The results of this size comparison study

are shown Figure 4.7. The study of the four quantities mentioned above is

assumed to be representative for the complete homogenisation procedure.

Moreover, the restriction of the RVE size study to the mode I case is assumed

to be appropriate since for the multiscale simulations considered in the up-

coming sections the macro cracks are observed to be mode I dominated.

82


The homogenised elastic and dielectric compliance components are ob-

served to converge in the sense that the coefficient of variation tends to go

to zero upon increasing the size of the polycrystal. For the 2μm model, coef-

ficient of variation of 3.2% and 0.2% are noticed for the elastic and dielectric

compliance component, respectively. When increasing the size to 8μm, these

coefficients decrease to 0.6% and 0.1%. Moreover, it is seen that for both quan-

tities the mean value hardly varies. The observations for the homogenised

ultimate traction and ultimate surface charge density differ from these for

the compliance components. The mean values are noticed to converge to an

asymptote when the microstructure size is increased to 8μm. The decrease

in ultimate values upon increasing the specimen size is explained by the fact

that more crack paths become available. Since the micro cracks choose the

path that requires minimum energy dissipation, the required load to rupture

a polycrystal decreases when more paths become available. Also the coeffi-

cient of variation is noticed to diminish from 6.6% to 4.2% for the ultimate

traction and from 0.9% to 0.6% for the ultimate surface charge density when

increasing the size from 4μm to 8μm. The coefficient of variation of the frac-

ture strength for the 2μm model is not following this trend. The reason for

this is that the actual randomness in microstructural geometry is constrained

due to the limited size of the microstructure.

Based on the RVE size study presented above, a representative volume size

of 8μm is selected. For this size both the compliance components and the ul-

timate traction and surface charge density tend to converge to a stable mean

value. Although a (slightly) bigger RVE size is desirable when considering the

coefficient of variation of the fracture strength, the 8μm RVE is considered

as an appropriate balance between computational costs and model accuracy.

Moreover, in order to appropriately represent the average microstructural

response, the realisation of the microstructure being closest to the mean ul-

timate traction is used. It should be noted that it is possible to use different

representative volume elements for different macroscale integration points.

Here, the same RVE is used everywhere in order to reduce the memory usage

of the multiscale computational framework.

4.4.2 Verification of the multiscale model using a full-resolution simulation

In the case of unlimited computational power the results of the multiscale

model could be verified using a full-resolution model in which the complete

granular microstructure is represented. However, due to the huge amount of

83


degrees of freedom required, such a computation cannot be performed. In

order to verify the multiscale framework, a macroscopic model is considered

in which the microstructure is only incorporated in the region of interest,

i.e. in the area where the crack runs (see Figure 4.8). The granular microstruc-

ture is modelled in a zone of 4× 100μm2 at an angle of 60 degrees with the

horizontal axis. In order to make a fair comparison, the crack in the multi-

scale model is also forced to run at this angle. Moreover, for consistency the

multiscale model is here employed with a micro model of 4× 4μm2.

Contour plots of the Cauchy stress in horizontal direction (see Figure 4.8)

show that the microscopic stress fluctuations are smeared out in the mul-

tiscale model. Comparison of the force-displacement curves for both sim-

ulations demonstrate that this averaging hardly affects the global response

of the system. The ultimate load is accurately predicted by the multiscale

model. The stiffness is slightly underestimated by the multiscale model. This

indicates that on average the homogenised stiffness of the micro model is

slightly lower than that of the considered representative volume element.

The fringing grains, observed in the full-resolution simulation but not in the

microscale representative volume element simulations, partially explain this

mismatch since they locally provide extra rigidity to the microstructure.

When considering the computational effort, a significant difference be-

tween the two methods is present. With 23214 elements (21049 linear trian-

gular bulk elements and 2165 four node linear interface elements) and 38868

degrees of freedom, construction of the force-displacement curve for the full-

resolution model requires approximately 20 times the computation time of

the multiscale model with 3811 macroscale linear triangular elements and

5892 macroscale degrees of freedom, when using a micro model with 268 el-

ements (192 linear triangular bulk elements and 76 four node linear interface

elements) and 600 degrees of freedom. For both simulations just over 150

steps are required for tracing of the equilibrium paths shown in Figure 4.8.

Moreover, it is emphasised that the multiscale model remains applicable in

the case that the crack trajectory is not defined a priori. For that situation

use of the full-resolution model is impractical.

4.4.3 Multiscale simulation

The multiscale model is now demonstrated using an 8μm micro model. On

the macroscale the same mesh as used in the previous subsection is con-

sidered. On the microscale, a mesh with 708 bulk elements, 292 interface

84


4μm60◦

100μm

Multiscale Full-resolution

×10−5

Displacement [mm]

F[N

]

453525155-5

3.5

3

2.5

2

1.5

1

0.5

0

Figure 4.8 Comparison of the horizontal stress contours and force-displacement curves obtained by the multiscale model (dashed curve) andby the full-resolution model (solid curve). The displacements in the con-tour plots are amplified by a factor of 10 and the force F acts as shown inFigure 2.8 and is plotted versus the downward displacement of the loadingpoint.

elements and a total of 2208 degrees of freedom is employed. The instance

and direction of propagation are based on the principal stress obtained by

equation (2.19) with the smoothing length lR taken as 5μm. The simulation

is performed on a single processor. It should be noted that the proposed

framework is suitable for parallel computing since the majority of the com-

putational effort is spent in solving the individual micro models.

The force-displacement curve for the experiment is shown in Figure 4.9.

The maximum load carried by the specimen equals 2.97 N. In Figure 4.10, the

horizontal stress contour is shown. It is observed that the cohesive zone

size is considerable larger than the size of the microstructural RVE. As a

consequence, the variation of the microscopic quantities over the size of the

RVEs is limited. The second RVE requirement as mentioned in Section 4.4.1

is therefore satisfied for the case considered here. For problems where the

separation of scales between the cohesive zone size and characteristic dimen-

sions of the microstructure is smaller, the second RVE requirement becomes

troublesome. The proposed framework can remain useful in such a situation,

but application of the method should be done with great caution.

In Figure 4.10 the electric potential over the specimen is shown. The cor-

responding solutions of the microscopic models are also depicted. As can

be seen, the macroscopic displacement jump is caused by a fractured grain

85


×10−5

Displacement [mm]

F[N

]

453525155-5

3

2.5

2

1.5

1

0.5

0

Figure 4.9 Force-displacement curve for the miniaturised three-point bend-ing test as modelled by the constitutive multiscale framework with 8μmrepresentative volume elements.

boundary. The traction-opening relations for the two microstructural mod-

els in Figure 4.10 are shown in Figure 4.11. The compatibility opening for

both micro models is in the order of magnitude of 1 nm. The corresponding

amount of dissipated energy is negligible compared to the total fracture en-

ergy of the micro models. As a consequence, the influence of the nonlinear

behaviour of the microscale prior to macroscopic crack nucleation is limited.

86


1680σ11 [MPa]

A

3224

Φ [V]

B

40

0 20 40 60 80 100

Figure 4.10 Contour plots of the horizontal Cauchy stress (top) and electricpotential field (bottom) for the miniaturised three-point bending specimenat the ultimate load. The displacements are 10 times magnified.

87


×10−4

Opening uM1 [mm]

Tra

ctio

ntM 1

[MPa]

1614121086420

35

30

25

20

15

10

5

0

Figure 4.11 Traction-opening curves in normal direction for the microstruc-tural representative volume elements A (solid) and B (dashed) shown in Fig-ure 4.10.

88

Chapter 5

Dissipation-based arc-length control forthe simulation of failure

Most of the results presented in the previous chapters required trac-

ing of the equilibrium path beyond the ultimate load. In some cases,

even the complete equilibrium path was determined. Tracing of the equi-

librium path is a subject which is not only encountered in this work, but in

quasi-static solid mechanics problems in general. The availability of a robust

method for stepwise determination of the required points on the equilibrium

path is therefore indispensable.

Several approaches have been proposed in the past. The pioneering work

of Riks (1979) is worthy of mention, together with the alternative formula-

tions by Ramm (1981) and Crisfield (1982). A comprehensive review of the

available techniques is provided by Geers (1999a,1999b). The traditional

approach consists of parametrising the equilibrium path with its own arc

parameter, i.e., with the norm of the incremental degree-of-freedom vector,

which also motivates the generic "arc-length" denomination for this kind of

methods. The arc parameter works properly for problems exhibiting geomet-

rical nonlinearities but often fails when material instabilities are involved that

would lead to localised failure process zones. A remedy for this problem is

to consider only the degrees of freedom involved in the failure process to be

coupled to the path parameter (de Borst, 1987). While such an approach is

robust in general, it is not applicable when the location or behaviour of the

failure process zone is not a priori predictable. This is the case when Monte-

Carlo simulations of failure of heterogeneous materials are carried out or

when crack propagation or interfacial delamination is the relevant dissipa-

tive phenomenon. Remedies for this problem have been proposed in Geers

89

Dissipation-based arc-length control for the simulation of failure

(1999a,1999b) and Alfano and Crisfield (2003), in which the path parameter

is coupled to smartly selected internal variables.

A path-following constraint based on the energy release rate in the case of

a geometrically linear continuum damage model was introduced by Gutiérrez

(2004). This method has the advantage that the dissipated energy is a global

quantity and therefore no a priori selection of degrees-of-freedom is required.

Moreover, as such a constraint is directly related to the failure process itself,

a stable convergence behaviour is observed even for far advanced stages of

the equilibrium path.

In this chapter, the path-following constraint as proposed by Gutiérrez

(2004) is extended to the case of geometrically linear plasticity computa-

tions and geometrically nonlinear damage computations. The applicability

of the constraint to the electromechanical problems considered in the previ-

ous chapters is also considered.

5.1 Path-following in quasi-static solid mechanics problems

Equilibrium in quasi-static nonlinear solid mechanics problems is generally

governed by nonlinear systems of equations of the form

fint(a) = ηf, (5.1)

in which fint is an internal force vector and η and f being a scalar load scale

and unit load vector, respectively. From (5.1) it is obvious that the product of

this load scale and unit load vector is referred to as the external load vector

fext. The nonlinearity of this system of equations is reflected by the (gen-

erally nonlinear) dependence of the internal force on the displacement field

a. Note that the discrete electromechanical equilibrium equations derived in

equation (2.11) and (3.4) are of the same form as equation (5.1).

For each value of the load factor η, a solution a of this system of equations

can be obtained. The collection of equilibrium points (a, η) of the system is

referred to as the equilibrium path. In practice, this equilibrium path is traced

in an incremental fashion. Given a point on the equilibrium path (a0, η0),

the next point on the path can be computed by solving the set of nonlinear

equations

fint (a0 +Δa) = [η0 +Δη]f, (5.2)

for the incremental displacement Δa and incremental load factor Δη. Since

this system of n equations has n + 1 unknowns (the degrees of freedoms a

90

Energy release rate path-following constraint

plus the load factor η), it is indeterminate. In order to solve it, an additional

constraint equation needs to be specified. Since this constraint prescribes

the steps made on the equilibrium path, it is normally referred to as the

path-following constraint. In general, this path following constraint can be

written as

g (a0, η0,Δa,Δη,τ) = 0, (5.3)

where τ is the prescribed path-parameter that determines the size of a step.

The equilibrium state of this well-posed, augmented system of n+1 equations

can be solved simultaneously from[fint

g

]=[ηf

0

]. (5.4)

This can be done in an iterative fashion by using a Newton-Raphson scheme.

The solution (a, η) at iteration number k+ 1 is equal to[Δak+1

Δηk+1

]=[Δak

Δηk

]+[

K −f

hT w

]−1 [rk

−gk], (5.5)

where K and r are the stiffness matrix and the residual, which are respectively

defined as

K = ∂fint

∂a;

r = ηf− fint.

(5.6)

The vector h and the scalar w in equation (5.5) are defined as

h = ∂g∂a

;

w = ∂g∂η.

(5.7)

5.2 Energy release rate path-following constraint

The only requirement to the choice of constraint equations of the form (5.3) is

that for a given set of unknowns (Δa,Δη) the path parameter τ is monoton-

ically strictly increasing. For the simulation of damage evolution, a natural

choice is to use a constraint based on the rate of energy dissipation (Gutiér-

rez, 2004). By virtue of the second law of thermodynamics, the rate of dissipa-

tion is non-negative. In the case of evolving damage, the rate of dissipation is

91


positive and hence it is suitable to be used as a path parameter. Alternatively,

when damage is not evolving, the rate of dissipation is equal to zero and it

cannot be used to trace the equilibrium path. Path-following constraints that

can trace such non-dissipative equilibrium paths (or parts thereof) are well-

developed. This work focuses on the development of a dissipation-based

path-following constraint. The treatment of non-dissipative parts of the equi-

librium path is further addressed in Section 5.3 and is illustrated with an

example in Section 5.4.1.

The rate of dissipation of a body G is equal to the exerted power P minus

the rate of elastic energy U

G = P − U, (5.8)

where � denotes the derivative with respect to time. In order to use the rate

of dissipation as a path-following constraint, it should be expressed in terms

of the nodal displacements a, the load factor η and the unit external force

vector.

The exerted power is defined as the applied external force times the nodal

velocity. In terms of the discretised model, this can be written as

P = fTexta = ηfTa. (5.9)

The expression for the elastic energy stored in the solid depends on the con-

stitutive behaviour of the material as well as on the kinematic formulation

that is used. The rate of elastic energy and the rate of dissipation will be

derived for three mechanical types of problems: (i) a geometrically linear

kinematic formulation with damage, (ii) a geometrically linear kinematic for-

mulation in combination with plasticity and (iii) a geometrically nonlinear

kinematic formulation in combination with damage. A path-following con-

straint will be derived for these three cases. In addition, a path-following

constraint is derived for the geometrically linear electromechanical case with

damage, as considered in the previous chapters of this thesis.

5.2.1 Geometrically linear mechanical model and damage

In Gutiérrez (2004) a path-following constraint based on the energy release

rate is proposed for a damage model in combination with a geometrically

linear kinematic formulation. An important assumption for such a model

is that the unloading behaviour is linear elastic and hence unloading occurs

along the secant, as shown in Figure 5.1. The elastic energy stored in the solid

92


can then be written as

U = 1

2

∫Ω

εTσdΩ, (5.10)

where ε is the engineering strain and σ is the Cauchy stress (both in Voigt

form). Assuming a small strain kinematic relation, the strain field can be

expressed in terms of the nodal displacements as

ε = Ba, (5.11)

where the strain-nodal displacement matrix B is independent of the nodal

deformation a. Substituting this expression into (5.10) in combination with

the expression for the definition of the internal force vector, the elastic energy

is written as

U = 1

2aT

∫Ω

BTσdΩ = 1

2aTfint. (5.12)

Assuming that the system is in equilibrium, relation (5.1) holds and the elastic

energy can be written in terms of the nodal displacement vector, unit load

vector and load scale as

U = 1

2ηaTf. (5.13)

Taking the derivative of this equation with respect to time, the rate of change

of the elastic energy equals

U = 1

2ηaTf+ 1

2ηaTf. (5.14)

Substituting this relation and the expression for the exerted power (5.9) into

equation (5.8) gives

G = 1

2fT (ηa− ηa) . (5.15)

A forward Euler discretisation is used to obtain the corresponding incremen-

tal path-following constraint of the form (5.3) as a function of the path pa-

rameter τ as

g = 1

2fT (η0Δa−Δηa0)− τ. (5.16)

Note that η0 and a0 are the converged load factor and the displacements from

the previous step, respectively. The derivatives required for the construction

of the consistent tangent (5.5) then read

∂g

∂a= 1

2η0f;

∂g

∂η= −1

2fTa0.

(5.17)

93


As can be seen the employed forward Euler time discretisation yields ad-

ditional consistent tangent terms that are independent of the displacement

increment, making it attractive from an algorithmic point of view. Other

time discretisations can be used (e.g. backward Euler), but will lead to more

complicated expressions. Moreover, no problems with the forward Euler dis-

cretisation were encountered in any of the sample problems considered.

In the one-dimensional case, the dissipation increment as formulated in

equation (5.16) can be illustrated as the shaded area in the force displacement

diagram shown in Figure 5.1. Note that when the complete equilibrium path

is followed, the dissipation increments add up to the maximum energy that

can be dissipated by the considered structure (i.e., the total area under the

force displacement diagram).

0 a0 a0 +Δa

ηf

η0f

[η0 +Δη]f

a

Figure 5.1 Schematic representation of the dissipation increment (shadedarea) in the case of secant unloading. The shaded area is equal to the energydissipation increment τ = 1

2 f (η0Δa − Δηa0), which is a one-dimensionalrepresentation of equation (5.16).

5.2.2 Geometrically linear mechanical model and plasticity

In the case of plasticity, unloading occurs along a path parallel to the elastic

tangent (Figure 5.3). The elastic energy stored in a solid can then be written

as

U = 1

2

∫Ω

εTeσdΩ, (5.18)

where εe is the elastic part of the strain. Since in the case of plasticity the

stress σ is linearly related to the elastic strain εe via the elastic stiffness

94


matrix De, the elastic energy can be rewritten as

U = 1

2

∫Ω

σTD−1e σdΩ. (5.19)

Making use of the symmetry of the elastic stiffness De, the rate of elastic

energy is derived as

U =∫Ω

σTD−1e σdΩ =

∫Ω

εTCTD−1e σdΩ, (5.20)

with C = ∂σ/∂ε being the consistent tangent, which is generally non-sym-

metric. Using the strain-nodal displacement matrix (5.11) the rate of elastic

energy is obtained as

U = aTf∗, (5.21)

where f∗ is a nodal force vector, which is defined as

f∗ =∫Ω

BTCTD−1e σdΩ. (5.22)

The energy release rate then follows from equation (5.8) as

G = aT

(ηf− f∗

), (5.23)

where use is made of expression (5.9) for the exerted power. Note that as

long as the deformation of all points in the domain Ω is elastic, the consistent

tangent C is identical to the elastic stiffness De. In this case, the nodal force

vector f∗ is equal to the internal force vector and the energy release rate G as

presented in (5.23) is equal to zero. Using a forward Euler time discretisation

the path-following constraint can be expressed as

g = ΔaT

(η0f− f∗0

)− τ (5.24)

and the derivatives that are required for the computation of the consistent

tangent (5.5) are obtained as

∂g

∂a=(η0f− f∗0

);

∂g

∂η= 0.

(5.25)

Note that, as a consequence of the forward Euler time discretisation, the ad-

ditional force vector f∗ only needs to be computed after each converged load

95


a

l

A

ηf

Figure 5.2 One dimensional bar loaded in tension

step. This implies that the energy release rate constraint can be applied at

the cost of only one more vector assembly per load step. For large systems of

equations that make use of the consistent tangent, the additional computa-

tional effort will therefore be negligible. An additional advantage of the fact

that the additional force vector is only required after each converged state is

that it does not depend on the increments of the nodal displacement vector.

In order to indicate the dissipation increment in the case of plasticity in

a graphical fashion, consider a one-dimensional bar loaded in tension (Fig-

ure 5.2). Assuming a uniform stress σ = ηf/A and uniform strain ε = a/l,the path parameter yields

τ = Δa(η0f − f∗0

)= Δa

(1− C0

De

)η0f ≈ Δapη0f , (5.26)

where use is made of

C0 ≈ Δσ

Δε= l

A

Δηf

Δa;

De ≈ Δσ

Δεe= l

A

Δηf

Δae.

(5.27)

The rate of dissipation in the case of plasticity (5.26) is graphically indicated

in Figure 5.3. It needs to be emphasised that the additive decomposition

of displacements (Δa = Δae + Δap) is only used for the one-dimensional

beam with uniform stress and strain fields. This decomposition is therefore

only used for illustrative purposes and not for the derivation of the multi-

dimensional constraint (5.24), which is based on the assumption of the addi-

tive decomposition of strains (ε = εe + εp).

96


0 aa0

ηf

η0f

[η0 +Δη]f

a0 +Δa

AlC0

1

1

AlDe

Δap

Figure 5.3 Schematic representation of the dissipation increment (shadedarea) in the case of elastic unloading. The shaded area equals

Δap

(η0 + 1

2Δη)f . For small increments (Δa,Δη � 1), this area is equal

to the dissipation increment τ = Δa(η0f − f∗0 ), which is a one-dimensionalrepresentation of equation (5.24).

5.2.3 Geometrically nonlinear mechanical model and damage

In the previous sections, small displacements and strains were assumed.

However, in many situations, such assumptions cannot be made. In this

section a damage description in combination with a large displacement but

small strain formulation is considered. In unloading, a linear elastic rela-

tion between the second Piola-Kirchhoff stress and Green-Lagrange strain is

assumed.

In such a finite deformation kinematic model, a distinction must be made

between the original and the current coordinate system. Using a Lagrangian

formulation (Bathe, 1996), the internal virtual work can be written as

δWint =∫Ωt+Δt

σTδεdΩt+Δt , (5.28)

where Ωt+Δt is the configuration of a body Ω at time t+Δt. Generally the inte-

gration is not performed over the current configuration but over a reference

configuration. In the case of a total Lagrangian formulation, the undeformed

configuration (Ω0) is taken as reference. In the case of an updated Lagrangian

formulation, the previously converged solution is taken as reference (Ωt).

Both formulations are suitable for modelling large displacements, large rota-

tions and large strains. The choice of one formulation rather than the other

97


is therefore generally made on the basis of arguments concerning numerical

efficiency or ease of implementation. In the case considered here, the total

Lagrangian formulation is most suitable since for that formulation the rate of

dissipation can easily be expressed in terms of the nodal displacements and

forces.

Using the total Lagrangian formulation, the virtual internal work is equal

to

δWint =∫Ω0

STδγdΩ0, (5.29)

where γ is the Voigt form of the Green-Lagrange strain tensor† and S is the

Voigt form of the second Piola-Kirchhoff stress tensor, both defined with re-

spect to the initial configuration. The Green-Lagrange strain tensor is defined

as

γij = 1

2

[(∂uk

∂xi+ δki

)(∂uk

∂xj+ δkj

)− δij

]. (5.30)

An increment of the Green-Lagrange strain can be related to an increment of

the nodal displacements via

δγ = Bδa. (5.31)

where the B is the geometrically nonlinear equivalent of the B matrix (5.11).

In contrast to the geometrically linear case, the matrix B depends on the

nodal displacement vector: B = B(a). Substitution of (5.31) in (5.29) then

yields the internal force vector

fint =∫Ω0B

TSdΩ0. (5.32)

Under the assumption of small strains (but large deformations and rotations)

it can be assumed that Hooke’s law can be applied to relate the second Piola-

Kirchhoff stresses to the Green-Lagrange strains in the case of elastic unload-

ing (Bathe, 1996). The internal elastic energy can then be expressed as

U = 1

2

∫Ω0γTSdΩ

0. (5.33)

The rate of change of elastic energy can then be derived as

U = 1

2

∫Ω0γTSdΩ

0 + 1

2

∫Ω0γTSdΩ

0. (5.34)

†In the Voigt form of the Green-Lagrange strain tensor, the shear strain is multiplied by a

factor of two, in order to satisfy STδγ = Sijδγij .

98


Using equation (5.31) to substitute the stain rate then yields

U = 1

2aTfint + 1

2

∫Ω0γTSdΩ

0, (5.35)

where use is made of (5.32). The latter term in this expression can be rewrit-

ten by using the symmetric material tangent

C = ∂S

∂γ, (5.36)

to yield

U = 1

2aTfint + 1

2

∫Ω0γTCγdΩ

0. (5.37)

Using equations (5.1) and (5.31) the rate of elastic energy can be formulated

as

U = 1

2aT

(ηf+ f∗

), (5.38)

in which

f∗ (a) =∫Ω0B

TCγdΩ0. (5.39)

The rate of dissipation can finally be determined using equation (5.8) as

G = 1

2aT

(ηf− f∗

). (5.40)

Note that in the case of a linear elastic material, the stress state S is equal

to the symmetrical tangent matrix C multiplied by the total strain γ. As a

result, the additional force vector f∗ in (5.39) is equal to the internal force

vector fint and the dissipated energy G, as formulated in (5.40) is equal to

zero. The path-following constraint is consequently obtained using a forward

Euler discretisation as

g (Δa,Δη) = 1

2ΔaT

(η0f− f∗0

)− τ. (5.41)

The derivatives of the constraint can directly be computed as

∂g

∂a= 1

2

(η0f− f∗0

);

∂g

∂η= 0.

(5.42)

As in the case of geometrically linear damage and geometrically linear plastic-

ity, the area that represents the dissipation increment can also be visualised

99


for the case of geometrically nonlinear damage computations. It is in this

case, however, not possible to represent the dissipation increment directly

by a simple area (with non-curved edges) in the force displacement diagram.

Alternatively, the dissipation increment can be visualised by an area in the

stress-strain curve. In order to reformulate the dissipation increment (5.41)

in terms of the Green-Lagrange strain and second Piola-Kirchhoff stress, the

one-dimensional bar loaded in tension (Figure 5.2) is again considered. In its

undeformed state, the bar has length l and cross-sectional area A.

The dissipation increment can in this one-dimensional case be derived

from equation (5.41) as

τ = 1

2Δa(η0f − f∗0

). (5.43)

Since the Green-Lagrange strain γ and corresponding B-matrix (in this one-

dimensional case this is actually a scalar) as defined in equation (5.30) and

equation (5.31) can be written as

γ(a) = 1

2

[(l+ al

)2

− 1

];

B(a) = l+ al2

,

(5.44)

the displacement increment can be approximated by

Δa ≈ δa

δγ

∣∣∣∣a=a0

Δγ = l2

l+ a0Δγ. (5.45)

The first term in between the parentheses in equation (5.43) can be reformu-

lated in terms of the stress and strain by using the internal force vector (5.32)

to yield

η0f = fint(a0) =∫Ω0

B(a0)S0dΩ0 = l+ a0

lAS0. (5.46)

The second term inside the parentheses in equation (5.43) can be obtained

from equation (5.32) as

f∗0 =∫Ω0

B(a0)C0γ0dΩ0 = l+ a0

lAC0γ0. (5.47)

Substitution of the equations (5.45), (5.46) and (5.47) in the expression for the

dissipation increment (5.43) and dividing by the volume of the bar then gives

100


the dissipation increment per unit volume as

τ

lA= 1

2[S0 − C0γ0]Δγ, (5.48)

which equals the shaded area in the stress-strain diagram as shown in Fig-

ure 5.4. As can be seen from Figure 5.4, the derived expression is indeed

equal to the dissipation increment.

γ

S

0 γ0 +Δγγ0

S0 +ΔS

S01C0

Figure 5.4 Schematic representation of the dissipation increment per unitvolume (shaded area), using a constitutive law in terms of the Green-Lagrange strain γ and second Piola-Kirchhoff stress S. Under the assump-tion that Δγ � 1, the shaded area equals 1

2 [S0 − C0γ0]Δγ, which is equal tothe dissipation increment per unit volume as formulated in equation (5.48).

5.2.4 Geometrically linear electromechanical model and damage

In the case of the electromechanical problems considered in the previous

chapters, formulation of the energy balance (5.8) is considerably more com-

plicated than in the purely mechanical case. An appropriate definition of the

rate of dissipation that serves as a robust path-following constraint is

G = P − Wint, (5.49)

in which Wint is the Gibbs free energy

Wint = 1

2

∫Ω

(σTε−DTE) dΩ. (5.50)

101


Note that the quasi-static equilibrium equations (2.8) resemble the minimi-

sation of this Gibbs energy. Using the finite element discretisation (2.6), the

Gibbs energy can be written as

Wint = 1

2η[

fT −gT

]( a

b

), (5.51)

which closely resembles the result as found in equation (5.13). The minus

sign in front of the electric unit external force vector g results from the fact

that the discrete electric equilibrium equation (2.11) was multiplied by -1.

The rate of Gibbs energy is obtained by differentiation of equation (5.51) with

respect to time

Wint = 1

2η[

fT −gT

]( a

b

)+ 1

2η[

fT −gT

]( a

b

). (5.52)

In accordance with the Gibbs energy, the electromechanical external power is

defined as

P =∫Γ

(tTu− qΦ) dΓ , (5.53)

which can be rewritten as

P =[

fT −gT

]( a

b

). (5.54)

Substitution of (5.52) and (5.54) in equation (5.49) yields the energy dissipa-

tion as

G = 1

2

[fT −gT

]{η

(a

b

)− η

(a

b

)}, (5.55)

which is discretised in time by a forward Euler discretisation to get

g = 1

2

[fT −gT

]{η0

(Δa

Δb

)−Δη

(a0

b0

)}− τ. (5.56)

The tangents required for the Newton-Raphson iterations (5.5) then follow as

∂g

∂a= 1

2η0f;

∂g

∂b= −1

2η0g;

∂g

∂η= −1

2

[fT −gT

]( a0

b0

).

(5.57)

102

Algorithmic aspects

Note that equation (5.56) closely resembles the result found for the purely

mechanical problem (5.16). Alternatively, the rate of dissipation can be based

on the Helmholtz free energy, which basically assumes the recoverable energy

to be equal to the sum of the elastic and dielectric energy (Verhoosel and

Gutiérrez, 2009a). The advantage of the formulation of the rate of dissipation

using the Helmholtz energy is that the minus sign in front of the electric force

vector in equation (5.55) disappears. Both the rate of dissipation in terms of

the Gibbs energy and in terms of the Helmholtz energy have the property

that they are equal to zero when the internal forces are linearly related to the

nodal degrees of freedom. Besides that, in the case that no electric loading

is applied (g = 0), both formulations coincide. In that case, the dissipation

increment can schematically be visualised as done in Figure 5.1.


The derived path-following constraints are suitable for incorporation in a fi-

nite element environment. In order to efficiently and flexibly incorporate

the constraints, some algorithmic aspects need further explanation. Here the

treatment of non-dissipative parts of an equilibrium path is treated, as well as

the step size adjustment algorithm. A detailed discussion on the incorpora-

tion of prescribed displacements is presented in Appendix A. A discussion on

the step size adjustment algorithm can be found in Verhoosel et. al (2009a).

As mentioned in Section 5.2, robust tracing of the complete equilibrium

path requires the path-following parameter to be non-negative. This require-

ment leads to problems in the case that non-dissipative parts exist on the

equilibrium path. Such parts appear for example when a materials initially

behaves elastically. Non-dissipative regions on the equilibrium path can, how-

ever, also occur in other situations. From a numerical point of view, the use of

the energy release rate constraint is also not attractive if the path parameter

gets close to the machine precision. Although the rate of dissipation might

then be non-negative, numerical errors can become critical.

For non-dissipative parts of the equilibrium path, alternative path-follow-

ing constraints must be used. In the case of a geometrically linear computa-

tion, a force control (or displacement control) constraint is suitable. In the

case of a geometrically nonlinear computation, snap-back can occur without

any energy being dissipated, making the use of force or displacement control

constraints impossible. In that case, the traditional arc-length constraints

103


(Riks, 1979) can be used to robustly trace the equilibrium path.

The implementation of a robust algorithm for switching from one con-

straint to another requires two issues to be addressed. The first is the defini-

tion of appropriate switching criteria. The second issue is the determination

of appropriate step sizes after switching. The treatment of these issues is

discussed in some detail in the first numerical experiment.


In this chapter, three examples are presented to demonstrate the dissipation-

based path-following constraints derived in Section 5.2. In order to empha-

sise the versatility of the method, a variety of finite element techniques has

been used. In the first example, fracture in a perforated beam is considered.

The emphasis here is put on the transition from a force controlled analysis to

the energy constraint analysis and vice versa. In the second example, the en-

ergy constraint for plasticity in combination with a linear kinematic relation

will be demonstrated by considering a polycrystal with softening plasticity in

its grain boundaries. Finally, the constraint for geometrically nonlinear dam-

age will be demonstrated by means of a buckling delamination experiment.

Numerous examples demonstrating the constraint for an electromechanical

formulation with a damage type of nonlinearity have been considered in the

previous chapters. Especially the simulations performed in Section 3.4 clearly

show the capability of the proposed path-following constraint to trace com-

plex equilibrium paths. Additional examples showing the versatility of the

proposed framework can be found in Verhoosel et. al (2009a).

5.4.1 Perforated cantilever beam

Consider the cantilever beam as shown in Figure 5.5. The beam is 7.5 mm

long and 1 mm thick. The beam is perforated across its entire length by holes

with a diameter of 0.2 mm. The spacing of the centre points of these holes

is 0.375 mm. The beam is made of a linear elastic material with Young’s

modulus Y = 100 N/mm2 and Poisson’s ratio ν = 0.3. The ultimate traction

for this material is set to tult = 1 N/mm2 and the fracture toughness is Gc =2.5·10−3 N/mm. The beam is loaded by two forces ηf as shown in the figure.

For the simulation, a plane strain condition is assumed.

Since the geometry of the specimen, its boundary conditions and the ap-

104


7.5 mm

0.375 mm

ηf

ηf

1.0 mm

v0.2 mm

x2

x1

Figure 5.5 Geometry and loading conditions of a perforated cantilever beam.

plied forces are symmetric with respect to the x1-axis, it may be assumed that

fracture takes place along this axis. In the finite element model, interface el-

ements as introduced in Chapter 3 have been placed here. In the derivations

carried out in Section 5.2 damage was supposed to occur in the continuum.

In the case considered here, damage occurs in a predefined interface. The

path-following constraint for this case is easily derived by considering an ad-

ditional elastic energy term such that equation (5.10) can be rewritten as

U = 1

2

∫Ω

εTσdΩ + 1

2

∫Γd

�u�Tt dΓd, (5.58)

in which t and �u� are the traction and displacement jump across a discon-

tinuity Γd. Following the derivation in Section 5.2.1 yields the same expres-

sion for the dissipation-based path-following constraint as obtained in equa-

tion (5.16).

From equation (5.58) it is obvious that a traction-opening law needs to be

supplemented. Because of symmetry of the considered problem, pure mode

I fracture is assumed. This is modelled by a bi-linear damage based cohesive

relation. In order to mimic a perfect bond prior to cracking, a linear dummy

stiffness of kinit = 1.0 · 104 N/mm3 is used. Note that because of the linear

elastic dummy stiffness, no energy is dissipated in the cohesive zones until

the traction reaches the ultimate value tult.

The considered problem is discretised using 9688 six-node triangles with a

seven-point Gauss integration scheme. The predefined interface is discretised

using 161 six-node interface elements with a three-point nodal integration

scheme. This discretisation results in a total of 40960 degrees of freedom.

105


An important aspect in this example is the fact that it is not possible to

use the energy constraint method throughout the complete simulation. This

is clearly shown in Figure 5.6, which displays the force-displacement diagram

for the perforated beam. Fracture of a segment between two holes will be

followed by a part without energy dissipation. In order to robustly trace the

equilibrium path, initially a force control constraint is used (indicated by cir-

cles ◦). When the dissipation increment becomes larger than 1 · 10−8 Nmm,

the simulation switches to the energy release constraint (indicated by solid

triangles �). While the energy release rate constraint is active, the step size

is adjusted by aiming for 5 Newton iterations per loading step. The maxi-

mum allowable dissipation increment τ and load step Δηf are respectively

taken as 1 · 10−5 Nmm and 0.011 N (corresponding to its value in the first

load step). The algorithm switches back to a force control constraint when

the scaled dissipation increment τ/|Δη| becomes smaller than 1·10−10 Nmm.

The load steps used for the force controlled part are taken as the average of

the absolute values of the 3 preceding force increments. During the force

controlled parts of the simulation, the step size is not adjusted. This switch-

ing algorithm is also used for the other numerical examples considered in

this section.

The shaded area in Figure 5.6 corresponds to half the amount of energy

that is dissipated during the complete fracture of a single segment between

two consecutive holes (since the external force is applied at two points), which

is equal to the length of these segments times the fracture toughness: 0.175·Gc = 4.375 · 10−4 Nmm. The dashed lines A, B and C represent the elastic

load-displacement curves for the cases in which the crack has propagated

across 1, 2 and 3 segments, respectively. The deformed specimen is shown

in Figure 5.7.

5.4.2 Polycrystalline structure with interface plasticity

The energy release rate constraint (5.24) for plasticity is demonstrated here

for softening plasticity. This type of plasticity is of particular interest since

global snap-back can occur. In order to regularise the problem, the plastic

flow is assumed to occur in predefined interfaces (Simo et. al, 1993). These

interfaces are discretised by interface elements that are inserted in the finite

element mesh before the computation. In order to determine the contribution

of the interface elements to the internal force vector, the traction t on an

106


C

B

A

Step 200

v [mm]

ηf

[N]

0.050.040.030.020.010

0.12

0.1

0.08

0.06

0.04

0.02

0

Figure 5.6 Load-displacement curve of the perforated cantilever beam.Force controlled steps are indicated by circles (◦) and energy release ratecontrolled steps by solid triangles (�). Note that the shaded area corre-sponds to half of the energy dissipated during the fracture of a single seg-ment between two consecutive holes. The dashed lines A, B and C representthe elastic load displacement curves for the cases in which the crack haspropagated across 1, 2 and 3 segments, respectively.

Figure 5.7 Deformation of the perforated beam (scaled by a factor of 5)after 200 steps.

107


u

25μm

ηf

25μ

m

Figure 5.8 Schematic representation of a geometrically periodic polycrystalgenerated using a Voronoi algorithm, subject to periodic boundary condi-tions.

interface has to be related to the interface opening �u� by

t = De

(�u�− υn

), (5.59)

where υ is the plastic multiplier and n is the normal to the yield surface. In

this case the von Mises yield criterion (von Mises, 1928) is used in combina-

tion with a work softening (negative hardening) relation. Under the assump-

tion of plane strain, the von Mises yield criterion can be written as

fyield =√

3J2 −σult(κ) =√t2n + 3t2

t −σult,0 − hκ = 0, (5.60)

where tn and ts are respectively the normal and tangent components of the

traction, σult is the yield stress, h the hardening parameter and κ the history

parameter that evolves according to

κ = υtTn. (5.61)

In each integration point the tractions are computed on the basis of the open-

ing by solving the set of nonlinear equations (5.59) and (5.60) using a return-

mapping algorithm (Simo and Taylor, 1986).

To demonstrate the suitability of the path-following constraint for plas-

ticity, a polycrystal of 25 by 25 micrometers (Figure 5.8) is considered. The

grains are generated using a Voronoi algorithm based on a set of random

108


nucleation points (see Section 6.2). Geometrical periodicity of the polycrys-

tal is enforced in order to apply periodic boundary conditions. The other

boundary conditions are shown in Figure 5.8, where a point load is applied to

the bottom-right support in order to deform the polycrystal. The grains are

modelled using plane strain continuum elements with modulus of elasticity

Y = 200 GPa and Poisson’s ratio ν = 0.3. In the interfaces, the yield stress

is σult,0 = 200 MPa and two possible choices for the hardening parameter are

considered, h = −55 · 103 mm−1 and h = −150 · 103 mm−1.

The polycrystal is discretised using 2022 linear triangular elements (i.e.

three nodes per element) with Gauss integration scheme. The interfaces are

modelled by 190 linear line elements with nodal integration scheme. This

discretisation leads to a total of 2700 degrees of freedom.

The response of the polycrystal is shown in Figure 5.9 for the case that

h = −55 · 103 mm−1. As can be seen, the polycrystal behaves linearly up to

80 percent of the yield stress, which allows for large load steps. At about

170 MPa some interfaces start to yield and energy is dissipated. The method

switches from force control to energy release rate control and the step size

is automatically adjusted. Near the maximum load the rate of dissipation

is large such that the energy release rate control uses small steps. In the

unloading part, initially less energy is dissipated and hence the step size

increases again. At some point (displacements larger than 3 · 10−5 mm) the

steps get smaller again. In Figure 5.9 it can be observed that this is caused

by the reduced path parameter. The reduction of this parameter is caused

by the fact that in this stage the maximum number of return-map iterations

is exceeded, due to a highly curved yield surface. Note that the small steps

required here are also required in the case that displacement control would

be used.

In the case of h = −150 · 103 mm−1 the snapback is more dramatic than

in the case of h = −55 · 103 mm−1, as can be seen from Figure 5.10. The

path-following constraint is, however, still capable of following the equilib-

rium path. In Figure 5.10 it can be seen that the maximum value for the

path parameter that is used, is approximately five times smaller than in the

previous case. The irregularity in the unloading path is caused by the fact

that the yielding initially localises in a single interface. When this interface

failed completely some other interfaces start to yield. The final part of the

unloading curve is comparable to that in the case of h = −55 · 103 mm−1.

109


×10−5

u [mm]

ηf

[N]

76543210

5

4

3

2

1

0

×10−6

Step

τ[m

J]

100806040200

4

3

2

1

0

Figure 5.9 Force-displacement diagram (left) and path parameter increment(right) for the polycrystal with hardening modulus h = −55 · 103 mm−1.

×10−5

u [mm]

ηf

[N]

32.521.510.50

5

4

3

2

1

0

×10−7

Step

τ[m

J]

1801501209060300

8

6

4

2

0

Figure 5.10 Force-displacement diagram (left) and path parameter in-crement (right) for the polycrystal with hardening modulus h = −150 ·103 mm−1.

110


Cohesive interface Traction free interface

a0

ηf

ηf

l

u1

P

Phu2

Figure 5.11 Double cantilever beam with an initial delamination a0 loadedin compression.

5.4.3 Delamination buckling of a double cantilever beam

The energy release rate constraint for a geometrically nonlinear model with

damage is demonstrated by considering the double cantilever beam shown

in Figure 5.11. The double cantilever beam has length l = 20 mm and semi

height h = 0.4 mm. An initial delamination of length a0 = 10 mm is present.

The beam is loaded by two compressive forces ηf placed in the midpoints

of the upper and lower part and two (small) perturbations P to trigger the

correct buckling mode.

Application of interface elements in a geometrically nonlinear setting has

been discussed in e.g. Allix and Corigliano (1999) and Wells et. al (2002).

Under the assumption that the crack opening remains small as long as the

interface has not fully cracked, the elastic energy can be formulated in the

undeformed configuration (Wells et. al, 2002) giving an additional integral

contribution to the elastic energy similarly as in equation (5.58). Following

the derivation as carried out in Section 5.2.3 yields an expression for the

energy release rate with the interface contributions included.

The bulk material is modelled using a plane strain assumption, with mod-

ulus of elasticity Y = 135 GPa and Poisson’s ratio ν = 0.18. The interfaces are

modelled using interface elements, where the Xu-Needleman (Xu and Needle-

man, 1993) constitutive law is used with an ultimate traction tult = 75 MPa.

Two settings for the fracture toughness are considered, Gc = 0.1 N/mm and

Gc = 0.01 N/mm.

The double cantilever beam is discretised using 1760 quadrilateral ele-

111


00.1 0.2 0.3 0.4 0.5

4

8

20

16

12

u1 [mm]

ηf[N]

0

20

16

4

8

12

12 164 8

ηf[N]

u2 [mm]

Figure 5.12 Force-displacement diagrams for the double cantilever beamwith fracture toughness Gc = 0.1 N/mm. The dashed-dotted lines representthe buckling loads for the original specimen without additional delamina-tion (Fcr = 17.77 N) and the fully delaminated specimen (Fcr = 4.44 N).

ments with 9 nodes per element, which leads to a total of 17620 degrees of

freedom. The cohesive interface is modelled using 400 quadratic interface

elements. The unit force f is set to 0.1 N and the perturbation forces P are

taken as 1.0 · 10−5 N.

The difficulty in this example is that there are two competing nonlinear

mechanisms. First the part of the specimen that is initially delaminated will

buckle locally due to the compressive load. Due to the lateral displacement,

the normal interface stress at the tip will increase which will result in a pro-

gressive delamination.

The force displacement curves for the case that Gc = 0.1 N/mm are shown

in Figure 5.12. As can be seen, the load is increased until the buckling load

is reached. At the moment that the interface starts to delaminate, the force

drops due to the softening in the interface. This softening continues until

the two bars are fully delaminated, after which the force again increases as

would be expected from the post-buckling behaviour.

In Figure 5.12 the upper dashed line indicates the analytical buckling load

of a beam with length a0 which can be computed by using the standard ex-

pressions for the critical loads of slender beams loaded in compression

Fa0cr = π

2Eh3

48a20

= 17.77 N, (5.62)

and the lower dashed line indicates the critical load in the case of the fully

112


u1 [mm]

0.005 0.0150.010

ηf[N]

0

12

16

20

4

8

0.2 0.4 0.6

12

16

20

8

4

0

ηf[N]

u2 [mm]

Figure 5.13 Force-displacement diagrams for the double cantilever beamwith fracture toughness Gc = 0.01 N/mm. The dashed-dotted lines repre-sent the buckling loads for the original specimen without additional delam-ination (Fcr = 17.77 N) and the fully delaminated specimen (Fcr = 4.44 N).

delaminated beam, which equals

Flcr =π2Eh3

48l2= 4.44 N. (5.63)

As can be seen, both theoretical buckling loads are approached in the compu-

tation. The deviations are caused by the fact that the structure is not slender

enough in order to neglect the effects of shear forces. This conclusion is sup-

ported by the observation that the deviation with respect to the analytical

solution is smaller in the case that a beam with length l is considered.

As can be seen from the displacement curve in u1-direction, this displace-

ment is monotonically increasing and hence a displacement controlled sim-

ulation could be used. This changes, however, in the case that the fracture

toughness is lowered to Gc = 0.01 N/mm. The force displacement curves

for this case are shown in Figure 5.13. As can be seen, the horizontal dis-

placement u1 is now no longer monotonically increasing. As a consequence

a displacement controlled computation would fail. The traditional arc-length

method would also fail, since it would not be able to deal with the localisa-

tion. Indirect displacement control would still offer a solution in this case,

but has the disadvantage that the relevant degrees of freedom need to be

selected a priori.

113

Chapter 6

Characterisation of microstructuralrandomness

Downscaling of components generally increases the influence of the

microstructure on the performance and reliability of devices. In the

previous chapters, a computational multiscale framework has been proposed

in which microstructural phenomena can be efficiently captured in numeri-

cal simulations of micro electromechanical systems (MEMS). In these simula-

tions, the size of the representative volume elements was chosen such that

the microstructural randomness caused by the variations in grain geometry

was homogenised. In reality, however, microstructural sources of random-

ness can exist with characteristic length scales considerably larger than the

characteristic size of the microstructure. These uncertainties are a result of

more difficult controllability of production processes for MEMS and can have

characteristic length scales in the order of the dimensions of the compo-

nent. As a consequence, the response of MEMS components can be far from

deterministic, making the characterisation of the microstructural sources of

randomness a relevant topic of interest.

In this chapter the randomness in the microstructure of a thick film of un-

poled lead zirconate titanate (PZT), produced using micro molding (Rosqvist

and Johansson, 2002; Bennekom et. al, 2009), is investigated (Verhoosel and

Gutiérrez, 2009b). This technique, which is commonly applied for macro-

scopic specimens, is relatively cheap compared to other existing methods for

producing piezoelectric films. However, the controllability of the production

process is difficult, resulting in specimens with relatively large imperfections

(see Figure 6.1). Further development of the production process will likely

improve the quality of the obtained specimens. It should, however, be men-

115

Characterisation of microstructural randomness

xcs

15.5μm

x2

x1

SubstratePZT sample4.5 mm

7.0 mm

Top-view

Cross-section

Figure 6.1 Schematic representation of the considered PZT thick film.

tioned that similar amounts of porosity are also observed in commercially

available PZT films, which illustrates the general problem of controllability of

MEMS production processes. Other manufacturing techniques, such as sput-

tering (Remiens et. al, 2002), yield considerably less randomness in the micro-

structure but are significantly more expensive and time consuming, making

them unattractive to use for commercial applications. It is emphasised that

the results presented in this chapter are restricted to the considered PZT

specimens. The methodology will, however, be relevant for a broad class of

micro systems.

The homogenisation framework employed in this chapter is outlined in

Figure 6.2. In Section 6.1, two-dimensional microscopic images are used

to obtain a statistical description of the microstructural geometry (Step 1)

from which random fields are then constructed using Karhunen-Loeve expan-

sions (Step 2). A three-dimensional microstructure is then reproduced using

a Voronoi tesselation (Step 3), as discussed in Section 6.2. In Section 6.3, the

homogenisation framework used to obtain the statistical description of the

material properties (Step 4) is discussed. Finally, the Karhunen-Loeve expan-

sion of the random vector field of material properties (Step 5) is discussed in

Section 6.4.

6.1 Characterisation of the microstructural geometry

In order to characterise the studied microstructure, the geometric properties

describing the microstructure need to be identified. Scanning electron mi-

croscopy (SEM) is used to obtain images of the microstructure on the top of

the specimen as well as of its cross-section (see Figure 6.1). Based on these

116

Characterisation of the microstructural geometry

Statistical properties of geometrical descriptors

Corr

elati

on

Distance

Statistical properties of material properties

Corr

elati

on

Artificially reproduced microstructure

Distance

Scanning Electron Microscopy (SEM) images

Random fields of material properties

Geo

met

ric

pro

per

ty

Location

Random fields of geometrical descriptors

MacroscaleMicroscale

1

3

1.

2.

3.

4.

5.

Karhunen-Loeve expansion

Image processing

Section 6.2Microstructure reproduction

Section 6.1

Section 6.1

Numerical homogenisationSection 6.3

Karhunen-Loeve expansionSection 6.4

4

5

2

Figure 6.2 Outline of the homogenisation framework.

117


microscopic observations, the microstructural descriptors are taken as:

◦ The porosity

◦ The average grain size

The porosity of the microstructure can be observed most clearly from the

cross-sectional images. For that reason, the porosity of the three-dimensional

microstructure is described using the area-ratio of pores in a cross-section,

φcs. The average grain size is observed best from top-view microscopic im-

ages and is described in terms of the average granular area, Atop.

6.1.1 Random field characterisation

In this work, the random microstructure of the PZT film is described using

random fields. Random fields are commonly used to describe random prop-

erties (Vanmarcke, 1983). The cross-sectional porosity for example is written

as

φcs = φcs(x, θ), (6.1)

where θ is an element of the space of all possible microstructural realisations

Θ. The elements in this set have a probability pθ(θ), which is known as the

probability density function in the case that the realisation set Θ is infinite di-

mensional. In the remainder of this work, the � is used to indicate quantities

defined in a probabilistic space.

Since the conditions for the fabrication process are isotropic (in the x1x2-

plane), also the random fields of the geometrical descriptors are assumed to

be isotropic. In addition, the random processes are assumed to be stationary,

which appears to be a reasonable assumption since the microstructure shows

similar characteristics in the complete specimen, also near the edges.

Ideally one would like to completely characterise the probabilistic space

describing the microstructural geometry by supplementing a functional space

with a probability density function. From a practical point of view, such

a probabilistic space can efficiently be represented by a few statistical mo-

ments. Here, both the random fields for the porosity and average grain size

are described using first and second-order statistical properties. Hence, char-

acterisation of the microstructural geometry (Step 1 in Figure 6.2) requires

determination of the mean, standard deviation and spatial correlation func-

tion (which is a single-variable function since isotropy is assumed) of both

microstructural descriptors.

118


6.1.2 Statistical moments of the microstructural descriptors

The statistical moments of both microstructural descriptors are determined

using a moving-window technique. The basic idea of this method is to grad-

ually move a window over microscopic images to obtain a finite dimensional

set of n� measurement points of either one of the microstructural descrip-

tors �. Making the ergodicity assumption, i.e. assuming that the statistical

averaging operator can be replaced by the spatial averaging operator, the

mean (μ�), standard deviation (σ�) and autocorrelation function (��) of the

microstructural descriptors can be obtained using

μ� = 1

n�

n�∑i=1

�i;

σ2� =

1

n� − 1

n�∑i=1

(�i − μ�

)2;

��

(Δ�

) = 1

σ2�

1

m� − 1

n�∑i=1

n�∑j=1

‖xi−xj‖=Δ�

(�i − μ�

) (�j − μ�

),

(6.2)

with m� being the number of measurement point pairs separated by a dis-

tance Δ�. Under the assumption of a stationary process, the validity of the

ergodicity hypothesis basically depends on the separation between the char-

acteristic length scales describing the random field and the size of the ob-

served image. The validity of this assumption will be commented upon at the

end of the following subsections. It should, however, be emphasised that the

ergodicity assumption is often inevitable from a practical point of view.

Cross-sectional pore ratio The statistical characterisation of the cross-sec-

tional pore ratio is accomplished by consideration of a microscopic image of

a cross-section of a PZT specimen. The observed cross-sectional image, of

which a small part is shown in Figure 6.3, is almost 3 mm long and 15.5μm

thick. The statistical data is here based on a single image.

As a first step in the processing of the image, the original SEM image is

transformed in a black-white image†. This is a non-trivial task, since pores

are not always easily distinguishable. This is mainly caused by the fact that

a cross-section gets damaged when breaking the specimen. The black-white

†The Matlab image processing toolbox is used for all image processing operations.

119


d

15.5μ

m

w

100μm

Figure 6.3 Small part of the SEM cross-sectional image before (top) and after(bottom) image processing. Some of the cavities in the top image are a resultof the breaking of the specimen, and were not observed in top view images.The pore ratio is determined from the lower figure by gradually shifting awindow of width w by a distance d.

image is partially shown in Figure 6.3. Comparison of this image with the

original SEM image shows that the pores can be identified. It should, how-

ever, be emphasised that its hard to distinguish pores from defects caused

by breaking the specimen. Alternative, preferably non-destructive, observa-

tion techniques should be used to increase the accuracy of the observations.

The cross-sectional pore ratio is determined using a window of width w, as

shown in Figure 6.3, with its centre positioned at the coordinate xcs. The pore

ratio is obtained as the fraction of white pixels inside the window. Gradually

shifting the window by a distance d then gives the pore ratio at many different

positions on the cross-section. In Figure 6.4, the cross-sectional pore ratio

measured using a 100μm window is shown. From this figure it is already

observed that variations with length scales in the order of magnitude of the

cross-section are observed, which is in agreement with observations on other

specimens.

One of the most important aspects in the determination of the statistical

characterisation of the microstructure using a moving-window technique, is

the selection of an appropriate window size. The influence of the window

size is investigated based on the correlation area, which is basically defined

as the area under the autocorrelation function (Graham-Brady et. al, 2003). In

the case of a single-variable correlation function, the integral under the curve

is referred to as the correlation length l�, with � being the associated random

process. In the case of a linearly decaying correlation function, the correlation

120


×10−2

Cross-sectional position xcs [μm]

φcs

[-]

300025002000150010005000

5

4

3

2

1

Figure 6.4 Cross-sectional pore ratio φcs observed using a moving-windowtechnique, with window width w = 100μm and d = 10μm.

length coincides with the distance over which no correlation is present. In

Figure 6.5, the correlation length is shown for various values of the window

size. In this figure, three regions (A, B and C) can be distinguished. To explain

the meaning of these three regions, two limiting cases of the window size are

considered.

The first limiting case is that in which the window size is so big that it

smoothens all spatial correlation (region C). This region is identified by com-

parison with the results obtained for an artificially produced specimen with

spatially uncorrelated pores as partially shown in Figure 6.6. In Figure 6.5 it

is observed that all spatial correlation is smoothened out for window sizes

bigger than 200μm. Note that the slope of the dashed line is equal to 1, since

the correlation length is linearly related to the window size. This window size

is therefore considered as an upper bound.

The second limiting case is that in which the window size gets in the or-

der of magnitude of the grains (region A). Since it is observed that voids can

be regarded as removed grains, a very strong correlation is observed for two

points within a single void, leading to a very small correlation length. This

lower bound for the window size can be identified by considering the depen-

dence of the standard deviation on the window size as shown in Figure 6.5.

In this figure, a rapid increase in the value of the standard deviation is ob-

served for window sizes smaller than 100μm. This is caused by the fact that

the definition of the porosity requires windows significantly larger than the

average pore (or grain) size. Therefore, the lower bound for the window size

is determined as 100μm.

121


CBA

Window size w [μm]

l φcs

[μm

]

1000100

1000

100

×10−2

Window size w [μm]

σφ

cs[-

]

1000750500250

2.5

2

1.5

1

0.5

0

Figure 6.5 Correlation length of the cross-sectional pore ratio lφcs as a func-tion of the window size (left), observed from the SEM images (solid line withtriangles) and artificial spatially uncorrelated sample (dashed line). Standarddeviation of the pore ratio σφcs as a function of the window size (right).

100μm

15.5μ

m

Figure 6.6 Cross-sectional image generated using a Voronoi tessellation.Spatially uncorrelated pores are created by randomly removing grains.

122


Cross-sectional pore ratio φcs [-]

Pro

bab

ilit

yd

ensi

ty[-

]

0.050.040.030.020.01

50

40

30

20

10

0

Distance Δφcs [μm]C

orr

elati

on�φ

cs[-

]

4003002001000

1

0.8

0.6

0.4

0.2

0

Figure 6.7 Probability density function for the cross-sectional pore ratio(left) obtained using the moving-window technique (histograms) and fittedusing a lognormal distribution (solid line). Autocorrelation function (right)based on the SEM images (�) and approximated using a linearly decayingcorrelation function (dashed line).

In Figure 6.5, region B is identified as the region in which the effect of

the window size on the correlation length is limited. An objective choice

for the window size can be made by selecting a window size in region B.

The window size used here is w = 100μm, which is an arbitrarily chosen

length within region B. The sensitivity of the results to the moving-window

parameter d, which was taken as 10μm, is observed to be limited. Using

the moving-window technique, the mean and standard deviation of the cross-

sectional pore ratio are determined as 0.0269 and 0.0094, respectively. In

Figure 6.7, the probability distribution is shown using histograms. The log-

normal distribution, based on the measured first two statistical moments is

also shown in Figure 6.7. As can be seen, the lognormal distribution rea-

sonably fits the moving-window data. The autocorrelation function obtained

using the moving-window technique is also shown in Figure 6.7. As can be

seen, this correlation function is appropriately approximated using a linearly

decaying correlation function.

The correlation length obtained using the moving-window technique is

equal to lφcs = 254μm, which is less than 10 percent of the studied cross-

section. Although there is a significant separation between these two lengths,

a larger cross-section is desirable for better satisfaction of the ergodicity as-

sumption. The fact that for large window sizes the theoretical unit slope is

found for the dependence of the correlation length on the window size is,

123


d

128μm

96μ

m

w

Figure 6.8 Top-view SEM image of the considered PZT specimen (left) andresult after image processing (right). A square window of width w is used toobtain the average granular area.

however, an indication that the correlation length is determined appropri-

ately.

Average grain size The average grain size is statistically characterised by

consideration of SEM images of the top-view of the specimen. Image process-

ing techniques are used to identify the grains in a 96× 128μm2 microscopic

image, as shown in Figure 6.8. In addition to this image, 20 × 20μm2 micro-

scopic images of the top view are taken on various places on the specimen.

Identification of the grain boundaries in the SEM image shown in Figure 6.8

requires a significant amount of manual input, which is primarily caused by

the reflections in the image. As for the cross-sectional porosity, a good in-

dication of the statistical properties of the grains can be obtained from the

image. In the case that high accuracy determination of the grain boundaries

is required, alternative observation techniques such as polarisation contrast

microscopy (Driggers, 2003) could be used. The performance of such tech-

niques has, however, not been assessed in this work.

The average grain area is obtained using a square window of width w, with

its centre positioned at x. The average grain area is computed as the total

granular area (i.e. the window area minus the void area) divided by the num-

ber of grains in the window. Grains falling on a single boundary of the win-

dow are counted as a half, and grains falling on two boundaries are counted

as a quarter. In Figure 6.9, the average grain area measured using a window

with a width of 20μm is shown. Visual inspection of this figure reveals signif-

icant variations in the average grain area over the specimen. Hence, from this

124


Ato

p[μ

m2]

8

7

6

5

x2[μ

m]

90

70

50

30

10

x1 [μm]

1109070503010

Figure 6.9 Average grain area Atop observed using the moving-window tech-nique, with w = 20μm and d = 4μm. Since the average grain area is com-puted in the centre of each window, it cannot be obtained in a border of halfthe window size. Although this reduces the effective image size, it is not afundamental problem for the moving-window technique.

image it appears that the length scale of the random process is significantly

smaller than the specimen size.

The influence of the window size on the statistical characterisation is

studied in Figure 6.10. As can be seen, the characteristic length scale of

the random process corresponds to that of the window size (�). From the

20×20μm2 images it is also observed that no larger length scales are involved

(◦, in top-right corner of Figure 6.10). For the considered window size, it is

observed that the spatial correlation of the random process is smoothened by

the considered window sizes. This is confirmed by comparison with results

obtained for an artificially generated, spatially uncorrelated, microstructure

(dashed line in Figure 6.10). This artificial microstructure, which is produced

using a Voronoi tessellation based on a set of spatially uncorrelated nucle-

ation sites, is shown in Figure 6.11. Considering the dependence of the stan-

dard deviation of the average grain area on the window size, as shown in

Figure 6.10, yields a similar result as observed for the cross-sectional poros-

ity. As can be observed, a rapid increase in the standard deviation is observed

for window sizes smaller than approximately 10μm. This increase is caused

125


Window size w [μm]

l Ato

p[μ

m]

10010

100

10

Window size w [μm]σA

top[μ

m2]

3020100

3

2

1

0

Figure 6.10 Average grain area correlation length lAtop as a function of thewindow size (left) determined using the moving-window technique for theSEM image (solid line) and artificially generated spatially uncorrelated sam-ple (dashed line). Standard deviation of the average granular area σAtop as afunction of the window size (right).

by the relatively poor definition of the average grain area in the case of win-

dow sizes in the order of magnitude of the grain sizes. The window size of

10μm is therefore considered as a lower bound.

From Figure 6.10 it is observed that no objective choice for the window

size is available in this case. In contrast to the case of the cross-sectional

porosity, the average grain area should be considered as a spatially uncorre-

lated random process. Although a spatial correlation length will be present,

its characteristic length is in the same order of magnitude as that of the mi-

128μm

96μ

m

Figure 6.11 Cross-sectional image generated using a Voronoi tessellation.Spatially uncorrelated pores are created by randomly removing grains.

126


Average grain area Atop [μm2]

Pro

bab

ilit

yd

ensi

ty[μ

m−2]

987654

0.8

0.6

0.4

0.2

0

Distance ΔAtop [μm]C

orr

elati

on�A

cs[-

]

403020100

1

0.8

0.6

0.4

0.2

0

Figure 6.12 Probability density function for the average granular area (left)obtained using the moving-window technique (histograms) and fitted usinga lognormal distribution (solid line). Autocorrelation kernel for the averagegranular area (right).

crostructural geometry features. For that reason, the random process can be

considered as spatially uncorrelated when looking at it from a macroscopic

point of view. The mean and standard deviation then still depend on the size

of the considered window. For a 20 × 20μm2 window, the mean and stan-

dard deviation are obtained as 6.253 and 0.628, respectively. The lognormal

probability density function for these two statistical moments is compared

with the moving-window data in Figure 6.12. As can be seen, the average

grain area can reasonably be represented by a lognormal distribution. In

this figure, it is also observed that the correlation length is governed by the

window size. For that reason, the average grain area is described using a spa-

tially uncorrelated lognormal process. It is important to note that although

no window-independent correlation length is found by the moving-window

analysis, there will likely be a spatial correlation in the order of magnitude

of 10μm. This is reflected by the relatively small (compared to the specimen

size) clusters of small and large grains observed in Figure 6.8. In this work,

the influence of this small spatial correlation is assumed to be negligible.

The validity of the ergodicity assumption here depends on the separation

between the SEM image size and window size. Since these two length scales

are separated by a factor of approximately five, the validity of the ergodic-

ity assumption is questionable. The unit slope found in Figure 6.10 and its

resemblance with the result for the Voronoi diagram is, however, a good indi-

cation that the ergodicity assumption allows for a reasonable approximation

127


of the correlation length. This is confirmed by comparison with a consider-

ably larger artificial specimen.

6.1.3 Random field parametrisation

The statistical characterisation of the cross-sectional porosity and average

grain area discussed above can be used to construct the random fields dis-

cussed in Section 6.1.1 (Step 2 in Figure 6.2). Since the average grain area

is described by a spatially uncorrelated lognormal process, its field is fully

described by the univariate lognormal distribution, with its coefficients ob-

tained using the moving-window technique. The spatially correlated lognor-

mal random field for the porosity requires parametrisation in terms of a finite

dimensional set of random variables. The parametrisation of such a station-

ary lognormal random field is obtained by defining an underlying stationary

Gaussian field

G(x, θ) = ln(φcs(x, θ)

), (6.3)

with the mean and standard deviation expressed in terms of those of the

cross-sectional porosity as

μG = ln(μφcs

)− 1

2ln(1+V 2

φcs

);

σ2G = ln

(1+V 2

φcs

),

(6.4)

in which V� is the coefficient of variation of the random variable �. The

correlation function of the underlying Gaussian field G(x, θ) can be exactly

related to the properties of the lognormal field (Der Kiureghian and Liu, 1986)

by

�G(x,y) =ln(1+ �φcs(x,y)V 2

φcs

)ln(1+V 2

φcs

) , (6.5)

with �φcs(x,y) being the correlation function based on the results of the

moving-window technique (see Figure 6.7). The transformation to an underly-

ing Gaussian field can also be performed for other distributions (e.g. Weibull).

An exact expression for the correlation of the underlying Gaussian field can-

not be obtained for most distributions other than lognormal. Approximate

expressions are, however, available for many other distributions.

The parametrisation of the Gaussian field G(x, θ) by a set of nz standard

normal random variables zi is obtained by the truncated Karhunen-Loeve (KL)

128


expansion (Huang et. al, 2001) as

G(x, θ) ≈ G(x, z) = μG +nz∑i=1

gi(x)zi, (6.6)

with gi(x) being the eigenfunctions of the covariance function �G(x,y), nor-

malised with respect to the square-root of the corresponding eigenvalues.

The KL-expansion is known to be the most efficient parametrisation of Gaus-

sian random processes, in the sense that it minimises the mean-square er-

ror of the approximated random field (Ghanem and Spanos, 1991). Here the

KL-eigenvalues and eigenfunctions are determined numerically using the fi-

nite element method to solve the governing Fredholm equation (Spanos and

Ghanem, 1989).

Since an nz-dimensional parametrisation of the Gaussian field G(x, z) is

obtained, a parametrisation of the lognormal field φcs(x, z) follows by appli-

cation of the inverse of the transformation (6.3) to yield

φcs(x, θ) ≈ φcs(x, z) = μφcs√1+V 2

φcs

nz∏i=1

exp(gi(x)zi

). (6.7)

Two realisations of this random field based on the exponential decaying cor-

relation function are shown on a 1 × 1 mm2 domain in Figure 6.13. The ran-

dom field is approximated using 20 × 20 KL-terms (nz = 400) discretised

using 50 × 50 bilinear elements. The capability of the KL-expansion to re-

produce the linearly decaying correlation function is shown in Figure 6.14,

where it is compared with a 10 × 10 terms expansion. It is clearly observed

that an increase in number of KL-terms improves the approximation. This is

confirmed by consideration of the approximated mean and standard devia-

tion. In the case of 10 × 10 KL-terms, the exact mean (0.0269) and standard

deviation (0.00937), are approximated as 0.0267 and 0.00878, respectively.

This approximation is significantly improved when considering 20×20 terms,

when the mean and standard deviation are obtained as 0.268 and 0.00900. In

Figure 6.14 it is observed that the largest error in the approximation of the

correlation function lies around the kinks in the correlation function. This is

a logical consequence of the smoothness of the Karhunen-Loeve terms. This

error can be reduced by further increasing the number of terms and conse-

quently also increasing the number of finite elements for the discretisation

of the KL-functions.

129


x2[μ

m]

1000

800

600

400

200

0

x1 [μm]

10008006004002000

φcs

[-]

0.06

0.05

0.04

0.03

0.02

0.01

x2[μ

m]

1000

800

600

400

200

0

x1 [μm]

10008006004002000

Figure 6.13 Two realisations of the KL-representation of the random fieldfor the cross-sectional porosity using 400 Karhunen-Loeve terms.

Distance Δφcs [μm]

Corr

elati

on�φ

cs[-

]

4003002001000

1

0.8

0.6

0.4

0.2

0

-0.2

Distance Δφcs [μm]

Corr

elati

on�φ

cs[-

]

4003002001000

1

0.8

0.6

0.4

0.2

0

-0.2

Figure 6.14 Linearly decaying correlation function (dashed lines) repro-duced using 10× 10 (left) and 20× 20 (right) KL-terms (dots).

130

Local representation of the microstructure

Figure 6.15 Schematic representation of the Voronoi growth model.

6.2 Local representation of the microstructure

In the previous sections, the random microstructure was defined by means

of random fields for the cross-sectional porosity and top-view average grain

area. Characterisation of the three-dimensional microstructure (Step 3 in

Figure 6.2) is, however, required to perform the homogenisation of material

properties, as will be discussed in the next section.

The three-dimensional representation of the microstructure is here gen-

erated using a Voronoi growth model (Okabe et. al, 1992). This geometri-

cal model mimics the process of isotropic grain growth, as schematically

visualised for five nucleation points in Figure 6.15. Standard algorithms†

are nowadays available to construct two- and three-dimensional Voronoi dia-

grams. Free or periodic surfaces can be produced by mirroring or translating

the nucleation points in the original domain to neighbouring cells (Verhoosel

and Gutiérrez, 2009a).

Comparison of Figure 6.8 and Figure 6.11 shows that the Voronoi diagram

has several characteristics in common with the experimentally obtained im-

age. For example, the (large) majority of the observed grains is convex, which

is similar to the convexity of Voronoi cells. In addition, the average number

of edges per grain is reasonably mimicked (4.9 for the Voronoi diagram ver-

sus 5.3 for the experimental observation) and agrees with results found in

literature Watson and Smith (1975).

Inspection of the distribution of grain areas in Figure 6.16 shows that

the grain area distribution of the SEM image is reasonably represented by

the Voronoi growth model, although it appears that the SEM images contains

†In this work the Matlab interface to the Qhull libraries is used for the construction of

Voronoi diagrams.

131


×10−3

Grain area [μm2]

Pro

bab

ilit

yd

ensi

ty[μ

m−2

]

2000150010005000

1.6

1.2

0.8

0.4

0

×10−2

Edge length [μm]Pro

bab

ilit

yd

ensi

ty[μ

m−1

]

6543210

7

6

5

4

3

2

1

0

Figure 6.16 Comparison of the grain area distribution (left) and edge lengthdistribution (right) for the SEM image (white bars) and Voronoi tessellation(filled bars).

more smaller grains. In Figure 6.16, the distribution of edge lengths shows

a more significant difference between the SEM image and the Voronoi dia-

gram. It appears that the standard deviation of the edge length is consider-

ably smaller for the SEM image than for the Voronoi diagram. Especially, the

absence of short edges (smaller than 0.6μm) is remarkable. This absence can

partially be explained by the image processing technique, since small edges

are just harder to identify and are instead recognised as junctions.

The physics of the grain-growth process might also play a role in the ab-

sence of small grain boundary edges, since the small edges might simply

diffuse during the sintering process. Capturing this effect would require the

use of a significantly more complex grain-growth model and is considered

beyond the scope of this work. A similar effect can, however, be obtained by

making a simple adjustment to the Voronoi growth model. Instead of adding

all nucleation points at once, the nucleation points are added one-by-one. Af-

ter the addition of a nucleation point, the minimum edge length is computed.

If this minimum edge length is larger than a specified threshold, the next nu-

cleation point is inserted. Otherwise, the nucleation point is shifted and the

Voronoi diagram is re-computed. This procedure is then repeated until the

required number of nucleation points is reached. An attractive side-effect of

avoiding small grain boundaries is that the mesh quality of the finite element

discretisation used for numerical homogenisation is significantly improved.

Suppose that a realisation of a volume of the microstructure needs to be

reproduced around a macroscopic point. The cross-sectional porosity and

132

Local representation of the microstructure

Figure 6.17 Schematic representation of the micro structure representativevolume element (RVE), used for numerical homogenisation.

average top-view grain area are obtained using the random fields discussed

in Section 6.1.1. To locally match the average grain area in the top-view,

nucleation points for the Voronoi tessellation are sequentially added. The

average grain area is computed after the addition of each nucleation point.

If the average grain area gets too small, the last nucleation point is shifted.

This procedure is repeated until the reproduced average grain area is within

a specified tolerance (typically 1 percent) of the required value.

The porosity is reproduced by randomly removing some of the grains. The

number of grains to be removed is determined by a Poisson distribution with

its parameter equal to the porosity times the number of grains in the Voronoi

tessellation (rounded of to the nearest smaller integer). The grains to be

removed are selected randomly. Once a set of grains is removed, the aver-

age cross-sectional porosity is computed by consideration of many different

cross-sections and computing the average porosity. If this average cross-

sectional porosity matches the target value with a specified tolerance (again

typically 1 percent), the removed grains are assumed to appropriately repre-

sent the local microstructure. If the porosity is not represented correctly, the

procedure is repeated by the removal of a new set of grains.

As an example, a volume of 20 × 20 × 14μm3 of the microstructure cor-

responding to a realisation of the random fields with φcs = 4.2% and Atop =5.55μm2 is shown in Figure 6.17. Some of the cross-sections used to compute

the pore ratio are shown as well.

133


6.3 Homogenisation of the random fields of material properties

Supplemented with a description of the kinematics and microscale constitu-

tive behaviour, the reproduced three-dimensional microstructural geometry

can be used for the homogenisation of the constitutive laws (Step 4 in Fig-

ure 6.2). The fundamental difference of this homogenisation procedure with

that discussed in Chapter 4 is that (part of) the microstructural randomness

needs to be conserved during the process of homogenisation. The way in

which this randomness is transported across the scales largely depends on

the homogenisation procedure.

6.3.1 Homogenisation procedure

In the case that computational homogenisation (an FEn-type approach) is

used, random micro models are employed in the macroscopic integration

points. The way in which the randomness is carried over from the micro to

the macroscale in that case depends on the stochastic finite element method

employed on the larger scale. In the case of a sampling-based simulation,

the homogenisation procedure as presented in Chapter 4 remains intact, but

requires the generation of new microscale models for every multiscale inte-

gration point for each realisation of the sample. In the case of a sensitivity-

based macroscale simulation, the sensitivities of the kinetic quantities need

to be homogenised, which can be a non-trivial task, since the microstructural

geometry varies randomly. Similar problems will be present in the case of

alternative stochastic methods, such as the spectral stochastic finite element

method.

In the case of numerical homogenisation, i.e. the case in which homogeni-

sation is used to derive the properties of the macroscopic problem, the mi-

crostructural randomness needs to be transformed into macroscopic random

fields for the bulk and cohesive properties. Since the most commonly used

stochastic finite element methods all rely on the presence of one or more ran-

dom fields of properties, this procedure does not depend on the macroscopic

analysis. The macroscopic random fields can be constructed from the ran-

dom macroscale using various methods, in which the sampling-based meth-

ods are conceptually the simplest. The moving-window generalised method

of cells (MWGMC) as proposed by Baxter et. al (2001), which is basically

a combination of the moving-window technique with deterministic numeri-

cal homogenisation methods, is a typical example of such a sampling-based

134

Homogenisation of the random fields of material properties

method.

From a computational effort point of view, numerical homogenisation is

considerably more attractive than computational homogenisation, especially

when considering sampling-based stochastic methods. Hybrid homogenisa-

tion procedures, in which for example computational homogenisation is only

considered for the cohesive behaviour (as done in Chapter 4), can be consid-

ered to reduce the involved computational effort. Nonetheless, consideration

of a FEn-type of homogenisation procedure in the context of the application

considered (a stochastic macroscale model with a three-dimensional random

microstructure) is impractical. Therefore MWGMC is used here, which comes

at the cost of a reduced accuracy of the homogenisation procedure.

6.3.2 Polycrystalline micro model

Before considering the homogenisation scheme, first the microscale finite el-

ement model is introduced. As in Chapter 4, intergranular fracture of a poly-

crystalline microstructure is considered. There are, however, also some dif-

ferences with the micro model introduced in Chapter 3. The microstructure

considered here and schematically shown in Figure 6.18 is three-dimensional

instead of two-dimensional and contains voids. Periodic boundary condi-

tions are used for the in-plane directions (light grey surfaces) and traction

free conditions are used for the free surfaces (dark grey surfaces). Loading

of the microstructural specimen is performed by application of a force F as

indicated in Figure 6.18. As a result of the periodic boundary conditions, this

force equals the integrated traction in x1-direction over the right surface.

Another important difference with the model considered for the deter-

ministic homogenisation, is that the current PZT material is unpoled, which

makes the electromechanical problem uncoupled. As a consequence, a purely

mechanical model can be used to determine the mechanical response of a mi-

crostructure. For the bulk material, isotropic linear material behaviour is as-

sumed, with modulus of elasticity Y = 82.3 GPa and Poisson’s ratio ν = 0.36.

The traction on the grain boundaries is related to the displacement jump over

the interfaces using the Xu-Needleman law, in which the shear opening is de-

fined as the length of the in-plane (orthogonal to the normal of the interface)

displacement jump vector. The fracture strength is taken as tult = 80 MPa

and the fracture toughness as Gc = 2.34 · 10−3 N/mm.

135


F

h

w

wI II

IIIIVVI

VIIVIII

x3

x2

x1

Ωm

Figure 6.18 Schematic representation of the micro model.

6.3.3 Determination of the representative volume element size

As in the case of deterministic homogenisation as considered in Chapter 4,

the selection of an appropriate representative volume element (RVE) size is

one of the most important topics of interest. The definition of the RVE, how-

ever, needs to be redefined if randomness needs to be preserved over the

scales. In order to redefine the definition of the representative volume ele-

ment, distinction is made between microscale randomness which influences

the macroscopic result and microscale randomness that has no influence on

the macroscopic result. This basically means that a distinction is made be-

tween randomness with large characteristic length scales and small charac-

teristic length scales, where large and small are defined relative to the macro-

scopic specimen size. Making this distinction in sources of microscale ran-

domness yields the representative volume element requirements as:

◦ The microstructural domain is large enough such that the homogenised

properties become independent of microstructural variations with char-

acteristic length scales small compared to the macroscopic specimen

size.

◦ The microstructural domain is small enough such that separation of

scales is guaranteed and in addition the microstructural randomness

with length scales in the order of magnitude of the macroscopic speci-

men is preserved.

In the specific case considered here, the RVE size should be chosen such that

the variations of the porosity and average grain size are preserved, but the

136


variation of the granular geometry is smeared out.

An additional problem for the selection of the RVE size is that a size study

as performed in Section 4.4.1 is impractical, since the RVE size would depend

on the porosity and average grain size. Consequently, the representative

volume element size should be considered as a random variable. Efficient

description of this random variable requires the consideration of stochas-

tic methods on the microscale. Such an analysis is considered beyond the

scope of this work, but is of crucial importance in the case that a computa-

tional homogenisation procedure is performed. Since in the present work,

the MWGMC is considered, the main requirement is that the RVE size should

be small compared to the window size. Based on the experience gained in the

representative volume size determination in Chapter 4 and moving-window

size determination in this chapter, 20 × 20 × 15.5μm3 RVEs are considered.

Note that in the thickness direction, the full thickness of the film is modelled.

6.3.4 Moving-window generalised method of cells

The moving-window generalised method of cells can be regarded as an ex-

tension of the moving-window techniques used for the description of the

random microstructural geometry. A window with fixed dimensions is grad-

ually moved over the microstructure. For every window position, the bulk

and cohesive properties are obtained using numerical homogenisation. In

the method as proposed in Baxter et. al (2001), this is done by taking the RVE

size equal to the window size. Here an alternative approach is taken. A re-

alisation of the material properties is first obtained using a realisation of the

microstructure. The moving-window technique is then applied to determine

the statistical characteristics of the random material property fields.

As for the microstructural geometry, isotropy is assumed for the material

property fields. The porosity field on which the considered microstructure

is based is shown in Figure 6.19 along with the approximation of the corre-

lation function using a 100μm moving-window. As can be seen, the 4 mm

long specimen reasonably resembles the correlation function. A consider-

ably larger specimen should be considered to have the ergodicity assumption

better satisfied. This is also confirmed by consideration of the coefficient of

variation of the mean and standard deviation of the porosity, which are both

in the order of magnitude of ten percent. For the ergodicity assumption to be

made satisfactorily, this value should be considerably smaller.

Using the finite element method to solve the quasi-static equilibrium equa-

137


×10−2


φcs

[-]

40003000200010000

6

5

4

3

2

1

Distance Δφcs [μm]C

orr

elati

on�φ

cs[-

]

4003002001000

1

0.8

0.6

0.4

0.2

0

-0.2

Figure 6.19 Realisation of the cross-sectional porosity based on theKarhunen-Loeve expansion of the measured random field (left) and autocor-relation function obtained using a moving-window technique (right) com-pared with the linear decaying function (dashed line).

tions for the microstructural boundary value problem in Section 6.3.2, yields

the response of the RVE at various points on the generated microstructure.

Some of these force-displacement curves are shown in Figure 6.20. A typi-

cal microstructural RVE is discretised using 60 thousand linear tetrahedron

volume elements and 7 thousand linear triangular interface elements, yield-

ing a systems of approximately 60 thousand degrees of freedom. From the

response of the microstructure, the bulk and cohesive macroscopic proper-

ties can be computed using numerical homogenisation. Since the numerical

homogenisation procedure considered here closely resembles the procedures

discussed in Chapter 4, the homogenisation laws for the various properties

will here only be discussed concisely.

Modulus of elasticity In the case of isotropic linear elasticity, the macro-

scopic modulus of elasticity can be written in terms of the homogenised

stress and strain defined in Section 4.1.1 as

Y M =(∂⟨εm

11

⟩Ωm

∂⟨σm

11

⟩Ωm

∣∣∣∣∣0

)−1

, (6.8)

where the �|0 indicates evaluation in the undeformed state. In the context of

a finite element model, as discussed in Section 4.2, the required strain and

138


×10−3

×10−5

Displacement aIII,1 [mm]

Forc

eF

[N]

86420

15

12

9

6

3

0

Figure 6.20 Force-displacement curves for various microstructural repre-sentative volume elements.

stress components can be written as

⟨εm

11

⟩Ωm = aIII,1

w;

⟨σm

11

⟩Ωm = F

wh,

(6.9)

which upon substitution in equation (6.8) yields

Y M = 1

h

∂F

∂aIII,1

∣∣∣∣0

. (6.10)

Hence, the modulus of elasticity can directly be related to the slope in the

force-displacement curves.

Poisson’s ratio The macroscopic Poisson’s ratio can be defined as

νM = − ∂⟨εm

33

⟩Ωm

∂⟨εm

11

⟩Ωm

∣∣∣∣∣0

, (6.11)

in the case of isotropic linear elasticity. In terms of the RVE quantities, this

is rewritten as

νM = −wh

∂aIII,3

∂aIII,1

∣∣∣∣0

. (6.12)

Fracture strength The macroscopic (mode I) fracture strength is taken as

the ultimate value of the traction on the right surface of the boundary value

139


problem in Figure 6.18. Using the homogenisation relations introduced in

Section 4.1.2, the fracture strength can then be defined as

tMult =

1

whF |ult , (6.13)

where the �|ult indicates that the force F is evaluated at the ultimate load.

Fracture toughness The macroscopic (mode I) fracture toughness is defined

in terms of the homogenised traction and displacement jump as

GMc =

∞∫0

tM1 d�uM

1� = 1

wh

∞∫0

Fd�uM

1�

daIII,1daIII,1. (6.14)

From the decomposition of the displacement of point I in an elastic displace-

ment and homogenised crack opening, it follows that

d�uM1�

daIII,1= 1− dF

daIII,1

(dF

daIII,1

∣∣∣∣0

)−1

. (6.15)

Substituting this expression in (6.14) then yields

GMc =

1

wh

∞∫0

F daIII,1 − 1

wh

(dF

daIII,1

∣∣∣∣0

)−1 ∞∫0

FdF

daIII,1daIII,1;

= 1

wh

∞∫0

F daIII,1 − 1

wh

(dF

daIII,1

∣∣∣∣0

)−1 [1

2F2

]∞0

;

= 1

wh

∞∫0

F daIII,1,

(6.16)

which states that the fracture toughness equals the work dissipated by the

micro model, which is directly related to the area under the force-displace-

ment curves shown in Figure 6.20.

Using the equations (6.10), (6.12), (6.13) and (6.16), the macroscopic bulk and

cohesive properties can be evaluated on the basis of the microstructural re-

sponses. The results for the modulus of elasticity and fracture strength, us-

ing a 100μm moving-window, are presented in Figure 6.21.

140



YM

[GPa]

40003000200010000

35

34

33

32

31

30


tM ult

[MPa]

40003000200010000

50

48

46

44

42

40

38

Figure 6.21 Variation of the modulus of elasticity (left) and fracturestrength (right) obtained using the moving-window generalised method ofcells with window size w = 100μm and d = 10μm.

Statistical moments of the material properties

μ� σ� V�

Young’s modulus Y M 33.1 GPa 0.880 GPa 2.7%

Poisson’s ratio νM 0.276 – –

Fracture strength tMult 43.8 MPa 1.91 MPa 4.4%

Fracture toughness GMc 1.18 N/m 0.166 N/m 14.1%

Cor. coef. ��(0) [-]

Y M tM

ult GMc

Y M 1 � �

tMult 0.69 1 �

GMc 0.22 0.32 1

Cor. len. l�� [μm]

Y M tM

ult GMc

Y M 259 � �

tMult 408 298 �

GMc 141 410 255

Table 6.1 Statistical properties of the bulk and cohesive properties obtainedusing the moving-window generalised method of cells.

141


Distance ΔtMult[μm]

Corr

elati

on�tM u

lt[-

]

4003002001000

1

0.8

0.6

0.4

0.2

Distance ΔYMtMult[μm]

Corr

elati

on�Y

MtM u

lt[-

]

4003002001000

1

0.8

0.6

0.4

0.2

0

Figure 6.22 Correlation functions for the modulus of fracture strength (left)and cross-correlation function of the fracture strength and modulus of elas-ticity (right) obtained using the moving-window generalised method of cellswith window size w = 100μm (�) and fitted using a Gaussian-shaped func-tion (dashed line).

As a final step, the mean, standard deviation and correlation function of

all four fields can be computed using the expressions in equation (6.2). The

computed statistical moments are presented in Table 6.1. The statistics are

obtained using a window size of 150μm, which is found to be an objective

choice for the window size. An exception for this is the determination of

the cross-correlation between the Young’s modulus and fracture toughness,

for which a 100μm window is found to be objective. The Poisson’s ratio

is observed to have a spatial correlation considerably smaller than that of

the other material properties, which implies that the Poisson’s ratio is sig-

nificantly influenced by the average grain size. In the macroscale stochastic

finite element simulations considered in this work, the spatially uncorrelated

randomness in the Poisson’s ratio is assumed to be homogenised. Hence, the

Poisson’s ratio will be considered as a deterministic quantity in the remainder

of this work.

In Figure 6.22, the correlation function of the fracture strength and cross-

correlation function of the fracture strength and modulus of elasticity are

shown. A least-squares approximation of the moving-window results is ob-

tained using the Gaussian-shaped function

��(Δ�) = ��(0) exp

(−πΔ2

�

l2�

), (6.17)

142

Parametrisation of the random fields of material properties

with the parameters ��(0) and l� assembled in Table 6.1. Note that the factor

π is introduced to let the integral over the real axis be equal to the correlation

at zero distance times the correlation length.

6.4 Parametrisation of the random fields of material properties

In the remainder of this work, the random material properties of the consid-

ered specimen are represented by the stationary lognormal random vector

field

α(x, θ) =

⎛⎜⎝ Y M(x, θ)

tMult(x, θ)

GMc (x, θ)

⎞⎟⎠ . (6.18)

The first- and second-order statistics of this field, i.e. the mean vector μαand covariance matrix Σα(x,y), are determined using the MWGMC as dis-

cussed in the previous section. For example, the correlation function �YMtMult

shown in Figure 6.22 is related to the covariance matrix by Σα,12(x,y) =σYMσtMult

�YMtMult(‖x− y‖).

In order to perform stochastic finite element analyses, the random vector

field of material properties (6.18) needs to be parametrised. A Karhunen-

Loeve expansion, as discussed in Section 6.1.3 for scalar valued fields, can

also be obtained in the case of a random vector field (Step 5 in Figure 6.2).

This is done by defining an underlying Gaussian random vector field G(x, θ).

The means and standard deviations of this Gaussian field can be obtained

using equation (6.4). The correlation functions are obtained similar as in

equation (6.5) by

�G,ij(x,y) =ln(1+ �α,ij(x,y)Vα,iVα,j

)√ln(1+V 2

α,i

)ln(1+V 2

α,j

) . (6.19)

A truncated Karhunen-Loeve expansion of the random vector field G(x, θ) is

then written as

G(x, θ) ≈ G(x, z) = μG +nz∑i=1

gi(x)zi. (6.20)

The eigenfunctions gi(x) are obtained by numerically solving∫yΣG(x,y)gi(y)dy = ζigi(x) (6.21)

143


Distance ΔtMult[μm]

Corr

elati

on�tM u

lt[-

]

4003002001000

1

0.8

0.6

0.4

0.2

Distance ΔYMtMult[μm]

Corr

elati

on�Y

MtM u

lt[-

]

4003002001000

1

0.8

0.6

0.4

0.2

0

Figure 6.23 Karhunen-Loeve approximation (dots) of the random vectorfield of material properties.

using a Galerkin projection (Gutiérrez and Krenk, 2004) and are normalised

according to ⟨gi,gi

⟩ = ∫x

gT

igi dx = ζi, (6.22)

with ζi being the corresponding KL-eigenvalue. Moreover, the eigenfunctions

are orthogonal with respect to the inner product 〈�,�〉 in equation (6.22). Fol-

lowing the derivations in Ghanem and Spanos (1991) it can also be demon-

strated that the error of the approximated random field measured by the

norm corresponding to this inner product is minimised by the Karhunen-

Loeve expansion. Note that in the case that the property fields are uncou-

pled, i.e. ΣG(x,y) is diagonal, equation (6.21) can be considered as (three)

independent scalar Fredholm equations. Since these properties are evidently

not uncoupled, the coupled system needs to be discretised (using e.g. the

FEM) and solved.

The correlation function of the fracture strength and its cross-correlation

function with the modulus of elasticity as approximated by a KL-expansion

are illustrated in Figure 6.23. The results presented are obtained on a 250 ×250μm2 domain, discretised using 40 × 40 elements and 16 KL-terms. In

this case the KL-eigenvalues are truncated below 1 percent of the maximum

eigenvalue magnitude.

144

Chapter 7

Partition of unity-based stochasticfracture modelling

Microstructural imperfections are found in any component, inde-

pendent of its size. In the case that a component is large compared

to the size of its microscopic imperfections, the influence of microstructural

randomness will be averaged out, such that the component appears to re-

spond deterministically. In the case that microscopic imperfections are of

the same order of magnitude as a component, they will also affect the perfor-

mance and reliability of the device. Since in that case the randomness is not

averaged out over the length scales, the response of the device appears to be

stochastic.

As already discussed in the previous chapters, micro electromechanical

systems (MEMS) are devices for which the separation between the dimensions

of the device and the microstructure is limited. In the previous chapter, it

was even demonstrated that the characteristic length scale of microstructural

sources of randomness, such as pores, might be in the same order of magni-

tude as the dimensions of the devices. As a consequence, MEMS are prone to

stochastic effects. In the case that a component is comprised of multiple re-

dundant MEMS components, this random effect can be effectively diminished.

A printerhead with several hundreds of inkjets is a typical example of this.

From a computational point of view, the challenge is to gain insight in

the stochastic response of components. This insight encompasses both qual-

itative and quantitative results. In the case of large devices, microstructural

randomness can be incorporated in the effective properties describing a com-

ponent, yielding a deterministic result. In the case of small devices, such as

MEMS, this is no longer yielding meaningful results. Stochastic methods are

145

Partition of unity-based stochastic fracture modelling

in that case required to transform random property fields into a random re-

sponse.

In brittle components, i.e. specimens in which the process zone is small

compared to the specimen size, weakest link models are commonly applied

to model failure (Bažant and Planas, 1998). These models are based on the

Weibull distribution, which is used to describe the fracture strength in a

component. Although these models are successfully applied to brittle compo-

nents, and are capable of describing the probabilistic size effect, they cannot

be used here due to the fact that for the considered specimens the process

zone is not small compared to the component sizes (see Chapter 4).

In this chapter, the stochastic response of miniaturised components is

modelled using the cohesize zone models described in prior chapters. Sto-

chastic finite element methods (SFEM) are used to incorporate the effect of

the random property fields. Another application of SFEMs for MEMS will be

considered in the next chapter.

7.1 Stochastic finite elements for ultimate load computations

In fracture mechanics problems, the ultimate load is a property of the re-

sponse which is of particular interest, since it gives information on the in-

tegrity of a component. In the case that the problem under consideration

is subject to random property fields, this ultimate load becomes a random

variable. Stochastic finite element analyses can be performed to quantify this

random variable.

In the case of quasi-static analyses, as performed in the preceding chap-

ters, generally two types of analysis are recognised: uncertainty analysis and

reliability analysis (Gutiérrez and Krenk, 2004). The goal of uncertainty anal-

ysis is to determine statistical moments of the random ultimate load, typi-

cally the mean and standard deviation. One could therefore say that such an

analysis focuses on the approximation of the centre of the probability distri-

bution of the ultimate load. In the case of reliability analysis, the probability

of occurrence of a rare event is studied. Hence, in that case the tail of the

probability distribution is considered.

A concise description of the various stochastic methods considered in this

work is presented in the remainder of this section. A detailed discussion of

the available methods is found in (Gutiérrez and Krenk, 2004) and references

therein. Consistently with the work presented in the previous chapters, the

146

Stochastic finite elements for ultimate load computations

I = 0

I = 1

z1

I = 0θ θ

I

θII

θIII

I = 1

z1

z2 z2

Figure 7.1 Two examples of a connected failure domain in a two-dimensional random space. A single design point is found in the left figure,whereas multiple design points are found in the right figure.

ultimate load scale ηult is considered as a measure for the ultimate load.

Recall that the � notation was introduced in the previous chapter to indicate

random quantities.

7.1.1 Sampling-based methods

In this work, the crude Monte-Carlo method (Kenney and Keeping, 1954;

Cochran, 1977) is considered for the computation of the statistical moment

of the ultimate load. The idea of this method is to generate a sample of re-

alisations of the ultimate load, using a sample of realisations of the input

random fields (i.e. the random variables zi as discussed in Section 6.4). The

mean and standard deviation of the ultimate load are then obtained using

sample statistics as introduced in equation (6.2). The sample size required

to achieve a specific confidence level† increases with increasing coefficient of

variation of the sampled quantity.

The probability of the occurrence of failure can also be computed using

crude Monte-Carlo sampling. This is done by defining the failure-indicator

function

I ={

0 ηult ≥ ηcr

1 ηult < ηcr

, (7.1)

which is equal to 1 (indicating failure) in the case that the random ultimate

†Let � be an estimator of �. The confidence level C� of this estimator is then defined by

Prh˛˛ �−�

�

˛˛ ≥ 1− C�

i≤ 1− C�.

147


load scale is smaller than a prescribed critical value ηcr and 0 otherwise.

The boundary between the failure domain and safe domain, as indicated in

Figure 7.1, is referred to as the failure boundary. Since the failure probability

is equal to the mean of the indicator function

Pr [ηult < ηcr] = E [I] =∫Rnz

I(z)pz dz, (7.2)

crude Monte-Carlo sampling can be applied directly to a sample of realisa-

tions of the indicator function. Since the coefficient of variation of the indi-

cator function is inversely proportional to the failure probability (Melchers,

1999), the required sample size increases with decreasing failure probabil-

ities. When considering finite element models, such as these discussed in

the preceding chapters, this makes crude Monte-Carlo simulations generally

impractical for reliability analysis.

The required sample size for obtaining the probability of failure can be

reduced significantly using variance reduction techniques. Importance sam-

pling (Melchers, 1989), which does not sample around the centre of the mul-

tivariate standard normal distribution z = 0 but around z = θ, is one of

the most commonly used techniques. Using importance sampling, the failure

probability as defined in equation (7.2) follows as

Pr [ηult < ηcr] =∫Rnz

(I(z)

pz

pθ

)pθ dθ = E [J] . (7.3)

Appropriate selection of the sampling offset vector θ, using e.g. sensitivity-

based methods, can make the coefficient of variation of the shifted indicator

function J significantly smaller than that of the original indicator function I.

Consequently, the required sample size for importance sampling is signifi-

cantly smaller than for crude Monte-Carlo reliability analysis.

Sampling-based methods are appreciated for being non-intrusive. In this

particular case, the deterministic partition of unity method as introduced in

Chapter 2 is considered for various realisations of the input random fields.

Although sampling-based methods, such as Monte-Carlo sampling, are at-

tractive from an implementation point of view, they also have some disad-

vantages. First and foremost, the computational effort involved to obtain

satisfactory accuracy makes the method impractical for many engineering

applications. Although multi-processor simulations can somewhat relieve

148


this computational burden, especially the applicability of the crude Monte-

Carlo method for the determination of small failure probabilities is practi-

cally impossible when a finite element simulation needs to be performed for

all realisations. An additional disadvantage of Monte-Carlo sampling is that

convergence studies are troublesome since the total error is comprised of

both discretisation errors and sampling errors. A discussion on such a con-

vergence study can be found in Verhoosel et. al (2009c), where the selection

of appropriate sample sizes is also considered.

7.1.2 Sensitivity-based methods

The most commonly used sensitivity-based method for uncertainty analysis,

the perturbation method, is considered in this work. In the perturbation

method, the random ultimate load is approximated by a multi-dimensional

Taylor-expansion around z�. In this work the second-order expansion

ηult(z) ≈ ηult(z�)+ ∂ηult

∂zi

∣∣∣∣z�

(zi − z�i

)+ 1

2

∂2ηult

∂zi∂zj

∣∣∣∣∣z�

(zi − z�i )(zj − z�j ) (7.4)

is used to derive expressions for the mean and standard deviation in terms

of the gradient and Hessian of the ultimate load with respect to the random

variables zi as

μηult = ηult(z�)− ∂ηult

∂zi

∣∣∣∣z�z�i +

1

2

∂2ηult

∂zi∂zj

∣∣∣∣∣z�

(δij + z�i z�j );

σ2ηult

= ∂ηult

∂zi

∣∣∣∣z�

∂ηult

∂zi

∣∣∣∣z�.

(7.5)

Note that in the determination of the variance σ2ηult

only the quadratic terms

of the random variables have been incorporated (Johnson and Kotz, 1970).

Furthermore, it is noticed from equation (7.5) that only the trace of the Hes-

sian is required in the case that a Taylor expansion around z� = 0 is consid-

ered, which is here referred to as the median-centred perturbation method,

since it corresponds to the median realisation of the lognormal input ran-

dom fields. In the case of other expansion points, the off-diagonal terms of

the Hessian are also required. The determination of other expansion points

on the basis of the mean and mode realisations of the input random fields,

here referred to as the mean-centred and mode-centred perturbation method,

is discussed in Verhoosel et. al (2009c).

149


The failure probability can also be approximated using the sensitivities of

the ultimate load scale. This is done by iterative determination of the local

maxima of the probability density function along the failure boundary. The

positions of these local maxima in the random space are referred to as the

design points θ as shown in Figure 7.1. Determination of the design points

is achieved by the Hashofer-Lind Rackwitz-Fiessler (HL-RF) algorithm, which

iteratively computes the design points using

θk+1i =

(∂ηult

∂zj

∣∣∣θkθkj − ηult + ηcr

)∂ηult

∂zi

∣∣∣θk

∂ηult

∂zj

∣∣∣θk

∂ηult

∂zj

∣∣∣θk

. (7.6)

In the case that multiple local maxima of the probability density function

exist on the failure boundary, the point to which this algorithm converges

depends on the starting point of the algorithm. Appropriate starting points

can be determined using e.g. directional searches of the failure boundary

(Gutiérrez, 1999).

Once the design points are determined, a first-order approximation of the

failure probability can be obtained (dashed line in Figure 7.1). In the case of

a single design point, the failure probability can be obtained using

Pr [ηult < ηcr] ≈ Pr [z < −‖θ‖] . (7.7)

The accuracy of this first-order approximation, obtained using the first-order

reliability method (FORM), can be improved by incorporation of a correction

factor for the curvature of the failure boundary (Breitung, 1984). This second-

order analysis is commonly referred to as the second-order reliability method

(SORM). Further improvement in accuracy can be obtained by using the design

point to initiate importance sampling.

In the case that multiple design points are present, the first-order approx-

imation of the failure boundary is given by

IFORM =

⎧⎪⎨⎪⎩

0⋂

j=I,II,...ziθ

ji ≤

∥∥∥θj∥∥∥2

1 otherwise

. (7.8)

Although analytical expressions for the failure probability can be derived for

a few design points, Monte-Carlo integration of the expectation of the approx-

imated failure-indicator function can be performed efficiently.

150


7.1.3 Spectral stochastic finite element method

The spectral stochastic finite element method (SSFEM) gained popularity for

performing uncertainty analysis of linear elastic solid mechanics problems

over the past decade (Ghanem and Spanos, 1991). The method was also suc-

cessfully applied to other problems, such as random eigenvalue problems

(Verhoosel et. al, 2006). Application to nonlinear fracture mechanics prob-

lems has been studied recently (Chung and Gutiérrez, 2007; Verhoosel and

Gutierrez, 2007). The main principle of the SSFEM is to project the random

ultimate load scale on a base of an expansion in terms of Hermite polynomi-

als† as

ηult(z) =nψ∑i=0

ηult,iψi(z). (7.9)

This expansion is advantageous compared to the Taylor expansion consid-

ered in the perturbation method, since the Hermite polynomials form a com-

plete orthogonal basis of all functions satisfying∫Rnz

f (z)2e−zTz

2 dz < ∞. (7.10)

From a practical point of view, only a few orders of the expansion can be

considered, making this advantage just theoretical. It should, however, be

emphasised that the spectral stochastic finite element method is generally

observed to have superior convergence properties compared to the perturba-

tion method (Ghanem and Spanos, 1991).

The coefficients ηult,i of the expansion (7.9) can be obtained using

ηult,i = E [ηult(z)ψi(z)]

E[ψj(z)ψj(z)

] , (7.11)

where the expectation in the numerator can be evaluated using Monte-Carlo

integration (Field et. al, 2000). In the case that crude Monte-Carlo simula-

tion is used for the spectral coefficient determination, the advantage of the

spectral approach over crude Monte-Carlo sampling is limited. Alternatively,

the numerator in equation (7.11) can be evaluated using the spectral expan-

sion of the equilibrium path (η, a). This expansion is found by projection of

†For a two-dimensional random space, the first six generalised Hermite polynomials are:ψ0 = 1, ψ1 = z1, ψ2 = z2, ψ3 = z2

1 − 1, ψ4 = z1z2 and ψ5 = z22 − 1. Note that here the

probabilists’ Hermite polynomials are used instead of the physicists’ Hermite polynomials.

151


the equilibrium equations and path-following constraint (5.4) on the Hermite

polynomial basis, yielding a system of n×nψ equations

E

[(fint

g

)ψi(z)

]= E

[(ηf

0

)ψi(z)

]∀i ∈ [0, nψ]. (7.12)

The solution procedure is elaborately described in Verhoosel and Gutierrez

(2007). Evaluation of the expressions in equation (7.12) requires subsequent

determination of the spectral expansions of the loading function, stress and

internal force vector. Significant problems appear in the determination of the

spectral expansion of the loading function, which is a discontinuous function.

Approximation of this function using the spectral expansion with low-order

expansions is inaccurate, making the use of the SSFEM for nonlinear fracture

mechanics problems debatable. Successful application of SSFEM to this kind

of problems requires extension of the Hermite basis with discontinuous func-

tions. Although research in this direction is ongoing (Nouy et. al, 2007), the

SSFEM in its current form is not an attractive option for performing uncer-

tainty analysis and will not be considered in the remainder of this work.

7.2 Sensitivities computation

The sensitivity-based methods discussed in the previous section require the

computation of the first- and second-order derivatives of the ultimate load

scale with respect to the random variables zi. Computation of these deriva-

tives for solid-mechanics problems has been discussed in detail in e.g. Gutiér-

rez and de Borst (1998). Here some aspects of the derivatives computation

that are specific to this work are discussed. On one hand, this discussion

encompasses the incorporation of the path-following constraint in the sensi-

tivity analysis. On the other hand, the sensitivity contributions specific to the

partition of unity formulation are addressed.

7.2.1 Sensitivity of the ultimate load

The ultimate load scale can be written as

ηult = η(τult(α, β), α, β

), (7.13)

in which τult is the dissipated energy at the ultimate load, α is the random

field of material properties as introduced in equation (6.18) and β is the ran-

dom parametrisation of the crack.

152

Sensitivities computation

The first-order sensitivity of the ultimate load scale is obtained by differ-

entiation equation (7.13) to the random variables zi to yield

∂ηult

∂zi=

�0

∂η

∂τ

∣∣∣∣ult

∂τult

∂zi+ ∂η

∂αk

∣∣∣∣ult

∂αk

∂zi+ ∂η

∂βk

∣∣∣∣ult

∂βk

∂zi, (7.14)

in which the derivative of the load scale with respect to the path-following

parameter is equal to zero when evaluated at the ultimate load. Further dif-

ferentiation of equation (7.14) yields the second-order derivative with respect

to the random variables as

∂2ηult

∂zi∂zj= ∂2η

∂τ2

∣∣∣∣pi

∂τult

∂zi

∂τult

∂zj+ ∂2η

∂τ∂αk

∣∣∣∣pi

(∂τult

∂zi

∂αk

∂zj+ ∂τult

∂zj

∂αk

∂zi

)+

+ ∂2η

∂τ∂βk

∣∣∣∣pi

(∂τult

∂zi

∂βk

∂zj+ ∂τult

∂zj

∂βk

∂zi

)+ ∂2ηult

∂αk∂αl

∂αk

∂zi

∂αl

∂zj+

+ ∂2ηult

∂βk∂βl

∂βk

∂zi

∂βl

∂zj+ ∂2ηult

∂αk∂βl

∂αk

∂zi

∂βl

∂zj+ ∂2ηult

∂βk∂αl

∂βk

∂zi

∂αl

∂zj+

+ ∂ηult

∂αk

∂2αk

∂zi∂zj+ ∂ηult

∂βk

∂2βk

∂zi∂zj,

(7.15)

in which again use is made of the fact that the derivative of the load scale with

respect to the path-parameter is equal to zero at the ultimate load. It is im-

portant to note that although the sensitivity of the ultimate path-parameter

is not required for the evaluation of the first-order ultimate load scale sen-

sitivities, it is required for the second-order sensitivities. Hence, the shifting

of the peak position is irrelevant for the first-order sensitivities, but not for

the second-order derivatives. From (7.15) it is observed that the second-order

sensitivities of the ultimate path-parameter are not required for the compu-

tation of the second-order derivatives of the ultimate load.

Computation of the first- and second-order sensitivities of the ultimate

load requires evaluation of various sensitivities. The computation of the

derivatives of the load scale with respect to the problem parameters α and β

and path-parameter τ are discussed in Section 7.2.2. The computation of the

sensitivities of the path-parameter at the ultimate load is discussed in Sec-

tion 7.2.3. The sensitivities of the material property fields α follow directly

from the Karhunen-Loeve expansion as discussed in Section 6.4. A discus-

sion on the sensitivities of the crack path parametrisation β is presented in

Section 7.2.4.

153


7.2.2 Sensitivity of the equilibrium path

The derivatives of the path-parameter are computed by differentiation of both

the equilibrium equation and path following constraint (5.4). In the case that

the first-order derivative with respect to the problem parameter αi is to be

computed, this yields (∂ fint

∂αi∂g∂αi

)=(

∂η∂αi

f

0

), (7.16)

where it is assumed that the unit load vector f is deterministic. Elaboration

of the expressions in equation (7.16) using equation (5.2) and (5.3) yields[K −f∂g∂a

∂g∂η

](∂a∂αi∂η∂αi

)=(

−∂ fint

∂αi

− ∂g∂a0

∂a0

∂αi− ∂g∂η0

∂η0

∂αi

), (7.17)

where the path-following parameter τ is assumed to be independent of the

random variables. The derivative of the internal force vector with respect

to the random fields αi are obtained by differentiation of equation (2.12) as

described by e.g. Gutiérrez and de Borst (1998).

The second-order sensitivities can be obtained by differentiation of (7.17)

to obtain[K −f∂g∂a

∂g∂η

]⎛⎝ ∂2a∂αi∂αj∂2η

∂αi∂αj

⎞⎠ =

⎛⎝ − ∂2fint

∂αi∂αj− ∂K∂αi

∂a∂zj− ∂K∂αj

∂a∂zi− ∂K

∂a∂a∂zi

∂a∂zj

− ∂g∂a0

∂2a0

∂αi∂αj− ∂g∂η0

∂η20

∂αi∂αj− . . .

⎞⎠ . (7.18)

The first- and second order sensitivities of the load scale with respect to the

problem parameter αi as required for the evaluation of the ultimate load scale

sensitivities are computed by solving the systems (7.17) and (7.18). Similarly,

the sensitivities of the load scale with respect to the crack parametrisation βjand path-parameter τ can be obtained.

7.2.3 Sensitivity of the ultimate path-following parameter

Although an explicit definition of the path parameter corresponding to the

ultimate load scale, τult, is generally unavailable, its sensitivities can be ob-

tained using an implicit definition. Here this implicit definition is based on

the derivative of the load scale with respect to the path-following parameter

written as∂η

∂τ

(τult(α, β), α, β

)= 0. (7.19)

154


xnuc

β1 β2 β3

Figure 7.2 Schematic representation of the crack path parametrisation.

Alternatively, an implicit definition of the ultimate load can be obtained con-

sidering the eigenvalues of the consistent tangent. This approach will be con-

sidered in the next chapter to obtain the derivatives of a stability point. The

sensitivities of the ultimate load path-parameter follows by differentiation of

equation (7.19) with respect to the random variable zi to obtain

∂2η

∂τ2

∣∣∣∣ult

∂τult

∂zi+ ∂2η

∂τ∂αj

∣∣∣∣∣ult

∂αj

∂zi+ ∂2η

∂τ∂βj

∣∣∣∣∣ult

∂βj

∂zi= 0, (7.20)

from which the sensitivity of the path-following parameter at the ultimate

load follows as

∂τult

∂zi= −

(∂2η

∂τ2

∣∣∣∣ult

)−1(

∂2η

∂τ∂αj

∣∣∣∣∣ult

∂αj

∂zi+ ∂2η

∂τ∂βj

∣∣∣∣∣ult

∂βj

∂zi

). (7.21)

In a similar way, the second-order sensitivities of the ultimate path-parameter

can be obtained. It should be noted, however, that these second-order sensi-

tivities are not required for the evaluation of the sensitivities of the ultimate

load in equation (7.14) and (7.15). For that reason, the second-order results

are not presented here.

7.2.4 Crack path sensitivities

In the considered formulation, each crack path is parametrised by a nucle-

ation point xnuc and per-element crack directions β, as shown schematically

in Figure 7.2. Since in the current work linear basis functions are used, the

stress state is uniform per element. As a consequence, the sensitivity of the

nucleation point is equal to zero (although it is random). The per-element di-

rections of the crack do, however, have a nonzero sensitivity. Since the crack

propagation direction coincides with the principal directions of the stress

tensor (see Section 2.3.3), these sensitivities can be related directly to the

sensitivities of the stress field.

155


Load

scale

(τult, ηult)

Path-parameter

τi−1 τi τi+1

ηi−1

ηi+1

ηi

Figure 7.3 Approximation of the equilibrium path around the ultimate load.

Incorporation of the crack path sensitivity in the ultimate load sensitivity

computation also requires the computation of the partial derivative of the in-

ternal force vector with respect to the crack parametrisation. Semi-analytical

relations for this sensitivity are derived in (Schimmel et. al, 2009). There it

is concluded that the influence of the crack path on the overall sensitivity is

small for problems where the randomness in the stress field is limited. As will

be demonstrated by comparison with finite difference derivatives, the same

observation is done for the numerical simulations considered in this thesis.

For that reason, the crack path sensitivities are not included in the presented

numerical simulations.

7.2.5 Approximation of the equilibrium path

Once the ultimate load scale is determined numerically, its sensitivities can

be computed using equation (7.14) and (7.15). However, from a computa-

tional effort point of view it is often unattractive to exactly determine the

ultimate load. In the case that the equilibrium path is anyway traced beyond

the ultimate load, an approximation of the equilibrium path around the max-

imum can be made to reduce the computational effort of the algorithm. An

additional advantage of this approach is that it makes the method less intru-

sive. In the case that the equilibrium path is not history-dependent, as for the

electrostatic pull-in problem considered in the next chapter, determination of

the exact ultimate load scale can be done more efficiently. Approximation of

the equilibrium path is then less beneficial than for the case considered here.

Consider the equilibrium path around the ultimate load as shown in Fig-

156


ure 7.3. The path is approximated using the quadratic polynomial

η(τ, α) ≈ a0(α)+ a1(α)τ + a2(α)τ2, (7.22)

with the coefficients ai satisfying⎛⎜⎝ a0

a1

a2

⎞⎟⎠ =

⎡⎢⎣ 1 τi−1 τ2

i−1

1 τi τ2i

1 τi+1 τ2i+1

⎤⎥⎦−1⎛⎜⎝ ηi−1

ηiηi+1

⎞⎟⎠ . (7.23)

Using this system of equations, the sensitivities of the coefficients ai can be

directly related to these of ηi. Note that since the influence of the crack path

on the sensitivity is neglected, the parameters ai are independent of the path

parametrisation β.

The first-order sensitivities of the ultimate load are obtained by evaluation

of equation (7.14) to yield

∂ηult

∂zi=(∂a0

∂αj− 1

2

∂a1

∂αj

a1

a2+ 1

4

∂a2

∂αj

a21

a22

)∂αj

∂zi. (7.24)

The first-order sensitivity of the ultimate path parameter and second-order

sensitivity of the ultimate load can be obtained by evaluation of the equa-

tions (7.21) and (7.15).


A stochastic analysis is performed for the specimen shown in Figure 7.4.

The set-up is a three-point bending test on a specimen with two off-centred

notches, loaded by a concentrated load F = η · 1 N. The material of the spec-

imen is described by the random material properties derived in the previous

chapter (see Table 6.1). A linear elastic isotropic plane-stress relation is used

to described the constitutive behaviour of the bulk material, and the traction-

opening law discussed in Section 2.3.2 is used for modelling the cohesive

behaviour.

The specimen is discretised using 9277 linear triangular elements with a

refinement near the notch tips, yielding a finite element model consisting of

9372 degrees of freedom (without enhancements). The equilibrium path is

traced using the dissipation-based path-following constraint for damage as

described in Chapter 5, with a maximum dissipation step of 1 · 10−9 J, which

157


100100

F , u25 213.75

477.5

100

125

Figure 7.4 Schematic representation of the studied micro component. Thedimensions of the specimen are in microns.

Displacement u [μm]

Forc

eF

[N]

0.50.40.30.20.10

1

0.8

0.6

0.4

0.2

0

Max.p

rin

cip

al

stre

ss[M

Pa]

0

10

20

30

50

40

Figure 7.5 Comparison of the symmetric (solid line, top picture) and un-symmetric (dashed line, bottom picture) failure modes.

158


is small enough to prevent cracks from propagating multiple elements per

dissipation step. Typically, the ultimate load is determined in 100 steps.

The deterministic response of the three-point bending specimen, using

the mean values of the random fields, is shown in Figure 7.5. The wiggles

in the force-displacement curves are caused by the overestimation of the tip-

stresses, due to the discrete load steps. In deterministic analyses, these wig-

gles are often removed using a traction continuity condition (see Section 2.4).

Adjustment of the fracture strength to ensure traction continuity is, however,

not an option in a stochastic analyses, since the numerical variation in frac-

ture strength would interfere with the prescribed random properties. As a

consequence, relatively fine meshes need to be considered in order to avoid

a too large influence of the wiggles.

In Figure 7.5, the force-displacement curve of the three-point bending

specimen is presented for two cases. In the first case (solid line), symmetric

deformation of the specimen is enforced using additional boundary condi-

tions. Although the response found for the symmetric case resembles the

exact solution of the considered boundary value problem, it is only found

when forcing the solution to be symmetric. In the case that the additional

symmetry boundary conditions are omitted (dashed line), an unsymmetric

failure pattern is found. This is a result of numerical errors causing one crack

to become dominant over the other. Comparison of the response curves for

the symmetric and unsymmetric case shows that the responses differ signif-

icantly. In particular, the unsymmetric response is observed to be consider-

ably more brittle. The ultimate load is found as 0.90 N in the unsymmetric

case. Mesh refinement shows that the relative error in the ultimate load is

approximately 0.5 percent.

Since tiny variations in the properties of the numerical model can cause

significant variations in the response, the deterministic problem can be con-

sidered as ill-posed. This ill-posedness is, however, not present if the proper-

ties are assumed to be random since in that case the variations causing one

crack to be dominant are physical. The chance of having a perfectly sym-

metric properties field is equal to zero (although the probability density is

finite), such that the problem of determining the statistics of the response is

well-posed.

The random fields are discretised using the Karhunen-Loeve expansion

for random vector fields as discussed in Section 6.4. An expansion of 20 KL-

terms is considered, which resembles truncation at 1 percent of the largest

159


×10−1

Second KL-termFirst KL-term×10−3

-1.2

1.2

-2.0

-1.2

-0.4

0.4

1.2

2.0

You

ng’s

mod

ulu

s

-4.0

4.0

Fra

ctu

reto

ugh

nes

sFra

ctu

rest

ren

gth

-0.72

0.36

-0.36

0.72

2.4

0.8

-0.8

-2.4

×10−2

Figure 7.6 Karhunen-Loeve functions for the random material properties.Note that the functions are eigenfunctions of the underlying Gaussian pro-cess.

160


μFult [N] σFult [N] VFult

Monte-Carlo 0.876 0.047 5.3%

Perturbation method 0.891 0.041 4.5%

Table 7.1 Mean, standard deviation and coefficient of variation of the ulti-mate load, computed using various stochastic methods.

in magnitude KL-eigenvalue. The first two terms of the KL-expansion, ob-

tained using the same mesh as for the problem discretisation, are shown in

Figure 7.6. Although the fields have similar mode shapes, it is observed that

they are not the same for the three property fields. This obviously is a result

of the cross-correlations of the fields as discussed in the previous chapter.

The unsymmetric perturbation coming from the second KL-mode obviously

dictates the position of the dominant crack.

7.3.1 Uncertainty analysis

The mean and standard deviation of the ultimate load are here computed

using two methods. A crude Monte-Carlo simulation is performed using a

sample size of 5000 realisations. The results of this method are presented in

Table 7.1. The confidence level of the result is estimated to be 99.8 percent

for the mean and 97.7 percent for the standard deviation. Considering the

computational effort for this simulation†, the practical usefulness of Monte-

Carlo sampling is limited.

A sensitivity-based computation of the first two statistical moments of

the ultimate load is also considered. It should, however, be noted that the

ill-posedness problem as outlined above also has implications for the com-

putation of the problem sensitivities. This is well illustrated by consider-

ing the variation of the ultimate load due to perturbations of the individual

Karhunen-Loeve modes, as shown in Figure 7.7. As can be seen, the fracture

load dependence on the first Karhunen-Loeve function is close to linear. Con-

sidering the variation of the ultimate load due to the second Karhunen-Loeve

mode as shown in Figure 7.7, however, reveals a problem. The sensitivity

with respect to the unsymmetric KL-mode does not exist at z2 = 0. The same

holds for higher-order unsymmetric KL-modes.

†One realisation takes about one hour on a 2.8 GHz Quad-Core Intel Xeon 5400 series

processor.

161


Random variable zi [−]

Ult

imate

loadF

ult

[N]

210-1-2

1

0.96

0.92

0.88

0.84

0.8

Figure 7.7 Variation of the ultimate load caused by variations in the first(z1, solid line) and second (z2, dashed line) Gaussian random variables, cor-responding to the Karhunen-Loeve functions shown in Figure 7.6.

Sensitivity∂Fult

∂z1

∂Fult

∂z2

Finite difference 0.030 -0.015

Semi-analytical 0.031 -0.018

Table 7.2 Comparison of some of the semi-analytically determined deriva-tives with finite difference approximations, evaluated for z = [0,2,0, . . . ,0].The finite difference results are computed using first-order accurate finitedifference approximations with Δzi = 1

2 .

162


Comparison of the semi-analytical sensitivities at z = [0,2,0, . . . ,0] with

finite difference approximations (see Table 7.2) shows that the derivatives of

the ultimate load with respect to the first random variable are well computed.

This confirms that the influence of the crack-path variation on the sensitivi-

ties is limited. The derivative of the ultimate load with respect to the second

random variable also reasonably resembles the finite difference approxima-

tion. Thereby it should be remarked that a determination of an accurate

finite difference result cannot be obtained due to the limited accuracy of the

method. For the same reason, verification of the second-order sensitivities is

troublesome.

As a consequence of the non-smoothness of the ultimate load (as a func-

tion of the random variables), application of the perturbation method with

centre z� = 0 does not yield reliable results. In order to circumvent this

problem, the perturbation analysis is performed for randomly generated ex-

pansion points z�. The mean and standard deviation are then obtained using

sampling statistics and are given in Table 7.1. With 100 expansion points, a

confidence level of 99.5% is obtained for the mean and 97.5% for the stan-

dard deviation. Comparison with the crude Monte-Carlo results shows that

the perturbation method requires considerably less computation time in or-

der to reach similar confidence levels.

Comparison of the results of the perturbation method with these from

the Monte-Carlo method, shows that the accuracy of the result is limited.

A 10 percent error is observed for the standard deviation of the ultimate

load. In addition, it is observed that the second-order derivatives are very

small, consequently not predicting the decrease in average ultimate load due

to the present randomness. Obviously, this is caused by the fact that the

symmetry of the ultimate load function with respect to the random variables

corresponding to the anti-symmetric KL-modes (such as the second mode) is

not reflected by the perturbation method.

Improvement of the results of the sensitivity-based methods for uncer-

tainty analysis is troublesome due the non-differentiability of the ultimate

load. Thereby it should be remarked that the position of the kink in the

ultimate load function is generally unknown.

7.3.2 Reliability analysis

The probability of the occurrence of an ultimate load less than 0.75 N is com-

puted using the first-order reliability method (FORM). Initial points for the

163


Random variable z1 z2 z7 z6

Directional search -4.18 ±5.62 ±12.9 17.3

Design point A A/B A/B A

Table 7.3 Initial points for the HL-RF algorithm determined using direc-tional searches, and corresponding design points.

Failure probability Pr[Fult < −0.75 N]

FORM 3.61 · 10−3

Importance sampling 6.0 · 10−3

Table 7.4 The probability of the occurrence of an ultimate load below 0.75 Nas determined by importance sampling and the first-order reliability method(FORM).

HL-RF algorithm are determined using directional searches. The results for

the four smallest in magnitude initial points are collected in Table 7.3. As can

be seen, two equal in magnitude but opposite in sign initial points are found

on the z2- and z7-axis, which is a result of the anti-symmetry of the corre-

sponding KL-modes. The initial points for the other KL-modes are observed

to be considerably larger, i.e. their probability of occurrence is significantly

smaller, and are therefore omitted in the analysis.

For the initial points in Table 7.3, the HL-RF algorithm is observed to con-

verge to two distinct design points, A and B. The corresponding realisations

of the modulus of elasticity, fracture strength and fracture toughness are

shown in Figure 7.8. As can be seen, the two design points are mirrored with

respect to the symmetry axis of the specimen. The determined failure modes

correspond to the unsymmetric failure mode shown in Figure 7.5, with the

crack emanating from either one of the notches. The absence of a symmetric

design point indicates that the symmetric failure mode is very unlikely (or

possibly even non-existent). When using the starting points on the z1- and

z6-axis, the HL-RF algorithm converges to design point A. It should, however,

be emphasised that this is a result of discretisation errors. The algorithm

could have equally well converged to design point B. Since both design point

are found anyway, this is not a problem for the analysis.

The first-order approximation of the probability of failure (shown in Ta-

ble 7.4) is computed by evaluation of equation (7.8) using Monte-Carlo sam-

pling with 1 · 108 realisations. The probability of an ultimate load smaller

164


33.0Design point A Design point B

×10−3

0.70.8

0.9

1.0

1.1

1.2

Fra

c.st

ren

.[M

Pa]

Fra

c.to

ugh

.[N

/mm

]

40.5

You

ng’s

mod

.[G

Pa]

41.1

41.7

42.3

42.9

43.5

32.0

32.2

32.4

32.6

32.8

Figure 7.8 Design points obtained using the HL-RF algorithm.

165


than 0.75 N is approximated as 3.61 · 10−3, with a confidence level of 95%.

Importance sampling around the design points is performed to improve the

accuracy of the result. With a sample size of 3600 realisations, the failure

probability is determined as 6.0 · 10−3 with a confidence level of 92.5 per-

cent.

Considering the significant difference between the result of the first-order

reliability method and importance sampling, it is concluded that for the con-

sidered application importance sampling is required to obtain quantitative

meaningful results. The reliability methods can, however, be useful to gain

insight in failure mechanisms, which can be of significant aid in the design of

structures.

166

Chapter 8

Stochastic analysis of the electrostaticpull-in instability

Micro electromechanical systems (MEMS) composed of piezoelec-

tric materials have been studied in the previous chapters. Another

class of MEMS are the systems in which electrostatic forces are used to de-

form (generally slender) parts of a micro system. This makes these sys-

tems suitable for sensing and actuating purposes, for example used in (high-

frequency) switches. One of the major issues regarding the reliability of such

components is the pull-in instability. This instability is characterised by the

fact that a critical loading of the system exists for which static equilibrium

becomes unstable. This instability is undesirable in many applications and

appropriate understanding of this phenomenon is of significant importance

for the development of this class of MEMS.

Numerical simulation of the pull-in instability has been studied exten-

sively, which has resulted in several analytical and numerical methods for

its prediction (Pamidighantam et. al, 2002; Rochus et. al, 2006). Generally,

these methods presume that the properties of a system are exactly known. In

reality, such properties are not known exactly due to the presence of material

imperfections, variations in manufacturing, material degradation and many

other sources of uncertainty. In practice, these uncertainties are taken into

account indirectly by using conservative safety factors. Direct incorporation

of uncertainties in the analysis of pull-in instabilities should therefore result

in more efficient designs.

Here the analysis of the pull-in instability with structural uncertainties is

performed using the stochastic finite element methods (SFEMs) introduced

in the previous chapter. The basic configuration of the considered elec-

167

Stochastic analysis of the electrostatic pull-in instability

L

h

V

X1

w(X1, h)X2 d

Y(X1, θ), I

Figure 8.1 Schematic representation of the considered pull-in problem.

trostatic problem is shown in Figure 8.1. Following the numerical exam-

ple presented in Rochus et. al (2007), a doubly clamped beam of length

L = 300μm and thickness 3μm is suspended above a rigid electrode at

height d = 1μm. The modulus of elasticity of the beam is described by a

stationary lognormal random field Y = Y(X1, θ) as described in Chapter 6

with mean μY = 77 GPa, standard deviation σY and Gaussian-shaped correla-

tion function (6.17) with correlation length lY = √π L10 . This random field of

elastic properties is considered for various values of the standard deviation.

An electric potential difference is applied over the vacuum (with permittivity

λ0 = 8.8542 · 10−12 N/V2) gap between the electrode and the beam by means

of a D.C. power generator. The voltage at which the static equilibrium be-

comes unstable is referred to as the pull-in voltage. Methods for determining

the statistical moments of this pull-in voltage, as well as the probability of the

occurrence of pull-in below a specified voltage, are proposed in the remainder

of this chapter.

8.1 Deterministic pull-in problem

A monolithically coupled electrostatic model is developed for the analysis

of the pull-in problem introduced in the previous section. The equilibrium

equations of the structure and the electric field are discretised using the fi-

nite element method, which allows for extension of the model by considering

complex geometries or more sophisticated descriptions of the structure.

In contrast to the electromechanical problems considered in the previous

chapters, the physical phenomenon of interest is here a consequence of geo-

metrical nonlinearities. Therefore, in order to capture this phenomenon, a ge-

ometrically nonlinear framework as schematically shown in Figure 8.2 is used.

The motion of the beam is governed by the Euler-Bernoulli equation, hence

168

Deterministic pull-in problem

Ω0

X2

Undeformed configuration Deformed configuration

x1

x2

Ω

X1

Figure 8.2 Schematic representation of the deformed and undeformed domain.

assuming small deformations and deformation gradients. Extension of the

formulation with non-linear kinematics for the beam is possible. However,

since these nonlinearities are not required for the simulation of the physical

phenomenon of interest (i.e. voltage driven pull-in for a slender beam with

small deflections, d� L), the kinematics are assumed to be linear. The posi-

tion of a point xi in the deformed electric domain is written in terms of the

undeformed configuration Xi as

xi = Xi + δi2w(X). (8.1)

Here the deformation field w(X) is considered to be of the form

w(X) =w(X1, h)X2

d, (8.2)

as schematically illustrated in Figure 8.2. According to this form of the dis-

placement field, it is equal to zero on the left, right and bottom boundary

of the domain and equals the vertical displacement of the beam at the top

boundary. Note that different choices for the form of the displacement field

are possible. Although such different choices will affect the conditioning of

the considered system, the obtained results will converge to the same solu-

tion.

8.1.1 Equilibrium formulation

The equilibrium of the electrostatic problem introduced above is described

by equating the rate of the internal Gibbs free-energy Wint to the externally

applied power P as

Wint = P. (8.3)

The term “external power” is used here to indicate the rate of external en-

ergy required to manifest a certain rate in the internal Gibbs energy. This

169


definition of the external power follows from the weak form of the electro-

static equilibrium equations as outlined in Section 2.2.2. For the considered

configuration, it is given by

P = −QV = −L∫

0

qΦ dX1, (8.4)

with Q being the accumulated charge on the beam, V the rate of change of

the applied voltage and q(X1) the surface charge density at the beam.

The Gibbs energy is given by

Wint = 1

2

∫ L

0YI

(∂2w

∂X21

)2

dX1 − 1

2

∫Ω0|J|DiEi dΩ

0, (8.5)

with Jij = ∂xi∂Xj

= δij + δi2 ∂w∂Xj being the Jacobian. Note that the � is dropped

since the modulus of elasticity Y(X1) is here considered deterministic. As-

suming linear constitutive behaviour for both the mechanical and electric

quantities, the rate of Gibbs energy is written as

Wint =∫ L

0YI∂2w

∂X21

∂2w

∂X21

dX1 −∫Ω0|J|DiEi dΩ

0 − 1

2

∫Ω0

∂ |J|∂t

DiEi dΩ0. (8.6)

Rewriting the rates of the electric field and Jacobian determinant using

Ei = ∂

∂t

(− ∂Φ∂xi

)= − ∂Φ

∂XjJ−1ji − E2

∂w

∂XkJ−1ki ;

∂ |J|∂t

= |J| J−1kl Jlk = |J| J−1

k2

∂w

∂Xk,

(8.7)

where the second relation is known as Jacobi’s formula, then yields

Wint =∫ L

0YI∂2w

∂X21

∂2w

∂X21

dX1 +∫Ω0|J|DiJ−1

ji

∂Φ

∂XjdΩ

0+

+∫Ω0|J|Di

(E2J

−1ki −

1

2J−1k2 Ei

)∂w

∂XkdΩ

0.

(8.8)

This equation expresses the rate of internal energy in terms of the rates of

the electric and mechanical fields, which is a useful form for the derivation

of a discretised system of equations as is discussed next.

170


8.1.2 Finite element formulation

The equilibrium formulation presented in the previous subsection is discre-

tised using the finite element method. The displacement of the beam, i.e. the

displacement field on the top boundary, is discretised by means of nX1 Her-

mite elements† as

w(X1, h) = ψ(X1) · a1 + υ(X1) · a2, (8.9)

with a1 and a2 being the nodal displacements and nodal rotations, respec-

tively. The displacement field and the electric potential field in the domain Ω

are discretised using nX1 ×nX2 linear quadrilateral elements as

w(X) = γ(X) · a1;

Φ(X) = γ(X) · a3,(8.10)

with a3 being the nodal value of the electric potential. Hence the state of

the discretised system is fully described by the n-dimensional state vector

a = [a1,a2,a3].

Substitution of (8.9) and (8.10) in the rate of Gibbs energy (8.8) then yields

Wint(a,α) = fint(a,α) · a, (8.11)

with fint(a,α) being the internal force vector and α a vector containing the

properties of the electrostatic problem. The components of the internal force

vector are elaborated in Appendix B. Using the discretisation (8.10), the ex-

ternally applied power (8.4) is written as

P(q,α) = −Q(q,α)V = fext(q,α) · a, (8.12)

which depends on both the surface charge density on the beam q and the

length of the beam, which is incorporated in the properties vector α.

Substitution of equation (8.11) and (8.12) in the continuous equilibrium

equation (8.3) then yields the discrete set of equilibrium equations

fint(a,α) = fext(q,α). (8.13)

The boundary conditions for the beam are enforced by setting both the nodal

displacements and nodal rotations at the clamping points equal to zero. Sim-

ilarly, the electric potential at the bottom electrode is set to zero by means

†Hermite elements make use of the Hermite cubic shape functions. These shape functions

are based on the displacements and rotations in the element vertices.

171


of the nodal potential vector. The loading of the system is performed by

gradually increasing the nodal potential values at the top boundary. These

boundary conditions are incorporated in the finite element program by means

of the decomposition

a = Caf + ap + ηa (8.14)

with af, ap and a being the free-, prescribed- and loading-degrees of freedom,

respectively. The matrix C is the constraint matrix and η is referred to as

the load factor. Note that in order to make the system non-singular, the

nodal displacements inside the electric domain are constrained according to

equation (8.2). A detailed discussion on this loading condition is found in

Appendix A.

8.1.3 Computation of the deterministic pull-in voltage

In the deterministic case, i.e. with σY = 0, the equilibrium path (η,a) is de-

termined as described in Chapter 5 by incrementally solving the nonlinear

set of equilibrium equations (8.13) in combination with the path-following

constraint

g = Q(a,α)− Q, (8.15)

proposed by Hannot and Rixen (2008). The equilibrium path, traced with 5 ·10−9 C charge steps, is shown in Figure 8.3. As can be seen, the applied

voltage increases upon an increase in charge and midpoint displacement until

a maximum value in the voltage is reached. After that the voltage needs to

decrease in order to let both the charge and midpoint displacement increase

further. This implies that after the maximum voltage is reached, the beam

displacement will keep increasing, even if the voltage is decreased. For that

reason, this maximum voltage is referred to as the pull-in voltage Vpi. The

corresponding accumulated charge at the beam is denoted by Qpi.

At the instance of pull-in, the rate of the load factor is equal to zero (η =0). Under this condition, the equilibrium equation in rate form can be written

as [CTK(api(α),α)C

]ap

(api(α),α

)= 0, (8.16)

with api the state vector at the instance of pull-in, which depends on the

properties vector α. From equation (8.16) it follows that the pull-in voltage

corresponds to the case in which the constrained consistent tangent CTKC

172


×10−1

Midpoint displacement [μm]

Volt

ageV

[V]

76543210

20

16

12

8

4

0 ×10−8

Charge Q [C]

Volt

ageV

[V]

876543210

20

16

12

8

4

0

Figure 8.3 Response of the pull-in problem traced with 5 · 10−9 C chargesteps (�). For illustration purposes, the solid lines are traced with consider-ably smaller charge steps.

×10−3

Voltage V [V]

Eig

envalu

eδ

1

201612840

1

0.5

0

-0.5

-1

-1.5

-2 ×10−8

×10−3

Charge Q [C]

Eig

envalu

eδ

1

876543210

1

0.5

0

-0.5

-1

-1.5

-2

Figure 8.4 Dependence of the eigenvalue δ1 on the voltage (left) and charge(right) for the pull-in problem. The eigenvalues are traced using 5 · 10−9 Ccharge steps (�). The solid lines are constructed using significantly smallersteps for illustration purposes.

173


has a zero eigenvalue, δi, according to the eigenvalue problem[CTKC

](api(α),α

)vi

(api(α),α

)= δi

(api(α),α

)vi

(api(α),α

), (8.17)

with i ∈ [1, n] and vi being the corresponding right eigenvector. This is

illustrated in Figure 8.4, in which the smallest in magnitude eigenvalue of

the consistent tangent is shown†. As can be seen, the smallest in magnitude

eigenvalue δ1 (since the eigenvalues are sorted as |δ1| < |δ2| < . . . < |δN|) is

indeed equal to zero at the pull-in voltage. Unfortunately, the derivative of

the critical eigenvalue with respect to the voltage is infinite at pull-in, which

makes direct iterative determination of the pull-in voltage troublesome. For

that reason, the pull-in voltage is determined indirectly by considering it to

be function of the accumulated charge and problem properties, leading to the

implicit definition of the charge at pull-in

δ1

(Qpi(α),α

) = 0. (8.18)

The dependence of the eigenvalue on the accumulated charge is shown in

Figure 8.4. The charge at pull-in Qpi is determined using a Newton-Raphson

procedure. The computation of the required tangent (the derivative of the

eigenvalue δ1 with respect to the charge Q) is discussed in Section 8.2. Once

the charge at pull-in is known, the corresponding pull-in voltage

Vpi = V(Qpi(α),α

)(8.19)

follows directly from the equilibrium solution, as shown in Figure 8.3.

It is important to note that for the determination of the pull-in voltage,

only the smallest in magnitude eigenvalue is required. This makes the use

of iterative algorithms for computing this eigenvalue (e.g. the inverse power

method) attractive. Application of the proposed method for large systems

therefore remains feasible. Moreover, it should be noted that the computa-

tion of the eigenvalues and eigenvalue derivatives is only required to per-

form the Newton-Raphson iterations. In order to limit the computational

effort, this iterative procedure is only started after the maximum has been

reached. This is checked by consideration of the finite difference estimate

of the derivative of the voltage with respect to the charge (once it becomes

negative, the Newton-Raphson procedure is started).

†Robust determination of the eigenvalues using iterative eigenvalue solvers requires scal-ing of the consistent tangent (Higham et. al, 2008). The shown eigenvalue is that of the matrix

CTSTKSC, with S a diagonal matrix containing the (estimated) matrix norms of K11, K22 andK33, corresponding to the degree of freedoms a1, a2 and a3, respectively, and evaluated in the

undeformed state.

174


2

1

×10−3

nX1

εV

pi(n

X1,n

X2)

100101

100

10

1

0.1

Figure 8.5 Mesh convergence plots of the pull-in voltage. The error εVpi

is defined in equation (8.20). The solution with nX1 = 128 and nX2 = 8 isconsidered as the fully resolved solution. The results for nX2 = 2 (solid line)and nX2 = 4 (�) coincide.

X1 [μm]

w(X

1,h)

[μm

]

300250200150100500

0

-0.2

-0.4

-0.6

-0.8

-1

Figure 8.6 Deformation of the beam at the pull-in voltage computed usingnX1 = 32 elements.

175


For the deterministic case, with μE = 77 GPa and σE = 0 GPa, the pull-in

voltage is computed as Vpi = 18.4 V on a mesh with nX1 = 128 and nX2 = 8.

This result compares favourably with the results reported by Pamidighantam

et. al (2002) and Rochus et. al (2007). The corresponding deformation of the

beam is shown in Figure 8.6. The pull-in voltage on a mesh with nX1 = 9

and nX2 = 3 is found as 18.6 V, which is also in agreement with the results

reported by Rochus et. al (2007).

A mesh convergence analysis is performed in order to obtain an error

estimate of the numerical result. The relative error of the pull-in voltage is

defined as

εVpi(nX1 , nX2) =Vpi(nX1 , nX2)− Vpi(nX1,res, nX2,res)

Vpi(nX1res, nX2,res), (8.20)

with nX1,res = 128 and nX2,res = 8 being the discretisation parameters corre-

sponding to the solution which is assumed to be fully resolved. This error is

depicted for various discretisations in Figure 8.5. As can be seen, the relative

error for the discretisations with nX2 = 2 and nX2 = 4 coincide for all val-

ues of the discretisation parameter nX1 , indicating that the relative error is

practically independent of nX2 . The constant slope in the loglog-plot, which

is equal to 2, indicates that the error falls in the asymptotic region of conver-

gence. As can be seen, the relative error on a mesh with nX1 = 32 and nX2 = 2

is less than 1 ·10−3. Since this error is considered to be acceptable this mesh

will be used in the remainder of this work.

8.2 Sensitivities computation

As discussed in the previous chapter, application of the various sensitivity-

based methods for uncertainty and reliability analysis to the pull-in voltage

requires the first- and second-order sensitivities of that voltage.

8.2.1 Sensitivities of the pull-in voltage

The sensitivity of the pull-in voltage is obtained by differentiation of equa-

tion (8.19) with respect to the random variable zi to yield

∂Vpi

∂zi=

�0

∂V

∂Q

∣∣∣∣pi

∂Qpi

∂zi+ ∂Vpi

∂αk

∂αk∂zi

, (8.21)

176


in which the derivative of the voltage with respect to the surface charge is

equal to zero since it is evaluated at the pull-in point, �|pi. The second-order

sensitivities of the pull-in voltage are obtained by subsequent differentiation

of this equation with respect to zj to get

∂2Vpi

∂zi∂zj= ∂2V

∂Q2

∣∣∣∣pi

∂Qpi

∂zi

∂Qpi

∂zj+ ∂2V

∂Q∂αk

∣∣∣∣pi

(∂Qpi

∂zi

∂αk

∂zj+ ∂Qpi

∂zj

∂αk

∂zi

)+

+ ∂2Vpi

∂αk∂αl

∂αk∂zi

∂αl∂zj

+ ∂Vpi

∂αk

∂2αk∂zi∂zj

.

(8.22)

The sensitivities of the voltage with respect to the accumulated charge and

properties vector are obtained as outlined in Section 7.2.2. The sensitivities

of the properties vector with respect to the random variables follows from

the definition of the random field as described in Section 6.1.3. Moreover,

in order to evaluate the second-order pull-in voltage sensitivities (8.22), the

derivatives of the accumulated charge at pull-in Qpi with respect to the ran-

dom variables zi are required.

8.2.2 Sensitivities of the accumulated charge at pull-in

The accumulated charge at pull-in is implicitly defined by equation (8.18). Its

sensitivities are obtained by differentiation of this equation with respect to

the random variable zi, resulting in

∂δ1

∂Q

∣∣∣∣∣pi

∂Qpi

∂zi+ ∂δ1

∂αj

∣∣∣∣∣pi

∂αj

∂zi= 0. (8.23)

Rewriting then yields the sensitivity of the charge at pull-in as

∂Qpi

∂zi= −

⎛⎝ ∂δ1

∂Q

∣∣∣∣∣pi

⎞⎠−1

∂δ1

∂αj

∣∣∣∣∣pi

∂αj

∂zi. (8.24)

For the evaluation of equation (8.22), the second-order derivatives of Qpi with

respect to zi is not required. From (8.24) it is observed that the first-order

derivatives of the charge at pull-in can be computed once the derivatives

of the smallest eigenvalue δ1 with respect to the accumulated charge and

random properties are known. In order to find these eigenvalue sensitivities,

the derivatives of the system matrices are required. These sensitivities can be

177


obtained as explained in Section 8.2.3. Using the system matrix sensitivities,

the eigenvalue derivatives are computed as elaborated in Plaut and Huseyin

(1973).

8.2.3 Sensitivities of the tangent matrix

As mentioned in the previous section, the first-order sensitivities of the con-

sistent tangent with respect to the charge and properties vector are required

for the evaluation of the eigenvalue sensitivities. Since a geometrically non-

linear model is considered, the consistent tangent is not only dependent on

the properties vector α, but also on the state vector a. Hence the first-order

sensitivities with respect to the accumulated charge and properties vector are

given bydK

dQpi= ∂K

∂aj

∂aj

∂Qpi;

dK

dαi= ∂K

∂aj

∂aj

∂αi+ ∂K

∂αi.

(8.25)

Determination of the derivatives of the state vector with respect to the charge

and properties vector is outlined in Section 7.2.2.

From equation (8.25) it is seen that the computation of the first-order

derivative of the pull-in voltage with respect to the random variables zi re-

quires the evaluation of the derivative of the stiffness matrix with respect to

the state vector. Although the storage requirements are not excessive since

the tangent derivatives are always contracted with the same vectors, the as-

sembly of these terms has a significant effect on the total computation time.


The stochastic methods introduced in the previous chapter are tested for the

uncertainty and reliability analysis of the pull-in problem. All computations

are performed on the mesh with nX1 = 32 and nX2 = 2, for which the de-

terministic pull-in voltage is obtained as 18.42 V with a relative error of less

than 1 · 10−3. In the subsequent sections, the results of the various methods

are compared on the basis of both accuracy and computational effort.

As discussed in the introduction, the mean modulus of elasticity μE is

taken as 77 GPa. The random field for the modulus of elasticity with the corre-

lation length lY =√π L

10 is discretised using 16 Karhunen-Loeve terms, which

178


Expectation of the pull-in voltage μVpi [V]

Young’s modulus c.o.v. VY 0.10 0.25 0.50

Crude Monte-Carlo 18.36 18.10 17.27

Mean-centred perturbation 18.37 18.10 17.26

Median-centred perturbation 18.37 18.10 17.27

Mode-centred perturbation 18.37 18.10 17.28

Standard deviation of the pull-in voltage σVpi [V]


Crude Monte-Carlo 0.416 1.02 1.87

Mean-centred perturbation 0.415 1.02 1.88

Median-centred perturbation 0.415 1.02 1.88

Mode-centred perturbation 0.415 1.01 1.87

Table 8.1 The expected value and standard deviation of the pull-in voltageas determined by crude Monte-Carlo simulations and the perturbation meth-ods for various coefficients of variations (c.o.v.) of the modulus of elasticity.

corresponds to the criterion that the KL-expansion is truncated for eigenval-

ues smaller in magnitude than two percent of the largest eigenvalue. Coeffi-

cients of variation of the stationary Young’s modulus field VY (= σY/μY ) up

to 50 percent are considered.

8.3.1 Uncertainty analysis

The mean and standard deviation of the pull-in voltage are determined us-

ing the sensitivity-based methods for uncertainty analysis introduced in the

previous chapter. The results of these analyses are presented in Table 8.1.

Sensitivity∂Vpi

∂z1

∂2Vpi

∂z21

Finite difference -0.2956 0.00533

Semi-analytical -0.2956 0.00533

Table 8.2 Comparison of some of the semi-analytically determined deriva-tives with finite difference approximations. The derivatives are evaluatedfor z = 0 and VY = 0.10. The finite difference results are computed usingsecond-order accurate finite difference approximations with Δz1 = 0.01.

179


The Monte-Carlo simulations have been performed with a sample size of

over 33 thousand, which is based on a confidence level of 99 percent for the

standard deviation. Since the discretisation errors are an order of magnitude

smaller than the sampling error, their effect on the confidence level for the

standard deviation is neglected. The confidence levels for the mean pull-

in voltage are 99.96%, 99.90% and 99.81% for coefficients of variation in the

modulus of elasticity of 10%, 25% and 50%, respectively. Hence for the mean

values, the sampling error is of the same order as the discretisation error.

Considering this, a further increase in sample size is not meaningful.

The sensitivities of the pull-in voltage, required for the perturbation meth-

ods are verified using finite difference approximations. As can be seen in

Table 8.2, the semi-analytical results match the finite difference results, in-

dicating the correctness of the sensitivities derived in Section 8.2. The re-

sults of all the perturbation methods are in excellent agreement with those

obtained using the crude Monte-Carlo simulation. Apparently, the relation

between the pull-in voltage and random variables can very well be approxi-

mated by a second-order Taylor-expansion. Since accurate perturbation re-

sults are obtained with only a small fraction of the computational effort in-

volved in crude Monte-Carlo sampling, the perturbation method is considered

to be more efficient than the crude Monte-Carlo method for the case studied

here. In addition, it should be remarked that for the current simulation, the

median-centred perturbation method (around z = 0) is favoured, since only

the diagonal of the Hessian matrix is required in that case.

In Figure 8.7 the dependence of the first two statistical moments of the

pull-in voltage on the coefficient of variation in the modulus of elasticity is

shown. The mean and standard deviation are well approximated by a second-

and first-order polynomial, respectively. This confirms that the pull-in volt-

age is well approximated by the second-order Taylor expansion used for the

perturbation results. Figure 8.7 shows a significant drop of the mean pull-in

voltage with increasing uncertainty in the modulus of elasticity. In the case

of 50 percent coefficient of variation of the Young’s modulus, the mean drops

with 6.4 percent. The assumed uncertainties reduce the pull-in voltage and

hence the deterministic results are non-conservative. This effect must nor-

mally be accounted for by ad-hoc safety factors. Reliability analysis, however,

can be used for the more precise definition of these safety factors.

180


Modulus of elasticity c.o.v. VY

Mea

nμV

pi[V

]

0.50.40.30.20.10

18.6

18.3

18

17.7

17.4

17.1

Modulus of elasticity c.o.v. VY

Std

.d

ev.σV

pi[V

]

0.50.40.30.20.10

2

1.6

1.2

0.8

0.4

0

Figure 8.7 Dependence of the mean and standard deviation of the pull-involtage on the coefficient of variation (c.o.v.) of the modulus of elasticity.The results computed by the mean-centred perturbation method (�) are fit-ted using a quadratic polynomial for the mean and a linear polynomial forthe standard deviation (dashed lines).

Pull-in probability Pr[Vpi < 15 V]


Importance sampling 3.68 · 10−19 4.77 · 10−4 10.6 · 10−2

FORM 2.82 · 10−19 3.71 · 10−4 8.46 · 10−2

SORM 4.74 · 10−19 3.87 · 10−4 8.84 · 10−2

Table 8.3 The probability of the occurrence of voltage pull-in below a volt-age of 15 V as determined by crude Monte-Carlo simulations, importancesampling and the first- and second-order reliability methods (FORM/SORM)for various coefficients of variations (c.o.v.) of the modulus of elasticity.

181


8.3.2 Reliability analysis

The results of the methods for reliability analysis are compared in Table 8.3,

where the probability of the occurrence of pull-in below a voltage of 15 V is

studied using importance sampling and the first- and second-order reliability

method.

Since obtaining acceptable confidence levels using crude Monte-Carlo sam-

pling requires impractical sample sizes, importance sampling is considered

as a benchmark method. The importance sampling simulations are per-

formed using 30 thousand realisations, which corresponds to confidence lev-

els of 95.95%, 97.36% and 98.23% for coefficients of variation of the modulus

of elasticity of 10%, 25% and 50%, respectively.

As can be seen in Table 8.3, the first- and second-order reliability meth-

ods give reasonable estimates for the probability of failure. In contrast to the

problem studied in the previous chapter, here only a single design-point is

found. The second-order reliability method (SORM) consistently gives better

accuracies than the first-order reliability method (FORM). In the case that no

high-precision determination of the probability of failure is required, SORM

turns out to be an appropriate method, since its computational cost is negli-

gible compared to the importance sampling method.

Besides quantifying failure probabilities, reliability methods are giving in-

sight in the influence of imperfections and can aid in improving the design of

structures. For example, consider the design point realisation of the random

field of elastic properties with a coefficient of variation of 25% as shown in

Figure 8.8. In this figure it is observed that the most likely failure mode is

caused by weakening of the modulus of elasticity in regions where the mag-

nitude of the second-order derivatives of the displacement field are high (in

the centre and on the sides). Reinforcement of the structure in these regions

can be considered as an appropriate measure to increase the reliability of the

structure.

182


X1 [μm]

Y(X

1)

[GPa]

300250200150100500

70

65

60

55

50

45

×10−5

X1 [μm]∂

2w(X

1,h)

∂X

2 1[μ

m−1

]

w(X

1,h)

[μm

] 10

5

0

-5

-10

-15300250200150100500

0

-0.2

-0.4

-0.6

-0.8

-1

Figure 8.8 Realisation of the modulus of elasticity at the design point (left)and corresponding shape of the beam at pull-in (right). In the right figure,the deformation is indicated by the solid line and the dashed line shows thesecond order derivative of the deformation.

183

Chapter 9

Conclusions and recommendations

Numerical methods for reliability prediction of miniaturised compo-

nents, such as micro electromechanical systems (MEMS), are of utmost

importance for the further development of these components. From a com-

putational point of view, modelling of MEMS poses several challenges. The

challenging computational aspects studied in this dissertation are subdivided

in three categories: multiphysical effects, multiscale effects and microstruc-

tural randomness.

Multiphysics The functional properties of MEMS are generally derived from

the coupling between mechanical and electric fields. This coupling can be

caused by a piezoelectric material, such as lead zirconate titanate (PZT), or

through capacitive effects.

The partition of unity method (PUM) can be applied to model discrete

crack growth in piezoelectric components (Ch. 2). This method is based on a

finite element discretisation of the quasi-static mechanical and electric equi-

librium equations. In order to solve the coupled electromechanical equilib-

rium equations, constitutive laws are required. In addition to a constitutive

law for the bulk behaviour, a cohesive law is required to related the traction

and surface charge density to the crack opening and potential jump. Such a

law can be derived by enhancing commonly used mechanical traction-opening

laws with electric effects based on relations for a parallel plate capacitor.

From numerical simulations it is concluded that the partition of unity

method is suitable for obtaining both qualitative and quantitative informa-

tion on the fracture behaviour of macroscale piezoelectric components. Par-

ticularly the dependence of the failure load on external electric fields is well

resembled by the model. Applying the PUM to macroscale piezoelectric spec-

185


imens, however, also has some downsides. Due to the brittleness of the

components a very fine mesh is required in the process zone, making the

method relatively expensive compared to a linear elastic fracture mechanics

approach. In addition it should be remarked that for the macroscale speci-

mens, the shape of the initial imperfection drastically influences the result.

The partition of unity method is a convenient approach to study the influence

of this imperfection shape. From a comparison of numerical and experimen-

tal results it is concluded that incorporation of nonlinear constitutive behav-

iour of the bulk material, by e.g. including domain switching, will increase the

accuracy of the numerical model. Although beyond the scope of this work,

such an extension can relatively easily be made in the considered numerical

framework.

The electromechanical cohesive law developed for the macroscale PUM

simulation can also be applied on the microscale (Ch. 3). On this scale an

interface element formulation is used to model inter- and transgranular frac-

ture of a piezoelectric polycrystal. Since quantitative experimental results are

unavailable for such polycrystalline configurations, no solid conclusions can

be drawn regarding the accuracy of the employed microscale model. Compar-

ison with qualitative results, however, shows that the model correctly mimics

effects like the transition from inter- to transgranular grain growth with in-

creasing grain size.

Capacitive electromechanical effects can also be modelled using the finite

element formulation (Ch. 8). An important aspect in these simulations is

the solution control technique. A generalised path-following constraint is

used throughout this thesis to robustly trace equilibrium paths (Ch. 5). The

energy dissipation is observed to be an appropriate path parameter in the

case that damage accumulation is considered. In the case of the micro bridge

studied in Chapter 8, the accumulated charge is successfully used to trace

the equilibrium path beyond the pull-in voltage.

For all simulations involving multiphysical effects it is recommended to

formulate the problems in a non-dimensional form, in order to avoid nu-

merical ill-conditioning due to the enormous separation in magnitude of the

elastic and dielectric properties.

Multiscale Downscaling of components inherently results in a decrease in

length scale separation between the micro- and macroscale. As a conse-

quence, the influence of the microstructure becomes increasingly more im-

186


portant when reducing the size of specimens. The incorporation of micro-

structural effects in analytical constitutive laws is often impractical, since

these laws would become too complex. Alternatively, computational ho-

mogenisation can be used to implicitly formulate the macroscale constitutive

behaviour in terms of a microscale model. Since in this work finite element

models are used on both scales, the considered approach is commonly re-

ferred to as a FE2-method.

Since the nonlinearty in the response of the considered macroscopic fi-

nite element models is dominated by the cohesive behaviour, a computa-

tional multiscale approach is only adopted for the cohesive law. Numerical

homogenisation is employed for the bulk constitutive behaviour. It is con-

cluded that a homogenised cohesive law can be formulated by defining the

macroscopic traction and surface charge density and using the Hill-Mandel

energy condition to derive an expression for the corresponding crack opening

and potential jump. The existence of a representative volume element (RVE)

using the proposed homogenisation scheme can be illustrated using a one-

dimensional example. Numerical treatment of the homogenisation scheme

results in solving a microscale finite element model, supplemented with a

generalised path-following constraint.

Numerical simulations show that the results of the computational ho-

mogenisation scheme closely match the results using the complete micro-

structure (around the crack path). From a computational effort point of view,

the homogenisation method is much more attractive than the full-resolution

model, making it an efficient model for the simulation of failure in MEMS.

An important limitation of the presented multi-scale model is the possible

insufficient scale separation between the microstructure (grain) size and the

cohesive zone size. The presented framework supposes the former to be

small compared to the latter. In the case that this scale separation is not

present, full-resolution modelling of the microstructure must be performed.

The proposed multiscale framework is only used here for the simulation

of cohesive failure of piezoelectric components. It should, however, be em-

phasised that the derivations are fully valid for the purely mechanical (or

purely electric) case. In addition, the formulation can easily be adjusted to

model the traction-opening behaviour of adhesive interfaces. Careful consid-

eration of the representative volume element definition is, however, recom-

mended in that case.

187


Microstructural randomness Downscaling of components complicates the

controllability of the production processes. As a consequence, MEMS are gen-

erally prone to microscopic imperfections. To worsen things, the influence

of these microstructural imperfections on the response of the device can be

significant. Where in the case of a relatively large component, the micro-

structural randomness would be filtered out across the length scales, the

randomness caused by microscopic defects might lead to stochastic compo-

nent behaviour in the case of miniaturised components.

The microstructural randomness is studied for a PZT specimen produced

using micro molding (Ch. 6). Microscopic images can be studied to describe

the microstructural geometry. For the considered specimen the porosity is

observed to be the dominant microstructural imperfection. Investigation of

the window-size used for the moving-window technique, illustrates that an

objective choice can be made for this parameter. Although a reasonable rep-

resentation of the microstructural randomness is obtained, several aspects

of the analysis require more attention. First, considerably larger specimens

should be studied in order to better satisfy the ergodicity assumption. Alter-

natively, the statistics should be based on multiple specimens. Second, better

microscopic images are required to increase the qualitative accuracy of the

microstructural randomness. Preferable, non-destructive techniques should

be employed to measure the porosity of the specimens.

A homogenisation method closely related to the moving-window gener-

alised method of cells (MWGMC) can be used to transform the random fields

for the microstructural geometry in random fields for the macroscopic ma-

terial properties. The latter random property fields are parametrised using

a vector field Karhunen-Loeve (KL) expansion, which takes into account the

(spatial) cross-correlations between the various property fields. Stochastic

finite element methods (SFEM) can then be used to gain insight in the ran-

dom failure behaviour of miniaturised components (Ch. 7). An important

conclusion of this work is that the microstructural randomness significantly

influences the response of MEMS. The relatively large characteristic length

scales of the random fields makes the use of SFEM attractive. Further devel-

opment of the spectral SFEM using partition of unity-based enhancements of

the random space is recommended, since it potentially yields considerably

better results for uncertainty analysis than the perturbation methods. Relia-

bility analysis can be efficiently performed using importance sampling based

on the design points obtained using a first-order reliability analysis (FORM).

188


Consideration of random material properties is also shown to be relevant

for the determination of pull-in voltages (Ch. 8) of capacitive MEMS. Since the

randomness decreases the pull-in instability, the deterministic result should

be considered non-conservative. Moreover, stochastic analyses can be used

to gain insight in design-improvements for microstructural devices.

It should be emphasised that the computational aspects discussed above

are all closely related. Especially the homogenisation of the microstructural

randomness is also a typical multiscale aspect. Combination of the pro-

posed computational homogenisation scheme with the stochastic finite ele-

ment methods is a logic extension of the work presented in this dissertation.

Finally, it is stressed once more that many improvements can be made to

the models used, especially on the microscale. The presented framework to

deal with the various computational aspects discussed in this work will, how-

ever, remain applicable for these improved models. The difficulties encoun-

tered in the problems discussed in this dissertation will be present in a broad

class of micro systems. The computational solutions offered are therefore

anticipated to be applicable for the analysis of many different miniaturised

components.

From a computational point of view, it is of utmost importance to realise

that methods that are successfully applied to classical structures might yield

unrealistic results for MEMS. Further development of microscale technologies

requires careful consideration of the computational aspects discussed in this

dissertation.

189

Appendix A

Path-following constraints forprescribed displacement problems

The derivations for the path-following constraints, discussed in Chap-

ter 5, have been carried out on the basis of the assumption that the load-

ing can be described using a scalable external force vector. Alternatively, a

specimen can be loaded by prescribing the displacement of a group of nodes.

The nodal displacements can then be decomposed as

a = Caf + ap + ηa, (A.1)

where af and ap are the free and prescribed nodal displacements, respectively.

Furthermore, η is a scalar load factor and a is a unit prescribed displacement

vector. In the case that np degrees of freedom are prescribed, the number of

free degrees of freedom equals nf = n−np. The constraint matrix C in (A.1)

has size n×nf.

The equilibrium equation (5.1) can be reformulated in terms of the con-

strained residue as

CTr (a) = −CTfint (af, η) = 0. (A.2)

Analogously to the derivations in Section 5.1, a Newton-Raphson scheme can

be derived to determine the solution of this equilibrium equation via[CTKC CTKa∂g∂a

C∂g∂a

a

][daf

dη

]=[CTr

gk

]. (A.3)

The equilibrium path can then be traced using one of the dissipation-based

path-following constraints (5.16), (5.24), (5.41), (5.56) or using the charge-

based constraint (8.15).

191

Path-following constraints for prescribed displacement problems

The augmented system of equations (5.5) in the case of loading by a scal-

able force vector can be solved efficiently using the Sherman-Morisson for-

mula (Gutiérrez, 2004). The Sherman-Morisson formula can also be applied

to determine the solution of the augmented system (A.3) in the case of pre-

scribed displacements (Verhoosel et. al, 2009a) using[da

dλ

]=[dI

g

]− 1

hTdII −w

[ (hTdI − g)dII

−hTdI + g (1+ hTdII −w)], (A.4)

with h and w as defined in equation (5.7). This result resembles that of the

Sherman-Morrison formula applied to the unconstrained augmented system.

This implies that the augmented system (A.3) can be solved using the uncon-

strained Sherman-Morrison formula under the condition that dI and dII are

obtained by solving the systems

KdI = r;

KdII = 0,(A.5)

subjected to the constraints

dI = CdIf;

dII = CdIIf + a.

(A.6)

192

Appendix B

Discretisation of the electrostatic pull-inproblem

In chapter 8 the discretisation of an electrostatic pull-in problem is con-

sidered. In equation (8.11), the internal force is defined as the nodal force

vector which upon multiplication with the rate of the nodal state vector yields

the rate of the Gibbs free energy. Also the consistent tangent is defined as the

derivative of the internal force vector with respect to the state vector. In this

appendix, the elements of the internal force vector and consistent tangent

are presented.

Using the discretisations (8.9) and (8.10), equation (8.8) can be written in

the form

Wint = fa1int · a1 + f

a2int · a2 + f

a3int · a3, (B.1)

such that the internal force vector is composed of three parts. The elements

of the internal force vector then follow as

fa1

int,i =∫ L

0YI∂2w

∂X21

d2ψi

dX21

dX1 +∫Ω0|J|λ0

∂γi∂Xk

(J−1kl E2 − 1

2J−1k2 El

)El dΩ

0; (B.2)

fa2

int,i =∫ L

0YI∂2w

∂X21

d2υi

dX21

dX1; (B.3)

fa3

int,i =∫Ω0|J|λ0

∂γi∂Xk

J−1kl El dΩ

0. (B.4)

In these expressions, the Jacobian and electric field are computed on the basis

of the discrete nodal quantities as

Jkl = δkl + δk2∂γm

∂Xla1,m;

El = −J−1kl

∂γm∂Xk

a3,m.

(B.5)

193

Discretisation of the electrostatic pull-in problem

The consistent tangent, which is required for the Newton-Raphson procedure

used to solve the system of nonlinear equations, follows as

Ka1a1

ij =∫ L

0YI

d2ψi

dX21

d2ψj

dX21

dX1 +∫Ω0

∂ |J|∂a1,j

λ0∂γi

∂Xk

(J−1kl E2 − 1

2J−1k2 El

)El dΩ

0+

+∫Ω0|J|λ0

∂γi∂Xk

(∂J−1kl

∂a1,jE2 − 1

2

∂J−1k2

∂a1,jEl + J−1

kl

∂E2

∂a1,j

)El dΩ

0+

+∫Ω0|J|λ0

∂γi∂Xk

(J−1kl E2 − J−1

k2 El) ∂El∂a1,j

dΩ0; (B.6)

Ka1a2

ij =∫ L

0YI

d2ψi

dX21

d2υj

dX21

dX1; (B.7)

Ka1a3

ij =∫Ω0|J|λ0

∂γi∂Xk

J−1kl

∂E2

∂a3,jEl dΩ

0+

+∫Ω0|J|λ0

∂γi

∂Xk

(J−1kl E2 − J−1

k2 El) ∂El∂a3,j

dΩ0; (B.8)

Ka2a1

ij = Ka1a2

ji ; (B.9)

Ka2a2

ij =∫ L

0YI

d2υi

dX21

d2υj

dX21

dX1; (B.10)

Ka2a3

ij = 0; (B.11)

Ka3a1

ij = Ka1a3

ji ; (B.12)

Ka3a2

ij = Ka2a3

ji ; (B.13)

Ka3a3

ij = −∫Ω0|J|λ0

∂γi

∂XkJ−1kl J

−1ml

∂γj

∂XmdΩ

0, (B.14)

with

∂ |J|∂a1,j

= |J| J−1lk

∂Jkl

∂a1,j; (B.15)

∂J−1kl

∂a1,j= −J−1

km

∂Jmn∂a1,j

J−1nl . (B.16)

194

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203

Summary



Clemens V. Verhoosel

Miniaturisation of components is a trend observed in many indus-

tries. On the one hand, this trend is caused by the reduction in cost

price per functional unit that can be achieved by means of miniaturisation.

On the other hand, downscaled components can show fundamentally differ-

ent behaviour, due to the fact that differing physical phenomena generally

scale dissimilarly with component dimensions. The increased difficulty of

production process controllability is considered as a downside of miniaturi-

sation. The resulting imperfections will generally influence the performance

and reliability of micro systems.

Computational techniques aid in the design of miniaturised components,

making the development of suitable numerical techniques a relevant research

topic. The numerical assessment of the reliability of micro electromechanical

systems (MEMS), which form an important class of micro systems, is the main

topic of interest of this work. Numerical modelling of MEMS poses several

computational challenges, which have been categorised into multiphysical

effects, multiscale effects, and the effects of microstructural randomness.

Piezoelectric ceramics, which are materials experiencing a coupling be-

tween mechanical and electric fields, are commonly used for MEMS. Piezo-

electric ceramics are brittle materials and susceptible to damage, making the

nucleation and growth of cracks an important failure mechanism. Cracks

initially lead to decreased component performance, and in the end to compo-

nent failure. In this work, a cohesive zone formulation for electromechanical

continua has been proposed. In an electromechanical solid, the displacement

205

Summary

field and electric fields are governed by the static equilibrium of stresses, in

combination with electrostatic equilibrium. Fracture has been modelled by

the provision of electromechanical cohesive laws. These laws have been de-

rived by enhancing commonly used mechanical traction-opening laws with

relations for parallel plate capacitors. Numerical simulations have been per-

formed on both the macroscale and microscale, demonstrating the suitability

of the proposed model for correctly simulating various experimentally ob-

served phenomena in a qualitative sense.

Miniaturisation of a component generally decreases the separation be-

tween the characteristic length scale of the specimen and that of its mi-

crostructural features (e.g. grains). As a consequence, the microstructure

can have a direct influence on the behaviour of a miniaturised component.

Correct incorporation of microstructural effects in a numerical model is a

challenging computational aspect, since fully resolving the microstructure is

impractical from a computational effort point of view. To efficiently incor-

porate the most important features of the microstructure in a macroscale

simulation, a computational homogenisation framework has been developed.

In this framework, a microscale electromechanical finite element model is

resolved at various positions on a macroscale crack, providing the macro-

scale with a micromechanically motivated cohesive law. A key feature of the

proposed multiscale framework is that it allows for the definition of repre-

sentative volume elements for softening materials.

Another multiscale aspect is encountered when considering the influence

of microscale imperfections, as observed in piezoelectric specimens, on the

performance of MEMS. Due to the small separation between micro and macro

length scales, these imperfections have a strong influence on the behaviour of

MEMS. Moreover, the random character of these imperfections causes MEMS

to respond stochastically. Appropriate incorporation of these random effects

has been achieved in two steps. In the first step, the microstructural random-

ness has been characterised using random fields for various bulk and cohe-

sive material properties. The statistical moments on which these fields are

based, have been determined by application of moving-window techniques.

In the second step, stochastic finite element methods (SFEM) have been em-

ployed to perform uncertainty and reliability analyses. These analyses have

been applied to various MEMS problems, and have aided in the quantification

of the effect of microstructural randomness. Quantification of the probability

of the occurrence of undesirable situations, has aided in the identification of

206

Summary

design improvements.

It is important to realise that the three classes of computational aspects

discussed above are interrelated for most of the problems considered in this

dissertation. Further development of numerical techniques for micro elec-

tromechanical systems requires these computational aspects to be addressed

carefully. Thereby, it is of utmost importance to realise that proven computa-

tional techniques for macroscale components will not necessarily yield mean-

ingful results for miniaturised systems. The further growth of micro systems

technologies will increase the demand for improved numerical techniques.

Careful consideration of the computational aspects addressed in this disser-

tation will be a prerequisite for the success of such innovative techniques.

207

Samenvatting

Multischaal en probabilistische modellering

van micro-elektromechanische systemen

Clemens V. Verhoosel

Miniaturisatie van componenten is een trend die wordt waargeno-

men in een groot aantal industrieën. Deze trend wordt veroorzaakt

door de reductie in kostprijs per functionele eenheid die door middel van

miniaturisatie kan worden bereikt. Ook kunnen geminiaturiseerde compo-

nenten fundamenteel verschillend fysisch gedrag vertonen, doordat verschil-

lende fysische fenomenenen in het algemeen anders schalen met de dimen-

sies van een component. De toenemende moeilijkheden met betrekking tot

de controleerbaarheid van productieprocessen wordt gezien als een nadeel

van miniaturisatie. De resulterende onvolmaaktheden zullen in het algemeen

de prestaties en betrouwbaarheid van microsystemen beïnvloeden.

Computermethoden worden gebruikt ter ondersteuning van het ontwerp-

proces van geminiaturiseerde componenten waardoor de ontwikkeling van

zulke numerieke technieken een relevant onderwerp van onderzoek is. De

numerieke bepaling van de betrouwbaarheid van micro-elektromechanische

systemen (MEMS), welke een belangrijke klasse van microsystemen vormen,

is het voornaamste onderwerp van onderzoek in dit werk. De numerieke mo-

dellering van MEMS werpt meerdere numerieke uitdagingen op, welke kunnen

worden gecategoriseerd in multifysica effecten, multischaal effecten en de ef-

fecten van microscopische onvolmaaktheden.

Piëzo-elektrische keramieken, materialen die een koppeling ervaren tus-

sen mechanische en elektrische velden, worden veelvuldig toegepast in MEMS.

Piëzo-elektrische keramieken zijn brosse materialen en zijn gevoelig voor

schade, waardoor het onstaan en voortplanten van scheuren een belangrijk

209

Samenvatting

bezwijkmechanisme is. In eerste instantie zorgen scheuren voor een afname

van de prestaties van een component en uiteindelijk leiden ze tot het bezwij-

ken van een apparaat. In een elektromechanische vaste stof worden het ver-

plaatsingsveld en elektrische veld bepaald door het evenwicht van spannin-

gen in combinatie met elektrostatisch evenwicht. Breuken zijn gemodelleerd

door middel van elektromechanische cohesieve wetten. Deze wetten zijn af-

geleid van veel gebruikte mechanische tractie-opening wetten door deze te

verrijken met relaties voor een parallele plaat condensator. Om het ont-

wikkelde model te testen, zijn numerieke simulaties uitgevoerd op zowel de

micro- als macroschaal. Deze berekeningen tonen aan dat het voorgestelde

model kan worden gebruikt voor het, in kwalitatieve zin, correct voorspellen

van verscheidene experimenteel waargenomen fenomenen.

Miniaturisatie van een component leidt in het algemeen tot een verkleinde

scheiding tussen de karakteristieke lengteschalen van een proefstuk en zijn

microstructuur (bijv. de korrels). Als gevolg hiervan kan de microstructuur

een directe invloed hebben op het gedrag van een geminiaturiseerde com-

ponent. Het correct opnemen van microstructurele effecten in een nume-

riek model is een uitdagend aspect, omdat het volledig oplossen van de mi-

crostructuur vanuit het oogpunt van rekentijd onpraktisch is. Om de voor-

naamste eigenschappen van de microstructuur efficiënt op te nemen in een

macroschaal berekening, is er een numeriek homogenisatie model ontwik-

keld. In dit model wordt een microschaal elektromechanisch eindige elemen-

ten model opgelost op verscheidene posities op een macroscopische scheur.

Daarmee wordt de macroschaal voorzien van een op micromechanica geba-

seerde cohesieve wet. Een voorname eigenschap van de voorgestelde homoge-

nisatie methode is dat het mogelijk is om representatieve volume elementen

te definiëren voor softening materialen.

Een ander multischaal aspect is de invloed van microschaal imperfecties,

zoals exerimenteel waargenomen in piëzo-elektrische proefstukken, op de

prestaties van MEMS. Door de geringe scheiding tussen de micro- en macro-

lengteschalen, hebben deze onvolmaaktheden een sterke invloed op het ge-

drag van MEMS. Daarbij komt dat het willekeurige karakter van deze imper-

fecties ertoe leidt dat MEMS zich stochastisch gedragen. Deze probabilisti-

sche effecten zijn door middel van twee stappen in de berekeningen opge-

nomen. In de eerste stap is de willekeurige microstructuur gekarakteriseerd

met stochastische velden voor de bulk en cohesieve materiaal eigenschappen.

De statistische momenten waarop deze velden zijn gebaseerd, zijn bepaald

210

Samenvatting

met behulp van moving-window technieken. In de tweede stap zijn stochas-

tische eindige elementen methoden (SFEM) toegepast om onzekerheids- en

betrouwbaarheidsanalyses uit te voeren. Deze analyses zijn toegepast op ver-

schillende MEMS problemen en hebben geholpen bij de kwantificering van de

door de willekeurige microstructuur veroorzaakte effecten. De kwantificering

van de kans van het plaatsvinden van ongewenste situaties heeft geholpen bij

het identificeren van ontwerpverbeteringen.

Het is belangrijk te realiseren dat de drie klassen van numerieke aspecten

die hierboven zijn besproken voor vrijwel alle problemen in dit proefschrift

met elkaar verwoven zijn. De verdere ontwikkeling van numerieke methoden

voor micro-elektromechanicshe systemen vereist dat deze numerieke aspec-

ten zorvuldig worden behandeld. Daarbij is het van het grootste belang te rea-

liseren dat numerieke methodes die zich hebben bewezen op de macroschaal

niet noodzakelijk zinvolle resultaten opleveren voor geminiaturiseerde sys-

temen. De verdere groei van microsysteem technologieën zal de vraag naar

verbeterde numerieke technieken doen toenemen. Een zorgvuldige overwe-

ging van de numerieke aspecten die zijn behandeld in deze dissertatie zal

een vereiste zijn voor het succes van zulke innovatieve technieken.

211

Curriculum Vitæ

Clemens Vitus Verhoosel

born July 15, 1982 in Diessen, The Netherlands

Since Sept. 2009: Post-doctoral researcher, Department of Me-

chanical Engineering, Eindhoven University of

Technology.

Sept. 2005 - Aug. 2009: Ph.D. candidate, Faculty of Aerospace Engineer-

ing, Delft University of Technology.

Sept. 2003 - Aug. 2005: Student assistant at the Faculty of Aerospace

Engineering, Delft University of Technology.

Sept. 2000 - Aug. 2005: M.Sc. in Aerospace Engineering, Delft Univer-

sity of Technology. Graduation thesis on sto-

chastic fluid-structure interactions.

Sept. 1994 - June 2000: Atheneum, St.-Odulphuslyceum, Tilburg.

213

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