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under the supervision of
Chair for Computation in Engineering
Technische Universitat Munchen
Phase-field modeling of brittlefracture with multi-level hp-FEM
and the Finite Cell Method
Master’s thesis
for the Master of Science programme Computational Sciencesin Engineering
Sindhu Nagaraja
Examiners: Prof. Dr.-Ing. Laura De Lorenzis
Prof. Dr.-Ing. habil. Manfred Krafczyk
Prof. Dr. rer. nat Ernst Rank
Supervisors: Mohamed Elhaddad, M.Sc.
Advisors: Dr.-Ing Stefan Kollmannsberger
Dr.-Ing Marreddy Ambati
Date of issue: 03. April, 2017
Date of submission: 29. September, 2017
2
Involved Organisations
Chair for Computation in Engineering
Department of Civil, Geo and Environmental Engineering
Technische Universitat Munchen
Arcisstraße 21
D-80333 Munchen
Institute of Applied Mechanics
Department of Civil Engineering and Environmental Science
Technische Universitat Braunschweig
Pockelsstraße 3 (Okerhochhaus)
38106 Braunschweig
4
Tasks
Implementation of a phase-field model for brittle fracture intoAdhoC++, the hp-
FEM and Finite Cell Code of the Chair for Computation in Engineering in two
dimensions.
Verification of the implementation on benchmark examples with homogeneous
material.
Examples for non-geometry conforming mesh using FCM.
Development of a heuristic refinement criterion.
Extension of the method for adaptively refined meshes.
Comparative studies for uniform and adaptively refined meshes.
Acknowledgements
This Master’s thesis has been written by me in September 2017 to fulfill one of the
requirements of my Graduate Program, Master of Science - Computational Sciences in
Engineering (CSE). The work has been carried out at the Chair for Computation in
Engineering, Technische Universitat Munchen. I would like to express my gratitude
towards Prof. Dr.-Ing. Laura De Lorenzis and Prof. Dr. rer. nat. Ernst Rank for
providing me this opportunity which broadened my research perspective.
This work is partly based on the research carried out by Dr.-Ing. Marreddy Ambati. I
would like to thank him for sharing his work. I am greatful for the guidance provided by
Dr.-Ing Stefan Kollmannsberger throughout my thesis. His cooperation, motivational
words and timely input played an important role in this work. I earnestly thank
Mohamed Elhaddad for mentoring me and supporting me at every stage, without which
it would have been impossible to accomplish this work. Furthermore, special thanks
to Phillip Kopp for his cooperation and inputs during the thesis. I also acknowledge
the hospitality of the members of the Chair of Computation in Engineering and thank
them for hosting me during the thesis.
Munchen, September 2017 Sindhu Nagaraja
Abstract
Phase-field approach for computational fracture mechanics is an elegant numerical
technique to predict fracture based failures in materials and components employed in
various engineering applications.
The multi-level hp-FEM has proven to yield an exponential convergence of the approx-
imation error even for problems with singular solution characteristics. This method
also enables a dynamically changing mesh which allows the refinement to stay local
near singularities or high gradients. This feature of the multi-level hp-FEM is ideally
suited to track sharp features such as propagation of cracks.
The ability to accurately predict fracture in industrial applications involving complex
geometries and loading conditions is gaining importance like never before in the
past decade. To achieve this, the conventional Finite Element Method considers a
mesh whose boundaries have to coincide with the boundaries of the geometry under
consideration. This makes the mesh generation process tedious. The Finite Cell
Method based on higher order elements, provides an alternate mesh generation process
and is an embedded domain method.
The research presented in this work focuses on integrating a two-dimensional phase-
field framework for both quasi-static and dynamic brittle fracture developed at the
Institut fur Angewandte Mechanik, TU Braunschweig with the multi-level dynamically
adaptive hp-framework and the Finite Cell Method developed at the Lehrstuhl fur
Computation in Engineering, TU Munchen. The key objective is to implement a phase-
field model into Adhoc++, the hp-FEM and the Finite Cell code of the Lehrstuhl fur
Computation in Engineering at TU Munchen and to develop a refinement criterion
that ensures a dynamically changing mesh which is adaptive in nature. The numerical
results presented in the thesis illustrate the potential of the application of the uniform
multi-level hp-refinement and the FCM in the context of phase-field models for both
quasi-static and dynamic brittle fracture.
Contents
List of Figures 2
List of Tables 4
List of Symbols 6
1 Introduction 8
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Fundamental concepts 12
2.1 Basics of fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.1 Modes of fracture . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.2 Linear elastic fracture mechanics (LEFM) . . . . . . . . . . . . 13
2.1.3 Griffith’s criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Phase field approximation of crack topology . . . . . . . . . . . . . . . 15
2.3 Multi-level hp-FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 The Finite Cell Method . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1 Basic concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.2 Application of non-boundary conforming boundary conditions . 21
3 Formulation 23
3.1 Quasi-static brittle fracture . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Extension to dynamic brittle fracture . . . . . . . . . . . . . . . . . . . 27
3.3 Numerical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.1 Quasi-static brittle fracture . . . . . . . . . . . . . . . . . . . . 29
3.3.2 Dynamic brittle fracture . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Finite Cell Method for phase-field quasi-static brittle fracture . . . . . 32
4 Numerical results 34
4.1 Single-edge notched tension test . . . . . . . . . . . . . . . . . . . . . . 34
4.1.1 Validation of implementation of the hybrid formulation . . . . . 35
4.1.2 Parametric influence of convergence behavior of the phase-field
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Single-edge notched shear test . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.1 Validation of implementation of the anisotropic Miehe formulation 42
4.2.2 Parametric influence of convergence behavior of the phase-field
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.3 Comparative study between uniform and adaptively refined meshes 48
1
Contents
4.3 Notched plate with hole . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3.1 Parametric influence of number of staggered iterations on crack
propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4 Crack propagation under compressive loads . . . . . . . . . . . . . . . . 54
4.5 Dynamic crack branching . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5 Summary and outlook 59
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Bibliography 61
2
List of Figures
1.1 Examples for fracture induced engineering failures . . . . . . . . . . . . 8
2.1 Modes of Fracture [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Phase-field approximation. a) Sharp crack topology in the domain of
the solid. b) Regularized crack surface using length scale parameter l . 16
2.3 One-dimensional shape functions . . . . . . . . . . . . . . . . . . . . . 18
2.4 Two-dimensional mode categories . . . . . . . . . . . . . . . . . . . . . 18
2.5 Refinement by superposition [10], [11] . . . . . . . . . . . . . . . . . . . 18
2.6 The multi-level hp-FEM concept [11] . . . . . . . . . . . . . . . . . . . 19
2.7 Basic concept of the Finite Cell Method [19] . . . . . . . . . . . . . . . 20
2.8 Quadtree integration [19] . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1 Schematic representation of the domain Ω with internal discontinuity Γ . 23
4.1 Geometry and boundary conditions for single-edge notched tension test 35
4.2 Single-edge notched tension test. Validation of crack evolution with
hybrid formulation against reference results. . . . . . . . . . . . . . . . 36
4.3 Single-edge notched tension test. Validation of load-displacement be-
havior with hybrid formulation against reference results. . . . . . . . . 36
4.4 Single-edge notched tension test. Multi-level hp-refinement for different
ansatz orders with k = 6. Crack phase-field at different displacements. 38
4.5 Single-edge notched tension test. Multi-level hp-refinement for different
ansatz orders with k = 6. Load-displacement curves. . . . . . . . . . . 39
4.6 Single-edge notched tension test. Multi-level hp-refinement for different
refinement depths with p = 3. Crack phase-field evolution at different
displacements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.7 Single-edge notched tension test. Multi-level hp-refinement for different
refinement depths with p = 3. Load-displacement curves. . . . . . . . . 41
4.8 Geometry and boundary conditions of single-edge notched shear test . . 42
4.9 Single-edge notched shear test. Validation of crack evolution with the
anisotropic Miehe formulation against reference results. . . . . . . . . . 43
4.10 Single-edge notched shear test. Validation of load-displacement behavior
with the anisotropic Miehe formulation against reference results. . . . . 43
4.11 Single-edge notched shear test. Multi-level hp-refinement for different
ansatz orders with k = 6. Crack phase-field at different displacements. 45
4.12 Single-edge notched shear test. Multi-level hp-refinement for different
ansatz orders with k = 6. Load-displacement curves. . . . . . . . . . . 46
4.13 Single-edge notched shear test. Multi-level hp-refinement for different
refinement depths with p = 5. Crack phase-field at different displacements. 47
3
List of Figures
4.14 Single-edge notched shear test. Multi-level hp-refinement for different
refinement depths with p = 5. Load-displacement curves. . . . . . . . . 48
4.15 Single-edge notched shear test. Comparison of crack phase-field for
uniformly and adaptively refined meshes with p = 1 at different dis-
placements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.16 Single-edge notched shear test. Comparison of load-displacement curves
for uniformly and adaptively refined meshes with p = 1. . . . . . . . . . 50
4.17 Geometry and boundary conditions of notched plate with hole . . . . . 51
4.18 Notched plate with hole. Crack pattern from the experiment . . . . . . 51
4.19 Notched plate with hole. Multi-level hp-refinement: p = 2, k = 4. Crack
phase-field at different displacements. . . . . . . . . . . . . . . . . . . . 52
4.20 Notched plate with hole. Multi-level hp-refinement for different number
of staggered iterations : p = 3, k = 4. Crack phase-field at different
displacements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.21 Geometry and boundary conditions of concrete block under compression. 54
4.22 Concrete block under compression. Crack pattern from the experiments
carried out by Dipl.-Ing. Gerald Schmidt-Thro, Lehrstuhl fur Massivbau,
TU Munchen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.23 Concrete block under compression. Multi-level hp-refinement: p = 1, k
= 4. Crack phase-field at different displacements. . . . . . . . . . . . . 55
4.24 Geometry and loading of dynamic uniaxial tension test. . . . . . . . . . 56
4.25 Dynamic crack branching under uniaxial tension. Multi-level hp-
refinement for different ansatz order with k = 5 using the anisotropic
Miehe formulation. Crack phase-field evolution over time. . . . . . . . . 57
4.26 Dynamic crack branching under uniaxial tension. Multi-level hp-
refinement for different ansatz order with k = 5. Plot of strain energy
overtime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.27 Dynamic crack branching under uniaxial tension. Multi-level hp-
refinement for different ansatz order with k = 5 using the hybrid
formulation. Crack phase-field evolution over time. . . . . . . . . . . . 58
4
List of Tables
4.1 Single-edge notched shear test. Parametric comparison for uniformly
and adaptively refined meshes with p = 1 and the hybrid formulation. . 50
5
List of Symbols
Abbreviations
Abbreviation Stands for
FEM Finite Element Methods
FCM Finite Cell Method
PDE Partial Differential Equation
SIF Stress Intensity Factor
LEFM Linear Elastic Fracture Mechanics
EPFM Elasto Plastic Fracture Mechanics
GP Gauß Points
DOFs Degrees Of Freedom
Symbols used in Formulae
Latin
Symbol Stands for Unit
l length scale parameter [m]
s phase field or crack field paramter [ ]
a crack length [m]
u displacement field [m]
L Lagrangian [J]
H history field [ Jm2 ]
E material stiffness [Pa]
Gc fracture toughness [Nm ]
h grid size in the square mesh [m]
p ansatz order of shape functions [ ]
k refinement depth in multi-level hp-refinement [ ]
∆u displacement increment [m]
dt temporal discretization or time step [s]
6
List of Tables
Greek
Symbol Stands for Unit
Ω the three dimensional domain (body) [m3]
Ωphy the three dimensional physical domain in the
context of the FCM
[m3]
Ωfict the three dimensional fictitious domain in the
context of the FCM
[m3]
Ω∪ union of the physical and the fictitious domain
in the context of the FCM
[m3]
Γ crack boundary [m2]
Ψkin kinetic energy [J ]
Ψpot potential energy [J ]
Ψe elastic strain energy density [ Jm3 ]
ε strain tensor [ ]
σ strain tensor [Pa]
λ Lame’s first parameter [Pa]
µ Lame’s second parameter [Pa]
η numerical paramter to model artificial stiffness [ ]
ρ mass density [ kgm3 ]
ν poisson’s ratio [ ]
α indicator function in the context of the FCM [ ]
7
1 Introduction
1.1 Motivation
Damage-based design of engineering structures has become one of the most important
design strategies in the past few decades. This ”prevention is better than cure” design
principle focuses mainly on inhibiting fracture-induced failure. The dramatic failure of
several Liberty ships during the World War II was a consequence of brittle fracture in
hull and decks of the ships that were made up of steel, see Fig:1.1a. The derailment
of the train ICE 884 at Eschede, Germany killing 101 people, depicted in Fig:1.1b,
is owed to the fatigue fracture of the wheel rim [3]. Several engineering structures
used in oil plants, reservoir construction, nuclear power plants are sensitive to the
presence of cracks. Since it is almost impossible to have a perfect crack free material,
it becomes necessary to study the performance and durability of such structures under
the given loading with a pre-existing crack. These serve as motivating examples for the
development of efficient fracture models and numerical simulation of fracture processes.
In this regard, a varitey of fracture models have been introduced in the research
community dedicated to damage mechanics. The most successful theory proposed for
brittle fracture is an energy based theory, proposed by Griffith. It relates the dissipated
energy during crack propagation to the energy required to form new crack surfaces
and is briefly introduced in section 2.1.2.
(a) Failure of Liberty ship S.S.Schenectady,1943 [1]
(b) ICE accident due to fracture ofwheel on June 3, 1998 [2]
Figure 1.1: Examples for fracture induced engineering failures
Most of the engineering problems are now solved numerically. Galerkin based Finite
Element Method (FEM) has become the most admired numerical methods to solve
partial differential equations (PDEs) and is undoubtedly the first choice of today’s
8
1 Introduction
engineers. One of the many complexities involved in fracture processess is to model the
crack. A crack is discontinuous by its very nature and this poses a challenge in modeling
a crack in finite element setting. Furthermore, it is necessary to track the crack as it
propagates. In the research dedicated to fracture mechanics, Extended Finite Element
Methods (XFEM), Cohesive Zone Methods, Cohesive Segment Methods, Discontinuous
Galerkin Method (DG), Phase-Field Modeling are a few popular computational meth-
ods. In this thesis, we study the phase-field approach for quasi-static and dynamic
brittle fracture. The phase-field method eliminates the necessity to algorithmically
track the discontinuous displacements across the crack path by modeling the discrete
crack as a smooth continuous function that elegantly transits from an undamaged
region to a fully cracked region.
Although FEM is the most sought after method to simulate structural engineering
problems, there is always a constant demand for simulating problems with higher
complexities and intricacies while using minimum resources and time with higher
accuracy in the numerical solution. This can be achieved by refinement techniques.
The most trivial one is the uniform h-refinement wherein the solution accuracy is
increased by increasing the number of elements. To this end, elements are bisected
in each direction, yielding four new elements. This is continued until convergence is
reached. Although this has been proven to yield good results, the rate at which the error
decreases is rather slow. For extremely accurate results, a very large number of degrees
of freedom might become necessary and hence a pure uniform h-refinement becomes
computationally expensive. Another method that has proven to yield exponential
convergence for smooth problems is the p-refinement technique. Here, higher order
shape functions are used which approximates the analytical solution better using lesser
number of degrees of freedom, compared to h-FEM. In the recent past, non-uniform
rational B-splines (NURBS) are being used for even better results and for higher
continuity in the solution in the context of Isogeometric Analysis (IGA).
The triumph of the above mentioned p-refinement technique starts deteriorating to an
algebraic convergence in the presence of discontinuities, singularities or high gradients
in the solution. The subject of this thesis has a discontinuity smoothened by the
phase-field function. Although this function is continuous, it has high gradients. This
is the trigerring point to use a refinement strategy in the study which yields excellent
convergence even in the presence of high gradients. A felicitous refinement technique
for this situation is the hp-FEM. In this refinement version, the advantages of both
h-version and p-version are exploited. While the elements at the vicinity of high
gradients are refined and lower order polynomials are used for solution approximation,
elements away from the sensitive zone remain large and higher order polynomials
are used to smoothly approximate the solution. This improves the solution accuracy
significantly, compared to h-FEM and p-FEM with comparable number of unknowns.
It is evident from the numerical analysis presented in Chapter 4 that the strong
gradients in the phase-field can be used to drive a dynamic refinement (a mesh that
changes over the course of the simulation) thereby recovering excellent convergence.
9
1 Introduction
While the convergence behavior of the hp-version is promising, it has not yet become
the most popular method in engineering applications owing to the challenges it poses,
the most challenging aspect being the hanging nodes generated in a local h-refinement.
These hanging nodes are to be constrained appropriately to yield atleast C0 continutiy
in the numerical solution. This process is tedious and algorithmically challenging.
Alternatively, hanging nodes which result in an irregular unstructured mesh can be
eliminated by appropriate refinement of the adjacent elements thus resulting in a
regular mesh. However, this method conflicts the idea of local refinement.
Another implementation of the hp-version of FEM adopts a refinement by superposition
rather than a refinement by replacement as described above. This is called the multi-
level hp-scheme. The idea of this is to superpose a base mesh with a finer overlay
mesh in such locations of the domain where there are singularities or high gradients
in the solution. The fundamental principle of this method to acheive inter-element
continuity is to apply homogeneous Dirichlet boundary conditions on the boundary
of the refinement zone. This method has proven to bring down the implementational
complexity related to hanging nodes while constructing the same Finite Element space
and thereby retaining the same approximation quality as that from a conventional
hp-FEM method. This refinement technique is used throughout in this work.
The Finite Cell Method (FCM) is an embedded or immersed boundary method which
is based on high order Finite Elements. It is a non-geometry conforming discretization
technique which does not resolve the geometry explicitly, rather embedds the physical
domain into a larger ficticious domain. This larger embedding domain is of simple
geometry and hence can be easily discretized thus evading the numerically expensive
and time consuming meshing procedure. The original geometry is recovered during
integration of the weak form by imposing a penalty on the integration points that
do not belong to the actual geometry. This idea not only enables complex geometry
representations but also is handy to use when the geometry under consideration is
changing with time. Section 4.3 considers an example of a notched plate with hole,
where the FCM has been applied successfully in the context of phase-field models.
1.2 Outline of the thesis
This thesis is organised into four chapters. The first chapter as already seen motivates
the under taken study. The second chapter introduces the concepts used in the thesis.
Groundwork required to understand a fracture problem, phase-field models, higher
order FEM, multi-level hp-FEM and FCM are laid here. The third chapter presents
the governing equations of a phase-field based brittle fracture model followed by its
numerical formulation in both, FEM and FCM. Firstly, the quasi-static model is
presented which is later extended to a dynamic model. Two different phase-field
formulations are used in this thesis, anisotropic Miehe and hybrid formulation. The
10
1 Introduction
last chapter is dedicated to the application of phase-field model, multi-level hp-FEM
and FCM to benchmark problems. A plane strain problem in which a plate with a
pre-existing crack subjected to two different loading conditions is simulated using multi-
level hp-FEM. This is followed by an example involving a non-geometry conforming
mesh solved with the FCM. To observe the behavior of Miehe and Hybrid formulations
for compression, a plane strain problem where a concrete block with no pre-existing
crack subjected to compressive loading is discussed. Finally, an attempt is made to
extend the implementation to a dynamic crack branching problem in which a plane
strain example under uniaxial tension is presented.
11
2 Fundamental concepts
2.1 Basics of fracture mechanics
Fracture Mechanics deals with the phenomenon of crack initiation, crack growth and
its propagation. These phenomena are sensitive and are influenced by various factors.
For instance, the process of fracture is highly dependent on the nature of the material.
Hence, there are different theories adopted for different materials. Brittle materials
are those which exhibit poor resistance to the external tensile loading. They undergo
failure without exhibiting significant deformations. Brittle fracture involves almost nil
or extremely small plastic deformation zones when compared to ductile materials and
is generally instantaneous. Hence, we may conclude that it is necessary to develop two
distinct models in order to represent features of ductile and brittle fracture. While
Linear Elastic Fracture Mechanics (LEFM) is suitable mostly for brittle materials since
they have very small plastic zone and hence have elastic response to the external applied
load, Elasto-Plastic Fracture Mechanics (EPFM) is to be used for materials that exhibit
evident plasticity. In addition to the material behavior, it is necessary to consider the
nature of the loading the component is subjected to. For example, dynamic problems,
creep, fatigue, etc are to be treated individually with suitable theories. Therefore,
given a load, material, geometry and a pre-exisiting crack, fracture mechanics helps to
estimate the maximum load a component can withstand before failure due to crack
propagation, also answering questions such as: Will the existing crack grow under the
applied external load? If it grows, with what speed does it? Will the crack growth be
arrested due to dissipative effects? When will the component fail?
2.1.1 Modes of fracture
External loads can be classified into three independent types, thus, leading to a
simplified scenario where the effect of each type can be determined individually. The
principle of superposition can then be applied in an appropriate manner to find the
combined effect of different loads. The three independent modes into which the loading
can be split are:
MODE 1 - Opening Mode - This mode arises due to a tensile loading perpendicular
to crack. It is the most common mode of fracture and also exhibits the highest
Stress Intensity Factor (SIF).
MODE 2 - Shearing Mode - Here, applied stress is shear in the in-plane direction
of the crack. In other words, applied stress is perpendicular to leading edge of
12
2 Fundamental concepts
the crack but in the plane of the crack.
MODE 3 - Tearing or Twisting Mode - Here, the applied external stress is out
of plane, i.e., the stress is parallel to the leading edge of the crack.
Figure 2.1: Modes of Fracture [4]
2.1.2 Linear elastic fracture mechanics (LEFM)
In fracture problems, the so-called process zone is the zone in which atomic bond
breaking occurs, thereby, leading to crack propagation. The size of this zone determines
the facet of the fracture process. Linear Elastic Fracture Mechanics deals with those
problems in which the process zone (also called plastic zone) is negligibly small. It
is assumed that the plastic zone around the crack tip is negligible when compared
to all other macroscopic dimensions of the physical domain. This is true for most
brittle materials. The other assumption of LEFM is that the material response is
linearly elastic throughout the fracture process. However, this assumption leads to
singularities in stresses at crack tip. (infinite stress at crack tip). In reality, the plastic
zone size is never a perfect zero and hence infinite stresses do not occur at the crack
tip. As long as this yielding zone is negligibly small, linear theory is still an acceptable
assumption. Hence, it is only logical to apply LEFM for brittle fracture, which is why
the formulation presented in this work assumes LEFM.
The characterizing parameter of LEFM is the Stress Intensity Factor. The SIF serves
as a crack propagation criteria for brittle fracture or in general LEFM. If the value of
the SIF at crack tip exceeds a critical SIF, then the crack propagates:
KI ≥ KIC (2.1)
13
2 Fundamental concepts
2.1.3 Griffith’s criteria
Griffith’s criteria is based on an energy balance. It is applicable for both elastic and
plastic cases. The fracture surface energy Γs is the energy necessary to create the
newly formed cracked surfaces during crack propagation. This energy is assumed to
be proportional to the area of the crack.
Γs = GcA (2.2)
The material constant Gc is called crack resistance, crack resistance force or fracture
toughness. It is the energy required to create a unit area of fracture surface. The
energy balance states that,
E + K + Γs = W +Q (2.3)
where E is the internal energy, K is the kinetic energy, W is the rate of work associated
to the external forces and Q is the rate of heat supply.
Assuming that no heat transfer occurs, we have
Γs = W − E − K (2.4)
If Γs ≥ 0 crack starts to propagate.
Now the quasi-static case is considered, where the contribution of kinetic energy is
negligible. Thus, the energy balance equation becomes
Γs = W − E (2.5)
Assuming a pure elastic case, the internal energy E and the external work W can
be expressed in terms of potentials. They take the forms E = Πint and W = −dΠext
dt .
Thus, the above equation simplifies to
dΠ
dt+dΓsdt
= 0 (2.6)
where Π = Πint + Πext is the total potential.
For a line crack of length a, the above further simplifies to[dΠ
da+dΓsda
]da
dt= 0 (2.7)
The energy release rate G is defined as G = −dΠda and the critical energy release rate
(or fracture toughness) Gc = −dΓs
da which leads to the final crack propagation criteria
as obtained using Griffith’s theory
(Gc −G)a = 0 (2.8)
Thus, we have G = Gc for initiation of crack propagation.
14
2 Fundamental concepts
2.2 Phase field approximation of crack topology
Fracture problems are discontinuous by nature. Methods such as XFEM, Cohesive Zone
Methods, Cohesive Segment Methods incorporate this discontinuity in the primary
field variable which is the displacement field. These are called discrete fracture models
since they represent the crack as a discontinuous entity. They make use of remeshing
strategies or enrichment of displacement fields. These methods involve algorithmically
tracking the crack during the course of the simulation which is complicated if the
crack geometry is intricate. Although these models represent crack propagation well
in two dimensions, it is numerically expensive and challenging to extend these in three
dimensions.
To overcome the difficulty of tracking crack surfaces, a new strategy called phase-
field modeling was developed in the late 1990’s. In this method, no discontinuity is
introduced to the displacement variable, instead a new field variable also called the field
order parameter, is introduced which ensures a smooth transition between discontinuity
and continuity. A phase-field model can model systems with sharp interfaces. In
fracture mechanics perspective, this field order parameter is called the crack field or
phase field and throughout this script it is denoted by s. This crack field is continuous
but is a scalar field variable. It describes a smooth transition between the undamaged
material and fully cracked material. This method is based on the minimization of the
total energy of the system with respect to the displacement field and the crack field.
s =
0 : Fully cracked material
1 : Undamaged material(2.9)
The evolution of this crack field under the given loading represents the process of
fracture. While an a-priori knowledge of the crack path becomes essential in discrete
fracture models, this becomes unnecessary for phase-field models since the evolution
of crack field represents the crack surface. Processes like crack propagation, kinking,
crack branching which involve complex crack surfaces are relatively simple to simulate
using phase-field models.
Mathematically, phase-field models are two coupled non-linear system of Partial
Differential Equations. The first equation is the balance equation obtained from
continuum mechanics and second one is the partial differential equation that describes
the evolution of the phase-field parameter s. Solutions to these coupled equations can
be obtained in two ways, using a monolithic scheme or a non-Monolithic or staggered
scheme. In the monolithic scheme, the two equations are solved simultaneously to
obtain solutions for the displacement field and the crack phase-field. In the staggered
scheme, first the displacement equation is solved for, assuming crack field to be constant
15
2 Fundamental concepts
and then using the obtained displacement field, the phase field equation is solved. The
obtained phase field solution is then used to solve again for displacement field. This
iterative process is continued until a certain accuracy is reached.
Yet another important aspect of phase-field models is the so-called length scale parame-
ter, denoted by l. This parameter controls the width of transition from the undamaged
to the fully cracked zone and can be interpreted as the width of the regularized crack,
see Fig:2.2. Phase-field models converge to Griffith’s theory for fracture of brittle
materials if the value of length scale parameter tends to zero [5]. On the outset, it
is appealing to set a very low value of l. But for small length scale parameter, the
spatial and in-turn the temporal discretization in the case of dynamic fracture, should
be fine which leads to a huge stiffness matrix thereby increasing the computational
cost. Recent advances in high performance computing (HPC) suggest, however, to
exploit parallel computing for solving problems on very fine discretizations that can
capture very small values of l [17].
Γ
l s = 0
s = 1
Ω
∂Ωgi
∂Ωhi
Γ
l
Ω
∂Ωgi
∂Ωhi
Figure 2.2: Phase-field approximation. a) Sharp crack topology in the domain of the
solid. b) Regularized crack surface using length scale parameter l
In the research dedicated to phase-field approach for fracture problems, there are
several models proposed by the mechanics community and also by the physicists, for
details refer [5] . In the scope of this work, followed is the regularized variational
formulation presented by Miehe et al [7] and the recently proposed hybrid formulation
by Ambati et al [5]. Both the formulations are presented in the next chapter.
2.3 Multi-level hp-FEM
In this section, the basic concept of the multi-level hp-FEM is introduced. As mentioned
earlier, this refinement technique combines the advantages of both h-version and p-
version ensuring exponential convergence even in the presence of high gradients or
16
2 Fundamental concepts
singularities in the solution field which is the case in fracture problems. While classical
FEM is generally implemented using Lagrange shape funtions which by nature are
nodal basis functions as shown in the figure Fig:2.3(a), the p-version makes use of
integrated Legendre polynomials. This is because, the condition number of the stiffness
matrix using Largrange basis funtions increase drastically for higher polynomial orders
making them unsuitable for p-version. Thus, the p-version as introduced by Babuska
et al [8] makes use of integrated Legendre polynomials such that it yields excellent
properties of the stiffness matrix. The Legendre polynomials have a unique property
that they are orthogonal. Since the stiffness matrix uses the derivatives of shape
functions, this orthogonality property needs to be transferred to the first derivative of
the shape functions. Integrated Legendre polynomials serve this purpose. All shape
functions except the linear ones are orthogonal to each other. As a result, the stiffness
matrices for 1D linear elastic problems are almost diagonal except the product of
the linear modes, thus improving its condition number. However, this orthogonolity
property does not hold in multiple dimensions. Despite this, the condition number
of the stiffness matrices using an Integrated Legendre basis are significantly better
compared to that using a Lagrange basis. Furthermore, the integrated Legendre shape
functions are hierarchical in nature. That implies, there is an addition of one shape
function for every order of the polynomial that is increased while all other shape
functions remain same. The one-dimensional integrated Legendre polynomials are
depicted in Fig:2.3(b) and are defined as,
N1(ξ) =1
2(1 + ξ)
N2(ξ) =1
2(1− ξ)
Ni(ξ) = Pi−1(ξ) i = 3, 4, ..., p+ 1
(2.10)
where, p is the polynomial order and Pi is defined in terms of the Legendre polynomials
Li:
Pi(ξ) =1√
4i− 2(Li(ξ)− Li−2(ξ)) i = 2, 3, 4, ... (2.11)
A tensor product of one dimensional basis functions is used to obtain higher dimensional
shape functions, see Eq:2.12. Following Zander et al [9], the two-dimensional tensor
product can be classified into three categories which plays an important role in the
definition of C0-continuous global basis functions: nodal modes, edge modes and
bubble modes, as depicted in Fig:2.4.
N2Di,j (ξ, η) = N1D
i (ξ)N1Dj (η)
N3Di,j,k(ξ, η, µ) = N2D
i,j (ξ, η)N1Dk (µ)
(2.12)
17
2 Fundamental concepts
(a)Lagrange shape functions (b) Integrated Legendre shape functions
Figure 2.3: One-dimensional shape functions
(a) Nodal mode (b) Edge mode (c) Bubble mode
Figure 2.4: Two-dimensional mode categories
Having had an overview of the hierarchical shape functions which forms an essential
ingredient of multi-level hp-FEM, the basic outline of multi-level hp-FEM is presented
here. The core idea of this refinement technique is to enhance the quality of the
solution by superposing finer overlay elements over those base elements which are to
be refined. The final approximation of the solution u is then the sum of base mesh
solution ub and the overlay solution u0 as illustrated in Fig:2.5. This enables an
optimal refinement using finer elements with low polynomial order in the vicinity of
singularities and strong gradients to ensure small discretization errors and a better
quality of solution while still using larger elements with higher polynomial orders where
the solution is smooth.
Base mesh solution ub Overlay solution u0 Final solution u
Figure 2.5: Refinement by superposition [10], [11]
18
2 Fundamental concepts
Unlike classical refine by replacement hp-refinement version, the multi-level hp-FEM
evades the algorithmic implementational difficulties by imposing two simple require-
ments for convergence. Namely, the compatibility and linear independence of basis
functions. Compatibility is achieved by imposing homogeneous Dirichlet boundary
conditions on each layer of superposed overlay mesh, thus maintaining C0 continuity.
This corresponds to deactivating those components which are connected to the bound-
ary of the overlay mesh. In one dimension, nodes, in two dimensions nodes and edges,
and in three dimensions nodes, edges and faces on the overlay boundary needs to be
deactivated. Linear independence of basis is achieved by deactivating all topological
components that have active sub-components. That is, high-order shape functions are
deactivated on those elements which have an underlying child element. This concept is
explained in Fig:2.6 where k is the refinement depth and p is the ansatz order. These
two impositions eliminate the burden of algorithmic constraining of hanging nodes.
Figure 2.6: The multi-level hp-FEM concept [11]
2.4 The Finite Cell Method
One of the fundamental ideas of the Finite Element Method is the isoparametric
mapping where same discretization is used to represent the solution field of the
PDE under consideration as well as the geometry. This imposes a requirement of
”good” elements to ensure adequate numerical accuracy, thus making mesh generation
a complex and time consuming step in Finite Element analysis. For complicated
geometries, this worsens. In order to conquer this limitation, fictitious or embedded
domain methods have emerged. These are non-boundary conforming discretization
methods. One of these is the Finite Cell Method (FCM) introduced in [12]. As
mentioned in the introduction, this is an immersed boundary method based on higher
19
2 Fundamental concepts
order finite elements wherein the physical domain Ωphy is embedded into a larger
embedding domain or fictitious domain Ωfict, such that the union Ω∪ of these yeilds
a simple geometry that can easily be meshed using a Cartesian grid as depicted in
Fig:2.7. The actual geometry is retrieved during the integration processes of the weak
form. If the integration is performed with sufficient accuracy, the FCM inherits the
convergence properties of the p-version. This section is dedicated to explain briefly
the basic concepts of the FCM.
Figure 2.7: Basic concept of the Finite Cell Method [19]
2.4.1 Basic concept
In Finite Cell Method, the process of mesh generation and solution approximation
are separated. This enables independent discretization of geometry and solution field.
Thus in FCM, there is a solution mesh which is used to approximate the solution
which does not resolve the geometry of the domain and an integration mesh which
is independent of the solution mesh and is used to resolve the geometry. Since the
integration mesh is used to represent the geometry and has nothing to do with the
solution, it does not introduce additional degrees of freedom.
Firstly, to recover the original boundary boundary value problem on Ω∪, on indicator
function α(x ) is defined as,
α(x ) =
1 : ∀x ∈ Ωphy
ε : ∀x /∈ Ωphy(2.13)
where ε is a small value used to ensure numerical stability.
The weak form is multiplied by this indicator function thus eliminating the contribu-
tions from the fictitious domain. This procedure for the phase-field problem under
consideration is formulated in the next chapter under section 3.4
20
2 Fundamental concepts
In FCM, the difficulty of geometry resolution is shifted from discretization level to
integration level. Therefore, in the second step, the original domain geometry is
recovered using a separate integration mesh. Due to the discontinuity introduced by
the indicator function α(x ) that is used to impose a penalty on the fictitious domain,
a standard Gauß-Legendre quadrature is not the optimal numerical integration scheme.
A suitable integration scheme is the space-tree approach where recursive sub-division
of cut cells is carried out to accurately integrate the discontinuous integrals. In 2D,
this space-tree subdivision method is called quadtree and octree in 3D. The results
presented in this thesis make use of quadtree integration scheme [19] as shown in
Fig:2.8. In this figure, k referes to the partitioning depth. Though quadtree approach is
robust, it generates a large number of quadrature points and hence more sophisticated
schemes have been developed for better efficiency: smart-octree [13], moment-fitting
[14], quadratic re-parameterization for terahedral finite cell method [15] and adaptively
weighted quadratures [16].
Figure 2.8: Quadtree integration [19]
2.4.2 Application of non-boundary conforming boundary
conditions
Since the Finite Cell Method modifies the actual physical domain to get a different
computational domain, boundaries of the physical domain no longer conform with
the Finite Element mesh. This makes application of boundary conditions challenging.
Application of the Neumann boundary condition involve evaluating a boundary integral
on the right hand side of the weak form. But since the boundary is not resolved by
21
2 Fundamental concepts
the mesh in FCM, Neumann boundary conditions are applied by an explicit surface
discretization.
On the other hand, enforcing the Dirichlet boundary conditions is involved. Mere
manipulation of entries related to Dirichlet boundary conditions in the stiffness matrix
cannot be done due to unresolved boundary, instead they are imposed weakly. To
meet this end, there are different ways in which Dirichlet boundary conditions can be
imposed, for example, the penalty method or Nitsche’s method [18]. These methods
extend the weak formulation to include the Dirichlet boundary conditions in a weak
sense. In this work, the penalty approach is followed and its formulation is presented
in section 3.4. Similar to the application of Neumann boundary conditions, the
constraining expressions are evaluated using a separate surface integration mesh which
does not introduce any additional degrees of freedom.
22
3 Formulation
Consider an arbitrary shaped domain Ω ∈ Rd with a boundary ∂Ω. Suppose that this
body has a crack boundary denoted by Γ as shown in Fig.3.1. Let u(x ,t) ∈ Rd be the
displacement of a point x ∈ Ω at time t. The displacement field satisfies the Dirichlet
boundary condition ui(x ,t) = gi(x ,t) on ∂Ωgi and Neuman boundary condition σijnj= hi on ∂Ωhi
. Note that i, j = 1,2,....d are the indices for components of vectors and
tensors represented in Einstein summation notation.
Γ
Ω
∂Ωgi
∂Ωhi
Figure 3.1: Schematic representation of the domain Ω with internal discontinuity Γ .
The following definition for strain tensor in the case of small deformation is used in
this work.
εij = ui,j =1
2
(∂ui∂xj
+∂uj∂xi
)(3.1)
The total potential energy of the body is the sum of elastic energy and fracture energy
and is given by
Ψpot(u ,Γ) =
∫Ω
Ψe(ε(u))dx +
∫Γ
Gcds (3.2)
where Ψe is the elastic energy density function and Gc is the fracture toughness as
introduced in Eq:(2.2).
The kinetic energy of the body is defined as
Ψkin(u) =1
2
∫Ω
ρuiuidx (3.3)
where u = ∂u∂t , ρ is the mass density of the material of the body.
23
3 Formulation
3.1 Quasi-static brittle fracture
As discussed in section 2.2, a phase-field parameter s is introduced to approximate
the discontinuous crack surface smoothly. Following the phase-field brittle fracture
model based on variational formulation introduced by Francfort and Marigo [22], the
quasi-static crack initiation and propagation process is governed by the minimization
of the free energy functional given by,
E(u ,Γ) = Ψpot(u ,Γ) =
∫Ω
Ψe(ε(u))dx +
∫Γ
Gcds (3.4)
For efficient numerical implementation, the energy functional is regularized by Bourdin
et al [21],
E(u , s) ≈∫
Ω
[Ψe(ε, s) +Gc
(1
4l(1− s)2 + l|∇,xs|2
)]dx (3.5)
where, the elastic strain energy density is degraded by a phase-field based penalty as
in Eq:(3.6) and the fracture energy is approximated as in Eq:(3.7).
Ψe(ε) ≈ Ψe(ε, s) = g(s)Ψe(ε) (3.6)
∫Γ
Gcds ≈∫
Ω
Gc
[1
4l(1− s)2 + l|∇,xs|2
]dx (3.7)
In Eq:(3.6), g(s) is called the stress degradation function. It is necessary for the stress
degradation function to satisfy the following conditions [24] :
g(0) = 0
g(1) = 1
g(s) > 0 for s 6= 0
g′(0) = 0
g′(1) > 0
As mentioned earlier, s is 0 at points which are fully damaged and 1 at which the
material is undamaged, see Eq:(2.9). To satisfy this, it is necessary to fulfill the first
two conditions. The third condition ensures that the phase field parameter goes to
zero if and only if the material point is fully damaged. The fourth condition is to
24
3 Formulation
be satisfied to keep intact the principle of conservation of energy. Lastly, the fifth
condition imposes that all the material points in the domain are undamaged at an
arbitrary initial state. It is important to satisfy the above conditions in order to
achieve Γ convergence as explained by Braides [25] and to ensure reasonable evolution
of stress and crack field. From literature, g(s) = (1 − η)s2 + η , which satisfies all
the requirements, has been used in this work (η is a dimensionless parameter used to
model an artificial stiffness of a completely damaged phase) [23].
Two ways of degrading the elastic strain energy density is followed in the field of
phase-field modeling. In isotropic formulation, the entire elastic strain energy density
is multiplied by the stress degradation function g(s) as mentioned in Eq:(3.6). The
second type of formulation is the one which has a tensile-compression split of strain
energy density and only the tensile part is degraded. In this work, one of the variants
with a tensile-compression split presented by Miehe et al [7] is followed.
Anisotropic Miehe formulation
The anisotropic Miehe formulation presented by Miehe et al [7] is considering a tensile
and compression split is anisotropic by nature which gives the formulation its name.
In this formulation, the free energy funtional defined in Eq:(3.5) is modified as,
E(u , s) =
∫Ω
[g(s)Ψ+
e (ε) + Ψ−e (ε) +Gc
(1
4l(1− s)2 + l|∇,xs|2
)]dx (3.8)
where, the strain energy density is approximated as,
Ψe(ε) ≈ Ψe(ε, s) = g(s)Ψ+e (ε) + Ψ−e (ε) (3.9)
Ψ+e and Ψ−e are the tensile and compressive components of elastic strain energy density,
respectively, obtained by the spectral decomposition of the strain tensor and are defined
as,
Ψe+ =
1
2λ〈tr(ε)〉2 + µtr[(ε+)2] (3.10)
and
Ψ−e =1
2λ[(tr(ε)− 〈tr(ε)〉)2
]+ µtr[(ε− ε+)
2] (3.11)
where λ and µ are Lame constants and
〈χ〉 =
χ : χ > 0
0 : χ ≤ 0(3.12)
25
3 Formulation
Let the strain tensor be defined as,
ε = PAPT (3.13)
where P consists of the orthonormal eigenvectors of ε and A is a diagonal matrix
of principal strains λi. The following definitions for tensile and compressive split of
strains are used for spectral decomposition.
ε+ = PA+PT (3.14)
ε− = PA−PT (3.15)
where, A+ = diag(〈λi〉) is a diagonal matrix and A− = A−A+.
This split in strain energy density allows degradation of tensile component of strain
energy density only, thereby inhibiting crack growth under compression. Selective
degradation of strain energy density is important for dynamic simulation in particular
since the stress waves reflecting from domain boundaries may create unphysical fracture
patterns otherwise, refer [23] for details.
The Euler-Lagrange equations of the minimization of the energy functional Eq:(3.8)
leads to the strong form where the governing equation for phase-field is driven by the
elastic strain energy density.
strong form
∂σij
∂xj+ ρb = 0[
4lGc
(1− η)H + 1
]s − 4 l2 ∆s = 1 on Ω
(3.16)
where, b is the body force and σij = g(s)∂Ψ+e
∂εij+ ∂Ψ−
e
∂εijand
Ht∈[0,τ ]
= max Ψ+e (ε, t) (3.17)
is called the history variable. It represents the maximum tensile elastic strain energy
density. This treatment enables decoupling of the two differential equations and hence
a staggered scheme can be used. Also, the irreversibility condition Γ(t) ⊆ Γ(t + dt)
at any time (psuedo-time in the case of quasi-static fracture) t is enforced using this
history variable H.
To sum up, the strong form for quasi-static brittle fracture problem is represented by
two partial differential equations Eq:(3.16) which are solved for u(x ) and s(x ) using
26
3 Formulation
staggered scheme as explained earlier. They are subjected to the following boundary
conditions,
boundary conditions
ui = gi on ∂Ωgi
σijnj = hi on ∂Ωhi
∂s∂xini = 0 on ∂Ω
(3.18)
Hybrid fomulation
The isotropic formulation where the entire elastic strain energy density is degraded
by the phase-field although numerically inexpensive due to its linear nature, leads to
unrealistic crack patterns since it allows crack propagation under compression. This is
overcome by anisotropic formulations by introducing a split in elastic strain energy
density which differentiates between tension and compression. However, this split is
anisotropic in nature. This means that within each staggered step, the momentum
equation is non-linear which is to be solved by incrementation techniques and hence is
numerically expensive. In order to take an advantage of the linear nature of the isotropic
formulation whilst preserving the tension-compression differentiation, Ambati et al [5]
proposed the so-called hybrid formulation. In this formulation, the tension-compression
split is retained for the calculation of strain energy density and thus Eq:(3.17) remains
the same. However, the decomposition that exists in stress calculation is let go,
σij = g(s)∂Ψe
∂εij(3.19)
This model hence retains the linear nature of the momentum equation and reduces
the computational cost significantly.
3.2 Extension to dynamic brittle fracture
The additional term to be considered for dynamic fracture is the kinetic energy
represented by Eq:(3.3). The Lagrangian for the fracture problem, Eq:(3.20), is
formed by combining the potential and kinetic energies incorporating the phase-field
approximations explained earlier. This Lagrangian based Euler-Lagrange equations
govern the motion of the body and the evolution of crack which are presented in this
27
3 Formulation
section and are adapted to phase-field setting. The fracture energy approximation
remains the same as shown in Eq:(3.7)
Lε(u , u , s) =
∫Ω
[12ρuiui− g(s)Ψ+
e (ε)−Ψ−e (ε)]dx −
∫Ω
Gc[ 1
4l(1− s)2 + l|∇,xs|2
]dx
(3.20)
The Euler-Lagrange equations of the minimization of the Lagrangian represented by
Eq:(3.20) leads to the strong form. Similar to the quasi-static case, the strong form for
dynamic brittle fracture problem is represented by two partial differential equations
which are solved for u(x , t) and s(x , t) using staggered scheme as explained earlier.
strong form
∂σij
∂xj+ ρb = ρui on Ω[
4lGc
(1− η)H + 1
]s − 4 l2 ∆s = 1 on Ω
(3.21)
These equations are subjected to the following boundary conditions and initial condi-
tions.
boundary conditions
ui = gi on ∂Ωgi
σijnj = hi on ∂Ωhi
∂s∂xini = 0 on ∂Ω
(3.22)
intial conditions
u(x , 0) = u0(x ) x ∈ Ω
u(x , 0) = v0(x ) x ∈ Ω(3.23)
3.3 Numerical formulation
In order to numerically solve the initial boundary value problem of the brittle fracture
process, it is necessary to derive a weak form and then discretize the same. This section
is dedicated to the weak formulation of the governing equations for both quasi-static
and dynamic cases, first in a continuous domain and then in Galerkin setting in order
to facilitate finite element implementation.
28
3 Formulation
3.3.1 Quasi-static brittle fracture
Continuous problem in the weak form
In the following, the necessary spaces for the derivation of the weak form are defined.
Suppose S be the trial space for displacement solution and S be the trial space for
phase-field solution. Similarly V and V are the spaces for the test functions.
S = u(x ) ∈ H1(Ω)| ui = gi on ∂Ωgi (3.24)
S = s(x ) ∈ H1(Ω) (3.25)
V = w(x ) ∈ H1(Ω)| wi = 0 on ∂Ωgi (3.26)
V = q ∈ H1(Ω) (3.27)
where H1 is a Hilbert space.
The weak form is obtained by multiplying the strong form by appropriate test functions
and then applying Green’s theorem. It states, given g(x ), h(x ), b(x ) , find u(x ) ∈S and s(x ) ∈ S such that,
weak form in continuous domain
(σ,∇w)Ω = (ρb,w)Ω + (h ,w)∂Ωh
([
4lGc
(1− η)H + 1]s, q)
Ω+ (4l2∇s,∇q)Ω = (1, q)Ω
(3.28)
where (., .)Ω is the L2 inner product on Ω. The above weak form holds ∀w ∈ V , q ∈ V
Galerkin form
The previously introduced spaces to which the solution and test functions belong are
infinite. For FE analysis, as the name suggests, it is necessary to logically reduce these
spaces to valid finite ones. Galerkin method is followed for this purpose. Let Sh ⊂ S,
Sh ⊂ S, Vh ⊂ V and Vh ⊂ V be the finite dimensional approximating spaces. The
Galerkin form states, find uh(x ) ∈ Sh and sh(x ) ∈ Sh such that,
Galerkin form
(σ,∇wh)Ωh = (ρbh,wh)Ωh + (hh,wh)∂Ωh
h
([
4lGc
(1− η)H + 1]sh, qh)
Ωh+ (4l2∇sh,∇qh)Ωh = (1, qh)Ωh
(3.29)
In finite element method, the continuous domain Ω is divided into finite number of
elements. This is denoted by Ωh. The discretized solutions, displacement uh(x ) and
phase field sh(x ) and the trial functions, wh(x ) for displacement field and qh(x ) for
29
3 Formulation
phase field are assumed to be linear combination of basis functions. Here, Bubnov-
Galerkin formulation is followed, where same basis functions are used for both solutions
and the trial functions.
u ≈ uhi =
nb∑A
NA(x )diA (3.30)
w ≈ whi =
nb∑A
NA(x )ciA (3.31)
s ≈ sh =
nb∑A
NA(x )φA (3.32)
q ≈ qh =
nb∑A
NA(x )χA (3.33)
where nb is the dimension of the discrete space Ωh, NA are the basis functions, i is the
spatial degree of freedom (nodes), diA, ciA, φA, χA are the nodal values.
Subtituting these in the Galerkin form in Eq:(3.29), the following system is obtained.
Kuu∆d = Fu (3.34)
Kuu = KAB,i (3.35)
KAB,i = (σjk, BijkA )
Ωh (3.36)
Fu = FA,i (3.37)
FA,i = (ρbh, NAe i)Ωh + (hh, NAe i)∂Ωhh
(3.38)
where e i is the i th Euclidean basis vector and
BijkA =
1
2
(∂NA∂x j
δik +∂NA∂x k
δij
)σjk = g(s)
∂Ψ+e
∂εjk+∂Ψ−e∂εjk
for the anisotropic Miehe formulation
σjk = g(s)∂Ψe
∂εjkfor the hybrid formulation
Similarly, the phase-field arrays are defined.
30
3 Formulation
Kss∆φ = Fs
Kss = KAB
KAB = ([ 4l
Gc(1− η)H + 1
]NB, NA)
Ωh
+ (4l2∂NB∂x i
,∂NA∂x i
)Ωh
Fs = FAFA = (1, NA)Ωh
In the case of the anisotropic Miehe formulation, due to the tension-compression
split of stresses, the governing equation system for the displacement field, Eq:(3.34),
is non-linear. Generally, such equations are solved using incrementational solution
techniques which required a consistent linearization of all the terms in the weak form.
In this work, followed is the Newton-Raphson method, see Ambati [26] for detailed
formulation.
3.3.2 Dynamic brittle fracture
The weak form for dynamic brittle fracture states, given g , h , u0, u0 and s0, find
u(x , t) ∈ S and s(x , t) ∈ S such that,
weak form in continuous domain
(ρu,w)Ω + (σ,∇w)Ω = (ρb,w)Ω + (h ,w)∂Ωh
([
4lGc
(1− η)H + 1]s, q)
Ω+ (4l2∇s,∇q)Ω = (1, q)Ω
(ρu(0),w)Ω = (ρu0,w)Ω
(ρu(0),w)Ω = (ρu0,w)Ω
(s(0), q)Ω = (s0, q)Ω
(3.39)
The semi-discrete Galerking form for numerical implementation states, find uh(x , t) ∈Sht and sh(x , t) ∈ Sht such that,
semidiscrete Galerkin form
(ρuh,wh)Ωh + (σ,∇wh)Ωh = (ρbh,wh)Ωh + (hh,wh)∂Ωh
h
([
4lGc
(1− η)H + 1]sh, qh)
Ωh+ (4l2∇sh,∇qh)Ωh = (1, qh)Ωh
(ρuh(0),wh)Ωh = (ρu0,wh)Ωh
(ρuh(0),wh)Ωh = (ρu0,wh)Ωh
(sh(0), qh)Ωh = (s0, qh)Ωh
(3.40)
The above is called semi-discrete since the discretization is only in space and not
in time. When compared to the Galerkin form of the quasi-static formulation, the
semi-discrete form is similar but with an additional mass matrix.
31
3 Formulation
Time discretization
As mentioned earlier, staggered time integration scheme is followed to solve the dynamic
problem. Recall that in staggered scheme, momentum and phase-field equations are
solved independently. The momentum equation is solved at a given time step, the
displacements in the phase-field equation is updated and then the equation is solved.
Assuming negligible body forces. the following definition of residuals are necessary for
this scheme.
Ru = RuA,i (3.41)
RuA,i = (h , NAe i)∂Ωhh− (ρuh, NAe i)Ωh − (σjk, B
ijkA )
Ωh (3.42)
Rs = RsA (3.43)
RsA = (1, NA)Ωh − ([ 4l
Gc(1− η)H + 1
]sh, NA)
Ωh
− (4l2∂sh
∂x i,∂NA∂x i
)Ωh
(3.44)
After linearization of the momentum equation, predictor-corrector method is then
applied to solve this system, see [23] for details.
3.4 Finite Cell Method for phase-field quasi-static
brittle fracture
As introduced in section 2.4, the Finite Cell Method is an embedded domain method.
The physical domain is recovered using the indicator function α as defined in Eq:(2.13).
In the following, presented is the weak formulation for phase-field quasi-static brittle
fracture modified for the Finite Cell Method implementation.
Momentum Equation
The momentum equation for quasi-static brittle fracture, Eq:(3.28), is modified for
the Finite Cell Method implementation and is as follows. Penalty method is used to
impose the Dirichlet boundary condition with β being the penalty parameter, typically
of the order 1010 to 1012.
(σ,∇w)Ωphy+ (εσ,∇w)Ωfict
+ (βu ,w)∂Ωg
= (ρb,w)Ωphy+ (h ,w)∂Ωh
+ (βg ,w)∂Ωg
(3.45)
(ε 4l2∇s,∇q)Ωfict
= (1, q)Ωphy(3.46)
32
3 Formulation
Phase-field Equation
Similar to the momentum equation, the weak form of the phase-field equation modified
for FCM is as presented below. Due to homogeneous Neumann boundary condition
throughout the boundary, enforcing the boundary conditions for this initial boundary
value problem is trivial.
([ 4l
Gc(1− η)H + 1
]s, q)
Ωphy
+ (ε[ 4l
Gc(1− η)H + 1
]s, q)
Ωfict
+ (4l2∇s,∇q)Ωphy+ (ε 4l2∇s,∇q)Ωfict
= (1, q)Ωphy
(3.47)
(Note that the Finite Cell formulation for dynamic brittle frature follows the same
procedure resulting in similar form with an additional mass matrix term related to the
kinetic energy.)
33
4 Numerical results
In this chapter, the numerical performance of phase-field fracture model combined
with the uniform multi-level hp-refinement and the Finite Cell Method is investigated.
Illustrative examples for quasi-static and dynamic brittle fracture are presented. Inte-
grated Legendre polynomials are used as basis functions for discretization and p+1
integration rule is adopted for numerical integration. In each example, history variable
is stored at geometric points using voxel domain geometry modeling technique and the
same is used for the definition of pre-existing crack. Two separate discretizations are
defined, one for the elastic problem and one for the phase-field problem. Two different
refinement strategies are followed for adaptive refinement of the two problems. In
every load or time step, the base mesh for the elastic problem is refined in the vicinity
of the crack. It is refined in those areas of the physical domain where the phase-field
values reach a certain threshold limit. The phase-field problem is then refined such
that it follows the discretization of the elastic problem.
The chapter starts off with two quasi-static benchmark problems, single-edge notched
tension test and single-edge notched shear test. The current implementation of the
hybrid formulation is validated by qualitative and quantitative comparison of the
obtained results for single-edge notched tension test to the results obtained by Ambati
et al [5] under similar settings. The anisotropic Miehe formulation is validated using
single-edge notched shear test. Further, for these two examples, convergence study for
both the anisotropic Miehe and the hybrid formulations are presented.
This is followed by a more complex example, a notched plate with hole, which demon-
strates the idea of combining Finite Cell Method with the concept of phase-field
modeling. A non-geometry conforming mesh is used in this case. This example is used
to study the effect of number of staggered iterations used within every displacement step.
As a concluding example for the quasi-static brittle fracture case, a non-pre-cracked
cocnrete block subjected to compressive load is studied.
The next section deals with a dynamic crack branching problem under uniaxial
tension. Numerical results obtained using both the anisotropic Miehe and the hybrid
formulations are discussed.
4.1 Single-edge notched tension test
Consider a two-dimensional sqaure plate of side 1mm with a pre-existing horizontal
crack at its mid height as shown in Fig:4.1. The pre-existing crack is modeled using the
history variable as in Eq:(4.1). This notched plate is subjected to a constant vertical
34
4 Numerical results
displacement on its top edge. The material and model parameters used are: E = 210
GPa, ν = 0.3, (λ = 121.15 kN/mm2 and µ = 80.77 kN/mm2) Gc = 0.0027 kN/mm, l
= 0.004mm, η = 10−6.
H(x ) = B
Gc
4l : 0 ≤ x1 ≤ 0.5, (0.5− l) ≤ x2 ≤ (0.5 + l)
0 : otherwise(4.1)
0.5mm
0.5mm
0.5mm 0.5mm
u
Figure 4.1: Geometry and boundary conditions for single-edge notched tension test
4.1.1 Validation of implementation of the hybrid formulation
To validate the current implementation of the hybrid formulation, consider a dis-
cretization with 4 × 4 elements, ansatz order p = 4 and a refinemenet depth k = 5.
Displacement control with a displacement increment of ∆u = 1 × 10−6mm and two
staggered iteration for every load step are used in oder to match the parameters of
the same example in Ambati et al [5]. The number of DOFs at the beginning of the
simulation is 12906 and 33258 at a displacement value of 7.0×10−3mm. The reference
result uses a fixed mesh with bilinear shape functions which is apriori refined in the
location where crack is known to develop and the crack has been defined using the
geometry instead of the history variable. The phase-field evolution representing crack
propagation is shown in Fig.4.2 and the corresponding load-displacement curve in
Fig.4.3. It can be observed that the crack patterns and the load-displacement curves
almost coincide, thus validating the implementation.
35
4 Numerical results
Ref
eren
cep
=4,k
=5
(a) u = 5.5×10−3mm (b) u = 5.7×10−3mm (c) u = 6.0×10−3mm
Figure 4.2: Single-edge notched tension test. Validation of crack evolution with hybrid
formulation against reference results.
0 1 2 3 4 5 6
10-3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
reference solution
p = 4, k = 5
Figure 4.3: Single-edge notched tension test. Validation of load-displacement behavior
with hybrid formulation against reference results.
36
4 Numerical results
4.1.2 Parametric influence of convergence behavior of the
phase-field problem
In the following, convergence behavior of the phase-field solution is studied under two
scenarios. To this end, firstly, three discretizations with linear, quadratic and cubic
ansatz functions and a grid of 4 × 4 base elements are chosen. The material and model
paramters remain the same except that a larger displacement increment of ∆u = 1 ×10−5mm is used in order to shorten the run time. The only disadvantage of using larger
displacement increments is that the sharp drop in the reaction forces at peak values,
i.e,. the catastrophic crack propagation phenomenon is not well captured. Fig.4.4
depicts snapshots of crack path at different displacements. A common observation in
the results of both the formulations is that for linear shape functions, a much higher
displacement value leads to crack propagation compared to that using higher order
shape functions. The same trend is reflected in the load-displacement plots in Fig.4.5
where the peak force value using linear basis is higher and is shifted towards a higher
displacement value. In Fig.4.5, a converging behavior is observed as the order of the
polynomial is increased, where error between the results using quadratic and cubic
basis is much lower than that between linear and quadratic basis.
p=
1p
=2
p=
3
Anisotropic Miehe formulation
(a) u = 6.5×10−3mm (b) u = 7.0×10−3mm (c) u = 7.3×10−3mm
37
4 Numerical results
p=
1p
=2
p=
3
Hybrid formulation
(a) u = 6.5×10−3mm (b) u = 7.0×10−3mm (c) u = 7.3×10−3mm
Figure 4.4: Single-edge notched tension test. Multi-level hp-refinement for different
ansatz orders with k = 6. Crack phase-field at different displacements.
0 1 2 3 4 5 6 7 8
10-3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
p = 1
p = 2
p = 3
reference solution
(a) Anisotropic Miehe formulation
38
4 Numerical results
0 1 2 3 4 5 6 7 8
10-3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
p = 1
p = 2
p = 3
reference solution
(b) Hybrid formulation
Figure 4.5: Single-edge notched tension test. Multi-level hp-refinement for different
ansatz orders with k = 6. Load-displacement curves.
To investigate the minimum required refinement depth for a given ansatz order, consider
a grid with 4 × 4 elements, ansatz order p = 3 and three different depths 4, 5 and
6. The results obtained for both the aniostropic Miehe and the hybrid formulation
are presented. Fig.4.6 depicts the phase-field crack evolution and Fig.4.7 shows the
corresponding load-displacement curves. At the outset, even though the crack path
is as expected for all cases, one can observe a significant difference in the phase-field
evolution for the lower refinement case. For a refinement depth of 4, crack propagation
is rather slow in the results of both the formulations. This is also reflected in the
load-displacement behavior where the load values with k = 4 show an overestimated
behavior. For both the formulations, excellent convergence can be observed as the
refinement depth is increased. It is worth noting that the solutions obtained using
multi-level hp-refinement for hybrid formulation almost coincide for the corresponding
results using the anisotropic Miehe formulation. Thus, it is clear from the above
two studies that, with different ansatz orders and with different refinement depths,
it is important to resolve the mesh carefully in order to avoid mesh related effects.
To conclude, it seems from the above parametric study that a dynamically adaptive
discretization with 4 × 4 elements, cubic shape functions and a refinement depth of 5
with around 7000 initial DOFs leading to around 18500 DOFs at the end of fracture
with run time of 50 minutes is sufficient to capture the fracture process. To obtain
similar quality results, a non-dynamically adaptive grid with 26058 DOFs has been
used in the reference results.
39
4 Numerical results
k=
4k
=5
k=
6
Anisotropic Miehe formulation
k=
4k
=5
k=
6
Hybrid formulation
(a) u = 6.5×10−3mm (b) u = 6.75×10−3mm (c) u = 7.0×10−3mm
Figure 4.6: Single-edge notched tension test. Multi-level hp-refinement for different
refinement depths with p = 3. Crack phase-field evolution at different
displacements. 40
4 Numerical results
0 1 2 3 4 5 6 7 8
10-3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
k = 4
k = 5
k = 6
reference solution
(a) Anisotropic Miehe formulation
0 1 2 3 4 5 6 7 8
10-3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
k = 4
k = 5
k = 6
reference solution
(b) Hybrid formulation
Figure 4.7: Single-edge notched tension test. Multi-level hp-refinement for different
refinement depths with p = 3. Load-displacement curves.
41
4 Numerical results
4.2 Single-edge notched shear test
0.5mm
0.5mm
0.5mm 0.5mm
u
Figure 4.8: Geometry and boundary conditions of single-edge notched shear test
4.2.1 Validation of implementation of the anisotropic Miehe
formulation
The same square plate with a pre-existing crack used for tension test is now subjected
to shear load as shown in Fig:4.8. The material and model parameters remain the
same as that for the tension test example. Firstly, implementation of the anisotropic
Miehe formulation is validated using this example. A displacement increment ∆u = 1
× 10−5mm is applied on a grid of 4 × 4 elements, ansatz order p = 4 and a refinement
depth k = 5 are used for this purpose. While the reference solution obtained by Ambati
et al [5] is using a relatively fine uniform mesh, the result obtained here is using an
adaptively refined mesh which changes over the course of the simulation from an intial
DOFs of 12906 to final DOFs of 41754. The crack patterns at different displacements
and the load-displacement curves are shown in Fig.4.9 and Fig.4.10, respectively. The
crack propagation path obtained using multi-level hp dynamically adaptive refinement
mostly coincides with that from the reference. However, a faster crack propagation
can be observed in this adaptive result in comparison to the reference results. This
relatively small difference, to the knowledge of the author, could be due to mesh related
effects. The reference solution is also a numerical one in which mesh related effects
cannot be completely eliminated.
42
4 Numerical results
Ref
eren
cep
=4,k
=5
(a) u = 0.012mm (b) u = 0.015mm (c) u = 0.020mm
Figure 4.9: Single-edge notched shear test. Validation of crack evolution with the
anisotropic Miehe formulation against reference results.
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
reference solution
p = 4, k = 5
Figure 4.10: Single-edge notched shear test. Validation of load-displacement behavior
with the anisotropic Miehe formulation against reference results.
43
4 Numerical results
4.2.2 Parametric influence of convergence behavior of the
phase-field problem
In this section, convergence study is carried out for different discretizations, similar to
the previous example. For this, a larger displacement increment of ∆u = 1 × 10−4mm
is used in order to shorten the computational time. Although this has no significant
effect on pre-peak part of the load-displacement curve, the post-peak curve varies. The
immediate drop of load values post-peak is not observed in this case similar to the
observation in the previous example.
A study of influence of ansatz order for a given refinement depth is carried out here.
Three case each, for both the anisotropic and the hybrid formulation, with ansatz
orders 1, 2 and 3, all with a refinement depth k = 6 and a grid with 4 × 4 elements are
studied. Result obtained by Ambati et al [5] for the same displacement increment value
is used as a reference. The crack pattern and the load-displacement curves for these
cases are depicted respectively in Fig.4.11 and Fig:4.12. The crack propagation path
seem to be almost the same for different ansatz orders. However, a careful observation
reveals small disturbances in the crack path with lower order polynomials. Similar to
the observation made in the tension test examples, the reaction force values are over
estimated at lower order ansatz functions. But excellent convergence is observed as
the order of the basis is increased.
p=
1p
=2
p=
3
Anisotropic Miehe formulation
(a) u = 0.012mm (b) u = 0.02mm (c) u = 0.025mm
44
4 Numerical results
p=
1p
=2
p=
3
Hybrid formulation
(a) u = 0.012mm (b) u = 0.02mm (c) u = 0.025mm
Figure 4.11: Single-edge notched shear test. Multi-level hp-refinement for different
ansatz orders with k = 6. Crack phase-field at different displacements.
0 0.005 0.01 0.015 0.02 0.0250
0.1
0.2
0.3
0.4
0.5
0.6
p = 1
p = 2
p = 3
reference solution
(a) Anisotropic Miehe formulation
45
4 Numerical results
0 0.005 0.01 0.015 0.02 0.0250
0.1
0.2
0.3
0.4
0.5
0.6
p = 1
p = 2
p = 3
reference solution
(b) Hybrid formulation
Figure 4.12: Single-edge notched shear test. Multi-level hp-refinement for different
ansatz orders with k = 6. Load-displacement curves.
To study the influence of refinement depth for a given ansatz order, consider a
discretization with 4 × 4 elements, ansatz order p = 5 and three different depths 3,
4 and 5. Both the aniostropic Miehe and the hybrid formulation are considered for
this study. Fig.4.13 depicts the phase-field crack evolution and Fig.4.14 shows the
corresponding load-displacement curves. Although the crack propagation path is well
defined and close to the expected one for the hybrid formulation, the displacement
required for the same amount of crack growth varies with refinement depth. This is
due to insufficient refinement which makes the solution badly resolved. The effect of
discretization on the solution is pronounced in the case of anisotropic Miehe formulation
where inadequate resolution leads to incorrect results. A crack can be seen propagating
at the boundaries which is unphysical.
Inference from the above two study cases is that for cubic shape functions, a refinement
depth of 6 with 21032 initial and 71834 final DOFs, approximate run time of 2 hours,
10 minutes or for a 5th order polynomial ansatz, a refinement depth of 5 with 20312
initial and 69032 final DOFs, with an approximate run time of 1 hour, 55 minutes,
eliminate major numerical artifacts.
46
4 Numerical results
k=
3k
=4
k=
5
Anisotropic Miehe formulation
k=
3k
=4
k=
5
Hybrid formulation
(a) u = 0.012mm (b) u = 0.02mm (c) u = 0.025mm
Figure 4.13: Single-edge notched shear test. Multi-level hp-refinement for different
refinement depths with p = 5. Crack phase-field at different displacements.
47
4 Numerical results
0 0.005 0.01 0.015 0.02 0.0250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
k = 3
k = 4
k = 5
reference solution
(a) Anisotropic Miehe formulation
0 0.005 0.01 0.015 0.02 0.0250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
k = 3
k = 4
k = 5
reference solution
(b) Hybrid formulation
Figure 4.14: Single-edge notched shear test. Multi-level hp-refinement for different
refinement depths with p = 5. Load-displacement curves.
4.2.3 Comparative study between uniform and adaptively refined
meshes
The objective of this section is to study the performance of a globally refined and locally
refined discretizations for the same order of the shape functions. For this, consider
the shear test specimen with the same material parameters as before. A displacement
increment of ∆u = 1 × 10−4mm is used. In order to resolve a length scale parameter
of l = 0.004 mm using a globally refined mesh, a grid with a minimum of 250 × 250
48
4 Numerical results
elements are required, resulting in 126002 degrees of freedom. The resources required
to run such cases are very high and time consuming. Hence, a larger length scale
parameter is chosen which results in a thicker crack due to the larger transition zone
between damaged and undamaged regions.
Firstly, the specimen is uniformly discretized into a grid of 128 × 128 elements with
linear ansatz such that the smallest element is 0.0078 mm × 0.0078 mm is size. This
descretization is sufficient to resolve a length scale parameter of l = 0.01 mm. In order
to compare the results obtained from uniform mesh to that from an adaptively refined
mesh using uniform multi-level hp-refinement, consider two cases with 32 × 32 with
2 multi-level hp-refinements and 16 × 16 with 3 multi-level hp-refinements. These
two discretizations ensure that the smallest element size remains the same as that
of the uniform grid. Fig.4.15 shows the crack evolution over displacements and the
corresponding load-displacement curve is shown in Fig.4.16. They clearly demonstrate
that the same quality of the solution can be achieved by using coarser base mesh if the
refinement depth is increased appropriately. In Table 4.1, the number of degrees of
freedom and the run time comparison for the three cases under consideration is done.
The result values reflect the increasing efficiency in terms of number of unknows and
the run time when the refinement is made more local at regions with crack while using
larger elements where away from the crack where the solution is mostly smooth.
128×
128
elem
s32×
32el
ems
16×
16el
ems
(a) u = 0.012mm (b) u = 0.015mm (c) u = 0.020mm
Figure 4.15: Single-edge notched shear test. Comparison of crack phase-field for uni-
formly and adaptively refined meshes with p = 1 at different displacements.49
4 Numerical results
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
128 X 128 elements
32 X 32 elements, k = 2
16 X 16 elements, k = 3
Figure 4.16: Single-edge notched shear test. Comparison of load-displacement curves
for uniformly and adaptively refined meshes with p = 1.
Case Initial DOFs Final DOFs Run time
128×128 elements 33828 33828 2 hours, 45 minutes
32×32 elements, 2 multi-level hp-ref 2678 6326 30 minutes
16×16 elements, 3 multi-level hp-ref 1168 4884 16 minutes
Table 4.1: Single-edge notched shear test. Parametric comparison for uniformly and
adaptively refined meshes with p = 1 and the hybrid formulation.
50
4 Numerical results
4.3 Notched plate with hole
0.55mm
0.65mm
uy = 0
0.2mm
0.2mm
0.51mm
1.2mm
0.365mm
0.2mm
0.2mm
0.1mm
x
y
0.2mm
0.1mm
0.1mm
ux = 0
ux = 0
uy = u
Boundary Conditions
Figure 4.17: Geometry and boundary conditions of notched plate with hole
Figure 4.18: Notched plate with hole. Crack pattern from the experiment
This example demonstrates the application of the FCM in the context of phase-field
models. Here, a notched plate with the geometry and boundary conditions as shown
in Fig.4.17 is simulated. The material and model properties used for this example are:
51
4 Numerical results
E = 6 GPa, ν = 0.22, (λ = 121.15 kN/mm2 and µ = 80.77 kN/mm2) Gc = 0.00227
kN/mm, l = 0.005mm, η = 10−6 and a displacement increment of ∆u = 5 × 10−4mm
is applied upto 0.1mm. A discretization with 15 × 30 elements, quadratic integrated
Legendre polynomials and refinement depth of 4 is used. The indicator function value
is set to 10−8 in the fictitious domain, the three holes in this example. A partitioning
depth of 3 is made use in numerical integration using quadtree integration scheme.
Crack patterns obtained for both the anisotropic Miehe and the hybrid formulations are
shown in Fig.4.19. It can be observed that the crack patterns are similar, except that
the anisotropic Miehe formulation predicts a larger crack growth at lower displacement
values when compared to the hrbrid formulation. Also, the crack pattern in the hybrid
formulation is inclined to the horizontal at a larger angle than that in the anisotropic
Miehe formulation. This inclined behaviour can be observed in the experimental results
obtained for a similar (scaled up in dimensions) specimen by Ambati et al [5], see
Fig.4.18.
Anisotropic Miehe formulation
Hybrid formulation
(a) u = 0.035mm (b) u = 0.05mm (c) u = 0.085mm (d) u = 0.1mm
Figure 4.19: Notched plate with hole. Multi-level hp-refinement: p = 2, k = 4. Crack
phase-field at different displacements.
52
4 Numerical results
4.3.1 Parametric influence of number of staggered iterations on
crack propagation
To study the influence of number of staggered iterations used per displacement incre-
ment step, consider the same example for three different staggered iteration value cases:
2 , 3 and 5. Hybrid formulation with cubic integrated Legendre shape functions are
used here, while rest of the parameters remain the same as explained above. Resulting
crack propagation is depicted in Fig.4.20. It can be observed that the number of
staggered iterations play an important role in crack propagation. This is because for
higher number of staggered iterations, the momentum equation and the phase-field
equation tend to attain equilibrium with each other. Hence, this necessitates the
definition of a proper stopping criterion which stops at a particular staggered iteration
that ensures equilibrium, refer [5] for details.
2st
agg
iter
3st
agg
iter
5st
agg
iter
(a) u = 0.035mm (b) u = 0.05mm (c) u = 0.085mm (d) u = 0.1mm
Figure 4.20: Notched plate with hole. Multi-level hp-refinement for different number
of staggered iterations : p = 3, k = 4. Crack phase-field at different
displacements.
53
4 Numerical results
4.4 Crack propagation under compressive loads
0.9m
0.35m
u0.045m
0.01m
Figure 4.21: Geometry and boundary conditions of concrete block under compression.
This example aims at studying the behavior of the phase-field model in the case of load
induced crack in a concrete block. Consider a crack-free concrete block with thickness
0.1m as shown in Fig.4.21. The material of the block has E = 38.992 GPa, ν = 0.18,
Gc = 148.5 N/m and is subjected to compression. This three dimensional scenario is
numerically solved as a two dimensional plane strain problem using phase-field model
with the following parameters: 5 staggered iterations, l = 0.01m, η = 10−6 and a
displacement increment of ∆ u = 5 × 10−5m. A discretization with linear 5 × 15
elements and 4 uniform multi-level hp-refinements are used. A pre-refinement is done
in the vicinity of the displacement boundary condition where the crack is expected to
initiate. Fig.4.22 and Fig.4.23 show the experimentally and numerically obtained crack
patterns, respectively. As evident, the numerical results are not similar to the ones
obtained from the experiment. Also, the results from the anisotropic and the hybrid
formulations do not comply with each other. With the hybrid formulation, no crack is
seem to be initiated. Larger displacements are observed to lead to unphysical results
and hence are not presented here. A few obersvations and remarks in this regard are
as follows:
In the anisotropic Miehe formulation, the number of Newton-Raphson iterations
taken for convergence is found to increase beyond 30 after a few load steps.
This worsens when body forces are added to the elastic initial boundary value
problem.
Despite using a large number of staggered iterations in order to ensure equilibrium
between the elastic and the phase-field problem, the convergence remains poor.
Not considering into account the body forces and the dynamic nature of the
problem could be one of the reasons for the propagation path of the crack to
54
4 Numerical results
deviate from the experimental one.
Further investigation needs to be done in order to study about applying the
phase-field model for such initial crack-free problems and is beyond the scope of
the thesis.
Figure 4.22: Concrete block under compression. Crack pattern from the experiments
carried out by Dipl.-Ing. Gerald Schmidt-Thro, Lehrstuhl fur Massivbau,
TU Munchen.
Anisotropic Miehe formulation
Hybrid formulation
(a) u = 0.5mm (b) u = 1mm (c) u = 1.5mm (d) u = 2.5mm
Figure 4.23: Concrete block under compression. Multi-level hp-refinement: p = 1, k =
4. Crack phase-field at different displacements.
55
4 Numerical results
4.5 Dynamic crack branching
Consider a rectangular specimen of length B = 0.1 m and H = 0.04 m with a pre-
existing horizontal crack at its mid height as shown in Fig:4.24. The specimen is
subjected to a constant uniaxial tension of σ(t). The material and model parameters
used for simulation are ρ = 2450 kg/m3, E = 32 GPa, Gc = 3 J/m2, ν = 0.2, η =
10−6. A discretization with 8 × 4 elements with two linear and quadratic ansatz, time
step of dt = 0.2µs and a length scale parameter of l = 2.5× 10−4m are used. Results
using two staggered iterations are presented in order to match the parameters used by
Borden et al [23] for the same benchmark example.
H
B
Initial crack, Γ
σ(t)
σ(t)
Boundary conditions toensure numerical stability
0.01m
Figure 4.24: Geometry and loading of dynamic uniaxial tension test.
Remark
When the problem is solved numerically, care should be taken while imposing
the boundary conditions in order to avoid artifacts. If rigid body motion of the
sample is not arrested, then the condition number of the stiffness matrix becomes
very high which might lead to a singular or a nearly singular matrix resulting in
unphysical behavior. In order to restrict the rigid body motion, in the example
presented here, the mid node on the right edge is fixed and a dirichlet boundary
condition is weakly imposed on a small segment on the mid-line as indicated in
Fig.4.24.
The predictor-corrector time integration scheme was found to convergence at a
very slow rate when the strong constrain at the mid node on the right edge was
imposed using a penalty.
56
4 Numerical results
The length of the segment along which the weak boundary condition is applied
has been empirically chosen to be 0.01m. Further optimization can be done in
this regard.
p=
1
k=
5
p=
2
k=
5
Anisotropic Miehe formulation
(a) t = 20µs (b) t = 50µs (c) t = 80µs
Figure 4.25: Dynamic crack branching under uniaxial tension. Multi-level hp-
refinement for different ansatz order with k = 5 using the anisotropic
Miehe formulation. Crack phase-field evolution over time.
0 10 20 30 40 50 60 70 800
0.05
0.1
0.15
0.2
0.25p = 1
p = 2
reference
Figure 4.26: Dynamic crack branching under uniaxial tension. Multi-level hp-
refinement for different ansatz order with k = 5. Plot of strain energy
overtime.
57
4 Numerical results
The numerical results obtained using the anisotropic Miehe formulation are presented in
Fig.4.25 and Fig.4.26. Similar to the observations made in quasi-static case, convergence
to the reference solution can be observed as the order of the shape functions are
increased. However, a sudden jump in the strain energy can be observed at around 5µs
while using quadratic ansatz. This could be due to the insufficient time step resolution
for the given discretization.
Remark
The same case is now solved using the hybrid formulation. The results for crack
propagation are shown in Fig.4.27. Two stagegred iterations could be insufficicent in
the case of the hybrid formulation to achieve equilibrium between the elastic and the
phase-field problems. However, this needs further investigation and is out of the scope
of this work.
p=
1
k=
5
p=
2
k=
5
Hybrid formulation
(a) t = 20µs (b) t = 50µs (c) t = 80µs
Figure 4.27: Dynamic crack branching under uniaxial tension. Multi-level hp-
refinement for different ansatz order with k = 5 using the hybrid formula-
tion. Crack phase-field evolution over time.
58
5 Summary and outlook
5.1 Summary
The phase-field model for quasi-static and dynamic brittle fracture has successfully been
integrated with a uniform multi-level hp-adaptive refinement scheme and the Finite
Cell Method for both the anisotropic Miehe and the hybrid formulations. Validation
of the implementation has been done through benchmark examples. A heuristic
refinement criteria to refine the mesh dynamically (over the course of the simulation)
has been developed to ensure h-refinement around the region with high gradients due
to the presence of crack. Taking advantage of phase-field models where the idea is to
have a diffusive crack which eliminates the necessity of algorithmic tracking of crack,
the heuristic criteria remains numerically simple and robust. Having developed the
adaptive refinement criteria, the benefits obtained from a local refinement scheme over
a global refinement one has been clearly established using the single-edge notched
shear test example. Importance of the number of staggered iterations used in solving
the two initial boundary value problems using a staggered approach has been studied
in the notched plate with hole example which also demonstrates the application of the
FCM for phase-field models. The dynamic uniaxial tension example is an evidence for
the robustness of the refinement criteria.
5.2 Outlook
The research in this work is not bounded. Undoubtedly, it provides a proof for the
applicability of the phase-field model in the context of uniform multi-level hp-adaptive
refinement and the Finite Cell Method but has also raised open questions. Following
addresses the same.
In the uniform multi-level hp-refinement scheme, the polynomial order of the
overlay meshes remain the same as the base mesh. This scheme has been
used in this work due to its robustness. However, crack has been studied as
a diffusive phenomenon in the phase-field model where the stress degradation
term is quadratic. Hence, it would be ideal to have a non-uniform hp-refinement
where the polynomial order is graded over the overlay mesh. This is done in
the non-uniform multi-level hp-refinement scheme. Implementation of such a
scenario remains open.
59
5 Summary and outlook
In the example where a concrete block subjected to compression is studied, the
behavior of the two formulations are not completely understood. This needs
further insight about the applicability of the formulations in such scenarios.
The results for dynamic crack branching using the hybrid formulations are not
the same as that from the anisotropic Miehe formulation. A further investigation
needs to be done in this regard.
In this work, a second-order phase-field theory with second derivative of the
phase-field has been applied in the strong form governing the fracture process.
Fourth-order theories with higher derivatives have been proposed to achieve
higher convergence rate when used with smooth bases. Hence, extension of this
work to a fourth order theory along with isogeometric framework could prove
beneficial.
Extension to three-dimensional problems, combining the elegant modeling capabil-
ities of phase-field models along with the above studied multi-level hp-refinement
technique to dynamically and locally refine the mesh only in the zone of interest
is a potential research area.
60
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