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Material and Computational Mechanics Group 1 Hardware Reliability for Microsystems - Mechanical Modelling Ragnar Larsson Material and Computational Mechanics Group Department of Applied Mechanics Chalmers University of Technology

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Page 1: Hardware Reliability for Microsystems - Mechanical Modellingragnar/micro_systems_home/literature/L1.pdf · Hardware Reliability for Microsystems - Mechanical Modelling ... Hardware

Material and Computational Mechanics Group 1

Hardware Reliability for Microsystems - Mechanical ModellingRagnar Larsson

Material and Computational Mechanics GroupDepartment of Applied MechanicsChalmers University of Technology

Page 2: Hardware Reliability for Microsystems - Mechanical Modellingragnar/micro_systems_home/literature/L1.pdf · Hardware Reliability for Microsystems - Mechanical Modelling ... Hardware

Material and Computational Mechanics Group 2

àGeneral purpose and contents

Main purpose (of this part): Give basic knowledge about some basic deformation phenomena and their modelling relatedto microsystem materials and components.

Note! Microsystem components subjected to high and sustained loading, e.g. cyclic temperature under normal working conditions.

In all real materials there are preexisting defects in microstructure fl Reduced strength corresponding to - inelastic deformation: creep, fatigue, fracture.

Constitutive models that describe these phenomena are reviewed, e.g. visco-elasticty,plasticity....

Computational aspects of constitutive modeling are emphasized!

àRequired knowledge to follow the course

Basic knowledge in strength of materials ....

Mathematics ...

Numerical analysis ....

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Material and Computational Mechanics Group 3

LEAssignment 2 (Extra time)2hLC

RLExtra time2hL5

LEAssignment 24hLC

RLPlasticity, yield criterion, flow rule, isotropic and kinematic hardening concepts.

2hL3

RLNon-linear viscoelasticity, transient and stationary creep, Norton’s creep law

2hL3

LEAssignment 12hP1

RLLinear viscoelasticity, creep relaxation, structural analysis

2hL2

RLIntroduction, Equilibrium of solids and kinematics, constitutive relations

2hL1

CommentsContentLecture

Schedule

Page 4: Hardware Reliability for Microsystems - Mechanical Modellingragnar/micro_systems_home/literature/L1.pdf · Hardware Reliability for Microsystems - Mechanical Modelling ... Hardware

Material and Computational Mechanics Group 4

àAdresses

Ragnar Larsson, tel. 772 5267, epost: [email protected] (lectures)Lisa Ekstrand, tel. 772 30 68, epost: [email protected] (problem classes)

Homepage: http://www.solid.chalmers.se/~ragnar/micro_systems_home/

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Material and Computational Mechanics Group 5

àLiterature

Ragnar Larsson, Hardware Reliability for Microsystems - Mechanical Modelling, Lecture notes (available via homepage)

Fundamentals of microsystems packaging, Tummala

àOrganization of lectures, problem sessions and lab classes

This part of the course comprises 8 hours of lectures (L1-L4 in the course outline), 2 hours of problem classes (P1), and 4 hours of computer lab (C1-2).

Lectures and problem sessions are located in rooms "ML" and the lab classes are located in rooms "MT", all in theM-building.

àAssignments: generic layered structure

Assignment 1: Elastic-thermal analysis of a microsystem interconnect

Assignment 2: Elastic-visco-thermal analysis of a microsystem interconnect

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Material and Computational Mechanics Group 6

L1. Characteristic behavior of solids

à Equilibrium and kinematics of solids: stress and strain

àDeformation and failure phenomena: Elasticity, inelasticity, creep, fatigue.

à The constitutive problem

üDifferent purposes and relevant models

üBasic question

üApproaches to constitutive modeling

ü "Typical" material behavior (metals and alloys)

üCharacteristics of material "fluidity"

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Material and Computational Mechanics Group 7

àEquilibrium and kinematics of solids: stress and strain

Consider mechanical relations for static, isothermal behavior of solid body:

Example: 1D problem

Kinematics: ε → u

Equilibrium: σ → f

σ↑ε

← f↑u

üMissing link: constitutive relation: σ → ε!

Example: 1D problem of elasticity

Consititutive relation of elasticity (Hooke's law): σ = E ε

E = Youngs modulus Nm2

Note! Material behavior is represented by constitutive relation under given conditons, e.g.temperature. No material is perfectly elastic!

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Material and Computational Mechanics Group 8

àDeformation and failure phenomena: Elasticity, inelasticity, creep, fatigue.

Important issues:

Temperature dependence

e.g. Enhanced creep in metals Reduced yield stressIncreased material ductility

Dependence on loading rate (strain rate):

increased strengthreduced ductility

Choice of constitutive model ?

Relevance ?Physical effect of interest (deform., life time ...)

Accuracy ?Application ? (building structure, microsystem component)

Computational aspects - complexity, reliability, costsHand calculation, commercial code ?

Note! Material behavior is essentially determined by material microstructure

Note again! Material behavior is represented by a constitutive model under given conditions - "a constitutive model is just a model".

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Material and Computational Mechanics Group 9

à The constitutive problem

üDifferent purposes and relevant models

Consider some concepts intuitively: elasticity, viscoelasticity, plasticity, viscoplasicity ...

Structural analysis under working load: Linear elasticity

Analysis of damped vibrations: Viscoelasticity

Calculation of limit load: Perfect plasticity

Accurate calculation of permanent deformation after monotonic cyclic loading: Hardeningelasto-plasticity

Analysis of stationary creep and relaxation: Perfect elastoviscoplasticity

Prediction of lifetime in high-cycle-fatigue: Damage coupled to elastic deformations

Prediction of lifetime in high-cycle-fatigue: Damage coupled to plastic deformations

Prediction of lifetime in creep and creep fatigue: Damage coupled to viscoplasticdeformations

Prediction of stability of a preexisting crack: Linear elasticity (singular stress field determinedfrom sharp cracks)

Prediction of strain localization in shear bands and incipient material failure: Softeningplasticity (or damage coupled to plastic deformation)

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Material and Computational Mechanics Group 10

üBasic question

Some phenomena and models listed above will be considered in the course!

Questions that should posed in regard to different model are:

- Is the model relevant for the current physical problem?

- Does the model produce sufficiently accurate predictions for the given purpose ?

- Is it possible to implement a robust numerical algorithm to obtain a truly operational algorithm?

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Material and Computational Mechanics Group 11

üApproaches to constitutive modeling

-Phenomenological approach (considered here!)

Continuum idealization of stress, strain, etc,Assumed homogeneous elementary testsNote ! microstructure processes represented by “internal” continuum variables

-Micromechanics (fundamental) approach

Control volume on micro structural scalee.g. steel (grains) 10-6 - 10-4 me.g. concrete stones 10-2 mmicromechanics considerations via homogenization Ø macroscopical relation

-Statistical approach

Variation of size, shape etc. specimen for “same”stress and strainMathematical distribution of strength

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Material and Computational Mechanics Group 12

üTypical behavior of metals and alloys

Consider 1) Monotonic loading

Creep and relaxation

- Temperature dependence- Identifiable stages with time fl

Strain rate dependence

- Static (slow) loading- Dynamic (rapid) loading fl “Higher stiffness and strength for larger loading rate”!!

Consider 2) Cyclic loading

Cyclic loading and High--Cycle--Fatigue (HCF)

- Elastic deformation (macroscopically) degradation of elasticity close to failure(Note! “weak” theoretical basis at present)

Cyclic loading and Low--Cycle--Fatigue (LCF)

Test modes: εa = const. or σa = const.

- Plastic deformation in each cycle

- Stages of fatigue process, stress control fl

fl Shakedown (=stabilized cyclic curve) or ratcheting behavior

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Material and Computational Mechanics Group 13

üCharacteristics of material "fluidity"

Creep: ε ≠ 0 when σ = 0

Relaxation: σ ≠ 0 when ε = 0

Stages of creep process:

- Transient (primary) ε decreasing- Stationary (secondary) ε ∽constant- Creep failure (tertiary) ε Ø ¶ when t → tRcf. bath-tub curve, Tummala

Creep/Relaxation

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Material and Computational Mechanics Group 14

Thanks for today!

Page 15: Hardware Reliability for Microsystems - Mechanical Modellingragnar/micro_systems_home/literature/L1.pdf · Hardware Reliability for Microsystems - Mechanical Modelling ... Hardware

Material and Computational Mechanics Group 15

àAssignments: generic layered structure

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Material and Computational Mechanics Group 16

StressesStresses and and strainsstrains, 1D, 1D

Small deformations:

εtrue = LogA LL0

E = L − L0L0

− HL − L0L22 L02

+ HL − L0L33 L03

+ O@L − L0D4

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Material and Computational Mechanics Group 17

Relation Relation betweenbetween loadload and and displacementdisplacement fieldfield, 1D, 1D

Equilibrium:

Constitutive model

Page 18: Hardware Reliability for Microsystems - Mechanical Modellingragnar/micro_systems_home/literature/L1.pdf · Hardware Reliability for Microsystems - Mechanical Modelling ... Hardware

Material and Computational Mechanics Group 18

Chip

Substrate

Interface with internal structures

k

lEk ici

12

2

Chip

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Material and Computational Mechanics Group 19

Types of constitutive modelsTypes of constitutive models

• Elasticity (reversible, time independent)ex. Hooke’s law, hyperelasticity: Neo-Hooke, Money-Rivlinmaterial: metals (small deformations), rubber

• Viscoelasticity (irreversible, time dependent)ex. Maxwell, Kelvin, Nortonmaterial: polymers, secondary creep in metals

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Material and Computational Mechanics Group 20

• Plasticity (irreversible outside elasticdomain, time dependent)material: metals

• Viscoplasticity (irreversible outside elasticdomain, time dependent)material: metals at high temperatures

TypesTypes of of constitutiveconstitutive modelsmodels, , cont’dcont’d

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Material and Computational Mechanics Group 21

Constitutive modelingConstitutive modeling

• Macroscopic (phenomenological) modeling: - Micro-structural processes represented as mean values of internal variables like plastic strain, damage.- Constitutive equations based on macroscopic experiments.

• Micromechanical modeling: - Representative volume of microstructure modeled in detail by mechanical models (e.g. crystal-plasticity).- Homogenization provides link to macroscopic level.- Computationally demanding

”The deeper you dive the darker it gets” Odqvist

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Material and Computational Mechanics Group 22

Plasticity 1DPlasticity 1D

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Material and Computational Mechanics Group 23

Isotropic/Isotropic/kinematickinematic hardeninghardening

Page 24: Hardware Reliability for Microsystems - Mechanical Modellingragnar/micro_systems_home/literature/L1.pdf · Hardware Reliability for Microsystems - Mechanical Modelling ... Hardware

Material and Computational Mechanics Group 24

Material behavior during cyclic (stress Material behavior during cyclic (stress controlled) load: controlled) load: shakedown shakedown -- ratchetingratcheting

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Material and Computational Mechanics Group 25

üCreep and relaxation behavior