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Material and Computational Mechanics Group 1
Hardware Reliability for Microsystems - Mechanical ModellingRagnar Larsson
Material and Computational Mechanics GroupDepartment of Applied MechanicsChalmers University of Technology
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 ....
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
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/
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
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"
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!
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".
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)
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?
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
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
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
Material and Computational Mechanics Group 14
Thanks for today!
Material and Computational Mechanics Group 15
àAssignments: generic layered structure
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
Material and Computational Mechanics Group 17
Relation Relation betweenbetween loadload and and displacementdisplacement fieldfield, 1D, 1D
Equilibrium:
Constitutive model
Material and Computational Mechanics Group 18
Chip
Substrate
Interface with internal structures
k
lEk ici
12
2
=θ
Chip
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
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
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
Material and Computational Mechanics Group 22
Plasticity 1DPlasticity 1D
Material and Computational Mechanics Group 23
Isotropic/Isotropic/kinematickinematic hardeninghardening
Material and Computational Mechanics Group 24
Material behavior during cyclic (stress Material behavior during cyclic (stress controlled) load: controlled) load: shakedown shakedown -- ratchetingratcheting
Material and Computational Mechanics Group 25
üCreep and relaxation behavior