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Integrated Computational Materials Science & Engineering (ICMSE )
Approaches to Problems with Evolving Domains
Workshop on Computational Methods for Problems with Evolving Domains and Discontinuities
AHPCRC, Stanford University, CA December 4-5, 2013
Somnath Ghosh Departments of Civil & Mechanical Engineering
Johns Hopkins University
Baltimore, Maryland USA
Integrated Materials Science & Engineering (ICMSE) Paradigm
J. Allison, D. Backman, and L. Christodoulou, "Integrated Computational Materials Engineering: A new paradigm for the global materials profession," JOM, pp. 25-27, 2006.
ICMSE philosophy “entails integration of information across length and time scales for all relevant materials phenomena and enables concurrent analysis of manufacturing, design, and materials within a holistic system”
Two Case Studies in the ICMSE Paradigm
1. Multi-scale model for ductile failure in heterogeneous metallic materials
2. Image-based modeling of fatigue failure in metallic alloys
Ductile Failure of Heterogeneous Metallic Materials
Automotive Engine Block
Microstructure: Cast Aluminum Alloy with Si Particulates and Intermetallics
Evolving Ductile Failure in Aluminum Microstructure
Stress-Strain plot showing ductility
• Ductile failure in heterogeneous materials typically initiates with
particle cracking or interfacial debonding.
• Voids grow near nucleated regions with deformation, and subsequently coalesce with neighboring voids to result in localized matrix failure.
• Evolution of matrix failure causes stress and strain redistribution in the microstructure that leads to ductile fracture at other sites.
• Eventually, the phenomena leads to catastrophic failure of the microstructure.
Fatigue in Aerospace Engine Materials
9 times during hold (2 min)
20 times during cycle
Stre
ss
time
1 sec 1 sec
Stre
ss
time
50 times during one cycle
9 times during hold (2 min)
20 times during cycle
Stre
ss
time
1 sec 1 sec
Stre
ss
time
50 times during one cycle
Dwell fatigue
Regular fatigue
• Crack initiation site is sub-surface • Initiation location depends on local microstructure • Initiation area is faceted with limited evidence of plasticity • Away from initiation site, crack growth is ‘normal’, i.e. striations • 2 min. dwell can lead to 2-10 x reduction in fatigue life
Effect of microstructure important in predicting fatigue life: e.g. nucleation at location of extreme values of grain morphology, orientation and misorientation, micro-texturing.
Structure-Material Interaction Challenges
• Modeling at the macroscopic scales cannot provide accurate
estimates of ductility and fatigue life • Lacks appropriate local geometric and thermo-mechanical
information of the incipient damage sites
• Modeling at the microstructural scales is computationally intractable
• Need appropriate multi-scale techniques in spatial and temporal domains that will uphold the efficiency of simulations, while not compromising the required resolutions
Adaptive Multi Spatial-Scale Modeling of Ductile Fracture in Heterogeneous Metallic
Materials
Case Study 1
• S. Ghosh, “Micromechanical Analysis and Multi-Scale Modeling Using the Voronoi Cell Finite Element Method”, CRC Press/Taylor & Francis, 2011, 729 pages.
• S. Ghosh and D. Paquet, “Adaptive Multi-Level Model for Multi-Scale Analysis of Ductile Fracture in Heterogeneous Aluminum Alloys”, Mechanics of Materials, (in press), 2013.
• S. Ghosh, J. Bai and D. Paquet, Jour. Mech. Physics Solids, Vol. 57, 2009.
• C. Hu, J. Bai and S. Ghosh, Modeling and Simulation in Materials Science and Engineering, Vol. 15, pp. S377-S392, 2007
Two-Way Coupled Adaptive Concurrent Multi-Scale Model
Modeling Error Introduce multiple-level hierarchy
Discretization Error Increase DOF e.g. by h-p-adaptation
RVE Homogenization
Level-0
B O T T O M
U P
T O P
D O W N
Localization
Level-1
Level-2
Physics-Based Reduced Ordered Models
Homogenization Theory-Based Swing Region for Error Analysis
Micromechanical Analysis in Critical Regions
Framework for Concurrent Multi-Scaling
1. Multi-Scale Characterization: Morphology-based Domain Partitioning and RVE Identification
2304 µm
48 µm
A
' ' '( ', ') ( ', ') ( ', ')g g ghrsm wvlt diffI x y I x y I x y= +
High Resolution Domain Reconstruction
Step 1. Wavelet interpolation of low res. images
Step 2. Correlation-based enhancement from limited high res. images
Recursive Refinement based on Morphological Characteristic Functions
Framework for Concurrent Multi-Scales
2. Micromechanical Analysis: Voronoi Cell FEM for Ductile Fracture
Optical micrograph
VCFEM
S. Ghosh, “Micromechanical Analysis and Multi-Scale Modeling Using the Voronoi Cell Finite Element Method”, CRC Press/Taylor & Francis, 2011, 729 pages.
VCFEM for undamaged particle
VCFEM for damaged particle
VCFEM for Matrix Cracking
( ) ( )
( ) ( )
( )
, , :
'
m c m c
e tm
c
ee
m m c c c
d d
d d
d
+ +Ω Ω
∂Ω Γ
∂Ω
∆ ∆ = − ∆ ∆ Ω − ∆ Ω
+ + ∆ ⋅ ⋅ ∆ ∂Ω − + ∆ ⋅ ∆ Γ
− + ∆ − − ∆ ⋅ ⋅ ∆ ∂Ω
∏ ∫ ∫
∫ ∫
∫
σ u B σ σ ε σ
σ σ n u t t u
σ σ σ σ n u
( )cr
c c cr d∂Ω
′′− + ∆ ⋅ ⋅∆ ∂Ω∫ σ σ n u
( )( )
, :s s
s
s s s s
s s
d d
d
Ω Ω
∂Ω
+ ∆ ∆ Ω + ∆ Ω
− + ∆ ⋅ ⋅∆ ∂Ω
∫ ∫∫
A σ ε σ ε
σ σ n u
VCFEM for damaged particle
VCFEM for Matrix Cracking
Stress function: /m m m c
poly recΦ = Φ + Φ c cpolyΦ = Φ/m cr
rec+Φ /c crrec+Φ
Voronoi Cell FEM Formulation for Particle and Matrix Cracking
VCFEM for undamaged particle
For local softening in stress-strain response: Higher-order displacement interpolated regions is embedded in the stress-interpolated VCFEM domain C. Hu and S. Ghosh, IJNME, 2008.
Assumed Stress-Hybrid FEM
Non-local void growth rate
Particle Cracking Nucleation: Weibull distribution based crack initiation criterion
Microstructural Particle and Matrix Cracking
Matrix Cracking Nucleation: Gurson-Tvergaard-Needleman (GTN) type Models
( )2
* *221 3
0 0
32 cosh 12σ σ
Φ = + − − +
q pq q f q f
( ) (1 )p p pnucleation growth kkdf df df A d f dε ε ε= + = + −
( )* *
c
u cc c c
F c
f for f ff f ff f f for f f
f f
≤= − + − > −
( ) ( ) ( ) ( )1localv
f f w dVW
= −∫x x x xx
( ) ( ) ( )2
21 12
32 cosh 1 0
2φ
Σ Σ = + − − =
eq hyd
f p f pQ f Q f
Y W Y W
( ) ( )1 pkkf f e A e e= − +
( )( ) ( )( ) ( )( ) ( )2 222 2eq p yy zz p zz xx p xx yy p xyF W G W H W C WΣ = Σ − Σ + Σ − Σ + Σ − Σ + Σ
Anisotropic yield surface in the GTN model
11 ;
(1 )= Σ = Σ + Σ + Σ
−
hyd xx yy zzinclusion
QQf
F, G, H and C: Anisotropic YS parameters calibrated from homogenization of micromechanics in principal material-damage coordinates
( ) ( ) ( ) ( )1localv
f f w dVW
= −∫x x x xx
3. Macroscopic Modeling: Homogenized Continuum Model for Plasticity and Damage Evolution with Heterogeneities
Framework for Concurrent Multi-Scales
Validation of Macroscopic HCPD Model
Stress–strain response by HCPD model and micromechanical solutions for non-proportional loading.
Evolution of anisotropy parameters F, G, H for RVE with 40 inclusions
Framework for Concurrent Multi-Scaling
3. Macroscopic Modeling: Homogenized Continuum Model for Plasticity and Damage Evolution with Heterogeneities
u
RVE1 RVE2 RVE3 RVE4 RVE5
xxΣ(GPa)
Void volume fraction
1 1 1 1 1 11 2 int
0δ δ δ δ δ δ+ + + + + +Ω Ω Ω Ω ΓΠ = Π + Π + Π + Π + Π =n n n n n n
het lo l l tr
Level-0
RVE Level-1
Level-2/tr
Coupling microscopic and macroscopic sub-domains using Relaxed Constraint method.
Framework for Concurrent Multi-Scaling 4. Adaptive Multi-Level Modeling: For Coupling Multiple Scales in Simulating Failure
, Obtained from LE-VCFEM.
, Obtained from the FEM implementation of the HCPD model.
Micromechanics simulation
Adaptive multi-scale simulation
Horizontal normal stress component σxx (GPa)
Adaptive Multi-level Model
Evolution of adaptive multi-level mesh
Uy=0.0μm Uy=7.8μm Uy=13.0μm Uy=13.7μm
Underlying microstructure and microscopic stress σyy (GPa)
Uy=13.0μm
Level-2/tr
Level-0 Level-1
Sealed
Adaptive Multi-level Model Tensile Deformation of Micro-specimen
Image-Based Modeling and Multi-Time
Scaling for Fatigue Problems in Ti and Mg Alloys
Case Study 2
• S. Ghosh and D. Dimiduk , “Computational Methods for Microstructure-Property Relations”, Springer NY, 2011, 790 pages.
• S. Ghosh and P. Chakraborty, Int. Jour. Fatigue, Vol. 48, pp. 231-246, 2013. • M. Anahid, M. Samal and S. Ghosh, Jour. Mech. Physics Solids, Vol. 59, 2011. • G. Venkatramani, S. Ghosh and M.J. Mills, Acta Materialia, Vol. 55, 2007. • D. Deka, D.S. Joseph, S. Ghosh, and M.J. Mills, Met. Mater. Trans. A, 2006.
• Image Based Crystal Plasticity FEM with Experimental
Validation
• 3D Polycrystalline Microstructure Simulation
• Fatigue Crack Initiation in Dwell Studies
• Multi-time Scale Models in Crystal Plasticity
Important Steps
1. Rate Dependent Crystal Plasticity
e=S EC
2 20
02( )
β βα αβ β β
α αβ β
αγ λ γ• • •
= +−∑ ∑
SSD GND
k G bg q hg g
0 1 1r
s s
g gh h signg g
β ββ β
β β
= − −
( )2 00
0
ss a s
s
h hh h sech h hβ β
β β β ββ β γ
τ τ −
= + − −
* p=F F FKinematics
Constitutive Relations
Flow rule
Slip System Deformation Resistance
Self Hardening Evolution (hcp)
Self Hardening Evolution (bcc)
, ( S)eT eeff kinα α α ατ τ τ τ= − ≡ ⊗: m nα αF F
1 m
eff signg
αα α
α
τγ γ τ
• •
=/
( )
α α α αkin kinτ cγ - d τ γ =
Back-stress Evolution
sα
mα
sα
mα
Fp
F* F
αα α= +
o
Kg gD
Grain size effect on Deformation Resistance
Acharya and Beaudoin, 2000
Experimental Data Processing
(ii) Statistically Equivalent Distribution and Correlation Functions
2. Methods of Virtual Microstructure Simulation
(i) CAD-Based Adaptive Non-Uniform Rational B-Spline (NURBS) functions for GB
Bhandari, Ghosh et, al. 2007 Groeber, Ghosh et. al, 2008
1-3. Image-Based Crystal Plasticity FEM Model for Ti-6242 Microstructure
Schmid Factor along a section
Local stress along a section
Deformation Twinning in Magnesium: Initiation and Evolution
Crack initiated from twin-grain boundary intersection.
Zigzag crack propagation at twin-twin interactions
crack propagation along twin boundaries
Micro-crack formation along twin boundaries
SEM observation of twin boundary cracking in fatigue test of Mg.
Twinning accumulation in fatigue test (D.K. Xu, E.H. Han, Scripta. Mat. 2013)
𝟏𝟏𝟓 cycles 𝟏𝟏𝟔 cycles
(Q. Yu et al, Mat. Sci. Eng. A, 2011)
Micro-crack formation was observed in fatigue samples in both twin-grain boundary intersection and inside grains along twin boundaries.
𝐸𝑖𝑖𝑖 → 𝐸𝑡𝑡 + 𝐸𝑟 + 𝐸𝑖𝑖𝑡 + 𝐸𝑓𝑓𝑓𝑓𝑡 −𝑊𝑒𝑒𝑡(𝜏)
Nucleation criteria: 𝐸𝑠𝑡𝑓𝑠𝑓𝑒 < 𝐸𝑖𝑖𝑖; 𝑑𝑠𝑡𝑓𝑠𝑓𝑒 > 2𝑟0
𝑡𝑡 = 𝜌𝑡𝑡𝑏𝑡𝑡𝑙𝑡𝑡𝑓exp∆𝐹 − 𝜏𝑉∗
𝐾𝐵𝑇
Twin dislocation propagation rate:
Energetic criteria of dislocation dissociation for twin nucleation
Modeling Deformation Twinning
Model for twin nucleation & propagation
CPFE simulation results
Twin formation mechanism
twin formation layer-by-layer twinning dislocations
Load Shedding Leading to Fatigue Crack Initiation
Soft grain (High Prism Schmid factor)
Dislocation pile up near the boundary of hard and soft grains
Hard grain (Low Prism Schmid factor)
Stress concentration in near boundary
3. A Nonlocal Crack Nucleation Model
2
8 (1 ) s
Gc Bπ υ γ
=−
2 2βπ
= + ≥ ⇒ceff n t
KT T Tc
= ≥eff cR T c R
/ π=c cR KSingle parameter to be calibrated
Models relating crack length c and opening B
• Micro-crack in hard grain due to dislocation pileup in soft grain
• Crack opening displacement corresponds to the closure failure along a Burger’s circuit surrounding the piled-up dislocations
• Traction across the micro-crack tip in hard grain opens up the crack.
Slippla
ne
c
Grain boundary
B=nb
T t
Tn
Pileup
length
Stroh (1964)
Experimental Calibration & Validations Dwell fatigue tests on Ti-6242
Test No
No. of cycles to crack initiation (experiments)
No. of cycles to crack initiation (simulation)
% Relative error
Calibrated at 80% life Calibrated at 80% life
I 550 cycles 620 cycles 12.7%
Microscopic features of predicted location of crack initiation
Experimentally observed
Sample 1 Sample 2
‘c’ axis orientation 0 - 30o 38.5o 25.2o
Prism Schmid factor 0 - 0.1 0.17 0.09
Basal Schmid factor 0.3 - 0.45 0.48 0.38
Crack Propagation in Crystalline Materials from MD Simulations
1. Characterization and Quantification of Mechanisms in Molecular Simulation
Dislocation Extraction (DXA) Deformation gradient for twins Crack surface
Dislocation DXA Dislocation density, Burgers vector Twin Deformation gradient Twin volume fraction
Crack surface Equivalent ellipse Crack length, opening
Dislocation segments colored by magnitude of Burgers vector
• Dislocation motion blunts crack tip
• Cross slip observed
• Dislocation from different slip systems interact forming immobile junctions (stair-rod dislocation)
Dislocation Evolution
Strain-Strain Response with Mechanisms Evolution
Energy Balance: 𝒅𝒅 = 𝒅𝑼𝐞𝐞 + 𝒅𝑼𝐢𝐢𝐞𝐞 + 𝒅𝒅
𝒅𝒅: work done by applied force 𝒅𝒅 : generated heat 𝒅𝑼𝐞𝐞 : elastic strain energy that can be recovered by unloading 𝒅𝑼𝐢𝐢𝐞𝐞: inelastic strain energy not recoverable, related to defect energy
Crack Evolution
4. Multi-Time Scale Modeling for Fatigue Analysis
Nf = 11,718
Nf = 43,180
Nf = 20,141
Nf = 24,241
Fp
F
Fe
sα
mα
sα
mα
Computational requirements for cycle by cycle complete polycrystalline microstructural fatigue analysis is prohibitive Extrapolation (often pursued) is grossly inaccurate
Field Data on Fatigue Life
Wavelet Decomposition of Nodal Displacements
( , ) ( ) ( )kk
ku N c Nτ ψ τ= ∑
Coefficients of wavelet basis •Depends on coarse cycle scale (N) •Independent of fine scale.
Wavelet basis function •Fine scale (τ) behavior •Independent of coarse scale (N)
Wavelet Transformation Based Multi-Time Scaling Methodology (WATMUS)
Haar Wavelet
Dilation
Translation
Multi-resolution basis functions: Translation and Dilation
• Compact Support : No spurious oscillations from truncation, e.g. Gibbs's instabilities • Orthogonal: Daubechies family • Multiresolution transformation: Space of basis functions for a resolution is well defined and finite. Reduced number of coefficients to characterize a waveform • Non-periodic: • Works for R=-1.
Properties and Advantages
21 0 10 ...... ... ( )mV V V L R− +⊂ ⊂ ⊂ ⊂
Higher Resolution Lower
Resolution
Wavelets and Multi-resolution
21 o 1 m,n m
n..... V V V ....... L ( ) with span V−⊂ ⊂ ⊂ ⊂ ⊂ φ =
Projection to Vm : Approximation of function at m-th resolution
2, 2 (2 )
mm
m n nφ φ τ= −Scaling Function
Dilation Translation
Orthogonal Basis for Wm (through translation) : ψ(τ) :mother wavelet 22 (2 )m
mmn nψ ψ τ= −
Detail space Wm : Orthogonal difference between resolutions m & m+1 1m m mVV W+ = ⊕
( ), ,( ) m nm n
m nf Cτ φ τ= ∑∑
Scaling Function Mother Wavelet
Daubechies-4 Wavelet, N=4
Basis functions: Square integrable functions projected into nested subspaces of varying resolution Vm
Evolution of State Variable Change of State Variable in a Cycle: Cycle Rate of Change of Cycle Scale State Variable (Coarse Scale Derivative): (independent of τ)
Coarse Scale Evolutionary Constitutive Equations: (Integration Point)
00
0 00
( ) ( , ) ( )
( , ( ), ) ( ( ), ( ))ε τ τ ε
= −
= =∫T
k k
dy N y N T y NdN
f y N d Y y N N
• Wavelet transformed of nodal displacements:
• Deformaton gradient
( , ) ( ) ( )ki i k
ku N c Nα ατ ψ τ= ∑
( , )m kij ij m i k
jT T
NF F t d c dX
αα
α
τ ψ τ ψ τ∂= =
∂∑ ∑∫ ∫
Element Level:
Wavelet Transformed Multi-Scale Methodology
( , ( )) ( , ( ), )ky f y t f y Nε ε τ= =( , , , )paccF gα αχ γ
00
( , ) ( ) ( , ( ), )τ
τ ε τ τ= + ∫ ky N y N f y N d
Adaptivity in the WATMUS Algorithm 2 Errors Sources
1. Truncation of higher order terms O(∆N3) in the numerical integration of coarse scale equations (ε0
p , g0) 3 2 3
33 2
1 ( 1) ( 1)6 ( 1) 1
prevNd y r r N rdN r N
∆+ − += ∆ =
+ − ∆
2 3 33
2 31 ( 1) ( 1), max6 ( 1) 1
el
pok k T
trunc trunc trunc poelV
d r r d Ff N dVdF r dN
σ + − +≤ ∆ =
+ −∫ B
Truncation Error from Residual in Equilibrium Equation due to Constitutive Integration
13
maxη
∈
∆ =
evol
truncstep k
trunck
fN
2. Ignore slowly varying displacement coefficients to reduce size
| / |kdc dN η≥-
evol evol add
non evol evol−
= ∪
=
Solve the element equilibrium residual components for evolving coefficients
Fp 0,2
2
Fp0,22 at a material point Fp
0,22 in the microstructure
WATMUS – Coarse and Fine Scale Response
σ22 at 300000th cycle
Evolution of stress along a material line with cycles
Summary
Conventionally implemented phenomenological models lack robust underlying physics-based mechanisms with little relation to actual micromechanical features.
Coupled with advanced modeling capabilities, provide the foundations for predictive science and technology with consequences in material design and processing to endure demanding mission profiles with improved reliability.
Comprehensive approaches, taking advantage of the emerging frontiers in computational and experimental science and engineering, are necessary for addressing this critical challenge.