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.