Wing Kam Liu* and Wei Chen, Northwestern University ... - SAMSI

Mathematical Science Methodologies for Materials by Design
Microspan Multiresolution Theory for the
Failure of Heterogeneous Microstructures
Statistics and Uncertainty Quantification
Wing Kam Liu* and Wei Chen, Northwestern University
Donald Estep, Colorado State University
*Walter P. Murphy Professor of Mechanical Engineering
Founding Director , NSF Summer Institute on Nano MechanicsMaterials and Nano/Micro Nano/Micro
Manufacturing
Founding Chair of the ASME NanoEngineering Council
World Class University Professor at SKKU
w‐liu@northwestern.edu
liu@northwestern.edu
http://www.tam.northwestern.edu/wkl/liu.html
http:// www.tam.northwestern.edu/wkl/liu.html
COLLABORATORS: KhaliI I. Elkhodary, Shan Tang, M. Steven Greene,
Adrian Kopacz, Stephaine Chan, Greg B. Olson, Ted Belystchko

Objectives
Mathematical Science (Multiresolution Mesoscale
Continuum) Methodologies for the Analysis and Design of
Multiscale/Multirate Microstructured Materials
•Facilitate the extraction of structure/property/performance
metrics for multiscale/multirate materials design
•Predictive dynamic modeling of fracture nucleation and
propagation due to contributions from all COUPLED scales
•Set up an approach that is valid across materials systems,
that is MATERIALS GENERIC field equations, and MATERIALS SPECIFIC
coupled multiscale constitutive laws
•Uncertainty Sources and Quantification
•Wish List for Multiscale Methodologies and UQ Discussions Topics

Multiresolution Modeling Scope: Processing/Structure/Property

Mathematical Science Foundation
Mathematically link spatial and temporal scales for continuous resolution of a
microstructure
Separate Data by Scales
with Help from
Materials Science
Quantification &
Variability of Data
Sets
Scale 1
Data
Refined
Scale 1
Data
Large Stochastic,
Multi-Scale Data Set
Scale 2
Data
Refined
Scale 2
Data
…
…
…
Scale n
Data
Refined
Scale n
Data

Mathematical Science Foundation, Continued
Reconstruction
of Data Set
Scale linking via testing and characterization of samples.
Refined
Scale 1
Data
Scale
Link
Refined
Scale 2
Data
Refined Data in Order of Scale
Refined
Scale n
Data
Keep in mind that the data is in 4D, (x,y,z,t), so the spatial and temporal
resolution axes are the 5 th and 6 th dimensions. Now the microstructure is
shown in continuous resolution.
To represent the stochastic nature of the data, another dimension will be
added for each parameter with an axis as the variance of that parameter.
…
Scale
Link
Resolution
Axes

Multiresolution Concept
t0
t
f
Macroscale Demonstration
Ice “Reinforced” Ice
Dropped Dropped
Newspaper
Fine resolution
material morphology
Structure dictates the bulk
properties
Properties dictate the bulk
performance

Multiresolution Concept: zoom into a microstructure in the same way that
modern satellite technology allows us to zoom into an image of the earth
a.
Small Number of pixels
per km 2 – suitable for a
global image
TiC
~5 um
0
u
Small Number of
degrees of freedom –
suitable for average
behavior
b.
Increasing Image Resolution
More resolution – cities
can be observed
1
u
Discrete behavior of TiN
primary inclusions
begins to be observed
Increasing Constitutive Resolution
c. d.
Increasing resolution –
buildings can be observed
~1 um ~0.1 um 10 nm
u
Discrete behavior of
TiC secondary
particles is observed
2
Northwestern University
- a fine institution
Macro Micro TiN Sub-Micro TiC Nano
3
u
Very fine resolution – material
interface is observed

Polymer Nanocomposite Material Design
Goals.
(1) Increase transportation materials performance,
(2) ballistic protection, and
(3) fracture toughness
Atomic bonds visible Aggregated structure,
interface morphology
clear
Random atomic
phase-space
Random secondary
microstructure,
interface morphology
Filler network visible,
detailed shape
information lost
Random aggregation,
mesostructure
Design materials: multiple monomers, fillers, interphase, interface
Filler shape invisible,
statistical homogeneity
Random primary
substructure, statistic
homogeneity
Increasing resolution, decreasing spatial scale, increasing uncertainty
Experiments at various spatial and temporal scales and its statistical analysis are important
Homogeneous medium
Random operating
conditions
Co-polymer design, interface and interphase design, experimental imaging at various scales (V2, V1)
The above is for matrix materials, now we need to add in structural materials to become polymer
matrix composites

High Strength Alloy Analysis & Uncertainty
Goal. Deliver high strength alloys for transportation applications
Primary ( microns) and Secondary (tens of nanometers) particles design to control strength & toughness
Quantum-scale design of interfacial strengths between particles and matrix
Random operating
(boundary) conditions
TiC
Small Number of degrees
of freedom – suitable for
average behavior
Primary microstructure
random
~5 μm ~1 μm ~0.1 μm 10 nm
Discrete behavior of TiN
primary inclusions begins
to be observed
Secondary microstructure
random
Macro Micro TiN Sub-Micro TiC Sub-Nano
Increasing resolution, decreasing spatial scale, increasing uncertainty
Discrete behavior of TiC
secondary particles is
observed
True system size: modeling only 1,000 primary particles, (about 2,500,000) secondary
particles > billions DOFs
Feasible system size: 120 primary particles, >30 million DOF
Destructive evaluation testing dictates the importance of statistical analysis
Random atomic phasespace
Very fine resolution – material
interface is observed

Multiscale Modeling & Simulations
Multiscale Continuum Strata
Computational
Theories
Scale-Bridging Theory
KEY
Control Flow
Information Flow
Atomic
Level
Free
Particle
Dislocation
Walls
Free
Atom
Space‐Time
Discretization
Specimen
Level
interface
Point
Defect
Void
Coalescence
Shear
Band
interphase Secondary
Phase
Dislocation Ensemble
of Defects
Collective
of Excitation
High Performance
Computing
Uncertainty
Multiresolution Theory
Fields
Rings
Groups

Equations of Archetype Multiresolution Theory
Sets of Equations
(1) Kinematics
(2) Kinetics
(3) Constitutive
(4) Compatibility
(5) Fracture
(6) Uncertainty
materials generic
Multiscale:
Lagrangian,
Computational Theory
materials specific
To be derived for all “building
blocks” as modules in
multiscale framework
Balance
Laws
Reference
Configuration
Constitutive
Laws
Compatibility
Laws
Final
Configuration
Deformation
Crack tip

Microstructural Hierarchy and Behavioral Complexity

More on the Theory: Motion Assumptions
Microstructural Kinematics
l*
*
( ( ), , Φ) ≡ ( ( ),0, Φ ) + ∫ ∂α ( ( ), , Φ)
0
x� p ξ l x� p ξ x� p ξ α dα
Beginning from the fundamental theorem of calculus on the second argument…
Meaning of the first term
�x(p(ξ),0,Φ)
Local (intrinsic) point behavior
Meaning of the second term
l*
∫ ∂ x� α ( p( ξ), α, Φ)
dα
0
Corrected influence at p due to
the inhomogeneous neighborhood
x
p

More on Kinematics…
Elemental Kinematics
The effective microstructural rate can be broken down as…
L eff 0 ( p(ξ),x(α ))
( p(ξ)) = Lij + Lij
ij
x(a 1 )
Each term representing
a relative rate for its span
or length scale, hence the
multiscale‐multirate double decomposition
x(0)
( p(ξ),x(α ))
+ Lij x(a 2 )
x(a 1 )
x
( p(ξ),x(α ))
+ ...+ Lij x(a n )
x(a n−1 )
p

More on the Theory: Variational Statement
E� ≡ S L + s L∇ dΩ
∫
1 M M
E� kin =∂ ∫ t ρv
⋅v dΩ
2 Ω
E�= ∫ f ⋅v dΩ
V M
imp i i
Ω
E�= ∫ t ⋅v dΓ
S M
imp i i t
Γ
t
Elemental Kinetics
( M M )
def ij ij ijk ijk
Ω
δE� − δE� = δE�
imp def kin
Elemental Domain ξ
Expanding V M and L M will express all elemental
Fields in terms of microstructural quantities …

More on the Theory: Strong Form
Euler‐Lagrange
Equations
σ −Σ + f = � γ
ij, j ijk , kj i i
Boundary Conditions
Σ ijk n k
= 0
σ ij n j −Σ ijk ,k n j = t i
Discretized Strong Form
δE� − δE� = δE�
imp def kin
[ ]
ρ ρ ρ
S ijk , k − s + B =Γ�ij , for ∀ρ∈ 1,..., n
ij ij
ρ
S ijknk = T ij
for ∀ρ ∈ 1,...,n ⎡⎣ ⎤⎦ Stress conjugate to relative deformation rates
Constitutive laws are needed for all archetypes…
x
p

Three Invariants (Strain rates, Inhomogeneity Length Scales,
Deformation Length Scales) Fracture Space in the Multiscale Context
Maximal stressed zone (m)
Strain rates (s ‐1 )
Materials-Generic Multiresolution Fracture Criterion
10 −2
10 −3
10 −4
10 −5
10 −6
10 −7
10 −8
10 −9
10 −10
10 2
10 3
10 4
10 5
10 6
10 7
10 8
10 1
Local Plastic
Deformation
10 9
Crack tip
Statistical Description
of Structural Inhomogeneity
Structure
Fragmentation
1 st Order Differential Form
Integral Form
(a) (b)
Essential Description
of Structural Inhomogeneity
10
Structural inhomogeneity (m)
−8
10 −7
10 −6
10 −5
10 −4
10 −2 10 −9
10 −3 10 −10
Dimension
(m)
Objects Realizing
Stressed Zone Size
Objects of Structural
Inhomogeneity
10 ‐2 Testing Sample Reinforcement
10 ‐3
10 ‐4
10 ‐5
10 ‐6
Zone at
Impact test
Zone at Fatigue
Fracture Samples
Zone at Cavitation.
Macrocrack
Mechanical Activation
by Processing
10 ‐7 Zone near Crack tip
Coarse Lamellar
Structures. Explosion
Fragments
Coarse Grains
Inclusions
Mn‐Dispersoids in
Aluminum
Al 2 Cu Precipitates
in Aluminum
10 ‐8 Zone at Irradiation Carbides in Steel
10 ‐9
Strain Rate
(s ‐1 )
Crystal Defects.
Nano particles.
Physical
Mechanisms
10 ‐3 Diffusion
10 3 ‐10 6 Wave interactions
10 9
Curl(F M ) = κ
M e p p
*
∫� ( F − R R U ) dx= δ
Shock Waves &
EOS
Atomic lattice
Deformations
Phenomenological
Inhomogeneity
Isothermal shear band
Void nucleation & growth
Adiabatic Localization of
shear, Spalling
Fragmentation, Pulverization
(c)

Multiscale Constitutive Physics of Microstructures
Schematic: Complex
Microstructure
(Al - 2139)
Volume Fractions
Phase Spacings
Phase Clusters
Variant Multiplicity
Residual Stresses
Interfaces and GBs
Porosities & Defects
Mass Densities
V1
V2
Structure and Property Metrics
Edge Crack
Model
Crystallography
� t
� n
� s
Slip System Multiplicity
Habit Planes, Orientations
Deformations
Grain Texture
Elastic/ Plastic Anisotropy
Lattice Curvature
Lattice Stretch, Rotation
Dislocation Densities

Illustrative Example: Two-Scale Elasticity
Dominant Edge Crack
Aluminum Matrix (Usual DOFs)
20% precipitates (Additional DOFs)
Stationary Crack, 5μm× 5 μm,
� ε ε
−1
nom = 20,000 s , nom = 14%
20% Volume fraction of precipitates
Illustration of Design of Materials
Strengthening Mechanisms
(a)Spatial Patterns (Normal Stress)
(b)Magnitudes & Signs (Shear stress)
(c)Curvatures, Nonlocality (Stress couples)
Nanosized
Ω Precipitate
Aluminum
Matrix

Illustrative Example: Two-Scale Elasticity, Continued
S 122(Imbedded Crystal) Σ 122(Matrix Crystal)
[ S, Σ]/
Eb
Stationary Crack, 5μm× 5 μm,
� ε ε
−1
nom = 20,000 s , nom = 14%
Materials Design Implications:
Matrix gradient term (curvature, nonlocality) is important for local fracture processes…
accounts for lattice curvature and accommodation processes

Two-Scale Concurrent Multiresolution Model for TiC (Secondary Particles) - Matrix
and Direct Numerical Simulation (DNS) of TiN (Primary Particles)
crack
Mod4330 Crack tip specimen #1:
Primary particles (yellow) near crack tip
Reconstruction area: 633x516x259 μm 3
Averaged: 2.857 μm
Spacing: 16 μm
Volume fraction: 1.756%
Ultra High Strength Alloys
Primary Inclusions
Primary Voids
Oxides
Submicron Carbides
Microvoids
Shearing
Mod4330 shear specimen:
Secondary particles (white dots) inside shear band
Reconstruction area: 70x15x28μm3 Averaged: 0.117035 μm
Spacing: 2 μm
Volume fraction: 0.0384%
Courtesy of Stephanie Chan (Olson’s group), and HJ Jou (QuesTek)
McVeigh & Liu, Linking microstructure & properties through a predictive multiresolution continuum, CMAME, 2008.
McVeigh & Liu, Multiresolution modeling of ductile reinforced brittle composites. J. Mech. Phys. Solids, 2009.
Liu et. al., Complexity science of multiscale materials via stochastic computations, IJNME, 2009.
Tian & Liu et al., “A Multiresolution Continuum Simulation of the Ductile Fracture Process,” JMPS (2010)

Multiscale Materials Modeling, High Performance Alloy
crack
Fracture test
Tomography
Fatigue
crack tip
Mod4330 crack tip specimen #1:
Primary particles (yellow) near crack tip
Reconstruction area: 633x516x259 μm 3
Averaged: 4.857 μm
Spacing: 16 μm
Volume fraction: 1.756%
Fracture Process Model
to be Validated
3μm
Large scale simulations even when blending TiC particles
Degrees of Freedom:
30 million
(132 particles)

Summary:
comparison with experiments
Experiments
Simulation
Pre‐crack
Shear bands
Mixed mode I/II fracture
Primary voids
Crack opening distance along fracture path
Experiments
Simulation
Crack Opening Displacment (μm)
Fracture path
20
15
10
5
0
-5
Fracture path
Experiments
Simulation
0 50 100 150 200 250 300 350
X μm
Experiments
Simulation
Contour plot of crack opening distance
Experiments
Void growth
Simulation
Final void radius (μm)
25
20
15
10
5
Δa
PZ
0
0 1 2 3 4 5 6 7 8 9 10
R0 inclusion radius(μm)

Formation of Multiscale and Multistage Fracture Process
Macro deformation (contour of macro effective strain)
(a) Void growth
(primary voids)
Micro deformation (contour of micro effective strain)
(a) Development of
micro deformation
(b) Intense deformation at
ligament between voids
(b) Microvoid sheeting (c) Microvoid shear
coalescence
(c) Initial macro crack (d) Macro crack propagation
Arising from the Strong Interaction between Micro and Macro Deformations
(d) Further Microvoid shear
coalescence
I. Multiresolution modeling across multi‐scale of microstructures (voids)
II. Concurrent modeling considering interaction across microstructures has been achieved
(Though there are many primary and secondary particles involved)

Animation of the Fracture Process
Initial Fatigue Crack Opening is about 3 microns, a load is
applied gradually until the material reaches the initial and then
final fracture toughness strengths followed by unloading to the
final state of deformation
Total experimental time is one minute
The below MOVIE shows the CONCURRENT interaction of macro and micro effective
strains due to submicron and micron voids growth, interaction, coalescence to form
Micro and macro defects (mini-cracks) that eventually define the fracture process zone

Examples of Archetypes and their Conformations
ALLOYS POLYMERS CERAMICS
PMC
Multiscale Phenomenology
Shear banding Pulverization
Cavitation Fragmentation
Conformation of archetypes (sub‐mesoscale)
Sub‐granular structures CNT through Epoxy Resin Polycrystalline Structure Layered Structure
Cortical Bone Lamellae
Archetypes (Space‐filling units)
Crystals
Polymer Chains AlON
CNT MicroCrack
Polymer
Nanocomposite
Fibers
BONES
Collagen Fibrils
Fracture
Path

Northwestern University Materials Design Loop
METRICS
Structural
Behavioral
Validation
Validation
Concurrent Mesoscale Physics
Multiscale Experiments & Characterization
Multiscale Modeling & Simulation
Scale‐
Bridging
Multiresolution Constitutive Coupling
Direct Numerical Modeling & Simulation
QM
MD
Computational
Theories
Dislocation
Dynamics
Advanced Experiments & Characterization
Sub - mesoscale Physics
High Performance
Computing
Phase
Fields
27

Multiphyics Extensions
Multiresolution
Mesoscale Manifold
Nonlocal
Conformation
Nanomorphic
Archetypes
Blending
Implanting
Cmcm I 4 / mcm I 4m2
Classical Forms Multiresolution
�
( q−p) i∇+
Q = μθ�
� �
pi∇− p+ R= � θ∇iI
Mechanical Balance
Thermal Balance
Electrodynamic Balance
Mass Diffusion Balance
N ⎛ n ⎞ �
⎜q−∑p ⎟i∇+
Q = μθ�
⎝ ⎠
p i∇− p + R = � θ ∇iI
n=
1
� �
n n n n n

Uncertainty Sources and Quantification
Inherent structural variation (batch to batch randomness, random field)
Uncertain multiscale boundary and initial conditions
Random parameters determined by experiment or behavior at other scales
Lack of knowledge, lack of information
Low fidelity physical descriptions
Sparse and limited experimental data
Missing models and/or data for sub-processes
Simulation and approximation errors
Discretization and finite sampling errors
Solver errors
Statistical modeling and representations
Model Complexity
STOCHASTIC PDE’s
Interactions between different sources of errors and uncertainty
Uncertainty feedback in coupled processes

General Stochastic Formulation
Write general partial differential equation governing evolution of some
system
( Ωℑ , ,P)
Ω
[ ] ( ,, t ω) [ ] ( ,, t ω)
Tvx + S+ S vx = 0
Temporal operator
( t ω)
[ B+ B ] v x, , = 0
ω
Probability space
Could be infinite dimensional
Governing System of Equations
Boundary condition
( , t = 0, ω) = ( , ω)
v x v x
RANDOM SOLUTION
0
ω
Nominal spatial operator
Initial condition
Random spatial operator
Representations
CDF and PDF
Finite collection of samples
Polynomial and spectral approximations
Equation Domain
in D( ω ) × (0, T ] ×Ω
Spatial domain
Time domain
Random domain
in ∂ D( ω)
× [0, T]
×Ω
Spatial boundary
in D( ω ) × { t = 0} ×Ω
Uncertainty
(1)
Stochastic differential operators.
Random coefficients
Material properties
Stochastic modeling of
complexity, chaos, fine
(2)
Uncertain environment, i.e. random
boundary or initial conditions.
Mechanical loads unknown
Temperature field unknown
Current state of system uncertain
(3)
Uncertain domain.
Rough edges from processing
Micro/tele scope finite resolution
Domain manifold subject to stress

Stochastic Mathematical Formulation
General
⎣ Z ⎦
( ,, t )
⎡T+S+ S ⎤v
x Z = 0
in D( ω ) × (0, T ] ×Ω
Solid Mechanics
Uncertainty parameterization technique required
Approximate representations are used
. e.g. KL, PC, finite collection of samples
Dimension reduction methodology
Composition operation
( v) = [ � � � ]( v)
⎡⎣T + S⎤⎦ −T
+ N f g h
[ T ]( • ) = � ∂ ( •)
acceleration t
• = balance law � ∇ ⋅ •
�
[ N ]( ) ( )
[ f ]( •) � constitutive law
[ g](
• ) = objectivity � sym ( •)
[ h](
• ) = compatibility ∇ ( •)
� �
Solution (vector) space
Tensor space
( ,, t )
v x Z
( ,, t )
∇v x Z
Symmetric tensor space
( t )
sym ⎡⎣ ∇v
x, , Z ⎤⎦
Stress tensor space
{ sym ⎡∇ ( , t,
) ⎤}
σ ⎣ v x Z ⎦
Balance Law
acts on stress space
Compatibility
conditions
Objectivity
requirements
Constitutive
law

Stochastic Formulation for Multiresolution Continua
Solid Mechanics
( v) ⎡ + �( ) � � ⎤(
v)
⎡T + S + S ⎤ = −T
N f + f g h
⎣ Z⎦ ⎣ Z ⎦
Intrinsic stress
( )
σ · ∇−Σ: ∇⊗∇ + f = � γ
ρ ρ ρ
S · ∇−s + B = Γ � , ρ = 1, 2, …,
N
Σ · n = 0
σ · n−Σ: ∇⊗n
S
ρ
Macro double stress
Spatial operator
Boundary conditions
( )
body force
= t
· n = T, ρ = 1,2, …,
N
acceleration
Temporal operator
( ( t ) ( t ) ( t ) )
n n n n
σβ , , β = f , f , f sym ∇v x, , Z , L x, , Z −L, ∇L
x, , Z
σ
Z(
s)
x1
β β
Random dimension
Z
x3
Z()
i
t1
x1
VELOCITY
x2
x3
(1)
t1
x2
x1
t2
x3
RANDOM SOLUTIONS
t1
x2
t2
MICRO-VELOCITY GRADIENT
t2
In stochastic collocation, Galerkin
projection, random dimension
discretized like space and time
Constitutive law an implicit function of a denumerable set of random
variables

Stochastic Constitutive Theory
Denumerable set of random
variables are structural
descriptors of microstructure
Unique stress distributions
averaged to provide
constitutive behavior
MACRO-SCALE SIMULATION
Greene, M.S., Y. Liu, et al. (2011).
SVE
l
δ =

Stochastic Constitutive Theory
Macroscale, e.g. Fine resolution statistical volume element realizations
d
( , , , T , t, , SA)
〈〉 S = f κ 〈〉〈〉〈 ε ε� 〉 …
( , 〈〉〈〉 , , T , t, , SA,
ω)
S = f κ ε ε� 〈 〉 …
ω
consider statistics of constitutive
behavior other than mean
For multiscale analysis, fine scale simulations create randomness
S
ε
l Realization 1 Realization 2
Deterministic homogenization
(standard constitutive modeling)
Transfer uncertainty from general
constitutive law to
phenomenological parameters
S
strain
strain rate
temperature
time
structural approx.
ε
General form of stochastic
constitutive theory
Specific form of
S
ε
Realization 3
Fit set of phenomenological
constitutive law parameters to SVE
simulation results
Treat parameters as multivariate
statistical distribution whose data
are each realization
stochastic constitutive theory
Sω = f ⎡⎣Κ( ω),
〈〉〈〉〈 ε , ε� , T〉 , t, …,
SA⎤⎦
Κ( θ ) ~ fK
( κ,
θ )

Desired Features Multiresolution Microspan Theory
• Unifies the three field effects: Nanomorphism, Conformation Complexity, Nonlocality.
• No need for cell modeling, or intermediate models of any kind.
• Strong coupling between length scales and 1st and 2nd order deformations.
• All archetype formulations are rate-dependent for dynamic applications.
• Conjugate stress measures have clear physical associations (archetype based).
• Asymmetric strain-rate matrices used: Micropolar effects, Interface effect captured
• No requirement on infinitesimality as was in micromorphic theory.
• Nonlocal sweeping permits holonomic velocity fields, simple compatibility conditions
• Derivation of a topologically three-invariant based multiscale fracture theory.
• Higher Order Boundary conditions are easier to interpret (as they are relative)
• No “relative densities” need be defined, so equations are more meaningful.
• No risk of double counting energies
• Continuous formulation, also accepts non-linear spatial fields

Research Challenges/Questions – UQ for Materials
• Materials specific UQ issues
– UQ of Microstructure
– Impact of uncertainty across multiple scales
– Relative role of microstructure and property uncertainties
– Life time prediction/extreme events
• General UQ methodologies
– Lack of data/Is extrapolation possible?
– Quantification of material model uncertainty
– Error analysis
• Tradeoffs
– Physics based vs. data-driven model refinement
– Concurrency of material, product, and process modeling for design
– When will UQ become intractable?