Predictive Multiscale Modeling for Decision Support in Design of - - PowerPoint PPT Presentation

Predictive Multiscale Modeling for Decision Support in Design of Hierarchical Alloy Systems

David L. McDowell

Woodruff School of Mechanical Engineering School of Materials Science & Engineering Georgia Institute of Technology Atlanta, GA USA Predictive Multiscale Materials Modeling Isaac Newton Institute, Cambridge University December 2, 2015

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• f the authors and do not necessarily reflect

• NSF PSU-GT Center for

Computational Materials Design

• NSF CMMI
• AFOSR, ARL
• QuesTek, NAVAIR
• DOE NEAMS
• Current Students: Shuozhi Xu,

Paul Kern, Aaron Tallman Former students and post docs

• Ryan Austin and Jeff Lloyd, ARL
• Craig Przybyla and Bill Musinski,

AFRL

• Gustavo Castelluccio, Sandia
• Conor Hennessey

Evolutionary responses (properties): Nonequilibrium, metastable

Essential for (i) mechanism ID, (ii) validation The Mesoscale Gap in Modeling Dislocations in Metallic Systems Length and time scales are both involved

Thermodynamics and near equilibrium kinetics

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Common Combined Strategy Bottom-Up limited but increasing Top-Down Objective Mechanisms, Validation Scale Specific

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A More General Perspective on Crystalline Plasticity

DDD, PFM Grain scale crystal plasticity Microscopic phase field models Generalized continua

Various problems demand a suite of models

Dislocation field mechanics Coarse-grained atomistics

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What are the Elements of Crystalline Plasticity?

• Plastic anisotropy via slip
• Elastic anisotropy
• Slip system interaction
• Multiplication and recovery (implicit or explicit)
• Thermally activated flow
• Lattice rotation via skew symmetric plastic velocity gradient
• Mixed character 3D dislocations (implicit or explicit)
• Dislocation core effects
• Crystal connection for elastic and plastic incompatibilities at

multi-resolution

• Long range elastic dislocation interactions
• Distinct nucleation, multiplication, and annihilation
• Junction formation and short range interactions
• Cross slip, climb

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Uncertainty in Multiscale Modeling

Uncertainty in Models at a Given Scale

• Assumed mechanism(s)
• Form of model/equation
• Model parameters
• Numerical algorithm and

implementation

• Solution convergence
• Sample sets analyzed and

spatial scales of simulation

• Randomness of structure

Uncertainty in Scale Linking Algorithms

• Model reduction (reduction of
• rder)
• Configuration of information

passing (e.g., handshaking vs. direct parameter estimation)

• Type of coupling – different

parameter spaces, discrete vs continuum, dynamic vs thermodynamic, etc.

• Lack of scale separation (time,

space)

• Forms of linking strategy
• Parameters passed

Not much attention in literature

Panchal, J.H., Kalidindi, S.R., and McDowell, D.L., Computer-Aided Design, Vol. 45, No. 1, 2013, pp. 4–25.

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Scale Bridging Methods – Mesoplasticity

Length Scale Time Scale Models Examples of Scale Bridging Approaches Primary Sources of Uncertainty 2 nm

200 nm

2 mm

20 mm

• bstacle interactions, increasing #

200 mm

Panchal, Kalidindi, McDowell, Computer-Aided Design, 2013

These are hierarchical two- scale transitions

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Material Selection

High Degree of Uncertainty

Structure Properties Performance G

• a

l s / m e a n s ( i n d u c t i v e ) C a u s e a n d e f f e c t ( d e d u c t i v e ) Processing Structure Properties Performance G

• a

l s / m e a n s ( i n d u c t i v e ) C a u s e a n d e f f e c t ( d e d u c t i v e ) Processing

Limitation in Inverse problem

G.B. Olson, Science, 29 Aug., 1997, Vol. 277

Multilevel Design & Development: Conceptualization Integrated materials & product design

Part System Assembly Continuum Quantum Mesoscale Atomistic

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Shift to Concurrent Product-Process-Material System Design

System Subsystems Components Parts Materials

System Specifications

Meso Macro Molecular Quantum

Material Specifications

Match the time frame

Penn State- GT CCMD NSF I/UCRC 2005-2013

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Mappings in Multilevel Design

Composition, initial microstructure a; To,to

Composition, actual microstructure, A,T,t,Ri Microstructure Attributes, A’,T,t,Ri * dist functions * explicit Properties,

• verlay on

A,T,t,Ri Properties,

• verlay on

A’,T,t,Ri M Performance Requirements Dimension of space: M + NA’’ or M + NP

PS SP PP

Properties P (NP)

Typical Materials Selection Emphasis on Materials Selection Typical Ashby Maps

Ranged sets of performance requirements and Pareto-optimal solutions

Structure Properties Performance Goals/means (inductive) C a u s e a n d e f f e c t ( d e d u c t i v e ) Processing Structure Properties Performance Goals/means (inductive) C a u s e a n d e f f e c t ( d e d u c t i v e ) Processing

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Multiscale Modeling Issues in Support of Multilevel Materials Design

Structure Properties Performance Goals/means (inductive) C a u s e a n d e f f e c t ( d e d u c t i v e ) Processing Structure Properties Performance Goals/means (inductive) C a u s e a n d e f f e c t ( d e d u c t i v e ) Processing



Some Key Issues: Properties are scale specific; the challenge is how to tailor at various scales of hierarchy (length and time) in the presence of scale coupling. Multiscale modeling can assist in addressing these questions.

• Modeling at selective scales of hierarchy to

provide decision support for materials development

• Uncertainties of models at various scales and

multiscale transitions are prevalent

• Uncertainties in initial conditions and process

history effects are ubiquitous

• Sensitivity of responses to microstructure

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Y X

Decision-Making with Uncertainty

• H. Choi et al, 2005.

McDowell, D.L., Panchal, J.H., Choi, H.-J., Seepersad, C.C., Allen, J.K. and Mistree, F., Integrated Design of Multiscale, Multifunctional Materials and Products, Elsevier, October 2009 (392 pages), ISBN-13: 978-1-85617-662-0

• Ranged sets of performance requirements
• Ranged sets of solutions, Pareto optimal

character

• Use set theory to facilitate top-down design

based on bottom-up simulation (IDEM)

• Type I: System variable

(noise) uncertainty

• Type II: Design variable

uncertainty

• Type III: Model

parameter/structure uncertainty

• Multi-level design: IDEM

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Key Enabling Elements: Multilevel Materials Design & Development under Uncertainty

• Approximate Inverse methods for property-

structure, structure-process relations based on data sciences and metamodeling

• High throughput strategies to accelerate

exploration and steer towards interesting potential solutions

• VVUQ for decision support
• Robustness– insensitivity to process variation and

material variability

• Addressing missing physics in many-body

problems (e.g., mesoscale) via learning techniques, e.g., Bayesian updating

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Uncertainty Examples in Today’s Talk

• I. Challenging prevailing understanding and

interpretation of mesoscale experiments based on high fidelity modeling

• II. Model form/structure uncertainty at

mesoscales

• III. Extreme value property estimates and MSC

fatigue crack growth

• IV. Combined bottom-up and top-down

strategies for model parameter estimation

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• I. Shock Physics: Experimental Uncertainty
• Dislocation velocity
• Thermally activated
• Drag dominated
• Relativistic damping

 

 

/ exp / 1

S S f G

h c h c v N v G k

a a a a a a

     

2

1 2

S eff

h B c b

a a a a a a a

       

Mean dislocation velocity as a function of shear stress

Mechanical threshold stress Athermal threshold Austin, R.A. and McDowell, D.L., Int. J. Plasticity, Vol. 27, No. 1, 2011, pp. 1-24. Austin, R.A. and McDowell, D.L., Int. J. Plasticity, Vol.32-33, 2012, pp. 134-154.

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Dislocation-Based Crystal Plasticity

 

tot m im m hom het mult ann trap im trap hom

1

m hom hom

N N N N N N N N N N N v N x N b b N

a a a a a a a a a a a a a a a

              

(1) Austin and McDowell, Int. Journ. Plast., 2011 (2) Austin and McDowell, Int. Journ. Plast., 2012

exp 1

g b N N k m                        

 

het het

N f a   

N  dislocation density

Reflect dislocation substructure evolution

Length scales down to sub 100 nm, time scales less than 1 ms

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Dislocation Substructure Evolution Rates and Nonequilbrium Stress

Steady wave analysis Dynamic shear stress

Highly nonequilibrium!

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Shock in polycrystal Al

• Elastic precursor decay rate is

much more rapid for polycrystal than for single crystal

• Shock broadening due to
• rientation spread

More plastic dissipation

Elastic Precursor

200 μm thick vapor-deposited Al samples, 4 GPa shock strength, Gupta et al.,JAP, 2009 Direct ablation experiments 0.72 μm thick vapor- deposited sample, Crowhurst et al., PRL, 2011 (40 GPa shock strength)

Lloyd, Clayton, Austin, McDowell, D.L., JMPS, 69, 2014, 14-32. Lloyd, Clayton, Becker, McDowell, D.L., Int. J. Plast. 60, 2014, 118-144. Single wave structure is not real, but an artifact of the experiment

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• II. Model Form Uncertainty:

Dislocation-GB Interactions

Slip transmission criteria (Lee-Robertson-Birnbaum criteria)

Lee et al., Scripta Mater., 1989

• Geometric condition

The angle 𝜄 between the lines of intersection of the incoming and

• utgoing slip planes with GB should be minimized.
• Resolved shear stress (RSS) condition

The RSS acting on the outgoing slip system from the incoming dislocation should be maximized.

• Residual GB dislocation condition

The magnitude of the Burgers of the residual dislocation 𝒄𝑠 should be minimized.

Abuzaid et al., JMPS, 2012

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Ex Situ Hi Res DIC and EBSD

Abuzaid et al., JMPS 60(6) 2012, 1201

Hastelloy X

What about uncertainty of progressive buildup of slip fields within grains? Ex situ experiments don’t give such information… 4D information regarding increments of slip transfer would be helpful

Ex situ Hi Res DIC and EBSD

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Model Form Uncertainty

Possible reactions

The dislocation-GB interaction is a complex process, which depends on

• Materials (FCC, BCC, HCP, different

stacking fault energy)

• Temperature
• Strain rate
• Resolved shear stress
• Type of dislocation (edge, screw, mixed)
• Type of GB (twin boundary, symmetric,

asymmetric, tilt, twist, etc) and misorientation, asymmetry

• Prior reactions with residual Burgers

vector

• Incoming gliding plane
• etc.

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Generalized Stacking Fault Energy

Illustration of 1NN and 2NN elements in CAC simulations (integration points (solid blue) and slave nodes (open blue). 1NN has 27 integration points and 2NN has 125. Xu, S., Che, R., Xiong, L., Chen, Y. and McDowell, D.L., “A Quasistatic Implementation of the Concurrent Atomistic-Continuum Method for FCC Crystals,” International Journal of Plasticity 72 (2015) 91-126.

• Sequential conjugate gradient energy

minimization, 0K

• Avoids overdriven MD results
• Can add quenched dynamics to

improve efficiency for evolving defects

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CAC Simulations: Slip Transfer of Mixed Character Dislocations across Σ𝟒 CTB in Cu and Al

Dislocation multiplication: Frank-Read source

For Cu

With Shuozhi Xu et al. – in progress

Quasistatic CAC: Xu et al., Int. J. Plast., 2015 Quenched Dynamics: Sheppard et al., J. Chem. Phys., 2008

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Scale and Geometry Dependent CTB Reactions

𝜏appl𝑗𝑓𝑒 = 2GPa

𝑀𝑦 = 260𝑐 = 74.47 nm (Al)

Dislocation recombination at CTB

𝑦[110] 𝑧[1 12] 𝑨[11 1]

Wide specimen, free surfaces

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Unresolved: Reduced Order Descriptions Etc… (including migration, initial conditions, etc.) Extended LRB criteria Mantle-Core

• Many potential variants of GB structure
• Single dislocations versus pileup – sequence of slip transfer

reactions

• 3D character of GBs
• Add to this multicomponent systems, impurities, segregants

– very high dimensional

• Can’t be solved without data sciences along with high

throughput experiments/many observations and modeling

?

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• III. Uncertainty in Extreme Value

Property Estimates and MSC Growth in Fatigue

• 2. Identify EV response of SVEs

via simulation

• 3. Characterize EV

distributions of key response parameters

• 4. Characterize correlated

microstructure attributes coincident with the EV response (EV marked correlation functions)

• 1. Generate multiple SVEs

based on predefined distributions of key microstructure attributes

• 5. Identify extreme value correlated

attributes key to response and rank microstructures 6(b). Select top candidates for experimental evaluation

1 1

, , ,

n

   

1 1

, , ,

n

   

Experimental Calibration/Validation

6(a). Iterate materials design

• C. Przybyla,

GT, 2010

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Microstructure-Sensitive Fatigue Problems

• Min. Length

Scale, L O(10-10 m) O(10-8 m) O(10-7 m) O(10-5 m) O (10-3 m) Atomistic Discrete Dislocation Polycrystal Macroscale dislocations patterns plasticity plasticity Statistical theories

Sub-micron Specimens; SEM MEMs regime Mech. Testing Lab scale TEM “TOP DOWN” Vacancies and Dislocation Slip banding and dislocation reactions substructures embryonic cracks

FIPs Crack nucleation and damage process zone

This Proposal

Mechanisms and validation Mesoscale DPZ

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Example of (a) the specimen notch and FE model, and (b) monotonic and cyclic plasticity for a four-point bending experiment on a polycrystalline ferritic steel. Sweeny … Dunne, J. Mech. Phys Solids 61 (2013) 1224-1240.

Experiment-Simulation Connectivity: Crack Formation and Early Growth

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Modeling Crack Formation and Early Growth: Fatigue Indicator Parameters

Crack formation due to intense shear along the slip band of Ti- 6Al-4V (Le Biavant, et. al, 2001). Slip band impingement on grain boundary of polycrystalline nickel (Morrison and Moosbrugger, 1997).

p max net n n y

1 k 2            

p max * max n y

1 k 2            

   

cyc cyc

1

p p ij ij V V

dV V     



O(10-100) mm3, depending on characteristic GS

DPZ

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Crystal Plasticity Model for Ti-6Al-4V

Drag stress

Da 

 

e p

F F F

   

Nsys p p p k k k k

ˆ

 

    



1 1

L F F s n

 

sgn D         

k k k k D

h h      

d     

k k k s s

   m  

y k k k k k CRSS s k

(0) D (0) D d         

Mayeur, Zhang, Bridier (2005-2009)

• 500
• 1000

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Validation of Fatigue Cracking

• Experiments are expensive for model calibration
• Few statistically significant datasets exist for distribution of local

slip to validate localization predictions We assume transgranular slip/failure modes here; GB dominated slip transfer is alternative approach (cf. Sangid et al., Acta Mater. 2011; JMPS, 2011). Bridier et al., IJP, 2009

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Extreme Value Statistics for Ti-6Al-4V

• Details of FE (ABAQUS) models:
• Ti-6Al-4V crystal plasticity model by Bridier et
• al. 2008
• Ellipsoid based microstructure generator
• Cycled at 0.6% strain at 0.2% s-1
• 0.400 mm x 0.400 mm 0.400 mm

microstructure block

• 100 instantiations for each microstructure
• Ten cycles in uniaxial tension (R=0)
• Periodic boundary conditions
• ODF with Random Texture

Microstructures Assumed Mean and St. Dev. for Grain Size Distributions α+β Colony Primary α Micro Name Transformed β Size (μm) Primary α Size (μm) Vol % Primary α μ σ μ σ A Fine bi-modal low α 50 10-50 30% 50 5 25 10 B Fine bi-modal high α 50 10-50 70% 50 5 25 10 C Coarse bi-modal low α 80 40-60 30% 80 10 50 5 D Coarse bi-modal high α 80 40-60 70% 80 10 50 5 Random Texture

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Extreme Value Fatigue Indicator Parameter Distributions in Duplex Ti-6Al-4V

• 2.00
• 1.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00 0.00E+00 1.00E-10 2.00E-10 3.00E-10 4.00E-10

ln(1/ln(1/p)) Extreme Value FS FIP

A B C D

A Fine bi-modal low α B Fine bi-modal high α C Coarse bi-modal low α D Coarse bi-modal high α

Gumbel Distribution (Type I):

 

 

exp

y u Y n

F y e

a  

     

• Ellipsoid based microstructure generator
• Cycled at 0.6% strain at 0.2% s-1, R = 0, 19

cycles, PBCs

• 0.400 mm x 0.400 mm 0.400 mm SVE
• 100 instantiations for each microstructure

ODF with Random Texture

Przybyla and McDowell, 2010

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mg=0.45 to 0.5 (Basal, primary α) mg’=0.45 to 0.5 (Basal, primary α) mg=0.45 to 0.5 (Basal, primary α) mg’=0.45 to 0.5 (Prismatic, primary α) mg=0.45 to 0.5 (Basal, primary α) mg’=0.45 to 0.5 (Pyramidal <a>, primary α) mg=0.45 to 0.5 (Basal, primary α) mg’=0.45 to 0.5 (Pyramidal <a+c>, primary α)

Validation of Extreme Value Failure Modes: Ti-6Al-4V, Random Texture

*S. K. Jha, J.

• M. Larsen,

VHCF-4, pp. 385-396, 2007 34

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Epistemic Uncertainty Fine scale (bottom-up)

• Slip system activation and hardening relations
• GB Interfaces – unstructured vs structured meshing – slip transfer

relations

• Heterogeneous slip localization, dislocation substructure and

mechanisms for slip irreversibilitySecond phase particle cracking, debonding

• Numerical implementation strategy (elements, meshing, order of

integration, and averaging domain for FPZ, etc.)

Coarse Scale (top-down)

• FIP definition
• Microstructure representation
• Grain size, shape and orientation distribution
• Grain boundary character distribution

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DPZ Band Averaging of FIPs

For transgranular MSC growth, FIPs are averaged along bands aligned with active slip planes.

FIP (FIP )

meso

Avg 

a a

per band

3D MSC Growth Modeling

Castelluccio, G.M., and McDowell, D.L., “A Mesoscale Approach for Growth of 3D Microstructurally Small Fatigue Cracks in Polycrystals,” Int. J. Damage Mechanics, 23(6), 2014, 791- 818. Castelluccio, G.M., and McDowell, D.L., "Mesoscale Modeling of Microstructurally Small Fatigue Cracks in Metallic Polycrystals," Mat.

• Sci. Eng. A, Vol. 598, No. 26, 2014, pp.

34-55.

RR 1000 Ni-base superalloy

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2

FIP 1 a FIP      

g

P

Initial value for band in next uncracked grain

Renormalized Sub-Grain FIP Evolution

FIP evolution during crack growth

Crack size based on equivalent area, as per Murakami, i.e.,

a A 

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Apply a few loading cycles Apply a few loading cycles, redistribute cyclic stress and plastic strain fields Extend the crack along the band with shortest MSC life along intersecting slip systems in adjacent grain Crack the band with shortest nucleation life

1st grain (Nucleation/Incubation)

 

N FIP d

a 

Subsequent grains (MSC) (analytical projection)

d D D   



Crack Growth Algorithm

c 1 N tanh d N c c c

         

A c 2d F d IPa  

c 2d c CTD  

• d

• th

1 FIP A CTD 1 a 2 da N N

            

= . 2FIP (1 0.5(a ) ) CTD

D D da d dN

     



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Validation: Physical Consistency with Observed Trends

1

= da c a dN

This is characteristic of our findings…

Note: we have not imposed anything remotely close to this law in simulations

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• C. H. Wang, K. J. Miller.
• Fat. Frac. of Eng. Mat. & Struc.

16, 181–198, 1993

Mean Stress Effect on Crack Nucleation – Uniaxial Loading

exp( )

C m a

N A B    

Differs from Morrow

• r Smith-Watson-

Topper (SWT)

 

2 g inc gr

N FIP d

 a a

a 

Validation: Physical Consistency with Observed Trends

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Cost/Imprecision Model Refinement Model utility based

• n Improvement

Potential Model development and execution cost Model imprecision; Epistemic uncertainty Now Eventual

A Note on Model Refinement

Includes:

• DPZ definition/size
• Subscale nucleation/growth models
• Unstructured meshing and mesh

refinement

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Top-Down

a

  

 

1 ; 0;

F s for s kT s for s

                                            

Patra, Zhu, McDowell IJP, doi10.1016/j.ijplas.2014.03.016.

Bottom-Up

Narayanan, McDowell, Zhu, JMPS 65, 2014

• IV. Fusing Bottom-Up and Top-Down

Information bcc Fe

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Parameters form Structural Bridge

1. Pre-exponential factor 2. Activation energy of dislocation glide at zero external stress 3. Shape factor p 4. Shape factor q 5. Thermal slip resistance at 0 K 1. Temperature 2. Driving stress 3. Athermal slip resistance

Reinterpretation of parameters!

Calibration Inputs Configuration Inputs Aaron Tallman, PhD work in progress

1. Thermal kink-based reference strain rate 2. Kinkpair activation energy at zero ext. stress 3. Both p, q describe stress 4. dependence of kinkpair activation energy 5. Thermal slip resistance at 0 K

Aaron Tallman, PhD Work in progress

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θ→x Squared Euclidean Distance

Loss function values Bottom up (x) Top down MLE (𝜾 ) Using Prior density function

0.792 1.032

Using Posterior density function

0.776 0.846

decrease in Bregman Divergence

2% 18%

Before data After data

These are the results for the test for requirement iii

Column 1: Agreement & Identifiability Column 2: Identifiability Only

   

2 max min

• space

|

i

i i i i

x Loss P Data d



  



                 

 

θ θ

x θ θ

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Multiscale Bayesian Calibration

 

, , , : Calibration Inputs , , , : Configuration Inputs , : Model Responses , : Experimental Data : Missing Physics Coefficients , : Missing Physics Functions

x w X Y X Y f X     θ φ x w

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Accumulation of Data

• Calibration applies data in a per datum formulation
• As new data are gathered, each new point can be

included in a new calibration

     

   

• ld data

, , 1 exp 2

Y Y Y Y Obj C  

                                       

 

x θ x x θ x θ

, : Calibration Inputs , : Configuration Inputs : Model Responses : Experimental Data : Standardizing Factor : std. dev. of exp. datum i

x Y Y C   θ x

The George W. Woodruff School of Mechanical Engineering School of Materials Science and Engineering

47

Recent TMS Study

On behalf of National Institute of Standards (NIST) Material Measurement Laboratory

John Ågren, KTH, Sweden Raymundo Arróyave, Texas A&M University Mark Asta, UC-Berkeley Corbett Battaile, Sandia Carelyn Campbell, NIST James Guest, Johns Hopkins Paul Krajewksi, GM Alexis Lewis, NSF Wing Kam Liu, Northwestern University David McDowell, Georgia Tech Tony Rollett, Carnegie Mellon University Dallas Trinkle, University of Illionis Peter Voorhees, Northwestern University

http://www.tms.org/multiscalestudy/

The George W. Woodruff School of Mechanical Engineering School of Materials Science and Engineering

48

Technical/Scientific Recommendations

• Recommendation T1: Develop initiatives that address uncertainty

quantification and propagation (UQ/UP) across multiple models describing a range of material length and time scales

• Recommendation T2: Develop strong coupling methods that allow

bidirectional communication between deformation and microstructural evolution models (i.e., methodologies to account for the co-evolution of microstructure and deformation)

• Recommendation T3: Devise methods and protocols for taking into

account rare events and extreme value statistical distributions

• Recommendation T4: Develop multi-resolution (or multiscale) multi-

physics free energy functions (and associated kinetic parameters) involving microstructure evolution, defect formation, and life prediction

• Recommendation T5: Develop and execute focused research efforts

addressing interfacial properties and nucleation effects, with particular emphasis on carrying out more systematic studies that couple theory, experiments, and simulations across length and time scales

• T6-T9…

The George W. Woodruff School of Mechanical Engineering School of Materials Science and Engineering

49

• Current Students: Shuozhi Xu,

Paul Kern, Aaron Tallman Former students and post docs

• Ryan Austin and Jeff Lloyd, ARL
• Craig Przybyla and Bill Musinski,

AFRL

• Gustavo Castelluccio, Sandia
• Conor Hennessey

• f the authors and do not necessarily reflect

• NSF PSU-GT Center for

Computational Materials Design

• NSF CMMI
• AFOSR, ARL
• QuesTek, NAVAIR
• DOE NEAMS

Th Thank anks