1. POLYCRYSTALS AND COMPUTATIONAL DESIGN: IS THE CONTROL OF MICROSTRUCTURE-SENSITIVE PROPERTIES FEASIBLE? Materials Process Design and Control Laboratory Shankar Ganapathysubramanian, Swagato Acharjee and Prof. Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: zabaras@cornell.edu URL: http://www.mae.cornell.edu/zabaras /

2. Materials Process Design and Control Laboratory LONG TERM RESEARCH FOCUS AND OBJECTIVES BROAD OBJECTIVES To develop a mathematically and computationally rigorous gradient-based optimization methodology for virtual materials process design that is based on quantified product quality and accounts for process targets and constraints

3. Materials Process Design and Control Laboratory MOTIVATION FOR MICROSTRUCTURE-BASED DESIGN Boeing 747 GE 90 Developing advanced materials for gas turbine engines is expensive – Is it possible to control material properties and product performance through deformation processes?  Reduce dependence on other expensive methods such as alloying  Especially important for critical hardware components in the aerospace, naval and automotive industry to reduce material utilization for reduced process cost, fuel consumption and higher mobility  Design processes to produce highly optimized microstructure with directional material properties Modern aircraft engine design and materials selection is an extremely challenging area. Desired directional properties include:  strength at high temperatures, R-values  elastic, creep, fatigue & fracture properties  thermal expansion, corrosion resistance  machinability properties

4.  Accurate modeling of deformation processes  Mathematically consistent & accurate continuum sensitivity finite element analysis  Gradient based optimization environment  Motivation towards microstructure sensitive design of deformation processes – Design across length scales Optimum deformation process Billet Product Minimal overall cost: force, energy, etc. Materials Process Design Simulator Tailored material properties in the final product Desired microstructural features Desired spatial distributions of state variables Controlled texture, recrystallization, fracture & porosity Desired shape with minimal material utilization Accelerated process sequence design APPROACH Interactive Optimization Environment Given process constraints & parameters Desired product properties Materials Process Design and Control Laboratory DEFORMATION PROCESS DESIGN FOR TAILORED PROPERTIES

5. Final shape Selection of stages Stage 1 Stage 2 Stage 3 Design of dies ? ? ? ? ? Thermal parameters Identification of stages Number of stages Preform shape Die shape Mechanical parameters DESIGN VARIABLES Processing temperature Press force Press speed Product quality Geometry restrictions CONSTRAINTS Given raw material and desired hardware material and desired hardware component performance, compute optimal compute optimal manufacturing process sequence Materials Process Design and Control Laboratory DESIGN OF MULTI STAGE DEFORMATION PROCESSES Design of preforms OBJECTIVES Material usage Energy, cost Near net shape Microstructure Residual stresses Uniform deformation Knowledge-based methods Design of Sequences Design of Preforms Design of Dies Shape and parameter sensitivity analysis Die and process parameter sensitivity analysis DesignObjective

6. 2D forming process design  Thermo-mechanical analyses for materials with ductile damage  Forming process design considering thermal effects in the die  Phenomenological modeling of grain growth and dynamic recrystallization Remeshing & data transfer  Effective remeshing based on geometric criteria  Accurate data transfer techniques  Assumed strain sensitivity methods Other important features  Very accurate and efficient computation of sensitivity fields (gradient calculation)  Accurate kinematics & contact modeling Polycrystal plasticity  Accurate, continuum framework  Reduced-order modeling scheme  Coupled macro-micro design framework Materials Process Design and Control Laboratory FEATURES & CAPABILITIES OF PHENOMENOLOGICAL MODELING The same material with an optimum design Lot more material - conventionaldesign Exact material in a conventionaldesign Noflash Fully filled cavity Unfilledcavity Flash Hugeflash

7. Materials Process Design and Control Laboratory FROM PHENOMENOLOGY TO POLYCRYSTAL PLASTICITY (1) Continuum framework (3) Desired effectiveness in terms of state variables (2) State variable evolution laws Initial configuration B o B F e F p F  F Deformed configuration Intermediatethermalconfiguration Stress free (relaxed) configuration PHENOMENOLOGY POLYCRYSTAL PLASTICITY The effectiveness of design for desired product properties is limited by the ability of phenomenological state-variables to capture the dynamics of the underlying microstructural mechanisms Initial configuration B o B F * F p F Deformed configuration Stress free (relaxed) configuration n0n0 s0s0 n0n0 s0s0 n s (1) Single crystal plasticity (3) Ability to tune microstructure for desired properties (2) State evolves for each crystal Polycrystal plasticity provides us with the ability to capture material properties in terms of the crystal properties. This approach is essential for realistic design leading to desired microstructure-sensitive properties

8. Materials Process Design and Control Laboratory EXTENDING TOWARDS POLYCRYSTAL PLASTICITY Crystal/lattice reference frame e1e1 ^ e2e2 ^ Sample reference frame e’ 1 ^ e’ 2 ^ crystal e’ 3 ^ e3e3 ^ RODRIGUES’ REPRESENTATION  Neo-Eulerian representation of orientation  Rotations about a fixed axis trace straight lines in parameter space  Set of orientations equidistant from 2 rotations is always a plane  Helps reduce symmetries to between a pair of planes Cubic crystal (FCC) zoom

9. Materials Process Design and Control Laboratory ORIENTATION DISTRIBUTION FUNCTION (ODF) EVOLUTION EQUATION FOR THE ODF DEFINITION OF VARIOUS TERMS v – re-orientation velocity --- how fast are the crystals re-orienting r – orientation of the crystal. A – is the ODF, a scalar field; Cubic crystal (FCC Cu) OTHER FEATURES  Neo-Eulerian representation of orientation (particularly Rodrigues’ parameterization)  Advantages – numerous (Dawson, 2000)  Fundamental region for FCC crystal shown in the figure to the right

10. PROCESS MODELING & DESIGN: A POLYCRYSTAL PLASTICITY BASED APPROACH Micro problem driven by the velocity gradient L Macro problem driven by the macro-design variable β B n+1 Ω = Ω (r, t; L) ~ Polycrystal plasticity x = x(X, t; β ) L = L (X, t; β ) L = velocity gradient F n+1 B0B0 Design variables (β) are macro design variables 1.Die shapes 2.Preform shapes 3.Processing conditions Etc. Design objectives are micro-scale averaged material/process properties Materials Process Design and Control Laboratory

11. MULTI-LENGTH SCALE CONTINUUM SENSITIVITY ANALYSIS The velocity gradient – depends on a macro design parameter Sensitivity of the velocity gradient – driven by perturbation to the macro design parameter A micro-field – depends on a macro design parameter (and) the velocity gradient as Sensitivity of this micro-field driven by the velocity gradient L Define the sensitivity of L with respect to infinitesimal changes to macro-design parameters  (e.g. die or preform shape). L, L The velocity gradient, L, links the two-length scales through the Taylor hypothesis. Define the sensitivity of a micro-field with respect to infinitesimal changes in L, which in turn depends on the macro-design parameters . Micro-sensitivity problem Macro-sensitivity problem 0 ~ Ω + Ω = Ω (r, t; L+ΔL) r – orientation parameter Ω = Ω (r, t; L) ~ I + (L s ) n+1 F n+1 + F n+1 o x + x = x(X, t; β+Δ β) o B n+1 L + L = L (X, t; β+Δ β) o F n+1 x = x(X, t; β) B n+1 L = L (X, t; β) L = velocity gradient B0B0 L s = design velocity gradient

12. Materials Process Design and Control Laboratory CONSTITUTIVE FRAMEWORK Impose the extended Taylor hypothesis, thus making no distinction between the crystal velocity gradient and the macroscopic velocity gradient Direct problem the continuum sensitivity analysisi.e. we make no distinction between the sensitivity of the crystal velocity gradient and the sensitivity of the macroscopic velocity Extend the Taylor hypothesis towards the continuum sensitivity analysis, i.e. we make no distinction between the sensitivity of the crystal velocity gradient and the sensitivity of the macroscopic velocitygradient. Sensitivity problem 0 0 0 Symmetric & spin components Re-orientation velocity x D = Stretch tensor = Lattice spin W= Spin tensor T = Schmid tensor w = Lattice spin vector v = Re-orientation velocity Symmetric & spin components Sensitivity of re-orientation velocity x FEATURES 1. Continuum sensitivity analysis 2. Framework based on the direct problem 3. Eulerian framework

13. Symmetric & spin components Materials Process Design and Control Laboratory POLYCRYSTAL SENSITIVITY ANALYSIS EVOLUTION EQUATION FOR THE SENSITIVITY OF THE ODF FEATURES  Continuum sensitivity analysis  Framework based on the direct problem  Eulerian framework Develop an extended Taylor hypothesis towards the continuum sensitivity analysis – i.e. we make no distinction between the sensitivity of the crystal velocity gradient and the sensitivity of the macroscopic velocity gradient. Sensitivity of reorientation velocity Gradient of the sensitivity of the velocity x

14. An important multi-length scale design problem Tune microstructure within the work-piece for control of material properties through macro processing design parameters (e.g. choice of dies, preforms, conditions, etc.) Enormous degrees of freedom & number of PDEs to be solved limits the scope of design Computational issues Microstructure-model reduction without significant loss of accuracy Possible Solution ? Materials Process Design and Control Laboratory COMPLEXITY OF MULTILENGTH SCALE DESIGN Dof per material point (ODF) ~800 Descritization of geometry ~250 Number of sensitivity problems ~6 Dof for 1 optimization problem ~1.4Mil Average # optimization problems required ~ 10

15. Materials Process Design and Control Laboratory INTRODUCE MICROSTRUCTURE-MODEL REDUCTION Suppose we had a collection of data (from experiments or simulations) for the ODF: such that it is optimal for the data represented as Is it possible to identify a basis POD technique – Proper Orthogonal Decomposition Solve the optimization problem Method of snapshots where Eigenvalue problem where N C C ij i

16. Materials Process Design and Control Laboratory REDUCED ORDER MODEL FOR THE ODF The reduced basis for the ODF ensemble has been evaluated, say Using this basis, ODF represented as follows This representation of the ODF leads to a reduced-model in the form of an ODE. Reduced model for the evolution of the ODF where Initial conditions

17. Materials Process Design and Control Laboratory REDUCED MODEL FOR THE SENSITIVITY ANALYSIS The reduced basis for the ODF ensemble has been evaluated as Using this basis, the sensitivity of the ODF is represented as follows: Observing that the basis is independent of the macro design parameters, we conclude that the basis generated in the direct analysis can be used for the sensitivity problem. Reduced model for the evolution of the sensitivity of the ODF where Initial conditions

18. Materials Process Design and Control Laboratory DECOMPOSITION OF THE VELOCITY GRADIENT Design vector α = {α 1, α 2, α 3, α 4, α 5, α 6, α 7, α 8 } T Uniaxial tension Plane strain compression Shear Spin α Design problem: Determine α so as to obtain desired properties in the final product.

19. Materials Process Design and Control Laboratory VALIDATION: Direct analysis Uniaxial tension test α = {1,0,0,0,0} T t = 0.1 s Reduced model (Basis II) Full model Reduced model (Basis III) Basis – II 3 POD modes from a uniaxial tension test with strain rate of 1s -1 for t=0.2 s Basis – III 9 POD modes from five tests ( α i = 1s -1 for the i th test, α j = 0, i ≠ j ) for t = 0.2 s

20. Same bases used for direct and sensitivity analysis Materials Process Design and Control Laboratory VALIDATION: Sensitivity problem Uniaxial tension test α = {1,0,0,0,0} T t = 0.1 s α = {1e -2,0,0,0,0} T Reduced (Basis II) Reduced (Basis III) Full model FDM solution

21. Desired ODF ODF obtained by optimization L =.001 0.853 -.001 Materials Process Design and Control Laboratory DESIGN THE VELOCITY GRADIENT – For desired ODF Orientation(rinradians) O r i e n t a t i o n d i s t r i b u t i o n f u n c t i o n -101 0.5 1 1.5 Initialguess Optimalsolution Iteration3 Iteration5,9,Optimal L = 0 0 0.866 0.0 L = 1.0 0.0 0.997 9.3e-4 -0.997 9.3e-4 L = Desired ODF ODF obtained by optimization

22. Design the velocity gradient applied for a fixed time on this piece, at a material point such that the difference in ODF obtained at the final time and the desired ODF is minimized. FCC Al workpiece Design parameter is the velocity gradient DEFORMATION PROCESS DESIGN OBJECTIVE Materials Process Design and Control Laboratory CONTROL OF TEXTURE IN FCC Al

23. Desired value: α = {0.02,0.86,0,0,0} T, Initial guess: α = {0,0.2,0,0,0} T Converged solution: α = {0.019,0.86,-0.018,0.006,-0.012} T, t = 0.1 s Reduced order solution using basis that contains data from plane strain compression, uniaxial tension and mixed deformation tests to ensure accurate representation of mixed deformation components in the desired and intermediate solutions. Materials Process Design and Control Laboratory DESIGN FOR MICROSTRUCTURE SENSITIVE PROPERTIES – Yield stress Normalized yield stress Normalized objective function Design Problem – – polycrystal Taylor factor (computed using Bishop-Hill analysis)

24. Materials Process Design and Control Laboratory DESIGN FOR MICROSTRUCTURE SENSITIVE PROPERTIES - R value - planar variation R value Normalized objective function Desired value: α = {1.2,0,0,0,0} T, Initial guess: α = {0.5,0,0,0,0} T Converged solution: α = {1.19,0.05,0.001,0,0} T, Reduced order solution at t = 0.1 s Design Problem: Ξ = {F,G,H,N} T (from Hill’s anisotropic yield criterion)

25. Desired value: α = {1,0,0,0,0} T Initial guess: α = {0.5,0,0,0,0} T Converged solution: α = {0.987,0.011,0,0,0} T 0.985 0.99 0.995 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 0102030405060708090 Angle from rolling direction Initial Intermediate Optimal Desired h Materials Process Design and Control Laboratory DESIGN FOR MICROSTRUCTURE BASED PROPERTY - Hysteresis loss Normalized hysteresis loss Normalized objective function Design Problem Hyteresis loss Crystal direction. Easy direction of magnetization – zero power loss External magnetization direction

26. Materials Process Design and Control Laboratory OPTIMIZE DIRECTIONAL DEPENDENCE OF ELASTIC MODULI Objective: Objective: Control of the distribution of Elastic moduli (away from the rolling direction) through a novel deformation process (design) Mathematical formulation: Desired distribution Distribution during different stages of optimization Reduced-order model: 17 modes (dof). Basis generated from an ensemble of 300. Full-order model: 800*8 = 6400 dof L = 0.6 0.0 -0.3 0.0 0.0 -0.3 0.0 Velocity gradient corresponding to the desired distribution L = 0.68 0.08 -0.2 0.03 0.01 0.03 -0.48 Velocity gradient corresponding to the optimum distribution

27.  New method of modeling and control of material properties.  Captures the essential dynamics of the problem – provided basis is ‘good’ enough.  Savings in computational time through reduced degrees of freedom  Open issues remain to be addressed  Non-uniqueness – Multiple paths  Basis selection – questions on optimal basis  Process selection – initial guess Materials Process Design and Control Laboratory HIGHLIGHTS OF THE REDUCED MODELING APPROACH Possible solutions  Support vector machines for classification  Universal basis – select basis adaptively as deformation proceeds  Pattern recognition and Machine vision

28. Materials Process Design and Control Laboratory WHAT DOES THE FUTURE HOLD?  Design for control of microstructure related properties – Design across length scales  Implementation of a general micro- macro process design framework (use design on the macroscale to affect microstructural features)  Generate a `universal’ snapshot basis for reduced-order modeling (including experimentally-obtained snapshots)  Develop algorithms for real-time microstructural reduced-order model mining  Design across length scales – an Elasto-Viscoplastic analysis 10% deformation 1100 Al Open die forging Total Lagrangian formulation Fundamental region = 150 elements

29. Materials Process Design and Control Laboratory INFORMATION RELEVANT PUBLICATIONS S. Ganapathysubramanian and N. Zabaras, “Design across length scales: A reduced-order model of polycrystal plasticity for the control of microstructure- sensitive material properties”, Computer Methods in Applied Mechanics and Engineering, submitted for publication. S. Acharjee and N. Zabaras, “A proper orthogonal decomposition approach to microstructure model reduction in Rodrigues’ space with applications to the control of material properties”, Acta Materialia, accepted. Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: zabaras@cornell.edu URL: http://www.mae.cornell.edu/zabaras / Prof. Nicholas Zabaras CONTACT INFORMATION