1. Materials Process Design and Control Laboratory Veera Sundararaghavan and Nicholas Zabaras Sibley School of Mechanical and Aerospace Engineering Cornell University Supported by AFOSR, ARO Design of materials with enhanced properties: A multi-length scale computational approach. Technical Presentation

2. Materials Process Design and Control Laboratory PRESENTATION OUTLINE Motivation of microstructure sensitive design Motivation of microstructure sensitive design Microstructure homogenization Microstructure homogenization Multi-scale deformation process simulation Multi-scale deformation process simulation Multi-scale sensitivity analysis Multi-scale sensitivity analysis Design results Design results

3. Development of a multi-scale continuum sensitivity method for multi-scale deformation problems Design processes and control properties using multi-scale modeling Materials Process Design and Control Laboratory DEFORMATION PROCESS DESIGN SIMULATOR Enhanced strength

4. RESEARCH OBJECTIVES Materials Process Design and Control Laboratory Info from NASA ALLSTAR network, 2005 -Materials design is a slower process than engineering design. -Replace empirical approaches to design with physically sound multi- scale approaches. -Integrate materials design into engineering design.

5. Materials Process Design and Control Laboratory MULTISCALE NATURE OF METALLIC STRUCTURES Grain/crystal Inter-grain slip Grain boundary Twins precipitates Atoms Meso Micro Nano Material-by-design Titanium armors with high specific strength.

6. Materials Process Design and Control Laboratory COMPUTATIONAL DESIGN OF DEFORMATION PROCESSES Press force Press speed Product shape Cost CONSTRAINTS OBJECTIVES Material usage Properties Microstructure Preform shape Die shape VARIABLES BROAD DESIGN OBJECTIVES Given raw material, obtain final product with desired microstructure and shape with minimal material utilization and costs Forging rate

7. Materials Process Design and Control Laboratory Crystal/lattice reference frame e1e1 ^ e2e2 ^ Sample reference frame e’ 1 ^ e’ 2 ^ crystal e’ 3 ^ e3e3 ^  Crystallographic orientation  Rotation relating sample and crystal axis  Properties governed by orientation during deformation POLYCRYSTALLINE MICROSTRUCTURES

8. Stress Evolution of slip system resistances Shearing rate Materials Process Design and Control Laboratory FCC SINGLE CRYSTAL RESPONSE TO IMPOSED DEFORMATION Athermal resistance (e.g. strong precipitates) Thermal resistance (e.g. Peierls stress, forest dislocations) If resolved shear stress < athermal resistance, otherwise Balasubramaniam and Anand, Int J Plasticity, 1998

9. Materials Process Design and Control Laboratory Crystallographic slip and re- orientation of crystals are assumed to be the primary mechanisms of plastic deformation Evolution of various material configurations for a single crystal as needed in the integration of the constitutive problem. Evolution of plastic deformation gradient The elastic deformation gradient is given by Rate-independent model (Anand and Kothari, 1996) B0B0 mm nn nn mm mm nn ^ nn mm nn mm ^ _ _ BnBn BnBn B n+1 _ _ FnFn FnFn FnFn F n+1 F trial p p e e e FrFr FcFc Intermediate configuration Deformed configuration Intermediate configuration Reference configuration SINGLE CRYSTAL CONSTITUTIVE ANALYSIS

10. Materials Process Design and Control Laboratory HOMOGENIZATION OF DEFORMATION GRADIENT Use BC: = 0 on the boundary Note = 0 on the volume is the Taylor assumption, which is the upper bound X x Macro Meso x = FX y = FY + w N n Macro-deformation is an average over the microscopic deformations (Hill, Proc. Roy. Soc. London A, 1972) Decompose deformation gradient in the microstructure as a sum of macro deformation gradient and a micro-fluctuation field Mapping implies that (Miehe, CMAME 1999). Macro Micro

11. Materials Process Design and Control Laboratory VIRTUAL WORK CONSIDERATIONS Hill Mandel condition: The variation of the internal work performed by macroscopic stresses on arbitrary virtual displacements of the microstructure is required to be equal to the work performed by external loads on the microstructure. (Hill, J Mech Phys Solids, 1963) Apply boundary condition Homogenized stresses Must be valid for arbitrary variations of  F Sundararaghavan and Zabaras, International Journal of Plasticity 2006. Macro Micro

12. Materials Process Design and Control Laboratory MICROSTRUCTURE DEFORMATION Thermal effects linking assumption Equate macro and micro temperatures  An equilibrium state of the microstructure is assumed  Updated Lagrangian formulation B n+1 x = x(X, t) F = F (X, t) F = deformation gradient F n+1 B0B0 X  Hexahedral meshing using CuBIT.

13. Materials Process Design and Control Laboratory IMPLEMENTATION Forming process Update macro displacements Boundary value problem for microstructure Solve for deformation field Integration of constitutive equations Dislocation plasticity Macro-deformation gradient Homogenized (macro) stress Micro-scale stress Micro-scale deformation gradient Macro Micro

14. Materials Process Design and Control Laboratory MICROSTRUCTURE RESPONSE VALIDATION Experimental Computed Experimental results from Anand and Kothari (1996)

15. Materials Process Design and Control Laboratory OPTIMIZATION OF MICROSTRUCTURE RESPONSE z: Deviation from desired property x: Process variable 1 y: Process variable 2 Initial guess Select optimal process parameters to achieve a desired property response. Need to evaluate gradients of objective function (deviation from desired property) with respect to process variables. Sundararaghavan and Zabaras, IJP 2006. Scale linking, homogenization Die shape Initial preform shape Forging rate Initial microstructure

16. Materials Process Design and Control Laboratory OPTIMIZATION FRAMEWORK Gradient methods  Finite differences (Kobayashi et al.)  Multiple direct (modeling) steps  Expensive, insensitive to small perturbations  Direct differentiation technique (Chenot et al., Grandhi et al.)  Discretization sensitive  Sensitivity of boundary condition  Coupling of different phenomena  Continuum sensitivity method (Zabaras et al.)  Design differentiate continuum equations  Complex physical system  Linear systems Continuum equations Design differentiate Discretize CSM -> Fast Multi- scale optimization Requires 1 Non-linear and n Linear multi- scale problems to compute gradients

17. Materials Process Design and Control Laboratory DESIGN DIFFERENTIATION (Badrinarayan and Zabaras, 1996)  Directional derivative

18. Materials Process Design and Control Laboratory MATERIAL POINT SENSITIVITY ANALYSIS Calculate such that x = x (x r, t, β, ∆β ) o o F r and x o o P r and F, o  o o Constitutive problem Kinematic problem Sensitivity of single crystal response Sensitivity of equilibrium equation L = L (X, t; β) I + L s x + x = x(X, t; β+Δ β) o F + F o L + L = L (X, t; β+Δ β) o F x = x(X, t; β) L = velocity gradient X Sensitivity linking assumption: The sensitivity of the deformation gradient at macro- scale is the same as the average of the sensitivities of deformation gradients in the microstructure.

19. Materials Process Design and Control Laboratory MATERIAL POINT SENSITIVITY ANALYSIS Solve for sensitivity of microstructure deformation field Integration of sensitivity constitutive equations Sensitivity of (macro) properties Perturbed Mesoscale stress Perturbed meso deformation gradient Perturbed macro deformation gradient SENSITIVITY DEFORMATION PROBLEM  Derive a weak form for the shape sensitivity of the equilibrium equation  Primary unknown of the weak form x – sensitivity of the deformed configuration

20. Materials Process Design and Control Laboratory SENSITIVITY OF THE CRYSTAL CONSTITUTIVE PROBLEM Sensitivity hardening law Sensitivity constitutive law for stress Derive sensitivity of PK-I stress Integration of sensitivity constitutive equations Sensitivity of (macro) properties Perturbed Mesoscale stress Perturbed meso deformation gradient Perturbed macro deformation gradient Solve for sensitivity of microstructure deformation field

21. Materials Process Design and Control Laboratory PROCESS DESIGN FOR STRESS RESPONSE AT A MATERIAL POINT b c Iterations Cost function Time (sec) Equivalent stress (MPa) (a)(b) (c) (d) Initial response Intermediate Final response Desired response Time (sec) Equivalent stress (MPa) Change in Neo- Eulerian angle (deg) Sundararaghavan and Zabaras, IJP 2006.

22. Materials Process Design and Control Laboratory Cubic crystal FIRST ORDER REPRESENTATION OF MICROSTRUCTURES RODRIGUES’ REPRESENTATION FCC FUNDAMENTAL REGION Crystal/lattice reference frame e2e2 ^ Sample reference frame e1e1 ^ e’ 1 ^ e’ 2 ^crystal e’ 3 ^ e3e3 ^ n Particular crystal orientation  Continuum representation  Orientation distribution function (ODF)  Handling crystal symmetries  Evolution equation for ODF Any property can be expressed as an expectation value or average given by Kumar and Dawson 1999, Ganapathysubramaniam and Zabaras, IJP 2005.

23. Materials Process Design and Control Laboratory EVOLUTION OF CRYSTAL VOLUME FRACTIONS WITH DEFORMATION Conservation principle Solve for evolution of the ODF with deformation Based on the Taylor hypothesis EVOLUTION EQUATION FOR THE ODF (Eulerian) v – re-orientation velocity: how fast are the crystals reorienting r – current orientation of the crystal. A – is the ODF, a scalar field; Constitutive sub-problem  Taylor hypothesis: deformation gradient (F) in each crystal of the polycrystal is same as the macroscopic deformation gradient.  Compute the reorientation velocity from the elastic deformation gradient Linking assumption Sundararaghavan and Zabaras, Acta Materialia, 2005

24. Materials Process Design and Control Laboratory Material: 99.987% pure polycrystalline f.c.c Aluminum Process: Simple shear motion VERIFICATION OF ODF EVOLUTION MODEL Experimental analysis addressed in Carreker and Hibbard, 1957 Benchmark problem in Balasubramanian and Anand 2002.

25. Materials Process Design and Control Laboratory MULTISCALE MODEL OF DEFORMATION USING ODF REPRESENTATION Largedef formulation for macro scale Update macro displacements ODF evolution update Polycrystal averaging for macro-quantities Integration of single crystal model Dislocation plasticity Macro-deformation gradient Homogenized (macro) stress Microscale stress Macro-deformation gradient Macro Micro Meso Parallel solver: PetSc (Argonne Labs) KSP-Solve Meso Micro Macro

26. Materials Process Design and Control Laboratory 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 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 Sensitivity thermal sub-problem Sensitivity constitutivesub-problem Sensitivity kinematic sub-problem Sensitivity contact & frictionsub-problem MULTI-LENGTH SCALE SENSITIVITY ANALYSIS

27. Materials Process Design and Control Laboratory MACRO-MICRO SENSITIVITY ANALYSIS Continuum problem Differentiate Discretize Design sensitivity of equilibrium equation Calculate such that x = x (x r, t, β, ∆β ) o o Variational form - F r and x o o o λ and x o P r and F, o  o o Constitutive problem Regularized contact problem Kinematic problem Material point sensitivity analysis

28. Materials Process Design and Control Laboratory EXTRUSION DESIGN PROBLEM Objective: Design the extrusion die for a fixed reduction such that the deviation in the Young’s Modulus at the exit cross section is minimized Material: FCC Cu Microstructure evolution is modeled using an orientation distribution function Minimize Youngs Modulus variation across cross-section Die design for improved properties

29. Materials Process Design and Control Laboratory DESIGN PARAMETERIZATION OF THE PROCESS VARIABLE Objective: Minimize Young’s Modulus variation in the final product by controlling die shape variations Identify optimal C i that results in a desired microstructure-sensitive property r(  )

30. Materials Process Design and Control Laboratory CONTROL OF YOUNGS MODULUS: ITERATION 1 121 114 115 116 117 118 119 120 122 Youngs Modulus (GPa) First iteration Objective function: Minimize variation in Youngs Modulus

31. Materials Process Design and Control Laboratory CONTROL OF YOUNGS MODULUS: ITERATION 2 121 114 115 116 117 118 119 120 122 Youngs Modulus (GPa) Intermediate iteration

32. Materials Process Design and Control Laboratory CONTROL OF YOUNGS MODULUS: ITERATION 4 121 114 115 116 117 118 119 120 122 Youngs Modulus (GPa) Optimal solution

33. Materials Process Design and Control Laboratory MULTISCALE EXTRUSION –VARIATION IN OBJECTIVE FUNCTION Objective function: variance (Young’s Modulus) Iteration number Die Shape Youngs Modulus (GPa) 121 114 115 116 117 118 119 120 122 Uniform Youngs modulus Small die shape changes leads to better properties

34. Materials Process Design and Control Laboratory DESIGN PROBLEM Objective: Design the initial preform such that the die cavity is fully filled and the yield strength is uniform over the external surface (shown in Figure below). Material: FCC Cu Uniform yield strength desired on this surface Fill cavity Multi-objective optimization Increase Volumetric yield Decrease property variation 

35. Materials Process Design and Control Laboratory UPDATED LAGRANGIAN SHAPE SENSITIVITY FORMULATION Sensitivity to initial preform shape

36. Materials Process Design and Control Laboratory MULTI-SCALE DESIGN FOR OPTIMUM STRENGTH: ITERATION 1 70 80 90 100 110 120 130 Large Underfill variation in yield strength Yield strength (MPa)

37. Materials Process Design and Control Laboratory 70 80 90 100 110 120 130 Yield strength (MPa) Smaller under-fill variation in yield strength MULTI-SCALE DESIGN FOR OPTIMUM STRENGTH: ITERATION 2

38. Materials Process Design and Control Laboratory 70 80 90 100 110 120 130 Underfill Yield strength (MPa) Optimal yield strength Optimal fill MULTI-SCALE DESIGN FOR OPTIMUM STRENGTH: ITERATION 7

39. Materials Process Design and Control Laboratory COMPARISON OF FINAL PRODUCTS AT DIFFERENT ITERATIONS Uniform yield strength Cost function:  (underfill,variance of yield strength) Initial preform design After forging Iteration number

40. Materials Process Design and Control Laboratory CONCLUSIONS First-ever effort to optimize macro-scale properties of materials using multi-scale design of deformation processes. Ability to relate process variables to microstructure evolution and directly control microstructure- dependent properties. Microstructure evolution Multi-scale optimization