1. Materials Process Design and Control Laboratory Babak Kouchmeshky Admission to Candidacy Exam Presentation Date April 22,2008 A multi-scale design approach for tailoring the macro-scale properties of polycrystalline materials

2. Outline Modeling HCP polycrystals deforming by slip and twinning Modeling HCP polycrystals deforming by slip and twinning A design approach for tailoring the processing parameters that lead to desired macro-scale properties A design approach for tailoring the processing parameters that lead to desired macro-scale properties A microstructure-sensitive design approach for controlling properties of HCP materials A microstructure-sensitive design approach for controlling properties of HCP materials Future work Future work Materials Process Design and Control Laboratory

3. I. Constitutive model Materials Process Design and Control Laboratory Theory

4. DEFORMATION TWINNING  Twinning produces a rotation of crystal lattice.  Important deformation mode for HCP materials  Dominant at room temperature. Twinning is lesser at high temperatures. Materials Process Design and Control Laboratory

5. SLIP AND TWIN PLANES Basal slip system Prismatic slip Pyramidal slip Pyramidal twin a 2 - axis c - axis a 1 - axis a 3 - axis e1e1 e2e2 e3e3 Orthogonal system used for modeling Materials Process Design and Control Laboratory

6. TWINS MODELED AS PSEUDO-SLIP Materials Process Design and Control Laboratory Dawson and Myagchilov (1999)

7. CONSTITUTIVE MODEL FOR SLIP AND TWINNING Velocity gradient Hardening law Isotropic flow rule to account for grain boundary accommodation For slip and twin systems no hardening Consistency condition (Anand,IJP 2003) Materials Process Design and Control Laboratory Condition for slip and twinning

8. Crystallographic slip, twinning 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. 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 p p e F trial e e FrFr FcFc Intermediate configuration Deformed configuration Intermediate configuration Reference configuration INCREMENTAL KINEMATICS Materials Process Design and Control Laboratory

9. Constitutive theory D = Macroscopic stretch = Schmid tensor = Lattice spin W = Macroscopic spin = Lattice spin vector Polycrystal plasticity Initial configuration B o B F * F p F Deformed configuration Stress free (relaxed) configuration n0n0 s0s0 n0n0 s0s0 n s (2) Ability to capture material properties in terms of the crystal properties (1) State evolves for each crystal

10. CONSTITUTIVE MODEL FOR SLIP AND TWINNING Solve for the reorientation velocity and the rate of change in volume fraction of twins Solve for shearing rates on slip and twin systems Rate of change of volume fraction Materials Process Design and Control Laboratory

11. Arresting of twinning systems Three stages of strain hardening obtained using current model. is a uniform distribution between 0.3 and 1.0. is supposed to be 0.2.

12. Materials Process Design and Control Laboratory Orientation distributions Any macroscale property can be expressed as an expectation value if the corresponding single crystal property χ (r,t) is known. Determines the volume fraction of crystals within a region R' of the fundamental region R Probability of finding a crystal orientation within a region R' of the fundamental region Characterizes texture evolution ORIENTATION DISTRIBUTION FUNCTION – A(r,t) – reorientation velocity ODF EVOLUTION EQUATION – EULERIAN DESCRIPTION

13. 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 Discrete aggregate of crystals (Anand et al.) Comparing & quantifying textures Continuum representation Orientation distribution function (ODF) Handling crystal symmetries Evolution equation for ODF Different methodologies in representing the texture Materials Process Design and Control Laboratory

14. Hexagonal crystal (HCP) Angle-axis and Rodrigues representation RODRIGUES REPRESENTATION  Neo-Eulerian representation of orientation  Rotations about a fixed axis trace straight lines in parameter space  Set of orientations equidistant from two rotations is always a plane  Helps reduce symmetries to between a pair of planes – fundamental region ANGLE AXIS REPRESENTATION Any orientation can be uniquely represented by a rotation about an axis n by an angle Φ Φ n

15. LATTICE REORIENTATION DUE TO TWINNING Twin plane normal angle Crystal Axis = h Twin mapping is a reflection Can be represented as a rotation of crystal axis about twin normal through 180 o In quaternion representation, T q = [0,n 1,n 2,n 3 ] 1)Convert crystal axis h to the quaternion representation h q 2)Perform quaternion product Q = T q h q 3)Project Q to the fundamental region (Q F ) based on crystal symmetries 4)Convert Q F to Rodrigues representation Materials Process Design and Control Laboratory We take advantage of Quaternions in here. They prove useful for coordinate transformations. The quaternion method is the natural choice when the coordinate systems keep moving.

16. ORIENTATION DISTRIBUTION FUNCTION (ODF) Conservation principle Texture can be described, quantified & compared Why continuum approach for ODF? EVOLUTION EQUATION FOR THE ODF J – Jacobian determinant of the reorientation of the crystals r – orientation of the crystal. A – is the ODF, a scalar field; Constitutive sub-problem Taylor hypothesis: deformation in each crystal of the polycrystal is the macroscopic deformation. Materials Process Design and Control Laboratory

17. ODF EVOLUTION EQUATION WITH TWINNING Calculate crystal reorientation Initial random ODF = 1.2158 at all nodal points HCP Fundamental region Source term due to twinning Volume fraction lost Volume fraction gained from other orientations Total Lagrangian formulation Materials Process Design and Control Laboratory

18. VOLUME FRACTION OF TWINS Volume fraction lost due to transfer of orientation from r to r k r rkrk Orientation space Materials Process Design and Control Laboratory Dawson and Myagchilov (1999)

19. Numerical results Materials Process Design and Control Laboratory

20. I. Tension mode Materials Process Design and Control Laboratory modeling tension on an initially textured Magnesium alloy AZ31B rod modeling tension on an initially textured Magnesium alloy AZ31B rod

21. Initial texture Materials Process Design and Control Laboratory initial texture in the experiment by anand and staroselsky, 2003 initial texture used in the simulation

22. Material properties Materials Process Design and Control Laboratory Elastic constants: Slip resistances: Slip and twining systems:

23. Final texture Materials Process Design and Control Laboratory Texture of the Mg rod at the tensile strain of 15% in the experiment by anand and staroselsky, 2003 Texture of the Mg rod at the tensile strain of 15% Comparison between stress- strain curve from experiment, this work and numerical simulation by anand and staroselsky, 2003

24. II. Compression mode Materials Process Design and Control Laboratory Modeling compression on an initially textured Magnesium alloy AZ31B rod Modeling compression on an initially textured Magnesium alloy AZ31B rod

25. Material properties Materials Process Design and Control Laboratory Elastic constants: Slip resistances: Slip and twining systems:

26. Final texture Materials Process Design and Control Laboratory Texture of the Mg rod at the tensile strain of 18% in the experiment by Anand and Staroselsky, 2003 Texture of the Mg rod at the tensile strain of 18%

27. Stress-strain curve Materials Process Design and Control Laboratory Comparison between stress- strain curve from experiment, this work and numerical simulation by anand and staroselsky,2003 3 different stages in the normalized strain hardening response

28. III. Shear mode Materials Process Design and Control Laboratory A shear mode is assumed. In this problem texture evolution and stress-strain curve is examined for Titanium.

29. Material properties Materials Process Design and Control Laboratory Elastic constants: Slip resistances: Slip and twining systems:

30. Result (cont.) Materials Process Design and Control Laboratory Experimentally measured texture of Ti at effective strain 1 for the shear mode Numerically predicted texture of Ti at effective strain 1 for the shear mode X. Wu et al, Acta Materialia 55 (2007) Experiment This work

31. Result (cont.) Materials Process Design and Control Laboratory Experiment is done by Wu et al. (2007)

32. IV. Plain strain compression mode Materials Process Design and Control Laboratory A plane strain compression mode is assumed. In this problem texture evolution is examined for Titanium.

33. Material properties Materials Process Design and Control Laboratory Elastic constants: Slip resistances: Slip and twining systems:

34. Result Materials Process Design and Control Laboratory Results obtained by Myagchilov and Dawson, Simul. Mater. Sci. Eng. 7 (1999)975-1004 6 5 4 3 2 1 0 Evolved texture at effective strain of 0.5

35. Materials Process Design and Control Laboratory CONCLUSION(part 1) 1)Continuous representation of texture Eliminates the need for splitting existing elements to account for new orientations caused by twinning Provides a natural tool for calculating sensitivities needed for design problem 2)Twinning is accounted through pseudo shear and reorientation of crystals 3)Twin saturation is phenomenologically accounted for. 4)ODF conservation equation modified to include the source and sink terms due to twinning. 5)The constitutive model is tested for Titanium and Magnesium alloy AZ31B.

36. Materials Process Design and Control Laboratory Enhancing properties of polycrystals through a sensitivity problem that spans macro and micro scales The orientation of crystals in a poly crystal sample has a direct influence on the properties of the specimen in the macro scale. Crystals reorient during the deformation. So macro scale processing parameters like velocity gradient affect the crystal reorientation.

37. Materials Process Design and Control Laboratory Definition of the problem The aim of this problem is to design the processing parameters in a sequence of two processes such that a micro structure with desired properties is obtained Desired qualities: High hardness and ductility Convex hull of B,G,B/G The hardness and ductility are presented by Bulk modulus (B), Shear modulus and B/G.

38. Materials Process Design and Control Laboratory Problem statement Sub problems: 1- Find the texture that provides the maximum hardness and ductility 2- Find the reduced order model for the processes 3- Find the optimum texture from process plane 4- Define the design problem as two coupled optimization problems where each represent a process. 5- Find the convex hull of textures obtainable from process 1 6- Define a supplementary problem for reducing the computational efforts needed for the inverse problem 7- Solve for the constrained optimization

39. Materials Process Design and Control Laboratory Optimizing the processes to get the optimum texture Step1: Find the design parameter L2 and initial texture A1 such that at the end of process 2 the desired texture is obtained. There will be a constraint on A1 based on the textures obtainable from texture 1. Step2: find the design parameter L1 in process 1 that leads to final texture A1. T 2T A1 A2 L2 L2 and L1 are the design parameters Process 1 is supposed to start from a random texture T A1 L1 0

40. Materials Process Design and Control Laboratory The sensitivity problem The sensitivity problem with respect to design parameters (L1,L2) The sensitivity problem with respect to the field A1

41. Materials Process Design and Control Laboratory Sensitivity of the reorientation velocity Constitutive sensitivity problem

42. Materials Process Design and Control Laboratory Definition of the adjoint problem Lagrange identity for obtaining the adjoint operator Gradient of the objective functional where is the solution of the following problem

43. Materials Process Design and Control Laboratory The texture that provides the maximum hardness and ductility Convex hull of B,G and B/G values obtainable for a single crystal B= 124.8 GPa, G=58.87 GPa B/G=2.12 The optimum texture from material space

44. Materials Process Design and Control Laboratory The reduced order model first process (simple tension, with random initial texture) second process (plain strain compression, with initial texture selected from the convex hull of all textures available from process 1.) Convex hull of textures for process 1

45. Materials Process Design and Control Laboratory 3- Find the optimum texture from process plane Prioritizing the objectives The emphasize is given on the ductility(B/G). Other parameters are treated by inequality constraints which forces them to be greater than two third of the maximum values obtainable.

46. Materials Process Design and Control Laboratory 3- Find the optimum texture from process space The optimum texture obtained from the process plane of the sequence of a tension process followed by a plain strain compression process Process space contains all the plausible textures obtainable from a sequence of processes Optimum texture from material space Optimum texture from Process space

47. Materials Process Design and Control Laboratory Verify From sensitivity analysis,1 st process t=5 sec From Finite difference, 1 st process t=5 sec Verify sensitivity problem Relative error: 0.3% Verify supplementary problem

48. Materials Process Design and Control Laboratory Verify To verify define t(sec) Relative error% 0.014 10 8 6 5 0.020 0.029 0.036 0.50 0.61 0.58 0.65 0.20840.20733 0.1982 0.1969 0.2429 0.2415 0.2617 0.2600

49. Materials Process Design and Control Laboratory Objective function for process 1 Objective function for process 2

50. Materials Process Design and Control Laboratory Conclusion (part 2) The problem of obtaining metallic alloys with optimum hardness and ductility through cold processing is addressed. The problem of finding optimum macro-scale properties was converted to that of polycrystalline texture through linear homogenization methodology. The optimum texture was projected from the material space to process space which contains all textures obtainable from the sequence of two parameters. Process parameters for a sequence of two deformation modes are optimized through two coupled optimization problems A functional optimization methodology is used for addressing the infinite dimensional optimization problem defined. Solution of an adjoint problem is used for calculating the gradient of the objective function with respect to a field parameter (initial texture of the second process).

51. Materials Process Design and Control Laboratory Forging Goals : Goals : 1.Minimal material wastage due to flash 2. Filling up the die cavity Material : Ti Why multi scale? Why multi scale? –The evolution of the material properties at the macro scale has a strong correlation with the underlying microstructure. Multi-scale polycrystal plasticity

52. Multi-length scale design environment Coupled micro-macro direct model Selection of the design variables like variables like preform parameterization Coupled micro-macro sensitivity model Reduced modeling of microstructure/tex ture evolution for meaningful and effective control. Design based on: Polycrystal plasticity, evolution of texture, multi-length scale analysis Deformation problem: -Updated Lagrangian framework -Connection to the micro scale through Taylor hypothesis - Microstructure represented as orientation distribution function (ODF) in Rodrigues space. Materials Process Design and Control Laboratory

53. Implementation of the direct problem Meso Macro formulation for macro scale Update macro displacements Texture evolution update Polycrystal averaging for macro-quantities Integration of single crystal slip and twinning laws Macro-deformation gradient microscale stress Macro-deformation gradient Micro

54. Materials Process Design and Control Laboratory THE DIRECT CONTACT PROBLEM r n Inadmissible region Reference configuration Current configuration Admissible region Impenetrability Constraints Augmented Lagrangian approach to enforce impenetrability

55. Polycrystal average of orientation dependent property Continuous representation of texture Materials Process Design and Control Laboratory REORIENTATION & TEXTURING

56. Materials Process Design and Control Laboratory A sample direct problem Equivalent stress (MPa) Equivalent stress for the direct problem in different time steps

57. Materials Process Design and Control Laboratory SCHEMATIC OF THE CONTINUUM SENSITIVITY METHOD Advantage : Fast Multi-scale optimization Requires 1 Non-linear and n Linear multi-scale problems for each step of the optimization algorithm. n: number of design parameters Equilibrium equation Design derivative of equilibrium equation Material constitutive laws Design derivative of the material constitutive laws Incremental sensitivity constitutive sub-problem Time and space discretized weak form Sensitivity weak form

58. Materials Process Design and Control Laboratory Curved surface parametrization – Cross section can at most be an ellipse Model semi-major and semi-minor axes as 6 degree bezier curves Design vector H Representing the preform shape

59. Materials Process Design and Control Laboratory DEFINITION OF PARAMETER SENSITIVITY state variable sensitivity contour w.r.t. parameter change X = X (Y;  s ) o F R + F R Y X X+X o x n +x n o xnxn o F n + F n FRFR FnFn BRBR BoBo I+L o x+x oo F r + F r x B x n + x n = x (Y, t n ;  s +   s ) o ~ Q n + Q n = Q (Y, t n ;  s +   s ) o ~ x = x (x n, t ;  s ) ^ B n x n = x (X, t n ;  s ) ~ Q n = Q (X, t n ;  s ) ~ I+L n Main Features Main Features Mathematically rigorous definition of sensitivity fieldsMathematically rigorous definition of sensitivity fields Gateaux differentials (directional derivatives) referred to fixed Y in the configuration B RGateaux differentials (directional derivatives) referred to fixed Y in the configuration B R o X + X= X (Y;  s +   s ) o ~ ~ FrFr x + x = x (x+x n, t ;  s +   s ) ^ oo

60. Materials Process Design and Control Laboratory SENSITIVITY KINEMATIC PROBLEM 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 Constitutive problem Regularized contact problem Kinematic problem Sensitivity of ODF evolution P r and F, o  o o

61. 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

62. 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

63. 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

64. Materials Process Design and Control Laboratory POLYCRYSTAL SENSITIVITY ANALYSIS EVOLUTION EQUATION FOR THE SENSITIVITY OF THE ODF Assumption: extended Taylor hypothesis for 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

65. Materials Process Design and Control Laboratory REDUCED ORDER MODEL FOR THE ODF SESNSITIVITY 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 o

66. Materials Process Design and Control Laboratory IMPLEMENTATION OF REDUCED MODEL -Regions suspected to come into contact (2 sets, before and after coming into contact) -Other part of the macro scale (1 set) Time varying boundary conditions during the deformation 3 different set of Reduced order modes for the microstructure are constructed for different regions in the macro scale. large number of reduced order modes

67. Materials Process Design and Control Laboratory IMPLEMENTATION OF REDUCED MODEL

68. Materials Process Design and Control Laboratory Validation of the reduced-order model: Sensitivity analysis The same set of regions are used for the sensitivity problem FDM solution Full model (Continuum sensitivity method) Reduced model ODF:22.6253.253.8754.5

69. Materials Process Design and Control Laboratory Design  Objective function

70. Materials Process Design and Control Laboratory Design First iteration of optimization Last iteration of optimization Preform shape for the first iteration Preform shape for the last iteration

71. Materials Process Design and Control Laboratory Distribution of texture for some points on macro scale

72. Materials Process Design and Control Laboratory Conclusion (part 3) A multi scale design methodology is applied for the problem of forging Ti alloy -The goal has been to fill the die cavity and minimize the wastage of material -Continuum sensitivity approach using extended Taylor hypothesis is used. -Reduced order modeling is used to address the large amount of computational effort needed

73. Materials Process Design and Control Laboratory Future direction 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 Reduced Order Modes Data mining techniques Multi-scale Computation Design variables (β) are macro design variables Processing sequence/parameters Design objectives are micro-scale averaged material/process properties Database

74. Materials Process Design and Control Laboratory Future direction In addition to preform shape, parameters like forging velocity and initial texture of the workpiece should be considered Obtaining optimized distribution of macroscale properties like Young modulus, Yield strength, etc. (with respect to design parameter )

75. Materials Process Design and Control Laboratory Plan for future work Extend the design methodology for sequence of processes to multi-scale polycrystal plasticity Extend the graphically based selection of processes to mathematically rigorous method using nonlinear model reduction.

76. Materials Process Design and Control Laboratory Publication and presentations since Aug. 2006 Journal: B. Kouchmeshky and N. Zabaras, "Modeling the response of HCP polycrystals deforming by slip and twinning using a finite element representation of the orientation space", Int. J. Plasticity, submitted. B. Kouchmeshky and N. Zabaras, "A designing approach with reduced computational effort for tailoring the processing parameters that lead to desired macro-scale properties in HCP polycrystals", in preparation. Conference: B. Kouchmeshky and N. Zabaras, "A microstructure-sensitive design approach for controlling properties of HCP materials", presented at the TMS Annual Meeting, New Orleans, Louisiana, March 9-13, 2008 B. Kouchmeshky and N. Zabaras, "A simple non-hardening rate- independent constitutive model for HCP polycrystals deforming by slip and twinning", presented at the TMS Annual Meeting, New Orleans, Louisiana, March 9-13, 2008