1. Materials Process Design and Control Laboratory Design and control of properties in polycrystalline materials using texture- property-process maps 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://mpdc.mae.cornell.edu/ V. Sundararaghavan and Prof. Nicholas Zabaras Supported by AFOSR, ARO

2. Materials Process Design and Control Laboratory MOTIVATION GRAPHICALLY REPRESENT THE SPACE OF MICROSTRUCTURES, PROPERTIES AND PROCESSES Applications: (i) Identify microstructures that have extremal properties. (ii) Identify processing sequences that lead to desired microstructures and properties. Taylor factor along RD Taylor factor along TD b 2. Rolling 3. Rolling followed by drawing 1. Drawing R C a d f e Process Property-process space Microstructure representations Process-structure space Process paths A 100 A 1000  A 80 Property-structure space

3. Materials Process Design and Control Laboratory Example: E <= E 1 x 1 + E 2 x 2 + E 3 x 3 (upper bound theory) E 1,2,3 are the Young’s modulus of each phase x 1,2,3 are the volume fractions of each phase Find microstructures with Young’s Modulus <= E x 1 + x 2 + x 3 = 1: Microstructure plane for 3 phase material x1x1 x2x2 x3x3 E <= E 1 x 1 + E 2 x 2 + E 3 x 3 Property plane Property Iso-line Microstructures with Youngs Modulus <= E FIRST ORDER STRUCTURE SPACE x1x1 x2x2 x3x3

4. Materials Process Design and Control Laboratory FIRST ORDER REPRESENTATION OF MICROSTRUCTURES Crystal/lattice reference frame e2e2 ^ Sample reference frame e1e1 ^ e’ 1 ^ e’ 2 ^crystal e’ 3 ^ e3e3 ^ n r = n tan(  /2) Cubic crystal RODRIGUES’ REPRESENTATION FCC FUNDAMENTAL REGION Particular crystal orientation r ODF representation for design Spectral (Adams, Kalidindi, Garmestani) FE discretization of RF space (Dawson) High dimensional space, difficult to visualize process-structure-property maps

5. Materials Process Design and Control Laboratory FINITE ELEMENT INTEGRATION IN RF SPACE Integration point [0.25,0.25,0.25] = r i, A i Normalization: = 1

6. Materials Process Design and Control Laboratory MATERIAL PLANE Applications: (i) Identify microstructures that have extremal properties. (ii) Identify closures of properties. SPACE OF ALL POSSIBLE ODFs Mathematical representation of all possible ODFs using FE degrees of freedom. Three constraints define the space of first order microstructural feature (ODF): Normalization, q T A = 1 Lower bound, A >= 0 Crystallographic Symmetry, r’ = Gr AA A 100 A 1000  ODF at  A 80 ~10 3 dimensions

7. Materials Process Design and Control Laboratory UPPER BOUND THEORY: LINEARIZATION Upper bound of a polycrystal property can be expressed as an expectation value or average given by A 100 A 1000  ODF at  A 80 ~10 3 dimensions

8. Materials Process Design and Control Laboratory LINEAR PROGRAMMING Geometrically, the linear constraints define a convex polyhedron, which is called the feasible region.convexpolyhedron -- all local optima are automatically global optima. ---optimal solution can only occur at a boundary point of the feasible region.

9. Materials Process Design and Control Laboratory EXTREMAL PROPERTY POINTS Constraints and objectives are linear in the ODF problem Identify microstructures that maximize properties in a particular direction (eg. C ) and lb = 0 Normalization Extremize property positiveness LINEAR PROGRAMMING Number of variables: 448 Number of linear inequality constraints: 448 Number of linear equality constraints: 1 For minima For maxima

10. Materials Process Design and Control Laboratory EXTREMAL TEXTURES Taylor factor calculated through Bishop-Hill analysis 500 400 300 200 100 0 500 400 300 200 100 0 (b) (c) (a) 0 3000 2000 1000 (d) 0 3000 2000 1000 ODF for maximum Taylor factor along RD (=3.668) ODF for maximum Taylor factor along TD (=3.668) ODF for maximum C 44 (=74.923 GPa) ODF for maximum C 55 (=74.923 GPa)

11. Materials Process Design and Control Laboratory UPPER BOUND PROPERTY CLOSURES 168 188 178 198 208 218 228 237 33 43 53 63 73 83 93 97 168 188 178 208 218 237 198 228 M along TD M along RD M at 45 degrees to RD 2.4 2.8 3.2 3.6 3.5 3.0 2.5 3.0 3.5 (b) C 11 (GPa) C 66 (GPa) C 22 (GPa) (a) Closure for stiffness constants (C 11,C 22,C 66 ) Closure for Taylor factor computed along RD, 45 o to RD and TD

12. Materials Process Design and Control Laboratory OPTIMIZATION ON MATERIAL PLANE Given Initial ODF find the closest ODF (A) that satisfies the desired property (d) Minimize scalar: r 0 Such that: Positiveness Uniform norm constraint Normalization A 100 A 1000 ODF at  A 80 ~10 3 dimensions Initial ODF roro Desired property

13. Materials Process Design and Control Laboratory BOUND PROBLEM Number of variables: 449 Number of linear inequality constraints: 1344 Number of linear equality constraints: 2 Solution: r 0 = 7.8569 (No other solution can be confined within A +/- r 0 ) Input ODF(A) C = 209.0696 GPa Optimized ODF C = 231.0 GPa (this is close to extreme value of 236.82 GPa)

14. Materials Process Design and Control Laboratory TEXTURE EVOLUTION Represent the ODF as Reduced model for the evolution of the ODF Initial conditions Viscoplastic rate dependent model, no hardening (Acharjee and Zabaras, 2003) Taylor hypothesis X-axis compression

15. Materials Process Design and Control Laboratory MODEL REDUCTION Suppose we had an ensemble of data (from experiments or simulations) for the ODF: such that it can represent the ODF as: Is it possible to identify a basis POD technique – Proper Orthogonal Decomposition Method of snapshots Eigenvalue problem where Other features Generated basis can be used in interpolatory as well as extrapolatory modes First few basis vectors enough to represent the ensemble data

16. Materials Process Design and Control Laboratory PROCESS PLANE    a 1   +a 2    a 3   a1a1 a2a2 a3a3 Equation of a plane Basis already includes symmetries. Normalization Lower bound Crystallographic Symmetry

17. Materials Process Design and Control Laboratory TEXTURE PLANES FOR SOME PROCESSES Plane strain compression Tension/compression X-Y shear Y-Z shear

18. Materials Process Design and Control Laboratory PROCESS PLANES FOR YIELD STRENGTH ALONG RD Plane strain compressionTension/compression Y-Z plane shearX-Y plane shear

19. Materials Process Design and Control Laboratory PROCESS PATH REPRESENTATION B B’ A A’ C’ C b b’ a’ a c c’ m m’ n n’ a1a1 a2a2 a3a3 a1a1 a2a2 a3a3 A A’ R R’ B B’ m m’ n n’ Process plane for x-axis tension (ensemble obtained by processing an initial random texture to 0.1 strain) Process plane for y-axis rolling followed by x-axis tension (initial random texture processed to 0.2 strain) 90% accurate reconstruction width Compression path Tension path initial textures Final textures 0.17 initial strain 0.07 initial strain

20. Materials Process Design and Control Laboratory BOUNDING IN A PROCESS BASIS Find the closest distance of a desired ODF in the material plane to an ODF in the process plane Desired property Deviation of the optimal ODF from the basis ODF space process basis Microstructure with the desired property Closest solution Process microstructures Minimize bound on the deviation Process plane equation Normalization

21. Materials Process Design and Control Laboratory PROCESS PLANE SOLUTION FOR DESIRED STIFFNESS (c) Normalization constraint (a) (b) (d) A A’ 1 2 a1a1 a2a2 a3a3 - Desired stiffness properties {c11 = 210.85 GPa, c22 = 210.42 GPa, c66 = 66.31 GPa}. - Process plane: x-axis tension preceded by y-axis rolling of a random texture to 0.1 strain. Optimal ODF Final ODF in the process plane Exact solution in the material plane Optimal process path in the second stage

22. Materials Process Design and Control Laboratory (a) (b) (c) PROCESS PLANE SOLUTION FOR DESIRED STRENGTH Desired Taylor Factors Design for obtaining a desired Taylor factor distribution. Three different process (x-axis rolling, tension and x-y shear) are tested. Provides the ability to select best processing paths Optimal ODF in Tension process Optimal ODF in rolling process

23. Materials Process Design and Control Laboratory PROPERTY CLOSURE OF A PROCESS PLANE Where, With normalization constraint Maximize (or minimize) properties And positiveness constraint Taylor factor along RD Taylor factor along TD R C Process plane Process-property plane

24. Materials Process Design and Control Laboratory CROSS-PLOTS (a) (b) a1a1 a2a2 M along RD (c) a1a1 a2a2 M along TD (d) Taylor factor along RD Taylor factor along TD R C Identify optimal textures from the intersections of property curves on the structure- property space

25. Materials Process Design and Control Laboratory PROCESS SELECTION Taylor factor along RD Taylor factor along TD b 4. z-axis rolling following y- axis rolling and x-axis tension 2. y-axis rolling 3. y-axis rolling followed by x-axis tension 1. x-axis tension R C a d f e Property space of Taylor factors for x- and y- direction loading corresponding to various process planes. Desired property is C and initial property is R Multiple processes can be identified by superposing the property closures of different process planes on the property space. R-C and R-a-b-C are two possible routes.

26. Materials Process Design and Control Laboratory NON-LINEAR PROPERTY SURFACES Lankford R parameter = 1.0253 surface (along RD) on the tension basis Youngs Modulus = 145.3GPa (along RD) surface on the tension basis Two solutions Solution on the reduced material plane Non-linearly related to ODF Linearization?

27. Materials Process Design and Control Laboratory FUTURE CHALLENGES Development of basis functions for complex strain paths Construction of property iso-surfaces in higher order feature spaces using statistical learning techniques Identification of error limits in reduced order models as part of design procedure

28. Materials Process Design and Control Laboratory CONCLUSIONS Linear analysis of texture–property relationships using process-based representations of Rodrigues space Acta Materialia, Volume 55, Issue 5, March 2007, Pages 1573-1587 Veera Sundararaghavan and Nicholas Zabaras The concept of a ‘material plane’ in Rodrigues space was employed to identify optimal or extremal ODFs A new concept of a ‘process plane’ was established that represents the space of reduced-order coefficients for a given process. The process plane is capable of extrapolating several different processing paths. Linear programming methods were constructed to solve problems involving identification of ODFs on the process plane that are as close as possible to desired ODFs on the ‘material plane’. Graphical solution to the process sequence selection problem was enabled through the identification of process paths on property closures of process planes.