1. Second-Order Microstructure Sensitive Design Brent L. Adams Department of Mechanical Engineering Brigham Young University Work sponsored by the Army Research Office. February 2006

2. Collaborations David Fullwood, Carl Gao, Jordan Cox Brigham Young University Surya Kalidindi, Gwenaelle Proust, Max Binci Drexel University

3. MSD Spaces Designed Processing Routes Microstructure Hull Property Closure FE Model Design g g’

4. First-Order Linkage Between MSD Spaces Texture Hull Property Closure Homogenization Techniques 1-point statistics: f(g) and local properties: C ijkl e.g. First-Order Bounds for Elastic Stiffness: Global properties: C * ijkl

5. First-order linkage Volume average, f(g), of each state (e.g. orientation, g) known Fourier representation of f(g) used to preserve the symmetry of the material Bounding theories give max and min for global properties Wide range of possible microstructure designs for each chosen f(g), may give wide range of properties

6. Second-Order Linkage Between MSD Spaces Microstructure Hull Property Closure Homogenization Techniques 2-point statistics: f(g,g’|r) and local properties: C ijkl Global properties: C * ijkl

7. Second-order linkage Two-point correlation function, f 2 (g,g’|r) gives probability of finding orientations at particular separation r Texture function, M(x,g), gives texture at each ‘point’ and used as basis for mapping Fourier representations used to discretize real and orientation spaces and the linking map An algebraic instantiation relation is found linking the Fourier coefficients of M and f 2. Perturbation theory used to give single property for each microstructure Optimization theory finds boundary of closure

8. Common Framework For Field Theories ConductivityElasticityRigid Plasticity Variables Constitutive Relations Conservation Continuity

9. Second-Order Theories Low contrast expansion (polycrystals) Non-local constitutive relations (e.g. elasticity) 2 nd order theory for texture design introduces the 2-point spatial correlations of local lattice orientation … combining crystallographic and morphologic texture.

10. Explicit form of the second-order correction Second-order correction: Green’s function convolution Correlation tensors / 2-point orientation correlation functions Full second-order correction for statistically homogeneous polycrystals

11. Recovery of 2- point spatial (pair) correlations of orientation

12. Experimental Recovery of the Pair Correlation Functions PCFs Scan 1

13. Experimental Recovery of the Pair Correlation Functions PCFs Scan 1

14. Typical (renormalized) Pair Correlation Functions The PCF can be normalized such that g.s. dist.

15. ‘Coherence Limit’ and the ‘Coherence Plateau’ rcrc CP The Coherence Plateau increases as grain size/sample volume increases, and as the tesselation of the fundamental zone coarsens.

16. Central Problem: Can we define the set (hull) of all physically-realizable 2-point orientation correlation functions? The problem of r-interdependence in statistically-homogeneous microstructures: Upshot: no method is known for proceeding directly to the hull of 2- point orientation correlation functions.

17. Introducing the Texture Function (TF) The texture function M(x,g) is the volume density of material in an infinitesimal neighborhood of material point x that associates with lattice orientation in an infinitesimal neighborhood of measure dg of orientation g.

18. Introduce piecewise continuous rectangular models The influence of the spatial correlations of components of microstructure is obtained by evaluating the Green’s function convolutions over pairs of cubical boxes. The overall effect is then obtained by summing over all pairs.

19. Primitive representations (piecewise- continuous) of the Texture Function Primitive basis: indicator functions

20. Fundamental relationship between the TF and the 2-point correlation functions In continuous form: In finite Fourier form: Note the algebraic (quadratic) dependence between the coefficients of the TFs and those of the PCFs! Instantiation relation ->

21. COMPLETENESS: Construction of the Hull of Texture Functions All possible microstructures are convex combinations of the eigen-texture functions Define eigen-texture functions Each sub-cell has orientation from only one sub-region

22. Geometrical Interpretation of the Hull of Texture Functions Set of S constraining hyper- planes intersecting the SN- dimensional hyper-cube to define an SN-S dimensional polytope, also known as a “pyramidal polytope” Defines a hyper-cube in the SN- dimensional Fourier space *The corners (extreme points) of the polytope are the N S eigen-texture functions

23. G H I J K A B D C V = 0 V = 1/4 V = 1/8 E V = 3/8 V = 1/2 V = 5/8 V = 3/4 V = 7/8 V = 1 L M N O P Sequence of Polytopes for S=4, N=2 Letters locate vertices (eigen-microstructures).

24. H C I J A B D E G K L M N O P F Eigen-textures for S=4, N=2

25. Explicit form of the second-order correction in the primitive basis The Aleph coefficients are constants defined by the basic elastic constants of the phase, and by the Green’s functions appropriate to the boundary conditions of the problem. * Note the quadratic dependence on the coefficients of the texture functions*

26. Link between Hull and Closure Properties are quadratic functions of discretized texture function: Standard optimization techniques may be used to determine closure

27. Exploring the closure Points near the property closure boundary are hard to find This figure shows properties for 10,000 random microstructures plotted on the full closure

28. Exploring the closure Generalized Pareto front techniques are used to find the sides of the closure These are new methods, and use quadratic programming (QP) and sequential QP. Generalized weighted sum Adaptive normal boundary interception

29. Second order closure First-order closure (o) is larger than second order (  ) The second order closure does not give rigorous bounds But some parts of the first order closure may not be realizable

30. Genetic Algorithms Genetic algorithms may be used to find the closure boundary The maximin algorithm has been used to spread out the points The ‘corner’ points must be found first and continually re-inserted into the search since random microstructures are rarely near the boundary This method is generally good for large problems

31. Linearizing the quadratic solution to find eigen-microstructures Eigen-micro- structures are found on the boundary The linearized solution ( o ) is close to the full solution (  )

32. Eigen-microstructures around C 11 / C 66 closure

33. Set of eigen-texture functions for a 4-component model of size 4 3 Many eigen- microstruct- ures on the boundary have strong symmetry

34. Hole in plate example Minimize stress concentration factor (assuming anisotropic material): P P

35. Design Optimization The closure is searched using convex combinations of boundary points Once the boundary points are available, this is a rapid process

36. Optimal Design

37. Summary and Conclusions Second-order homogenization relations give rise to coupling of crystallographic and morphologic texture. Microstructure hulls of “texture functions” eliminate the complex r-interdependence of the PCFs. An algebraic instantiation relation is obtained, defining the set of RVEs associated with fixed PCFs. When expressed in terms of the primitive representations, the microstructure hull is a convex polytope, whose vertices are eigen-texture functions. The boundaries of the properties closure are readily explored by Pareto front optimization methods facilitated by QP and SQP methods.