1. Uncertainty quantification in multiscale deformation processes Babak Kouchmeshky Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 101 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 URL: http://mpdc.mae.cornell.edu/ Materials Process Design and Control Laboratory

2. Problem definition -Obtain the effect of uncertainty in initial texture on macro- scale material properties Uncertain initial microstructure

3. Materials Process Design and Control Laboratory Deterministic multi-scale deformation process

4. Materials Process Design and Control Laboratory Implementation of the deterministic 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

5. 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

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

7. Materials Process Design and Control Laboratory Evolution of texture 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(s,t) ODF EVOLUTION EQUATION – LAGRANGIAN DESCRIPTION

8. Materials Process Design and Control Laboratory Constitutive theory D = Macroscopic stretch = Schmid tensor = Lattice spin W = Macroscopic spin = Lattice spin vector Reorientation velocity Symmetric and spin components Velocity gradient Divergence of reorientation velocity 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

9. Materials Process Design and Control Laboratory Convergence of the deterministic problem Bulk modulus

10. Materials Process Design and Control Laboratory Convergence of the deterministic problem Young modulus

11. Materials Process Design and Control Laboratory Convergence of the deterministic problem Shear modulus

12. Materials Process Design and Control Laboratory Convergence of the deterministic problem

13. Materials Process Design and Control Laboratory Convergence of ODF

14. Materials Process Design and Control Laboratory Convergence of ODF

15. Materials Process Design and Control Laboratory Stochastic multi-scale deformation process

16. Materials Process Design and Control Laboratory The effect of uncertainty in the initial geometry of the work- piece on the macro-scale properties

17. Materials Process Design and Control Laboratory H Curved surface parametrization – Cross section can at most be an ellipse Model semi-major and semi-minor axes as 6 degree bezier curves Random parameters Deterministic parameters The effect of uncertainty in the initial geometry of the work- piece on the macro-scale properties

18. STOCHASTIC COLLOCATION STRATEGY Use Adaptive Sparse Grid Collocation (ASGC) to construct the complete stochastic solution by sampling the stochastic space at M distinct points Two issues with constructing accurate interpolating functions: 1)What is the choice of optimal points to sample at? 2) How can one construct multidimensional polynomial functions? Materials Process Design and Control Laboratory 1.X. Ma, N. Zabaras, A stabilized stochastic finite element second order projection methodology for modeling natural convection in random porous media, JCPA stabilized stochastic finite element second order projection methodology for modeling natural convection in random porous media 2.D. Xiu and G. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comp. 24 (2002) 619-644The Wiener-Askey polynomial chaos for stochastic differential equations 3.X. Wan and G.E. Karniadakis, Beyond Wiener-Askey expansions: Handling arbitrary PDFs, SIAM J Sci Comp 28(3) (2006) 455-464Beyond Wiener-Askey expansions: Handling arbitrary PDFs Since the Karhunen-Loeve approximation reduces the infinite size of stochastic domain representing the initial texture to a small space one can reformulate the SPDE in terms of these N ‘stochastic variables’

19. Materials Process Design and Control Laboratory Mean(G) Var(G) Mean(B) Var(B) Mean(E) Var(E) The effect of uncertainty in the initial geometry

20. Materials Process Design and Control Laboratory Error of Mean(B) Error of Var(B) Comparison with Monte-Carlo Error of Mean(E) Error of Var(E) Error of Mean(G) Error of Var(G)

21. Materials Process Design and Control Laboratory Reduced order model for a stochastic microstructure Current method where are modes strongly orthogonal in Rodrigues space and are spatial modes weakly orthogonal in space 1- D. Venturi, X. Wan, G.E. Karniadakis, J. fluid Mech. 2008, vol 606, pp 339-367 (1)

22. Materials Process Design and Control Laboratory Reconstructing a stochastic microstructure Step1: Construct the autocorrelation using the snapshots Step2: Obtain the eigenvalues and eigenvectors: ;

23. Materials Process Design and Control Laboratory Reconstructing a stochastic microstructure Step3: Obtain the spatial modes Step4: Decompose the spatial modes using the polynomial Chaos: are in a one to one correspondent to the Hermite polynomials.

24. Materials Process Design and Control Laboratory B E G Comparison between the original microstructure and the reduced order one

25. Materials Process Design and Control Laboratory B E G Comparison between the original microstructure and the reduced order one

26. Materials Process Design and Control Laboratory B E G Comparison between the original microstructure and the reduced order one

27. Materials Process Design and Control Laboratory B E G Comparison between the original microstructure and the reduced order one

28. Materials Process Design and Control Laboratory B E G Comparison between the original microstructure and the reduced order one

29. Materials Process Design and Control Laboratory Mean(G) Mpa Mean(B) Mpa Mean(E) Mpa Original Reconstructed Mean(G) Mpa Mean(B) Mpa Mean(E) Mpa Comparison between the original microstructure and the reduced order one

30. Materials Process Design and Control Laboratory Var(G) Var(B) Var(E) Original Reconstructed Var(G) Var(B) Var(E) Comparison between the original microstructure and the reduced order one

31. Materials Process Design and Control Laboratory The effect of uncertainty in the initial texture of the work- piece on the macro-scale properties

32. Materials Process Design and Control Laboratory Conclusion A reduced order model for quantifying the uncertainty in multi-scale deformation process has been provided