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An information-theoretic approach for property prediction of random microstructures Nicholas Zabaras 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: [email protected] URL: http://mpdc.mae.cornell.edu/ CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory NEED FOR UNCERTAINTY ANALYSIS Uncertainty is everywhere From NIST Porous media From Intel website Silicon wafer From GE-AE website Aircraft engines From DOE Material process Variation in properties, constitutive relations Imprecise knowledge of governing physics, surroundings Simulation based uncertainties (irreducible) CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory UNCERTAINTY AND MULTISCALING Uncertainties introduced across various length scales have a non-trivial interaction Current sophistications – resolve macro uncertainties Micro Physical properties, structure follow a statistical description CORNELL U N I V E R S I T Y Meso Macro Use micro averaged models for resolving physical scales Imprecise boundary conditions Initial perturbations Materials Process Design and Control Laboratory UNCERTAINTY IN METAL FORMING PROCESSES Material Process Model Forging rate Stereology/Grain texture Yield surface changes Die/Billet shape Dynamic recrystallization Friction Phase transformation Isotropic/Kinematic hardening Cooling rate Phase separation Stroke length Internal fracture Billet temperature Other heterogeneities Softening laws Rate sensitivity Internal state variables Dependance Nature and degree of correlation Forging velocity Small change in preform shape could lead to underfill Die shape Die/workpiece friction Initial preform shape Material properties/models CORNELL U N I V E R S I T Y Texture, grain sizes Materials Process Design and Control Laboratory RANDOM VARIABLES = FUNCTIONS ? Math: Probability space (W, F, P) Sample space Probability measure Sigma-algebra Random variable F W W W : Random variable W : (W) A stochastic process is a random field with variations across space and time X : ( x, t , W ) CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory SPECTRAL STOCHASTIC REPRESENTATION A stochastic process = spatially, temporally varying random function X : ( x, t , W ) CHOOSE APPROPRIATE BASIS FOR THE PROBABILITY SPACE HYPERGEOMETRIC ASKEY POLYNOMIALS GENERALIZED POLYNOMIAL CHAOS EXPANSION SUPPORT-SPACE REPRESENTATION PIECEWISE POLYNOMIALS (FE TYPE) SPECTRAL DECOMPOSITION KARHUNEN-LOÈVE EXPANSION COLLOCATION, MC (DELTA FUNCTIONS) SMOLYAK QUADRATURE, CUBATURE, LH CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory KARHUNEN-LOEVE EXPANSION X ( x, t , ) X ( x, t ) X i ( x, t )i ( ) i 1 ON random variables Deterministic functions Stochastic Mean process function Deterministic functions ~ eigen-values , eigenvectors of the covariance function Orthonormal random variables ~ type of stochastic process In practice, we truncate (KL) to first N terms X ( x, t , ) fn( x, t , 1, CORNELL U N I V E R S I T Y ,N ) Materials Process Design and Control Laboratory GENERALIZED POLYNOMIAL CHAOS Generalized polynomial chaos expansion is used to represent the stochastic output in terms of the input X ( x, t , ) fn( x, t , 1, ,N ) Stochastic input Z ( x, t , ) Z i ( x, t ) i (ξ( )) i 0 Stochastic output Askey polynomials in input Deterministic functions Askey polynomials ~ type of input stochastic process Usually, Hermite, Legendre, Jacobi etc. CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory SUPPORT-SPACE REPRESENTATION Any function of the inputs, thus can be represented as a function defined over the support-space A ξ (1 , , N ) : f (ξ) 0 FINITE ELEMENT GRID REFINED IN HIGH-DENSITY REGIONS X Xˆ L2 2 ˆ ( X (ξ) X (ξ)) f (ξ)dξ A Ch q 1 JOINT PDF OF A TWO RANDOM VARIABLE INPUT CORNELL U N I V E R S I T Y – SMOLYAK QUADRATURE OUTPUT REPRESENTED ALONG SPECIAL COLLOCATION POINTS – IMPORTANCE MONTE CARLO Materials Process Design and Control Laboratory UNCERTAINTY DUE TO MATERIAL HETEROGENEITY State variable based power law model. State variable – Measure of deformation resistance- mesoscale property Material heterogeneity in the state variable assumed to be a second order random process with an exponential covariance kernel. Eigen decomposition of the kernel using KLE. Initial and mean deformed config. Eigenvectors f 0 s n r b 2 s ( p) s0 (1 i i vi ( p)) R ( p1 , 0, p2 , 0) 2 exp i 1 CORNELL U N I V E R S I T Y V1 0.409396 0.395813 0.38223 0.368646 0.355063 0.341479 0.327896 0.314313 0.300729 0.287146 V2 0.339819 0.239033 0.138247 0.0374605 -0.0633257 -0.164112 -0.264898 -0.365684 -0.466471 -0.567257 Materials Process Design and Control Laboratory UNCERTAINTY DUE TO MATERIAL HETEROGENEITY Dominant effect of material heterogeneity on response statistics Load vs Displacement SD Load vs Displacement 14 1.6 12 1.4 1.2 8 SD Load (N) 10 Load (N) Homogeneous material Heterogeneous material Mean 6 1 0.8 0.6 4 0.4 2 0.2 0 0 0.1 0.2 0.3 0.4 0.5 Displacement (mm) CORNELL U N I V E R S I T Y 0.6 0.7 0.8 0 0 0.1 0.2 0.3 0.4 0.5 Displacement (mm) 0.6 0.7 0.8 Materials Process Design and Control Laboratory NISG - FORMULATION Parameters of interest in stochastic analysis are the moment information (mean, standard deviation, kurtosis etc.) and the PDF. For a stochastic process g ( x, t , ) g ( x, t , ) x X , t T , W Definition of moments M p ( g ( x, t , )) p f ( )d W NISG - Random space W discretized using finite elements to W h M h p nel n h p h ( g ( x , t , )) f ( ) d w ( g ( x , t , )) f i ( ie ) i e ie h Wh nel nint p h M h p wi ( g eih ( x, t )) p f eih e 1 i 1 e 1 i 1 Deterministic evaluations at fixed points ie Output PDF computed using local least squares interpolation from function evaluations at integration points. CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory NISG - DETAILS Finite element representation of the support space. True PDF Interpolant Inherits properties of FEM – piece wise representations, allows discontinuous functions, quadrature based integration rules, local support. Provides complete response statistics. Decoupled function evaluations at element integration points. FE Grid Convergence rate identical to usual finite elements, depends on order of interpolation, mesh size (h , p versions). CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR Material heterogeneity induced by random distribution of micro-voids modeled using KLE and an exponential kernel. Gurson type model for damage evolution 2 fˆ ( p) fˆ0 (1 i n vi ( p)) Initial i 1 Final Mean Uniform 0.02 SD Eq. strain SD-Void fraction 0.268086 0.0098 0.242507 0.0096 0.216928 0.0094 0.191349 0.0092 0.16577 0.0091 0.140191 0.0089 0.114612 0.0087 0.0890327 0.0634537 0.0378747 Void fraction 0.0419 0.0388 0.0357 0.0325 0.0294 0.0263 0.0231 SD-Void fraction 0.0186 0.0172 0.0158 0.0143 0.0129 0.0115 0.0101 Using 6x6 uniform support space grid CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR Load displacement curves Mean 0.7 6 0.6 SD Load (N) Load (N) 5 4 3 2 0.5 0.4 0.3 0.2 Mean +/- SD 0.1 1 0.1 0.2 0.3 Displacement (mm) CORNELL U N I V E R S I T Y 0.4 0 0.1 0.2 0.3 0.4 Displacement (mm) Materials Process Design and Control Laboratory PROCESS UNCERTAINTY Random ? friction Random ? Shape Axisymmetric cylinder upsetting – 60% height reduction (Initial height 1.5 mm) Random initial radius – 10% variation about mean (1 mm)– uniformly distributed Random die workpiece friction U[0.1,0.5] Power law constitutive model Using 10x10 support space grid CORNELL U N I V E R S I T Y Void fraction: 0.002 0.004 0.007 0.009 0.011 0.013 0.016 Materials Process Design and Control Laboratory PROCESS STATISTICS Force CORNELL U N I V E R S I T Y SD Force Materials Process Design and Control Laboratory PROCESS STATISTICS Final force statistics Parameter Monte Carlo Support (20000 LHS space 10x10 samples) Mean 2.2859e3 2.2863e6 SD 297.912 299.59 m3 -8.156e6 -9.545e6 m4 1.850e10 1.979e10 3.50E-04 3.00E-04 Convergence study Relative Error Relative error 2.50E-04 2.00E-04 1.50E-04 1.00E-04 5.00E-05 0.00E+00 0 2 4 6 8 10 12 14 Grid resolution (Number of elements per dimension) CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory CURSE OF DIMENSIONALITY As the number of random variables increases, problem size rises exponentially. 1E+20 Function evaluations 1E+16 1E+12 1E+08 10000 1 0 5 10 15 20 25 No. of variables (assume 10 evaluations per random dimension) CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory PROPOSED SOLUTIONS ADAPTIVE DISCRETIZATION BASED ON OUTPUT STOCHASTIC FIELD • Refine/Coarsen input support space grid based on output defined control parameter (Gradients, standard deviations etc.) • Applicable using standard h,p adaptive schemes. Support-space of input CORNELL U N I V E R S I T Y Importance spaced grid Materials Process Design and Control Laboratory PROPOSED SOLUTIONS DIMENSION ADAPTIVE QUADRATURE (Gerstner et. al. 2003) 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 Full grid Scheme 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 Sparse grid Scheme 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1 -0.8-0.6 -0.4-0.2 0 0.2 0.4 0.6 0.8 1 Dimension adaptive Scheme Very popular in computational finance applications. Has been used in as high as 256 dimensions. CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Idea Behind Information Theoretic Approach Basic Questions: 1. Microstructures are realizations of a random field. Is there a principle by which the underlying pdf itself can be obtained. 2. If so, how can the known information about microstructure be incorporated in the solution. 3. How do we obtain actual statistics of properties of the microstructure characterized at macro scale. Information Theory Statistical Mechanics CORNELL U N I V E R S I T Y Rigorously quantifying and modeling uncertainty, linking scales using criterion derived from information theory, and use information theoretic tools to predict parameters in the face of incomplete Information etc Linkage? Information Theory Materials Process Design and Control Laboratory MAXENT as a tool for microstructure reconstruction Input: Given average and lower moments of grain sizes and ODFs Obtain: microstructures that satisfy the given properties Constraints are viewed as expectations of features over a random field. Problem is viewed as finding that distribution whose ensemble properties match those that are given. Since, problem is ill-posed, we choose the distribution that has the maximum entropy. Microstructures are considered as realizations of a random field which comprises of randomness in grain sizes and orientation distribution functions. CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory The MAXENT principle E.T. Jaynes 1957 The principle of maximum entropy (MAXENT) states that amongst the probability distributions that satisfy our incomplete information about the system, the probability distribution that maximizes entropy is the least-biased estimate that can be made. It agrees with everything that is known but carefully avoids anything that is unknown. MAXENT is a guiding principle to construct PDFs based on limited information There is no proof behind the MAXENT principle. The intuition for choosing distribution with maximum entropy is derived from several diverse natural phenomenon and it works in practice. The missing information in the input data is fit into a probabilistic model such that randomness induced by the missing data is maximized. This step minimizes assumptions about unknown information about the system. CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory MAXENT : a statistical viewpoint MAXENT solution to any problem with set of features gi ( I ) is i Parameters of the distribution gi ( I ) Input features of the microstructure Fit an exponential family with N parameters (N is the number of features given), MAXENT reduces to a parameter estimation problem. Commonly seen distributions No information provided (unconstrained optimiz.) The uniform distribution Mean, variance given Mean provided 1-parameter exponential family (similar to Poisson distribution) Gaussian distribution 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 -2 0 2 4 6 8 CORNELL U N I V E R S I T Y 10 12 Materials Process Design and Control Laboratory Microstructural feature: Grain sizes Grain size obtained by using a series of equidistant, parallel lines on a given microstructure at different angles. In 3D, the size of a grain is chosen as the number of voxels (proportional to volume) inside a particular grain. CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Microstructural feature : ODF e^ 3 Crystal/lattice reference frame ^ e’ 3 CRYSTAL SYMMETRIES? Sample reference frame Same axis of rotation => planes ^ e’ crystal e^ 1 ^ e’ 1 e^ 2 2 Each symmetry reduces the space by a pair of planes RODRIGUES’ REPRESENTATION FCC FUNDAMENTAL REGION n ORIENTATION SPACE Euler angles – symmetries Neo Eulerian representation Rodrigues’ parametrization CORNELL U N I V E R S I T Y Particular crystal orientation Cubic crystal Materials Process Design and Control Laboratory MAXENT as an optimization problem Find feature constraints Subject to features of image I Lagrange Multiplier optimization Lagrange Multiplier optimization Partition Function CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Equivalent log-linear model Equivalent log-likelihood problem Find that minimizes Kuhn-Tucker theorem: The that minimizes the dual function L also maximizes the system entropy and satisfies the constraints posed by the problem A comparison CORNELL U N I V E R S I T Y Direct models Log-linear models Concave Concave Constrained (simplex) Unconstrained “Count and normalize” (closed form solution) Gradient based methods Materials Process Design and Control Laboratory Gradient Evaluation • Objective function and its gradients: • Infeasible to compute at all points in one conjugate gradient iteration • Use sampling techniques to sample from the distribution evaluated at the previous point. (Gibbs Sampler) CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Parallel Gibbs sampler algorithm Improper pdf (function of lagrange multipliers) continue till the samples converge to the distribution Start from a random microstructure. Each processor goes through only a subset of the grains. … Processor 1 CORNELL U N I V E R S I T Y Go through each grain of the microstructure and sample an ODF according to the conditional probability distribution (conditioned on the other grains) Processor r Materials Process Design and Control Laboratory Optimization Schemes Convergence analysis with stabilization Convergence analysis w/o stabilization Noise in function evaluation increases as step size for the next minima increases. This ensures that the impact on the next evaluation is reduced. CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Voronoi structure Voronoi cell tessellation : Division of n-D space into non-overlapping regions such that the union of all regions fill the entire space. Sn { p1, p2 ,..., pn } k {p1,p2,…,pk} : generator points. Cell division of k-dimensional space : Division of k into subdivisions so that for each point, pi there is an associated convex cell, Voronoi tessellation of 3d space. Each cell is a microstructural grain. Ci {x k : j i, d ( x, pi ) d ( x, p j )} CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Mathematical representation OFF file representation (used by Qhull package) Initial lines consists of keywords (OFF), number of vertices and volumes. Next n lines consists of the coordinates of each vertex. The remaining lines consists of vertices that are contained in each volume. Brep (used by qmg, mesh generator) Dimension of the problem. A table of control points (vertices). Its faces listed in increasing order of dimension (i.e., vertices first, etc) each associated with it the following: 1.The face name, which is a string. 2.The boundary of the face, which is a list of faces of one lower dimension. 3.The geometric entities making up the face. its type (vertex, curve, triangle, or quadrilateral), • its degree (for a curve or triangle) or degree-pair (for a quad), and • its list of control points CORNELL U N I V E R S I T Y Volumes need to be hulled to obtain consistent representation with commercial packages Convex hulling to obtain a triangulation of surfaces/grain boundaries Materials Process Design and Control Laboratory Preprocessing: stage 1 Growth of big grains to accommodate small grains entrenched in-between Compute volumes of all grains Adjust vertices of neighboring grains so that the new voronoi tessellation fills the volume of initial grain Recompute surfaces and planes of the new geometry CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Preprocessing: stage 2 Steps Obtain input voronoi representation in OFF format. Obtain the convex hull of the volumes/grains so that each surface is a triangle (triangulation of surfaces). Use ANSYSTM to convert this representation to the universal IGES (Initial Graphics Exchange specification) format. • Surface database : To ensure non-duplication of surfaces, a database consisting of previously encountered hyper-planes is searched. When a new surface is created, if it is already in the database and if all the vertices of the surface were not present in a previous grain, no new surface is made. Domain smoothing: The regions of the microstructure inside the region [0 1]3 is chosen. Edges are smoothed so that the boundaries represent edges of a k-dimensional cube of unit side. CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Meshing Tetrahedral element meshed. Grain boundaries conform with the mesh shapes. Frame 001 17 Nov 2005 Z X Y Pixel based meshing scheme. Boundary is distorted since element shapes and sizes are fixed. CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Mesh refinement Tetrahedral mesh Hexahedral mesh Input to homogenization tool to obtain plastic property and eventually property statistics CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory (First order) homogenization scheme How does macro loading affect the microstructure (b) 1. Microstructure is a representation of a material point at a smaller scale 2. Deformation at a macro-scale point can be represented by the motion of the exterior boundary of the microstructure. (Hill, R., 1972) CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Homogenization of deformation gradient How does macro loading affect the microstructure Microstructure without cracks Use BC: = 0 on the boundary Note w = 0 on the volume is the Taylor assumption, which is the upper bound CORNELL U N I V E R S I T Y (a) Materials Process Design and Control Laboratory Implementation Largedef formulation for macro scale Update macro displacements Macro-deformation gradient Macro Homogenized (macro) stress, Consistent tangent Boundary value problem for microstructure Solve for deformation field Consistent tangent formulation (macro) Meso (b) meso deformation gradient Mesoscale stress, consistent tangent Integration of constitutive equations Micro Continuum slip theory Consistent tangent formulation (meso) CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Homogenized properties 60 Equivalent stress (MPa) 50 40 Simple shear 30 Plane strain compression 20 10 Z 0 0.000 Y ) 0.010 0.020 0.030 0.040 0.050 0.060 Equivalent plastic strain X (a)Z (b) (b) Z Y X X Y Z X Y Equivalent Strain: 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.4 0.6 X 0.8 Equivalent Stress (MPa): 20 30 40 50 60 70 80 (d) Equivalent Stress (MPa): CORNELL U N I V E R S I T Y (c) 19 27 36 45 53 62 70 79 Equivalent Stress (MPa): 20 30 40 50 60 70 80 (d) Materials Process Design and Control Laboratory 2D random microstructures: evaluation of property statistics Problem definition: Given an experimental image of an aluminium alloy (AA3302), properties of individual components and given the expected orientation properties of grains, it is desired to obtain the entire variability of the class of microstructures that satisfy these given constraints. Polarized light micrograph of aluminium alloy AA3302 (source Wittridge NJ et al. Mat.Sci.Eng. A, 1999) CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory MAXENT distribution of grain sizes Grain sizes: Heyn’s intercept method. An equidistant network of parallel lines drawn on a microstructure and intersections with grain boundaries are computed. 0.2 0.18 0.16 probability 0.14 Input constraints in the form of first two moments. The corresponding MAXENT distribution is shown on the right. <Gsz>=10.97 <Gsz 2>=124.90 0.12 0.1 0.08 0.06 0.04 0.02 0 0 2 4 6 8 10 12 14 16 18 20 Grain Size( m) CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Assigning orientation to grains Given: Expected value of the orientation distribution function. To obtain: Samples of orientation distribution function that satisfies the given ensemble properties Input ODF (corresponds to a pure shear deformation, Zabaras et al. 2004) Orientation distribution function Orientation distribution function 0.14 Ensemble properties of ODF from reconstructed distribution 0.14 0.12 0.12 0.1 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0 0.1 0.02 0 50 100 Orientation angle (in radians) CORNELL U N I V E R S I T Y 150 0 -2 -1 0 1 Orientation angle (in radians) 2 Materials Process Design and Control Laboratory Evaluation of plastic property bounds Orientations assigned to individual grains from the ODF samples obtained using MAXENT. Bounds on plastic properties obtained from the samples of the microstructure 80 Equivalent Strain (MPa) 70 60 Bounding plastic curves over a set of microstructural samples 50 40 30 0 0.05 0.1 0.15 0.2 Equivalent Stress CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Motivation Uncertainties induced due to nonuniformities in grain growth patterns CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Input uncertainties Problem inputs: Microstructures obtained using monte-carlo grain growth model at different stages of the growth. Sources of uncertainty: Anything that changes the driving force for grain growth (curvature driven, reduction in surface energy) (e.g) ambient conditions not exactly same in microstructures near surface and in the bulk. Problem parameters: 1. 10 input microstructures used that constraint the input information 2. Time lag of ~50 MC steps between each sample. 3. Simulated on a 9261 point grid CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Maximum-entropic distribution of grain sizes 0.01 0.009 0.008 <Gsz>=383.4967 <std(Gsz)>=41.4490 Probability 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 0 100 200 CORNELL U N I V E R S I T Y 300 400 Grain size ( m3) 500 600 Materials Process Design and Control Laboratory Sampling technique employed 0.2 0.18 0.16 Weakly consistent scheme 0.14 Probability 0.12 0.1 0.08 0.06 0.04 0.02 0 CORNELL U N I V E R S I T Y 0 5 10 15 Grain size 20 25 30 Materials Process Design and Control Laboratory ODF reconstruction using MAXENT Input ODF Problem inputs/algorithm parameters: 1. 145 degrees of freedom 2. MaxEnt algorithm using Brent’s line search method 3. Eighty Gibbs iteration through each grain of the microstructure CORNELL U N I V E R S I T Y Some representative ODF samples from the MaxEnt distribution Materials Process Design and Control Laboratory Ensemble properties Input ODF CORNELL U N I V E R S I T Y Ensemble properties of reconstructed samples of microstructures Materials Process Design and Control Laboratory pdf Final uncertainty representation Microstructures sampled as points from the joint pdf space CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Microstructure models & meshes Tetrahedral meshes CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Obtaining statistics of non-linear properties Different microstructural models of a polycrystal Aluminium microstructure is obtained by sampling the resultant distribution. Each of these specimens is subject to a pure axial tension along the x direction. Plots of the resultant stress-contour and the resulting homogenized stress-strain curves are plotted for different realizations CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Homogenized stress fields on the microstructure Hexahedral meshing Pixel based meshing Equivalent Stress (MPa) 84.8819 80.7536 76.6254 72.4971 68.3689 64.2406 60.1124 55.9841 51.8559 47.7276 43.5994 CORNELL U N I V E R S I T Y Equivalent Stress (MPa) 125 115 105 95 85 75 65 55 45 35 25 Materials Process Design and Control Laboratory Homogenized stress fields on the microstructure Equivalent Stress (MPa) 84.4691 80.7536 77.0382 73.3228 69.6074 65.8919 62.1765 58.4611 54.7457 51.0302 47.3148 43.5994 CORNELL U N I V E R S I T Y Equivalent Stress (MPa) 125 115 105 95 85 75 65 55 45 35 25 Materials Process Design and Control Laboratory Comparison of pixel based versus hexahedral meshing schemes The pixel based meshing scheme distorts grain boundaries and not only increases their area but also twists their shape which leads to a higher degree of stress localization as viewed in previous plot. Equivalent stress (MPa) 50 40 Hexahedral mesh Pixel based mesh 30 20 10 0 0.001 0.002 0.003 Equivalent strain CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Plots of homogenized stress-strain curves 45 40 Equivalent stress (MPa) A plot showing three different samples of the stress-strain plots obtained for different statistical models of the microstructure generated using the MaxEnt scheme. 35 30 25 20 15 10 0 0.0005 0.001 0.0015 0.002 Equivalent strain CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Stress contours across grain boundaries and triple junctions Orientation 0.2071 -0.4142 0.0858 Orientation 0.4142 0.0858 -0.2071 Orientation 0.4142 -0.2071 -0.0858 Orientation 0.4142 0.0858 -0.2071 Orientation -0.2929 -0.4142 0.2929 Extreme sharp variation in texture across the triple junction. Hence, leads to a large degree of stress localization CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory Applications (many …) Z X Y Equivalent Stress (MPa): 20 30 40 50 60 70 80 Statistics of plastic properties Z X Y Equivalent Stress (MPa): 20 30 40 50 60 70 80 Z X Y Equivalent Stress (MPa): 20 CORNELL U N I V E R S I T Y 30 40 50 60 70 80 Materials Process Design and Control Laboratory Discussion • A statistical distributions of mictrostructure was obtained incorporating variability in grain sizes and grain orientations. • Stress field distributions show a significant difference between the pixel based mesh and the hexahedral mesh. One possible reason may be attributed to the fact that grain boundaries are distorted as a result of which the localized stresses near the grain boundaries are felt in some regions in the bulk of the grain. Also, for the hexahedral grid 21960 elements were used while for the pixel based grid, 13824 elements were used. We are currently performing convergence studies with respect to the mesh sizes but the number of elements used were roughly equivalent. Also, sharp changes in the field were noticed in the vicinity of the grain boundaries due to steep variations in texture. • Statistical samples of microstructure model were used to obtained different samples of homogenized stress-strain curves. CORNELL U N I V E R S I T Y Materials Process Design and Control Laboratory MODELING GRAIN BOUNDARY PHYSICS Equivalent stress contours CORNELL U N I V E R S I T Y –Include failure mechanisms –Grain boundary properties –Local stress concentrations develop to cause the emission of a few partial dislocations from grain boundaries, and these high stresses drive the partial dislocations across the grain interiors –MD studies indicate that this is the major mechanism of the limited inelastic deformation in the grain interiors of nanocrystalline materials. Materials Process Design and Control Laboratory