Microstructure Models and Material Response by Extreme Value Theory

Abstract

We develop (1) probabilistic models for the compliance tensor $A(x)$ of linear elastic random microstructures and (2) a method for estimating the distributions of large material responses from relatively small sets of response samples by using concepts of the extreme value theory (EVT). Two models are considered for the compliance tensor $A(x)$. The first constructs the compliance tensor from random fields defining its eigenvalues and eigenvectors. The second represents $A(x)$ by products of triangular matrices and their transposes. Numerical examples are presented to illustrate the implementation of the models of $A(x)$. The EVT-based method for estimating properties of large material responses constitutes an efficient and accurate alternative to classical Monte Carlo simulation. It is applied to estimate the distribution of the largest stress $\Sigma_{\rm max}$ in a rectangular plate with random properties under uniform tension. The proposed estimates of the distribution of $\Sigma_{\rm max}$ are robust even if based on relatively small sets of response samples and can be used to extend empirical distributions of $\Sigma_{\rm max}$ beyond data.

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