Maximum Value Distribution of Micromechanical Response Quantities

Abstract

The maximum value of a random field over a subdomain is a random variable whose probability distribution can be determined from the first-order marginal probability density function and autocorrelation function of the random field given that certain assumptions hold. It is shown in this work that this formulation, which traditionally has been applied to wind engineering problems, can express the probability distributions of the maximum values of mechanical response quantities of structures subjected to the boundary conditions applied in computational homogenization. Once the expression for the maximum value distribution is determined, the convergence of the maximum value to a deterministic value as a function of structure size can be easily computed. This may have implications in determining the representative volume element for mechanical properties driven by the extremes of the response quantities from which they are derived, such as in upscaling damage parameters. The concept is demonstrated by comparing the results using the maximum value formula to brute-force Monte Carlo simulation for a stochastic bar.