Statistical modeling of heterogeneous micro-beams

Statistical characteristics (averages and variances) of heterogeneous, small size beams (micro-beams), composed of micro-elements (grains) of randomly distributed stiffness (isotropic) is studied. Only longitudinal variations are considered. Element to beam size (length) ratio is not negligible and the use of an equivalent homogeneous structure with the classical effective material properties is not sufficient. Using standard statistical tools (probability density and correlation functions), it is shown that for indeterminate cases, even average deflections (obtained explicitly) are load dependent, and are not identical with their corresponding homogeneous case. Bounds for the statistical dispersion of deflections and reaction forces are found analytically for some cases of weak heterogeneity. It is shown that dispersion parameters for indeterminate problems can be obtained by superposition (using both averages and dispersion data) of corresponding determinate cases, leading to reciprocal type relations. (C) 2001 Elsevier Science Ltd. All rights reserved.