Scale and boundary conditions effects in elastic properties of random composites

We study elastic anti-plane responses of unidirectional fiber-matrix compos ites. The fibers are of circular cylinder shape, aligned in the axial direc tion, and arranged randomly, with no overlap, in the transverse plane. We a ssume that both fibers and matrix are linear elastic and isotropic. In part icular. we focus on the effects of scale of observation and boundary condit ions on the overall anti-plane (axial shear) elastic moduli. We conduct thi s analysis numerically, using a two-dimensional square spring network, at t he mesoscale level. More specifically, we consider finite "windows of obser vation, which we increase in size. We subject these regions to several diff erent boundary conditions: displacement-controlled, traction-controlled, pe riodic. and mixed (combination of any of the first three) to evaluate the m esoscale moduli. The first two boundary conditions give us scale-dependent bounds on the anti-plane elastic moduli. For each boundary condition case w e consider many realizations of the random composite to obtain statistics. In this parametric study we cover a very wide range of stiffness ratios ran ging from composites with very soft inclusions (approximating holes) to tho se with very stiff inclusions (approaching rigid fibers), all at several vo lume fractions.