Comparison of microscopic and macroscopic instabilities in a class of two-dimensional periodic composites

Abstract

O f interest in the present work is the quantitative comparison between the microscopic and the corresponding macroscopic failure in a two-dimensional elastic medium with known periodic microstructure. This comparison establishes theoretical limits for the validity for the averaged (homogenized) response of microstructured elastic media. Attention is focussed on an approximation of a fiber reinforced composite modeled as an infinite periodic grillage of axially compressed beams with an average shear stiffness. The particular beam model was chosen so that the problem would be analytically tractable, yet of sufficient complexity to exhibit nontrivial microscopic and macroscopic failure modes. A comparison of the stresses at the onset of the microscopic failure (taken to be the first bifurcation away from the principal periodic solution) to the stresses at the onset of the macroscopic failure (which corresponds to the loss of ellipticity for the incremental response of the homogenized model) allows one to identify whether the first failure mode, also termed the critical mode, is local (microscopic) or global (macroscopic) in nature. An extensive investigation of the influence of various model parameters on the failure modes of the composite has been undertaken. The results obtained show the importance of the interstitial stiffness in deciding the nature of the critical mode. The presentation is concluded by a discussion of the results with suggestions for future work.