COMPARISON OF MICROSCOPIC AND MACROSCOPIC INSTABILITIES IN A CLASS OF2-DIMENSIONAL PERIODIC COMPOSITES

OF INTEREST in the present work is the quantitative comparison between the microscopic and the corresponding macroscopic failure in a two-di mensional elastic medium with known periodic microstructure. This comp arison establishes theoretical limits for the validity for the average d (homogenized) response of microstructured elastic media. Attention i s 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 complex ity to exhibit nontrivial microscopic and macroscopic failure modes. A comparison of the stresses at the onset of the microscopic failure (t aken to be the first bifurcation away from the principal periodic solu tion) to the stresses at the onset of the macroscopic failure (which c orresponds to the loss of ellipticity for the incremental response of the homogenized model) allows one to identify whether the first failur e mode, also termed the critical mode, is local (microscopic) or globa l (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 inte rstitial stiffness in deciding the nature of the critical mode. The pr esentation is concluded by a discussion of the results with suggestion s for future work.