On the local and global instability of a class of two-dimensional periodic composite structures.
William clark Schnaidt · Deep Blue (University of Michigan) · 1995
This work examines the effect of scale size on the onset of failure in microstructured media. More specifically, it presents quantitative comparisons between the onset of the first instability in a two dimensional elastic medium with known periodic microstructure and the onset of instability predicted by the macroscopic description of the medium. The comparisons establish theoretical limits for the validity of failure predictions based on the averaged (homogenized) response of microstructured elastic media. An approximation of a fiber reinforced composite modeled as a periodic grillage of axially compressed beams with an average shear stiffness was chosen for examination so the problem would be analytically tractable, yet exhibit nontrivial microscopic failure modes. In the first part of this work, the effect of boundaries is neglected, i.e., the model is assumed to be infinite. The stress level at the first bifurcation away from the principal periodic solution is compared to that corresponding to the loss of ellipticity for the incremental response of the homogenized model. An extensive investigation of the influence of various model parameters has been undertaken on the above two results. Whether the results are the same (the critical mode is thus macroscopic, i.e., global) or the former is smaller than the latter (microscopic, i.e., local failure) is solely dependent upon the model average shear stiffness. In the second part of this work, the effect of finite boundaries is taken into account. The microstructure scale size is defined to be the ratio of the size of the unit cell of the medium to the overall size of the model. It is found that, for a wide range of models, if a local failure is predicted for the infinite medium, any finite medium with the same unit cell properties will fail in the same manner. However, failure modes of finite models possessing properties for which the equivalent infinite models predict global failure are strongly dependent on the exact nature of the applied boundary constraints. Even when the scale size tends toward zero, these models do not necessarily fail according to macroscopic criteria.