Scale Effects in Media With Periodic and Nearly Periodic Microstructures, Part II: Failure Mechanisms

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Abstract
Using the nonlinearly elastic planar lattice model presented in Part I, the influence of scale (i.e., the size of the representative volume, relative to the size of the unit cell) on the onset of failure in periodic and nearly periodic media is investigated. For this study, the concept of a microfailure surface is introduced—this surface being defined as the locus of first instability points found along radial load paths through macroscopic strain space. The influence of specimen size and microstructural imperfections (both geometric and constitutive) on these failure surfaces is investigated. The microfailure surface determined for the infinite model with perfectly periodic microstructure, is found to be a lower bound for the failure surfaces of perfectly periodic, finite models, and an upper bound for the failure surfaces of finite models with microstructural imperfections. The concept of a macrofailure surface is also introduced—this surface being defined as the locus of points corresponding to the loss of ellipticity in the macroscopic (homogenized) moduli of the model. The macrofailure surface is easier to construct than the microfailure surface, because it only requires calculation of the macroscopic properties for the unit cell, at each loading state along the principal equilibrium path. The relation between these two failure surfaces is explored in detail, with attention focused on their regions of coincidence, which are of particular interest due to the possible development of macroscopically localized failure modes.