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In a previous paper, a mathematical model that captures the effects of microstructure on the macromechanical response of granular materials was presented. The overall response of the particle assembly derives the response of a macroscopic point using the conventional hypothesis in cmputational plasticity, i.e. a given overall displacement gradient determines the overall stress response. Numerical simulations with uniform as well as non-uniform assemblies of two-dimension circular disks capture, at least qualitatively, such mechanical responses as structural anisotropy, pressure dependency and strain-softening effects due to microstructural changes. The present paper attempts to re-analyze the problem from an opposite point of view, i.e. the overall imposed stresses determine the overall strain response. This formulation allows the model to capture naturally the volume-change characteristics of granular materials such as compaction as well as dilation, since the strains now result from the analysis rather than act as a prescribed force that drives the problem. The new formulation also requires the derivation of the so-called overall algorithmic tangential moduli from the consistent linearization of the elasto-plastic constitutive equation. The task becomes complicated because the linearization permeates down to the microstructural level, thus necessitating the extraction of the so-called algorithmic concentration tensor. To circumvent this difficulty, we approximate the algorithmic tensor of overall moduli with a secant approximation that retains the essential properties of the exact description. We then use this approximation to solve the inverse problem as well as perform some stability analyses that could be useful in predicting the onset of localized deformation in granular materials. |
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