Effective elasto-(visco)plastic coefficients of a bi-phasic composite material with scale-dependent size effects

We employ the theory of asymptotic homogenization (AH) to study the elasto-plastic behavior of a composite medium comprising two solid phases, separated by a sharp interface and characterized by mechanical properties, such as elastic coefficients and “initial yield stresses” (i.e., a threshold stress above which remodeling is triggered), that may differ up to several orders of magnitude. We speak of “plastic” behavior because we have in mind a material behavior that, to a certain extent, resembles plasticity, although, for biological systems, it embraces a much wider class of inelastic phenomena. In particular, we are interested in studying the influence of gradient effects in the remodeling variable on the homogenized mechanical properties of the composite. The jump of the mechanical properties from one phase to the other makes the composite highly heterogeneous and calls for the determination of effective properties, that is, properties that are associated with a homogenized “version” of the original composite, and that are obtained through a suitable averaging procedure. The determination of the effective properties results convenient, in particular, when it comes to the multiscale description of inelastic processes, such as remodeling in soft or hard tissues, like bones. To accomplish this task with the aid of AH, we assume that the length scale over which the heterogeneities manifest themselves is several orders of magnitude smaller than the characteristic length scale of the composite as a whole. We identify both a fine-scale problem and a coarse-scale problem, each of which characterizes the elasto-plastic dynamics of the composite at the corresponding scale, and we discuss how they are reciprocally coupled through a transfer of information from one scale to the other. In particular, we highlight how the coarse-scale plastic distortions influence the fine-scale problem. Moreover, in the limit of negligible hysteresis effects, we individuate two viscoplastic effective coefficients that encode the information of the two-scale nature of the composite medium in the upscaled equations. Finally, to deal with a case study tractable semi-analytically, we consider a multilayered composite material with an initial yield stress that is constant in each phase. Such investigation is meant to contribute to the constitution of a robust framework for devising the effective properties of hierarchical biological media.