Continuum Micromechanics: Survey

Abstract

The foundations of classical homogenization techniques, which aim at predicting the overall behavior of heterogeneous materials from that of their constituents, are reviewed. After introductory definitions and a methodological preamble, attention is focused on linear elasticity, for which the basic principles of estimating and bounding the overall properties are introduced and illustrated. In this context, special recourse is made for that to the solution of the inclusion and inhomogeneity problems as reported by Eshelby in 1957. Approaches proposed recently to account in a better way for the structural morphology of the considered materials are briefly mentioned. The case of linear elasticity with eigenstrains is then discussed: several applications, including heterogeneous thermoelasticity, can be investigated within this framework. Finally, outlines of nonlinear micromechanics are briefly reported from a historical point of view: from rate-independent elastoplasticity to nonlinear elasticity and viscoplasticity, examples of a fruitful interaction between the search for new estimates and the derivation of rigorous bounds are given and the crucial question of the description of intraphase heterogeneity is emphasized. Viscoelastic coupling and rate-dependent effects are briefly discussed in conclusion.

The role of disc-type crystal shape for micromechanical predictions of elasticity and strength of hydroxyapatite biomaterials

“…Also, such approaches are typically restricted to the linear elastic regime of material behaviour (FE-based biomaterial strength predictions are rarely documented-if at all). Quite recently, continuum micromechanics formulations (Hill 1963;Zaoui 2002) have turned out as a valuable alternative for predicting porosity-mechanical property relationships on the basis of microstructural mechanical interactions in ceramic bone biomaterials (Fritsch et al , 2009Malasoma et al 2008): such models do not resolve all details of the materials' microstructure (which, as a rule, are not known anyway), but consider essential morphological features (shapes) of homogeneous subdomains (phases) within a representative volume element (RVE) of the investigated biomaterial. In our case, both the intercrystalline pores and the ceramic crystals are modelled as ellipsoidal inclusions in an infinite matrix with stiffness properties of the overall biomaterial (self-consistent approach ;Hershey 1954;Kröner 1958).…”

Self-Consistent Channel Approach for Upscaling Chloride Diffusivity in Cement Pastes

“…We here introduce a representative volume element (RVE) fulfilling the standard "separation of scales"-requirement needed for random homogenization methods or continuum micromechanics (Zaoui 2002;Drugan and Willis 1996;Dormieux and Kondo 2005). Inside the RVE, microscopic ionic fluxes j fulfill the mass conservation law (Dormieux and Lemarchand 2001) …”

“…with A pore as the second-order "downscaling" or localization tensor (alternatively also referred to as concentration tensor) related to the concentration gradient of chloride ions encountered in the pore space; such localization tensors have been originally defined for linear elasticity (Zaoui 1997(Zaoui , 2002, and later also for pressure gradients driving Darcy-type fluid flow Kondo 2004, 2005). Derivation of A pore has been dealt with in great detail in (Abdalrahman et al 2015), based on Eshelby's famous inhomogeneity problem (Eshelby 1957), see also the work of Dormieux and Kondo (2005), eventually yielding…”

A multiscale analytical approach for bone remodeling simulations: Linking scales from collagen to trabeculae

“…2) is assumed as the mechanical stimulus sensed by the osteocytes regulating the bone remodeling process. It is defined by means of an Eshelby-problem micromechanics model [25], similar to that having undergone extensive experimental validation in Ref. [7] and to that used for micromechanics-supported finite element models of human mandibles to simulate the effects of atrophy on the bone density distribution [26].…”