Computational homogenization of the elastic properties of polycrystalline fcc metals within Mindlin’s second strain-gradient theory

Abstract

The lack of interest in Mindlin’s second strain-gradient theory, despite its success in characterizing various phenomena for which there is no explanation in the classical elasticity, is predominantly due to the numerous higher-order elastic constants involved therein. As an attempt to overcome this shortcoming, a computational homogenization method for determining such elastic constants of polycrystalline face-centered cubic (fcc) metals is proposed in the present study. In this homogenization method, using the Voigt-type averaging scheme, a polycrystalline material is modeled by an isotropic aggregate of randomly oriented single crystals without accounting for the grain size effect and the complexities due to grain boundaries and junctions. Subsequently, analytical expressions for the strain energy due to certain modes of loading are determined, and molecular simulations of a few fcc metals under such modes of loading are performed. Then, by fitting the corresponding analytical expressions to the results obtained from these simulations, the effective elastic constants and consequently the effective characteristic lengths of the fcc metals in their polycrystalline form are determined. Moreover, the free-surface-induced reconstruction in a thin layer of a solid is addressed both analytically and by molecular simulations, as a result of which the effective moduli of cohesion of the fcc metals in their polycrystalline form are calculated. In addition, the complete set of conditions for the positive-definiteness of the strain-energy-density function of isotropic materials within the adopted theory is derived, and subsequently a discussion of whether or not the numerical values obtained for the effective elastic moduli of the polycrystalline fcc metals satisfy such conditions is provided.