On the Relations between Direct and Energy Based Homogenization Approaches in Second Gradient Elasticity

Abstract

In this chapter we consider Eshelby integral formulas in the framework of second gradient elasticity theory (SGET). Based on these formulas we provide a comparison between direct and energy based homogenization approaches for the second gradient elastic media containing inhomogeneities. These approaches are widely used in micromechanics for the estimation of the effective properties of composites assuming the equivalence of averaged field variables (direct approach) or the equivalence of strain energy (energy based approach) in heterogeneous and equivalent homogeneous media. Equivalence between these approaches in classical elasticity can be proven by using Eshelby formulas. In the present work we use these formulas generalized for SGET to show, that the direct and energy based homogenization approaches are equivalent in SGET only in the case of homogeneous boundary conditions, while for the more general type of loading these approaches may provide different predictions, resulting, e.g., in different values of the effective high-order material constants estimated for the composite materials and mechanical metamaterials.