Mindlin second-gradient elastic properties from dilute two-phase Cauchy-elastic composites Part I: Closed form expression for the effective higher-order constitutive tensor

TL;DR

This paper derives a closed-form expression for the effective higher-order elastic tensor of a dilute two-phase composite, extending previous work to anisotropic phases and providing a perfect energy match between heterogeneous and homogenized materials.

Contribution

It provides a new, simple formula for the second-gradient elastic properties of dilute composites, generalizing prior results to anisotropic phases and independent boundary conditions.

Findings

Effective second-gradient elastic tensor derived in closed form.

Result valid for anisotropic phases with spherical or ellipsoidal inclusions.

Perfect energy equivalence between heterogeneous and homogenized materials.

Abstract

It is shown that second-order homogenization of a Cauchy-elastic dilute suspension of randomly distributed inclusions yields an equivalent second gradient (Mindlin) elastic material. This result is valid for both plane and three-dimensional problems and extends earlier findings by Bigoni and Drugan (Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. J. Appl. Mech., 2007, 74, 741-753) from several points of view: (i.) the result holds for anisotropic phases with spherical or circular ellipsoid of inertia; (ii.) the displacement boundary conditions considered in the homogenization procedure is independent of the characteristics of the material; (iii.) a perfect energy match is found between heterogeneous and equivalent materials (instead of an optimal bound). The constitutive higher-order tensor defining the equivalent Mindlin solid is given in a surprisingly simple formula. Applications, treatment of material symmetries and positive definiteness of the effective higher-order constitutive tensor are deferred to Part II of the present article.