Estimation of material properties for linear elastic strain gradient effective media

In this paper materials with microstructures, composed of linear elastic constituents are considered. Spherical volume elements with spherical inclusions are used to obtain Voigt and Reuss bounds for the material properties of a corresponding strain gradient effective continuum. The bounds are derived following the general line of argumentation well established for simple effective continua presenting in detail the major differences arising in the case of a strain gradient continuum. Furthermore, an alternative method for the approximation of the material properties is developed, where the volume element is exposed entirely to a kinematically admissible quadratic displacement field. Since the resulting stresses do not fulfill the equilibrium conditions, a second strain field is superimposed which is approximated by a third order polynomial. The coefficients of this polynomial are determined from an energy principle together with a constraint equation in order to fulfill the equilibrium conditions in an integral sense and to assure that the resulting effective constitutive relations are insensitive to rigid body motions. Based on this approximate solution, the components of the sixth-order material tensor which relates the gradients of the macroscopic strain with the corresponding higher order stresses can be approximated analytically. In spite of the fact that these estimations are no bounds in general, useful results for the effective material parameters can be derived. This is shown by comparing the obtained approximations with the corresponding numerical results from finite element solutions of the boundary value problem for a spherical volume element considering different void volume fractions.