Derivation of strain gradient length via homogenization of heterogeneous elastic materials

We present explicit upper bound estimates of the microstructural length used in simple gradient elasticity. Our model is a two dimensional composite made of circular hard inclusions randomly dispersed in a soft matrix. Both inclusions and matrix are described by isotropic linear elastic constitutive laws. The composite, however, is described by an isotropic gradient elastic law. The elastic modulus and the Poisson’s ratio are given by the exact classic analysis of Christensen. The in-plane microstructural length is estimated by energy optimization, based on solutions of the gradient elastic hollow cylinder. It was shown that the microstructural length decreases with the composition of the particles, taking high values at low particle composition. Naturally, the microstructural length is proportional to the particle diameter and increases with the stiffness of the particles. It was shown that there can be no microstructural prediction for particles that are softer than the matrix. This interesting result seems to be complementary to the result of Bigoni and Drugan who found that, for the couple-stress composite model, there can be no prediction for the microstructural length when the particles are stiffer than the matrix.

► The internal length used in strain gradient theory was estimated via homogenization.
► Estimates were found for inclusions stiffer than the matrix.
► Internal length is between 0.5 to 7 times the inclusion radius for small composition.
► Upper bound estimates for the internal length for the case of rigid inclusions.
► No prediction was possible for inclusions less stiff then the matrix.