A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium

Homogenization may change fundamentally the constitutive laws of materials. We show how a heterogeneous Cauchy continuum may lead to a non Cauchy continuum. We study the effective properties of a linear elastic medium reinforced periodically with thin parallel fibers made up of a much stronger linear elastic medium and we prove that, when the Lamé coefficients in the fibers and the radius of the fibers have appropriate order of magnitude, the effective material is a second gradient material, i.e. a material whose energy depends on the second gradient of the displacement.