Second gradient homogenization of multilayered composites based on the method of oscillating functions

The present paper aims at developing a homogenization scheme for the identification of strain-gradient elastic moduli, based on the mathematical approach of homogenization introduced by L. Tartar. We expose in the first part of this paper the needed mathematical apparatus in view of the derivation of the effective first and second gradient mechanical properties of two-phase composite materials, focusing on a one-dimensional situation. Each of the two phases is supposed to obey a second gradient linear elastic constitutive law. Application of the method of oscillating functions to the homogenization of multilayer materials showing strong strain gradients like composites with a strong contrast of properties of their constituents leads to closed form expressions of the effective first and second gradient moduli, including the coupling coefficients between first and second gradient terms. These results are then applied to the situations of multilayers with interfaces considered as the locus of strong strain gradients, and stratified materials. Analytical expressions of the effective moduli versus the first and second elasticity properties of the plies within the unit cell are obtained in both situations