Determination of closed form expressions of the second-gradient elastic moduli of multi-layer composites using the periodic unfolding method

Abstract

The present paper aims at introducing a homogenization scheme for the identification of strain–gradient elastic moduli of composite materials, based on the unfolding mathematical method. We expose in the first part of this paper the necessary 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 unfolding method to the homogenization of multi-layer materials provides closed form expressions of all effective first- and second-gradient elastic moduli as well as coupling moduli between first- and second-gradient elasticity. A comparison between the unfolding method and the method of oscillating functions shows that both methods, despite their differences, deliver the same effective second-gradient elastic constitutive law for stratified materials.