Extended Hill’s lemma for non-Cauchy continua based on a modified couple stress theory

Abstract

Two extended versions of Hill’s lemma for non-Cauchy continua are provided using a modified couple stress theory, which includes a constraint relation between the displacement gradient and the micro-rotation, unlike the Cosserat (micro-polar) elasticity theory. The first version is used to obtain the classical elasticity tensor that relates the Cauchy (force) stress to the strain, while the second version is employed to determine the couple stress elasticity tensor that links the couple stress to the curvature. The kinematic (displacement- or strain-based) boundary conditions used in each version of the extended Hill’s lemma are then modified to accommodate composites with periodic microstructures by introducing periodic parts to the displacement and micro-rotation fields and reconstructing the Hill–Mandel condition. To illustrate the two newly proposed versions of the extended Hill’s lemma, homogenization analyses are performed for a two-phase composite using a meshfree radial point interpolation method incorporating the extended Hill’s lemma and the modified couple stress theory, which is newly developed in the current study.