Two versions of the extended Hill’s lemma for non-Cauchy continua based on the couple stress theory

Two versions of the extended Hill’s lemma for non-Cauchy continua satisfying the couple stress theory are proposed. Each version can be used to determine two effective elasticity (stiffness) tensors: one classical and the other higher order. The classical elasticity tensor relates the symmetric part of the force stress to the symmetric strain, whereas the higher-order elasticity tensor links the deviatoric part of the couple stress to the non-symmetric curvature. Four sets of boundary conditions (BCs) are identified using Version I, and three sets of BCs are obtained from Version II of the extended Hill’s lemma, which can all satisfy the Hill–Mandel condition. For each BC set selected, admissibility and average field requirements are checked. Furthermore, the equilibrium is examined for the cases with the kinetic BCs, and the compatibility is checked for the cases with the kinematic BCs. To illustrate the two newly proposed versions of the extended Hill’s lemma, a homogenization analysis is conducted for a two-phase composite using a meshfree radial point interpolation method. The effective elastic constants obtained in this analysis are compared with those predicted by the Voigt and Reuss bounds and computed through a finite element model constructed using COMSOL, which verifies and supports the current method.