An extended Hill's lemma for non-Cauchy continua based on the modified couple stress and surface elasticity theories.

An extended Hill's lemma is provided for non-Cauchy continua satisfying the modified couple stress theory in the bulk and the surface elasticity theory in the surface layer. Based on the Hill–Mandel condition, four sets of boundary conditions (BCs) are identified. It is shown that each of these four sets of BCs satisfies the admissibility and average field requirements. In addition, the set of kinetic BCs meets the force and moment balance conditions, and the set of modified kinematic BCs satisfies the displacement and micro-rotation compatibility constraint. By applying the newly extended Hill's lemma, a homogenization analysis of a two-phase composite is performed, in which the average strain energy of the composite is computed using a finite element model constructed using COMSOL. The effective elastic constants obtained in this analysis are found to satisfy the Voigt and Reuss bounds. [ABSTRACT FROM AUTHOR]