Finite volume based asymptotic homogenization theory for periodic materials under anti-plane shear

A finite volume based approach is employed in the solution of unit cell problems at different orders of the asymptotic field expansion to construct a homogenization theory for anti-plane shear loading of unidirectional fiber-reinforced periodic structures. This new construction complements and further extends our recent contribution to asymptotic homogenization based on locally-exact elasticity unit cell solutions, He and Pindera (2020), to unit cells with multiple inclusions of arbitrary shapes. The present approach builds upon the previously developed finite-volume direct averaging micromechanics theory applicable under uniform strain fields, and extends it to account for strain gradients and non-vanishing microstructural scale relative to structural dimensions. The unit cell problems at different orders of the asymptotic field expansion are solved by satisfying local equilibrium equations in each subvolume of the discretized microstructure in a surface-averaged sense. This facilitates construction of local stiffness matrices at the subvolume level and subsequent assembly into the global stiffness matrix for the unit cell response under uniform and gradient strain fields. Comparison of the calculated microfluctuation functions and associated stress fields under uniform and gradient strain fields with those reported in the literature verifies the finite volume asymptotic solutions. The new theory's ability to accurately recover local fields is further illustrated through comparison with the direct numerical solution of a periodic structure with varying number of inclusions under gradient loading. The proposed homogenization approach is an efficient and accurate alternative to current numerical techniques for the analysis of periodic materials experiencing strain gradients regardless of microstructural scale, inclusion shape and number, demonstrated by analyzing arrays characterized by elliptical and square inclusions and multi-inclusion unit cells.