Finite Volume-Based Asymptotic Homogenization of Periodic Materials Under In-Plane Loading

The previously developed finite volume-based asymptotic homogenization theory (FVBAHT) for anti-plane shear loading (He, Z., and Pindera, M.-J., "Finite-Volume Based Asymptotic Homogenization Theory for Periodic Materials Under Anti-Plane Shear," Eur. J. Mech. A Solids (in revision)) is further extended to in-plane loading of unidirectional fiber reinforced periodic structures. Like the anti-plane FVBAHT, the present extension builds upon the previously developed finite volume direct averaging micromechanics theory applicable under uniform strain fields and further accounts for strain gradients and non-vanishing microstructural scale relative to structural dimensions, albeit with multidimensional in-plane loadings incorporated. The unit cell problems at different orders of the asymptotic field expansion are solved by satisfying local equilibrium equations and displacement and traction continuity in a surface-averaged sense which is unique among the existing asymptotic homogenization schemes, leading to microfluctuation functions that yield homogenized stiffness tensors at each order for use in macroscale problems. The newly extended multiscale theory is employed in the analysis of a structural boundary-value problem under in-plane loading, illustrating pronounced boundary effects. A combination approach proposed in the literature is subsequently employed to mitigate the boundary layer effects by explicitly accounting for the microstructural details in the boundary region. This combination approach produces accurate recovery of the local fields in both regions. The extension to in-plane problem marks FVBAHT as an alternative, self-contained asymptotic homogenization tool, with documented advantages relative to current numerical techniques, for the analysis of periodic materials in the presence of strain gradients produced by three-dimensional loading regardless of microstructural scale.