Locally exact asymptotic homogenization of periodic materials under anti-plane shear loading

A new implementation of the asymptotic homogenization theory for anti-plane shear loading is constructed based on locally exact elasticity solutions of unit cell problems at different orders in the asymptotic field expansion. The approach is a generalization of the previously developed locally exact homogenization theory for the determination of homogenized elastic moduli under uniform stress fields. The new theory has been developed for periodic materials in the presence of stress gradients and non-vanishing microstructural scale relative to the structural dimensions. The local interior unit cell problems are solved exactly up to the third order using Fourier series representations of the microfluctuation functions. The exterior unit cell periodic boundary value problems at different orders are tackled by an asymptotic extension of the previously introduced balanced variational principle for periodic materials. The effectiveness of the theory is verified by solving a structural problem with varying number of inclusions directly, and comparing the local fields with those reconstructed by the locally exact asymptotic homogenization approach. The theory is also employed to assess the accuracy of an uncoupled homogenization-localization approach in the reconstruction of local stress fields in periodic materials with different levels of microstructural refinement in the presence of stress gradients. The present contribution provides the foundation for further extension of the new locally exact asymptotic homogenization theory to in-plane loading problems of unidirectionally reinforced periodic materials.