Locally exact asymptotic homogenization of viscoelastic composites under anti-plane shear loading

Abstract

The recently developed elasticity-based locally exact asymptotic homogenization theory is extended to accommodate non-ageing linearly viscoelastic composites under anti-plane shear loading. The theory is based on transforming the problem to the Laplace domain using the elastic-viscoelastic correspondence principle, solving it in the transformed domain and transforming it back to time domain using an efficient and accurate inversion technique. In the Laplace domain, the microscale unit cell interior problems are solved exactly up to the third order through Fourier series expansions, whereas the inseparable exterior periodic boundary value problems are tackled by asymptotic extensions of a balanced variational principle. Numerical examples of microscale solutions are presented and compared with the finite element method in support of the theory's accuracy. The effectiveness of the theory is assessed through a solution of a viscoelastic composite structural problem with finite-sized microstructures, illustrating good comparison with the results of a direct numerical solution. The theory is also applied to the microscale analysis of composites reinforced by viscoelastic fibers with different relaxation rates rarely reported in the literature. The analytical solution methodology in the Laplace domain is unique among the existing higher-order asymptotic homogenization methods employed in the solution of viscoelastic composite structural problems with non-vanishing microstructural scales. It serves as a gold standard for validating strictly numerical approaches.