Quasi-periodic asymptotic homogenization in the framework of strain gradient mechanics

We develop in this contribution a theory for quasi-periodic media, assuming that the initial non-periodic geometry can be mapped to a periodic parent domain. The asymptotic homogenization method is employed to compute the quasi-periodic elastic properties of such quasi-periodic continuous media. We consider situations with a fast grading of the geometry or topology across neighbouring unit cells, which entail mutual interactions between unit cells requiring the incorporation of higher-order terms into the homogenized constitutive law, reflected in a strain gradient effective model. The theoretical background underlying quasi-periodic asymptotic homogenization in the context of linearized anisotropic elasticity is exposed in the framework of strain gradient mechanics. Different methodologies for evaluating the tensors of effective quasi-periodic properties are developed, based on either solving the unit cell problems in the periodic domain or in the quasi-periodic domain to expand the average microscopic energy, leading then to the identification of the tensors of effective moduli. A simple strategy consists of ignoring the spatial variation of the microscopic fields within the unit cell, directly mapping the macroscopic kinematic strain tensor to the quasi-periodic unit cell. The situation of a one-dimensional quasi-periodically repeated two-phase unit cell within a bar proves convenient to exemplify the different homogenization strategies; moreover, it proves able to deliver closed-form expressions of both Cauchy and strain gradient effective tensile moduli versus the microstructural parameters.