Higher gradient homogenization of quasi-periodic media and applications to inclusion-based composites

This paper introduces quasi-periodic homogenization schemes for quasi-periodic media, those without strict periodicity, but that can be mapped to a parent periodic medium. Quasi-periodic homogenization relies on the conceptual idea of mapping a non-periodic domain to a reference periodic one through a point mapping of material points within the domain of an identified repetitive unit cell. The theoretical background of quasiperiodic homogenization introduced in the first part of this paper relies on expressing the microscopic position of micropoints within a physical unit cell as a sum of the macroscopic position (the center of area of the unit cell) and the relative position of micropoints with respect to the center of area. This decomposition parameterized by the small-scale parameter entails a corresponding additive decomposition of the tangent map defining the geometrical transformation of the periodic UC into the quasi-periodic one in terms of an additive decomposition into macroscopic and microscopic contributions. The quasi-periodic homogenized effective moduli are then determined, starting from the average of the microscopic energy, those being expressed in terms of the periodic moduli and a perturbation term, both expressed in a volumetric format as surface integrals over the reference unit cell domain in a 2D context. In the second part of this contribution, a surface formulation of the quasi-periodic moduli is derived, based on the notion of shape derivative of the total potential energy stored within the unit cell. This approach relies on introducing a shape velocity field at the boundary of the periodic unit cell to change its design, driven by the normal projection of Eshelby stress onto both the internal and external boundaries of the unit cell. This second scheme offers comparatively to the first one a simpler way to compute the quasi-periodic moduli as it only requires the evaluation of the mechanical fields on the unit cell boundaries. Application of the proposed homogenization schemes are done for inclusion-based composites, underlining the importance of strain gradient energetic contribution in the situation of a micrograding of the unit cell geometry.