Homogenization of quasi-periodic conformal architectured materials and applications to chiral lattices

Abstract

In this study, we propose to extend asymptotic periodic homogenization for non-periodic continuous microstructured media, assuming that the non-periodic geometry (called quasi-periodic) can be designed by a conformal planar transformation of a periodic parent domain architectured media with periodically disposed unit cells. Conformal transformations are shown to play a privileged role in the design of circular macroscopic heterogeneous domains tessellated with non-periodic unit cells, obtained from a periodic parent domain architectured with these unit cells. The conditions for conformal invariance are established, leading to the general form of conformal transformation in their dependencies upon the periodic coordinates. It is shown that any conformal map can be decomposed into the product of an isotropic dilatation function of the first periodic spatial position of decreasing exponential type and a rotation characterized by an angular function linear in the second periodic position. A general theory of quasi-periodic homogenization in the framework of conformal transformations is established for the first time, leading to an expression of the tensor of quasi-periodic moduli which is fully evaluated from the solution of the elasticity boundary value problem posed over the periodic unit cell. The influence of microcurvature distortion of individual unit cells on their effective properties is evaluated. Closed-form solutions are confronted to numerical examples issued from the implementation of circular periodicity in a finite element solver, showing overall a good agreement with the identified homogenized moduli.