Scale-size dependent multi-continuum homogenization of complex bodies

Abstract

We derive effective equations of a periodically heterogeneous Cosserat material encompassing intrinsic lengths modelling scale-size effects. The resultant homogenized material supports internal body torques and leads to an asymmetric effective stress providing a connection to the theory of odd elasticity. Furthermore, a link to the classical Cauchy stress is given. Moreover, the corresponding local problem exhibits asymmetry as well, due to the micropolar couple modulus inherited from the original microscopic Cosserat problem. We validate our results by conducting numerical simulations using the finite element method on circularly perforated square and rectangular unit cells, highlighting the impact, of not only volume fraction but also of internal body torques on effective coefficients. Additionally, we numerically quantify the “amount” that the body can torque internally.