On the micro-macro relation for the microdeformation in the homogenization towards micromorphic and micropolar continua

Abstract

The modeling of size effects requires to employ generalized continuum theories. For instance, the micromorphic theory introduces the microdeformation χ̰ as additional kinematic degree of freedom. Such generalized theories require additional, i. e. non-classical, constitutive relations. The experimental determination of the corresponding non-classical constitutive parameters is cumbersome, even for isotropic and linear-elastic material. Homogenization approaches aim in providing the macroscopic constitutive relations from the behavior of the microscopic constituents of a material. For that purpose, micro-macro relations are required for all kinematic quantities. Though, different micro-macro relations have been used in literature for the microdeformation χ̰.The present contribution investigates the effect of the chosen micro-macro relation for χ̰ on the predicted macroscopic behavior. In particular, a material with pores or inclusions at the microscale is considered for the special case of plane micropolar (Cosserat) elasticity. It is discussed that the coupling modulus should vanish if the material is homogeneous at the micro-scale in order to avoid the artificial prediction of size effects. This goal is reached if the deformation of microscopic heterogeneities is used for the micro-macro relation of the microdeformation. Classical and non-classical micropolar moduli of a material with pores or inclusions are derived in closed form.