Entropy Correctors in Homogenization of Microheterogeneous Rigid Bodies

<div> We propose an effective heat equation for an enhanced heat conduction problem that can account for thermal scale-size effects in miniaturized or nanofabricatied metamaterials. The derivation of the proposed enhanced heat conduction problem is based on contributions to the global energy balance in the form of Gurtin's microforce balance introduced in (M.E. Gurtin, Phys. D, 92(3-4):178-192, '96 ) and was first proposed by (S. Forest, M. Amestoy, C.R. Méc., 336(4):347-353, '08 ). The resulting effective heat equation retains its classical form with temperature as the macroscopic variable, however, the effective conductivity tensor is constructed using the local entropy and the local gradient of entropy instead of the local temperature. </div>