Heat Transport Equations in Elastic Nanosystems

The paper is devoted to some modelling equations for heat propagation in nanosystems. The purpose is to investigate the thermodynamic consistency so that appropriate constitutive restrictions are established. First a second-order differential equation is considered, with two phase lags, that fits experimental data on heat propagation in graphene. Next a non-local model is examined through a rate-type equation with second-order spatial derivatives of the heat flux. For generality the body is allowed to be elastically deformable. The thermodynamic consistency leads to some simplifications and relates the whole model to free energy function of temperature, strain, and heat flux.