Energy equation of micromorphic continuum model based on lattice dynamics

In order to model a solid as an assembly of atoms, the mesodomain concept is effective to relate microscopic quantities of atoms with macroscopic quantities of solids. In previous papers, mechanical balance equations were derived on the basis of the equations of motion of atoms without the assumption of a constrained gradient. In the present paper, microscopic expressions of heat flux and internal energy are discussed, expressing the balance of energy with microscopic quantities in the mesodomains. The energy equation is formulated using averaged values of microscopic quantities over the domain by dividing the velocity of an atom into macroscopic motion and thermal motions. Heat flux is defined as an area-averaged value of thermal motions of atoms. Internal energy is described as the sum of the kinematic and potential energies of atoms in the mesodomain. Moreover, it is indicated from the microscopic expressions that the higher-order couple stress power and the moment of inertia are equivalent to the heat flux and the internal energy, respectively.