Balance equations for higher-order materials have been widely studied. In this paper, Gibbs equations for higher-order materials are formulated introducing a concept of the equivalence between higher-order stress powers and heat flux. Gibbs equations for higher-order materials and simple materials are expressed with microscopic quantities by use of the microscopic expressions such as stress power, higher-order stress powers and internal energy. The results obtained from the simulation of molecular dynamics can be substituted into the microscopic expressions of Gibbs equations. The values of entropies for higher-order materials and simple materials can be calculated in this procedure. It is shown that the difference in entropies between higher-order materials and simple materials becomes notable in an example of the calculation.
The essence of macroscopic quantities in solid mechanics can be grasped by expressing these quantities in terms of kinematic and mechanical quantities of atoms. In this paper, a method is proposed for obtaining the microscopic definitions of internal forces of continua such as stress, higher-order stresses and heat flux. Moreover, the relation between higher-order stress power and heat flux is discussed expressing the first law of thermodynamics with microscopic quantities in the mesodomain. Comparing heat flux with higher-order stress power, it is clarified that the divergence of heat flux is equivalent to the total of each order power due to higher-order stresses.
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