Volume 33, Issue 3

In this work we analyze the relation between the multiplicative decomposition F = F e F p of the deformation gradient as a product of the elastic and plastic factors and the theory of uniform materials. We prove that postulating such a decomposition is equivalent to having a uniform material model with two configurations – total φ and the inelastic φ 1 . We introduce strain tensors characterizing different types of evolutions of the material and discuss the form of the internal energy and that of the dissipative potential. The evolution equations are obtained for the configurations (φ, φ 1 ) and the material metric g . Finally, the dissipative inequality for the materials of this type is presented. It is shown that the conditions of positivity of the internal dissipation terms related to the processes of plastic and metric evolution provide the anisotropic yield criteria.

Dynamic degrees of freedom and internal variables are treated in a uniform way. The unification is achieved by means of the introduction of a dual internal variable. This duality provides the corresponding evolution equations depending on whether the Onsager–Casimir reciprocity relations are satisfied or not.

A study is made of the mesoscopic dynamics of granular materials by modeling a dry granular medium as a mixture with components, specified by a proposed continuous field variable. By treating a representative volume element as a macroscopic point and dividing granular particles into small pieces, a macroscopic point in this model owns an internal degree of freedom, which can be described by the position vectors of these small pieces relative to the center of mass of those particles within the representative volume element. At the macroscopic level, the field quantities for the granular medium are defined and the evolution equations of this system are discussed. It is found that a reduction of these equations for granular materials produces the balance equations of a micro-continuum. Finally, this study decomposes the combination of the balance equation of momentum moment and the macroscopic equation for the second moment of distribution function into a volume-stretch equation and a shearing equation. The derivation shows that the volume-stretch equation is analogous to the famous Goodman–Cowin equation.

The optimal configuration of a class of endoreversible heat engines with fixed compression ratio and generalized radiative heat transfer law [ q ∝ Δ( T n )] is determined in this paper. The optimal cycle that maximizes the power output of the engine has been obtained using optimal control theory. The respective differential equations are solved by Taylor series expansions. It is shown that the optimal cycle for maximum power output has eight branches including two isothermal branches, four maximum power branches, and two isochoric branches but no adiabatic branches. The interval of each branch has been obtained, as well as the solutions of the temperatures of heat reservoirs and working fluid. Numerical examples for the optimal configurations with n = −1, 1 … 4 are given, respectively. The results obtained are compared with each other and with those results obtained without the fixed compression ratio constraint for maximum power output.