Thermodynamics of monopolar continuum of grade $n$

Abstract

Contrary to the theory of the simple material it is assumed that the values of physical quantities at a point are arrected by the deformation history of a finite neighborhood of the point. In the case of the monopolar continuum of grade $n$, the physical quantities are in a functional dependence on the temperature and, moreover, on 4n$ deformation gradients which are found from a single shift function. Equations of equilibrium and the boundary values for all $n$ stress tensors are evaluated on the basis of the First Law of Thermodynamics. Introducing the Hilbert space with the norm which expresses the fading of the memory it is possible to derive the system of constitutive equations from the Second Law of Thermodynamics. These equations enable us to evaluate the entropy as well as all stress tensors provided the functional dependence of the free energy on the history of $n$ deformation gradients and on the history of temperature is given.