ABSTRACT
The total energy increase in a body, including internal energy and kinetic energy, is equal to the corresponding work done on the body and the heat added to the body. In rate form,
(7.1)
in which: Ξ is the internal energy with density ξ
(7.2a)
is the rate of mechanical work, satisfying
(7.2b)
is the rate of heat input, with heat production h and heat flux q, satisfying
(7.2c)
is the rate of increase in the kinetic energy,
(7.2d)
It has been tacitly assumed that all work is done on S, and that body forces do no work. Invoking the divergence theorem and balance of linear momentum furnishes
(7.3)
The inner bracketed term inside the integrand vanishes by virtue of the balance of linear momentum. The relation holds for arbitrary volumes, from which the local form of balance of energy, referred to undeformed coordinates, is obtained as
(7.4)
To convert to undeformed coordinates, note that
(7.5)
In undeformed coordinates, Equation 7.3 is rewritten as
(7.6a)
furnishing the local form
(7.6b)
7.1.2 ENTROPY PRODUCTION INEQUALITY
Following the thermodynamics of ideal and non-ideal gases, the entropy production inequality is introduced as follows (see Callen, 1985):
(7.7a)
in which H is the total entropy, η is the specific entropy per unit mass, and T is the absolute temperature. This relation provides a “framework” for describing the irreversible nature of dissipative processes, such as heat flow and plastic deformation. We apply the divergence theorem to the surface integral and obtain the local form of the entropy production inequality:
(7.7b)
The corresponding relation in undeformed coordinates is
(7.7c)
7.1.3 THERMODYNAMIC POTENTIALS
The Balance of Energy introduces the internal energy Ξ, which is an extensive variableits value accumulates over the domain. The mass and volume averages of extensive variables are also referred to as extensive variables. This contrasts with intensive, or pointwise, variables, such as the stresses and the temperature. Another extensive variable
is the entropy H. In reversible elastic systems, the heat flux is completely converted into entropy according to
(7.8)
(We shall consider several irreversible effects, such as plasticity, viscosity, and heat conduction.) In undeformed coordinates, the balance of energy for reversible processes can be written as
(7.9)
We call this equation the thermal equilibrium equation. It is assumed to be integrable, so that the internal energy is dependent on the current state represented by the current values of the state variables E and η. For the sake of understanding, we can think of T as a thermal stress and η as a thermal strain. Clearly, if there is no heat input across the surface or generated in the volume. Consequently, the entropy is a convenient state variable for representing adiabatic processes.
