This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved. It was named after the German physicist
Rudolf Clausius and French physicist
Pierre Duhem.
Clausius–Duhem inequality in terms of the specific entropy
The Clausius–Duhem inequality can be expressed in
integral form as
In this equation is the time, represents a body and the
integration is over the volume of the body, represents the surface of the body, is the
massdensity of the body, is the specific
entropy (entropy per unit mass), is the
normal velocity of , is the
velocity of particles inside , is the unit normal to the surface, is the
heatflux vector, is an
energy source per unit mass, and is the absolute
temperature. All the variables are functions of a material point at at time .
In
differential form the Clausius–Duhem inequality can be written as
where is the time derivative of and is the
divergence of the
vector.
Proof
Assume that is an arbitrary fixed
control volume. Then
and the
derivative can be taken inside the integral to give
Clausius–Duhem inequality in terms of specific internal energy
The inequality can be expressed in terms of the
internal energy as
where is the time derivative of the specific internal energy (the internal energy per unit mass), is the
Cauchy stress, and is the
gradient of the velocity. This inequality incorporates the
balance of energy and the
balance of linear and angular momentum into the expression for the Clausius–Duhem inequality.
Proof
Using the identity
in the Clausius–Duhem inequality, we get
is called the
dissipation which is defined as the rate of internal
entropy production per unit volume times the
absolute temperature. Hence the Clausius–Duhem inequality is also called the dissipation inequality. In a real material, the dissipation is always greater than zero.
^Truesdell, Clifford (1952), "The Mechanical foundations of elasticity and fluid dynamics", Journal of Rational Mechanics and Analysis, 1: 125–300.
^Truesdell, Clifford & Toupin, Richard (1960), "The Classical Field Theories of Mechanics", Handbuch der Physik, vol. III, Berlin: Springer.
^Frémond, M. (2006), "The Clausius–Duhem Inequality, an Interesting and Productive Inequality", Nonsmooth Mechanics and Analysis, Advances in mechanics and mathematics, vol. 12, New York: Springer, pp. 107–118,
doi:
10.1007/0-387-29195-4_10,
ISBN0-387-29196-2.