The Clausius–Duhem inequality[1][2] is a way of expressing the second law of thermodynamics that is used in continuum mechanics.
This inequality is particularly useful in determining whether the constitutive relation of a material is thermodynamically allowable.
[3] 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.
The Clausius–Duhem inequality can be expressed in integral form as In this equation
represents a body and the integration is over the volume of the body,
represents the surface of the body,
is the mass density of the body,
is the velocity of particles inside
is the unit normal to the surface,
is the heat flux vector,
is an energy source per unit mass, and
All the variables are functions of a material point at
In differential form the Clausius–Duhem inequality can be written as where
is an arbitrary fixed control volume.
and the derivative can be taken inside the integral to give Using the divergence theorem, we get Since
is arbitrary, we must have Expanding out or, or, Now, the material time derivatives of
Hence, 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),
This inequality incorporates the balance of energy and the balance of linear and angular momentum into the expression for the Clausius–Duhem inequality.
in the Clausius–Duhem inequality, we get Now, using index notation with respect to a Cartesian coordinate system
, Hence, From the balance of energy Therefore, Rearranging,
The quantity is called the dissipation which is defined as the rate of internal entropy production per unit volume times the absolute temperature.
In a real material, the dissipation is always greater than zero.