Clausius–Duhem inequality

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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.

Clausius–Duhem inequality in terms of the specific entropy

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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 mass density 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 heat flux 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

 

Using the divergence theorem, we get

 

Since   is arbitrary, we must have

 

Expanding out

 

or,

 

or,

 

Now, the material time derivatives of   and   are given by

 

Therefore,

 

From the conservation of mass  . Hence,

 

Clausius–Duhem inequality in terms of specific internal energy

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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

 

Now, using index notation with respect to a Cartesian coordinate system  ,

 

Hence,

 

From the balance of energy

 

Therefore,

 

Rearranging,

 

Q.E.D.

Dissipation

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The quantity

 

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.

See also

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References

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  1. ^ Truesdell, Clifford (1952), "The Mechanical foundations of elasticity and fluid dynamics", Journal of Rational Mechanics and Analysis, 1: 125–300.
  2. ^ Truesdell, Clifford & Toupin, Richard (1960), "The Classical Field Theories of Mechanics", Handbuch der Physik, vol. III, Berlin: Springer.
  3. ^ 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, ISBN 0-387-29196-2.
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