Clausius–Duhem inequality

Clausius–Duhem inequality

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

The Clausius–Duhem inequality can be expressed in integral form as

   \cfrac{d}{dt}\left(\int_\Omega \rho~\eta~\text{dV}\right) \ge
   \int_{\partial \Omega} \rho~\eta~(u_n - \mathbf{v}\cdot\mathbf{n})~\text{dA} - 
   \int_{\partial \Omega} \cfrac{\mathbf{q}\cdot\mathbf{n}}{T}~\text{dA} + 
    \int_\Omega \cfrac{\rho~s}{T}~\text{dV}.

In this equation t\, is the time, \Omega\, represents a body and the integration is over the volume of the body, \partial \Omega\, represents the surface of the body, \rho\, is the mass density of the body, \eta\, is the specific entropy (entropy per unit mass), u_n\, is the normal velocity of \partial \Omega\,, \mathbf{v} is the velocity of particles inside \Omega\,, \mathbf{n} is the unit normal to the surface, \mathbf{q} is the heat flux vector, s\, is an energy source per unit mass, and T\, is the absolute temperature. All the variables are functions of a material point at \mathbf{x} at time t\,.

In differential form the Clausius–Duhem inequality can be written as

     \rho~\dot{\eta} \ge - \boldsymbol{\nabla} \cdot \left(\cfrac{\mathbf{q}}{T}\right) 
        + \cfrac{\rho~s}{T}

where \dot{\eta} is the time derivative of \eta\, and \boldsymbol{\nabla} \cdot (\mathbf{a}) is the divergence of the vector \mathbf{a}.

Clausius–Duhem inequality in terms of specific internal energy

The inequality can be expressed in terms of the internal energy as

     \rho~(\dot{e} - T~\dot{\eta}) - \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} \le 
       - \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T}

where \dot{e} is the time derivative of the specific internal energy e\, (the internal energy per unit mass), \boldsymbol{\sigma} is the Cauchy stress, and \boldsymbol{\nabla}\mathbf{v} 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.


The quantity

   \mathcal{D} := \rho~(T~\dot{\eta}-\dot{e}) + \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v}  
       - \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T} \ge 0

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


  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, 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, 12, New York: Springer, pp. 107–118, doi:10.1007/0-387-29195-4_10, ISBN 0387291962 .

External links

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Clausius theorem — Thermodynamics …   Wikipedia

  • Continuum mechanics — Continuum mechanics …   Wikipedia

  • Entropy — This article is about entropy in thermodynamics. For entropy in information theory, see Entropy (information theory). For a comparison of entropy in information theory with entropy in thermodynamics, see Entropy in thermodynamics and information… …   Wikipedia

  • Martin Feinberg — Born April 2, 1942 (1942 04 02) (age 69) New York City, New York …   Wikipedia

  • Chemical thermodynamics — is the study of the interrelation of heat and work with chemical reactions or with physical changes of state within the confines of the laws of thermodynamics. Chemical thermodynamics involves not only laboratory measurements of various… …   Wikipedia

  • Josiah Willard Gibbs — Infobox Scientist box width = 300px name = J. Willard Gibbs image size = 300px caption = Josiah Willard Gibbs birth date = birth date|1839|2|11|mf=y birth place = New Haven, Connecticut, USA death date = death date and… …   Wikipedia

  • Heat engine — Thermodynamics …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”