To some extent modern continuum thermodynamics amounts to a collection of “ thermodynamical theories” sharing common premisses and common. sources on Ωt. Total entropy: units [J/K], defined up to a constant by. dS = dQ. T. Clausius-Duhem inequality: mathematical form of the 2nd law: DS. Dt. ≥. ∫. Ωt. sθ is the specific dissipation (or internal dissipation) and is denoted by the symbol ϕ. The Clausius-Duhem inequality can simply be written as.

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Using the divergence theoremwe get.

## Continuum mechanics/Clausius-Duhem inequality for thermoelasticity

In this equation is the time, represents a body and the integration is over the claussius of the body, represents the surface of the body, is the mass density of the body, is the specific entropy entropy per unit massis the normal velocity ofis clausiua velocity of particles insideis the unit normal to the surface, is the heat flux vector, is an energy source per unit lnequality, and is the absolute temperature. In differential form the Clausius—Duhem inequality can be written as.

Surface tension Capillary action. Views Read Edit View history. Rheology Viscoelasticity Dhem Rheometer. In a real material, the dissipation is always greater than zero.

This inequality is particularly useful in determining whether the constitutive relation of a material is thermodynamically allowable. This inequality incorporates the balance of energy and the balance of linear and angular momentum into the expression for the Clausius—Duhem inequality. This inequality incorporates the balance of energy and the balance of linear clausous angular momentum into the expression for the Clausius—Duhem inequality.

The inequality can be expressed in terms of the internal energy as. Then and the derivative can be taken inside the integral to give Using the divergence theoremwe 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 the Clausius—Duhem inequality is also called the dissipation inequality.

### Clausius–Duhem inequality

Laws Conservations Energy Mass Momentum. Now, using index notation with respect to a Cartesian coordinate system. By using this site, you agree to the Terms of Use and Privacy Policy.

From the conservation of mass.

The Clausius—Duhem inequality can be expressed in integral form as. This page was last edited on 9 Augustat Hence the Clausius—Duhem inequality is also called the dissipation inequality. Using the divergence theoremwe get.

This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved.

In differential form the Clausius—Duhem inequality can be written as. The Clausius—Duhem inequality [1] [2] is a way of expressing the second law of thermodynamics that is used in continuum mechanics. From Wikipedia, the free encyclopedia. Clausius—Duhem inequality Continuum mechanics.

Surface tension Capillary action. In a real material, the dissipation is always greater than zero. Since is arbitrary, we must have. This inequality is particularly useful in determining whether the constitutive relation of a inequaljty is thermodynamically allowable.

All the variables are functions of a material point at at time. Using the identity in the Clausius—Duhem inequality, we get Now, using index notation with respect to a Cartesian coordinate systemHence, From the balance of energy Therefore, Rearranging. Retrieved from ” https: Now, the material time derivatives of and are given by. Rheology Viscoelasticity Rheometry Rheometer.

The Clausius—Duhem inequality can be expressed in integral form as. Assume that is an arbitrary fixed control volume. This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved. The inequality can be expressed in terms of the internal energy as.

inequaality The Clausius—Duhem inequality [1] [2] is a way of expressing the second law of thermodynamics that is used in continuum mechanics. From the balance of energy. From the balance of energy.

Laws Conservations Energy Mass Momentum.