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

Laws Conservations Energy Mass Momentum.

The Clausius—Duhem inequality [1] [2] is a way of expressing the second law of thermodynamics that is used in continuum mechanics. Using the divergence theoremwe get. Rheology Viscoelasticity Rheometry Rheometer. Surface tension Capillary action.

This inequality incorporates the balance of energy and the balance of linear and angular momentum into the expression for the Clausius—Duhem inequality. Now, the material time derivatives of and are given by. This inequality is particularly useful in determining whether the constitutive relation of a material is thermodynamically allowable.

Hence the Clausius—Duhem inequality is also called the dissipation inequality. In differential form the Clausius—Duhem inequality can be written as.

Hence the Clausius—Duhem inequality is also called the dissipation inequality. Clausius—Duhem inequality Continuum mechanics. This inequality is particularly useful in determining whether the constitutive relation of a material is thermodynamically allowable. Now, using index notation with respect to a Cartesian coordinate system.

All the variables are functions of a material point at at time.

In a real material, the dissipation is always greater than zero. Inequaliry inequality incorporates the balance of energy and the balance of linear and angular momentum into the expression for the Clausius—Duhem inequality. This page was last edited on 9 Augustat Views Read Edit View history.

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

### Clausius–Duhem inequality

In a real material, the dissipation is always greater than zero. From Wikipedia, the free encyclopedia. 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.

## Continuum mechanics/Clausius-Duhem inequality for thermoelasticity

In this equation is the time, represents inequapity 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 massis the normal velocity ofis the velocity of particles insideis the unit normal to the surface, is the heat flux vector, is an energy source per unit mass, and is the absolute temperature.

From the conservation of mass. The Clausius—Duhem inequality can be expressed in integral form as. Assume that is an arbitrary fixed control volume. The inequality can be expressed in terms of the internal energy as. The inequality can be expressed in terms of the internal energy as.

Surface tension Capillary action. Laws Conservations Energy Mass Momentum. In differential form the Clausius—Duhem inequality can be written as. Retrieved from ” https: Using the inequslity 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.

Using the divergence theoremwe get.

### Continuum mechanics/Clausius-Duhem inequality for thermoelasticity – Wikiversity

Ineqality Viscoelasticity Rheometry Rheometer. 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.

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The Clausius—Duhem inequality can be expressed in integral form as. Since is arbitrary, we must have.