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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## Continuum mechanics/Clausius-Duhem inequality for thermoelasticity

In a real material, the dissipation is always greater than zero. The Clausius—Duhem inequality [1] [2] is a way of expressing the second law of thermodynamics that is used in continuum mechanics. Now, the material time derivatives of and are given by.

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

Laws Conservations Energy Mass Momentum. By using this site, you agree to the Terms of Use and Privacy Policy. The Clausius—Duhem inequality can be expressed in integral form as. Views Read Edit View history.

Assume that is an arbitrary fixed control volume. Hence the Clausius—Duhem inequality clusius also called the dissipation inequality. In differential form the Clausius—Duhem inequality can be written as.

This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved. Now, using index notation with respect to a Cartesian coordinate system. Laws Conservations Energy Mass Momentum. This inequality incorporates the balance of energy and the balance of linear and angular momentum ineqiality the expression for the Clausius—Duhem inequality.

## Clausius–Duhem inequality

Retrieved from ” https: The inequality can be expressed in terms of the internal energy as. 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. Hence the Clausius—Duhem inequality is also called the dissipation inequality.

### Clausius–Duhem inequality – Wikipedia

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.

Rheology Viscoelasticity Rheometry Rheometer. This page was last edited on 9 Augustat Rheology Viscoelasticity Rheometry Rheometer. Since is arbitrary, we must have. This inequality is particularly useful in determining whether the constitutive relation of a material is thermodynamically allowable. In differential form the Clausius—Duhem inequality can be written as.

Using the divergence theoremwe get. From the balance of energy.

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

The inequality can be expressed in terms of the internal energy as. This inequality incorporates the balance of energy and the balance of linear and angular momentum into the expression for the Clausius—Duhem inequality.

All the variables are functions of a material point at at time. The Clausius—Duhem inequality can be expressed in integral form as. In a real material, the dissipation is always greater than zero.

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 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.