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.
The Clausius–Duhem inequality can be expressed in integral form as
| d | |
| dt |
\left(\int\Omega\rhoηdV\right)\ge \int\partial\rhoη\left(un-v ⋅ n\right)dA-\int\partial
| q ⋅ n | |
| T |
~dA+\int\Omega
| \rhos | |
| T |
~dV.
t
\Omega
\partial\Omega
\rho
η
un
\partial\Omega
v
\Omega
n
q
s
T
x
t
In differential form the Clausius–Duhem inequality can be written as
\rho
| η |
\ge-\boldsymbol{\nabla} ⋅ \left(
| q | |
| T |
\right)+
| \rho~s | |
| T |
| η |
η
\boldsymbol{\nabla} ⋅ (a)
a
The inequality can be expressed in terms of the internal energy as
| \rho~(e |
-T~
| η |
)-\boldsymbol{\sigma}:\boldsymbol{\nabla}v\le-\cfrac{q ⋅ \boldsymbol{\nabla}T}{T}
| e |
e
\boldsymbol{\sigma}
\boldsymbol{\nabla}v
The quantity
l{D}=\rho~(T~
| η | - |
| e |
)+\boldsymbol{\sigma}:\boldsymbol{\nabla}v-\cfrac{q ⋅ \boldsymbol{\nabla}T}{T}\ge0