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The Clausius-Duhem inequality is a local expression of the Second law of thermodynamics for a simple continuum whose thermodynamic entropy flux is heat flux divided by positive absolute temperature. With specific entropy $s$, outward heat flux $\mathbf{q}$ and heat supply $r$ per unit mass, it readsCombining this with the local internal energy balance and writing the specific Helmholtz free energy as $\psi =e-Ts$ givesHere $D$ is the rate-of-strain tensor and the dot denotes a material derivative.^[ref]

The inequality constrains constitutive equations, including both mechanical dissipation and thermal conduction. For Fourier's law $\mathbf{q}=-k\nabla T$ with $k\ge 0$, the thermal contribution is $k|\nabla T{|}^2/T\ge 0$. This direct sign check explains why passive thermal conduction carries heat down a temperature gradient. More general continua can require a different thermodynamic entropy flux, so the stated assumption matters.

Requiring the Clausius-Duhem inequality for independent reversible mechanical and thermal variations constrains a constitutive equation. For $U(F,\eta ,\xi )$ it gives $P_{Ii}={\rho }_0{U}_{,F_{iI}}$ and $\theta =U_{,\eta }$. Combining the Lagrangian internal energy balance with the Lagrangian entropy inequality leaves $f_r{\dot{\xi }}_r-q_I{\theta }_{,I}/\theta \ge 0$. The independence assumptions are part of the constitutive argument.

A constitutive equation specifies a material's response beyond universal balances such as mass conservation and momentum conservation. In a simple elastic test, Hooke's law $\sigma =E\epsilon$ relates axial stress to strain, with Young's modulus $E$ describing the specimen's elastic response. A Newtonian fluid instead relates viscous stress to the rate-of-strain tensor.

A useful constitutive equation states its variables and regime: temperature, deformation rate and loading history can change the response. Plasticity in continuum mechanics therefore uses internal variables as well as current strain, whereas an ideal elastic material stores and returns mechanical energy. Experiments identify parameters and test whether the model predicts other load paths. Physical consistency also imposes restrictions, including invariance under changes of observer and the Clausius-Duhem inequality. Merely fitting one measured curve does not establish validity for arbitrary deformation.

Continuum thermodynamics applies the first law of thermodynamics and Second law of thermodynamics locally to a deforming material. A region can gain internal energy through mechanical work, heat flux through its boundary and internal heat supply. Those mechanisms must be distinguished from transport caused by the region itself moving.

With specific internal energy $e$, mean mass density $\rho$, symmetric stress tensor $\sigma$, rate-of-strain tensor $D$ and outward heat flux $\mathbf{q}$, a simple continuum satisfiesHere $r$ is heat supplied per unit mass per unit time, and the dot is a material derivative. The Clausius-Duhem inequality additionally requires nonnegative local entropy production. Together the two laws test proposed constitutive equations: an equation can satisfy mechanical balance while incorrectly predicting heat flow from cold to hot without compensating work. Elastic energy storage, viscous dissipation and thermal conduction enter the same accounting framework.

A dissipation potential is a function used to generate dissipative rates or their conjugate driving forces. In a force-based convention, a potential $\Phi (X)$ specifies an internal variable rate through $\dot{\alpha }={\partial }_X\Phi$. For example, $\Phi =X^2/(2\eta )$ with $\eta >0$ gives $\dot{\alpha }=X/\eta$ and dissipation $X\dot{\alpha }=X^2/\eta \ge 0$.

If $\Phi$ is differentiable and convex with a minimum at $X=0$, the convexity inequality gives $X{\partial }_X\Phi \ge \Phi (X)-\Phi (0)\ge 0$. This supplies a direct route to nonnegative dissipation. A rate-based convention instead generates forces from rates, so its variables must be stated explicitly. The potential is useful for organizing constitutive equations, but the complete model must still satisfy the Clausius-Duhem inequality, including thermal terms and the chosen energetic sign convention.

Take any fixed material subvolume $B_0$ of the reference configuration, with outward normal vector $N$, current particle velocity $v_i={\dot{x}}_i$, and positive temperature $\theta$. The first law of thermodynamics, including kinetic energy, isThe heat-flux sign is negative because $N_I{q}_I^0$ is outward. The Second law of thermodynamics supplies the corresponding integral Lagrangian entropy inequality:The temperature in each supply or flux term is the local temperature, not a common temperature pulled outside the integrals.

Apply the divergence theorem and use the reference momentum conservation equation ${\rho }_0{\dot{v}}_i=P_{Ii,I}+{\rho }_0{g}_i$. In the total-energy equation, the kinetic derivative cancels the body-force work and the $v_i{P}_{Ii,I}$ term. Since $v_{i,I}={\dot{F}}_{iI}$, the remaining local Lagrangian internal energy balance and Lagrangian entropy inequality areThey hold pointwise because $B_0$ was arbitrary. In particular,

The specific Helmholtz free energy is $\psi =u-\theta \eta$. Substitution yields its balanceand elimination of the heat supply yields the Clausius-Duhem inequalityFor $\psi =\psi (F,\theta ,{\xi }_r)$, expand the derivative to obtainThe usual Coleman-Noll procedure admits independent reversible choices of $\dot{F}$ and $\dot{\theta }$ at a fixed state. Their coefficients must vanish, since either sign and arbitrary magnitude would otherwise violate the inequality. ThusInternal-variable evolution and heat conduction must satisfy this remaining inequality. Substituting the constitutive relations back into the energy balance givesThe last printed equation is missing a factor ${\rho }_0$ on its internal-variable term. The PDF defines $f_r=-{\psi }_{,{\xi }_r}$ with $\psi$ per unit mass, so the expression above is the dimensionally consistent consequence. Equivalently define the volumetric thermodynamic force conjugate to an internal variable ${\hat{f}}_r=-{\rho }_0{\psi }_{,{\xi }_r}$; the term is then ${\hat{f}}_r{\dot{\xi }}_r$ with no extra density. The two conventions must not be mixed.

For thermoelasticity with a temperature-dependent constraint, admissible rates obeyThe reversible coefficient functional need now vanish only on this tangent hyperplane. Hence it can be a scalar multiple $q$ of the constraint normal:ThereforeThe reaction terms cancel on admissible rates, leaving the same internal-variable/conduction inequality and entropy heating equation. The multiplier $q$ is determined by equilibrium and boundary conditions rather than by the free energy alone.

For linear thermoelasticity with prescribed thermal volume, write $F=I+H$, $H_{ij}=u_{i,j}$, and $\vartheta =\theta -{\theta }_0$. Hold the internal variables fixed in this reversible linearization, and evaluate all derivatives below at $(I,{\theta }_0)$ and zero reaction. The stress-free reference state has ${\psi }_{,F}=0$. Since $\partial \det F/\partial F=I$ there, first-order nominal and Cauchy stress tensors agree. Their constitutive laws becomewhere equality of mixed free-energy derivatives givesThe linearized constraint is $u_{i,i}=h^{\prime }({\theta }_0)\vartheta$. Thus volume is thermally prescribed; strict temperature-independent incompressibility corresponds to $h^{\prime }=0$. The coefficient $C_e$ is an entropy slope; the corresponding volumetric heat capacity at fixed deformation is ${\theta }_0{C}_e$.

Together with ${\rho }_0{\ddot{u}}_i={\sigma }_{ji,j}+{\rho }_0{g}_i$, the first-order thermal balance is ${\theta }_0(C_e\dot{\vartheta }-{\beta }_{ji}{\dot{u}}_{i,j}+h^{\prime }({\theta }_0)\dot{q})={\rho }_{0}r-q_{I,I}^0$. A heat-flux constitutive law completes the evolution problem. The paired signs of the coupling tensor follow from the same free energy, rather than from separately postulated mechanical and thermal equations.

Use the material-first nominal stress tensor convention $P_{Ii}$, with $F_{iI}$ the deformation gradient. For any fixed reference material subvolume $V_0$ with outward unit normal $N$, conservation of energy isHere $b$ is body force per mass, $r$ is heat supply per mass and $q$ is nominal heat flux. Using momentum conservation to remove the kinetic energy balance gives the integral Lagrangian internal energy balance:The integral Lagrangian entropy inequality iswhere $\theta >0$ is temperature and $q/\theta$ is the thermodynamic entropy flux.

Since the subvolume is arbitrary, the divergence theorem givesExpand $\dot{U}=U_{,F_{iI}}{\dot{F}}_{iI}+U_{,\eta }\dot{\eta }+U_{,{\xi }_r}{\dot{\xi }}_r$ and eliminate the heat supply between these relations. The Clausius-Duhem inequality becomesUnder the usual local constitutive independence assumptions, the reversible mechanical and thermal variations can have either sign. The Coleman-Noll procedure therefore imposesDefine the thermodynamic force conjugate to an internal variable by $f_r=-{\rho }_0{U}_{,{\xi }_r}$. The energy balance and the remaining entropy constraint now give, respectively,

A thermodynamic force conjugate to an internal variable is the energetic driving quantity paired with that variable's rate in the dissipation. Its sign and density normalization must be specified. With specific internal energy $U(F,\eta ,\xi )$, reference mass density ${\rho }_0$ and the convention used here,Equivalently, if the specific Helmholtz free energy is $\psi (F,T,\xi )$, then $f_{\xi }=-{\rho }_0(\partial \psi /\partial \xi {)}_{F,T}$. The negative sign makes positive force-rate product correspond to free-energy decrease at fixed external variables.

For an isothermal free energy per reference volume $W=E(\epsilon -\xi {)}^2/2$, the driving force is $f_{\xi }=E(\epsilon -\xi )$. Choosing $\dot{\xi }=f_{\xi }/{\eta }_v$ with ${\eta }_v>0$ yields $f_{\xi }\dot{\xi }=f_{\xi }^2/{\eta }_v\ge 0$. The internal variable relaxes toward the imposed strain, reducing stored elastic energy. Such a law supplies the mechanical part of a Clausius-Duhem inequality check; the thermal contributions still need their own consistent description.

The admissible rates obey ${\varphi }_{,F}:\dot{F}=h^{\prime }(\theta )\dot{\theta }$. A reaction multiplier $q$ in the reversible nominal stress tensor therefore has an accompanying specific entropy contribution:Their reaction contributions cancel in the Clausius-Duhem inequality on admissible rates. Omitting the entropy term would generally violate the constraint's power balance.