Continuum thermodynamics

Let an interface move at signed speed $c$ in a reference unit normal direction $n$, and set $[a]=a^{+}-a^{-}$ with $n$ pointing from minus to plus. For continuous reference density and no surface sources, distributional momentum conservation, conservation of energy and the Lagrangian entropy inequality giveFor a level-set interface $\varphi =0$, ${\varphi }_t=-c|\nabla \varphi |$, the singular part of ${\partial }_{t}a$ is $-c[a]{\delta }_S$ and that of $\mathrm{Div}b$ is $n\cdot [b]{\delta }_S$, proving these formulas. Using $V=-c$ reverses the speed terms. Density jumps require jumps of the complete conserved densities.

Thermoelasticity couples elasticity and temperature through a specific Helmholtz free energy $\psi (F,\theta )$. Without constraints its reversible laws are $P_{Ii}={\rho }_0{\psi }_{,F_{iI}}$ and $\eta =-{\psi }_{,\theta }$. Mixed derivatives give the same coupling tensor in the thermal stress response and the strain dependence of specific entropy, with opposite signs.

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.

Linearize $\det F=h(\theta )$ at $F=I$, $\theta ={\theta }_0$, $h({\theta }_0)=1$, and zero reaction. With displacement gradient $H$ and temperature increment $\vartheta$, one has $\mathrm{tr}H=h^{\prime }({\theta }_0)\vartheta$ andwhere $C_{jilk}={\rho }_0{\psi }_{,F_{ij}{F}_{kl}}$, ${\beta }_{ji}={\rho }_0{\psi }_{,F_{ij}\theta }$ and $C_e=-{\rho }_0{\psi }_{,\theta \theta }$ at the reference state. The temperature factor converts this entropy coefficient to a volumetric heat capacity: ${\theta }_0{C}_e$. Strictly constant volume is the case $h^{\prime }=0$.

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.

The local entropy balance has nonnegative production, with heat supply ${\rho }_{0}r/\theta$ and thermodynamic entropy flux $q/\theta$ at positive temperature $\theta$. Its integral version holds on every material reference subvolume.

For the material-first nominal stress tensor, the displayed local conservation of energy law balances mechanical work, heat supply and outgoing heat flux. Integrating over a fixed reference material volume and using the divergence theorem gives its integral form. The full total-energy balance includes kinetic energy and external mechanical work; momentum conservation reduces it to this internal-energy balance.

An internal variable records a material state beyond its instantaneous deformation gradient and temperature, such as a relaxing microstructural state. A constitutive equation specifies its evolution and its contribution to the free energy.

A force-dependent potential specifies internal-variable rates by differentiation. Convexity with a minimum at zero force can ensure nonnegative force-rate dissipation. The name alone does not ensure the Clausius-Duhem inequality for an arbitrary potential.

The dissipation potential $\Omega =\alpha \Vert Q{\Vert }^{n+1}/(n+1)-\mathrm{tr}(AQ)/\tau$, with the Frobenius norm, gives $\dot{A}+A/\tau =\alpha \Vert Q{\Vert }^{n-1}{Q}^T$. An integrating factor proves the displayed memory formula. For $n>0$, its nonlinear driving term is defined continuously as zero at $Q=0$. Mechanical dissipation is $\mathrm{tr}(Q\dot{A})=\alpha \Vert Q{\Vert }^{n+1}-\mathrm{tr}(AQ)/\tau$, which must be nonnegative on thermodynamically admissible histories.

The derivative of internal energy at fixed deformation gradient and entropy, or of specific Helmholtz free energy at fixed temperature, defines this force. Its product with the internal-variable rate contributes mechanical dissipation to the Clausius-Duhem inequality.