Restricted gradient constitutive dissipative model — Examples

Take a state point $(v_0,{\sigma }_0)$ of the selected model's supplied material domain with $v_0>0$ and $T_0=e_{{\sigma }_p}(v_0,{\sigma }_0)>0$. Its state map is constant, with $n_0=1\mathrm{/}{v}_0$, $s_0=n_0{\sigma }_0$, ${\epsilon }_0=n_{0}e(v_0,{\sigma }_0)$, and $p_0=-e_v(v_0,{\sigma }_0)$ as in the state conventions of Exact entropy identity in the selected bulk branch. Let ${\zeta }_0,{\tau }_0>0$ be constant coefficients with units Pa s and s, and take ${\mathrm{\Pi }}_0$ in Pa. On $M={\mathbb{R}}^4$ use global length coordinates $(x^0,x^1,x^2,x^3)=(ct,x^1,x^2,x^3)$, Minkowski metric $g=-(d{x}^0{)}^2+{\sum }_{i=1}^3(d{x}^i{)}^2$, and future direction ${\mathrm{\partial }}_{x^0}$. The frame is the observer congruence $u={\mathrm{\partial }}_{x^0}$ in that chart. For some $\delta >0$ seconds, take the open region $O=(-c\delta ,\mathrm{\infty })\times {\mathbb{R}}^3$, put $t=x^0\mathrm{/}c$, and define $\mathrm{\Pi }(t)={\mathrm{\Pi }}_0{e}^{-t\mathrm{/}{\tau }_0},t>-\delta \mathrm{.}$ Assume the application's open near-equilibrium regime contains these constant zero-gradient reference data at $\mathrm{\Pi }=0$; choose ${\mathrm{\Pi }}_0$ small enough that the field and first-derivative data remain in that regime throughout $O$. Then the source-free bulk balances and the corrected relaxation law hold. For every finite $t\ge 0$ the entropy production is ${\mathrm{\nabla }}_a{S}^a=\frac{{\mathrm{\Pi }}_0^2{e}^{-2t\mathrm{/}{\tau }_0}}{{\zeta }_{0}c{T}_0}\mathrm{.}$ It vanishes when ${\mathrm{\Pi }}_0=0$ and is strictly positive when ${\mathrm{\Pi }}_0\mathrm{\ne }0$. This is a constructed model solution, not a reported experiment.