Gibbs cancellation and entropy-production identity

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$: every sequence $(X_n{)}_{n\in \mathbb{N}}$ of nonempty sets has a function $f$ with $f(n)\in X_n$ for every $n$. Let $U\subset (0,\mathrm{\infty })\times \mathbb{R}$ be open and let $e:U\to \mathbb{R}$ have continuous total derivatives through order two. For $(v,s)\in U$, $v$ is the specific-volume coordinate, $s$ the specific-entropy coordinate, and $e(v,s)$ the specific-energy value; set $T=e_s(v,s)$ and $p=-e_v(v,s)$, with each state partial derivative holding the other coordinate fixed.

Let $I\subset \mathbb{R}$ be a nonempty open interval. Let $B\subset {\mathbb{R}}^3$ be a nonempty bounded open set with $C^1$ boundary, meaning that near every boundary point it is the graph of a continuously differentiable function after a rigid coordinate change. Let $N\subset {\mathbb{R}}^3$ be open with $\bar{B}\subset N$, and let $\chi \in C^2(I\times N,{\mathbb{R}}^3)$, where $C^r$ means that all coordinate partial derivatives through total order $r$ exist and are continuous. For each $t\in I$, assume ${\chi }_t(a):=\chi (t,a)$ is a bijection from $N$ onto an open set with a $C^2$ inverse and $\det D_a\chi (t,a)>0$. Define ${\mathrm{\Omega }}_t:={\chi }_t(B)$, $\mathrm{\Omega }:=\{(t,x):t\in I,x\in {\mathrm{\Omega }}_t\}$, and $u(t,x):={\mathrm{\partial }}_t\chi (t,{\chi }_t^{-1}(x))$; this swept set $\mathrm{\Omega }$ is open. For a scalar or vector field $f$ on $\mathrm{\Omega }$, set $D_{t}f:={\mathrm{\partial }}_{t}f+{\sum }_{j=1}^3{u}_j{\mathrm{\partial }}_{x_j}f$ componentwise. For each nonempty bounded open $C^1$ subdomain $V$ with $\bar{V}\subset B$, set $P_t:={\chi }_t(V)$; here a $C^1$ domain is locally on one side of a $C^1$ boundary graph.

Let $\rho ,v,s,T,p,q,r,b$ be fields on $\mathrm{\Omega }$ with continuous first partial derivatives, with $\rho >0$, $(v,s)\in U$, $v=1\mathrm{/}\rho$, $T=e_s(v,s)>0$, and $p=-e_v(v,s)$. Let $e$ on $\mathrm{\Omega }$ denote the composition $e(v,s)$. Let $\sigma$ be a continuously differentiable $3\times 3$ stress matrix field, let $\tau$ be a continuously differentiable $3\times 3$ matrix field, and suppose $\sigma =-pI+\tau$, where $I$ is the $3\times 3$ identity. For a scalar field $h$, define $(\mathrm{\nabla }h{)}_i:={\mathrm{\partial }}_{x_i}h$; define $L:=\mathrm{\nabla }u$ by $L_{ij}:={\mathrm{\partial }}_{x_j}{u}_i$, $D:=(L+L^T)\mathrm{/}2$, and $\theta :=\mathrm{tr}D={\sum }_{i=1}^3{\mathrm{\partial }}_{x_i}{u}_i$. Define $A:B:={\sum }_{i,j=1}^3{A}_{ij}{B}_{ij}$ for matrices, $\mathrm{div}q:={\sum }_i{\mathrm{\partial }}_{x_i}{q}_i$, and $\mathrm{div}(q\mathrm{/}T)$ componentwise. Put $a_m:={\mathrm{\partial }}_{t}u+(u\cdot \mathrm{\nabla })u$, $k:=\mathrm{\mid }u{\mathrm{\mid }}^2\mathrm{/}2$, and $E:=e+k$. The fields $u,q,b$ are vector-valued and $r$ is scalar-valued.

Assume there is a contact traction $t_c(t,y,n)\in {\mathbb{R}}^3$ depending on a surface only through its point $y$ and outward unit normal $n\in S^2$, continuous in $(y,n)$ and uniformly bounded near each point. Assume its local force balance ${\int }_{\mathrm{\partial }P}{t}_c(t,y,{\nu }_P(y))d{S}_y+{\int }_P\rho (t,y)(b(t,y)-a_m(t,y))dy=0$ for every nonempty bounded open $C^1$ subdomain $P$ whose closure is compactly contained in ${\mathrm{\Omega }}_t$. Surface integrals use Euclidean surface measure with chart area density $\sqrt{\det (D{X}^{T}DX)}$. Suppose the unique Cauchy stress supplied by this contact balance satisfies $t_c(t,y,n)=\sigma (t,y)n$ and has the stated $C^1$ regularity; the contact result alone does not imply that regularity.

Assume the pointwise continuity equation ${\mathrm{\partial }}_t\rho +\mathrm{div}(\rho u)=0$ and, for every such material subdomain $V$ and every $t\in I$, the integral balances $\frac{d}{dt}{\int }_{P_t}\rho udx={\int }_{P_t}\rho bdx+{\int }_{\mathrm{\partial }{P}_t}\sigma \nu dS,$ $\frac{d}{dt}{\int }_{P_t}x\times (\rho u)dx={\int }_{P_t}x\times (\rho b)dx+{\int }_{\mathrm{\partial }{P}_t}x\times (\sigma \nu )dS,$ $\frac{d}{dt}{\int }_{P_t}\rho Edx={\int }_{P_t}\rho b\cdot udx+{\int }_{\mathrm{\partial }{P}_t}(\sigma \nu )\cdot udS-{\int }_{\mathrm{\partial }{P}_t}q\cdot \nu dS+{\int }_{P_t}\rho rdx,$ where $\nu$ is the outward unit normal, $\times$ is the ordinary vector cross product, and all fields are continuous up to the swept closures. The angular-momentum balance gives $\sigma ={\sigma }^T$, and the local internal-energy equation supplied by these hypotheses is $\rho D_{t}e=\sigma :D-\mathrm{div}q+\rho r$. Define the entropy residual $\mathcal{G}:=\rho D_{t}s+\mathrm{div}(q\mathrm{/}T)-\rho r\mathrm{/}T$. Then $\mathcal{G}=\frac{\tau :D}{T}-\frac{q\cdot \mathrm{\nabla }T}{T^2}\mathrm{.}$ At any state with $e_{ss}>0$, a unique local $C^1$ branch $s(v,T)$ satisfies $e_s(v,s(v,T))=T$; the local Helmholtz function $a(v,T):=e(v,s(v,T))-Ts(v,T)$ then satisfies $a_T=-s$ and $a_v=-p$. The entropy identity does not assert $\mathcal{G}\ge 0$ and makes no empirical claim. In the physical SI interpretation, $\rho$ is in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, $u$ in $\mathrm{m}{\mathrm{s}}^{-1}$, $b$ in $\mathrm{m}{\mathrm{s}}^{-2}$, $v$ in ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}$, $e,k,E$ in $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$, $s$ in $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$, $T$ in $\mathrm{K}$, $p,\sigma ,\tau ,t_c$ in $\mathrm{P}\mathrm{a}$, $q$ in $\mathrm{W}{\mathrm{m}}^{-2}$, and $r$ in $\mathrm{W}\mathrm{k}{\mathrm{g}}^{-1}$.

Rohan Abeyaratne, Continuum Mechanics, Volume II, Chapter 5 §§5.1–5.2, printed pp. 147–151