Local momentum balance: $\rho D_{t}u=\rho b+\mathrm{div}\sigma$ from the integral balance on admissible material subregions

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as used by the transport and divergence theorems. Let $I\subseteq \mathbb{R}$ be a nonempty open interval, let $B\subseteq {\mathbb{R}}^3$ be a material region, let $\chi :I\times O\to {\mathbb{R}}^3$ be a motion of $B$ on $I$ with spatial velocity $u$ and material derivative $D_t$ (Continuum motions, spatial velocity, material acceleration and swept regions, Euclidean spatial calculus: gradient, divergence and material derivative), and let $J_0\subseteq I$ be an open interval with $\overline{J_0}\subseteq I$. Define the swept region $\mathrm{\Sigma }=\{(t,x):t\in \overline{J_0},x\in {\chi }_t(\bar{B})\}$.

Write ${\mathcal{D}}_{\chi }:=\{(t,x):t\in I,x\in {\chi }_t(O)\}$ for the open domain of the spatial velocity $u$ (Continuum motions, spatial velocity, material acceleration and swept regions), and take $U\subseteq {\mathcal{D}}_{\chi }$ open and containing $\mathrm{\Sigma }$. Let $\rho :U\to (0,\mathrm{\infty })$ be a $C^1$ density field, let $\sigma :U\to M_3(\mathbb{R})$ be a $C^1$ field (the stress field of Traction fields and stress matrices representing them), and let $b:U\to {\mathbb{R}}^3$ be continuous. Suppose:

(K) $\rho$ satisfies ${\mathrm{\partial }}_t\rho +\mathrm{div}(\rho u)=0$ at every $(t,x)$ with $t\in J_0$ and $x\in {\chi }_t(B)$;

(L) for every material subregion $B^{\mathrm{\prime }}$ (nonempty bounded open with $\overline{B^{\mathrm{\prime }}}\subseteq B$) and every $t\in J_0$ for which ${\chi }_t(B^{\mathrm{\prime }})$ is a bounded $C^1$ domain (Bounded C1 domains and their outward normals) , the balance is

$\frac{d}{dt}{\int }_{{\chi }_t(B^{\mathrm{\prime }})}\rho (t,x)u(t,x)dx={\int }_{{\chi }_t(B^{\mathrm{\prime }})}\rho (t,x)b(t,x)dx+{\int }_{\mathrm{\partial }{\chi }_t(B^{\mathrm{\prime }})}\sigma (t,y)\nu (y)dS(y),$

where $\nu$ is the outward unit normal and the surface integral is that of Surface integration on compact C1 hypersurfaces.

Define the row divergence of $\sigma$ componentwise by

$g(t,x):=({\sum }_j{\mathrm{\partial }}_j{\sigma }_{1j},{\sum }_j{\mathrm{\partial }}_j{\sigma }_{2j},{\sum }_j{\mathrm{\partial }}_j{\sigma }_{3j})(t,x)\mathrm{.}$

Then for every $(t,x)\in J_0\times {\chi }_t(B)$,

$\rho (t,x)D_{t}u(t,x)=\rho (t,x)b(t,x)+g(t,x)\mathrm{.}$

• The conclusion is conditional on the integral balance (L) and the continuity equation (K); it is a localization theorem, not a derivation of the balance law. Hypothesis (L) also states the boundary regularity that makes the surface integral and the divergence theorem meaningful; without it the surface term is not defined.

• The row divergence convention $g_i={\sum }_j{\mathrm{\partial }}_j{\sigma }_{ij}$ is exactly the vector whose component $i$ appears in the divergence theorem applied to the $i$-th row of $\sigma$; it is the matrix divergence consistent with the traction convention $t(n)=\sigma n$ of Traction fields and stress matrices representing them. For a symmetric stress matrix the row and column divergences coincide; their agreement is not assumed here.

• No symmetry of $\sigma$, no constitutive relation and no boundary condition are used or asserted. The equation is local on the current region only.