Adopted balance principles of the continuum model: force and couple-free torque balance on every admissible region

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the Lebesgue volume and surface integration used in these general balances. The pointwise field and open-region predicates are unaffected.

Adopt for every model of the continuum branch, together with the force description of Adopted force model of the continuum branch: continuous body forces and Cauchy contact tractions and the fields of The density field of the continuum model and the mass of a material subregion and Traction fields and stress matrices of the continuum model, with units and sign conventions, the following balance principles.

(i) Setting. Let $U\subseteq \mathbb{R}\times {\mathbb{R}}^3$ be an open set containing the swept region of the motion over its time window. The density $\rho$, the spatial velocity $u$ and the material acceleration $a=D_{t}u$ are continuous on $U$ (Euclidean spatial calculus: gradient, divergence and material derivative, Continuum motions, spatial velocity, material acceleration and swept regions), and $b$ and $t$ are the body-force and traction fields of Adopted force model of the continuum branch: continuous body forces and Cauchy contact tractions. For $s$ in the time window the current region is ${\mathrm{\Omega }}_s={\chi }_s(B)$.

(ii) Admissible regions. An admissible region at time $s$ is a bounded piecewise $C^1$ Euclidean domain with a specified finite presentation with $D\subseteq {\mathrm{\Omega }}_s$, whose closure lies in the open domain of the fields at time $s$ (Specified finite piecewise C1 boundary presentations), with outward unit normal ${\nu }_D$ off its edge set and boundary term the sum of the surface integrals over its faces (Surface integration on compact C1 hypersurfaces).

(iii) Balance of forces. For every time $s$ and every admissible region $D$ at time $s$,

$$\begin{array}{cccc}& {\int }_{\mathrm{\partial }D}t(s,y,{\nu }_D(y))dS(y)+{\int }_D(\rho b-\rho a)(s,y)dy=0.& & \text{(F)}\end{array}$$

Read as a momentum balance, (F) says that the resultant contact force on $D$ equals ${\int }_D\rho ady-{\int }_D\rho bdy$; equivalently contact plus body force equals ${\int }_D\rho ady$, i.e. the total applied force equals the mass times acceleration of the region, with the inertial term written as a volume force.

(iv) Balance of torques, with no couples. For every time $s$, every admissible region $D$ at time $s$, and every fixed origin $x_0\in {\mathbb{R}}^3$,

$$\begin{array}{cccc}& {\int }_{\mathrm{\partial }D}(y-x_0)\times t(s,y,{\nu }_D(y))dS(y)+{\int }_D(y-x_0)\times (\rho b-\rho a)(s,y)dy=0,& & \text{(T)}\end{array}$$

the cross product being that of ${\mathbb{R}}^3$ (The cross product in ${\mathbb{R}}^3$). The balance is imposed with the same volume force density $\rho b-\rho a$ as in (F); there are no body couples and no surface couples. Since the identity holds for every origin, the choice of $x_0$ is immaterial given (F).

(v) Exclusions and status. These principles are not adopted for: bodies with couple stresses or surface couples; systems exchanging momentum with electromagnetic or thermal fields (the momentum of those fields would have to be accounted for separately); bodies with mass sources; and descriptions in non-inertial frames (where additional inertial terms appear and are excluded by the choice of the fixed inertial frame of Continuum body data, the fixed inertial frame and the dimensions of the fields). The principles are adopted model assumptions, not derived here; comparisons of predicted motions with observations can support them under stated apparatus assumptions, and no measurement is asserted or used by this postulate. In particular, the existence of a representing stress matrix, its continuity and its symmetry are not assumed here; they are conditional conclusions of the theorem items of this page, whose arguments apply the balance principles exactly as stated.