Adopted force model of the continuum branch: continuous body forces and Cauchy contact tractions

Adopt for every model of the continuum kinematic branch (Adopted continuum kinematic model: body data, smooth motions as families of label worldlines, the continuum idealization and frame covariance) the following description of the forces acting on the body, read in the fixed inertial frame and with the dimensions of Continuum body data, the fixed inertial frame and the dimensions of the fields.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the Lebesgue volume and surface integrals below. Let $\rho$ be the positive $C^1$ density of The density field of the continuum model and the mass of a material subregion. Use a common open field domain $U\subseteq \mathbb{R}\times {\mathbb{R}}^3$ containing the full swept closure $\{(s,x):s\in I,x\in {\chi }_s(\bar{B})\}$. At time $s\in I$, write $U_s:=\{x:(s,x)\in U\}$ and ${\mathrm{\Omega }}_s:={\chi }_s(B)$. An admissible region at time $s$ is a nonempty bounded open $D\subseteq {\mathrm{\Omega }}_s$ with $\bar{D}\subseteq U_s$ and a specified finite piecewise $C^1$ boundary presentation in Specified finite piecewise C1 boundary presentations. Its faces $S_j$ carry the outward unit normal ${\nu }_D$ off the surface-null edge set. Boundary integration means the finite sum of the face integrals of Surface integration on compact C1 hypersurfaces, componentwise for vectors; volume integration uses the Lebesgue integral over $D$ (Integral over a measurable subset), with signed integrals defined by positive and negative parts in Integrable real and complex functions, and their integrals. Compact closure, continuous fields and finite face areas make these integrals finite.

(i) Body forces. There is a continuous field $b:U\to {\mathbb{R}}^3$ on the common open field domain $U$, the body force per unit mass, of dimension $[b]=\mathrm{\ell }{\tau }^{-2}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). The body force exerted on an admissible region $D$ at time $s$ is ${\int }_D\rho (s,x)b(s,x)dx$, of dimension $\mu \mathrm{\ell }{\tau }^{-2}$ (force).

(ii) Contact forces and Cauchy's hypothesis. There is a traction field $t:U\times S^2\to {\mathbb{R}}^3$ of class $C^1$, hence continuous, in the sense of Traction fields and stress matrices of the continuum model, with units and sign conventions, force per unit area, with dimension $[t]=\mu {\mathrm{\ell }}^{-1}{\tau }^{-2}$. The definition of a traction field already encodes Cauchy's hypothesis: the contact force exerted across a surface at a point depends on the surface only through the value of its unit normal at that point, so two surfaces through the same point with the same unit normal experience the same traction. The contact force exerted across the boundary of an admissible region $D$ at time $s$ is ${\int }_{\mathrm{\partial }D}t(s,y,{\nu }_D(y))dS(y)$, with the outward unit normal ${\nu }_D$ and the finite face-sum integral defined above. No dependence of $t$ on the curvature of the surface, on higher derivatives of the surface, or on the distant shape of the body is introduced; no surface couples, no traction singularities and no contact forces on curves are introduced.

(iii) Exclusions. This force description is not adopted, and no statement of this page covers: couple stresses or distributions of surface couples; contact interactions that depend on the geometry of the surface beyond its tangent plane at the point (thin boundary layers, capillarity at contact lines, adhesion laws with curvature dependence); nonlocal contact interactions; electromagnetic, thermal, chemical and radiative forces; and body couples. A model with such features requires an extended force description and its own balance laws.

(iv) Status. These are adopted assumptions about how forces are represented in the model. No balance of forces is asserted by this postulate; the balance principles are the subject of the next postulate, and the existence of a representing stress matrix is a conditional conclusion of the theorem that applies Cauchy's tetrahedron argument to those balances. The fields $b$ and $t$ are model quantities; no measured force is asserted, and the frame-covariance requirement of the kinematic branch applies to them.