Traction fields and stress matrices of the continuum model, with units and sign conventions

Fix body data $(B,O,I)$ and a motion $\chi$ of the model with current regions ${\mathrm{\Omega }}_t={\chi }_t(B)$ and $\mathcal{U}=\{(s,x):s\in I,x\in {\chi }_s(O)\}$. Let $U\subseteq \mathcal{U}$ be open and contain the swept closure $\{(s,x):s\in \overline{J_0},x\in {\chi }_s(\bar{B})\}$, where $J_0$ is an open interval with compact closure in $I$ (Adopted continuum kinematic model: body data, smooth motions as families of label worldlines, the continuum idealization and frame covariance, Continuum motions, spatial velocity, material acceleration and swept regions). Use on $U$ the restriction of the positive $C^1$ density slot $\rho :\mathcal{U}\to (0,\mathrm{\infty })$ of Continuum body data, the fixed inertial frame and the dimensions of the fields, and denote that restriction again by $\rho$. Let $S^2=\{n\in {\mathbb{R}}^3:\mathrm{\parallel }n{\mathrm{\parallel }}_2=1\}$ be the unit sphere.

A traction field of the model is a map

$t:U\times S^2⟶{\mathbb{R}}^3,$

read with the convention of Traction fields and stress matrices representing them: $t(s,x,n)$ is the force per unit area exerted by the material on the side towards which $n$ points upon the material on the other side, at the point $x$ of a surface whose unit normal is $n$, at time $s$. Its dimension is that of force per area, $[t]=\mu {\mathrm{\ell }}^{-1}{\tau }^{-2}$, read in $\mathrm{N}{\mathrm{m}}^{-2}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework).

A $C^1$ field of matrices $\sigma :U\to M_3(\mathbb{R})$ represents the traction field when

$t(s,x,n)=\sigma (s,x)n\text{for every }(s,x,n)\in U\times S^2,$

the product being matrix-vector multiplication. The existence of such a $\sigma$ for a given traction field is not asserted here: it is a conditional conclusion of the balance items of this page. When it exists, $\sigma (s,x)$ is the stress matrix of the model at that event, and its entries read ${\sigma }_{ij}=(t(s,x,e_j){)}_i$: the $i$-th component of the traction across the surface with normal $e_j$.

The isotropic matrix $-\lambda I_3$ with $\lambda \in \mathbb{R}$ represents the normal traction $t(n)=-\lambda n$: under the convention above, positive $\lambda$ is a compressive normal traction and $\lambda$ is called the pressure in this sign convention. Within this isotropic family, $\lambda =0$ gives the zero stress matrix, a pressure-free unstressed state. The body force per unit mass is a separate object: a continuous field $b:U\to {\mathbb{R}}^3$ with the dimension of acceleration, $[b]=\mathrm{\ell }{\tau }^{-2}$. The body-force density per unit volume is $\rho b:U\to {\mathbb{R}}^3$, of dimension $\mu {\mathrm{\ell }}^{-2}{\tau }^{-2}$. A traction field need not satisfy $t(-n)=-t(n)$ and its representing matrix, when one exists, need not be symmetric; both properties are separate conditional conclusions.

• Cauchy's hypothesis is a modelling assumption. The definition lets the traction depend on the surface only through its unit normal at the point. Contact interactions that depend on the curvature of the surface, on a surface couple distribution, or on a thin boundary layer would require a richer object; the counterexample item of the companion page shows that the global symmetry $t(-n)=-t(n)$ alone, without a balance law, does not even force the existence of a representing matrix.

• Units and interpretation. The values of $t$ and $\sigma$ are model quantities in the fixed frame of Continuum body data, the fixed inertial frame and the dimensions of the fields. A measured force per area is related to them only through a measurement model and is not asserted by this definition; no measured stress is recorded on this page.

• Sign convention. The convention is the one of Traction fields and stress matrices representing them: $t(n)$ acts on the material on the side opposite to that towards which $n$ points. With this convention a fluid whose total stress is assumed to lie in the isotropic family defined above has $\sigma =-p{I}_3$, so positive $p$ gives compression in the direction $-n$. Positive pressure alone does not impose this isotropic stress assumption.