= Solution
{c}
At an oriented internal surface with unit <normal vector> $\mathbf n$, the <traction> $\mathbf t(\mathbf x,t;\mathbf n)$ is the limiting contact <force> per unit <surface area> exerted by the material on the positive-$\mathbf n$ side on the material on the negative-$\mathbf n$ side:
$$
\mathbf t(\mathbf n)=\lim_{\Delta A\to0}\frac{\Delta\mathbf F}{\Delta A}.
$$
For a boundary of a chosen <material volume>, this convention gives the external <traction> when $\mathbf n$ is outward. The <Cauchy stress tensor> has components
$$
\sigma_{ij}=\mathbf e_i\cdot\mathbf t(\mathbf e_j),
$$
so the second index specifies the surface orientation and the first specifies the <force> component. The vectors $\mathbf e_i$ form the standard <Cartesian coordinate system> basis of <unit vectors>.
<Cauchy's stress theorem> states that the dependence of <traction> on the <normal vector> is linear:
$$
\boxed{\mathbf t(\mathbf n)=\boldsymbol\sigma\mathbf n,
\qquad t_i=\sigma_{ij}n_j.}
$$
Repeated indices are summed according to <Einstein notation>. In particular, reversing the <normal vector> reverses the <traction>, and knowing the <tractions> on three mutually perpendicular coordinate planes determines the <traction> on every plane. Symmetry of the <Cauchy stress tensor>, for an ordinary fluid without body couples, follows separately from <angular momentum conservation>.
Back to article page
