= Solution
In the classical smooth setting, a <deformation in continuum mechanics> is a sufficiently regular, <injective> map $y:\Omega\to\mathbb R^3$ with $\det Dy>0$. <Injectivity> excludes material interpenetration, and a positive <determinant> preserves orientation and local <volume>. We use $C^1$ regularity for these kinematics and additional differentiability where stress-divergence identities are applied.
<Cauchy's stress theorem> states that the contact <force> on an oriented surface element at a current point depends on the point and its <unit normal>, rather than on the shape of the chosen subbody. Thus the contact <force> on the <boundary> of a current subbody $v$ has the form $\int_{\partial v}t(z,n)\,da$. Static <force> balance, for a <body force> $b$ per unit current <volume>, is
$$
\int_{\partial v}t(z,n)\,da+\int_vb(z)\,dv=0
$$
for every subbody. A shrinking pillbox gives $t(z,-n)=-t(z,n)$; a shrinking tetrahedron then gives linear dependence on the normal, $t(z,n)=S(z)n$. Applying the <divergence theorem> and using arbitrary subbodies gives
$$
\boxed{t(z,n)=S(z)n,\qquad\operatorname{div}S+b=0.}
$$
If <body force> is specified per unit mass instead, replace $b$ by $\rho b$. <Symmetry> of $S$ follows additionally from <conservation of angular momentum> in the absence of body couples; it is not a consequence of <force> balance alone.
Back to article page
