= Solution
<Cauchy's stress hypothesis> says that <traction> across a cut depends locally on its point, time and oriented normal, rather than on the global shape of the cut. Here all areas and normals are referential: $E$ is a fixed material region in reference coordinates, $\rho_R$ its mass <density> per reference volume, $v$ the current particle velocity, $b$ <body force> per unit mass, and $s_R$ force per unit reference area. The time derivative on the left is total material momentum rate; the two right terms are contact and <body forces>.
A shrinking pillbox gives $s_R(-N)=-s_R(N)$. In a shrinking tetrahedron the volume contributions scale as its linear size cubed, whereas face forces scale as size squared. Their leading balance expresses <traction> on an arbitrary face as the normal-component-weighted sum of <tractions> on three coordinate faces. Defining those three <tractions> as tensor columns gives $\boxed{s_R(N)=T_RN}$. Applying the <divergence theorem> to <conservation of momentum>[momentum balance] and localizing over arbitrary $E$ then yields
$$
\boxed{\rho_R\dot v=\operatorname{Div}T_R+\rho_Rb.}
$$
The referential tensor is the <first Piola-Kirchhoff stress tensor>[first Piola-Kirchhoff stress] and need not be <symmetric>; it should not be silently identified with the spatial <Cauchy stress tensor>[Cauchy stress].
<Incompressibility> is $\boxed{\det F=1}$. For a <homogeneous hyperelastic material> the referential <constitutive equation>[constitutive relation] becomes
$$
\boxed{T_R=DW(F)-p\operatorname{cof}F=DW(F)-pF^{-\mathsf T}\quad(\det F=1),}
$$
where the undetermined scalar <pressure> $p$ enforces the <constraint>. Homogeneity removes explicit reference-position dependence from $W$, not the possible spatial variation of that <pressure>.
For the <neo-Hookean solid>[neo-Hookean] energy, $DW=2\alpha F$. For the positive diagonal ansatz the nominal principal stresses are $t_i=2\alpha v_i-p/v_i$. Since $F$ is constant, <equilibrium> requires $\operatorname{cof}F\nabla p=0$, so $p$ is constant. Stress-free endfaces give $\boxed{p=2\alpha v_3^2>0}$. The side normals include $(0,-1,0)$ and $\pm(\sqrt3/2,1/2,0)$, so their prescribed <tractions> force $\boxed{t_1=t_2=T}$.
For fixed positive $p$, the scalar function $2\alpha v-p/v$ is <strictly increasing> on $v>0$. Thus equality of the two nominal stresses implies $v_1=v_2=v$. <Incompressibility> gives $v_3=v^{-2}$, and substitution gives
$$
\boxed{T=2\alpha(v-v^{-5}),\qquad p=2\alpha v^{-4}.}
$$
The load function has derivative $2\alpha(1+5v^{-6})>0$, tends to minus infinity as $v\downarrow0$ and to infinity as $v\to\infty$. It therefore determines exactly one positive $v$ for every specified load, with $v\geq1$ for outward <tension>[tensile] $T\geq0$. The resulting constant stress satisfies <equilibrium> and every prescribed face <traction>, proving existence and <uniqueness> within the requested positive diagonal class. It does not claim <uniqueness> among all possible nonhomogeneous <deformations>. This is the <equibiaxial dead loading of a neo-Hookean prism>.
Back to article page
