= Dynamic continuation of a static incompressible elastic deformation
{title2=$T=T_0-\rho\xi I+C(t)I$}
Let $\mathbf x_0(\mathbf X;\tau)$ be a family of <static elastic equilibrium> solutions for a homogeneous material with <incompressibility> under a fixed spatial <body force>. Replacing $\tau$ by time gives a dynamic solution of the elastic field equations exactly when the resulting <material acceleration> has a global <acceleration potential> $\xi$. At the same <deformation gradient>, the <Cauchy stress tensor> can change only by the scalar <pressure> reaction. Subtracting static and dynamic <balance of linear momentum> gives $\nabla q=\rho\mathbf a$, and constant <mass density> then gives $q=-\rho\xi+C(t)$. Surface <tractions> must match the adjusted <Cauchy stress tensor>.
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