 Energy uniqueness for traction-driven elasticity

The difference of two solutions with the same body force, boundary traction and initial data satisfies a homogeneous elastic equation with zero boundary work. Multiplication by its velocity and integration by parts conserve the displayed positive energy. Zero initial energy forces zero velocity for all times, and zero initial displacement field then forces zero displacement field. Rigid zero-strain motions do not cause nonuniqueness after initial data are fixed. This assumes positive mass density, stable real elastic moduli and sufficient regularity for the energy identity.