= Energy uniqueness for a linear elastic traction problem
{title2=$\int_B2W(e(v))\,dV=0\implies v=a+\omega\times x$}
On a connected isotropic <linear elasticity>[linear elastic] body, assume $\mu>0$ and $3\lambda+2\mu>0$. Then $W(e)=\mu|e^{\rm dev}|^2+(3\lambda+2\mu)(\operatorname{tr}e)^2/6$ is positive for every nonzero symmetric <infinitesimal strain tensor>[strain tensor]. If two equilibria have the same <body force> and boundary <traction>, their difference $v$ satisfies zero loading. Multiplication of equilibrium by $v$ and the <divergence theorem> yield $\int_B2W(e(v))\,dV=0$. Hence $e(v)=0$, and the kernel of the symmetric gradient consists of infinitesimal translations and rotations: $v=a+\omega\times x$.
The conclusion is uniqueness modulo these modes, not existence for arbitrary tractions. Existence additionally requires zero total resultant force and torque. On a disconnected body each connected component has its own rigid modes; fixing enough displacements removes those modes and can produce full uniqueness.
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