Dissipation potential (${\dot{\xi }}_r=\partial \Omega /\partial f_r$)

A force-dependent potential specifies internal-variable rates by differentiation. Convexity with a minimum at zero force can ensure nonnegative force-rate dissipation. The name alone does not ensure the Clausius-Duhem inequality for an arbitrary potential.

The dissipation potential $\Omega =\alpha \Vert Q{\Vert }^{n+1}/(n+1)-\mathrm{tr}(AQ)/\tau$, with the Frobenius norm, gives $\dot{A}+A/\tau =\alpha \Vert Q{\Vert }^{n-1}{Q}^T$. An integrating factor proves the displayed memory formula. For $n>0$, its nonlinear driving term is defined continuously as zero at $Q=0$. Mechanical dissipation is $\mathrm{tr}(Q\dot{A})=\alpha \Vert Q{\Vert }^{n+1}-\mathrm{tr}(AQ)/\tau$, which must be nonnegative on thermodynamically admissible histories.