Internal variable (${\xi }_r$)

An internal variable records a material state beyond its instantaneous deformation gradient and temperature, such as a relaxing microstructural state. A constitutive equation specifies its evolution and its contribution to the free energy.

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.

The derivative of internal energy at fixed deformation gradient and entropy, or of specific Helmholtz free energy at fixed temperature, defines this force. Its product with the internal-variable rate contributes mechanical dissipation to the Clausius-Duhem inequality.