 Thermodynamic force conjugate to an internal variable

A thermodynamic force conjugate to an internal variable is the energetic driving quantity paired with that variable's rate in the dissipation. Its sign and density normalization must be specified. With specific internal energy $U(F,\eta ,\xi )$, reference mass density ${\rho }_0$ and the convention used here,Equivalently, if the specific Helmholtz free energy is $\psi (F,T,\xi )$, then $f_{\xi }=-{\rho }_0(\partial \psi /\partial \xi {)}_{F,T}$. The negative sign makes positive force-rate product correspond to free-energy decrease at fixed external variables.

For an isothermal free energy per reference volume $W=E(\epsilon -\xi {)}^2/2$, the driving force is $f_{\xi }=E(\epsilon -\xi )$. Choosing $\dot{\xi }=f_{\xi }/{\eta }_v$ with ${\eta }_v>0$ yields $f_{\xi }\dot{\xi }=f_{\xi }^2/{\eta }_v\ge 0$. The internal variable relaxes toward the imposed strain, reducing stored elastic energy. Such a law supplies the mechanical part of a Clausius-Duhem inequality check; the thermal contributions still need their own consistent description.