= Internal variable
{title2=$\xi_r$}
An <internal variable> records a material state not determined by instantaneous <deformation> and <temperature> alone. Accumulated <plastic strain>, a relaxing internal strain and an evolving damage parameter are examples. Two specimens at the same overall <strain> and <temperature> can therefore carry different <stresses> because their loading histories leave different internal states.
For an isothermal scalar illustration, let the <free energy> per unit <volume> be $\psi(\varepsilon,\alpha)=E(\varepsilon-\alpha)^2/2$. Set $\sigma=\partial_\varepsilon\psi=E(\varepsilon-\alpha)$ and $\dot\alpha=\sigma/\eta$ with $\eta>0$. Then the mechanical dissipation is $\sigma\dot\varepsilon-\dot\psi=\sigma^2/\eta\geq0$. The hidden strain $\alpha$ evolves even when the total <strain> is held fixed, so the <stress> relaxes. This shows why an <internal variable> needs both an energetic role and an evolution law in a complete <constitutive equation>.
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