Table 1.

Subset of currently available relaxation functions $g(\mathbf{F}(t^v),t-t^v)$ for nonlinear reactive viscoelasticity in febio

Name	$g(\mathbf{F}(t^v),t-t^v)$	Material constants (units)
Exponential	$e^{-(t-t^v)/\tau }$	$\tau \hspace{0.17em}\left(\text{time}\right)$
Exp-distortion	$\begin{array}{c}\text{exponential}\\ \tau (K_2(t^v))={\tau }_0+{\tau }_1{K}_2^{\alpha }(t^v)\end{array}$	$\begin{array}{c}{\tau }_0(\text{time})\\ {\tau }_1(\text{time})\\ \alpha (\text{time})\end{array}$
Power	${(1+\frac{t-t^v}{\tau })}^{-\beta }$	$\begin{array}{c}\tau (\text{time})\\ \beta (\text{time})\end{array}$
Power-dist-user	$\begin{array}{c}\text{power}\\ \tau (K_2(t^v))=\text{user}-\text{specified}\\ \beta (K_2(t^v))=\text{user}-\text{specified}\end{array}$	$\begin{array}{c}\tau (\text{time})\\ \beta (\text{time})\end{array}$
Malkin	$\frac{(\beta -1)t^{1-\beta }}{{\tau }_1^{1-\beta }-{\tau }_2^{1-\beta }}\left[\Gamma \left(\beta -1,\frac{t}{{\tau }_2}\right)-\Gamma \left(\beta -1,\frac{t}{{\tau }_1}\right)\right]$	$\begin{array}{c}{\tau }_1(\text{time})\\ {\tau }_2(\text{time})\\ \beta (\text{time})\end{array}$
Malkin-distortion	$\begin{array}{c}\text{Malkin}\\ {\tau }_1={\tau }_{10}+{\tau }_{11}\hspace{0.17em}\text{exp}\left(-\frac{K_2(t^v)}{s_1}\right)\\ {\tau }_2={\tau }_{20}+{\tau }_{21}\hspace{0.17em}\text{exp}\left(-\frac{K_2(t^v)}{s_2}\right)\end{array}$	$\begin{array}{c}{\tau }_{10}(\text{time}){\tau }_{20}(\text{time})\\ {\tau }_{11}(\text{time}){\tau }_{21}(\text{time})\\ s_1(\text{time})s_2(\text{time})\end{array}$

In these relations, $K_2(t^v)$ is the second invariant of the natural (Hencky) strain tensor, which represents a kinematic measure of distortion [32]. Specifically, $K_2=\Vert \text{dev}\mathit{\eta }\Vert$ where $\mathit{\eta }=\ln \mathbf{V}$ is the spatial natural strain tensor (also known as the left Hencky strain tensor) and V is the left stretch tensor. Here, K₂ is evaluated at the time t^v when $u-$ generation bonds start breaking. In febio user-specified functions, such as those needed for $\tau (K_2)$ and $\beta (K_2)$, may be given either as a mathematical formula or a curve (piecewise linear or cubic) passing through data points. The function $\Gamma (s,x)$ represents the upper incomplete gamma function.