The question now remains how to evaluate the internal variables. From equation (5.4-4) it appears that we have to integrate over the entire time domain. However, we can find a recurrence relationship that will allow us to evaluate the internal variables at a time $t+\Delta t$ given the values at time $t$ . $$\begin{equation} \begin{aligned}\mathbf{H}^{\left(i\right)}\left(t+\Delta t\right) & =\int\limits _{-\infty}^{t+\Delta t}\exp\left[-\left(t+\Delta t-s\right)/\tau_{i}\right]\frac{d\mathbf{S}^{e}}{ds}\,ds\\ & =\exp\left(-\Delta t/\tau_{i}\right)\int\limits _{-\infty}^{t}\exp\left[-\left(t-s\right)/\tau_{i}\right]\frac{d\mathbf{S}^{e}}{ds}\,ds+\int\limits _{t}^{t+\Delta t}\exp\left[-\left(t+\Delta t-s\right)/\tau_{i}\right]\frac{d\mathbf{S}^{e}}{ds}\,ds\\ & =\exp\left(-\Delta t/\tau_{i}\right)\mathbf{H}^{\left(i\right)}\left(t\right)+\int\limits _{t}^{t+\Delta t}\exp\left[-\left(t+\Delta t-s\right)/\tau_{i}\right]\frac{d\mathbf{S}^{e}}{ds}\,ds\,. \end{aligned} \label{eq490} \end{equation}$$ The last term can now be simplified using the midpoint rule to approximate the derivate. In that case we find the recurrence relation: $$\begin{equation} \mathbf{H}^{\left(i\right)}\left(t+\Delta t\right)=\exp\left(-\Delta t/\tau_{i}\right)\mathbf{H}^{\left(i\right)}\left(t\right)+\frac{1-\exp\left(-\Delta t/\tau_{i}\right)}{\Delta t/\tau_{i}}\left(\mathbf{S}^{e}\left(t+\Delta t\right)-\mathbf{S}^{e}\left(t\right)\right)\,.\label{eq491} \end{equation}$$ The following procedure can now be applied to calculate the new stress. Given $\mathbf{S}_{n}^{e}$ and $\mathbf{H}_{n}^{\left(i\right)}$ corresponding to time $t$ , find $\mathbf{S}_{n+1}^{e}$ and $\mathbf{H}_{n+1}^{\left(i\right)}$ corresponding to time $t+\Delta t$ :