# Damage Behaviour of Quasi-Brittle Composites: Mathematical and Computational Aspects
## 3 A Model Problem
### The First Step: A Viscoelastic Model
In particular, select \(v=u_{s}^{m}-u_{s-1}^{m}\) in (7) and \(\gamma=\tau_{s}^{m}\) in (9). Then, the sum of (7) and (9) yields  
\[ \begin{split}\left((\tau_{s}^{m},\mathcal{C}^{-1}(\tau_{s}^{m}- \tau_{s-1}^{m})/\alpha\right)\right)+h\big{(}(\tau_{s}^{m},\mathcal{C}^{-1} \tau_{s}^{m}/\beta)\big{)}+\left((e(u_{s}^{m}-u_{s-1}^{m}),\mathcal{C}e(u_{s}^{ m}))\right)\\ =(u_{s}^{m}-u_{s-1}^{m},f_{s}^{m})+\left\langle u_{s}^{m}-u_{s-1} ^{m},\mathcal{S}_{s}^{m}\right\rangle.\end{split}  \tag{13}\]  
The formal modification of (13) leads to  
\[ \begin{split}\left((\tau_{s}^{m},\mathcal{C}^{-1}\tau_{s}^{m}/ \alpha)\right)/2-\left((\tau_{s-1}^{m},\mathcal{C}^{-1}\tau_{s-1}^{m}/\alpha )\right)/2\\ +\left((\tau_{s}^{m}-\tau_{s-1}^{m},\mathcal{C}^{-1}(\tau_{s}^{m} -\tau_{s-1}^{m})/\alpha)\right)/2\\ +h\big{(}(\tau_{s}^{m},\mathcal{C}^{-1}\tau_{s}^{m}/\beta)\big{)} +\left((\varepsilon(u_{s}^{m}),\mathcal{C}e(u_{s}^{m}))\right)/2\\ -\left((\varepsilon(u_{s-1}^{m}),\mathcal{C}\varepsilon(u_{s-1}^{ m}))\right)/2+\left((\varepsilon(u_{s}^{m}-u_{s-1}^{m})\mathcal{C}\varepsilon(u_{s}^{ m}-u_{s-1}^{m}))\right)/2\\ =(u_{s-1}^{m},f_{s-1}^{m}-f_{s}^{m})+(u_{s}^{m},f_{s}^{m})-(u_{s-1 }^{m},f_{s-1}^{m})\\ +\left\langle u_{s-1}^{m},g_{s-1}^{m}-g_{s}^{m}\right\rangle+ \left\langle u_{s}^{m},g_{s}^{m}\right\rangle-\left\langle u_{s-1}^{m},g_{s-1} ^{m}\right\rangle.\end{split}  \tag{14}\]  
Now, we can sum up all equations (14) for \(s\in\{1,\ldots,r\}\), multiplied by \(2\), where \(r\) is an arbitrary fixed element from \(\{1,\ldots,m\}\). The result is as follows:  
\[ \begin{split}\left((\tau_{r}^{m},\mathcal{C}^{-1}\tau_{r}^{m}/ \alpha)\right)+h\sum_{s=1}^{r}\left((\tau_{s}^{m}-\tau_{s-1}^{m},\mathcal{C}^{- 1}(\tau_{s}^{m}-\tau_{s-1}^{m})/\alpha\right)\right)\\ +2h\sum_{s=1}^{r}\left((\tau_{s}^{m},\mathcal{C}^{-1}\tau_{s}^{m }/\beta)\right)+\left((\varepsilon(u_{r}^{m}),\mathcal{C}\varepsilon(u_{r}^{m }))\right)\\ +\left.\sum_{s=1}^{r}\left((\varepsilon(u_{s}^{m}-u_{s-1}^{m}), \mathcal{C}\varepsilon(u_{s}^{m}-u_{s-1}^{m}))\right)\\ =2(u_{s}^{m},f_{r}^{m})-2h\sum_{s=1}^{r}(u_{s-1}^{m},Df_{s}^{m} )+2\langle u_{s}^{m},g_{r}^{m}\rangle-2h\sum_{s=1}^{r}\langle u_{s-1}^{m},D _{S}^{m}\rangle\end{split}  \tag{15}\]  
exploiting the notation of relative differences \(Df_{s}^{m}=(f_{s}^{m}-f_{s-1}^{m})/h\), \(D_{S}^{m}=(g_{s}^{m}-g_{s-1}^{m})/h\), usual in numerical differentiation. Moreover, for any positive \(\eta\) and \(\tilde{\eta}\), we have  
\[ \begin{split} 2(u_{s}^{m},f_{r}^{m})&=\eta(u_{s}^{m},u_{s} ^{m})+\eta^{-1}(f_{r}^{m},f_{r}^{m})-(\eta u_{s}^{m}-\eta^{-1}f_{r}^{m},\eta u_ {s}^{m}-\eta^{-1}f_{r}^{m})\,,\\ 2\langle u_{s}^{m},g_{r}^{m}\rangle&=\tilde{\eta}(u_ {s}^{m},u_{s}^{m})+\tilde{\eta}^{-1}(g_{r}^{m},g_{r}^{m})-\langle\tilde{\eta} u_{s}^{m}-\tilde{\eta}^{-1}g_{r}^{m},\tilde{\eta}u_{s}^{m}-\tilde{\eta}^{-1}g_{r}^{m} \rangle\end{split}  \tag{16}\]  
and also  
\[ \begin{split}-2h\sum_{s=1}^{r}(u_{s-1}^{m},Df_{s}^{m})& =\eta\sum_{s=1}^{r}(u_{s-1}^{m},u_{s-1}^{m})+\eta^{-1}\sum_{s=1}^{r}(Df _{s}^{m},Df_{s}^{m})\\ &-\sum_{s=1}^{r}(\eta u_{s-1}^{m}+\eta^{-1}f_{s}^{m},\eta u_{s-1} ^{m}+\eta^{-1}f_{s}^{m})\,,\\ -2h\sum_{s=1}^{r}\langle u_{s-1}^{m},D_{S}^{m}\rangle& =\tilde{\eta}\sum_{s=1}^{r}\langle u_{s-1}^{m},u_{s-1}^{m}\rangle+\tilde{\eta}^{ -1}\sum_{s=1}^{r}\langle D_{S}^{m},D_{S}^{m}\rangle\\ &-\sum_{s=1}^{r}\langle\tilde{\eta}u_{s-1}^{m}+\tilde{\eta}^{-1}g_ {s}^{m},\tilde{\eta}u_{s-1}^{m}+\tilde{\eta}^{-1}g_{s}^{m}\rangle\,.\end{split}