# Damage Behaviour of Quasi-Brittle Composites: Mathematical and Computational Aspects
## 3 A Model Problem
### The Second Step: Incorporation of Damage
Then, the generalized version of (4), involving certain damages, boils down to  
\[\sigma=\tau+(1-\mathcal{D}(u))\mathcal{C}\varepsilon(u)\,. \tag{25}\]  
For sake of making the implementation of \(\mathcal{D}(u)\) simple, brief, transparent and comparable with the preceding step, we do not change any of the other constitutive equations. From the point of view of the scheme from Figure 1, we modify only \(\mathcal{C}\) in its lower branch by means of \((1-\mathcal{D}(u))\mathcal{C}\). It is obvious that more advanced extensions of this scheme as well as the related constitutive equations may be useful in practical settings of real material parameters. We venture to repeat all of the investigations from the preceding step, except the unchanged formulae, namely (2), (3), (6), (8), and (9). Instead of (5), we replace (4) by (25) and obtain  
\[\left((\varepsilon(v),\tau)\right)+\left((\varepsilon(v),(1-\mathcal{D}(u)) \mathcal{C}\varepsilon(u))\right)=(v,f)+\left\langle v,g\right\rangle, \tag{26}\]  
Consequently, (7) must be replaced by  
\[\left((\varepsilon(v),\tau_{s}^{m})\right)+\left((\varepsilon(v),(1-\mathcal{ D}(u))\mathcal{C}\varepsilon(u_{s}^{m}))\right)=(v,f_{s}^{m})+\left\langle v,g_{s}^{m} \right\rangle. \tag{27}\]  
Although the theorem of Lax-Milgram cannot be directly applied at this point, there is fortunately another classical result regarding the solvability of (27) and (8) together with (6),which is available in [71]. Instead of (18), the sum of (27) and (9) with \(v=u_{s}^{m}-u_{s-1}^{m}\) and \(\gamma=\tau_{s}^{m}\) yields  
\[ \begin{split}\big{(}(\tau_{r}^{m},\mathcal{C}^{-1}\tau_{r}^{m}/ \alpha)\big{)}+h\sum_{s=1}^{r}((\tau_{s}^{m}-\tau_{s-1}^{m},\mathcal{C}^{-1}( \tau_{s}^{m}-\tau_{s-1}^{m})\alpha))\\ +2h\sum_{s=1}^{r}((\tau_{s}^{m},\mathcal{C}^{-1}\tau_{s}^{m}/ \beta))+\big{(}(\varepsilon(u_{r}^{m}),\mathcal{C}\varepsilon(u_{r}^{m})))\\ +\sum_{s=1}^{r}((\varepsilon(u_{s}^{m}-u_{s-1}^{m}),(1-\mathcal{ D}(u_{s}))\mathcal{C}\varepsilon(u_{s}^{m})))\\ +\sum_{s=1}^{r}(\eta u_{s-1}^{m}+\eta^{-1}f_{s}^{m},\eta u_{s-1 }^{m}+\eta^{-1}f_{s}^{m})\\ +\sum_{s=1}^{r}(\widetilde{\eta}u_{s-1}^{m}+\widetilde{\eta}^{- 1}g_{s}^{m},\widetilde{\eta}u_{s-1}^{m}+\widetilde{\eta}^{-1}g_{s}^{m})\\ =\eta h\sum_{s=1}^{r}(u_{s-1}^{m},u_{s-1}^{m})+\widetilde{\eta} h\sum_{s=1}^{r}(u_{s-1}^{m},u_{s-1}^{m})\\ +\eta^{-1}h\sum_{s=1}^{r}(D_{s}^{m},D_{s}^{m})+\widetilde{\eta}^{ -1}h\sum_{s=1}^{r}\langle D_{s}^{m},D_{s}^{m}\rangle\,.\end{split}  \tag{28}\]  
The details of the following approach can be omitted because only the fifth left-hand-side additive term of (28) needs to be handled in another way, analogous to the first step. Namely,  
\[ \begin{split} 2\sum_{s=1}^{r}&((\varepsilon(u_{s}^{m}-u_{s-1} ^{m})(1-\mathcal{D}(u_{s}))\mathcal{C}\varepsilon(u_{s}^{m})))\\ &=\sum_{s=1}^{r}((\varepsilon(u_{s}^{m}),(1-\mathcal{D}(u_{s})) \mathcal{C}\varepsilon(u_{s}^{m})))\\ -\sum_{s=1}^{r}&((\varepsilon(u_{s-1}^{m}),(1- \mathcal{D}(u_{s-1}))\mathcal{C}\varepsilon(u_{s-1}^{m})))\\ +\sum_{s=1}^{r}&((\varepsilon(u_{s}^{m}-u_{s-1}^{ m})(1-\mathcal{D}(u_{s}))\mathcal{C}\varepsilon(u_{s}^{m}-u_{s-1}^{m})))\\ +\sum_{s=1}^{r}&((\varepsilon(u_{s-1}^{m})(\mathcal{ D}(u_{s}^{m})-\mathcal{D}(u_{s-1}^{m}))\mathcal{C}\varepsilon(u_{s-1}^{m})))\\ =&((\varepsilon(u_{r}^{m}),(1-\mathcal{D}(u_{r})) \mathcal{C}\varepsilon(u_{r}^{m})))\\ +\sum_{s=1}^{r}&((\varepsilon(u_{s}^{m}-u_{s-1}^{ m})(1-\mathcal{D}(u_{s}))\mathcal{C}\varepsilon(u_{s}^{m}-u_{s-1}^{m})))\\ +\sum_{s=1}^{r}&((\varepsilon(u_{s-1}^{m})(\mathcal{ D}(u_{s}^{m})-\mathcal{D}(u_{s-1}^{m}))\mathcal{C}\varepsilon(u_{s-1}^{m})))\,. \end{split}  \tag{29}\]  
Since \(\mathcal{D}(u_{s}^{m})\geq 1-\mathcal{D}_{*}>0\) holds everywhere for any \(s\in\{1,\ldots,r\}\), only the last right-hand-side additive term of (29) deserves a special mention. Clearly, \(\mathcal{D}(u_{s}^{m})-\mathcal{D}(u_{s-1}^{m})\) is always non-negative due to (24) formally (from the mechanical point of view, no damage recovery is allowed); thus, this term can be estimated by zero from below.