# Damage Behaviour of Quasi-Brittle Composites: Mathematical and Computational Aspects
## 3 A Model Problem
### The Second Step: Incorporation of Damage  
We demonstrate this approach on our slightly upgraded model problem. We can multiply \(\mathcal{C}\), as shown in the lower branch of Figure 1, by a certain scalar factor \(1-\mathcal{D}(u)\), with positive irreversible real non-negative damage measure \(\mathcal{D}(u)\) lesser than a certain constant \(\mathcal{D}_{*}\) with values in \(\mathcal{Z}\) between 0 and 1. In particular, \(\mathcal{D}_{*}=0\) (no damage) is allowed; thus, we come back to our original model problem, but (at least) \(\mathcal{D}_{*}<1\) is required everywhere to avoid the transfer up to stage d) without adequate physical and mathematical analysis. The inputs for evaluation of \(\mathcal{D}_{*}\) should be some invariant strain or stress values, to preserve the objectivity of such an analysis. More exactly, we can work with the five-step gradual evaluation of \(\mathcal{D}\) here, as developed in [42]:  
1. For any fixed \(t\in\mathcal{I}\), compute three scalar principal values \(\epsilon_{i}\), \(i\in\{1,2,3\}\), corresponding to \(\epsilon(u)\), the computations being made on \(\Omega\). Denoting by \(I\) the unit matrix in \(\mathbb{R}^{3\times 3}\), it is necessary to find three eigenvalues \(\epsilon_{i}\) satisfi\[\det\left(\varepsilon(u)-\varepsilon_{i}(u)I\right)=0\,. \tag{20}\]
2. Evaluate an equivalent strain \[\widetilde{\varepsilon}(u)=\omega(-\varepsilon_{1-}(u),\epsilon_{1+}(u),- \varepsilon_{2-}(u),\epsilon_{2+}(u),-\varepsilon_{3-}(u),\epsilon_{3+}(u))\] (21) for \(\varepsilon_{i+}(u)\) and \(\varepsilon_{i-}(u)\) with \(i\in\{1,2,3\}\) again, by \(\varepsilon_{i+}(u)\) and \(\varepsilon_{i-}(u)\) in (21) denoting the positive and negative parts of \(\varepsilon_{i}(u)\) from (20). A bounded continuous function \(\omega\) of six real non-negative arguments (at most three non-zero ones) must be prescribed; an instructive example of such a function \(\omega\) can be found in [63]. 3. Introduce the nonlocal form of \(\varepsilon(u)\) from (21), motivated by [64], as \[\widetilde{\varepsilon}(u(x,t))=\int_{\Omega}\mathcal{K}(x,\widetilde{x})\, \widetilde{\varepsilon}(u(\widetilde{x},t))\,\mathrm{d}\widetilde{x}\,,\qquad \int_{\Omega}\mathcal{K}(x,\widetilde{x})\,\mathrm{d}\widetilde{x}=1\] (22) for all \(x\in\Omega\), with a fixed \(t\in\mathcal{I}\). A regularising kernel \(\mathcal{K}\in L^{2}(\Omega\times\Omega)\), as suggested by [65], is required in (22) as well. The following approach avoids the problems connected with the non-existence of the solution in the sense of examples from [66; 67]. Using the Dirac measure \(\delta\), the seemingly simple choice \(\mathcal{K}(x,\widetilde{x})=\delta(x-\widetilde{x})\), followed by \(\widetilde{\varepsilon}(u)=\widetilde{\varepsilon}(u)\), is not allowed, in contrast to the Gaussian kernel from [68] or to the various classes of radial basis function approximations from [69; 70]. 4. Applying the result of (22), evaluate a trial value of damage \[\widehat{\mathcal{D}}(u)=\omega(\widetilde{\varepsilon}(u))\,,\] (23) using a non-decreasing real continuous function \(\omega\). 5. Force the irreversibility of damage by (23), i.e., the non-decreasing damage factor in time, using the formula \[\mathcal{D}(u(.,t))=\max_{0\leq I\leq t}\widehat{\mathcal{D}}(u(.,\widetilde{ t}))\,.\] (24)  
A practical design of \(\omega\), \(\mathcal{K}\), and \(\omega\) may be difficult since any reasonable setting of material parameters obtained from experiments leads to a non-trivial sensitivity and inverse problems, which are frequently ill-conditioned. On the other hand, accenting the mathematical point of view presented in [58], Part 2.2, where two different versions of the proof are presented, entails the compactness of any operator from (22) acting over \(L^{2}(\Omega)\).