# Damage Behaviour of Quasi-Brittle Composites: Mathematical and Computational Aspects
## 3 A Model Problem
### The First Step: A Viscoelastic Model  
Let us consider an arbitrary test function \(v\in\mathcal{V}\), which is the well-known concept of the vector of virtual displacements used in engineering mechanics. Further, consider \(\tau\in\mathcal{H}\), which can be viewed as a symmetric matrix of certain virtual stresses where all of the values of \(t\in\mathcal{I}\) are allowed. Then, the weak formulation of the linear momentum conservation reads  
\[\big{(}(\varepsilon(v),\sigma)\big{)}=(v,f)+\langle v,g\rangle\,. \tag{2}\]  
The upper branch of Figure 1, stemming from the Maxwell model, corresponds to the integral constitutive relation  
\[\big{(}(\gamma,\mathcal{C}^{-1}\tau/\alpha)\big{)}+\big{[}((\gamma,\mathcal{C }^{-1}\tau/\beta))\big{]}=\big{(}(\gamma,\varepsilon(u))\big{)}\,. \tag{3}\]  
To involve the lower branch of Figure 1, it is sufficient to directly evaluate  
\[\sigma=\tau+\mathcal{C}\varepsilon(u) \tag{4}\]  
on \(\Omega\). This will be generalised in the second step. Inserting (4) into (2), we obtain  
\[\big{(}(\varepsilon(v),\tau)\big{)}+\big{(}(\varepsilon(v),\mathcal{C} \varepsilon(u))\big{)}=(v,f)+\langle v,g\rangle\,. \tag{5}\]  
Then, (5) together with (3) represents a weak formulation of a system of two partial differential equations of parabolic type, the unknowns of which are \(u\) and \(\tau\). Both of them are time-dependent, in contrast to \(v\) and \(\gamma\). The Neumann boundary condition is incorporated in the last additive right-hand-side term of (2) and (5). The Dirichlet boundary condition is involved in the definition of \(\mathcal{V}\) by (1); natural Cauchy initial conditions are  
\[u=o\,,\qquad\tau=\mathcal{O} \tag{6}\]  
on \(\Omega\). At this point, \(\mathcal{O}\) denotes the zero-valued matrix from \(\mathbb{R}^{3\times 3}\). Nevertheless, the unknowns \(u\) and \(\tau\) of this kind are elements of an infinite-dimensional abstract function space, and their analytical or at least semi-analytical form (expressed e. g., in the form of sums of certain a priori known function sequences) is rarely available. Thus, we need some discretisation technique applicable for practical computations. We come out from the method of the discretisation in time, following [57], Part 8. The time interval \(\mathcal{I}\) will be covered by a finite number \(m\) of equidistant subintervals \(\mathcal{I}^{m}_{s}\), introduced as the sets of all \(t\) satisfying \((s-1)h<t\leq sh\) for \(s\in\{1,\ldots,m\}\). Here, \(h=\varsigma/m\) (this dependence of \(h\) on \(m\) is not expressed explicitly for brevity of notations); the analysis limit passage \(m\to\infty\) is necessary. Let us remind the reader that for \(t=0\) (not contained in \(\mathcal{I}^{m}_{s}\)), we have (6). Nevertheless, the resulting sequence of partial differential equations of elliptic type, obtained step-by-step for individual \(s\), is time-independent only formally, since they are still defined in the infinite-dimensional function space. From this point of view, the convergence considerations are still not constructive for practical evaluations, since finite-dimensional function spaces are required. This will be discussed in more detail in Section 5. On the other hand, it is useful to see that the convergence of algorithms can be guaranteed without any additional assumptions on the discretisation of \(\Omega\), \(\Theta\), and \(\Gamma\). Let us remind the reader of the concurrent approach, which is not preferred here. Namely, it is the Fourier multiplicative decomposition of unknown functions and the primary discretisation of \(\Omega\), \(\Theta\), and \(\Gamma\). We refer to the so-called method of lines, which leads to the sparse (but still finite) systems of ordinary differential equations. Such systems require effective generalised eigenvalue analysis or an additional time discretisation. For a detailed overview, see [60].