# Damage Behaviour of Quasi-Brittle Composites: Mathematical and Computational Aspects
## 3 A Model Problem
### Mathematical Preliminaries  
In the first step, we conduct an analysis of the weak solvability of the linear parabolic (and later also hyperbolic, cf. Section 3.2) initial and boundary value problem. In the second step, we add a nonlocal and nonlinear term. The analysis of the elliptic linear problems can be found in [53, 54], and it is adapted to parabolic and hyperbolic problems in [55]. A generalisation covering a large class of nonlinear problems can be found in [56]. In the present paper, we follow the notations and results from the monograph [57], completed by the regularisation results from [58]. Some ideas in proofs related to our model problem are similar to those from [41], where the classical viscoelastic Kelvin model is implemented. Nevertheless, we shall see that the generalisation of these results, needed in engineering applications like [46], is not straightforward and requires a separate mathematical analysis. A deformable body is assumed to occupy a certain domain \(\Omega\) with Lipschitzian boundary \(\partial\Omega\) in the three-dimensional Euclidean space \(\mathbb{R}^{3}\), supplied by a Cartesian coordinate system \(x=(x_{1},x_{2},x_{3})\), with \(\partial\Omega\) consisting of two disjoint parts \(\Gamma\) (for Neumann boundary conditions, inhomogeneous in general) and \(\Theta\) (for homogeneous Dirichlet boundary conditions). An additional assumption of the non-zero Hausdorff measure of \(\Theta\) on \(\partial\Omega\) enables us to avoid the existence of infinitely many solutions (for equilibrium loads caused by an insufficient support) as well as the non-existence of any solution (for non-equilibrium loads caused by missing dynamic considerations). A strain and stress development on \(\Omega\) will be considered for any time \(t\) belonging to a time interval \(\mathcal{I}=[0,\varsigma]\). The finite real \(\varsigma\) opens the door to potential studies of a limit passage \(\varsigma\to\infty\), which is not discussed in detail. An initial condition can be prescribed for \(t=0\). Let us consider the following function spaces introduced in [57], Part 1.2: \(\mathcal{H}=L^{2}(\Omega)^{3}\), \(\mathcal{E}=L^{2}(\Omega)^{3\times 3}\), \(\mathcal{G}=L^{2}(\Gamma)^{3}\), \(\mathcal{Z}=L^{\infty}(\Omega)\), \(\mathcal{M}=L^{2}(\Omega)^{(3\times 3)\times(3\times 3)}_{\text{sym}}\) and  
\[\mathcal{V}=\left\{v\in W^{1,2}(\Omega)^{3}:\;v=o\text{ on }\Theta\right\}, \tag{1}\]  
where \(o=(0,0,0)\). In the last definition, the strict support in all directions excludes any rigid motion; for the relation to the line geometry, cf. [59]. For an unknown variable \(u(x,t)\) belonging to \(L^{2}(\mathcal{I},\mathcal{V})\) in the sense of [57], Part 7.1, which was introduced as a vector of displacements related to the initial configuration at \(t=0\), we are allowed to assume \(u(x,0)=o\) for any \(x\in\Omega\). As is usual in the linear theory of elasticity, we can introduce a strain tensor \(\varepsilon(v)\in\mathcal{E}\) for any \(v\in\mathcal{V}\) with components \(\varepsilon_{ij}(v)=(\partial v_{i}/\partial x_{j}+\partial v_{j}/\partial x_{ i})/2\), where \(i,j\in\{1,2,3\}\). As for the material characteristics, which are considered to be time-independent, it is natural to suppose that \(\alpha,\beta\in\mathcal{Z}\) and \(\mathcal{C}\in\mathcal{M}\) satisfy \(\alpha(x)\geq c\), \(\beta(x)\geq c\), and \(a^{T}\mathcal{C}(x)a\geq ca^{T}a\) for any \(x\in\Omega\), any matrix \(a\in\mathbb{R}^{3\times 3}_{\text{sym}}\), and some positive constant \(c\). All displacements \(u\) introduced above as well as the strains\(\varepsilon(u)\) and stresses \(\sigma,\tau\in L^{2}(\mathcal{I},\mathcal{E})\) (see Figure 1) are caused by the volume loads \(f\) on \(\Omega\) and surface loads on \(\Gamma\). To avoid unpleasant technical difficulties in proofs and to obtain stronger convergence results, we assume the smoothness in the sense of \(f\in W^{1,2}(\mathcal{I},\mathcal{H})\) and \(g\in W^{1,2}(\mathcal{I},\mathcal{G})\).