# Damage Behaviour of Quasi-Brittle Composites: Mathematical and Computational Aspects
## 2 Physical Background and Engineering Simplifications  
As announced in Section 1, from the macroscopic point of view, it is natural to work with the conservation of scalar quantities such as mass, linear and angular momentum components, and/or energy. As for our model problem, we can work only with the conservation of the linear momentum formulated for the Boltzmann continuum under the \begin{table}
\begin{tabular}{c l} \hline \hline
**Step** & **Topics** \\ \hline
1 & weak formulation of a model problem in appropriate Bochner–Sobolev spaces \\ \hline
2 & time-discretised formulation of 1 with a finite number of \(m\) steps \\ \hline
3 & step-by-step verification of solvability of 2 \\ \hline
4 & a priori bounds for sequences of approximate solutions, independent of \(m\) \\ \hline
5 & weak and strong convergence properties of sequences by 4, \\ \hline
6 & identification of limits by 5 with solution of 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The rough scheme of the proofs in Section 3. Figure 1: Engineering scheme of an SLS model. usual small strain simplification as follows. All quantities are related to the initial state, for which no irreversible damage occurs. The later remarks in Section 4 will be devoted to potential generalisations of such an approach. The principal challenge of computational modelling of fracture in solid materials and structures is in the incorporation of complicated crack topologies. The fracture evolution can be attributed to the interplay between the crack topology and the distribution of deformation energy. Two main groups of numerical approaches, reviewed by [49], are as follows: (i) topology-modifying approaches and (ii) field-modifying approaches. In (i), it is necessary to make some steps identical or similar to remeshing, while in (ii), additional field(s) representing the spatial location of the fracture are required. Unfortunately, both (i) and (ii) are computationally expensive, namely for three-dimensional simulations with crack branching and interactions. In addition to that, their reliability strongly depends on the design and identification of material parameters from appropriate experiments. Roughly speaking, we can recognise four stages of damage development from the engineering point of view. They are as follows: (a) a formation of microfractured zones, (b) an initiation of macroscopic cracks and their gradual growth, (c) a formation of systems of macroscopic cracks and their coalescence, (d) the total destruction of a considered material specimen or structure. From the point of view of solid-state physics, it is challenging to study the development from (b) to (c) and (d) thoroughly. Nevertheless, engineering practice needs, most of all, to incorporate (a) and evaluate the risk of progressive crack growth in (b), without extremely long and expensive distributed, parallel, etc., computations. This justifies the following approach, starting with a viscoelastic model based on the Kelvin and Maxwell chains, as introduced by [50], with later implementation of stiffness reduction due to (a), as demonstrated in Section 3, and potential activity of (b), sketched in Section 4.2. For the sake of illustration, Figure 1 shows quite a simple three-component viscoelastic scheme. It is the well-known SLS (standard linear solid) or the Zener scheme, viewed as the analogy of serial, parallel, etc., schemes in the theory of electrical circuits. The related symbols \(\mathcal{C}\), \(\alpha\), \(\beta\), \(\sigma\), and \(\tau\) will be introduced in Section 3 properly. For the preliminary information, \(\mathcal{C}\) refers here to a symmetric matrix of material parameters for the Hooke law of purely elastic material behaviour, with up to 21 independent elements in the general case. This can be reduced to a certain function of two parameters in the particular isotropic case. The parameters are the well-known Lame constant or, alternatively, the Young's modulus and the Poisson ratio, apart from potential material inhomogeneity.