# Damage Behaviour of Quasi-Brittle Composites: Mathematical and Computational Aspects
## 1 Introduction  
Numerous engineering materials can be characterised as composites, i.e., materials produced from two or more constituents. The classification of [1] distinguishes among (i) miscellaneous classes of ductile materials (see [2, 3]); (ii) brittle materials, cf. [4], with potential transition to quasi-brittle behaviour in some cases, cf. [5]; and (iii) quasi-brittle materials, discussed in [6, 7, 8]. Apart from various mechanical characteristics of their constituents and their interfaces, typical particularly for matrices with stiffening fibres, there may appear some other phenomena commonly observable, particularly for materials from (iii). They are the initial, micro-fractured zones with growth potential that result in visible macro-cracks. Let us recall the general review of composite structures [9], considered as various kinds of anisotropic and heterogeneous structures, including composite-like laminates and fibre-reinforced matrices. Connecting this with the formal and variational asymptotic methods can be useful for further investigations, although the damage and failure analysis is generally viewed as a method or direction to be developed in the future. The computations at electronic, atomistic, microscopic, mesoscopic, etc., scales, together with various approaches to scale bridging, are analysed in [10]. In recent studies, scale bridging appears as an inspiring method. It is applied on various levels, from working with multiscale discrete damages (see [11]) up to the molecular dynamics applied for sufficiently simple material structures in [12]. An advanced non-local deterministic multiscale damage model is presented in [13]. A survey and a mutual comparison of various computational approaches to fractures and damages of quasi-brittle materials, like the discrete crack and the smeared crack techniques, molecular dynamics, etc., considered in context with their history and perspectives, can be found in [14]. Nevertheless, the detailed data are rarely known in sufficient detail. This is the case of cementitious composites with steel reinforcement, significant in design and construction in civil engineering. Thus, qualitative and/or quantitative information on smaller scales are exploited as the motivation for the formulation of semi-empiric constitutive relations, e.g., those between strain and stress fields. On the other hand, the primary formulation of computational models relies on the conservation principles of classical thermodynamics, compatible with [15; 16]. As a representative example of such a scale-bridging approach, the model of [17] introduces concrete as a two-phase composite material consisting of (i) coarse aggregates and (ii) mortar matrix; microcracks are allowed in (ii). The stress and strain fields are evaluated at the mesoscale, exploiting the classical Eshelby solution (see [18]) for equivalent inclusions. The crack density parameter is then interpreted as a scalar damage variable at the macroscopic scale. As an alternative to such an approach, predominantly probabilistic computational techniques of different types are considered in [19; 20; 21]. Due to the usefulness of such methods for civil engineering designs and constructions, the prediction of the damage behaviour of cementitious composites, including concretes with various kinds of reinforcement, is discussed in numerous studies. As for the progress in this field made in the last three decades, we mention the sequence of upgrades of the so-called Mazars model documented by [22; 23; 24; 25; 26; 27]. The original Mazars model was designed for the description of the elastic damage behaviour of concrete. The model is three-dimensional in general and is tailored for isotropic materials. It works with a damage criterion expressed in terms of strain, taking the traction-compression asymmetry into account. Its later improvements cover the restoration of stiffness forced by the re-closing of cracks, viscous and/or plastic deformation effects and influences, etc.