## 5 Analytical Modelling
### Description of the ACK and the Stochastic Cracking Models  
The Aveston, Cooper and Kelly theory [15; 16] for fibre-reinforced composites was originally developed to describe the tensile stress-strain curve of brittle matrix composites, reinforced with quasi-unidirectional fibres characterised by a volume fraction greater than the critical one. The response (see Figure 14) is described by means of a three stages curve: pre-cracking zone (Stage I), multiple cracking zone (Stage II) and post-cracking zone (Stage III). The basic assumptions of this model are:  
* The fibres are considered capable of carrying loads only along their longitudinal axis. * The matrix-fibre bond is assumed to be weak. * Once the matrix and the fibres debond, a pure frictional shear stress \(\tau_{0}\) is considered. This shear stress is assumed constant along the debonded interface. * Poisson effects (transverse contraction) of both the fibres and the matrix are neglected. * Global load sharing is used for fibres. * In a section orthogonal to the applied load, matrix normal stresses are considered uniform. The first branch of the response is linear elastic and the bond between the matrix and the fibres is assumed to be perfect. In this stage, the stiffness of the composite \(E_{c,I}\) is function of the fibres and the matrix volume fractions (\(V_{f}\), \(V_{m}\)) and their stiffness (\(E_{f}\), \(E_{m}\)) and it is computed by means of the well-known rule of mixtures, via the following equation:  
\[E_{c,I}=E_{m}V_{m}+E_{f}V_{f}. \tag{6}\]  
Figure 14: Typical stages of the tensile response in the ACK and Stochastic cracking models and illustrative cracking patterns. The two volume fractions can be decreased by means of fibres and matrix efficiency factors (\(\eta_{f}\), \(\eta_{m}\)) connected to imperfect matrix-fibre adhesion, warping or misalignment of the unidirectional fibres and eventual presence of inclusions or air voids in the matrix [24]. The modification of the ACK theory is described as follows:  
\[E_{c,I} =E_{m}V_{m}^{*}+E_{f}V_{f}^{*} \tag{7}\] \[V_{f}^{*} =V_{f}\cdot\eta_{f}\] (8) \[V_{m}^{*} =V_{m}\cdot\eta_{m}. \tag{9}\]  
According to the ACK model, the matrix has a deterministic tensile failure stress \(\sigma_{mu}\). When this value is reached, the composite shows multiple cracking (Stage II). The composite multiple cracking stress \(\sigma_{mc}\) is computed as:  
\[\sigma_{mc}=\frac{E_{c1}\cdot\sigma_{mu}}{E_{m}}. \tag{10}\]  
When the first crack appears, matrix and fibres debond and the pure constant frictional shear stress is considered, \(\tau_{0}\) provides the stress transfer between fibres and matrix. From the equilibrium in the longitudinal direction, it is possible to obtain the transmission length \(\delta_{0}\), equal to the distance from a crack face at which the matrix reaches again the stress \(\sigma_{mu}\):  
\[\delta_{0}=\frac{V_{m}\ r\ \sigma_{mu}}{V_{f}\ 2\ \tau_{0}}. \tag{11}\]  
where \(r\) is the equivalent fibres radius. Since, according to the ACK theory, the matrix has a unique tensile failure stress, multiple parallel cracks are simultaneously introduced in the specimen, until saturation is reached. In Stage II, the internal stress leads to a final state where the distance among cracks is between \(\delta_{0}\) and \(2\delta_{0}\) (with an average value equal to \(X=1.337\cdot\delta_{0}\)[25]). In Stage III, when the multi-cracking phenomenon is over, only fibres contribute to the stiffness of the composite:  
\[E_{c,III} =E_{f}\cdot V_{f}^{*}. \tag{12}\]  
An extension of the original ACK theory to E-glass fibre reinforced cementitious composites was proposed by [PERSON] and [PERSON] under the name of stochastic cracking model [17]. They considered the stochastic nature of the tensile strength of the matrix, through the use of a two-parameter Weibull distribution function [26] and they assumed the same distribution for the crack spacing.