## 2 Materials and Methods
### Experimental Methods and Testing Set-up
#### 2.2.3 Calculation of Fracture Parameters  
Ideally, brittle materials are the ones that can be characterized by the propagation of one single crack with the elastic zones near the crack. For fracture analysis of such materials, the linear elastic fracture mechanics (LEFM) theory can be applied. LEFM indicates that the fracture instability of brittle materials in Mode-I or opening mode can be defined by only one parameter, i.e., the critical stress intensity factor (K\({}_{\text{IC}}\)) or the specific fracture energy (G\({}_{\text{f}}\)) [15]. Back to Griffith [29], a crack propagates unstably when the rate of change in the elastic energy released by the system for a unit crack extension (U), is equal or higher than the energy required for the crack to propagate within the material, for a unit crack extension (W):  
\[G_{IC}=\ \frac{\partial U}{\partial a}=\ \frac{\partial W}{\partial a}=R \tag{3}\]  
Figure 3: (**a**) Representative three-dimensional (3D) topography of a substrate profile after sandblasting and prior to the repair application and (**b**) two-dimensional (2D) profiles used to calculate crack deviation in monolithic, material tests. where \(\rm G_{IC}=Mode-I\) critical strain energy release (or specific fracture energy \(\rm G_{f}\)), a = crack length and R = crack growth resistance. The stress intensity factor, \(\rm K_{I}\), that defines the elastic stress field intensity ahead of the crack is then sufficient to characterize the whole stress field around the crack. \(\rm K_{I}\) reaches \(\rm K_{IC}\) (critical value) when enough energy is supplied to the system and unstable crack propagation occurs. The LEFM theory is applicable in materials where the zone around the crack that undergoes inelastic deformations remains small and is therefore neglected. However, in the case of cement-based materials (typically referred to as quasi-brittle materials), a large zone of damaged material, known as fracture process zone (FPZ), subject to high stress concentration, develops around the crack tip. Micro-cracking along the FPZ improves the energy absorption capability because it causes less energy flux released into the crack tip and increased cracking surface area [30]. The extension of FPZ affects toughness of quasi-brittle materials as well as tortuosity and bridging effects of aggregate and unbroken ligaments [31; 32]. There are various, well-calibrated classic methods for analysing the behavior of the FPZ, including fracture models proposed by [PERSON] [33], [PERSON] [34], [PERSON] [35], [PERSON] [36], [PERSON] [37], [PERSON] [30], and [PERSON] [38]. In the analysis of the CDCB test, a methodology was developed [15; 21; 39] for the computation of crack growth resistance curves based on the effective crack model. With this approach, the effect of the FPZ is accounted for by treating the quasi-brittle material as a brittle material with an increased, fictitious crack length, called effective crack length (\(\rm a_{eff}\)). By treating the material as brittle, a Modified Linear Elastic Fracture Mechanics (MLEFM) based on adaptation of LEFM [32] equations can still be utilized. The typical crack-growth-resistance curves of brittle and quasi-brittle materials (or interfaces, in case of bi-material systems) are represented in Figure 4a. In brittle systems, the crack propagates unstably since formation and, hence, its resistance to propagation (here represented by the stress intensity factor) is a straight, horizontal line, unrelated to crack extension. In the case of quasi-brittle systems, the toughening mechanisms occurring after crack nucleation allow for a stage of stable (nonlinear) crack growth, represented by an initial rising trend (BOP-B), followed by a flat branch of unstable propagation after critical crack length and critical stress intensity factor are reached, point B of Figure 4a. A typical Splitting Load (SL)-Crack Mouth Opening Displacement (CMOD) response curve obtained from the CDCB test of quasi-brittle materials or bi-material interfaces (for bond tests) is represented in Figure 4b.