## 2 Materials and Methods
### Experimental Methods and Testing Set-up
#### 2.2.3 Calculation of Fracture Parameters
Three main stages can be identified, namely: (i) Initial linear elastic behavior (A-BOP), (ii) stable, subcritical crack growth (BOP-B) upon crack formation, here identified at the curve Bend Over Point (BOP), and (iii) unstable crack propagation (after the critical crack length is achieved at point B). These stages are related to the aforementioned stages depicted in Figure 4a. In order to  
Figure 4: (a) Typical crack-growth-resistance curves of brittle and quasi-brittle materials and (b) typical response of quasi-brittle materials under CDCB test [15]. compute the crack growth resistance curves (Figure 4a) from the experimental curves (Figure 4b), a procedure based on experimental and theoretical compliances [40] was employed. The compliance (C) is defined as the ratio between the elastic component of the CMOD (CMODe, Figure 4b) and the corresponding splitting load (SL):  
\[C=\frac{CMODe}{SL} \tag{4}\]  
To calculate the experimental compliance based on the experimental CMOD and SL data, it is possible to neglect the permanent deformation during the subcritical crack growth (CMODe = CMOD, Figure 4b), by assuming a return to the origin upon unloading. Considering MLEFM, each specimen arm at one of the crack sides can be considered as a simple cantilever beam made of a perfectly elastic material. With reference to the strength-of-materials approach, the theoretical compliance can be calculated as follows by considering bending and shear deformations of the beam:  
\[C_{th}=\ \frac{2\Delta}{SL}=\ \frac{CMOD}{SL}=\ \frac{24}{EB}\ \int_{0}^{a} \frac{x^{2}}{H^{3}}dx+\ \frac{6(1+\
u)}{EB}\int_{0}^{a}\frac{1}{H}dx \tag{5}\]  
where E = Young's Modules, \(\Delta\) = arm displacement, B = beam width (specimen depth), \(\
u\) = Poisson's ratio and a = crack length. Since the beam height is variable, a constant equivalent height equal to the mean height is considered:  
\[H_{c}=\ \frac{H_{o}+\ H_{a}}{2}=\ H_{o}+\ \frac{ma}{2} \tag{6}\]  
where Ho is the width of the top of the beam and Ha is the beam height at distance a from the loading point. Equation 5 was written based on the assumption that the end of the cantilever beam is fixed. However, the actual rotation of the bottom edge around the supporting hinge (Figure 1) needs to be considered. Based on the findings by [41], this can be amended by increasing the beam length by adopting an equivalent, increased crack length equal to a + 0.6 Hc. As a result, Equation 5 can be re-written as follows:  
\[C_{th}(a)=\ \frac{24}{EB}\left\{\frac{\left[a+0.6H_{c}(a)\right]^{3}}{3H_{c}^{3}(a) }+\ \frac{0.3a}{H_{c}(a)}\right\} \tag{7}\]  
By equating the initial experimental compliance to the initial theoretical compliance prior to BOP (Figure 4b), the effective modulus of elasticity (Eeff) is determined. The initial experimental compliance (Cexp,i) is calculated at the BOP, that is the first point of nonlinearity. Considering that the crack length before the BOP is equal to the length of the notch (a0), the initial theoretical compliance is given by:  
\[C_{th,i}=\ C_{th}\ (a_{0}) \tag{8}\]  
Once Eeff is known, the effective crack length (aeff) is calculated be equating the experimental compliance to the theoretical one at each point of the SL-CMOD curve. Hence, the Mode-I SIF (KI) for a CDCB sample is determined from :  
\[K_{I}^{2}=\ \eta^{2}SL^{2}B^{-2}H_{a}^{-3}\Big{(}a^{2}+1.4 aH_{a}+0.5H_{a}^{2} \Big{)} \tag{9}\]  
where \(\eta\) is a function of the slope and here is equal to 3.1 for m = 0.222 41. Considering that one B parameter in B\({}^{2}\) stands for beam depth and one for crack depth, when crack depth and beam depth are not equal to each other (in the case of monolithic specimens, Figure 1a) one of the B parameters has to be changed to b (crack depth, Figure 1a).