## 2 Materials and Methods
### Static Fracture Tests
#### 2.4.3 Direct Evaluation of Mechanical Fracture Parameters from \(F-CMOD\) Diagrams  
The double-\(K\) fracture (2K) model ([PERSON], 2013) was used for the evaluation of the \(F-CMOD\) diagrams to determine selected fracture parameters. The parameters describing different phases of the fracture process were determined using this fracture model. The unstable fracture toughness \(K_{\text{KL}}\textsuperscript{un}\) is defined as the critical stress intensity factor corresponding to the maximum load, and it represents the phase of unstable crack propagation. This parameter is similar to effective fracture toughness (Equation (8)) used in the effective crack model by [PERSON] (2013). The equivalent elastic crack length \(a_{c}\) was determined from the following equation ([PERSON], 2013):  
\[CMOD_{F_{\max}}=\frac{6F_{\max}Sa_{c}}{BW^{2}E}V(a_{c}) \tag{11}\]  
where \(CMOD_{F\max}\) is \(CMOD\) corresponding to peak load \(F_{\max}\), and  
\[V(a_{c})=0.76-2.28\alpha_{c}+3.87\alpha_{c}{}^{2}-2.04\alpha_{c}{}^{3}\frac{0. 66}{(1-\alpha_{c})^{2}} \tag{12}\]  
where \(\alpha_{c}=(a_{c}+H_{0})/(W+H_{0})\); \(H_{0}\) is the thickness of blades fixed on the bottom surface of the specimens between which the strain gauge was placed. When the equivalent elastic crack length \(a_{c}\) is known, the \(K_{\text{KL}}\textsuperscript{un}\) was determined according to Equation (8), when the \(a_{c}\) is substitute by \(a_{c}\) and geometry function was in this case ([PERSON], 2013):  
\[Y(\alpha_{c})=\frac{1.99-\alpha_{c}(1-\alpha_{c})\big{(}2.15-3.93\alpha_{c}+2. 70\alpha_{c}{}^{2}\big{)}}{(1+2\alpha_{c})(1-\alpha_{c})^{3/2}}, \tag{13}\]where \(\alpha_{c}=a_{c}/W\). The cohesive softening function has to be known to calculate cohesive fracture toughness \(K_{\text{lc}}{}^{c}\), which can be interpreted as an increase in the resistance to crack propagation caused by the bridging of aggregate grains and other toughening mechanisms in the FPZ. This function describes the relationship between the cohesive stress and effective crack opening displacement. Based on the author's previous studies, it can be stated that the type of softening function (bilinear or non-linear) had no significant effect on the calculated values of fracture parameters. The input parameters of the softening function are more important, especially the way of estimation of tensile strength. Commonly, the compressive strength values are used for the estimation of the tensile strength of the materials. It is more appropriate to determine the tensile strength directly from the tensile test performed in the configuration similar to the loading of the real structural element. Therefore, the tensile strength \(f_{\text{t,ID}}\) was identified from the measured \(F\)-\(d\) diagrams (see Section 2.4.2). In this study, the results obtained using the non-linear softening function according to [PERSON] [34] are presented. Then, the cohesive stress \(\sigma(CTOD_{c})\) at the tip of an initial notch at the critical state can be obtained from this softening function:  
\[\sigma(CTOD_{c})=f_{\text{t}}\Bigg{\{}\Bigg{[}1+\left(c_{1}\frac{CTOD_{c}}{ CDO_{c}}\right)^{3}\Bigg{]}\exp\left(-c_{2}\frac{CTOD_{c}}{COD_{c}}\right)-\frac{ CTOD_{c}}{COD_{c}}\left(1+c_{1}{}^{3}\right)\exp(-c_{2})\Bigg{\}} \tag{14}\]  
where \(f_{\text{t}}\) is the tensile strength, \(CTOD_{c}\) is the critical crack tip opening displacement according to [PERSON] and [PERSON] [35]:  
\[CTOD_{c}=CMOD_{c}\Bigg{(}\left(1-\frac{a_{0}}{a_{c}}\right)^{2}+\left(1.081-1. 149\frac{a_{c}}{W}\right)\left(\frac{a_{0}}{a_{c}}-\left(\frac{a_{0}}{a_{c}} \right)^{2}\right)\Bigg{)}^{\frac{1}{2}} \tag{15}\]  
\(COD_{c}\) is the critical crack opening displacement, and \(c_{1}\) and \(c_{2}\) are the material constants, which were taken from [36].