## 2 Materials and Methods
### Numerical Analysis
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \begin{tabular}{c} **Number** \\ **of Dowels** \\ **(no.dowels)** \\ \end{tabular} & \begin{tabular}{c} **Span** \\ **\(L\)** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **Connection-Support** \\ **Distance** \\ **(mm)** \\ \end{tabular} & **Series** & \begin{tabular}{c} **Number** \\ **of** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **Density** \\ **of** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **Moisture** \\ **Content** \\ **(0\%)** \\ \end{tabular} & \
\begin{tabular}{c} **Connection** \\ **(5\%)** \\ \end{tabular} &
\begin{tabular}{c} **\(\alpha\)** \\ **(1\({}_{c}\)** \\ \end{tabular} \\ \hline \multirow{7}{*}{**Group 1**} & \multirow{7}{*}{1D} & 450 & 225 & 100/1D/450/0.5L & 4 & 747 & 9.7 & \multirow{7}{*}{1D} & \multirow{7}{*}{1D} \\  & & 475 & 100/1D/950/0.5L & 4 & 734 & 11.1 & & & \\  & & 360 & 100/1D/950/0.38L & 3 & 708 & 10.4 & & & \\  & & 240 & 100/1D/950/0.25L & 3 & 727 & 10.4 & & & \\  & & 120 & 100/1D/950/0.13L & 3 & 730 & 10.1 & & & \\ \cline{2-10}  & & 450 & 225 & 100/2D/450/0.5L & 4 & 727 & 10.2 & & & \\ \cline{2-10}  & & 475 & 100/2D/950/0.5L & 4 & 710 & 10.7 & & & \\  & & 360 & 100/2D/950/0.38L & 3 & 748 & 10.7 & 230 & 0.64 & \\  & & 240 & 100/2D/950/0.25L & 3 & 738 & 11.5 & 470 & & \\  & & 120 & 100/2D/950/0.13L & 3 & 717 & 10.9 & 710 & & \\ \hline \multirow{7}{*}{**Group 2**} & \multirow{7}{*}{1D} & 450 & 225 & 200/1D/450/0.5L & 4 & 756 & 9.7 & \multirow{7}{*}{1D} & \multirow{7}{*}{1D} \\  & & 475 & 200/1D/950/0.5L & 3 & 744 & 10.0 & & & \\ \cline{1-1}  & & 360 & 200/1D/950/0.38L & 3 & 716 & 9.7 & & & \\ \cline{1-1}  & & 950 & 240 & 200/1D/950/0.25L & 3 & 730 & 10.1 & & \\ \cline{1-1}  & & 120 & 200/1D/950/0.13L & 3 & 758 & 9.9 & & & \\ \cline{1-1} \cline{2-10}  & & 450 & 225 & 200/2D/450/0.5L & 4 & 730 & 9.6 & - & \\ \cline{1-1}  & & 475 & 200/2D/950/0.5L & 4 & 713 & 10.0 & & & \\ \cline{1-1}  & & 360 & 200/2D/950/0.38L & 3 & 741 & 10.0 & 230 & 0.32 & \\ \cline{1-1}  & & 950 & 240 & 200/2D/950/0.25L & 3 & 742 & 10.9 & 470 & \\ \cline{1-1}  & & 120 & 200/2D/950/0.13L & 3 & 716 & 10.3 & 710 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experimental test program. A bilinear cohesive law model with a linear softening relationship was used to reproduce the fracture process in mixed-mode I + II (Figure 4) [57; 58]. The proposed damage model is characterised by two branches. The first one describes the elastic behaviour of the undamaged material until the stress limit (\(f_{\text{u,}\dot{\
u}}\), \(i\) = I, II, m) is reached, and the second one defines the softening behaviour under damage (Figure 4). Under mixed\(-\)mode I + II, an equivalent relative displacement combining the two loading modes is defined (Figure 4) according to Equation (1):  
\[\delta_{\text{m}}=\sqrt{\delta_{1}^{2}+\delta_{\text{II}}^{2}} \tag{1}\]  
which can be written as  
\[\delta_{\text{m}}=\delta_{\text{I}}\sqrt{1+\beta^{2}} \tag{2}\]  
being  
\[\beta=\frac{|\delta_{\text{II}}|}{\delta_{\text{I}}} \tag{3}\]  
In the first branch, the constitutive relationship is given by  
\[f_{\text{m}}=k\delta_{\text{m}} \tag{4}\]  
where \(f_{\text{m}}\) is the resulting mixed\(-\)mode I + II traction and \(k\) is the initial interface stiffness usually known as the penalty parameter which is set by the user. In this study, the value of \(10^{6}\) N/mm\({}^{3}\)[54; 59] was adopted in all modes (I, II, mixed) to avoid undesirable interpenetration of the crack faces and numerical problems. A quadratic stress criterion is assumed to simulate damage onset under mixed-mode loading:  
\[\left(\frac{f_{\text{I}}}{f_{\text{u,I}}}\right)^{2}+\left(\frac{f_{\text{II} }}{f_{\text{u,II}}}\right)^{2}=1 \tag{5}\]  
where (\(f_{\text{L}},f_{\text{II}}\)) are the current mode I and II loading components and (\(f_{\text{u,I}},f_{\text{u,II}}\)) are the corresponding local strengths (Figure 4).