## 3 Material and Numerical Models
### Material Models
#### 3.2.3 Model of Contact (Interface) Elements  
Contact elements were used in the numerical models. They provided more accurate representations of the model behaviour and the accuracy of obtained results. The contact elements were applied at the interface between the masonry units in head joints, between the masonry units with mortar in bed joints, and between the masonry units and reinforced concrete and steel elements. The initial shear stiffness \(K_{\text{tt}}\) and the initial normal stiffness \(K_{\text{nn}}\) were decisive parameters for the contact elements. Constitutive relations for the flat state, regarding horizontal and standard displacements, were expressed in the following way:  
\[\left[ \begin{array}{c}\tau\\ \sigma\end{array} \right]=\left[ \begin{array}{cc}K_{\text{tt}}&0\\ 0&K_{\text{nn}}\end{array} \right]\left[ \begin{array}{c}\Delta u\\ \Delta v\end{array} \right], \tag{12}\]  
where \(\tau\) = shear stress, and \(\sigma\) = normal stress. The failure surface was expressed using the Mohr-Coulomb criterion (Figure 11) for normal compressive stresses (13) and elliptical cap at the side subjected to tension (14), in the following form:  
\[|\tau|\leq f_{\text{vo}}-\text{tg}\alpha\sigma,\text{ when }\sigma\leq 0, \tag{13}\]  
\[\tau=\tau_{o}\sqrt{1-\frac{\left(\sigma-\sigma_{c}\right)^{2}}{\left(f_{ \text{t}}-\sigma_{c}\right)^{2}}},\,\tau_{o}=\frac{1}{\sqrt{1-\frac{\sigma_{ c}^{2}}{\left(f_{\text{t}}-\sigma_{c}\right)^{2}}}},\,\sigma_{c}=\frac{-f_{ \text{t}}^{2}\text{tg}\alpha}{f_{\text{vo}}-2f\text{tg}\alpha} \tag{14}\]  
For normal stress greater than tensile strength, the criterion was expressed as  
\[\tau=0,\text{ when }\sigma>f_{\text{t}}. \tag{15}\]  
For shear stresses greater than limit values described by relationships (13) and (14), the boundary surface was reduced to the residual surface, corresponding to the dry friction surface described by the following equations:  
\[|\tau|\leq\text{tg}\alpha\sigma\text{ when }\sigma\leq 0, \tag{16}\]  
\[\tau=0\text{ when }\sigma>f_{\text{t}}. \tag{17}\]  
Figure 11: Failure criterion taken for contact elements: 1—boundary surface, 2—residual surface. A cap material was used in the interval of tensile stresses with the Mohr-Coulomb criterion replacing straight lines and with the ellipse intersecting the axis of normal stresses at the point corresponding to tensile strength \(f_{\mathrm{t}}\). The intersection point between the ellipse and the axis of shear stresses \(\tau\) corresponded to cohesion \(c\), and was tangential at that point to the Mohr-Coulomb line (Figure 11). Therefore, parameters of interface elements had to meet the following conditions:  
\[f_{\mathrm{t}}<\frac{f_{\mathrm{vo}}}{\mathrm{tg}\alpha},f_{\mathrm{t}}<f_{ \mathrm{vo}}, \tag{18}\]  
\[c>0,f_{\mathrm{t}}>0,\ \mathrm{tg}\alpha>0. \tag{19}\]  
The relationships of shear stress-displacement, and normal stress-displacement until achieving the failure surface by shear stress were linear, defined by the initial shear stiffness \(K_{\mathrm{tt}}\) and the initial normal stiffness \(K_{\mathrm{nn}}\) (Figure 12). Due to there being zero thickness of the contact elements, stiffness values should be greater than those of adjacent finite elements, but at the same time low enough to avoid numerical instability. Additionally, stiffness values \(K_{\mathrm{tt,min}}\) and \(K_{\mathrm{nn,min}}\) were required to obtain numerical solutions for the softening phase. Values were chosen iteratively, decreasing initial stiffness values \(K_{\mathrm{tt}}\) and \(K_{\mathrm{nn}}\). Multi-section relationships could be defined to describe the behaviour of the contact element subjected to tension and shearing. Two-section softening laws, shown in Figure 13, were the simplest solutions.