## 3 Material and Numerical Models
### Material Models
#### 3.2.2 Model of the Wall and Lintels--Elastic-Plastic-Based Degradation Model
Figure 8: The Rankine criterion in Haigh–Westergaard space: (**a**) the space of principal stresses, (**b**) axiatoric section, (**c**) deviatoric section. Figure 7: Menétrey–Wilam criterion in Haigh–Westergaard space: (**a**) the space of principal stresses, (**b**) hydrostatic cross-section, (**c**) deviatoric section. The required material parameters were obtained from the laboratory tests and validation of the model described in the paper [42]. Table 4 presents material parameters for the wall and lin  
Figure 9: The relative position of the Rankine and M-W-3 surfaces at \(\lambda_{\mathrm{t}}=2\): (**a**) view of surfaces in principal stress space, (**b**) axiatoric sections, (**c**) deviatoric sections; \(I\)—Rankine surface, \(2\)—M-W-3 surface at \(k=1\) (yield strength), \(3\)—M-W-3 surface at \(k=k_{0}\) (end of elastic stage). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Test Formula or Results** & **Masony** & **Lintel** \\ \hline Uniaxial compressive strength/\(f_{\mathrm{v}}\), N/mm\({}^{2}\) & Assumed from the tests & 4.04 & 3.71 \\ Plastic strain under compression \(\epsilon_{\mathrm{c}}\)g & Assumed from the tests & \(3.33\times 10^{-4}\) & 3.771 \\ Uniaxial tensile strength \(f_{\mathrm{b}}\), N/mm\({}^{2}\) & Assumed from the tests & 0.61 & 0.61 \\ Initial modulus of elasticity \(\mathrm{c}_{\mathrm{c}}\), N/mm\({}^{2}\) & Assumed from the tests & 2204 & 2198 \\ Poisson’s ratio & Assumed from the tests & 0.200 & 0.179 \\ Fracture energy \(G_{\mathrm{v}}\), MN/mm & Assumed from the tests & \(1.07\times 10^{-5}\) & \(1.602\times 10^{-5}\) \\ Weakering function at tension & Assumed softening described by the exponential function & – & – \\ Displacement \(w_{\mathrm{c}}\) under tension [m] & Displacements were calculated from the equation \(w_{\mathrm{c}}=5.14\frac{\sigma}{f_{\mathrm{v}}}\) & \(4.36\times 10^{-4}\) & – \\ Crack spacing \(g_{\mathrm{max}}\) [m] & Assumed constant value & 0.5 & 0.5 \\ Coefficient of tensile strength reduction at the softening phase \(c_{\mathrm{v}}\) & Assumed constant value for unconfected material & 0 & 0 \\ Model of cracks & developing in uniform directions & – & – \\ Critical displacement under compression, m & & \(-5.0\times 10^{-4}\) & \(-5.0\times 10^{-4}\) \\ Reduction of compressive strength caused by cracking \(\epsilon_{\mathrm{c},\mathrm{min}}\) & & 0.8 & 0.8 \\ Compressive stiffness of cracks \(g_{\mathrm{r}}\) & & 20.0 & 20.0 \\ Size of aggregate particles [m] & Determined on the basis of macroscopic observations of the masonry units & 0.02 & 0.02 \\ Eccentricity of elliptical function \(e\) & Determined from the tests & 0.5 & 0.5 \\ Direction of plastic flow & Assumed as for incompressible material & \(\beta=0\) & \(\beta=0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of the plastic and elastic model used in calculations. Therefore, the stress state corresponding to tension and slight compression was expressed by the Rankine failure surface. The failure surface at higher compressive values of hydrostatic stresses was described by the M-W-3 surface. Displacement of the M-W-3 surface along the hydrostatic axis was caused by changes in the parameter \(\lambda_{\mathrm{t}}\). The surface could expand, simulating the stages of hardening or softening, depending on the parameter \(\kappa\), related to dilatational, non-dilatational or plastic strains:  
\[d\kappa=d\varepsilon_{0}^{p}=d\varepsilon_{1}^{p}+d\varepsilon_{2}^{p}+d \varepsilon_{3}^{p}. \tag{9}\]  
A temporary shape and location of the M-W-3 surface at the hardening phase was defined by the hardening function (\(\kappa\)), which depended on the hardening/softening parameters [50; 51].