## 5 Masonry Mechanical Properties and Constitutive Law
### Constitutive Material Law for Unreinforced Masornry  
Unreinforced masonry, like other quasi-brittle materials, exhibits strain-softening in tension and an initial strain-hardening and subsequent softening in compression, which leads to a loss of loading capacity [22]. The concrete damage plasticity (CDP) model captures this behaviour and is suitable for quasi-brittle materials undergoing cyclic and/or dynamic actions [77]. It has been successfully used for conducting nonlinear dynamic analyses of unreinforced masonry structures subjected to seismic actions [17; 78; 79; 80; 81; 82]. The CDP constitutive law in Abaqus/CAE 2019 [23] is based on the work by [PERSON] et al. [83] and [PERSON] [84]. The model assumes that the material is isotropic and describes its behaviour through four parameters: the damage evolution, hardening/softening law, yield criterion, and flow rule. Based on the classical plasticity theory, the CDP model decomposes the total strain \(\varepsilon\) into elastic strain \(\varepsilon^{el}\) and plastic strain \(\varepsilon^{pl}\)[85]:  
\[\varepsilon=\varepsilon^{el}+\varepsilon^{pl} \tag{4}\]  
and simulates the elastic stiffness degradation with an isotropic scalar damage variable \(0\leq d\leq 1\), where zero corresponds to an undamaged state and one to a completely damaged condition. The stress state \(\sigma\) is then related to the damage variable through:  
\[\sigma=(1-d)\mathbf{D}_{0}^{el}:\left(\varepsilon-\varepsilon^{pl}\right) \tag{5}\]  
where \(\mathbf{D}_{0}^{el}\) is the initial elasticity matrix and \(\varepsilon\) and \(\varepsilon^{pl}\) are the total strain and plastic strain tensors, respectively. The hardening/softening behaviour of the material is characterized by the uniaxial stress-strain curves under tension and compression. Considering the model's assumption of isotropy, these behaviours can be further extended to multiaxial loading conditions [77]. The degraded stiffness is then defined by two independent uniaxial damage variables, namely \(d_{c}\) for compression and \(d_{t}\) for tension. These variables are related to the stress in compression \(\sigma_{c}\) and tension \(\sigma_{t}\) through the following relationship:  
\[ \begin{split}\sigma_{c}&=(1-d_{c})E_{0}\Big{(} \varepsilon_{c}-\varepsilon_{c}^{pl}\Big{)}\\ \sigma_{t}&=(1-d_{t})E_{0}\Big{(}\varepsilon_{t}- \varepsilon_{t}^{pl}\Big{)}\end{split}  \tag{6}\]  
where \(E_{0}\) is the initial [PERSON]'s modulus. The CDP model assumes that the elastic stiffness starts degrading right after the material reaches its maximum strength, i.e., when it is unloaded from any point along the softening branch [77], as shown in Figure 7. Thus, the values for \(d_{c}\) and \(d_{t}\) are a function of the plastic strains. However, values between 0.9 and 0.95 are chosen as maximums to avoid numerical issues [22]. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**E [MPa]** & **Poisson’s Ratio** & \(f_{c}\) **[MPa]** & \(f_{t}\) **[MPa]** & \(G_{c}\) **[N/mm]** & \(G_{t}\) **[N/mm]** & **Specific Weight [kh/m\({}^{3}\)]** \\ \hline
1050 & 0.2 & 1.5 & 0.15 & 4.17 & 0.05 & 19 \\ \hline \end{tabular}
\end{table} 
Table 1: Mechanical properties for irregular masonry. For this study, the nonlinear compressive behaviour of masonry is described by a stress-strain curve (Figure 8). Table 1 provides the values used to construct the curve using the equations proposed by [PERSON] [86], which transforms the fracture energy \(G_{c}\) into strains for the softening branch. When a stress-strain curve is used as input for tension, the response is highly sensitive to the mesh refinement. The fracture energy proposal overcomes this problem. With this approach, the brittle behaviour is characterized by a stress-displacement response rather than a stress-strain response. In this case, [PERSON] transforms the fracture energy \(G_{t}\) into a stress-displacement curve with a linear loss, applying Equation (7) [83,87]:  
\[u_{t,u}=\frac{2G_{t}}{\sigma_{t,max}} \tag{7}\]  
where \(u_{t,u}\) is the ultimate displacement.