# Parametric Analyses of the Response of Masonry Walls with Reinforced Plaster
## 3 Numerical Simulations
### Mechanical Properties and Constitutive Laws  
The mechanical properties adopted in the numerical analyses are collected in Table 1, where \(\gamma\) is the weight density, E is the Young modulus, \(\upsilon\) is the Poisson ratio, and \(\sigma_{\text{cu}}\) and \(\sigma_{\text{t0}}\) are the failure stress in the compression and tension, respectively. The values for the masonry, concrete, and steel were defined based on the data available from the experimental tests and according to EC2 [19]. All the unavailable mechanical data were quantified by calibration on the experimental results or on the basis of studies published by other researchers on analogous materials. A concrete damage plasticity (CDP) model [37], already implemented in the Abaqus code, was assumed for the mortar, blocks, and concrete, while elastic-plastic behavior with strain hardening was considered for the steel, with yielding stress of 450 MPa, ultimate stress of 550 MPa, and plastic strain equal to 0.14. The CDP was developed by [PERSON] [37]  
Figure 6: (**a**) DC test set-up; (**b**) FE model of the wall with load and boundary conditions. to describe the non-linear response of concrete, but it was found to be suitable for all applications with materials that exhibit brittle/quasi-brittle behavior and damage. Examples of the application of the CDP model to masonry structures may be found in [18; 38; 39]. The main features of the CDP model are the following: (1) linear and isotropic behavior in the elastic regime; and (2) elastic-plastic damageable behavior in the nonlinear range, taking into account the difference between the compressive and tensile strengths and the softening behavior once the strength of the material is reached. The adopted plasticity-based damage model assumes that the two main failure mechanisms are tensile cracking and compressive crushing. Under uniaxial tension, the stress-strain response follows a linear elastic relationship until the value of the failure stress, \(\sigma_{\text{t0}}\), is reached. The failure stress corresponds to the onset of micro-cracking in the material. Beyond the failure stress, the formation of micro-cracks is represented macroscopically with a softening stress-strain response, which induces strain localization in the material's structure. Under uniaxial compression, the response is linear until the value of the initial yield, \(\sigma_{\text{c0}}\). In the plastic regime, the response is typically characterized by stress hardening followed by strain softening beyond the ultimate stress, \(\sigma_{\text{cu}}\). It is assumed that when the material is unloaded from any point on the softening branch, the response is characterized by a reduced elastic stiffness (damage). The degradation \"d\" of the elastic stiffness is governed by two damage variables, \(\text{d}_{\text{c}}\) and \(\text{d}_{\text{t}}\), which are assumed to be functions of the plastic strains. These variables may assume values from zero (undamaged material) to one, which represents the total loss of stiffness. Figure 7 reports, for the mortar, brick, and plaster, the stress-strain behavior of the compression, the post-failure stress-displacement curves of the tension, and the values of the damage factor of the tension and compression. A Drucker-Prager (DP) strength domain was assumed; its main parameters for the different materials are reported in Table 2. K is a parameter whose role is to smoothly distort the DP surface in the deviatoric plane from a circle to a surface more similar to a Mohr-Coulomb one; the adopted K value was the one suggested by the user's guide to closely approximate a Mohr-Coulomb failure criterion. The regularization of the tensile corner was also adopted by means of a correction parameter, referred to as eccentricity e, expressing the rate at which the plastic flow potential approached the linear Drucker-Prager function at high confining pressure stress. A default value equal to 0.1 was adopted, which ensured that the dilation angle would remain constant over a wide range of confining pressures.