# Numerical Modeling Technique of Damage Behavior of
## 2 Introduction of XFEM
### Constitutive Relations  
In order to precisely model the nonlinear behavior of masonry-infilled RC frame structures, the rational bulk material constitutive relation (Equation (12)) and discrete interface constitutive relation (Equation (13)) are required to properly consider the distributed crushing and cracking behavior of the concrete and masonry blocks. For this purpose, two of constitutive relation subroutines called \"umatmises\" and \"umatinterface\" are built. Firstly, an elastic-plastic constitutive model called J2-plasticity model [53] is employed in this study to model the crushing process of concrete and masonry blocks, as depicted in Figure 6. In this model, a yield criterion combining the von [PERSON] yield criterion and tension cutoff criterion are adopted, as displayed in Figure 6a. The von [PERSON] yield criterion can be written in the following format:  
\[\mathrm{J}_{2}-\frac{1}{3}\sigma_{\mathrm{e}}^{2}(\varepsilon_{\mathrm{P}})= \frac{1}{2}\mathrm{s}_{\mathrm{ij}}\mathrm{s}_{\mathrm{ij}}-\frac{1}{3} \sigma_{\mathrm{e}}^{2}(\varepsilon_{\mathrm{P}})=0 \tag{27}\]  
where \(\mathrm{J}_{2}\) is second invariant of the deviatoric stress. For plane stress state, \(\sigma_{13}=\sigma_{23}=\sigma_{33}=0\). Therefore, the second invariant of the deviatoric stress tensor can be expressed as [54]:  
\[\mathrm{J}_{2}=\mathrm{s}_{\mathrm{ij}}\mathrm{s}_{\mathrm{ij}}=\sigma^{ \mathrm{T}}\mathrm{P}\sigma \tag{28}\]  
Figure 5: Subdivision of sub-quadrilaterals penetrated by the crack. Moreover, \(\sigma_{e}\) is effective stress and \(\varepsilon_{p}\) is effective plastic strain. The effective stress is a function of the effective plastic strain, which governs the isotropic strain hardening/softening law. The function is described as a parabolic function followed by an exponential decay function, which is determined by the material parameters \(f_{0,}f_{c}\), \(\varepsilon_{lp}\) and \(\varepsilon_{2p}\), as shown in Figure 6b. In this analysis, the implementation of the constitutive relation is in the rate form, and it can be described as follows: upon discretization in time, the stress vector and effective plastic strain in the current step are given; then in the next step, a new stress vector and effective plastic strain are updated in a prescribed strain rate. To be specific, elastic predictor stress vector is firstly obtained through the current stress vector, stiffness matrix and strain rate. Together with the current effective plastic strain, the value of the yield function is then calculated. If the value is negative, the elastic predictor stress vector is the new admissible stress vector for the next step; or an iteration procedure is preformed to correct the predictor stress vector and effective plastic strain, until the yield function is equal to zero within a permissible tolerance. Finally, the stress vector and effective plastic strain at the next step are updated. The discrete interface constitutive relation in Equation (13) usually plays an important role in modeling the failure behavior of masonry-infilled frames due to the weak performance of cracks. In this study, a precise discrete constitutive relation [28; 36; 38; 55] is employed to model the cracking behavior of the concrete and masonry blocks, and it is able to simulate the degradation of tensile strength and tensile stiffness, normal compaction, shear dilatation and decreases in friction surface roughness under combined normal and shear tractions. The interface displacement discontinuity w is decomposed into three displacement modes, i.e., the conventional elastic and plastic displacements, as well as a geometric part to account for the reversible dilatation caused by the wedging effect of asperity surfaces. A hyperbolic yield criterion [28] is adopted to deal with the yielding of the crack surface in both tension-shear and compression-shear regions.