# Numerical Modeling Technique of Damage Behavior of
## 2 Introduction of XFEM
### Incremental Equilibrium Equation  
Consider a two-dimensional domain \(\Omega\) bounded by \(\Gamma\), as shown in Figure 1. The boundary \(\Gamma\) includes three complementary sets: \(\Gamma_{\text{t}}\), \(\Gamma_{\text{u}}\) and \(\Gamma_{\text{c}}\). The traction is prescribed on \(\Gamma_{\text{u}}\), the displacement is prescribed on \(\Gamma_{\text{c}}\), and the internal discontinuity surface is represented by \(\Gamma_{\text{c}}\), which consists of \(\Gamma_{\text{c}}^{+}\) and \(\Gamma_{\text{c}}^{-}\). The equilibrium equation in the strong form and corresponding boundary conditions can be written as:  
\[\
abla\cdot\sigma+b=0 \text{in}\ \Omega \tag{4a}\] \[\sigma\cdot n=\bar{t} \text{on}\ \Gamma_{t}\] (4b) \[u=\overline{u} \text{on}\ \Gamma_{u}\] (4c) \[\sigma\cdot n^{+}=-\sigma\cdot n^{-}=t^{+}=-t^{-}=t \text{on}\ \Gamma_{c} \tag{4d}\]  
where \(\sigma\) is the Cauchy stress tensor in the bulk material, b is body force per unit volume, and \(\bar{t}\) and \(\overline{u}\) are prescribed tractions and displacements, respectively. \(t^{+}\) and \(t^{-}\) are interface tractions acting upon the two sides of \(\Gamma_{c}\). n is the unit outward normal. It is assumed that the displacement remains small and the strain field can be expressed as the derivation of the displacement field:  
\[\varepsilon=\varepsilon(u)=\
abla_{s}u \tag{5}\]  
The magnitude of displacement discontinuity \(w\) across \(\Gamma_{c}\) can be computed as the displacement separation on two sides of the crack surface:  
\[w=u^{-}-u^{+} \tag{6}\]  
To describe the nonlinearity involved in solving the problem, the incremental equilibrium equation in weak form is necessary. With the admissible variational displacement fields, it can be given as:  
\[\int_{\Omega}\dot{\sigma}\big{(}\dot{u}\big{)}:\varepsilon(v)dx+\int_{\Gamma_ {c}}\dot{t}\big{(}\dot{u}\big{)}\cdot w(v)ds=\int_{\Omega}\dot{b}\cdot vd \Omega+\int_{\Gamma_{t}}\dot{\bar{t}}\cdot vd\Gamma\ \ \ \forall v\in U_{0} \tag{7}\]  
where \(w(v)=v^{-}-v^{+}\). Figure 1: A cracked domain with boundary conditions. In XFEM, as described in Equation (1), apart from the classical degrees of freedom, there are several additional enriched degrees of freedom that should be introduced for the displacement field description, and these enriched degrees of freedom are also associated with the corresponding element nodes [40]. To be more specific, two enriched degrees of freedom at each node are needed for the modified Heaviside enrichment, and eight enriched degrees of freedom at each node are built for the crack tip enrichment. In this study, two types of extended finite element modules are built in FEAP using a user subprogram called \"ELMTnn\": one is programed to simulate the behavior of concrete for columns and beams, and the other is used for the masonry blocks in the infill panel. The difference in enrichment type for two modules leads to different numbers of required nodal degrees of freedom. For masonry elements, two enriched degrees of freedom per node associated with the modified Heaviside enrichment are added. For concrete elements, ten enriched degrees of freedom per node associated with both the modified Heaviside and crack tip enrichments are used. It should be pointed out that multiple cracks are allowed in the modeling. Specifically, if either the modified Heaviside or crack tip enrichment is activated, degrees of freedom numbered 3, 4 or 5-12 are correspondingly used to perform the subsequent work. If two modified Heaviside enrichments for two cracks are activated at the same node, the degrees of freedom 3-6 are used. If modified Heaviside and crack tip enrichments associated with two different cracks are activated, the degrees of freedom 3-12 are all adopted. Two crack tip enrichments at the same node and more than one crack intersecting in the same element are not considered in this paper. If needed, the number of degrees of freedom should be appropriately set according to the desired analysis type.