## 3 Modeling Procedures of Extended Finite Element Method (XFEM)-Based Multiscale Analysis
### Multiscale Modeling Methodology
\(\varepsilon_{c}^{pl}\) and \(\varepsilon_{t}^{pl}\) are equivalent compressive plastic strain and equivalent tensile plastic strain. The XFEM-based multiscale simulation proposed in this study is exhibited in Figure 5b. By contrast with the two-scale modeling approach with conventional solid elements, the two-scale model herein is composed of XFEM-based solid elements and fiber elements. The multiscale models of RC columns are established with fiber beam elements, XFEM-based solid elements and XFEM-based mesoscale concrete constructed with mortar, the ITZ and randomly distributed aggregates. The material properties of the steel reinforcement, the XFEM parameters of homogeneous concrete, the ITZ, mortar and the aggregates are detailed in Table 2.  
For reducing computational complexity and avoiding severe convergence problems in 3D XFEM, the numerical models in Figure 5b are simplified as 2D cases. Similar to CDP-based two-scale and multiscale models, the interface between fiber elements and plane stress elements is connected utilizing a kinematic coupling method. The steel reinforcements are simulated with the truss element T2D2, and the four-node plane stress element CPS4R with reduced integration is adopted to model homogeneous concrete, mortar, ITZ and aggregates. In order to improve the calculation accuracy, the bond-slip between steel reinforcements and concrete is considered in XFEM models. The element nodes of steel bars and concrete are connected with spring connector elements. The bond-slip effect is neglected at the top half of RC columns. It has been reported that the mesostructure variation of concrete leads to fluctuation of material strength and Young's modulus of concrete subjected to axial tension and compression [3]. In order to investigate the influence of mesoscale structure of concrete on the cracking pattern of tested RC \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & **[PERSON]’s** & **[PERSON]’s** & **Compressive** & **Density** & **Fracture Energy** & **Maximum** \\  & **Modulus (GPa)** & **Ratio** & **Strength (MPa)** & **(kg/m\({}^{3}\))** & **(N/m)** & **Principal Stress** \\ \hline Aggregates & 50.0 & 0.16 & / & 2733 & 288.0 & 6.0 \\ \hline Mortar & 24.7 & 0.20 & 27.2 & 2400 & 168.0 & 3.5 \\ \hline ITZ & 21.2 & 0.20 & 19.3 & 2400 & 115.2 & 2.4 \\ \hline Concrete & 31.3 & 0.20 & 40.9 * & 2400 & 216.0 & 4.5 \\ \hline Stirrup & 200.0 & 0.30 & \
\begin{tabular}{c} 371.5 * \\ (yielding stress) \\ \end{tabular} & 7800 & — & — \\ \hline Longitudinal steel bar & 200.0 & 0.30 &
\begin{tabular}{c} 379.6 * \\ (yielding stress) \\ \end{tabular} & 7800 & — & — \\ \hline \hline \end{tabular} Note.* denotes the material parameters direly measured from material strength tests.  
\end{table} 
Table 2: Material parameters. columns, the multiscale numerical models are constructed utilizing a random aggregate method (RAM). The modeling procedure of the RAM has been extensively reported [3; 4; 27], which is excluded in this section. Besides, the randomly-distributed aggregates are simulated using polygonal, circular and elliptical aggregates generated by the random aggregate generation and delivering program compiled in this study. As illustrated in Figure 5, three polygonal samples are designed to investigate the influence of distribution variation of aggregates. Meanwhile, elliptical and circular aggregate samples are considered to discuss the effect of aggregate geometry variation on the cracking evolution process and global structural response of tested RC columns. It is challenging to find a unified criterion for the definition of ITZ. The actual thickness of the ITZ is about 20-50 \(\upmu\)m, and it is impossible to establish elements in numerical models with such a tiny thickness [28]. Taking 1.0 mm as the thickness of the ITZ is accepted in most of the literature related to mesoscale modeling of concrete [28].