# Meso-Scale Simulation of Concrete Based on Fracture and Interaction Behavior
## 4 Tension Simulation Result  
The complete tensile stress-strain curves of the specimens are shown in Figure 7. A mean curve is plotted to show the integral trend of the curves. Assume that \(\sigma_{i}(\varepsilon)\) is the stress of i-th specimen at the global strain \(\varepsilon\). The points \((\overline{\sigma},\varepsilon)\) on the mean curve can be calculated as follows:  
\[\overline{\sigma}=\frac{1}{N}\sum_{i=1}^{N}\sigma_{i}(\varepsilon) \tag{7}\]  
Through the calibration, the mean curve agrees well with the curve of the experimental result [32]. The curves of standard deviation and standard deviation coefficient that are calculated with simulation data are also plotted in Figure 7. The curves represent the absolute and relative errors of the simulation data, respectively. It is worth noting that the standard deviation almost remains constant in the softening stage. The stable deviation indicates that the specimens have a close damage evolution rate at the same global deformation level in spite of their different aggregate distributions. As shown in Figure 8, there are two typical tension failure modes: the specimen fracture with one or two dominant cracks perpendicular to the tension direction. In the failure mode with only one dominant crack, the crack extends through the whole cross-section as the experimental result. The majority of the tension failures are this mode. In the other failure mode, the two dominant cracks at different horizontal positions extend from both sides. With the crack propagation, the horizontal projections of the two cracks intersect each other. Afterward, one of the crack's propagation rates \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Parameter** & **Aggregate** & **Mortar** & **ITZs** & **MII** \\ \hline Elastic Modulus, \(E\) (GPa) & 72 & 28 & 24 & 26 \\ Poisson’s ratio in plane, \(\
u_{12}\) & 0.16 & 0.2 & - & - \\ Poisson’s ratio normal to plane, \(\
u_{13}\) and \(\
u_{23}\) & 0 & 0 & - & - \\ Shear modulus \(G\) (GPa) & 31.0 & 11.7 & 10.0 & 10.8 \\ Maximum nominal stress in normal direction, \(t_{n}^{0}\) (MPa) & - & - & 2.6 & 4 \\ Maximum nominal stress in shear direction, \(t_{n}^{0}\) and \(t_{t}^{0}\) (MPa) & - & - & 10 & 30 \\ Normal Mode fracture energy, \(G_{n}^{C}\) (N/mm) & - & - & 0.025 & 0.1 \\ Shear mode Fracture energy, \(G_{s}^{C}\) (N/mm) & - & - & 0.625 & 2.5 \\ B-K criterion material parameter, \(\eta\) & - & - & 1.2 & 1.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Material properties of aggregate particles, mortar particles, interfacial transition zones (ITZs), and mortar internal interface (MII). Figure 7: Statistical characteristic of global stress-strain curve under tension. declines, and the other crack extends through the whole cross-section. Figure 9 shows the strain-stress curves of the specimens with one or two dominant cracks. The strength and energy dissipation of the specimen with two dominant cracks is higher than that of the specimen with one dominant crack  
The specimen with one dominant crack in Figure 8 is selected for the deformation process analysis. In order to observe the whole deformation process, the deformation of the specimen is enlarged geometrically, as shown in Figure 10. The specimen's original deformation in the direction of tension is multiplied by a deformation scale factor \(D_{\mathrm{v}}\) to make the aspect ratio become 2:1 from 1:1. Eight characteristic states are selected to describe the deformation process. The corresponding points of the states are plotted on the stress-strain curve in Figure 9.  
After the specimen is loaded, before state A, the deformation evenly distributes at the aggregates and mortar, as shown in Figure 10a. The obvious plastic deformation appears since state B. Although the global stress hasn't reached the damage initial stress of the ITZ elements, a few ITZ elements begin sustaining damage due to the stress concentration caused by the deformation difference between mortar and aggregates.