# Lattice Fracture Model for Concrete Fracture Revisited: Calibration and Validation
## 4 Validation
### Double-Edge-Notched Beam Shear Test  
The DEN beam shear tests performed by [PERSON] [32] were analyzed to simulate the pure shear zone and produce a shear crack. An experimental sample with span of 400 mm, depth of 37.5 mm, and height of 150 mm was simulated. The experiments were performed under two different boundary conditions (fixed-supported and freely-rotating), which was achieved by mounting diagonal bars to the loading frame in the experiments. The value of crack mouth opening displacement (CMOD) and the crack mouth sliding displacement (CMSD) at the top and the bottom notch on the front of the specimen was used as a feedback signal for displacement in the experiment (see Figure 11a,b). In numerical analyses, the fixed-support was modeled with a constrain in \(x\), \(y\) and \(z\) directions for nodes on the lower surface, and in \(z\)-direction for the nodes on the upper surface while in \(x\) and \(y\), a prescribed load was applied with the ratio of 1:15 (Figure 11a). For freely-rotating supports, nodes in the left support were constrained with respect to \(x\), \(y\) and \(x\) directions, and nodes in the right support were fixed in \(z\)-direction. Furthermore, the prescribed load was applied to the nodes on the upper surface without any other constraints (Figure 11b). Furthermore, the loading area was utilized in the numerical analyses to avoid stress concentrations below the loading points. Figure 10: (**a**) Crack patterns for the smallest notch ratio, (**b**) crack patterns for the model with different notch ratio (red-cracked element) (**c**) load and displacement curve for the models with different notch ratio. Local mechanical properties of lattice elements were calibrated by the experiment with fixed supports; the properties are given in Table 8. For fixed supports, both cracks continued to propagate until they were arrested in the compression zone. Due to the limitation of horizontal support, the double-curved cracks were arrested due to the new configuration of the horizontal forces in the model and stop from propagating further; then a splitting crack arose at the center of the beam and the system eventually failed, in which the final crack pattern in Figure 11(a) is the same as that observed in the experiment (see Figure 11(b)). Figure 11(c) shows the final crack patterns for the sample with free rotating supports. At the beginning, two cracks occurred and grew at both notches and propagated towards the respective opposite specimen sides. Then, one crack stopped propagating and the other one became dominant and continued until reaching the lower surface. The formed crack was a typically curved crack that grew outside the shear zone and ended at the outer side of the middle support (see Figure 11(c)), which is close to the experimental results (see Figure 11(d)). Figure 11: Setup for DEN (double-edge-notch) beam shear test (a) fixed supports, (b) freely rotating. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(E\) (GPa) & \(f_{\epsilon}\) (MPa) & \(f_{t}\) (MPa) & **Cell Size (mm)** & \(\alpha_{\text{N}}\) & **Poisson’s Ratio** & \(\alpha_{\text{M}}\) \\ \hline
20 & 60 & 6 & 2.5 & 1 & 0.2 & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Values of input parameters in the numerical analyses about DEN beam shear test. Figure 12: Cont. A comparison between the numerical and experimental load-displacement curve is plotted in Figure 13 with high and low friction. The agreement between experimental and numerical results is good up to the peak load. For the post-peak branch in the numerical analysis, the load increased with the increased displacement, opposite to the experimental result for fixed boundary condition. This can be explained by the fact that ideal (i.e., high friction) boundary conditions cannot be achieved in the real experiments so that the load-displacement curve shows the decreasing tendency for the softening part.