## 4 Calculation
### Finite Element Simulation of the Three-Point Bending Test
Figure 16: Deformed shape with (unaveraged) equivalent plastic strain contour (strain is unitless) for the final load step with midspan displacement of 3.5 mm. Figure 15: The deformed shape and visible mesh deformation with vertical displacements contour (displacements in mm) for the final load step with midspan displacement of 3.5 mm. The highest equivalent plastic strain was not calculated at the midspan (symmetry plane) but at the location of the corner edge of the notch, which is a geometrical stress concentration point. However, due to the relatively small mesh size of 5 mm (relative to the prism size), this discrepancy did not affect the results. The finite element results in terms of load-displacement relations are given in Figure 17.  
The results in Figure 17 show post-crack softening behavior, but the slope of the decreasing part of the load-displacement curve for larger displacements (greater than 0.5 mm) is relatively small. The vertical displacements and CMOD and CTOD values were read at points on the symmetry plane, as depicted in Figure 18.  
Table 10 shows the finite element analysis results of the CMOD, CTOD, vertical displacement (\(\delta\)), and loads (F) for the corresponding load steps. The CMOD and CTOD values were calculated by multiplying the horizontal displacement results of the marked points in Figure 18 by two (due to symmetry). The load value was calculated by multiplying the vertical support force by four (as one-quarter of the prism was modeled). The first row in Table 10 corresponds to the last load step with no equivalent plastic strain. The load value is relatively low and results from the geometric stress concentration \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**CMOD** & **CTOD** & **F** & \(\delta\) & **CMOD/CTOD** \\
**[mm]** & **[mm]** & **[kN]** & **[mm]** & **[/]** \\ \hline
0.0032 & 0.0007 & 3.01 & 0.01 & 4.79 \\
0.0417 & 0.0271 & 14.09 & 0.07 & 1.54 \\
0.5036 & 0.4113 & 7.15 & 0.45 & 1.22 \\
2.4978 & 2.0694 & 6.98 & 2.12 & 1.21 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Finite element analysis results for CMOD, CTOD, vertical displacement (\(\delta\)), and load (F) (the row values correspond to the same load step). Figure 17: Finite element analysis load-displacement relation. Figure 18: Finite element analysis load-displacement relation. region at the notch, which means that different geometric representations of the notch and corresponding meshes would impact the load value of the crack formation. The second row in Table 10 corresponds to the load step with the maximum load. The comparison of the CMOD and CTOD values from the finite element analysis with the CMOD and CTOD values from the calculation based on the moment-curvature relations in Table 7 shows large differences for the loads (or load steps) before reaching the maximum load. An additional reason (to the previously mentioned stress concentration in the finite element model) for this discrepancy is the difference in the stress-strain relations in the tensile stress region up to the tensile strength. While linear elastic behavior up to the tensile strength was considered in the finite element analysis, the calculation based on the moment-curvature relations considered a more realistic stress-strain relation in the same region (see Figure 7b). However, the CMOD/CTOD ratio (see Table 10), again, as expected, for larger CMOD values approaches the EN14651 standard [19] value for this ratio (1.2). The normal stress (in the global X-direction, as per Figure 13) distributions on a path (Figure 19 (left)) starting at the notched edge, for the same load steps as the row results in Table 10, are given in Figure 19 (right). As equivalent plastic strains first occur at point 1 in Figure 19 (left), it was chosen as the starting point of the stress path. The normal stress results are averaged nodal results.