## 4 Calculation
### Finite Element Simulation of the Three-Point Bending Test
The properties (input parameters) of the used material can be found in Table 9.  
The material properties in Table 9 mostly coincide with those in Table 6. However, to better match the experimental and numerical load-displacement curve after cracking, a lower value was considered for the mode 1 area specific fracture energy (compared with the value resulting from Equation (8)). It was found that a value resulting from the CEB-FIP Model Code 1990 [31] equation for the fracture energy gave a better match. The equation used was:  
\[\mathrm{G_{F}}=\mathrm{G_{F0}}\cdot\left(\frac{\mathrm{f_{cm}}}{\mathrm{f_{ cm}}}\right)^{0.7}, \tag{15}\]  
Figure 14: Finite element mesh in the region of the notch. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Property** & **Unit** & **Value** \\ \hline Young’s modulus & MPa & \(3.26\times 10^{4}\) \\ Poisson’s ratio & / & \(2.00\times 10^{-1}\) \\ Bulk modulus & MPa & \(1.81\times 10^{4}\) \\ Shear modulus & MPa & \(1.36\times 10^{4}\) \\ Uniaxial compressive strength & MPa & \(3.71\times 10^{1}\) \\ Uniaxial tensile strength & MPa & \(2.82\times 10^{0}\) \\ Biaxial compressive strength & MPa & \(4.31\times 10^{1}\) \\ Plastic strain at uniaxial Compressive strength & / & \(1.69\times 10^{-3}\) \\ Plastic strain at transition from power law to exponential softening & / & \(3.04\times 10^{-3}\) \\ Relative stress at start of nonlinear hardening & / & \(3.66\times 10^{-1}\) \\ Residual relative stress at transition from power law to exponential softening & / & \(5.11\times 10^{-1}\) \\ Residual compressive relative stress & / & \(2.00\times 10^{-1}\) \\ Mode 1 area specific fracture energy & N/m & \(7.51\times 10^{1}\) \\ Residual tensile relative stress & / & \(2.66\times 10^{-1}\) \\ Dilatancy angle & 0 & 9 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Material properties for the Menetrey-Willam material model. where \(\rm G_{Fo}\) is the base fracture energy dependent on the maximum aggregate size. In our case, the maximum aggregate size was 16 mm, and the base fracture energy was equal to 30 N/m. The parameter \(\rm f_{cmo}\) is equal was 10 MPa. It must be noted that the fracture energy impacts the calculation in Chapter 4.1 only in the sense of the strain at which the fiber action starts to dominate. In contrast, for the Menetrey-Willam material model, the fracture energy is the most important parameter for defining the exponential yield function in tension. Moreover, in addition to the fracture energy, the other important material property is the residual tensile relative stress. The experiments showed post-crack hardening behavior, meaning the residual tensile stress values increased with the strains (after the crack). The limitation of the Menetrey-Willam material model, where the residual tensile stresses can only be considered as constant values, was the deciding factor for considering the residual tensile stress \(\rm f_{Fbs}\) (as the lower value of \(\rm f_{Fbs}\) and \(\rm f_{Ftu}\)) in Table 6 as the residual tensile stress in the Menetrey-Willam material model. In this sense, the finite element analysis results can be considered a lower-bound approximation of the structural behavior. The decision to employ the arc-length method was made to tackle the finite element problem. A total of 361 load steps were calculated as part of this approach. This choice and the extensive sub-step calculations were undertaken to enhance the precision and robustness of the solution. The deformed shape for a midspan displacement of 3.5 mm can be seen in Figure 15.  
It can be seen from the deformed shape in Figure 15 that most of the displacement is the result of the deformation of the finite element layer at the notch. The 5 mm wide layer at the notch acts as a crack band over which the crack is smeared. This is even clearer when looking at the equivalent plastic strain results in Figure 16 (for a midspan displacement of 3.5 mm).