# Lattice Fracture Model for Concrete Fracture Revisited: Calibration and Validation
## 3 Calibration  
In order to simulate the crack initiation and propagation during fracture of cementitious materials, attention should be paid to the determination of input parameters. Recently, research concerning the calibration and validation in LDPM [42; 43] shows great success in lattice modeling. LBM input parameters necessary for fracture simulation are described below, together with their effect on the macroscopic model response. * Elastic modulus, \(E\), and shear modulus, \(G\), govern the response in the elastic regime. In LBM, the Poisson's ratio is determined by the randomness. Based on the 0.2 randomness, the Poisson's ratio of 0.18 is derived in the study. * Tensile strength, \(f_{\mathrm{t}}\), and compressive strength, \(f_{\mathrm{c}}\), of lattice elements govern the macroscopic strength of the model. In this paper, the compressive strength of the lattice element is assumed as 10 times higher than the tensile strength, considering the fact that cementitious materials have a higher resistance to the compressive loading than tensile loading [3]. * Normal force factor, \(\alpha_{\mathrm{N}}\), and bending influence factor, \(\alpha_{\mathrm{M}}\), govern the crack pattern. The macroscopic model response, including the elastic regime and softening part, is also influenced by these two factors. In this paper, the \(\alpha_{\mathrm{N}}\) is set to 1, which is in line with the literature [18; 32; 44]. The \(\alpha_{\mathrm{M}}\) is determined through the calibration procedure that follows. A calibration procedure that relies on a combination of two uniaxial tensile tests is proposed to determine input parameters of lattice elements--tensile strength, compressive strength, elastic modulus, and \(\alpha_{\mathrm{M}}\). The calibration procedure proposed in Figure 2 was used for determining these input parameters. Herein, the uniaxial tensile test performed by [PERSON] and co-authors [45] was used for parameter calibration. In their experiments, uniaxial tensile tests of dog-bone specimens of different sizes with rotating boundaries were performed. A special hinge construction was developed to ensure the free rotations of specimens in two orthogonal directions and the point of rotation is situated exactly at the glue layer between the specimen and the loading platen. To model this loading boundary, the displacement in the vertical direction was fixed and applied as prescribed displacement in the simulation. Detailed information about the sample size can be found in Figure 3a and Table 1. In the experiment, linear variable displacement transducers (LVDTs) with a measuring length of 60 mm were placed on both the front and the back sides of the two specimens specified in Table 1. In the numerical analysis, the measured displacement can be obtained by recording the vertical displacements of node \(A\) and node \(B\) (see Figure 3b). Following the calibration flowchart, the local mechanical property \(E_{f}f_{\mathrm{c}}\),\(f_{\mathrm{t}}\) and \(\alpha_{\mathrm{M}}\) can be calibrated by fitting the load-displacement curve for these two models. The best-fitting results are shown in Table 2. Figure 3b,c show the numerical and experimental crack patterns for the final failure mode with regards to the uniaxial tensile test in dog-bone shape and Figure 3d shows a comparison between the experimental and numerical results. The numerical results agree well with the experimental results before the peak load. After the peak load, the numerical results show more brittle behavior compared to the experimental data. Concerning the crack patterns, a typical failure mode can be found and the main crack occurs in the middle of the specimen, which agrees well with the experimental results [45]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & \(D\) **(mm)** & \(R\) **(mm)** & \(t\) **(mm)** & **Cell Size (mm)** \\ \hline No.1 & 50 & 36.25 & 100 & 2.5 \\ No.2 & 100 & 72.5 & 100 & 2.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Detailed information about the specimens. _Appl. Sci._**2020**, _10_, 4822