# Wedge-Splitting Test on Carbon-Containing Refracories at High Temperatures
## 4 Results and Discussion
### Inverse Evaluation  
In order to determine other parameters of interest, such as the tensile strength\(f_{t}\) and the total specific fracture energy \(G_{f}\) and to consider the influence of the creep behavior, an inverse evaluation method was applied on a two-dimensional finite element model of the wedge splitting test developed earlier [10]. The commercial software package ABAQUS (6.13, Dassault Sys Simulia Corp, Providence, RI, USA) was used for modeling. The mode I fracture was considered with an exponential strain-softening behavior applied in the ligament area (Figure 8) (Equation (4)). \[\sigma=f_{t}\left(1-\left(\frac{1-e^{-\varphi}\frac{x}{a_{\text{int}}}}{1-e^{- \varphi}}\right)\right) \tag{4}\] \begin{table}
\begin{tabular}{c c c c c} \hline  & **20 \({}^{\circ}\)C (RT)** & **1100 \({}^{\circ}\)C** & **1370 \({}^{\circ}\)C** & **1470 \({}^{\circ}\)C** \\ \hline \(F_{Vmax}\) [N] & 301 & 288 & 439 & 440 \\ \(F_{Hmax}\) [N] & 1720 & 1646 & 2508 & 2510 \\ Specific fracture energy \(G_{f}^{\prime}\) [N/m] & 240 & 125 & 362 & 513 \\ Nominal notch tensile strength \(\sigma_{NT}\) [N/mm\({}^{2}\)] & 2.71 & 3.13 & 4.02 & 4.35 \\ \(G_{f}^{\prime}\)/\(\sigma_{NT}\) [mm] & 0.09 & 0.04 & 0.09 & 0.12 \\ \hline \end{tabular}
\end{table} 
Table 1: Fracture mechanical parameters determined from tests performed at room temperature (RT), 1100 \({}^{\circ}\)C, 1370 \({}^{\circ}\)C, and 1470 \({}^{\circ}\)C.  
Figure 7: Sample after a wedge-splitting test at 1470 \({}^{\circ}\)C.  
In this equation \(f_{t}\) is the tensile strength of the material, the parameter \(x_{ult}\) is the ultimate crack opening and \(\varphi\) is a non-dimensional parameter that defines the rate of damage evolution [11]. Cohesive elements were applied to the interfaces of the two half specimens to simulate the mode I fracture [11]. Through an adaptive nonlinear least-squares algorithm, the fracture parameters are inversely calculated by fitting the results of finite element simulations to the results of laboratory wedge splitting tests. The following Figures 9-11 show the load-displacement diagrams and the determined strain-softening behavior for experiments between 1100 \({}^{\circ}\)C and 1470 \({}^{\circ}\)C. The green curves in the left diagrams (a) are the measured ones from the wedge splitting tests. The blue curves (a) are the curves simulated by finite element simulations. The red curves in the diagrams show the results with consideration of the creep behavior and are explained in the following chapter. The (b) figures show the strain-softening behavior determined via inverse evaluation from parameters \(f_{t}\), \(x_{ult}\), and \(\varphi\). Results from (b) correspond to (a). Figure 8: Finite element model geometry and mesh. Figure 9: Load-displacement diagram (**a**) and strain-softening behavior (**b**) at 1470 \({}^{\circ}\)C.  
The inversely calculated curve (a) shows only a slight deviation from the tested one. The minimization procedure is automatically terminated when a minimum of residuals is reached [10]. The tensile strength and \(x_{ult}\) show a decrease with decreasing temperature when the creep is not considered. Table 2 shows the inversely evaluated parameters and brittleness indicators. With increasing temperature, the specific fracture energy \(G_{f}\) increases. The tensile strength follows the same trend. At 1470 \({}^{\circ}\)C and 1370 \({}^{\circ}\)C, the maximum horizontal forces are similar but the specific fracture energy is lower at 1370 \({}^{\circ}\)C because of lower horizontal displacement. Additionally, the ratio between \(\sigma_{\text{NT}}\) and \(f_{\text{t}}\) was calculated. The difference between \(\sigma_{\text{NT}}\) and \(f_{\text{t}}\) is expected. Reasons are the action of the notch on the one hand and on the other hand the dependence on the brittleness number [12]. As well as \(G_{f}^{\prime}\)/\(\sigma_{\text{NT}}\), the ratio between the real specific fracture energy \(G_{f}\) and the tensile strength \(f_{\text{t}}\) is an indicator of the brittleness.