4.1. Finite Element Model
A finite element method (FEM) was used to simulate the thermo-mechanical test process. It is worth mentioning that the use of a robust FE model to represent the phenomenon could be instrumental to reduce the time available for the experimental tests, and especially the financial costs associated to full-scale experimental trials. A full-size three-dimensional model of the UHTC specimen and the adhesive layer was built using the finite element analysis (FEA) software (ANSYS, Version 16.1), as shown in Figure 6
a. The FE model was assumed to be a combination of two quarter-spheres and a half-cylinder, approximately. The detailed dimensions of the adhesive layer and UHTC specimen used in this study are shown in Table 2
. The CCG sensor was ignored in the FE model due to its small size. The material properties of the UHTC and adhesive are shown in Table 3
, Table 4
and Table 5
. The properties not listed in the tables at other temperatures can be estimated by linear interpolation. Heat transfer analyses were carried out with three dimensional twenty-node DC3D20 elements. Thermal-mechanical analyses were carried out with three dimensional twenty-node C3D20 elements. A mesh convergence study had been performed, and showed that the mesh size was sufficient for the stress within the critical region to converge.
The mechanical and thermal boundary conditions are illustrated in Figure 6
b,c. The displacements of two side faces of the specimen along the vertical direction were fixed to the constraints representing the horizontal vice. The temperature history curve measured during the experimental test (Figure 7
) was applied to the bottom surface of the specimen as the temperature boundary condition. The total heating time was 2570 s. Heating area was a 25 mm diameter circular area, which is the same as that of the laser heater spot. Other areas of the bottom and top surface were specified as the radiation boundary condition with a surface emissivity of 0.85. Natural convection was not considered in this study. Both the UHTC and adhesive layer were assumed be isotropic materials and remained in the linear elastic range during analysis, and they were always perfectly bonded to each other. The experimental simulation was carried out based on the sequential coupled thermo-mechanical analysis with a Newton-Raphson method.
4.2. Results and Discussion
The temperature distributions at the end of the thermal analysis are given in Figure 8
. It can be seen that the maximum temperature of the specimen at the centre reached the value of 695 °C while the minimum temperature was 681.2 °C and localised at the corners of the specimen. The large thermal conductivity value of the UHTCs material made the maximum temperature difference lower than 14 °C.
The temperature field distribution has been used as a thermal load to predict the internal stresses in the structure. The transient stress analysis results showed that the primary stress on the specimen was a compressive one (vertical to the CCG), and was caused by the constraints provided by the horizontal vice. By comparing the thermal stress contours at different times, it is possible to draw the same conclusion about the specimen’s stress along the y-direction: stresses are always negative (i.e., compressive), and the minimum compressive stress in absolute value is located at the bonding region between the UHTC and the adhesive layer, while the maximum compressive stress (magnitude) is located in the area around the boundary of the bonding area. The compressive stress distributions along the y-direction without an adhesive layer at 1900 s and 2000 s are shown in Figure 9
a,b. The compressive stress at the four borders reached 1053 MPa at 1900 s (Figure 9
a) and exceeded the UHTCs’ compressive tensile strength at that specific temperature (446 °C).
According to the FE analysis, the temperature of the border after 2000 s was around 484 °C, and the corresponding compressive strength was 1048 MPa. Two strip-shaped areas near the connection region exceeded their compressive strength (Figure 9
b). If we compare the locations of the failure with the fracture point shown in Figure 9
c, it is apparent that the fracture occurred in the location corresponding to the maximum compressive stress. The FEA results provide evidence to justify the observed experimental results. The micro-cracks present in the structure may have been generated as a result of the constrained thermal expansion after 2000 s, followed by crack propagation until fracture occurred.
The strain on the specimen can also be obtained from the FEA analysis. We selected all the nodes along the line where the CCG was bonded at the UHTCs’ adhesive surface, then calculated the average value of the nodal mechanical strains along the x-direction and obtained the strain versus time and temperature curves. These numerical values were then compared with the strain values measured by the CCG sensor (Figure 10
a). By comparing the finite element simulation results with the experimental data we found that the results of the two methods are in good agreement. Table 6
provides the relative error of the two results, which decreased for increasing temperatures. The relative error was however larger in the low temperature regime (less than 100 °C), due to the strain value being itself very small. Above 200 °C the relative error was less than 15%, and the relative error between the two methods at the final temperature of 695 °C was only 6.71%.
Within the elastic range we can use the strain to calculate the corresponding stresses, according to Hooke’s law. As shown in Figure 10
b, the structural compressive stress curves were calculated over time by using the strain of the FEA and the CCG sensor. At the same time, the compressive strength of the UHTC at the corresponding temperature was also obtained (see Figure 10
b). Both compressive stresses were greater than the material compressive strength in the 1900 s–2000 s range, and the temperature range of the corresponding times were roughly 446 °C–483 °C, respectively. This was in agreement with the experimental results.
The discrepancies between simulations and experimental results were mainly derived from the representation of the boundary conditions and assumptions like the use of adiabatic boundary conditions. In the actual experiment, it was difficult to completely insulate the specimen from the surrounding environment and to keep the applied displacement constraints constants. Nonetheless, the results show that the FE model is capable to represent the physics and provide some robust predictions about the thermo-mechanical coupling occurring inside the structure.