## 5 Results  
### Multi-Objective Optimization  
#### 5.1.1 Multi-Objective Optimization Model  
Due to the significant role played by energy-absorbing structures in train collisions, the optimal design of structural crashworthiness has been broadly applied. Commonly used crashworthiness indicators of cutting-type energy-absorbing structures include mean crash force, energy absorption, initial peak force, etc. Tool failure also has a non-negligible impact on the reliability of cutting energy-absorbing structures. In order to consider the reliability and crashworthiness of cutting-type energy-absorbing structures simultaneously, this paper adopts steady-state temperature (\(STT\)) and mean cutting force (\(MCF\)) as optimization objectives, which are defined as follows:  
\[MCF=\frac{\int_{0}^{d}F(s)ds}{d} \tag{7}\]  
where \(F(s)\) is the instantaneous cutting force at the distance \(s\), and \(d\) is the total displacement.  
The \(STT\) is taken as the steady-state value of the maximum cutting temperature of the tool. It should not exceed 770 K, which can be considered a reasonable value for ensuring that the cutting temperature does not exceed the heat-resistant temperature within a tube length of 180 mm in the tool-tube model.  
Considerations for determining the range of the three design variables, cutting depth, width, and tool rake angle, are as follows. The cutting depth is taken from its typical range in machining (approximately 1 to 5 mm). Considering that energy-absorbing structures are usually used on a one-off basis, increasing the depth appropriately to enhance energy absorption by sacrificing its partial durability is acceptable. The depth of cut is therefore set at 2 to 4 mm. Variations in cutting width do not significantly affect the tool's performance; therefore, only the match to the current structure size usually needs to be considered. For example, in the case of a typical round cross-section energy-absorbing tube, an excessive cutting width at a defined depth of cut will result in thin thickness at both ends, to the detriment of a stable cutting process. In addition, it is necessary to ensure that there should be sufficient spacing between the individual tools, etc. Therefore, the cutting widths are set at 15 to 26.5 mm. The setting of the rake angle takes into account the tool and tube material and the cutting conditions. An increased rake angle means sharper tools and lower cutting temperatures. However, an excessive angle reduces the rigidity and strength of the tool, and bad heat dissipates, resulting in severe wear and breakage. In addition, a rake angle less than zero (negative rake angle) increases cutting force fluctuations and is detrimental to the stability of the energy absorption process. Therefore, in combination with the typical tool rake angle (approximately \(-10\sim 30^{\circ}\)) in machining and the careful consideration of the collision scenario and performance requirements of the energy-absorbing structure, the tool rake angle is set to 0-30\({}^{\circ}\). Therefore, the mathematical equation of the optimization problem in the multi-objective framework is  
\[\left\{\begin{array}{l}\min\left[-MCF(d,\,w,\,\alpha),\,\,STT(d,\,w,\,\alpha )\right]\\ s.t.\,\,STT\leq 770\,\,K\\ 2\leq d\leq 4\,\,mm\\ 15\leq w\leq 26.5\,\,\mathrm{mm}\\ 0\leq\alpha\leq 30^{\circ}\end{array}\right. \tag{8}\]