## 2 Materials and Methods
Let us calculate the next example of how many time steps are necessary to solve a bus body element task with the next boundary conditions:  
* [PERSON]'s modulus \(E=2\times 10^{11}\) Pa;
* Density \(\rho=\)7850 kg/m\({}^{3}\);
* Characteristic size of the element \(h=50\times 10^{-2}\) m;
* Required task analysis time is \(T=0.2\) s.  
\[c=\sqrt{\frac{E}{\rho}}=\sqrt{\frac{2\times 10^{11}}{7850}}=5048\frac{\text{m}}{ \text{s}}, \tag{3}\]  
\[\Delta t=f\cdot\frac{h}{c}=0.9\cdot\frac{50\cdot 10^{-2}}{5048}=8.91\times 10^{-6} \text{ s}, \tag{4}\]  
\[n=\frac{T}{\Delta t}=\frac{0.2}{8.91\times 10^{-6}}=2243. \tag{5}\]  
where  
\(n\)--time step number required for this explicit analysis of the element with a characteristic size of \(50\cdot 10^{-2}\) m.  
Therefore, a time step scale factor (TSSF) of 0.9 was chosen in the Ansys Explicit Dynamics module to ensure that each step is smaller than the critical time step [20]. Equation (9) could also be presented in an extended formulation:\[c=\sqrt{\frac{E(1-\
u)}{\rho(1+\
u)(1-2\
u)}}, \tag{6}\]  
where: \(\
u\)--Poisson's ratio. To reduce the computational time required to perform extensive explicit analyses of the bus body model, mass scaling is a simulation technique used. Before modeling, a series of tests are conducted on the material to determine the yield point in order to determine the accurate stress values at which plastic deformation begins. The strain gauge serves as a measuring device for determining elastic and plastic deformation along the measuring length. Under conditions of significant strain, a high strain rate, and possibly high temperatures, the Johnson-Cook model can represent the relationship between stress and strain in a metallic material. The stress flow can be expressed using the subsequent equation [20]:  
\[\sigma_{y}\big{(}\epsilon_{p},\,\dot{\epsilon}_{p},T\big{)}=\Big{[}A+B\big{(} \epsilon_{p}\big{)}^{n}\Big{]}[1+C\ln\Big{(}\dot{\epsilon_{p}}^{*}\Big{)}] \big{[}1-(T^{*})^{m}\big{]}, \tag{7}\]  
where  
\(\epsilon_{p}\)--the equivalent plastic strain, mm/mm;  
\(\dot{\epsilon}_{p}\)--the normalized equivalent plastic strain rate (speed of deformation),  
\(s^{-1}\); A, B, C--the constants of the material,  
\(n\)--the exponential of the mechanical hardening;  
\(m\)--the exponential of the thermal softening. The homologous temperature \(T^{*}\) and normalized strain rate could be defined as follows [20]:  
\[\dot{\epsilon_{p}}^{*}=\frac{\dot{\epsilon_{p}}}{\dot{\epsilon_{p0}}};\,T^{*}= \frac{(T-T_{0})}{(T_{m}-T_{0})}, \tag{8}\]  
where:  
0--represents initial values (room temperature);  
\(\dot{\epsilon_{p}}^{*}\)--effective total strain rate (is normalized by a quasi-static threshold rate)--non-dimensional by the reference strain rate;  
\(\dot{\epsilon_{p0}}^{*}\)--reference strain rate;  
\(T_{m}\)--fusion temperature, \({}^{\circ}\)C;  
\(T\)--current specimen temperature, \({}^{\circ}\)C.  
According to [PERSON] and [PERSON]'s theory, fracture strain is typically influenced by the ratio of triaxial stress, strain rate, and temperature. The fracture model can be expressed in the following manner [21]:  
\[\epsilon_{f}=[D_{1}+D_{2}exp\Big{(}D_{3}\Big{(}\frac{\sigma_{m}}{\sigma_{eq}} \Big{)}\Big{)}\Big{[}1+D_{4}\ln\Big{(}\dot{\epsilon}_{p}^{*}\Big{)}\Big{]}[1 +D_{5}T^{*}], \tag{9}\]  
where:  
\(D_{1}\) to \(D_{5}\) are the constants of failure model;  
\(\sigma_{m}\)--mean stress;  
\(\sigma_{eq}\)--equivalent stress. The material's strength decreases during deformation due to the occurrence of damage, and the mathematical relationship representing this damage, also known as the \"constitutive relationship,\" can be expressed as follows:  
\[\Delta t\leq f\cdot\left[\frac{h}{c}\right]_{min}, \tag{1}\]0  
where\(\sigma_{D}\)--represents the stress in the damaged state, while stress triaxiality and equivalent stress can be derived from the material in the undamaged state.