## 2 Analysis Method
### Numerical Analysis Method
The accelerations were integrated through time using the central difference rule, and this change in velocity was added to the velocity from the middle of the previous increment to determine the velocities at the middle of the current increment:  
Figure 3:5  
where \(\dot{u}\) is the nodal velocity, and \(\Delta t\) is the time increment; \(i-\frac{1}{2}\) and \(i+\frac{1}{2}\) refer to the mid-increment values. The velocities were integrated through time and were added to the nodal displacements \(u\) at the beginning of the increment to determine the displacements at the end of the increment:  
Figure 3:6  
To estimate the stable time increment size, an approximation to the stability limit is often written as the shortest transit time of a dilatational wave across any of the elements in the mesh:  
Figure 3:7  
where \(L_{min}\) is the smallest element dimension in the mesh and \(c_{d}\) is the dilatational wave speed in terms of \(\lambda_{0}\) and \(\mu_{0}\), defined below. Figure 3:8  
In isotropic material, the effective Lame parameters can be defined as follows:  
Figure 3:9  
\[\rho J=\rho_{0}, \tag{1}\]0where \(\hat{\lambda}\) and \(\hat{\mu}\) are referred to as [PERSON]'s first parameter and [PERSON]'s second parameter, respectively. \(E\) is the [PERSON]'s modulus, and \(v\) is [PERSON]'s ratio. The element calculations included determining the element strains and applying material constitutive relationships (element stiffness) to determine the element stresses, and, consequently, the internal forces. Then, the Cauchy stress tensor can be determined as follows:  
\[\sigma=\left[ \begin{array}{ccc}\sigma_{xx}&\sigma_{xy}&\sigma_{xz}\\ \sigma_{yx}&\sigma_{yy}&\sigma_{yz}\\ \sigma_{zx}&\sigma_{zy}&\sigma_{zz}\end{array} \right] \tag{19}\]  
\[\sigma_{xy}=\sigma_{yx},\sigma_{yz}=\sigma_{zy},\sigma_{zx}=\sigma_{xz}. \tag{20}\]  
In material science and engineering, the six components of stress can describe the state of stress at a point in the general 3D domain. \[\sigma_{v}=\sqrt{\frac{1}{2}\left[\left(\sigma_{xx}-\sigma_{yy}\right)^{2}+ \left(\sigma_{yy}-\sigma_{zz}\right)^{2}+\left(\sigma_{yy}-\sigma_{zz}\right) ^{2}+6\left(\sigma_{xy}{}^{2}+\sigma_{yz}{}^{2}+\sigma_{zx}{}^{2}\right) \right]}, \tag{21}\]  
where \(\sigma\) is the Cauchy stress tensor, \(\sigma_{v}\) is the von Mises stress, and it can also be called the von Mises yield criterion or equivalent tensile stress. \(\sigma_{xx}\), \(\sigma_{yy}\), and \(\sigma_{zz}\) are the normal stress, while \(\sigma_{xy}\), \(\sigma_{yz}\), and \(\sigma_{zx}\) are the shear stress. In physics, the strain energy is the energy stored by a system undergoing deformation. For a linear elastic material, the strain energy is  
\[U=\int\,U_{0}\mathrm{d}V, \tag{22}\]  
where \(U_{0}\) is the strain energy density, and \(V\) is the volume. For a 3D problem, the strain energy density is given by  
\[U_{0}=\frac{1}{2}\Big{(}\sigma_{xx}\varepsilon_{xx}+\sigma_{yy}\varepsilon_{yy }+\sigma_{zz}\varepsilon_{zz}+\sigma_{xy}\varepsilon_{xy}+\sigma_{yz} \varepsilon_{yz}+\sigma_{zx}\varepsilon_{zx}\Big{)}, \tag{23}\]  
\[\varepsilon_{xy}=\varepsilon_{yx},\varepsilon_{yz}=\varepsilon_{zy}, \varepsilon_{zx}=\varepsilon_{xz}, \tag{24}\]  
where \(\varepsilon_{xx}\), \(\varepsilon_{yy}\), and \(\varepsilon_{zz}\) are the normal strain, while \(\varepsilon_{xy}\), \(\varepsilon_{yz}\), and \(\varepsilon_{zx}\) are the shear strain.