## 4 Results and Discussion  
For the examples of numerical calculations, let us clarify the physical meaning and possible values of the constants \(S_{0}\), \(S_{1}\), \(S_{2}\), \(a_{15}\), \(a_{25}\), \(b_{15}\) that remain in solutions (23)-(25). From (21), it can be seen that the constants \(a_{15}\), \(b_{15}\) are related to the scalar Lame potential \(\phi_{1}\), and therefore, they define the dilatational strain field around the tip of the crack. Conversely, constant \(a_{25}\) is related to the vector potential \(\mathbf{\phi}_{2}\) and defines the rotations in the vicinity of a crack. Similar observations for two constants that persist in the asymptotic solution for the stationary cracks within SGET are given in Ref. [18]. Note that constant \(b_{15}\) stays before the modified Bessel function in Equation (21), and mostly defines the high-order effects in the distribution of dilatation (interconnected with terms \(\sim r^{7/2}\) and higher). The influence of \(b_{15}\) on the crack tip field is negligible, and in further examples of calculation, we will use \(b_{15}=0\), so only two independent amplitudes of the leading terms \(a_{15}\) and \(a_{25}\) will be considered. The amplitudes of the lower-order crack-tip fields \(S_{i}\) in the case of the static problem (\(m_{i}=0\)) have the physical meaning of the constant normal strains that can be represented in the Cartesian coordinates as \(\varepsilon_{11}=S_{0}+S_{1}+S_{2}\) and \(\varepsilon_{22}=S_{0}-S_{1}-S_{2}\) (see [18]). These components of the asymptotic solution do not make a contribution to the energy release rate and to the crack tip open displacement [18; 19]. For the nonzero speed of crack propagation constants, \(S_{i}\) corresponds to the distorted lower-order strain fields around the growing crack, whose distribution is defined by functions \(\beta_{ij}\) (Appendix A). For the examples of calculations, we will define the lower-order crack tip fields, assuming that they correspond to the uniform tension along the \(x_{2}\)-axis in the quasi-static limit:  
\[\varepsilon_{22}=-\varepsilon_{11}/\
u=\varepsilon_{0}\]  
Then, the values of amplitudes can be chosen as  
\[S_{0}=\varepsilon_{0}(1-\
u)/2,\quad S_{1}=S_{2}=-\varepsilon_{0}(1+\
u)/4\]  
which can be provided by using the following definitions for the constants of the Lame potentials (21):  
\[b_{10}=\varepsilon_{0}(1-\
u),\quad a_{12}=a_{22}=-\varepsilon_{0}(1+\
u)/8, \quad b_{12}=b_{22}=0\]  
and the last constant \(b_{20}\) is used to provide the fulfilment of traction-free boundary conditions. Note that the case of the loaded crack faces can be also considered, and the corresponding boundary conditions can be satisfied by the appropriate choice of constants \(b_{12}\), \(b_{22}\), \(b_{20}\), though this case is out of consideration in this study. Thus, for the following analysis, we can retain only four nondimensional parameters: Poisson ratio \(\
u\), Mach number \(m_{2}\) (or \(m_{1}\)), and the ratios between the amplitudes of local dilatation and rotation fields to the amplitude of uniform tension denoted by \(k_{\theta}=-a_{15}/\varepsilon_{0}\) and \(k_{\omega}=-a_{25}/\varepsilon_{0}\), respectively. In definitions for \(k_{\theta}\) and \(k_{\omega}\), we use the minus sign due to the condition of the open mode-I crack that requires \(u_{\theta}(r,\pi)<0\) when \(\varepsilon_{0}>0\). In the following, unless otherwise stated, we use the values of Poisson's ratio \(\
u=0.3\), Mach number \(m_{2}=0.1\), and amplitude ratios \(k_{\theta}=0.1\) and \(k_{\omega}=0.1\) with the prescribed uniform tension amplitude \(\varepsilon_{0}=1\) (some of these quantities are varied, and this will be denoted on the plots). All spatial dimensions, coordinates, and displacements are normalized with respect to the length scale parameter \(l\). Stresses and surface tractions are normalized with respect to the Young's modulus of material multiplied with the prescribed amplitude of tensile strain \(\varepsilon_{0}\).