## 4 Results and Discussion
These normalized quantities will be denoted with a hat symbol (\(\hat{t}_{\theta}\), etc.). The stresses and surface tractions are calculated by using given solution (21) and (23) and relations (14)-(17). All evaluations are performed by using the initial definition for the solution with Bessel functions (21) to provide the correct values of their derivatives and taking into account definitions for the scaled coordinates (20). Asymptotic solutions for the crack tip fields are found for the resulting relations using the series expansion for the Bessel functions around \(r=0\), and retaining only the leading terms. Illustrations for the displacement field that is realized in the vicinity of a crack are presented in Figure 2. Here, we show the deformations of a small circle with radius \(r=0.1l\) around the tip of a crack in SGET (left) and in classical [12] (right) solutions. It can be seen that in contrast to the classical solution with \(r^{1/2}\) asymptotics, the smooth closure profile is realized in SGET, which can be related to the presence of the fracture process zone in quasi-brittle materials [20]. The dependence of crack tip open displacement on Poisson's ratio and the Mach number are similar in both solutions: the opening amplitude becomes higher for the lower values of Poisson's ratio and for the high Mach number (though in the classical solution, we have to use higher values of \(m_{2}\) to obtain a more visible effect). At the same time, the deformed state ahead of the crack tip varies differently in the classical and SGET solutions. In the classical solution, a tension along the \(x_{1}\)-axis always arises in this zone. In SGET, there arises a compression along the \(x_{1}\)-axis ahead of the crack tip for the low values of Poisson's ratio and for all considered values of the Mach number. The influence of the amplitude ratios \(k_{\theta}\), \(k_{\omega}\) on the displacement field around the crack tip is shown in Figure 3. It can be seen that the increase in dilatational amplitude \(k_{\theta}\) results in a higher crack opening and a more intensive compression ahead of the crack tip (Figure 3a). The influence of the rotational amplitude is much less pronounced, and can be seen only for relatively large values of \(k_{\omega}\), which results in the higher curvature of the deformed crack face and additional distortion along the \(x_{1}\) axis of the plotted circles (Figure 3b). The negative values of \(k_{\omega}\) lead to the double curvature of the crack face (red curve in Figure 3b). Note that in the vicinity of a crack tip, the change in \(k_{\omega}\) does not influence the crack tip open displacement (all curves coincide at the crack tip in Figure 3b), so the rotational effects in the SGET asymptotic solution could be negligible for Mode-I loading. An illustration of the fulfilment of the traction-free boundary conditions by the developed asymptotic solution is presented in Figure 4. In Figure 4a, we show the typical angular distribution of normalized tractions (17) and double tractions (18) that take zero values at the crack face \(\theta=\pi\). The contour plots for the in-plane distribution of normalized hoop traction \(t_{\theta}\) are presented in Figure 4b. This traction has a singular behavior \(\sim r^{-3/2}\) and growth rapidly close to the crack tip. Note that the angular distribution and dependence on the Mach number of the tractions in the SGET solution are very similar to those in  
Figure 3: Influence of dilatational (**a**) and rotational (**b**) amplitudes on the deformed state around the tip of crack. Figure 2: Influence of Poisson’s ratio (**a**) and Mach number (**b**) of the deformations of small circles around the crack tip in SGET (left) and classical (right) asymptotic steady-state solutions. classical solutions (Figure 5), though their definitions are much more complicated (17). Hoop traction \(t_{\theta}\) has a maximum at about \(\theta=60\) deg. (Figure 5a), where the radial traction \(t_{r}\) changes the sign (Figure 5b).