## 4 Results and Discussion
Similarly to the classical theory, in SGET, the maximum values of \(t_{\theta}\) and \(t_{r}\) increase with the increase in Mach number (solid lines in Figure 5). Recently, it was proposed that the nominal strength of materials with cracks can be evaluated within SGET based on the regularized solutions for the stresses [28,41]. In SGET, stress and strain are regular at the crack tip, and one can use their finite values to define the criteria of the crack growth initiation and the direction of growth. A description of the known experimental data for the quasi-static and fatigue tests with brittle and quasi-brittle materials within SGET is given in Refs. [27,28,41]. It was shown that the size effect for the nominal strength and transition between short and long crack regimes can be described within SGET [27,28]. Thus, in this study, we evaluate the effects that can be captured by using the regularized asymptotic solution for the steady-state crack in SGET. Note that since the leading terms for stresses (5) vary as \(\sim r^{1/2}\), they vanish exactly at the crack tip, and only the low-order fields with the terms \(\sim r^{0}\) play a role in the fracture initiation if we adopt the viewpoint of regularized analysis within SGET. This is illustrated in Figures 5 and 6. In Figure 6, we show the angular distribution of stress tensor components at different distances from the crack tip. It is seen that all components vary differently in the full range of angles, though from the side of the crack face (\(\theta=\pi\)), they reach the constant values that are not affected by the distance \(\hat{r}\) at which they are evaluated. Note that these nonzero stresses at crack faces are not restricted by the formulation of SGET, since the tractions (17) are defined with additional terms related to the gradients of double stresses (these terms are equilibrated by the stresses so that the traction-free boundary conditions are fulfilled, as can  
Figure 4: Typical dependence of normalized tractions and double tractions on angular coordinate (**a**) and in-plane distribution of normalized hoop traction \(\hat{t}_{\theta}\) ((**b**), crack is shown by thick black line) in SGET asymptotic solution. Figure 5: Angular distribution of normalized hoop traction (**a**) and radial traction (**b**) for different values of Mach number and Poisson’s ratio. be seen in Figures 3 and 4). Thus, the non-zero stresses at the crack faces can be treated as the cohesive forces that naturally arise in the SGET solutions for crack problems [27]. Evaluating the normalized maximum principal stress \(\hat{\sigma}_{I}\) and maximum shear stress \(\hat{\tau}_{max}\) according to their standard definitions with components of stress tensor \(\boldsymbol{\sigma}\) (5), we can check the criterion for the direction of the crack growth within the regularized analysis. In Figure 7a,b, we show the angular distribution of \(\hat{\sigma}_{I}\) and \(\hat{\tau}_{max}\). It is seen that the maximum principal stress always arises at the crack tip from the side of the crack face (\(\theta=\pi\)), while the maximum stress has lower values and has a maximum at about \(\theta=\pi/2\). Therefore, adopting the maximum principal stress criterion for the direction of crack growth, we found that the crack should propagate straightly along the \(x_{1}\)-axis direction with the permanent destruction of cohesive bonds in the fracture process zone in the crack tip at \(\theta=\pi/2\), where the maximum principal stress should reach the critical values of the cohesive stress. Note that a similar observation is given based on the atomistic simulations in Ref. [31], where it was shown that the maximum normal stress arises behind the position of the virtual crack tip inside the cohesive zone. Note that the obtained SGET results do not contain paradoxes of the classical elasticity solution, where the assumed straight propagation of the crack cannot be validated by the criteria of the crack propagation direction [12].