## 5 Conclusions  
In this paper, we derived an asymptotic solution for the problem of steady-state propagation of Mode-I cracks in the quasi-brittle materials governed by SGET. A simplified variant of SGET and the particular case of the same values of the elastic and microinertia length scale parameters were considered. The form of asymptotic solution was derived based on the Lame potential approach, and its approximate variant for the moderate values of Mach number were used for the analysis of the deformations and the stress state around the moving crack tip. It was shown that the typical smooth crack opening profile with \"cohesive zone\" effects are predicted by the dynamic solution of SGET. Moreover, from the given assessments, it follows the possibility for the description of the straight direction of the Mode-I crack growth without paradoxes of classical theory. The increases in the dynamic fracture toughness of the quasi-brittle material that were observed in the experiments can be also described within the regularized analysis and maximum principal stress criterion of SGET.  
The presented results are given for the not very high Mach numbers of crack propagation and particular relations between the length scale parameters of SGET. Specifically, the main limitation of the presented results is the assumption of the same values of the length scale parameters for the strain (\(l\)) and for the inertia (\(g\)) gradient effects and the limited number of terms used in the definition of the angular distribution of asymptotic solution (21). Therefore, in future work, the presented analysis should be extended for the subsonic and intersonic regimes. The more general constitutive equations of SGET should be also considered and compared to the full-field numerical simulations.  
This research was funded by Russian Science Foundation grant number 22-79-10228 issued to Moscow Aviation Institute.  
Not applicable.  
No data were generated.  
The author declares no conflict of interest.  
The following abbreviations and symbols are used in this manuscript:  
\begin{tabular}{l l} SGET & strain gradient elasticity theory \\ \(\mathbf{e}_{i}\) & unit vectors of coordinate system \\ \(\mathbf{I}\) & second-order identity tensor \\ \(\mathbf{m}\) & double traction vector \\ \end{tabular}  
where \(Q_{i}=\sqrt{\cos^{2}\theta+\alpha_{i}^{2}\sin^{2}\theta}\) and \(\alpha_{i}=\sqrt{1-m_{i}^{2}}\).  
Note that for the quasi-static limit \(m_{i}=0\), we have \(\alpha_{i}=1\), \(Q_{i}=1\), \(\theta_{i}=\theta\), \(r_{i}=r\) (\(i=1,2\)), and the displacement solution (25) can be reduced to the following form:  
\[\begin{split}\tilde{u}_{r}&=\bar{r}\,S_{0}+\bar{r} \,(S_{1}+S_{2})\cos 2\theta+\bar{r}^{3/2}\sum_{i=1}^{2}\Bigl{(}a_{i1}\kappa_{i1} \cos\tfrac{\theta}{2}+c_{i11}\kappa_{i2}\cos\tfrac{3\theta}{2}+(a_{i5}\kappa_ {i3}+b_{i5}\kappa_{i4})\cos\tfrac{5\theta}{2}\Bigr{)}\\ \tilde{u}_{\theta}&=-\bar{r}\,(S_{1}+S_{2})\sin 2 \theta+\bar{r}^{3/2}\sum_{i=1}^{2}\Bigl{(}a_{i1}\kappa_{i5}\sin\tfrac{ \theta}{2}+c_{i11}\kappa_{i6}\sin\tfrac{3\theta}{2}+(a_{i5}\kappa_{i7}+b_{i5 }\kappa_{i8})\sin\tfrac{5\theta}{2}\Bigr{)}\end{split} \tag{26}\]  
where \(\kappa_{ij}\) become the constants, whose values follow from (25).  
Then, by using the appropriate choice of constants \(a_{11}\), \(a_{21}\), \(c_{111}\), \(c_{211}\), \(a_{15}\), \(a_{25}\), \(b_{15}\), \(b_{25}\) we can reduce (26) to the known static asymptotic solution of SGET for a Mode-I crack that contains only two amplitude factors [18; 19]. The choice of the constants is not unique. Moreover, it should be noted that in these derivations, constants \(c_{111}\), \(c_{211}\) can be treated as independent since they are proportional to \(b_{11}\), \(b_{21}\) (22) which do not arise in the given representation for the displacement field.