## 3 Asymptotic Solution for Growing Crack
Based on the analysis of the behavior of functions \(\phi_{i}^{(n)}\) around \(r=0\), we found that the following terms should remain for the asymptotic solution:  
\[\phi_{i}=\phi_{i}^{(1/2)}+\phi_{i}^{(5/2)}+\phi_{i}^{(0)}+\phi_{i}^{(2)}, \qquad(i=1,2) \tag{23}\]  
whose regularity around \(r=0\) can be provided by the following choice of constants:  
\[b_{i1}=-\sqrt{\frac{\pi}{2}}a_{i1} \tag{24}\]  
where for the shortness, we adopt the following notation for the fractional indexes: \"n\" instead of \"n/2\" for odd values of \(n\) (i.e., \(a_{i1}\) means \(a_{i(1/2)}\), \(b_{i1}\) means \(b_{i(1/2)}\), etc.). This notation is also used in the following derivations. Note that apart from the mentioned functions in (23), all functions \(\phi_{i}^{(n/2)}\) with the negative index \(n/2<0\) also contain the terms with behavior \(\sim r^{3/2}\). However, all these functions allow for regularity for the displacement field around \(r=0\) only if both constants are zero: \(a_{in}=b_{in}=0\). Therefore, these functions are excluded from the solution. The high-order terms with \(n>5/2\) are also out of consideration within the asymptotic solution for the crack tip fields only. Separately, within SGET, one should consider the case that corresponds to the particular solution with vanishing strain gradients around the crack tip. Such a solution can be obtained by using the Lame potentials \(\phi_{1}^{(0)}\), \(\phi_{i}^{(2)}\) that produce the displacement field with asymptotic behavior \(\sim r^{1}\). For this case (\(n=0\) and \(n=2\)), the energy integrability condition can be relaxed, and functions \(\phi_{1}^{(0)}\), \(\phi_{i}^{(2)}\) can be included in (23). These lower-order terms can be related to the so-called generalized T-stress field that also arises in the quasi-static SGET solutions [18; 19; 20]. Substituting definitions for \(\phi_{i}^{(n)}\) (21) into (23) and taking into account (24), one can obtain the final form of the Lame potentials. Using these potentials in (14) and evaluating the limit at \(r\to 0\), one can find the SGET asymptotic solution for the displacement field around the tip of the propagated crack in the following form:  
\[ \begin{split}\tilde{u}_{r}&=r\,S_{0}\beta_{10}+r \sum_{i=1}^{2}S_{i}\beta_{i1}\cos 2\theta_{i}+m_{2}^{2}F_{r}(r,\theta_{i})\\ &+\tilde{r}^{3/2}\sum_{i=1}^{2}\Bigl{(}a_{i1}\kappa_{i1}\cos \frac{\theta_{i}}{2}+c_{i11}\kappa_{i2}\cos\frac{3\theta_{i}}{2}+(a_{i5}\kappa _{i3}+b_{i5}\kappa_{i4})\cos\frac{5\theta_{i}}{2}\Bigr{)}\\ \tilde{u}_{\theta}&=-r\sum_{i=1}^{2}S_{i}\beta_{i2} \sin 2\theta_{i}+m_{1}^{2}F_{\theta}(r,\theta_{i})\\ &+\tilde{r}^{3/2}\sum_{i=1}^{2}\Bigl{(}a_{i1}\kappa_{i5}\sin \frac{\theta_{i}}{2}+c_{i11}\kappa_{i6}\sin\frac{3\theta_{i}}{2}+(a_{i5}\kappa _{i7}+b_{i5}\kappa_{i8})\sin\frac{5\theta_{i}}{2}\Bigr{)}\end{split}  \tag{25}\]where \(\bar{r}=r/l\), \(\bar{a}_{r}=u_{r}/l\), \(\bar{a}_{\theta}=u_{\theta}/l\) are the normalized radial distance and components of the displacement field, respectively; \(c_{in1}\) are defined by (22); \(\kappa_{ij}\), \(\beta_{ij}\) are the functions of the angular coordinate, whose explicit form is given in Appendix A; the amplitudes for the lower-order crack tip fields are defined by \(S_{0}=b_{10}/2\), \(S_{1}=2a_{12}+b_{12}/4+2c_{101}\), \(S_{2}=2a_{22}+b_{22}/4-2c_{201}\); and functions \(F_{r}\), \(F_{\theta}\) are introduced to define the terms that arise only in the steady-state problem and that vanish in the quasi-static limit \(m_{i}=0\) (see Appendix A). Solution (25) contains six unknown constants that define the leading terms: \(a_{11}\), \(a_{15}\), \(b_{15}\), \(a_{21}\), \(a_{25}\), \(b_{25}\). Three of these constants (e.g., \(a_{11}\), \(a_{21}\), \(b_{25}\)) can be determined from the boundary conditions (17) and (18) to provide their fulfilment in the asymptotic sense. The representation for these constants can be easily found in analytical form by using a symbolic algebra system, and they are not presented here for clarity.