## 1 Introduction  
The application of high-grade continuum theories for the simulations of materials and structures allows us to provide a refined analysis of their mechanical and physical properties [1, 2]. The influence of the size effects is one important feature that arises in nanostructured materials and can be effectively captured by generalized continuum theories [3, 4]. In this study, we consider the variant of generalized theory that is known as the strain gradient elasticity theory (SGET) [5], which was efficiently applied for the analysis of the size effects in quasi-brittle fractures [6, 7] for the nonsingular description of dislocation cores [8, 9], for the wave dynamics analysis in granular materials and metamaterials [10, 11], etc. The subject of this study is the steady-state problem of a Mode-I crack propagated in isotropic quasi-brittle material with an internal structure. The classical asymptotic solution for this problem that does not contain the microstructural characteristic length parameters is well known [12]. Within gradient theories, this problem has been considered using numerical simulations [13]. The Wiener-Hopf technique for full-field analysis was applied within the couple stress theory for mode-II [14, 15] and the mode-III [16, 17] cracks. Our goal is to show that the strain gradient elasticity (in its dynamic formulation, here called second gradient elastodynamics) allows us to predict the regularized stress state and smooth opening profile for growing Mode-I cracks. For quasi-static problems, such effects in simplified SGET (also known as dipolar gradient elasticity) with a single length scale parameter have been shown based on the [PERSON] asymptotic method in Refs. [18; 19]. A more general solution within [PERSON]'s Form II of SGET with five length scale parameters is presented in Ref. [20]. A discussion on the correct methods of the derivation of asymptotic solutions within gradient theories accounting for traction-free boundary conditions is presented in Ref. [21]. In this work, we consider the simplified SGET and develop an asymptotic solution by using the Lame potential method [22; 23]. This method is usually used for moving crack problems in classical elasticity [15], as well as in gradient theories [14]. The important feature of this method (in comparison with the [PERSON] asymptotic technique used for static problems [18; 19]) is that it allows us to satisfy the motion equations exactly, so that only boundary conditions and additional bounded energy conditions should be involved in further analysis. To the best of the author's knowledge, the steady-state in-plane problems of crack tip fields within the simplified SGET and the more general gradient theories have not been considered previously. It was shown that the couple stress theory (which can be considered an incomplete SGET [24]) allows us to describe the stabilizing effect [25] and the crack tip shielding effect [14] due to microstructural contributions. However, the couple stress theory does not allow us to describe the regular stress state at the crack tip for in-plane loading [14; 26], so more general gradient theories can be preferable for such analysis, with assessments on the fracture process zone effects [20]. Notably, the presented smoothed solution for moving cracks can be related to the phenomenological description of cohesive zone effects, as was discussed previously for the static problems in SGET [20; 27]. Although the size effects known for the quasi-brittle materials [28] cannot be captured within the asymptotic analysis, the nonclassical effects in stress distribution can be evaluated within the considered method. As will be shown, the SGET solution allows us to avoid the classical paradox of stable crack propagation under in-plane loading when no standard criteria can be used to validate the initial assumption about straight growth of a crack [12].