## 2 Second Gradient Elastodynamics
### Plane Strain Steady-State Problem  
Let us consider the plane strain problem, assuming that the displacement component \(u_{3}\) is zero and that the other components are independent on \(X_{3}\), i.e.,(\(i=1,2\)). For the plane problems, the vector Lame potential \(\mathbf{\phi}_{2}\) in (7) and (8) can be defined by using the corresponding scalar potential \(\phi_{2}(X_{1},X_{2},t)\) as follows:  
\[\mathbf{\phi}_{2}=\phi_{2}\mathbf{e}_{3},\qquad(1-l^{2}\
abla^{2})c_{2}^{2}\
abla^{2} \phi_{2}=(1-g^{2}\
abla^{2})\ddot{\phi}_{2} \tag{9}\]  
In the following analysis, we will also assume that the crack propagates at a constant speed \(V\) along the \(X_{1}\)-axis (see Figure 1). In this case, it is convenient to introduce the moving coordinate system with the center at the tip of the crack: \(x_{1}=X_{1}-Vt\), \(x_{2}=X_{2}\), \(x_{3}=X_{3}\), so that the following time-derivative rule for any field variable \(f\) is valid:  
\[\dot{f}=-V\frac{\partial f}{\partial x_{1}} \tag{10}\]  
Using (10) in (8) and (9), we obtain the governing equations for the Lame potentials in the following form:  
\[(1-l^{2}\
abla^{2})\
abla^{2}\phi_{i}=m_{i}^{2}(1-g^{2}\
abla^{2})\frac{ \partial^{2}\phi_{i}}{\partial x_{1}^{2}},\qquad(i=1,2) \tag{11}\]  
where \(\
abla^{2}=\frac{\partial^{2}}{\partial x_{1}^{2}}+\frac{\partial^{2}}{ \partial x_{2}^{2}}\) is the two-dimensional Laplace operator; \(m_{1}=V/c_{1}=cm_{2}\) and \(m_{2}=V/c_{2}\) are the Mach numbers; \(c=c_{2}/c_{1}=\sqrt{(1-2v)/(2(1-v))}<1\) is the ratio between the dilatational and the shear wave speeds; and \(v\) is the Poisson ratio. We restrict the following analysis for the case of the same values of the length scale parameters \(g=l\). This case is generally admissible and corresponds to the materials with an almost nondispersive nature of the elastic waves [36]. Moreover, this case can be treated as a zero-order approximation for the materials with close values of the length scale parameters \(g=l+\epsilon\), \(|\epsilon|\ll 1\). The governing Equation (11) can be reordered then as follows:  
\[\left(1-l^{2}\
abla^{2}\right)\left(\
abla^{2}\phi_{i}-m_{i}^{2}\frac{ \partial^{2}\phi_{i}}{\partial x_{1}^{2}}\right)=0,\qquad(i=1,2) \tag{12}\]  
To the best of the author's knowledge, the general solution for Equation (12) is not available as of yet (as well as that for the more complicated one in Equation (11)). Moreover, the possibility of the separation of the variables in these equations is not obvious and has not been proven yet. A variant of such a solution in the asymptotic sense is presented in the next section. Figure 1: Illustration for the growing crack problem with global and local (moving) coordinate systems. Opening mode under remotely applied loading is considered. Solution is found in the vicinity of a crack tip, where \(r\ll 1\), i.e., \(r\ll l\). Notably, that classical governing equations for the steady-state problem can be obtained from (12), assuming \(l=0\) (see [12]):  
\[\
abla^{2}\phi_{i}-m_{i}^{2}\frac{\partial^{2}\Phi_{i}}{\partial x_{1}^{2}}=0, \qquad(i=1,2) \tag{13}\]  
Introducing the polar coordinate system related to the moving framework (\(r=\sqrt{x_{1}^{2}+x_{2}^{2}}\) and \(\theta=\arctan(x_{2}/x_{1})\)), the components of the displacement vector (7) and the stress tensors (5) can be represented as follows [19]:  
\[u_{r}=\frac{\partial\Phi_{1}}{\partial r}+\frac{1}{r}\frac{\partial\theta_{2} }{\partial r},\qquad u_{\theta}=\frac{1}{r}\frac{\partial\Phi_{1}}{\partial \theta}-\frac{\partial\Phi_{2}}{\partial r} \tag{14}\]  
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