## 2 Materials and Methods  
We considered an isotropic cantilever strip of \(2L_{3}\) length that was rigidly fixed at left end. Its width and thickness were designated \(2h\) and \(2\overline{b}\), respectively. A Cartesian coordinate system was placed in symmetry axes of the strip as shown in Figure 1. The strip was weakened by a through crack of \(2l\) length, non-symmetrically placed on the \(Ox\)- axis about the \(Oy\)- axis. Parameter \(d\) denotes the distance from the left edge of the strip to the crack line (\(d<L_{3}\)). It was assumed that the crack length was essentially less than the width of the strip. The strip was loaded by a concentrated force \(Q\), applied at its end, perpendicularly to the \(Oy\)- axis. Crack tips were in the points \(A(d_{1},0)\) and \(B(d_{2},0)\) (\(|d_{i}|<1\)). \begin{table}
\begin{tabular}{c c} \hline \hline
**Mark** & **Mark meaning** \\ \hline \(L_{3}\) & the distance from the left edge of the strip to the crack line \\ \hline \(2h\) & crack width \\ \hline \(2\overline{b}\) & crack thickness \\ \hline \(A(d_{1},0)\) and \(B(d_{2},0)\). & crack tips \\ \hline \(2l\) & crack length \\ \hline \(L_{1}\) & unloaded zone \(AC\) of the crack \\ \hline \(L\) & crack faces are smooth closed in area \(CB\) \\ \hline \(\sigma_{yy}\) and \(y_{aby}\) & components of stress tensor \\ \hline \(\
u\) & the second component of displacement vector of the beam point \\ \hline \(\Phi(z),\Omega(z)\) & complex potentials \\ \hline \(\mu\) & shear modulus \\ \hline \(k\) & Mushelishvili’s constant \\ \hline \(u\) & the first component of displacement vector of the beam point \\ \hline \(\Phi_{0}(z),\Omega_{0}(z)\), & crack potentials \\ \hline \(b_{1}\) and \(b_{0}\) & unknown constants \\ \hline \(\lambda\) & length of non-contacted zone of the crack \\ \hline \(c_{j}(j=\overline{0},\overline{3})\) & unknown coefficients \\ \hline \(K_{1}^{\text{a}},K_{2}^{\text{a}}\) & stress intensity factors \\ \hline \(Q^{\text{a}}\) & the ultimate value of the force causing beam fracture for contacted crack tip, where sign “+” corresponds to tip \(A\) and “-” to tip \(B\). \\ \hline \(K_{1C},K_{2C}\) & constants responsible for fracture strength of material \\ \hline \(\widetilde{Q}^{\text{a}}\) & the ultimate force causing beam fracture for non-contacted crack tip, where sign “+” corresponds to tip \(A\) and “-” to tip \(B\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mark meaning. We considered three possible cases of stress-strain state of the cracked beam. Case 1. Crack with particularly contacting faces. We assume that under external load, crack faces are smooth and enclosed in area \(CB\), denoted by \(L\). Point \(C\) has coordinates \(C(\lambda,0)\), where \(\lambda\) is unknown parameter, responsible for length of contact zone. The unloaded zone \(AC\) of the crack is denoted by \(L_{1}\). According to the problem statement, there are the following boundary conditions at the crack faces  
\[\sigma_{yy}^{+}=\sigma_{yy}^{-}\sigma^{+}-\overline{\
u}^{-}=0,x\in L\sigma_{ yy}^{\pm}=0,x\in L_{1}; \tag{1}\]  
\[\sigma_{bky}^{\pm}=0,x\in L+L_{1}, \tag{2}\]  
where \(\sigma_{yy}\) and \(y_{bky}\) are components of stress tensor; signs \"+\" and \"-\" stand for limit values of appropriate magnitude as \(\sigma\rightarrow\pm 0\); \(\
u\) is the second component of displacement vector of the beam point. Stresses and displacements may be expressed in terms of complex potentials \(\Phi(z)\), \(\Omega(z)\) according to [13]  
\[\sigma_{yy}-i\sigma_{xy}=\Phi(z)+\Omega(\overline{z})+(z-\overline{z}) \overline{\Phi^{\prime}(z)}, \tag{3}\]  
\[2\mu(u+iv)^{\prime}_{th}=\kappa\Phi(z)-\Omega(\overline{z})-(z-\overline{z}) \overline{\Phi^{\prime}(z)}, \tag{4}\]  
where \(\mu\) is shear modulus; \(k-\) Mushelishvili's constant; \(f^{\prime}=\frac{\partial f}{\partial x};u\) is the first component of displacement vector of the beam point, \(z=x+iy\), \(i=\sqrt{-1}\).