## 2 Materials and Methods
Next, we introduce complex potentials \(\Phi_{0}(z)\) and \(\Omega_{0}(z)\)[7, 8] that describe the stress-strain state in the cantilever-uncracked strip, under the same load. For large values of \(|z|\) they are  
\[\Phi_{0}(z)=-ia\overline{z}^{2}+bz,\Omega_{0}(z)=5 ia\overline{z}^{2}+3 bz-i4 ahl ^{2}, \tag{5}\]  
where  
\[a=Q/8I,\ I=\frac{4}{3}b\overline{h}^{3},\ b=-2a(2L_{3}-d). \tag{6}\]  
We determine the stress-strain state of cantilever-cracked strip by satisfying boundary conditions (1), (2) at the crack and by demanding that in the considerable distance from the crack potentials \(\Phi_{0}(z)\) and \(\Omega_{0}(z)\) have the form  
\[\Phi(z)=\Phi_{0}(z)+O\Big{(}1/z^{2}\Big{)},\Omega(z)=\Omega_{0}(z)+O\Big{(}1/ z^{2}\Big{)}. \tag{7}\]  
Figure 1: Scheme of loading of the cracked beam. Parameter \(\lambda\) is length of the non-contacted zone of the crack. Since point \(B\) lies at a greater distance from origin O than \(\lambda\) (see Figure 1), it is follows from (19) that the crack faces are in particular contact if \(d_{2}<-\frac{1}{3}d_{1}\). In such a way, we have one equation (15) in two functions \(F(z)+\overline{F}(z)=(b_{1}z+b_{0})\overline{X}(z)\). In order to obtain missing the equation, we consider the following boundary condition  
\[\sigma_{y}^{+}+\sigma_{y}^{-}=0,x\in L+L_{1}.\]  
Taking into account the relations (3), (11), and (12), we obtain the linear conjunction problem  
\[\sigma_{yy}^{+}=\sigma_{yy}^{-}\sigma^{+}-\overline{\
u}^{-}=0,x\in L\sigma_{ yy}^{\pm}=0,x\in L_{1}; \tag{1}\]0  
Its solution is  
\[\sigma_{yy}^{+}=\sigma_{yy}^{-}\sigma^{+}-\overline{\
u}^{-}=0,x\in L\sigma_{ yy}^{\pm}=0,x\in L_{1}; \tag{1}\]1  
where \(c_{j}(j=\overline{0,3})\) are unknown coefficients and  
\[\sigma_{yy}^{+}=\sigma_{yy}^{-}\sigma^{+}-\overline{\
u}^{-}=0,x\in L\sigma_{ yy}^{\pm}=0,x\in L_{1}; \tag{1}\]2  
Then, the function \(X^{-1}(z)\) for large values of \(|z|\) has a series expansion  
\[\sigma_{yy}^{+}=\sigma_{yy}^{-}\sigma^{+}-\overline{\
u}^{-}=0,x\in L\sigma_{ yy}^{\pm}=0,x\in L_{1}; \tag{1}\]3  
Taking into account (5), (7), (12), and (22), by expanding both sides (20) into a series at large \(|z|\) and equating the coefficients at the same degrees of \(z\) after some transformations, we obtain formulas for the unknown coefficients \(c_{j}(j=\overline{0,3})\)  
\[\sigma_{yy}^{+}=\sigma_{yy}^{-}\sigma^{+}-\overline{\
u}^{-}=0,x\in L\sigma_{ yy}^{\pm}=0,x\in L_{1}; \tag{1}\]4  
Adding (20) and (15), we find the function  
\[\sigma_{yy}^{+}=\sigma_{yy}^{-}\sigma^{+}-\overline{\
u}^{-}=0,x\in L\sigma_{ yy}^{\pm}=0,x\in L_{1}; \tag{1}\]5  
Function \(\Phi(z)\) is found from (12) using (16), (21) and (24)  
\[\sigma_{yy}^{+}=\sigma_{yy}^{-}\sigma^{+}-\overline{\
u}^{-}=0,x\in L\sigma_{ yy}^{\pm}=0,x\in L_{1}; \tag{1}\]6  
Stress intensity factors are determined on the base of formula [14]  
\[ \begin{array}{l}K_{1}^{+}-iK_{2}^{+}=2\lim\limits_{x\to d_{1}+0}\Big{[}\sqrt {2(x-d_{1})}\Phi(x)\Big{]},\\ K_{1}^{-}-iK_{2}^{-}=2\lim\limits_{x\to d_{2}-0}\Big{[}\sqrt{2(d_{2}-x)}\Phi(x )\Big{]}.\end{array}  \tag{26}\]  
Replacing the function \(\Phi(x)\) by expression (25) in (26) we obtain  
\[ \begin{array}{l}K_{1}^{+}=\sqrt{2(d_{1}-\lambda)}(b_{1}d_{1}+b_{0}),K_{2}^{ +}=-\frac{\sqrt{2}}{\sqrt{d_{1}-d_{2}}}\sum\limits_{j=0}^{3}\overline{c_{j}}d _{1}^{j},\\ K_{1}^{-}=0,K_{2}^{-}=\frac{\sqrt{2}}{\sqrt{d_{1}-d_{2}}}\sum\limits_{j=0}^{ 3}\overline{c_{j}}d_{1}^{j},c_{j}=i\overline{c_{j}}.\end{array}  \tag{27}\]According to (18), (23), Formula (27) can be expressed as  
\[K_{j}^{\pm}=q\widetilde{K}_{j}^{\pm}. \tag{28}\]  
In (28) we use the following notifications  
\[