## 2 Materials and Methods
Then, from the boundary condition  
\[\left(v^{+}-v^{-}\right)^{\prime}_{x}=0,x\in L,\]  
taking into account (4) and (11), we obtain another linear conjunction problem  
\[\left[F(kh)+\overline{F}(kh)\right]^{+}-\left[F(kh)+\overline{F}(kh)\right]^{ -}=0,\ \ x\in L.\]  
By solving this problem, we get  
\[F(z)+\overline{F}(z)=4 bz. \tag{38}\]  
Next, using Formulas (3), (11), and (12), and satisfying the boundary condition  
\[\sigma_{xy}^{+}+\sigma_{xy}^{-}=0,\ \ x\in L,\]  
we come to the linear conjunction problem. Its solution is given by (20). Adding (38) and (20), we find the function  
\[F(z)=2 bz+\frac{1}{2X(z)}\sum_{j=0}^{3}c_{j}z^{j}. \tag{39}\]  
Substituting (39) into (12) gives  
\[\Phi(z)=2 bz+\frac{1}{2}D(z)+\frac{1}{2X(z)}\sum_{j=0}^{3}c_{j}z^{j}. \tag{40}\]  
Considering (40) and (26), we find the stress intensity factors  
\[\widetilde{K}_{1}^{+}=\frac{\sqrt{l}l}{2}\big{(}2\widetilde{L}_{3}-\widetilde{d} \big{)}\big{(}2\widetilde{d}_{1}-\widetilde{l}\big{)}\widetilde{K}_{1}^{-}= \frac{\sqrt{l}l}{2}\big{(}2\widetilde{L}_{3}-\widetilde{d}\big{)}\big{(}2 \widetilde{d}_{1}-3\widetilde{l}\big{)}.\]0  
where \(q\), \(\widetilde{K}_{2}^{\pm}\) are determined according to the Formula (29). The ultimate value of the force causing beam failure is obtained from the formula  
\[\widetilde{K}_{1}^{+}=\frac{\sqrt{l}l}{2}\big{(}2\widetilde{L}_{3}-\widetilde{d} \big{)}\big{(}2\widetilde{d}_{1}-\widetilde{l}\big{)}\widetilde{K}_{1}^{-}= \frac{\sqrt{l}l}{2}\big{(}2\widetilde{L}_{3}-\widetilde{d}\big{)}\big{(}2 \widetilde{d}_{1}-3\widetilde{l}\big{)}.\]1It is known from scientific literature that the energy criteria of the fracture of cracked bodies do not always give satisfactory results. Therefore, we use an improved energy fracture criterion presented in [20]. \[\widetilde{K}_{1}^{+}=\frac{\sqrt{l}l}{2}\big{(}2\widetilde{L}_{3}-\widetilde{d} \big{)}\big{(}2\widetilde{d}_{1}-\widetilde{l}\big{)}\widetilde{K}_{1}^{-}= \frac{\sqrt{l}l}{2}\big{(}2\widetilde{L}_{3}-\widetilde{d}\big{)}\big{(}2 \widetilde{d}_{1}-3\widetilde{l}\big{)}.\]2  
where \(K_{1C}\) and \(K_{2C}\) are constants responsible for the fracture strength of the material and  
\[\widetilde{K}_{1}^{+}=\frac{\sqrt{l}l}{2}\big{(}2\widetilde{L}_{3}-\widetilde{d} \big{)}\big{(}2\widetilde{d}_{1}-\widetilde{l}\big{)}\widetilde{K}_{1}^{-}= \frac{\sqrt{l}l}{2}\big{(}2\widetilde{L}_{3}-\widetilde{d}\big{)}\big{(}2 \widetilde{d}_{1}-3\widetilde{l}\big{)}.\]3  
Using (43) and (44), the ultimate force causing beam fracture for a non-contacted crack tip is determined as  
\[\widetilde{Q}^{\star\pm}=\frac{2\sqrt{2}Q}{h^{3/2\widetilde{b}}\sqrt{\pi}K_{1 C}}=\frac{8\sqrt{2}}{3\sqrt{\pi}}\frac{1}{\sqrt{\left(\widetilde{K}_{1}^{ \pm}\right)^{2}+\left(\widetilde{K}_{2}^{\pm}/\  
In Figure 3, curve 1 corresponds to coordinate \(\widetilde{d}_{1}=d/h=0,3\), while curve 2 corresponds to coordinate \(\widetilde{d}_{1}=0,4\). It is seen that for a fixed crack length, the ultimate force \(Q^{*}\) decreases with increasing distance between the crack center and the beam axis. Dependences in Figures 4 and 5 stand for cracks with particularly contacting faces at \(\widetilde{l}>(2\widetilde{d}_{1})/3\). In Figure 4, curve 1 corresponds to contacting tip \(B\) and curve 2 - to non-contacting crack tip \(A\). These curves were built using Formula (36). Curves 3 and 4 were built using Formula (44). Curve 3 corresponds to \(\alpha=\sqrt{\frac{2}{3}}\) and curve 4 - to \(\alpha=\frac{3}{4}\). It is clear that beam failure begins from a non-contacting tip. As shown in Figures 2b and 4b, the energy criterion for the fracture of cracked bodies gives an overestimated value of the force, in comparison with improved fracture criterion (42). In addition, with increasing crack length, the ultimate force decreases. The curves in Figure 5 were constructed for non-contacting crack tip, \(A\). Curve 1 corresponds to \(\widetilde{d}_{1}=0,03\), curve 2 - to \(\widetilde{d}_{1}=0,3\), curve 3 - to \(\widetilde{d}_{1}=0,4\).