# A Probable Approach to Reliability Assessment of Reinforced Plates
## 2 Materials and Methods
Therefore, in [21], stress \(\sigma_{r}\) is assumed to be responsible for the occurrence of brittle fracture. It is also hypothesized that the strength of the contact layer between the inclusion and the matrix is higher than the strength of the matrix. Similarly to the cracks theory, the next condition is proposed for the failure criterion  
\[\max_{\theta}\Psi(P,\eta,\alpha,l,\theta,\varkappa)=K_{0}, \tag{3}\]  
where \(\Psi(P,\eta,\alpha,l,\theta,\varkappa)=\lim\limits_{r\to 0}\sqrt{r}\, \sigma_{r}\), the constant \(K_{0}\) characterizes the material's resistance to local failure. This characteristic is determined from the experiment (or expressed in terms of the energy density of a given material failure). It is taken into consideration that the matrix material is assumed to be homogeneous, isotropic, and does not have the property of strength anisotropy. According to Expressions (1)-(3), the formulas for calculating the failure loading (a crack forms in the vicinity of a rigid rectilinear inclusion) of the plate under the conditions of plane stress state are presented in the following general form [22]:  
\[|P|=K_{0}l^{-0.5}\Phi_{j}(\alpha,\eta,\varkappa)\;\;(i=1,2), \tag{4}\]  
Figure 1: Model of a plate reinforced with random rigid linear inclusions. where the analytical form of the function \(\Phi_{i}(\alpha,\eta,\varkappa)\) is:  
\[ \begin{array}{l}\Phi_{1}(\alpha,\eta,\varkappa)=(\varphi_{1}(\eta,\varkappa)+ \varphi_{2}(\eta,\varkappa)cos2\alpha)^{-1}\,;\\ \Phi_{2}(\alpha,\eta,\varkappa)=(\varphi_{3}(\eta,\varkappa)+\varphi_{4}( \eta,\varkappa)cos2\alpha)^{-1};\end{array}  \tag{5}\]  
where  
\[\varphi_{1}(\eta,\varkappa)=\frac{(1+\eta)(\varkappa-1)(3+\varkappa)}{8\sqrt{ 2}\varkappa}\,;\ \ \ \ \ \varphi_{2}(\eta,\varkappa)=\frac{(1-\eta)(3+\varkappa)}{4\sqrt{2}\varkappa}\,; \tag{6}\]  
\[\varphi_{3}(\eta,\varkappa)=\frac{(1+\eta)(\varkappa-1)}{12\varkappa}\sqrt{ \frac{(3\varkappa-1)^{3}}{12(2\varkappa+1)}};\ \ \ \ \ \ \varphi_{4}(\eta,\varkappa)=\frac{(1-\eta)}{6\varkappa}\sqrt{\frac{(3 \varkappa-1)^{3}}{12(2\varkappa+1)}}\,. \tag{7}\]  
Expressions (4)-(7) and relations [22] allow us to write the formulas for calculating the failure loading \(P\). For the case \(P>0,\ \eta\leq 1\) (biaxial tension, or tension-compression)  
\[P=K_{0}l^{-0.5}\Phi_{1}(\alpha,\eta,\varkappa),\ \ \begin{cases}|\alpha|\leq\pi/2,\ \frac{3- \varkappa}{1+\varkappa}\leq\eta\leq 1\,,\\ |\alpha|\leq\alpha_{1},\ -\infty<\eta\leq\frac{3-\varkappa}{1+\varkappa}; \end{cases}  \tag{8}\]  
\[P=-K_{0}l^{-0.5}\Phi_{2}(\alpha,\eta,\varkappa),\ \ |\alpha|\geq\alpha_{1},\ - \infty<\eta\leq\frac{3-\varkappa}{1+\varkappa}\,. \tag{9}\]  
For the case \(P<0,\ \eta\geq 1\) (biaxial compression)  
\[P=-K_{0}l^{-0.5}\Phi_{2}(\alpha,\eta,\varkappa),\ \ \begin{cases}|\alpha|\leq\pi/2,\ 1 \leq\eta\leq\frac{1+\varkappa}{3-\varkappa}\,,\\ |\alpha|\geq\alpha_{1},\ \frac{1+\varkappa}{3-\varkappa}\leq\eta<\infty\,; \end{cases}  \tag{10}\]  
\[P=K_{0}l^{-0.5}\Phi_{1}(\alpha,\eta,\ \varkappa),\ \ |\alpha|\leq\alpha_{1},\ \frac{1+ \varkappa}{3-\varkappa}\leq\eta<\infty\,. \tag{11}\]  
The angle value \(\alpha_{1}\) is determined as follows:  
\[\alpha_{1}=0.5 arccos\frac{(1+\eta)(1-\varkappa)}{2(1-\eta)}\,. \tag{12}\]  
We assume that the random half-length of inclusions can be of arbitrary values \(0\ \leq\ l\ <\ \infty\). This assumption simplifies mathematical calculations and the material model and is adopted in several papers as well. In addition, the values of the failure loading \(0\leq\ |P|\ <\ \infty\) are also possible. The minimum value of the failure loading is \(P_{\min}=0\) (the strength threshold value). This means that even with arbitrarily small loading \(P\) and \(Q\), there exists a non-zero probability of failure.