# A Probable Approach to Reliability Assessment of Reinforced Plates
## 2 Materials and Methods  
The failure of the composites with rigid inclusion under conditions of a complex stress state was extensively studied recently. In papers [14; 15], various methods for solving the inverse problem of determining a rigid inclusion in linear elastic bodies are considered. The optimal inequality of three spheres was presented as a mathematical apparatus for solving the inverse problem of determining a rigid inclusion in a thin plate [16; 17]. The article [18] considers the problem of separating a rigid inclusion from an infinite elastic plate. Many results have been obtained to determine the limiting state of such materials depending on their physical and mechanical properties, loading conditions, plate geometry, and parameters of individual defects in them. In this case, these factors are considered given (deterministic), although they can be arbitrary. In particular, in [19], an experimental and numerical analysis of the short fiber-reinforced composite material's behavior during failure was carried out. We will utilize a technique that provides for joint consideration of defectiveness and stochastic of the composite materials structure. We study an elastic rectangular plate made of a brittle material reinforced with rigid linear (rod) inclusions of random size and placement which do not interact with each other. The strength of the contact layer between the inclusions and the matrix (base material) is higher than the strength of the matrix. The plate is considered in conditions of comprehensive tension compression under the action of uniform loading \(P\) and \(Q=\eta P\) (Figure 1), where \(\eta\) is the loading biaxial coefficient. The values \(P\) and \(Q\) are introduced as the principal stresses in the plane stress state. The geometric parameters of the inclusions are the next: \(2l\) is the length and \(\alpha\) is the angle between the direction of the inclusion and the action of the loading \(P\), \(\alpha\in(-\pi/2;\pi/2)\). They are defined as independent random variables. The laws of the probability distribution for these random variables because of certain physical reasons would be formulated assuming that the material of the plate is isotropic. In the case of a surface crack that is larger, the thickness of the plate should also be taken into account. Similarly, in the case of through cracks in limited (small) plates, the magnitude of the failure loading will be influenced by the linear dimensions of the plate. This algorithm can be applied to the study of fiber concrete failure with metal reinforcing fibers as one of the components. The introduction of such fibers with arbitrary orientation into the concrete composition helps to increase its tensile and compressive strength, increase crack resistance, impact toughness, and wear resistance. It was shown in the paper [20] that in the small vicinity of the inclusion ends, which is not restrained from turning under the action of external loading, the components of the stress tensor in the local polar coordinate system are written as follows:  
\[\left\{ \begin{array}{l}\sigma_{r}=\frac{K_{4}}{\sqrt{r}}\Big{(}5\cos\frac{ \theta}{2}+(2\varkappa+1)\cos\frac{3}{2}\theta\Big{)}+O(1);\\ \sigma_{\theta}=\frac{K_{4}}{\sqrt{r}}\Big{(}3\cos\frac{\theta}{2}-(2\varkappa+ 1)\cos\frac{3}{2}\theta\Big{)}+O(1);\\ \tau_{r\theta}=\frac{K_{4}}{\sqrt{r}}\Big{(}\sin\frac{\theta}{2}-(2\varkappa+ 1)\sin\frac{3}{2}\theta\Big{)}+O(1),\end{array} \right. \tag{1}\]  
where the value \(K_{4}\) is analytically represented:  
\[K_{4}=\frac{P\sqrt{l}}{8\sqrt{2}\varkappa}\bigg{(}\frac{1+\eta}{2}(\varkappa- 1)+(1-\eta)\cos 2\varkappa\bigg{)}, \tag{2}\]  
\(r,\theta\) are polar coordinates centered at the crack tip (\(r\) is the radial coordinate, \(\theta\) is the polar angle), \(\varkappa=\frac{3-\
u}{1+\
u}\) is the elastic constant (the case of plane stress state), and \(\
u\) is the Poisson's ratio of the plate material. It follows from Expressions (1) and (2) that the inequality \(\sigma_{r}>\sigma_{\theta}\) is true in the vicinity of the inclusion vertex.