# Stress Analysis in Damaged Pipeline with Composite Coating
## 2 Materials and Methods
### Effect of Composite Coating on Stresses near Surface Cracks  
Suppose that an elastic isotropic ring with the radius of the middle surface \(R_{0}\) and the thickness \(h_{0}\), weakened by a deep radial crack with the depth \(l\), is covered by a thin coating that is in perfect mechanical contact with the ring. We studied the influence of \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Layer** & & \multicolumn{2}{c}{\(\boldsymbol{P_{repair}=0}\)} & \multicolumn{2}{c}{\(\boldsymbol{p_{repair}=0.5p=5\text{MPa}}\)} & \multicolumn{2}{c}{\(\boldsymbol{p_{repair}=p=10\text{ MPa(Without Coating)}}\)} \\ \cline{2-10}
**Number** & \(\boldsymbol{\sigma_{x}^{(0)}}\) & \(\boldsymbol{\sigma_{\varphi}^{(0)}}\) & \(\boldsymbol{\tau_{x\varphi}^{(0)}}\) & \(\boldsymbol{\sigma_{x}^{(0)}}\) & \(\boldsymbol{\sigma_{\varphi}^{(0)}}\) & \(\boldsymbol{\tau_{x\varphi}^{(0)}}\) & \(\boldsymbol{\sigma_{x}^{(0)}}\) & \(\boldsymbol{\sigma_{\varphi}^{(0)}}\) & \(\boldsymbol{\tau_{x\varphi}^{(0)}}\) \\ \hline
0 & 114.3 & 383.2 & 0 & 117.4 & 392.3 & 0 & 120.4 & 401.3 & 0 \\
1 & 15.8 & 39.5 & 0 & 7.9 & 19.8 & 0 & 0 & 0 & 0 \\
2 & 17.1 & 113.7 & 7.7 & 8.6 & 56.9 & 3.9 & 0 & 0 & 0 \\
3 & 0.02 & 0.02 & 0 & 0.01 & 0.02 & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Stresses (MPa) in the pipe and in the composite coating (\(p=10\) MPa and \(R=610\) mm). the coating on the plane deformation of the pipe with a defect caused by the action of the internal operating pressure \(p\). We also studied cases of application of the coating on the ring, pre-loaded with an internal repair pressure \(p_{repair}\leq p\). The elastic boundary value problem for a two-component ring with a radial cut corresponds to the formulated problem:  
* Equilibrium equation in the region: \[ \begin{array}{l}\frac{\partial\tau_{r}}{\partial r}+\frac{\sigma_{r}-\sigma_ {\varphi}}{r}+\frac{1}{r}\frac{\partial\tau_{\varphi p}}{\partial\varphi}=0, \frac{\partial\tau_{\varphi p}}{\partial r}+\frac{\tau_{\varphi p}}{r}+\frac{1 }{r}\frac{\partial\sigma_{\varphi}}{\partial\varphi}=0,\\ (r,\varphi)\in(R_{0}-h_{0}/2,R_{0}+h_{0}/2+h)\times[-\tau_{r},\pi]\backslash L ;\end{array} \] (9)
* Physical relations for a plane-deformed sectionally homogeneous medium: \[\sigma_{r}=(\lambda+2\mu)\varepsilon_{r}+\lambda\varepsilon_{\varphi},\; \sigma_{\varphi}=\lambda\varepsilon_{r}+(\lambda+2\mu)\varepsilon_{\varphi},\; \tau_{r\varphi}=2\mu\varepsilon_{r\varphi};\] (10)
* Cauchy relations: \[\varepsilon_{r}=\frac{\partial u_{r}}{\partial r},\;\varepsilon_{\varphi}= \frac{1}{r}\frac{\partial u_{\varphi}}{\partial\varphi}+\frac{u_{r}}{r}, \varepsilon_{r\varphi}=\frac{1}{2}\bigg{(}\frac{1}{r}\frac{\partial u_{r}}{ \partial\varphi}+\frac{\partial u_{\varphi}}{\partial\tau}\bigg{)};\] (11)
* Boundary conditions on the circular surfaces: \[\sigma_{r}(R_{0}+h_{0}/2+h,\varphi)=0,\;\tau_{r\varphi}(R_{0}+h_{0}/2+h, \varphi)=0,\;\varphi\in[-\tau,\pi];\] (12) and on the radial section: \[\sigma_{\varphi}(r,\;\pm\tau)=0,\;\tau_{\varphi\varphi}(r,\;\pm\pi)=0,\;u_{ \varphi}(r,+\pi)-u_{\varphi}(r,-\pi)\geq 0,\;r\in L.\] (13) In ratios (9)-(13), \(\sigma_{r},\sigma_{\varphi},\sigma_{r\varphi}\) are the stress tensor components, \(\varepsilon_{r},\varepsilon_{\varphi},\varepsilon_{r\varphi}\) are the components of the strain tensor, \(u_{r},u_{\varphi}\) are the displacement vector components, \(\lambda=\frac{Ev}{(1+\
u)(1-2\
u)}\), \(\mu=\frac{E}{2(1+\
u)}\) are Lame constants, and \[E=\left\{ \begin{array}{rl}E_{0},&r\in[R_{0}-h_{0}/2,R_{0}+h_{0}/2)\\ E_{c},&r\in(R_{0}+h_{0}/2,R_{0}+h_{0}/2+h]\end{array} \right\},\
u=\left\{ \begin{array}{rl}v_{0},&r\in[R_{0}-h_{0}/2,R_{0}+h_{0}/2)\\ v_{c},&r\in(R_{0}+h_{0}/2,R_{0}+h_{0}/2+h]\end{array} \right\}\] are [PERSON]'s modulus and [PERSON]'s ratio of the shell materials (index \"0\") and coating (index \"c\"), respectively. The numerical analysis of the formulated problem ((9)-(13)) was performed using the finite element method [45] and the MSC Marc software.