PSI - Issue 40

Kirill E. Kazakov et al. / Procedia Structural Integrity 40 (2022) 201–206 Kirill E. Kazakov / Structural Integrity Procedia 00 (2022) 000 – 000

202

2

features of applying such a coating (additive manufacturing, for example, — see in Parshin (2020, 2021)). Rigid inserts, the outer diameter of which can also be variable, can be used to strengthen or connect different sections of the pipe. Due to the tension with which the pipe is put on the insert, stresses arise in it, the values of which depend, among other things, on the profiles of the contacting surfaces. In works by Manzhirov and Chernysh (1988) or Arutyunyan and Manzhirov (1999), analytical solutions were obtained for finding the distribution of contact stresses in the contact region in the case when the coatings are homogeneous, the coating thickness is constant, and the insert is cylindrical. In article by Kazakov (2021), an analytical solution is obtained for a similar problem with an inhomogeneous coating. In article by Kazakov and Kurdina (2021), the applied approaches are generalized to cases when the insert has a variable external radius. This article is devoted to constructing a solution for the case when the coating is homogeneous, but both its thickness and the outer radius of the insert are variable. 2. Key assumptions and problem formulation Consider a cylindrical pipe made of viscoelastic aging material. The properties of such a pipe change over time. Inside, such a pipe is covered with a thin layer made of another viscoelastic aging material with less rigidity. It is assumed that the thickness of the coating depends on the axial coordinate. The layer is applied in such a way that there is a smooth contact between it and the main layer of the pipe. At time  0 , a rigid insert is placed in such a pipe. It has axial symmetry, but its outer radius depends on the longitudinal coordinate that coincides with the axis of the pipe. A smooth contact is carried out between insert and pipe. Since the outer radius of the rigid insert exceeds the inner radius of the coated pipe, the pipe is deformed. The values of stresses and strains change over time, since the pipe and the coating are made of viscoelastic aging materials. Fig. 1 shows the described interaction.

Fig. 1. Contact between insert and coated tube.

Finding an analytical solution (stresses, deformations, displacements) for all points of the pipe is very problematic, but analytical methods can be used to find stresses in the contact area. Generalizing the mathematical models given in works by Manzhirov and Chernysh (1988) or Arutyunyan and Manzhirov (1999), it is possible to obtain a mathematical model of the problem considered in this article. It is an integral equation with operators of various types. The only unknown function in it is the distribution of contact pressure in the area of interaction between the insert and the coated pipe. It has the form