PSI - Issue 50

Kirill E. Kazakov et al. / Procedia Structural Integrity 50 (2023) 125–130 Kirill E. Kazakov / Structural Integrity Procedia 00 (2022) 000 – 000

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the foundation of a building, study of the reliability of clamping joints on a printed circuit board, assessment of stress levels in the interaction of medical instruments and biological tissues are based on studies of the contact interaction of bodies. To increase the service life of products, as well as to provide thermal or electrical insulation, coatings are often used whose properties differ from the properties of the body on which they are applied. Surface layers formed due to surface treatment of parts of a mechanism or the influence of any physical fields on it can also be considered as coatings, because they have properties different from the main body. The coatings can have a rather complex structure, be non-uniform, porous, contain inclusions, etc. The coating (external layers) thickness is usually small compared to the thickness of the main body and, together with the dimensions of the contact area, it can however be variable, which affects the stress levels in the areas of contact with other structural elements. Many articles have recently been devoted to the search for analytical solutions to problems of contact interaction for the cases where one or more bodies have complex surface properties and shapes or are covered with coatings. For example, problems for the case where an additional layer is modeled at the contact point are studied in the work of Argatov and Chai (2020), problems of discrete contact for rough bodies are considered by Lyubicheva et al. (2018), problems for bodies coated by complex layers are described in papers of Kazakov (2017), Manzhirov and Kazakov (2017, 2018), and Kazakov and Kurdina (2020). This paper describes the statement and solution of the interaction problem for a system of various axisymmetric rigid bushes and a pipe with a thin outer coating. A peculiar feature of the problem statement is that we have different aging viscoelastic layers, and the thickness of the external thin layer and the profiles of the bushes are variable. 2. Problem formulation and mathematical model Consider a long viscoelastic aging pipe made at time  in . Its inner radius is r in , and its outer radius is r out . A coating of another viscoelastic aging material made at time  out is applied to such a pipe. The stiffness of the coating material does not exceed the stiffness of the pipe material. It is assumed that there is a smooth contact between the coating and the pipe. The coating has thickness h ( z ) that depends on the axial coordinate z . At time  0 , several different rigid bushes are simultaneously put on the pipe so that there is no gap between the inner surfaces of the bushes and the pipe (see Fig. 1). All bushes have axial symmetry, but their inner profiles depend on the coordinate z , i.e. the internal radii are described by functions g i ( z ), where i varies from 1 to n and n is the number of bushes. Note that g i ( z )  r out + h ( z ). As a result of this interaction, the pipe is deformed. It is necessary to determine the contact pressure distributions in the interaction areas between the pipe and the bushes. The solution to the problem will be constructed under the following assumptions: 1) the coating thickness is much less than the length of any bush and the inner radius of the pipe; 2) the boundaries of the bushes and the coating are slightly curved (see, for example, Soldatenkov (2010)), i.e. | ( ) | 1   g z i , | ( ) | 1   h z , i n 1,  ; 3) there is a smooth contact between the bushes and the coating; 4) the bushes are located at a sufficient distance from the ends of the pipe, therefore, their effect on the stress-strain state under the bushes can be neglected. This work does not take into account the presence of plastic deformations.

Fig. 1. Contact interaction of coated pipe and bushes.