Mathematical modeling in the calculation of reinforcing elements

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Authors:

T.Kagadiy, Dr. Sc. (Phys.-Math.), Assoc. Prof., orcid.org/0000-0001-6116-4971, Dnipro University of Technology, Dnipro, Ukraine,e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.

A.Shporta, orcid.org/0000-0002-1260-7358, Dnipro University of Technology, Dnipro, Ukraine,e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.

Abstract:

Purpose. Determination of stress distribution laws for the hard stamp and an elastic plate interaction with cylindrical anisotropy. Simulation of contact interaction tasks in order to determine the processes of wear, strength, destruction and structures durability. Development of analytical methods for calculating contact interactions of structures taking into account various material properties.

Methodology. The mathematical model spatial problem of a hard stamp and a circular sector with cylindrical anisotropy interaction has been compiled. To study the model, an asymptotic method has been proposed, which allows dividing the stress-strain state of an infinite circular sector into two components and reducing the solution of the elasticity theory problem to the sequential solution of potential theory problems.

Findings. A concrete contact problem was investigated, for which the asymptotic method was used. The solution takes into account the friction that occurs in the interaction process between the rigid stamp and the elastic plate. The considered task is new and rather difficult. It causes significant difficulties when considered. Therefore, the obtained analytical solution is a useful result for further analysis or numerical calculations. The pressure values under the stamp were found, the influence of friction was taken into account.

Originality. The previously proposed method is generalized to the case of material cylindrical anisotropy. An analytical solution has been obtained for a new complex spatial contact problem.

Practical value. The asymptotic method proposed in the paper allows us to move from mechanics solving complex mixed problems to solving sequential boundary problems of potential theory – the most developed section of mathematical physics. The solutions obtained by the proposed method make it possible to analyze the stress-strain state that occurs when a hard stamp is pressed into the free face of an elastic orthotropic infinite circular sector with cylindrical anisotropy, the edges of which are fixed. The following problem is considered: in the free edge of an infinite circular sector, which is elastic, orthotropic, and also its material possesses the properties of cylindrical anisotropy, a rigid stamp is pressed. The edges of the circular sector are fixed. Application of the obtained results is possible at the calculation and design of various types of fastenings. The results can be used in calculating and designing various types of mounts.

References.

1. Kagadiy, T. S., & Shporta, A. H. (2015). The asymptotic method in problems of the linear and nonlinear elasticity theory. Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu, 3, 76-81.

2. Kaminsky, A. O., & Selivanov, M. F. (2017). Slow crack growth with contact area. Reports of the National Academy of Sciences of Ukraine, 1, 38-43. DOI:10.15407/dopovidi2017.01.038.

3. Kriven, I., & Yavorskaya, M. (2010). Development of plastic strips in the vicinity of the vertices of the two arcs of symmetric cut out under antiplastic deformation. Phys.-Math. modeling and inform. Technology, (11), 91-96.

4. Dziubik, A. R., Nykolyshyn, T. M., & Porokhovskyi, Yu. V. (2016). Influence of residual stresses on the boundary equilibrium of a pipeline with an internal split of arbitrary configuration. Physical-chemical mechanics of materials, 52(1), 83-90.

5. Bomba, A. Ya., Prysiazhniuk, I. M., & Prysiazhniuk, O. V. (2013). Asymptotic method for solving a class of model singularly perturbed problems of the mass transfer process in heterogeneous media. Reports of the National Academy of Sciences of Ukraine, 3, 28-34.

6. Kliuchnyk, I. (2014). Integration of a system of differential equations with a small parameter in the part of derivatives / I. Glyu­chnyk. Bulletin of the Donetsk National University. Ser A: Natural sciences, (2), 21-25.

7. Staroselsky, A., Acharya, R., & Cassenti, B. (2019). Phase field modeling of fracture and crack growth. Engineering Fracture Mechanics, 205, 268-284. DOI:10.1016/j.engfracmech.2018.11.007. 

8. Kolosov, D., Dolgov, O., & Kolosov, A. (2014). Analytical determination of stress-strain state of rope caused by the transmission of the drive drum traction. In V. Bondarenko, I. Kovalevs’ka, K. Ganushevych (Eds.), Progressive Technologies of Coal, Coalbed Methane, and Ores Mining (1st ed., pp. 499-504). CRC Press.

9. Kelby, B., & Baka, Z. (2019). Annual crack in an elastic half-space. International Journal of Engineering Science, (134), 117-147. DOI:10.1016/j.ijengsci.2018.10.007. 

10. Ostrik, V. I. (2008). Injection of semi-non-stamp in elastic band in the presence of friction and adhesion. Math. methods and physical fur. Fields, 51(1), 138-149.

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ISSN (print) 2071-2227,
ISSN (online) 2223-2362.
Journal was registered by Ministry of Justice of Ukraine.
Registration number КВ No.17742-6592PR dated April 27, 2011.

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