Exact solution to the problem of stage-by-stage deformation of a multilayer cylinder made of incompressible hypoelastic material
https://doi.org/10.22405/2226-8383-2022-23-4-262-271
Abstract
The work is devoted to one of the problems of the theory of superimposed large deformations.
An algorithm for the exact solution of the problem of forming an infinite circular compound cylinder from a certain finite number of hypoelastic layers is presented. The problem is formulated in a quasi-static statement. The hypoelasticity model corresponding to the material of the cylindrical layers is described by the equations of state with the participation of the corotational Dienes derivative. When attached, each successive layer undergoes two phases of deformation over some time intervals. The first phase of deformation is the radial expansion or contraction of the cylindrical layer. The second phase of deformation is torsion. Each successive layer is attached to the composite hypoelastic cylindrical body after the deformation of the
previous layer is completed. At the same time, the deformation of each hypoelastic layer affects the general state of the composite cylinder, that is, all internal layers. It is required to determine the stress field in a composite nonlinearly elastic cylinder. The paper describes the notation and coordinate systems used in solving the problem. All the main steps for solving the problem are described, including the calculation of the stress tensor components. The formulas for the axial
force and torque of a compound cylinder are also given. Numerical studies have been carried out. The results of numerical studies - graphs of the dependence of the axial force and torque on the deformation parameters - are presented at the end of the work.
About the Authors
Vladimir Anatol’evich LevinRussian Federation
doctor of physical and mathematical sciences, professor
Anatoliy Victorovich Vershinin
Russian Federation
doctor of physical and mathematical sciences, professor
Konstantin Moiseevich Zingerman
Russian Federation
doctor of physical and mathematical sciences, professor
Danila Ruslanovich Biryukov
Russian Federation
postgraduate student
References
1. Martynova, E.D. 2019, ”Torsion processes of cylindrical samples from incompressible viscoelastic
2. materials of the Maxwell type”. Prikladnaya Matematika i mekhanika (J. of Applied Mathematics
3. and Mechanics) vol. 83, Issue 1, pp. 95–106.
4. Ovchinnikova, N.V. 2020 ”The problem of torsion of a hypoelastic incompressible cylinder”.
5. Mathematicheskoe modelirovanie i eksperimental’naya mekhanika deformiruemogo tverdogo
6. tela: Mezhv. sbornik nauch. trudov. Vypusk 3. Tver: Tverskoy Gosudarstvenny Tekhnicheskiy
7. Universitet. 2020, pp. 65-72.
8. Levin V.A., Zubov L.M., Zingerman K.M.: ”Torsion of a Composite Nonlinear Elastic Cylinder
9. with Inclusion at Large Initial Deformations”. International Journal of Solids and Structures,
10. , V. 51, No. 6, pp. 1403-1409.
11. Levin, V.A., Zubov, L.M., Zingerman, K.M.: ”Torsion of a Composite Nonlinear Elastic Cylinder
12. with a Prestressed Inclusion”, 2013, Doklady Physics, V. 58, No. 12, pp. 540-543.
13. Levin, V.A., Zubov, L.M., Zingerman, K.M. 2015. ”An exact solution for the problem of flexure
14. of a composite beam with preliminarily strained layers under large strains”, International
15. Journal of Solids and Structures. vol. 67–68. pp. 244-249. https://www.sciencedirect.com/
16. science/article/pii/S0020768315001973?via
17. Levin, V.A., Zingerman, K.M. 2008, ”A class of methods and algorithms for the analysis of
18. successive origination of holes in a pre-stressed viscoelastic body. Finite strains”, Communications
19. in Numerical Methods in Engineering. vol. 24, Issue 12. pp. 2240-2251. https://doi.org/
20. 1002/cnm.1080
21. Lurie, A.I. 1980, ”Nonlinear theory of elasticity” Nauka, Moscow, 1980, 512 p.
22. Truesdell K. ”A first course in rational continuum mechanics”. Johns Hopkins University,
23. Baltimore, Maryland, 1972, 592 p.
24. Dienes, J.K. 1979. On the analysis of rotation speed and stress in deformable bodies. Acta
25. Mech. vol. 32. pp. 217-232.
26. Dienes, J.K. 1986, ”A discussion of material rotation and stress rate”. Acta Mech. vol. 65. pp.
27. –11.
28. Brovko G.L.: ”Constitutive relations of continuum mechanics”, Development of the mathematical
29. apparatus and foundations of the general theory, 2017, 431 p.
30. Brovko G.L.: ”Some approaches to the construction of constitutive relations of plasticity at
31. large deformations”, Elasticity and inelasticity, M.: Mosk. un-t, 1987, pp. 68–81.
32. Brovko G.L.: ”Objective tensors and their mappings in classical continuum mechanics”,
33. Mechanics of Solids, 2021, Vol. 56, no. 1., pp. 65-83.
34. Finoshkina A.S.: ”Using new objective derivatives in the simplest models of hypoelasticity and
35. plastic flow with kinematic hardening”, Tula State Univ., 2000, no. 6, pp. 160-166.
36. Finoshkina A.S.: ”On the construction of models of plasticity at finite deformations based on
37. constitutive relations known at small deformations”, Elasticity and Inelasticity: Proceedings of
38. the International Scientific Symposium on the Problems of the Mechanics of Deformable Bodies
39. Dedicated to the 95th Anniversary of A.A. Ilyushin. M.: LENAND, 2006, pp. 256–264.
40. Konovalov, D., Vershinin, A., Zingerman, K., Levin, V. 2017. ”The implementation of spectral
41. element method in a CAE system for the solution of elasticity problems on hybrid curvilinear
42. meshes”, Modelling and Simulation in Engineering, vol. 2017, paper no. 1797561.
43. Karpenko, V.S., Vershinin, A.V., Levin, V.A., Zingerman, K.M. 2016. ”Some results of mesh
44. convergence estimation for the spectral element method of different orders in FIDESYS
45. industrial package”. IOP Conference Series: Materials Science and Engineering, vol. 158, no.
46. , paper no. 012049.
Review
For citations:
Levin V.A., Vershinin A.V., Zingerman K.M., Biryukov D.R. Exact solution to the problem of stage-by-stage deformation of a multilayer cylinder made of incompressible hypoelastic material. Chebyshevskii Sbornik. 2022;23(4):262-271. https://doi.org/10.22405/2226-8383-2022-23-4-262-271
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