Geodesic
Exact solution to the problem of stage-by-stage deformation of a multilayer cylinder made of incompressible hypoelastic material
Čebyševskij sbornik, Tome 23 (2022) no. 4, pp. 262-271

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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.
Keywords: hypoelastic material, multilayer cylinder, tension-compression, torsion, superposition of large strains, exact analytical solution. 
@article{CHEB_2022_23_4_a22,
     author = {V. A. Levin and A. V. Vershinin and K. M. Zingerman and D. R. Biryukov},
     title = {Exact solution to the problem of stage-by-stage deformation of a multilayer cylinder made of incompressible hypoelastic material},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {262--271},
     publisher = {mathdoc},
     volume = {23},
     number = {4},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2022_23_4_a22/}
}
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V. A. Levin; A. V. Vershinin; K. M. Zingerman; D. R. Biryukov. Exact solution to the problem of stage-by-stage deformation of a multilayer cylinder made of incompressible hypoelastic material. Čebyševskij sbornik, Tome 23 (2022) no. 4, pp. 262-271. http://geodesic.mathdoc.fr/item/CHEB_2022_23_4_a22/