One approach to the axisymmetric problem of impact of fine shells of the S.P. Timoshenko type on elastic half-space
DOI:
https://doi.org/10.32347/tit2021.42.0302Keywords:
impact, elastic, elastic-plastic, half-space, axisymmetric problem, fine, spherical shell, S.P. TimoshenkoAbstract
Refined model of S.P. Timoshenko makes it possible to consider the shear and the inertia rotation of the transverse section of the shell. Disturbances spread in the shells of S.P. Timoshenko type with finite speed. Therefore, to study the dynamics of propagation of wave processes in the fine shells of S.P. Timoshenko type is an important aspect as well as it is important to investigate a wave processes of the impact, shock in elastic foundation in which a striker is penetrating. The method of the outcoming dynamics problems to solve an infinite system of integral equations Volterra of the second kind and the convergence of this solution are well studied. Such approach has been successfully used for cases of the investigation of problems of the impact a hard bodies and an elastic fine shells of the Kirchhoff-Love type on elastic a half-space and a layer. In this paper an attempt is made to solve the axisymmetric problem of the impact of an elastic fine spheric shell of the S.P. Timoshenko type on an elastic half-space using the method of the outcoming dynamics problems to solve an infinite system of integral equations Volterra of the second kind. It is shown that this approach is not acceptable for investigated in this paper axisymmetric problem. The discretization using the Gregory methods for numerical integration and Adams for solving the Cauchy problem of the reduced infinite system of Volterra equations of the second kind results in a poorly defined system of linear algebraic equations: as the size of reduction increases the determinant of such a system to aim at infinity. This technique does not allow to solve plane and axisymmetric problems of dynamics for fine shells of the S.P. Timoshenko type and elastic bodies. This shows the limitations of this approach and leads to the feasibility of developing other mathematical approaches and models. It should be noted that to calibrate the computational process in the elastoplastic formulation at the elastic stage, it is convenient and expedient to use the technique of the outcoming dynamics problems to solve an infinite system of integral equations Volterra of the second kind.
References
Bogdanov V.R., 2018. Impact a circular cylinder with a flat on an elastic layer. Transfer of Innovative Technologies. Vol.1(2), 68-74.
Bogdanov V.R., 2017. Impact of a hard cylinder with flat surface on the elastic layer. Underwater Technologies, Vol.05, 8-15.
Bogdanov V.R., Lewicki H.R., Pry-hodko T.B., Radzivill O.Y., Sam-borska L.R., 2009. The planar problem of the impact shell against elastic layer. Visnyk NTU, Kyiv, Nr. 18, 281-292 (in Ukrainian).
Kubenko V.D., Bogdanov V.R., 1995. Planar problem of the impact of a shell on an elastic half-space. International Applied Me-chanics, 31, No. 6, 483-490.
Kubenko V.D., Bogdanov V.R., 1995. Axe-isym¬metric impact of a shell on an elastic halfspace. International Applied Mechanics, 31, No. 10, 829-835.
Kubenko V.D., Popov S.N., Bogdanov V.R., 1995. The impact of elastic cylindrical shell with the surface of elastic half-space. Dop. NAN Ukrainy, No. 7, 40-44 (in Ukrainian).
Kubenko V.D., Popov S.N., 1988. Plane prob¬lem of the impact of hard blunt body on the sur¬face of an elastic half-space. Pricl. Mechanika, 24, No. 7, 69-77 (in Russian).
Popov S.N., 1989. Vertical impact of the hard circular cylinder lateral surface on the elastic half-space. Pricl. Mechanika, 25, No. 12, 41-47 (in Russian).
Bogdanov V.R., Sulim G.T., 2016. Determine-na¬tion of the material fracture toughness by numerical analysis of 3D elastoplastic dy-namic defor¬mation. Mechanics of Solids, 51(2), 206-215; DOI 10.3103/S0025654416020084.
Bogdanov V.R., 2015. A plane problem of im¬pact of hard cylinder with elastic layer. Bulletin of University of Kyiv, Mathematics. Mechanics, No. 34, 42-47 (in Ukrainian).
Bogdanov V.R., Sulym G.T., 2013. Plain deformation of elastoplastic material with profile shaped as a compact specimen (dynamic load¬ing). Mechanics of Solids, May, 48(3), 329–336, DOI 10.3103/S0025654413030096.
Bogdanov V.R., Sulym G.T., 2013. A modeling of plastic deformation's growth under impact, based on a numerical solution of the plane stress deformation problem. Vestnik Moskovskogo Aviatsionnogo Insti-tuta, Vol. 20, Issue 3, 196-201 (in Russian).
Bogdanov V.R., 2009. Three dimension prob¬lem of plastic deformations and stresses concen¬tration near the top of crack. Bulletin of University of Kyiv, Series Physics & Mathematics, No. 2, 51-56 (in Ukrainian).
Bogdanov V.R., Sulym G.T., 2012. The plane strain state of the material with station-ary crack with taking in account the process of unloading. Mathematical Methods and Physicomechanical Fields, Lviv, 55, No. 3, 132-138 (in Ukrainian).
Bogdanov V.R., Sulym G.T., 2010. The crack growing in compact specimen by plas-tic-elastic model of planar stress state. Bulle-tin of Univer¬sity of Kyiv, Series Physics & Mathematics, No. 4, 58-62 (in Ukrainian).
Bogdanov V.R., Sulym G.T., 2010. The crack clevage simulation based on the numerical mod¬elling of the plane stress state. Bulletin of Uni¬versity of Lviv, Series Physics & Mathematics, No. 73, 192-204 (in Ukrainian).
Bohdanov V.R., Sulym G.T., 2011. Evaluation of crack resistance based on the numerical mod¬elling of the plane strained state. Material Sci¬ence, 46, No. 6, 723-732.
Bogdanov V.R., Sulym G.T., 2011. The clevage crack simulation based on the nu-merical model¬ling of the plane deformation state. Scientific collection Problems of Calculation Mechanics and Constructions Strength, Dnepropetrovsk, No. 15, 33-44 (in Ukrainian).
Bogdanov V.R., Sulym G.T., 2010. Destruction toughness determination based on the numerical modelling of the three dimension dynamic prob¬lem. International scientific collection Strength of Machines and Constructions, Kyiv, No. 43, 158-167 (in Ukrainian).
Bogdanov V.R., Sulym G.T., 2012. A three di¬mension simulation of process of growing crack based on the numerical solution. Scientific col¬lection Problems of Calculation Mechanics and Constructions Strength, Dnepropetrovsk, No. 19, 10-19 (in Ukrainian).
Bogdanov V.R., Sulym G.T., 2012. The crack clevage simulation in a compact specimen based on the numerical modelling of the three dimen¬sion problem. Scientific col-lection Methods of Solving Applied Prob-lems in Solid Mechanics, Dnepropetrovsk, No. 13, 60-68 (In Ukrainian).
Bogdanov V.R., 2011. About three dimensional deformation of an elastic-plastic material with the profile of compact shape. Theoretical and Applied Mechanics, Donetsk, No. 3 (49), 51-58 (in Ukrainian).
Fedotenkov G.V., 2001. Cylinder shell impact along elastic semi-plane. Moscow Aviation Institute, 100 p. (In Russian).
Mihailova E.Y., Tarlakovsky D.V., Vahterova Y.A., 2018. generalized linear model of the dynamics of thin elastic shells. Scientific Notes of Kazan University, Series of Physics and Mathematics, 160(3), 561-577 (in Russian).
Lokteva N.A., Serduk D.O., Skopintsev P.D., Fedotenkov G.V., 2020. Non-stationary stress-deformed state of a composite cylindrical shell. Mechanics of Composite Materials and Structures, 26(4), 544-559; DOI: 10.33113/mkmk.ras.2020.26.04.544_559.08 (in Russian).
Vestyak A.V., Igumnov L.A., Tarla-kovskii D.V., Fedotenkov G.V., 2016. The influence of non-stationary pressure on a thin spherical shell with an elastic filler. Computational Continuum Mechanics. 9(4), 443-452 (in Russian); DOI: 10.7242/1999-6691/2016.9.4.37.
Afanasyeva O.A. Mikhailova E.Y. Fe-dotenkov G.V., 2012. Random phase of con-tact interaction of a spherical shell and elas-tic half space. Problems of Computer Mechanics and Strength of Structures, 20, 19-26 (in Russian).
Michailova E.Y., Tarlakovsky D.V., Fedotenkov G.V., 2014. Non-stationary con-tact of a spherical shell and an elastic half-space. Works MAI. 78, 1-26 (in Russian).
Igumnov L.A., Okonechnikov A.S., Tarlakovskii D.V., Fedotenkov G.V. 2013. Plane nonsteady-state problem of motion of the surface load on an elastic half-space. Mathematical Methods and Physicomechani-cal Fields, Lviv, 56, No. 2, 157-163 (In Rus-sian).
Kuznetsova E.L., Tarlakovsky D.V., Fedotenkov G.V., Medvedsky A.L., 2013. Influence of non-stationary distributed load on the surface of the elastic layer. Works MAI. 71, 1-21. (In Russian)
Fedotenkov G.V., Tarlakovsky D.V., Vahterova Y.A., 2019. Identification of Non-stationary Load Upon Timoshenko Beam. Lobachevskii Journal of Mathematics, 40(4), 439-447.
Vahterova Y.A., Fedotenkov G.V., 2020. The inverse problem of recovering an unsteady linear load for an elastic rod of finite length. Journal of Applied Engineering Science, 18(4), 687-692, DOI:10.5937/jaes0-28073.
Gorshkov A.G., Tarlakovsky D.V., 1985. Dynamic contact problems with moving boundaries. Nauka, Fizmatlit, 352 (in Russian).
Sagomonian A.J., 1985. Stress waves in a continuous medium. Moscow University Publishing House, 416 (in Russian).
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2021 Vladislav Bogdanov
This work is licensed under a Creative Commons Attribution 4.0 International License.
Our journal abides by the CREATIVE COMMONS copyright rights and permissions for open access journals.
Authors, who are published in this journal, agree to the following conditions:
1. The authors reserve the right to authorship of the work and pass the first publication right of this work to the journal under the terms of a Creative Commons Attribution License, which allows others to freely distribute the published research with the obligatory reference to the authors of the original work and the first publication of the work in this journal.
2. The authors have the right to conclude separate supplement agreements that relate to non-exclusive work distribution in the form in which it has been published by the journal (for example, to upload the work to the online storage of the journal or publish it as part of a monograph), provided that the reference to the first publication of the work in this journal is included.
