Axially symmetric elasticity problems for the hollow cylinder with the stress-free ends. Analytical solving via a variational method of homogeneous solutions

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dc.citation.epage63
dc.citation.issue1
dc.citation.spage48
dc.contributor.affiliationIнститут прикладних проблем механiки i математики iм. Я. С. Пiдстригача НАН України
dc.contributor.affiliationКуявсько-Поморський Унiверситет у Бидгощi
dc.contributor.affiliationPidstryhach Institute for Applied Problems of Mechanics and Mathematics,National Academy of Sciences of Ukraine
dc.contributor.affiliationKujawy and Pomorze University in Bydgoszcz
dc.contributor.authorЧекурін, В. Ф.
dc.contributor.authorПостолакі, Л. І.
dc.contributor.authorChekurin, V. F.
dc.contributor.authorPostolaki, L. I.
dc.date.accessioned2023-03-06T12:28:23Z
dc.date.available2023-03-06T12:28:23Z
dc.date.created2020-01-01
dc.date.issued2020-01-01
dc.description.abstractРозглянуто осесиметричну задачу для порожнистого цилiндра iз ненавантаженими основами. На внутрiшнiй i зовнiшнiй цилiндричних поверхнях задано нормальнi i тангенцiальнi навантаження. Задачу зведено до бiгармонiчного рiвняння з вiдповiдними крайовими умовами. За допомогою методу вiдокремлення змiнних отримано однорiдну крайову задачу для звичайного диференцiального рiвняння. Використовуючи власнi функцiї цiєї задачi, побудувано систему однорiдних розв’язкiв вихiдної бiгармонiчної задачi. Її розв’язок, який поданий як розвинення за цими функцiями, залежить вiд чотирьох безмежних послiдовностей невизначених дiйсних констант. Для визначення невiдомих констант застосовано варiацiйний метод, згiдно з яким пiдпорядкування розв’язку крайовим умовам, що заданi на цилiндричних поверхнях, здiйснюється не поточково, а “в середньому” — за нормою L2. З цiєю метою введено функцiонал, який визначає середньоквадратичне вiдхилення розв’язку вiд крайових умов, що заданi на цилiндричних поверхнях. У результатi отримано безмежну систему алгебраїчних рiвнянь, яку розв’язано за допомогою методу редукцiї. Проведенi кiлькiснi дослiдження пiдтвердили добру збiжнiсть методу.
dc.description.abstractAn axially symmetric problem for a hollow cylinder with unloaded bases is considered. On the inner and outer cylindrical surfaces, the normal and tangential loads are prescribed. The problem is reduced to a biharmonic equation with corresponding boundary conditions. Application of the method of variables separation results in a homogeneous boundary value problem for the ordinary differential equation. Its eigenfunctions have been used to construct an infinite system of homogeneous solutions for the initial biharmonic problem. Its solution, represented as a series expansion in terms of homogeneous solutions, depends on four infinite sequences of real constants. To determine them, the variational method has been applied, in which the subordination of the solution to the boundary conditions, given on cylindrical surfaces, is performed in the norm L2. It brings to an infinite system of algebraic equations which has been solved by the reduction method. The quantitative studies have confirmed the good convergence of the method.
dc.format.extent48-63
dc.format.pages16
dc.identifier.citationChekurin V. F. Axially symmetric elasticity problems for the hollow cylinder with the stress-free ends. Analytical solving via a variational method of homogeneous solutions / Chekurin V. F., Postolaki L. I. // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2020. — Vol 7. — No 1. — P. 48–63.
dc.identifier.citationenChekurin V. F., Postolaki L. I. (2020) Axially symmetric elasticity problems for the hollow cylinder with the stress-free ends. Analytical solving via a variational method of homogeneous solutions. Mathematical Modeling and Computing (Lviv), vol. 7, no 1, pp. 48-63.
dc.identifier.doiDOI: 10.23939/mmc2020.01.048
dc.identifier.urihttps://ena.lpnu.ua/handle/ntb/57520
dc.language.isoen
dc.publisherВидавництво Львівської політехніки
dc.publisherLviv Politechnic Publishing House
dc.relation.ispartofMathematical Modeling and Computing, 1 (7), 2020
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dc.relation.references[13] Chekurin V. F., Postolaki L. I. A variational method for the solution of biharmonic problems for a rectangular domain. Journal of Mathematical Sciences. 160 (3), 386–399 (2009).
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dc.relation.references[15] Chekurin V. F., Postolaki L. I. Theoretical-experimental determination of residual stresses in plane joints. Material Science. 45, 318–328 (2009).
dc.relation.references[16] Chekurin V. F., Pokhmurs’ka H. V. Field of residual stresses in a coating with crack. Materials Science. 42(2), 233–242 (2006).
dc.relation.references[17] Chekurin V. F. Inverse problem of nondestructive control of the level of hardening of sheet glass. Mechanics of solids. 33 (3), 68–77 (1998).
dc.relation.references[18] Chekurin V. F. Variational method for the solution of the problems of tomography of the stressed state of solids. Materials Science. 35 (5), 623–633 (1999).
dc.relation.references[19] Chekurin V. F., Kravchyshyn O. Z. Inverse problem for acoustical tomography of stress fields in piecewisehomogeneous strip. In Proceedings of 8th International Seminar. Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory DIPED (2003).
dc.relation.references[20] Chekurin V. F., Postolaki L. I. Variational method of homogeneous solutions in axisymmetric elasticity problems for a semiinfinite cylinder. Journal of Mathematical Sciences. 201 (2), 175–189 (2014).
dc.relation.references[21] Chekurin V. F., Postolaki L. I. A variational method of homogeneous solutions for axisymmetric elasticity problems for cylinder. Mathematical modeling and Computing. 2 (2), 128–132 (2015).
dc.relation.references[22] Chekurin V., Postolaki L. Application of the least square method in axisymmetric biharmonic problems. Mathematical Problems in Engineering. 2016, Article ID 3457649, 1–9 (2016).
dc.relation.references[23] Chekurin V., Postolaki L. Residual stresses in a finite cylinder. Direct and inverse problems and their solving using the variational method of homogeneous solutions. Mathematical Modeling and Computing.5 (2), 119–133 (2018).
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dc.relation.references[26] NowackiW. Theory of Elasticity. Moscow, Mir (1975), (in Russian).
dc.relation.references[27] Aben H. Integrated Photoelasticity. New York, McGraw-Hill (1979).
dc.relation.references[28] Chekurin V. F. Integral photoelasticity relations for inhomogeneously strained dielectrics. Mathematical Modeling and Computing. 1 (2), 144–155 (2014).
dc.relation.referencesen[1] Collin F., Caillerie D., Chambon R. Analytical solutions for the thick-walled cylinder problem modeledwith an isotropic elastic second gradient constitutive equation. International Journal of Solids and Structures.46 (22–23), 3927–3937 (2009).
dc.relation.referencesen[2] Zang W. The Elastic Solution of a Radial Heterogeneous Cylinder Subjected to Non-Uniform Distributed Normal and Tangential Loads. Mathematical Problems in Engineering. 2018, Article ID 9230609, 8 p.(2018).
dc.relation.referencesen[3] Chatzigeorgiou G., Charalambakis N., Murat F. Homogenization problems of a hollow cylinder made of elastic materials with discontinuous properties. International Journal of Solids and Structures. 45 (18–19), 5165–5180 (2008).
dc.relation.referencesen[4] Theotokoglou E. E., Stampouloglou I. H. The radially nonhomogeneous elastic axisymmentric problem. International Journal of Solids and Structures. 45 (25–26), 6535–6552 (2008).
dc.relation.referencesen[5] Avci A., Bulu A., Yapici A. Axisymmetric smooth contact for an elastic isotropic infinite hollow cylinder compressed by an outer rigid ring with circular profile. Acta Mechanica Sinica. 22 (1), 46–53 (2006).
dc.relation.referencesen[6] Tokovyy Yu. V., Ma Chien-Ching. Analysis of residual stresses in a long hollow cylinder. International Journal of Pressure Vessels and Piping. 88 (5–7), 248–255 (2011).
dc.relation.referencesen[7] Ramesh M., Kailas S. V., Simha K. R. Y. Axisymmetric fretting analysis in coated cylinder. Sadhana. 33,299–327 (2008).
dc.relation.referencesen[8] Grinchenko V. T. The axisymmetric problem of the theory of elasticity for a thick-walled cylinder of finite length. Soviet Applied Mechanics. 3 (8), 65–70 (1967).
dc.relation.referencesen[9] Bazarenko N. A. The contact problem for hollow and solid cylinders with stress-free faces. Journal of Applied Mathematics and Mechanics. 72, 214–225 (2008).
dc.relation.referencesen[10] Lurie S. A., Vasiliev V. V. The biharmonic problem in the theory of elasticity. Gordon and Breach, Australia etc. (1995).
dc.relation.referencesen[11] Meleshko V. V. Selected topics in the history of the two-dimensional biharmonic problem. Applied Mechanics Reviews. 56 (1), 33–85 (2003).
dc.relation.referencesen[12] Chekurin V. F. Variational method for solving direct and inverse problems of elasticity for a semi-infinite strip. Izv. Akad. Nauk. Mekh. Tverd. Tela. 2, 58–70 (1999). [Mech. Solids (Engl. Transl.) 34 (2), 49–59(1999)].
dc.relation.referencesen[13] Chekurin V. F., Postolaki L. I. A variational method for the solution of biharmonic problems for a rectangular domain. Journal of Mathematical Sciences. 160 (3), 386–399 (2009).
dc.relation.referencesen[14] Kantorovich L. V., Krylov V. I. Approximate methods of higher analysis. Translated from the 3rd Russian Edition by C. D. Benster. New York, Interscience Publ. (1958).
dc.relation.referencesen[15] Chekurin V. F., Postolaki L. I. Theoretical-experimental determination of residual stresses in plane joints. Material Science. 45, 318–328 (2009).
dc.relation.referencesen[16] Chekurin V. F., Pokhmurs’ka H. V. Field of residual stresses in a coating with crack. Materials Science. 42(2), 233–242 (2006).
dc.relation.referencesen[17] Chekurin V. F. Inverse problem of nondestructive control of the level of hardening of sheet glass. Mechanics of solids. 33 (3), 68–77 (1998).
dc.relation.referencesen[18] Chekurin V. F. Variational method for the solution of the problems of tomography of the stressed state of solids. Materials Science. 35 (5), 623–633 (1999).
dc.relation.referencesen[19] Chekurin V. F., Kravchyshyn O. Z. Inverse problem for acoustical tomography of stress fields in piecewisehomogeneous strip. In Proceedings of 8th International Seminar. Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory DIPED (2003).
dc.relation.referencesen[20] Chekurin V. F., Postolaki L. I. Variational method of homogeneous solutions in axisymmetric elasticity problems for a semiinfinite cylinder. Journal of Mathematical Sciences. 201 (2), 175–189 (2014).
dc.relation.referencesen[21] Chekurin V. F., Postolaki L. I. A variational method of homogeneous solutions for axisymmetric elasticity problems for cylinder. Mathematical modeling and Computing. 2 (2), 128–132 (2015).
dc.relation.referencesen[22] Chekurin V., Postolaki L. Application of the least square method in axisymmetric biharmonic problems. Mathematical Problems in Engineering. 2016, Article ID 3457649, 1–9 (2016).
dc.relation.referencesen[23] Chekurin V., Postolaki L. Residual stresses in a finite cylinder. Direct and inverse problems and their solving using the variational method of homogeneous solutions. Mathematical Modeling and Computing.5 (2), 119–133 (2018).
dc.relation.referencesen[24] Saad H. M. Elasticity. Theory, Applications and Numerics. Academic Press (2005).
dc.relation.referencesen[25] Arfken G. "Hankel Functions." §11.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 604–610 (1985).
dc.relation.referencesen[26] NowackiW. Theory of Elasticity. Moscow, Mir (1975), (in Russian).
dc.relation.referencesen[27] Aben H. Integrated Photoelasticity. New York, McGraw-Hill (1979).
dc.relation.referencesen[28] Chekurin V. F. Integral photoelasticity relations for inhomogeneously strained dielectrics. Mathematical Modeling and Computing. 1 (2), 144–155 (2014).
dc.rights.holder©2020 Lviv Polytechnic National University CMM IAPMM NASU
dc.subjectадачi теорiї пружностi
dc.subjectпорожнистий цилiндр
dc.subjectварiацiйний метод однорiдних розв’язкiв
dc.subjectelasticity problem
dc.subjecthollow cylinder
dc.subjectvariational method of homogeneous solutions
dc.subject.udc35J30
dc.titleAxially symmetric elasticity problems for the hollow cylinder with the stress-free ends. Analytical solving via a variational method of homogeneous solutions
dc.title.alternativeОсесиметрична задача теорії пружності для порожнистого циліндра з ненавантаженими основами. Аналітичне розв’язування із використанням варіаційного методу однорідних розв’язків
dc.typeArticle

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Mathematical Modeling And Computing. – 2020. – Vol. 7, No. 1