Email this article Construction of elastic fields in the problem from the action of body forces of a cyclic nature
- Authors: Ivanychev D.A.1, Levina E.Y.2
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Affiliations:
- Lipetsk State Technical University
- Bauman Moscow State Technical University
- Issue: Vol 28, No 1 (2024)
- Pages: 59-72
- Section: Mechanics of Solids
- URL: https://journal-vniispk.ru/1991-8615/article/view/310999
- DOI: https://doi.org/10.14498/vsgtu2064
- EDN: https://elibrary.ru/IVANRN
- ID: 310999
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Abstract
The paper presents a method for determining the stress-strain state of transversely isotropic bodies of revolution under the action of non-axisymmetric stationary volumetric forces. This problem involves the use of boundary state method definitions. The basis of the space of internal states is formed using fundamental polynomials. The polynomial is placed in any position of the displacement vector of the plane auxiliary state, and the spatial state is determined by the transition formulaes. The set of such states forms a finite-dimensional basis according to which, after orthogonalization, the desired state is expanded into Fourier series with the same coefficients. Series coefficients are scalar products of vectors of given and basic volumetric forces. Finally, the search for an elastic state is reduced to solving quadratures.
The solutions of problems of the theory of elasticity for a transversely isotropic circular cylinder from the action of volumetric forces given by various cyclic laws (sine and cosine) are analyzed. Recommendations are given for constructing the basis of internal states depending on the form of the function of given volumetric forces. The analysis of the series convergence and the estimation of the solution accuracy in graphical form are given.
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##article.viewOnOriginalSite##About the authors
Dmitriy A. Ivanychev
Lipetsk State Technical University
Author for correspondence.
Email: lsivdmal@mail.ru
ORCID iD: 0000-0002-7736-9311
SPIN-code: 5839-4063
Scopus Author ID: 57201671293
ResearcherId: 57198777321
https://www.mathnet.ru/person153196
Cand. Phys. & Math. Sci.; Associate Professor; Institute of Mechanical Engineering and Transport
Russian Federation, 398055, Lipetsk, Moskovskaya st., 30Ekaterina Yu. Levina
Bauman Moscow State Technical University
Email: hensi-l@yandex.ru
ORCID iD: 0000-0001-6193-9036
SPIN-code: 8100-5287
Scopus Author ID: 57198777321
https://www.mathnet.ru/person209459
Associate Professor, Associate Professor, Department of Physics
Russian Federation, 105005, Moscow, 2-ya Baumanskaya st., 5References
- Vestyak V. A., Tarlakovskii D. V. Unsteady axisymmetric deformation of an elastic thickwalled sphere under the action of volume forces, J. Appl. Mech. Techn. Phys., 2015, vol. 56, no. 6, pp. 984–994. EDN: WPQXUN. DOI: https://doi.org/10.1134/S0021894415060085.
- Fukalov A. A. Problems of elastic equilibrium of composite thick-walled transversely isotropic spheres under the action of mass forces and internal pressure, and their applications, In: Proc. of XI Russian Congress on Fundamental Problems of Theoretical and Applied Mechanics. Kazan, 2015, pp. 3951–3953 (In Russian). EDN: UXGYNX.
- Zaitsev A. V., Fukalov A. A. Exact analytical solutions of problems on the equilibrium of elastic anisotropic bodies with central and axial symmetry located in the field of gravitational forces, and their applications to problems of geomechanics, In: Proc. of Russian Conference on Mathematical Modeling in Natural Sciences. Perm, 2015, pp. 141–144 (In Russian). EDN: UMDSHT.
- Agakhanov E. K. On the development of complex methods for solving problems of the mechanics of a deformable solid body, Vestn. Dagestan Gos. Tekhn. Univ. Tekhn. Nauki, 2013, no. 2, pp. 39–45 (In Russian). EDN: SCMJQR.
- Sharafutdinov G. Z. Functions of a complex variable in problems in the theory of elasticity with mass forces, J. Appl. Mech. Techn. Phys., 2009, vol. 73, no. 1, pp. 48–62. EDN: WRUJZL. DOI: https://doi.org/10.1016/j.jappmathmech.2009.03.008.
- Struzhanov V. V. On the solution of boundary value problems of the theory of elasticity by the method of orthogonal projections, Vestn. Perm. Gos. Tekhn. Univ. Matem. Model. Sist. Prots., 2004, no. 12, pp. 89–100 (In Russian). EDN: PBHAIN.
- Kuz’menko V. I., Kuz’menko N. V., Levina L. V., Pen’kov V. B. A method for solving problems of the isotropic elasticity theory with bulk forces in polynomial representation, Mech. Solids, 2019, vol. 54, no. 5, pp. 741–749. EDN: YXZLRX. DOI: https://doi.org/10.3103/S0025654419050108.
- Pen’kov V. B., Levina L. V., Novikova O. S. Analytical solution of elastostatic problems of a simply connected body loaded with nonconservative volume forces: theoretical and algorithmic support, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2020, vol. 24, no. 1, pp. 56–73 (In Russian). EDN: IUYYDV. DOI: https://doi.org/10.14498/vsgtu1711.
- Ivanychev D. A. A method of boundary states in a solution to the first fundamental problem of the theory of anisotropic elasticity with mass forces, Vestn. Tomsk. Gos. Univ. Matem. Mekhanika [Tomsk State Univ. J. Math. Mech.], 2020, no. 66, pp. 96–111 (In Russian). EDN: DFNLEW. DOI: https://doi.org/10.17223/19988621/66/8.
- Ivanychev D. A. The method of boundary states in the solution to the second fundamental problem of the theory of anisotropic elasticity with mass forces, Vestn. Tomsk. Gos. Univ. Matem. Mekhanika [Tomsk State Univ. J. Math. Mech.], 2019, no. 61, pp. 45–60 (In Russian). EDN: GLTGDZ. DOI: https://doi.org/10.17223/19988621/61/5.
- Ivanychev D. A. The contact problem Solution of the elasticity theory for anisotropic rotation bodies with mass forces, PNRPU Mechanics Bulletin, 2019, no. 2, pp. 49–62 (In Russian). EDN: XMFCKQ. DOI: https://doi.org/10.15593/perm.mech/2019.2.05.
- Ivanychev D. A., Levina E. Yu. Solution of thermoelasticity problems for solids of revolution with transversal isotropic feature and a body force, J. Phys.: Conf. Ser., 2019, vol. 1348, 012058. EDN: CTQIWR. DOI: https://doi.org/10.1088/1742-6596/1348/1/012058.
- Ivanychev D. A. The method of boundary states in solving problems of thermoelasticity in the presence of mass forces, In: Proc. of International Conference on Control Systems, Mathematical Modelling, Automation and Energy Efficiency. Lipetsk, 2019, pp. 83–87. DOI: https://doi.org/10.1109/SUMMA48161.2019.8947505.
- Aleksandrov A. Ya., Solov’ev Yu. I. Prostranstvennye zadachi teorii uprugosti (primenenie metodov teorii funktsii kompleksnogo peremennogo) Spatial Problems in the Elasticity Theory (Application of Methods of the Theory of Functions of a Complex Variable). Moscow, Nauka, 1978, 464 pp. (In Russian)
- Lur’e A. I. Three-dimensional Problems of the Theory of Elasticity. New York, Interscience Publ., 1964, xii+493 pp.
- Pen’kov V. B., Pen’kov V. V. Boundary conditions method for solving linear mechanics problems, Far Eastern Math. J., 2001, vol. 2, no. 2, pp. 115–137 (In Russian). EDN: EQVLZP.
- Satalkina L. V. Increasing the basis of the state space under strict restrictions to the energy intensity of calculations, In: Collection of Abstracts of the Scientific Conference. Lipetsk, Lipetsk State Univ., 2007, pp. 130–131 (In Russian).
- Lekhnitsky S. G. Teoriia uprugosti anizotropnogo tela [Theory of Elasticity of Anisotropic Body]. Moscow, Nauka, 1977, 416 pp. (In Russian)
- Levina L. V., Novikova O. S., Pen’kov V. B. Full-parameter solution of the problem of the theory of elasticity of a simply connected bounded body, Vestn. Lipetsk. Gos. Tekhn. Univ., 2016, no. 2, pp. 16–24 (In Russian). EDN: WEEWJN.
- Vikharev D. Yu., Rodin N. A. Model of implicit pole electric machine based on mathematical formulation of magnetic field in air gap, Vestnik IGEU, 2021, no. 6, pp. 27–37 (In Russian). EDN: SSJQHM. DOI: https://doi.org/10.17588/2072-2672.2021.6.027-037.
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