Email this article Torsion of prismatic orthotropic elastoplastic rods
- Authors: Burenin A.A.1, Senashov S.I.2, Savostyanova I.L.2
-
Affiliations:
- Khabarovsk Federal Research Center of the Far Eastern Branch of the Russian Academy of Sciences
- Reshetnev Siberian State University of Science and Technology
- Issue: Vol 22, No 1 (2021)
- Pages: 8-17
- Section: Section 1. Computer Science, Computer Engineering and Management
- URL: https://journals.eco-vector.com/2712-8970/article/view/562756
- DOI: https://doi.org/10.31772/2712-8970-2021-22-1-8-17
- ID: 562756
Cite item
Abstract
Conservation laws were introduced into the theory of differential equations by E. Noether more than 100 years ago and are gradually becoming an important tool for the study of systems of differential equations. They not only allow us to qualitatively investigate the equation, but, as shown by the authors of this article, they allow us to find exact solutions to boundary value problems. For the equations of the isotropic theory of elasticity, the conservation laws were first calculated by P. Olver. For the equations of the theory of plasticity in the two-dimensional case, the conservation laws were found by one of the authors of this article and used to solve the main boundary value problems of the plasticity equations. Later it turned out that the conservation laws can also be used to find the boundaries between elastic and plastic zones in twisted rods, bent beams, and deformable plates. In this paper, we find conservation laws for equations describing the orthotropic elastic-plastic state of a twisted rectilinear rod. It is assumed that the conserved current depends linearly on the components of the voltage tensor. In this paper, we find an infinite series of conservation laws, which allows us to find the elastic-plastic boundary that occurs when an orthotropic rod is twisted.
About the authors
Anatoly A. Burenin
Khabarovsk Federal Research Center of the Far Eastern Branch of the Russian Academy of Sciences
Author for correspondence.
Email: sen@sibsau.ru
Corresponding Member of the Russian Academy of Sciences, Dr. Sc., Professor; Chief Researcher
Russian Federation, 51, Turgenev St., Khabarovsk, 680000Sergei I. Senashov
Reshetnev Siberian State University of Science and Technology
Email: sen@sibsau.ru
Dr. Sc., Professor, Head of the Department of IES
Russian Federation, 31, Krasnoiarskii Rabochi Prospekt, Krasnoyarsk, 660037Irina L. Savostyanova
Reshetnev Siberian State University of Science and Technology
Email: savostyanova@sibsau.ru
Cand. Sc., Associate Professor of the Department of IES
Russian Federation, 31, Krasnoiarskii Rabochi Prospekt, Krasnoyarsk, 660037References
- Kiryakov P. P., Senashov S. I., Yakhno A. N. Prilozhenie simmetrij i zakonov sohraneniya k resheniyu differencial'nyh uravneniy [Application of symmetries and conservation laws to the solution of differential equations]. Novosibirsk; Nauka Publ., 2001, 192 p.
- Senashov S. I., Vinogradov A. M. Symmetries and conservation laws of 2-dimensional ideal plasticity Proc. Edinburgh Math. Soc. 1988, P. 415–439.
- Vinogradov A. M., Krasilshchik I. S., Lychagin V. V. Simmetrii i zakony sohraneniya [Symmetries and conservation laws]. Moscow, Factorial Publ., 1996, 380 p.
- Annin B. D., Bytev V. O., Senashov S. I. Gruppovye svojstva uravnenij uprugosti i plastichnosti [Group properties of equations of elasticity and plasticity]. Novosibirsk, Nauka Publ., 1983, 239 p.
- Senashov S. I., Gomonova O. V., Yakhno A. N. Matematicheskie voprosy dvumernyh uravnenij ideal'noj plastichnosti [Mathematical problems of two-dimensional equations of ideal plasticity]. Krasnoyarsk, 2012, 139 p.
- Senashov S. I., Vinogradov A. M. Symmetries and conservation laws of 2-dimensional ideal plasticity Proc. Edinburg Math.Soc. 1988, P. 415–439.
- Olver P. Conservation laws in elasticity 1. General result. Arch. Rat. Mech. Anal. 1984, No. 85, P. 111–129.
- Olver P. Conservation laws in elasticity 11. Linear homogeneous isotropic elastostatic. Arch. Rat. Mech. Anal. 1984, No. 85, P. 131–160.
- Senashov S. I., Savostyanova I. L. Elastic state of a plate with holes of arbitrary shape Vestnik CHuvashskogo gosudarstvennogo pedagogicheskogo universiteta im. I. YA. Yakovleva. Seriya: Mekhanika predel'nogo sostoyaniya. 2016. No. 3 (29), P. 128–134. (In Russ.)
- Senashov S. I., Kondrin A. V. Development of an information system for finding the elastic-plastic boundary of rolling profile rods. Vestnik SibGAU. 2014, No. 4(56), P. 119–125. (In Russ.)
- Senashov S. I., Filyushina E. V., Gomonova O. V. Construction of elastic-plastic boundaries with the help of conservation laws. Vestnik SibGAU. 2015, Vol. 16, No. 2, P. 343–359. (In Russ.)
- Senashov S. I., Cherepanova O. N., Kondrin A.V. On elastic-plastic torsion of the rod Vestnik SibGAU. 2013, Vol. 3(49), P. 100–103. (In Russ.)
- Senashov S. I., Cherepanova O. N., Kondrin A.V. Elastoplastic Bending of Beam. J. Siberian Federal Univ., Math. & Physics. 2014, No. 7(2), P. 203–208.
- Senashov S. I., Cherepanova O. N., Kondrin A.V. On Elastoplastic Torsion of a Rod with Multiply Connected Cross-Section J. Siberian Federal Univ., Math. & Physics. 2015, No. 7(1), P. 343–351.
- Senashov S. I., Gomonova O. V. Construction of elastoplastic boundary in problem of tension of a plate weakened by holes International Journal of Non-Linear Mechanics. 2019, Vol. 108, P. 7–10.
- Lekhnitsky S. G. Teoriya uprugosti anizotropnogo tela [Theory of elasticity of an anisotropic body]. Moscow, Nauka Publ., 1977, 416 p.
Supplementary files
