Научная электронная библиотека ULRICH'S Periodicals Directory

Сибирский государственный университет
науки и технологий имени академика М.Ф. Решетнева

Сибирский аэрокосмический журнал
ISSN 2712-8970 (Print) ISSN 2782-5760 (on-line)

Сведения об образовательной организации

UDK 539.374
CONSTRUCTION OF ELASTO-PLASTIC BOUNDARIES using CONSERVATION LAWS
S. I. Senashov*, E. V. Filyushina, O. V. Gomonova
Reshetnev Siberian State Aerospace University 31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation *E-mail: sen@sibsau.ru
The solution of elasto-plastic problems is one of the most complicated and actual problems of solid mechanics. Traditionally, these problems are solved by the methods of complex analysis, calculus of variations or semi-inverse methods. Unfortunately, all these methods can be applied to a limited number of problems only. In this paper, a technique of conservation laws is used. This technique allows constructing analytical formulas to determine the elasto-plastic boundary for a wide class of problems. As a result, the elasto-plastic boundaries were constructed for twisted straight rods with cross sections limited by piecewise smooth contour, for flexible consoles with constant cross-sections, as well as for anti-plane problems. Computer programs for construction of elasto-plastic boundaries for twisted straight rods were written using obtained technique. In this work, the elasto-plastic boundary arising during the torsion of a straight beam of arbitrary cross section, which is limited by a piecewise smooth contour is constructed; and the elasto-plastic boundaries for the problems of a consol bending and anti-plane deformation are found. The plan of the paper is the following. In the first section the basic equations of elasticity and boundary problems are considered; in the second section the basic equations of the theory of ideal plasticity of von Mises are given; in the third section the conditions on the boundaries of the elastic and plastic domains are formulated. The fourth section is devoted to torsion of elastic prismatic rods; the fifth one describes elastic bending of bars; in the sixth section the plane problem of theory of elasticity is given. The seventh section covers an anti-plane problem of elasticity theory; in the eighth section, conservation laws for the equations of elasticity are constructed; in the ninth one, conservation laws of two-dimensional equations of plasticity are discussed. In the tenth section an elasto-plastic boundary of a twisted straight rod is found; in the eleventh one an elasto-plastic boundary in the bended console is given; and finally, in the twelfth section a method for the construction of elasto-plastic boundaries for large areas is described.
conservation laws, elasto-plastic boundary, exact solutions, elasticity, plasticity, elasto-plasticity.
References
  1. Annin B. D., Cherepanov G. P. Uprugo-plasticheskaya zadacha [The Elasto-plastic Problem]. Novosibirsk, Nauka Publ., 1983, 238 p.
  2. Galin L. A. [The plane elasto-plastic problem], PMM. 1946, Vol. 10, No. 3, P. 367–386 (In Russ.).
  3. Galin L. A. Uprugoplasticheskie zadachi [Elastoplastic problems]. Moscow, Nauka Publ.,1984, 232 p.
  4. Kiryakov P. P., Senashov S. I., Yahno A. N. Prilozhenie simmetriy I zakonov sokhraneniya k resheniyu differentsialnyikh uravneniy [Application of symmetries and conservation laws to solving of differential equations]. Novosibirsk, Izd-vo SO RAN Publ., 2001, 190 p.
  5. Senashov S. I., Gomonova O. V., Yahno A. N. Matematicheskie voprosyi dvumernyikh uravneniy idealnoy plastichnosti [Mathematical problems of the two-dimensional equations of ideal plasticity]. Krasnoyarsk, SibGAU Publ., 2012, 137 p.
  6. Senashov S. I., Yakhno A. N. Some Symmetry Group Aspects of a Perfect Plane Plasticity System. J. Phys. A: Math. Theor. 2013, No. 46, P. 355202.
  7. 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.
  8. Senashov S. I. Cherepanova O. N., Kondrin A. V. [An elasto-plastic torsion of rod]. Vestnik SibGAU, 2013, No. 3(49), P. 100–103 (In Russ.).
  9. Senashov S. I., Cherepanova O. N., Kondrin A. V., Filyushina E. V. Raschet napryazhennogo sostoyaniya vo vnutrennikh tochkakh uprugoplasticheskogo sterzhnya postoyannogo secheniya [The Calculation of the Stress State in the Interior points of Elasto-plastic Rod of Constant Cross-section]. Certificate of state registration of the computer number 2013618484, 2013, P. 1.
  10. Senashov S. I., Cherepanova O. N., Kondrin A. V., Yahno A. N., Filyushina E. V. Postroenie uprugoplasticheskoy granitsi, voznikayuschiy pri kruchenii pryamolineynogo sterzhnya s secheniem pryamougolnoy formy [Construction of the Elastoplastic Boundary Arising under the Torsion of a Straight Rod with a Rectangular Cross-section]. Certificate of state registration of the computer number 20146616472, 2014, P. 1.
  11. Arutyunyan N. H., Abramyan B. L. Kruchenie uprugikh tel [Torsion of Elastic Bodies]. Moscow, Fizmatgiz Publ., 1963, 688 p.
  12. Senashov S. I., Filyushina E. V. [Conservation Laws of the Plane Theory of Elasticity]. Vestnik SibGAU, 2014, No. 1(53), P. 79–81 (In Russ.).
  13. Senashov S. I., Kondrin A. V. [Development of an information system for determination of elastic-plastic boundary of rods of rolled profiles]. Vestnik SibGAU, 2014, No. 4(56), P. 119–125 (In Russ.).
  14. Senashov S. I., Yakhno A. N. Conservation Laws, Hodograph Transformation and Boundary Value Problems of Plane Plasticity. SIGMA. Special Issue “Geometrical Methods in Mathematical Physics”. 2012, Vol. 8, 16 p. Available at: http://dx.doi.org/10.3842/ SIGMA.2012.071.
  15. Senashov S. I., Yakhno A. N. Group analysis of Slutions of 2-dimensional Differential Equations. Lie Groups: New Research. Nova science publishers, New York, 2009, P. 123–138.
  16. Senashov S. I., Yakhno A. N. Deformation of Characteristic Curves of the Plane Ideal Plasticity Equations by Point Symmetries. Nonlinear analysis. 2009, No. 71. P. 1274–1284.
  17. Senashov S. I. [Conservation Laws in the Problem of the Longitudinal Plane Wave Loaded in the Elasto-plastic Rod]. Vestnik SibGAU, 2011, No. 3(36), P. 82–85 (In Russ.).

Senashov Sergei Ivanovich – Dr. Sc., professor, head of Department of information and economic systems, Reshetnev Siberian State Aerospace University. Е-mail: sen@sibsau.ru

Filyushina Elena Vladimirovna – Docent of Department of Information and economic systems, Reshetnev Siberian State Aerospace University. E-mail: filyushina@sibsau.ru

Gomonova Olga Valer’evna – Cand. Sc., Docent of Higher Methimatics Department, Reshetnev Siberian State Aerospace University. E-mail: gomonova@mail.ru