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
ABOUT TORSION OF PARALLELEPIPED AROUND THREE AXIS
S. I. Senashov*, I. L. Savostyanova, E. V. Filyushina
Reshetnev Siberian State University of Science and Technology 31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation *Е-mail: sen@sibsau.ru
The theory of limit state deals with statically determinate condition of solids. In this case the system is closed due to extreme conditions, such properties of matter such as viscosity, elasticity, etc. cannot influence the limit state. In other words, when reaching the limit state the nature of the relationship between stress and strain has no effect on the ultimate state. The study of such systems has been consistently pursued by D. D. Ivlev and his coauthors. To the equilibrium equations they attached two or an equation relating the components of the stress tensor. This led to the closure of the system of equilibrium equations. In the theory of plasticity equations, which are closed with a single yield stress are studied well. The most well-known system describing the ultimate state of deformable bodies are well-studied equations describing the torsion of the plastic bodies, the two-dimensional stationary problem of the theory of plasticity. The article discusses some other systems of equations which are closed only by one equation of flow, which corresponds to the classical theory of plasticity. It is assumed that the components of the velocity vector depend only on two spatial coordinates. In addition, for the component of velocity vector conditions of deformations compatibility are performed identically. The constructed systems can be used to describe the twisting of the parallelepiped around the three orthogonal axes. For the constructed system of equations point group symmetries, conservation laws have been found. It is shown that the system allows 8 -dimensional Lie algebra. On the basis of the symmetry group some classes of invariant solutions of rank 1 have been constructed. They depend on arbitrary functions of one variable. It is shown that these solutions can be used to describe plastic torsion of a parallelepiped around three orthogonal axes. It is shown that the system admits infinite series of conservation laws. The concluding paragraph describes the construction of elastic solutions to the problem. It is shown that it boils down to finding three harmonic functions.
Keyword: plasticity theory, limit state, the exact solutions.
References

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2. Ivlev D. D. Teoriya ideal’noy plastichnosti [Theory of ideal plasticity]. Moscow, Nauka Publ., 1966, 232 p.

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5. Ovsyannikov L. V. Group Analysis of Differential Equations. Academic Press, NewYork, 1982.

6. Senashov S. I., Yakchno A. N. Reproduction of solutions for bidimensional ideal plasticity. Journal of Non-Linear Mechanics. 2007, Vol. 42, P. 500–503.

7. Senashov S. I., Yakchno A. N. Deformation of characteristic curves of the plane ideal plasticity equations by point symmetries. Nonlinear analysis. 2009, Vol. 71, P. 1274–1284.

8. Senashov S. I., Cherepanova O. N. [New classes of solutions of the minimal surface equations]. Journal of Siberian Fed. Univ., Math. & Ph. 2010, No. 3(2), P. 248– 255 (In Russ.).

9. Senashov S. I Conservation Laws, Hodograph Transformation and Boundary Value Problems of Plane Plasticity. SIGMA. 2012, Vol. 8, P. 16.

10. Senashov S. I., Yakchno A. N. Some symmetry group aspects of a perfect plane plasticity system, J. Phys. A: Math. Theor. 2013, Vol. 46, No. 355202.

11. Senashov S. I., Yakchno A. N. Conservation Laws of Three-Dimensional Perfect Plasticity Equations under von Mises Yield Criterion. Abstract and Applied Analysis. 2013, Article ID 702132, 8 p.

12. Senashov S. I, Kondrin A. V., Cherepanova O. N. On Elastoplastic Torsion of a Rod with Multiply Connected Cross-Section. J. Siberian Federal Univ., Math. & Physics. 2015, No. 7(1), P. 343–351.

13. Senashov S. I., Cherepanova O. N., Kondrin A. V. Elastoplastic Bending of Beam. J. Siberian Federal Univ., Math. & Physics. 2014, Vol. 7(2), P. 203–208.

14. Senashov S. I., Filyushina E. V., Gomonova O. V. Sonstruction of elasto-plastic boundaries using conservation laws. Vestnik SibGAU. 2015, Vol. 16, No. 2, P. 343–360 (In Russ.).

15. Senashov S. I., Vinogradov A. M. Symmetries and Conservation Laws of 2-Dimensional Ideal Plasticity. Proc. of Edinb. Math. Soc. 1988, Vol. 31, P. 415–439.


Senashov Sergey Ivanovich – Dr. Sc., professor, head of Department of Nuclear power engineering, Reshetnev

Siberian State University of Science and Technology. E-mail: Sen@sibsau.ru.

Sevostyanova Irina Leonidovna – Cand. Sc., Docent, IES department, Reshetnev Siberian State University of

Science and Technology. E-mail: Sen@sibsau.ru.

Filyushina Elena Vladimirovna – Cand. Sc., Docent, Department of Nuclear Power Engineering, Reshetnev

Siberian State University of Science and Technology. E-mail: Filyushina@sibsau.ru.