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 Doi: 10.31772/2587-6066-2019-20-3-320-326
SYSTEM ANALYSIS OF DYNAMIC PROBLEMS OF ANISOTROPIC PLASTICITY THEORY
S. I. Senashov, I. L. Savostyanova, O. N. Cherepanova
Reshetnev Siberian State University of Science and Technology, 31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation; Siberian Federal University, 79, Svobodniy Av., Krasnoyarsk, 660041, Russian Federation. E-mail: sen@sibsau.ru
Dynamic problems are the least studied area of plasticity theory. These problems arise in various fields of engineering and science, but the complexity of the original differential equations do not allow to develop accurate solutions and correctly solve numerical boundary value problems. This is even more typical of dynamic equations of anisotropic plasticity. Anisotropy reduces the group of symmetries allowed by the equations, and therefore narrows the number of invariant solutions. One-dimensional dynamic plasticity problems are well studied, but two-dimensional problems cause insurmountable mathematical difficulties due to the nonlinearity of the basic equations, even in the isotropic case. The study of the symmetries of the plasticity equations allowed us to find some exact solutions. The most known solution was found by B. D. Annin, who described the unsteady compression of a plastic layer made of isotropic material by rigid plates. Annin's solution is linear in two spatial variables, however, it includes arbitrary functions of time. Symmetries are also used in the proposed work. Point symmetries are first calculated for dynamic plasticity equations in the anisotropic case and are presented in the paper. The Lie algebra generated by the found symmetries appeared to be infinite-dimensional. This circumstance made it possible to apply the method of constructing new classes of nonstationary solutions. Symmetry can transform the exact solution of stationary dynamic equations in non-stationary solutions. The framed solutions include arbitrary functions and arbitrary constants. The outline of the article is as follows: according to the method of Lie group of point symmetries allowed by the equations of anisotropic plasticity is calculated. Two classes of new stationary invariant solutions are framed. These stationary solutions, by means of transformations generated by point symmetries, are transformed into new non-stationary solutions. In conclusion, a new selfsimilar solution of unsteady equations of anisotropic plasticity is framed; Annin's solution is generalized for the anisotropic case. The framed solutions can be used to describe the compression of plastic material between rigid plates, as well as to test programs, designed to solve anisotropic plastic problems.
Keywords: anisotropic plasticity, dynamics, symmetries, exact solutions.
References

1. Ivlev D. D., Maksimova L. A., Nepershin R. I. Predelnoe sostoyanie deformirovannykh tel i gornykh porod
[The ultimate state of deformed bodies and rocks]. Moscow, Fizmatlit Publ., 2008, 832 p.
2. Polyanin A. D., Zaittsev V. F. Handbook of nonlinear partial differential equations. CRC Press. London,
New York, Second Editional, 2012, 1876 p.
3. Novatskiy V. K. Volnovye zadachi teorii plastichnosti [Wave problems of the theory of plasticity]. Moscow,
Mir Publ., 1978, 307 p.
4. Zadoyan M.A. Prostranstvennye zadachi teorii plastichnosti [Spatial problems of the theory of plasticity]
Moscow, Nauka Publ., 1992. 382 p.
5. Ishlinskiy A. Yu., Ivlev D. D. Matematicheskaya teoriya plastichnosti [Mathematical theory of plasticity].
Moscow, Fizmatlit Publ., 2001, 704 p.
6. Annin B. D., Bytev V. O., Senashov S. I. Gruppovye svoystva uravneniy uprugosti i plastichnosti [Group
properties of equations of elasticity and plasticity]. Novosibirsk, Nauka Publ., 1985, 142 p.
7. Senashov S. I., Savostyanova I. L. [New solutions of the dynamic equations of perfect plasticity]. Sibirskiy
zhurnal industrial'noy matematiki. T. 22, No. 4(80), Р. 89–91.
8. Kiryakov P. P., Senashov S. I., Yakhno A. N. Prilozhenie simmetriy i zakonov sokhraneniya k
resheniyu differentsial'nykh uravneniy [The application of symmetries and conservation laws to the solution of differential
equations]. Novosibirsk, Izd-vo SO RAN Publ., 2001, 193 p.
9. Senashov S. I., Gomonova O. V., Yakhno A. N. Matematicheskie voprosy dvumernykh uravneniy
ideal'noy plastichnosti [Mathematical problems of twodimensional ideal plasticity equations]. Krasnoyarsk, Sib.
gos. aerokosmich. un-t Publ., 2012, 139 p.
10. Senashov S. I., Yakchno A. N. Deformation of characteristic curves of the plane ideal plasticity equations
by point symmetries Nonlinear analysis, 2009, No. 71, P. 1274–1284.
11. Senashov S. I., Savostyanova I. L., Filyushina E. V. [About torsion of parallelepiped around three axis] Sibirskiy
zhurnal nauki i tekhnologiy, 2017, Т. 18, No. 3, Р. 545–551.
12. Senashov S.I., Yakchno A.N. Some symmetry group aspects of a perfect plane plasticity system. J. Phys.
A: Math. Theor. 2013, No. 46, 355202.
13. Senashov S. I., Yakchno A. N. Conservation Laws, Hodograph Transformation and Boundary
Value Problems of Plane Plasticity SIGMA 8 (2012), 071, 16 pages http://dx.doi.org/10.3842/SIGMA.2012.071
Special Issue “Geometrical Methods in Mathematical Physics”.
14. Senashov S. I., Yakhno A. N., Yakhno L. V. [The construction of new solutions and their characteristics for
two-dimensional perfect plasticity with the help of symmetries]. Vestnik ChGPU. 2010, No. 2(8), P. 473–491.
15. Senashov S. I., Savost'yanova I. L. [A new threedimensional plastic flow, corresponding to a homogeneous
stress state] Sibirskiy zhurnal industrial'noy matematiki. 2019, tom KhKh11. No. 3(71), P.114–117.
16. Babskiy V. G., Kolpachevskiy N. D., Myshkis A. D. et al . Gidrodinamika nevesomosti [Hydrodynamics of
weightlessness]. Moscow, Nauka Publ., 1975.
17. Senashov S. I. [About one class of exact solutions of ideal plasticity equations]. Zhurn. prikl. mekhaniki i
tekhn. fiziki. 1986, No. 1, P. 139–142.


Senashov Sergey Ivanovich – Dr. Sc., Professor, Head of the Department of Nuclear Power Engineering; Siberian
State University of Science and Technology. E-mail: sen@sibsau.ru.

Savostyanova Irina Leonidovna – Cand. Sc., Associate Professor of the Department of IES, Siberian State University
of Science and Technology. E-mail: savostyanova@sibsau.ru.

Cherepanova Olga Nikolaevna – Сand. Sc., associate Professor, acting Director of the Institute of mathematics
and fundamental Informatics, Siberian Federal University. E-mail: cheronik@mail.ru.


  SYSTEM ANALYSIS OF DYNAMIC PROBLEMS OF ANISOTROPIC PLASTICITY THEORY