One class of solutions to the equations of ideal plasticity
- Authors: Senashov S.I.1, Savostyanova I.L.1, Cherepanova O.N.2, Lukyanov S.V.1
-
Affiliations:
- Reshetnev Siberian State University of Science and Technology
- Siberian Federal University
- Issue: Vol 25, No 2 (2024)
- Pages: 182-188
- Section: Section 1. Computer Science, Computer Engineering and Management
- URL: https://journals.eco-vector.com/2712-8970/article/view/634614
- DOI: https://doi.org/10.31772/2712-8970-2024-25-2-182-188
- ID: 634614
Cite item
Abstract
Much attention is given to the study and solution of nonlinear differential equations in the modern mathematical literature. Despite this, there are not many methods for researching and solving such equations. These are point and contact transformations of equations, various methods of separating variables, the method of differential connections, the search for various symmetries and their use to construct solutions, as well as conservation laws. The paper considers a nonlinear differential equation describing the plastic flow of a prismatic rod. A group of point symmetries is found for this equation. The optimal system of one-dimensional subalgebras is calculated. Conservation laws corresponding to Noetherian symmetries are given, and it is also shown that there are infinitely many non-Noetherian conservation laws. Several new invariant solutions of rank one, i. e. depending on one independent variable, are constructed. It is shown how classes of new solutions can be constructed from two exact solutions, passing to a linear equation. Thus, in this short article, almost all methods of modern research of nonlinear differential equations are involved.
Full Text
Introduction
Solving and studying differential equations is still one of the most important problems of modern mathematics. In linear differential and integro-differential equations, questions of solvability of mixed nonlocal boundary value and inverse problems containing real parameters and differential operators of mathematical physics are studied [1; 2].
Exact solutions for nonlinear differential equations are known only in exceptional cases. To search for them, methods of generalized separation of variables, methods of group analysis, the method of differential connections and some others are used. A large list of solved equations and a review of methods for solving them are given in the fundamental work [3]. Recently, conservation laws have begun to be used to solve boundary value problems for nonlinear differential equations [4–7]. Previously, they most often played a supporting role. Methods of using group analysis to various equations that arise in physics and mechanics can be seen in works [8–15] .
Formulation of the problem
In [1], a solution is given that describes the purely plastic stress state of a prismatic rod
(1)
where are constants, is function determined from a system of equations.
(2)
The condition for the compatibility of these relations gives a second-order elliptic equation
(3)
In this case, the components of the stress tensor are identically equal to zero, and
(4)
where is plastic constant.
The purpose of the work is to study some properties of equation (3) and construct its solution, provided that
.
In this case, we obtain the following nonlinear differential equation
(5)
The index below means differentiation by the corresponding argument; all functions are assumed to be smooth.
Some properties of the equation (5) where .
- Equation (5) can be derived from the variational principle, and it is the minimum of the functional
- The group of point symmetries of equation (5) is generated by the following operators:
(6)
To construct various invariant solutions, it is necessary to construct an optimal system of subalgebras. For the Lie algebra generated by operators (6), it has the form
(7)
where α is arbitrary constant. Different values of this constant correspond to dissimilar subalgebras.
Уравнение (5) приведем к виду
(8)
- Definition. Let us call the conservation law for equation (8) an expression of the form
(9)
where is linear differential operator that is not identically equal to zero.
For equations derived from the variational principle, each operator admitted by the equation corresponds, according to Noether’s theorem [3], to a certain conservation law. Let us use this theorem for equation (5). We get five conservation laws.
The conservation law corresponds to the operator
.
The conservation law corresponds to the operator
The conservation law corresponds to the operator
The conservation law corresponds to the operator
The conservation law corresponds to the operator
Note that equation (8) has other conservation laws that are different from the previous laws. Let's point out some. Let , then from (9) we have
From here we easily obtain two equations for determining the conserved current
It follows that equation (8) admits an infinite series of conservation laws. This follows, in particular, from the linearity of the reduced system with respect to a conserved current.
- The Legendre transformation allows us to linearize equation (8) and bring it to the form
This transformation is determined by the relations
Exact solutions (5). All these solutions are invariant solutions built on subalgebras (7) :
a) We look for a solution to equation (8) in the form
(10)
Substituting (10) into (8), we get
(11)
Here the prime means the derivative with respect to the corresponding argument.
From (11) we obtain
(12)
Integrating (12), we obtain
(13)
Integrating (13), we obtain
(14)
Considering solution (10) can be written in a more convenient form
;
b) we write equation (5) with in polar coordinate system . We have
(15)
We are looking for a solution to this equation in the form
From (15) we obtain an ordinary differential equation of the second order, which could be integrated only in the case when . In this case we get
(16)
c) we look for a solution to equation (5) in the form
substituting this relation into (5) we get
;
d) we look for a solution to equation (5) in the form
substituting this relation into (5), we obtain
;
e) we look for a solution to equation (5) in the form
substituting this relation into (5) we get
Homotopy of two solutions to equation (5)
We will show how from two solutions (14) and (16) one can obtain a whole series of new exact solutions to equation (5). To do this, we apply the Legendre transformation to these solutions.
The solution to the Legendre equation corresponds to the solution(we define it as ), which is determined by the formula
From here we get
(17)
Finally we get
We will do the same with solution (14), denoting it by , and we denote its image via . We have
(18)
The Legendre equation corresponding to equation (7) is linear, so for it we have
(19)
where is an arbitrary constant is again a solution to the same equation. At the same time the solution coincides with the solution , and with coincides with the solution . It follows that formula (19) allows one to continuously transform solution (16) into solution (14) by changing from zero to one. The action algorithm is as follows: we write solutions (17) and (18) in variables , we add them according to formula (19), and then write their linear combination in terms . Thus, we obtain
a series of solutions to equation (7) for each fixed value .
Conclusion
In the work, a group of point transformations is found that are allowed by equation (5) in the Lie–Ovsyannikov sense. This group has dimension five. It is generated by three translations in spatial variables and the desired function, stretching in these same variables, circular rotation in the plane ОХY. New classes of exact solutions of this equation are found, depending on arbitrary functions from the class . Based on point symmetries, four conservation laws for equation (5) are found. A new infinite series of conservation laws is presented, which was found by direct calculation. The new solutions obtained complement other invariant solutions given in [1].
All solutions constructed in this work can be used to describe the stress-strain state of a straight rod subjected to tension along the axis OZ and torsion around this axis by a pair of forces.
About the authors
Sergey I. Senashov
Reshetnev Siberian State University of Science and Technology
Email: sen@sibsau.ru
Dr Sc., Professor, Professor of the IES Department
Russian Federation, 31, Krasnoyarskii rabochii prospekt, Krasnoyarsk, 660037Irina L. Savostyanova
Reshetnev Siberian State University of Science and Technology
Author for correspondence.
Email: ruppa@inbox.ru
Cand. Sc., Associate Professor of the Department of IES
Russian Federation, 31, Krasnoyarskii rabochii prospekt, Krasnoyarsk, 660037Olga N. Cherepanova
Siberian Federal University
Email: OCherepanova@sfu-kras.ru
Cand. Sc., Director of the Institute of Mathematics and Fundamental Informatics
Russian Federation, 79, Svobodny Av., Krasnoyarsk, 660014Sergey V. Lukyanov
Reshetnev Siberian State University of Science and Technology
Email: lukyanovsv@sibsau.ru
post-graduate student
Russian Federation, 31, Krasnoyarskii rabochii prospekt, Krasnoyarsk, 660037References
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