One class of solutions to the equations of ideal plasticity

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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

$u=\frac{1}{4}A(y^2-x^2-2z^2)-\frac{1}{2}Bxy-\frac{1}{2}Cx+Dyz,$

$v=\frac{1}{4}B(-y^2+x^2-2z^2)-\frac{1}{2}Axy-\frac{1}{2}y+Dxz,$

$w=\psi (x,y)+Axz+Byz+Cz,$ (1)

where $A, B, C, D$ are constants, $\psi (x,y)$ is function determined from a system of equations.

$\frac{\partial \psi }{\partial x}=-Dy\mp \frac{\sqrt{3}(Ax+By+C)f_y}{\sqrt{1-f_x^2-f_y^2}},$ $\frac{\partial \psi }{\partial y}=Dy\pm \frac{\sqrt{3}(Ax+By+C)f_x}{\sqrt{1-f_x^2-f_y^2}}.$ (2)

The condition for the compatibility of these relations gives a second-order elliptic equation

$\frac{\partial }{\partial x}\left(\frac{(Ax+By+C)f_x}{\sqrt{1-f_x^2-f_y^2}}\right)+\frac{\partial }{\partial y}\left(\frac{(Ax+By+C)f_y}{\sqrt{1-f_x^2-f_y^2}}\right)\pm \frac{2D}{\sqrt{3}}=0.$ (3)

In this case, the components of the stress tensor ${\sigma }_x,{\sigma }_y,{\tau }_{xy}$ are identically equal to zero, and

${\sigma }_z=\pm \sqrt{3}k\sqrt{1-f_x^2-f_y^2},{\tau }_{xz}=-k{f}_y,{\tau }_{yz}=k{f}_x,$ (4)

where $k$ is plastic constant.

The purpose of the work is to study some properties of equation (3) and construct its solution, provided that

$A=B=0$.

In this case, we obtain the following nonlinear differential equation

${\partial }_x\left(\frac{f_x}{\sqrt{1-f_x^2-f_y^2}}\right)+{\partial }_y\left(\frac{f_y}{\sqrt{1-f_x^2-f_y^2}}\right)=K,K=\mp \frac{2D}{\sqrt{3}}.$ (5)

The index below means differentiation by the corresponding argument; all functions are assumed to be smooth.

Some properties of the equation (5) where $K=0$.

1. Equation (5) can be derived from the variational principle, and it is the minimum of the functional
   $Z\left(w\right)=\iint \sqrt{w_x^2+w_y^2}dxdy=\iint Ldxdy.$
2. The group of point symmetries of equation (5) is generated by the following operators:

$X_1={\partial }_x,$ $X_2={\partial }_y,$ $X_3={\partial }_f,$ $X_4=x{\partial }_x+y{\partial }_y+f{\partial }_f,$ $X_5=y{\partial }_x-x{\partial }_y.$ (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

$X_1+\alpha X_3,\alpha X_3+X_5,X_5+\alpha X_4,X_3,X_4,$ (7)

where α is arbitrary constant. Different values of this constant correspond to dissimilar subalgebras.

Уравнение (5) приведем к виду

$(1-f_y^2)f_{xx}+2f_x{f}_y{f}_{xy}+(1-f_x^2)f_{yy}=0.$ (8)

3. Definition. Let us call the conservation law for equation (8) an expression of the form

$A_x+B_y=\Delta \left[\right(1-f_y^2)f_{xx}+2f_x{f}_y{f}_{xy}+(1-f_x^2\left)f_{yy}\right]=\mathrm{0,}$ (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 $X_1$

$D_x\left(L-f_x\frac{\partial L}{\partial f_x}\right)+D_y\left(-f_y\frac{\partial L}{\partial f_y}\right)=0$.

The conservation law corresponds to the operator $X_2$

$D_x\left(-f_y\frac{\partial L}{\partial f_x}\right)+D_y\left(L-f_y\frac{\partial L}{\partial f_y}\right)=0.$

The conservation law corresponds to the operator $X_3$

$D_x\left(\frac{f_x}{\sqrt{1-f_x^2-f_y^2}}\right)+D_y\left(\frac{f_y}{\sqrt{1-f_x^2-f_y^2}}\right)=0.$

The conservation law corresponds to the operator $X_4$

$D_x\left(Lx+(f-x{f}_x-y{f}_y)\frac{\partial L}{\partial f_x}\right)+D_y\left(Ly+(f-x{f}_x-y{f}_y)\frac{\partial L}{\partial f_y}\right)=0.$

The conservation law corresponds to the operator $X_5$

$D_x\left(Ly+(y{f}_x-x{f}_y)\frac{\partial L}{\partial f_x}\right)+D_y\left(-Lx+(y{f}_x-x{f}_y)\frac{\partial L}{\partial f_y}\right)=0.$

Note that equation (8) has other conservation laws that are different from the previous laws. Let's point out some. Let $A(f_x,f_y),B(f_x,f_y)$ , then from (9) we have

$\frac{\partial A}{\partial f_x}{f}_{xx}+\frac{\partial A}{\partial f_y}{f}_{xy}+\frac{\partial B}{\partial f_x}{f}_{xy}+\frac{\partial B}{\partial f_y}{f}_{yy}=\Delta \left[\right(1-f_y^2)f_{xx}+2f_x{f}_y{f}_{xy}+(1-f_x^2\left)f_{yy}\right]=0.$

From here we easily obtain two equations for determining the conserved current

$(1-f_x^2)\frac{\partial A}{\partial f_x}=(1-f_y^2)\frac{\partial B}{\partial f_y},\frac{\partial A}{\partial f_y}+(1-f_x^2)\frac{\partial B}{\partial f_x}=2f_x{f}_y\frac{\partial B}{\partial f_y}.$

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.

4. The Legendre transformation allows us to linearize equation (8) and bring it to the form

$(1-{\eta }^2)w_{\eta \eta }-2\xi \eta w_{\xi \eta }+(1-{\xi }^2)w_{\xi \xi }=0.$

This transformation is determined by the relations

$f_x=\xi ,f_y=\eta ,x=f_{\xi },y=f_{\eta },w+f=x\xi +y\eta .$

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

$f=g\left(x\right)+h\left(y\right).$ (10)

Substituting (10) into (8), we get

$(1-h{\text{'}}^2)g\text{'}\text{'}+(1-g{\text{'}}^2)h\text{'}\text{'}=0.$ (11)

Here the prime means the derivative with respect to the corresponding argument.

From (11) we obtain

$\frac{h\text{'}\text{'}}{1-h{\text{'}}^2}=-\frac{g\text{'}\text{'}}{1-g{\text{'}}^2}=\lambda -\text{const}.$ (12)

Integrating (12), we obtain

$\frac{1}{2}\ln \left|\frac{1+h\text{'}}{1-h\text{'}}\right|=2\lambda +\ln C_1,$ $\frac{1}{2}\ln \left|\frac{1+g\text{'}}{1-g\text{'}}\right|=-2\lambda +\ln C_2.$ (13)

Integrating (13), we obtain

$h=-x+\frac{1}{\lambda }\ln (1+C_1\exp 2\lambda x)+C_3,$ $g=-y-\frac{1}{\lambda }\ln (1+C_2\exp (-2\lambda y\left)\right)+C_4.$ (14)

Considering $C_1=C_2=1$ solution (10) can be written in a more convenient form

$h\text{'}=th\lambda x,g\text{'}=th\lambda y,h=\ln ch\lambda x,g=-\ln ch\lambda y,f=\ln \frac{ch\lambda x}{ch\lambda y}$;

b) we write equation (5) with in polar coordinate system  . We have

$\frac{\partial }{\partial r}\left(\frac{r^2{f}_r}{\sqrt{r^2-r^2{f}_r^2-f_{\theta }^2}}\right)+\frac{\partial }{\partial \theta }\left(\frac{f_{\theta }}{\sqrt{r^2-r^2{f}_r^2-f_{\theta }^2}}\right)=0.$ (15)

We are looking for a solution to this equation in the form $f=rf\left(t\right),t=r\exp \left(\alpha \theta \right),\alpha -\text{const}.$

From (15) we obtain an ordinary differential equation of the second order, which could be integrated only in the case when $\alpha =0$. In this case we get

$f=\ln \left|r+\sqrt{r^2+c^2}\right|=\text{Arcsh}\frac{r}{c},c-\text{const}$ (16)

c) we look for a solution to equation (5) in the form

$f=\alpha y+\phi \left(x\right).$

substituting this relation into (5) we get

$\phi =\frac{\sqrt{1-{\alpha }^2}}{K}\sqrt{1+{(Kx+C)}^2}$;

d) we look for a solution to equation (5) in the form

$f=\alpha \theta +\phi \left(r\right).$

substituting this relation into (5), we obtain

$\phi =K\int \sqrt{\frac{r^2-{\alpha }^2}{r^2+K^2}}\frac{dr}{r}$;

e) we look for a solution to equation (5) in the form

$f=r\phi \left(\theta \right).$

substituting this relation into (5) we get

 $f=r\sin \sqrt{1+K^2{\theta }^2}.$

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 $U$ corresponds to the solution(we define it as $F$), which is determined by the formula

$F_x=\xi ,F_y=\eta ,x=F_{\xi },\text{ }y=F_{\eta },U+F=x\xi +y\eta .$

From here we get

$U=-F+x\xi +y\eta =-F+x{F}_x+y{F}_y=\ln (r+\sqrt{r^2+c^2})+\frac{x^2}{\sqrt{x^2+y^2}}+\frac{y^2}{\sqrt{x^2+y^2}}.$ (17)

Finally we get

$U=\ln (r+\sqrt{r^2+c^2})+\sqrt{x^2+y^2}.$

We will do the same with solution (14), denoting it by $G$, and we denote its image via $V$. We have

$V=-G+x\xi +y\eta =-G+x{G}_x+y{G}_y=-x-y+$

$+\frac{1}{2\lambda }\ln \frac{1+C_1\exp 2\lambda x}{1+C_2(-2\lambda y)}+x\frac{C_1\exp 2\lambda x-1}{C_1\exp 2\lambda +1}+y\frac{C_2\exp (-2\lambda y)-1}{C{}_2\mathrm{e}xp(-2\lambda y)+1}.$ (18)

The Legendre equation corresponding to equation (7) is linear, so for it we have

$w=aU+(1-a)V,$ (19)

where $a$ is an arbitrary constant is again a solution to the same equation.  At the same time $a=0$ the solution coincides with the solution $V$, and with $a=1$ coincides with the solution $U$. It follows that formula (19) allows one to continuously transform solution (16) into solution (14) by changing $a$ from zero to one. The action algorithm is as follows: we write solutions (17) and (18) in variables $\xi , \eta$, we add them according to formula (19), and then write their linear combination in terms $x,y$. Thus, we obtain
a series of solutions to equation (7) for each fixed value $a$.

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 $C^2$. 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.

作者简介

Sergey Senashov

Reshetnev Siberian State University of Science and Technology

Email: sen@sibsau.ru

Dr Sc., Professor, Professor of the IES Department

俄罗斯联邦, 31, Krasnoyarskii rabochii prospekt, Krasnoyarsk, 660037

Irina Savostyanova

Reshetnev Siberian State University of Science and Technology

编辑信件的主要联系方式.
Email: ruppa@inbox.ru

Cand. Sc., Associate Professor of the Department of IES

俄罗斯联邦, 31, Krasnoyarskii rabochii prospekt, Krasnoyarsk, 660037

Olga Cherepanova

Siberian Federal University

Email: OCherepanova@sfu-kras.ru

Cand. Sc., Director of the Institute of Mathematics and Fundamental Informatics

俄罗斯联邦, 79, Svobodny Av., Krasnoyarsk, 660014

Sergey Lukyanov

Reshetnev Siberian State University of Science and Technology

Email: lukyanovsv@sibsau.ru

post-graduate student

俄罗斯联邦, 31, Krasnoyarskii rabochii prospekt, Krasnoyarsk, 660037

参考

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