Solution of the first boundary value problem of plane elasticity theory using conservation laws

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Introduction

The solution to boundary value problems for equations of elasticity theory in a plane stationary case is described in a huge number of articles and monographs. The classic work in this direction is a book written by G. V. Kolosov’s student [1]. Despite the long history of solving such problems, the interest in their solution has not waned. This is due to the fact that the classical formulae by G. V. Kolosov allow solving equations of elasticity theory not for all boundary value problems arising in science and technology. The main limitation is caused by the smoothness of the boundary and some features of complex variable functions. The other methods involved in decomposing the desired functions into series for different types of special functions also have natural limitations, which are related to the similarity of the series being used as well as the complexity of the results obtained.

Let us describe some results of the recent studies on the theory of elasticity. The paper [2] presents a brief historical review of the studies devoted to the bending theory of elastic plates. In [3] the author considers the problem of detecting and identifying elastic inclusions in an isotropic linearly elastic plane. The article [4] considers a complicated version of the well-known Lame problem posed in 1852 describing the solution to the static equilibrium of a parallelepiped with free lateral surfaces exposed to the action of opposite end forces, and also for the case of impact effects of end forces. In paper [5] the construction of fundamental solutions for harmonic oscillations equations in elastic anisotropic elastic media theory is carried out; the fundamental solution to oscillations equations for isotropic medium in closed form is constructed. The work [6] presents a general solution to the problems of the elastic theory for anisotropic half-planes and bands with arbitrary holes and cracks using complex potentials of the plane problem of the elastic theory of an anisotropic body. The article [7] is devoted to the study of the functions of tensions, which allow satisfying exactly the equations of equilibrium of the classical theory of elasticity and to obtain a solution in tensions. In order to obtain dependencies between voltages and voltage functions, a mathematical apparatus of general relativity is used. In [8] terms of complex-significant displacements, the system of equations of the axisymmetric theory of elasticity is written; the fundamental solution to this system of equations is a general representation of the field of displacements in an axisymmetric case, similar to the formulae by Kolosov-Muskhelishvili in a plane problem. The basic equations of linear moment elasticity theory are presented in [9]. The determining ratios are written for the case of general anisotropy in the form of linear equations. Some simplified options are considered, in particular with constricted rotation and plane deformation when only shear stresses are present. The work [10] is devoted to the analysis of a boundary value problem with an unknown area of contact, describing the equilibrium of two-dimensional elastic bodies with a thin slightly curved web. The paper [11] explores the evolution of the wave pattern in a multi-modular elastic half-space with a non-stationary one-axis piecewise-linear motion of its boundary in the mode “tension – compression – stop”.

In this work the laws of conservation of differential equations of elasticity are being used. The laws, which allow reducing the presence of tensor component of the tensions at the point to the contour integral along the boundary of the area being considered, have been presented. In these conditions the boundary of an area requires only partial smoothness. Note that earlier some conservation laws were presented in works [12; 13] but they were not used to solve any problems.

Statement of the problem

Let us consider the equations describing plane elastic deformation.

The relations between strain tensor components and displacement vector components in the case of small deformations have the following form

${\epsilon }_x=\frac{\partial w_1}{\partial x},{\epsilon }_y=\frac{\partial w_2}{\partial y},{\epsilon }_{xy}=\frac{\partial w_1}{\partial y}+\frac{\partial w_2}{\partial x}.$                                                        (1)

The Hooke’s will be written as follows:

${\epsilon }_x=\frac{1}{E}({\sigma }_x-\nu {\sigma }_y),{\epsilon }_y=\frac{1}{E}({\sigma }_y-\nu {\sigma }_x),{\epsilon }_{xy}=\frac{2(1+\nu )}{E}\tau .$                                             (2)

Strain compatibility conditions:

$\frac{{\partial }^2{\epsilon }_x}{{\partial }^{2}y}+\frac{{\partial }^2{\epsilon }_y}{{\partial }^{2}x}=\frac{{\partial }^2{\epsilon }_{xy}}{\partial x\partial y}$.                                                                                        (3)

By putting (2) into (3), we obtain

$\frac{{\partial }^2({\sigma }_x-\nu {\sigma }_y)}{{\partial }^{2}y}+\frac{{\partial }^2({\sigma }_y-\nu {\sigma }_x)}{{\partial }^{2}x}=2(1+\nu )\frac{{\partial }^2\tau }{\partial x\partial y}$                                                  (4)

Equilibrium equations:

$\frac{\partial {\sigma }_x}{\partial x}+\frac{\partial \tau }{\partial y}=\mathrm{0,}\frac{\partial \tau }{\partial x}+\frac{\partial {\sigma }_y}{\partial y}=0.$

From hence we obtain

$\frac{{\partial }^2{\sigma }_y}{{\partial }^{2}y}+\frac{{\partial }^2{\sigma }_x}{{\partial }^{2}x}=-2\frac{{\partial }^2\tau }{\partial x\partial y}.$                                                                             (5)

From (4) and (5) we obtain

Here are strain tensor components;  are voltage tensor components;  are displacement vector components;  are elastic constants.

The system has the following final form:

$\frac{\partial {\sigma }_x}{\partial x}+\frac{\partial \tau }{\partial y}=\mathrm{0,}\frac{\partial \tau }{\partial x}+\frac{\partial {\sigma }_y}{\partial y}=\mathrm{0,} \Delta ({\sigma }_x+{\sigma }_y)=0.$                                       (6)

Let us put the first boundary value problem for the system (6):

${\sigma }_x{n}_1+\tau n_2{|}_L=X(x,y),\tau n_1+{\sigma }_y{n}_2{|}_L=Y(x,y).$                                               (7)

Here  are components of an outer normal vector to a piecewise smooth contour limiting the end area

Let us find the solution to the problems (6) and (7) in the form

${\sigma }_x+{\sigma }_y=p-\text{const, }\mathit{\text{p}}\text{ ≠0. }$                                                                            (8)

Let us introduce new variables:

$u=\raisebox{1ex}{${\sigma }_x$}\!\left/ \!\raisebox{-1ex}{$p,v$}\right.=\raisebox{1ex}{$\tau $}\!\left/ \!\raisebox{-1ex}{$p,{\sigma }_y^{\text{'}}$}\right.=\raisebox{1ex}{${\sigma }_y$}\!\left/ \!\raisebox{-1ex}{$p,f$}\right.=\raisebox{1ex}{$X$}\!\left/ \!\raisebox{-1ex}{$p,g$}\right.=\raisebox{1ex}{$Y$}\!\left/ \!\raisebox{-1ex}{$p$}\right..$                                            (9)

Then the problem (6), (7) is written in the following form

$F_1=u_x+v_y=\mathrm{0,}{F}_2=u_y-v_x=\mathrm{0,}$                                                                 (10)

$u{n}_1+v{n}_2=f,v{n}_1-u{n}_2=g-n_2,$                                                                 (11)

here and then the index at the bottom means a derivative of this variable.

Thus, it is necessary to solve the boundary value problem (11) for the system of equations (10) with the use of conservation laws.

Conservation laws of the system of equations (10)

Definition. The conservation law for the system of equations (10) will be called the following expression

$A_x+B_y={\omega }_1{F}_1+{\omega }_2{F}_2,$                                                                          (12)

where  are linear differential operators that are simultaneously nonzero identical,

$A={\alpha }^{1}u+{\beta }^{1}v+{\gamma }^1,B={\alpha }^{2}u+{\beta }^{2}v+{\gamma }^2,$                                                   (13)

 are some smooth functions that only depend on $x,y.$

Remark. A more general definition of the conservation law suitable for the arbitrary systems of equations can be found in [14; 15].

From (12) taking into account (13) we obtain

${\alpha }_x^{1}u+{\alpha }^1{u}_x+{{\beta }^1}_{x}v+{\beta }^1{v}_x+{{\gamma }^1}_x+{\alpha }_y^{2}u+{\alpha }^2{u}_y+{\beta }_y^{2}v+{\beta }^2{v}_y+{\gamma }_y^2={\omega }_1(u_x+v_y)+{\omega }_2(u_y-v_x)=0.$      (14)

From (14) it follows

${\alpha }_x^1+{\alpha }_y^2=\mathrm{0,}{\beta }_x^1+{\beta }_y^2=\mathrm{0,}{\alpha }^1={\omega }_1,{\beta }^1=-{\omega }_2,{\alpha }^2={\omega }_2,{\beta }^2={\omega }_1,{\gamma }^1+{\gamma }^2=0.$

From hence we obtain

${\alpha }^1={\beta }^2,{\alpha }^2=-{\beta }^1.$                                                                                       (15)

Therefore

${\alpha }_x^1-{\beta }_y^1=\mathrm{0,}{\alpha }_y^1+{\beta }_x^1=0.$                                                                              (16)

From the given formulae it follows that the system of equations (10) allows infinitely many laws of conservation. Only those that can solve the problem will be listed below.

Therefore, the conserved current is:

$A={\alpha }^{1}u+{\beta }^{1}v+{\gamma }^1,B=-{\beta }^{1}u+{\alpha }^{1}v+{\gamma }^2.$

From (13) we obtain

$\underset{S}{\iint }(A_x+B_y)dxdy=\underset{L}{\oint }-Ady+Bdx=\mathrm{0,}$                                                 (17)

where S is the area bounded by the L curve.

Solution to the problem (10), (11)

In order to find the  values within the S area, it is necessary to construct solutions to the Cauchy – Riemann system (16) having singularities at the arbitrary point $(x_0,y_0)\in S$.

The first of these solutions has the folloing form:

${\alpha }^1=\frac{x-x_0}{{(x-x_0)}^2+{(y-y_0)}^2},{\beta }^1=-\frac{y-y_0}{{(x-x_0)}^2+{(y-y_0)}^2},{\gamma }^1={\gamma }^2=0.$     (18)

Remark. If mass forces are included in the equilibrium equation, then ${\gamma }^1,{\gamma }^2$ will no longer be equal to zero.

At the point $(x_0,y_0)\in S$ the functions ${\alpha }^1,{\beta }^1$ have singularities, thus let us circle this point with the circumference $\epsilon :{(x-x_0)}^2+{(y-y_0)}^2={\epsilon }^2.$

Then we obtain from the formula (17)

$\underset{L}{\oint }-Ady+Bdx+\underset{\epsilon }{\oint }-Ady+Bdx=\mathrm{0,}$                                                      (19)

Let us calculate the second integral in the formula (19). We have

$\underset{}{\oint }-Ady+Bdx=\underset{\epsilon }{\oint }-(\frac{u(x-x_0)}{{(x-x_0)}^2+{(x-x_0)}^2}-\frac{v(y-y_0)}{{(x-x_0)}^2+{(x-x_0)}^2})dy+$

$+\left(-\frac{u(y-y_0)}{{(x-x_0)}^2+{(y-y_0)}^2}-\frac{v(x-x_0)}{{(x-x_0)}^2+{(y-y_0)}^2}\right)dx.$

Let us introduce the new coordinates $x-x_0=\epsilon \cos \phi ,y-y_0=\epsilon \sin \phi ,$ we obtain

$$\begin{gathered} \begin{array}{c}\underset{\epsilon }{\oint }-Ady+Bdx=\underset{0}{\overset{2\pi }{\int }}[-(u\cos \phi +v\sin \phi )\cos \phi -(u\sin \phi +v\cos \phi \left)sin\phi \right]d\phi =\\ \\ =-\underset{0}{\overset{2\pi }{\int }}ud\phi =-2\pi u(x_0,y_0)\end{array} \end{gathered}$$.              (20)

The last equality is obtained by the mean-value theorem at $\epsilon \to 0$.

For the final construction of the solution, let us find the values $u,v$ on the L boundary. From the formula (11) we obtain

$u=f{n}_1-g{n}_2+n_2^2,v=f{n}_2+g{n}_1+n_1{n}_2^{}.$                                                            (21)

Let us put (21) into (20); and taking into account (19) we obtain

$\begin{array}{c}2\pi u(x_0,y_0)=2\pi {\sigma }_x(x_0,y_0)/p=\\ \underset{L}{=\oint }-(\frac{(f{n}_1-g{n}_2+n_1^2)(x-x_0)}{{(x-x_0)}^2+{(y-y_0)}^2}-\frac{(f{n}_2+g{n}_1-n_1^{}{n}_2)(y-y_0)}{{(x-x_0)}^2+{(y-y_0)}^2})dy+\\ -\left(\frac{(f{n}_1-g{n}_2+n_1^2)(y-y_0)}{{(x-x_0)}^2+{(y-y_0)}^2}+\frac{(f{n}_2+g{n}_1-n_1^{}{n}_2)(x-x_0)}{{(x-x_0)}^2+{(y-y_0)}^2}\right)dx.\end{array}$

The second solution to the system of equations (16) is taken as

${\alpha }^1=\frac{y-y_0}{{(x-x_0)}^2+{(y-y_0)}^2},{\beta }^1=\frac{x-x_0}{{(x-x_0)}^2+{(y-y_0)}^2},$                                               (22)

Having made calculations similar to those made with the solution (18), we obtain

$\begin{array}{c}2\pi v(x_0,y_0)=2\pi \tau (x_0,y_0)/p=\\ =\underset{L}{\oint }-(\frac{(f{n}_1-g{n}_2+n_1^2)(y-y_0)}{{(x-x_0)}^2+{(y-y_0)}^2}+\frac{(f{n}_2+g{n}_1-n_1^{}{n}_2)(x-x_0)}{{(x-x_0)}^2+{(y-y_0)}^2})dy+\\ -\left(-\frac{(f{n}_1-g{n}_2+n_1^2)(x-x_0)}{{(x-x_0)}^2+{(y-y_0)}^2}+\frac{(f{n}_2+g{n}_1-n_1^{}{n}_2)(y-y_0)}{{(x-x_0)}^2+{(y-y_0)}^2}\right)dx.\end{array}$

Conclusion

This paper proposes a new method for solving the first boundary value problem for the equations of plane elasticity theory in a stationary case. This method makes it possible to find the value of the component of a strain tensor at each point of the area being studied. In this case, the stress calculations are limited only to the calculation of contour integrals along the boundaries of the area.

作者简介

Olga Pashkovskaya

Reshetnev Siberian State University of Science and Technology

编辑信件的主要联系方式.
Email: pashkovskaya@sibsau.ru
ORCID iD: 0009-0003-2529-4105

Cand. Sс., associate Professor, Reshetnev

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

Sergei Lukyanov

Reshetnev Siberian State University of Science and Technology

Email: lukyanovsv@sibsau.ru

postgraduate student

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

参考

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