Solving boundary value problems of equations of two-dimensional elasticity theory using conservation laws

Introduction. Usually, the equations of twodimensional static elasticity are solved using the complex variable theory methods. These methods were developed by G. V. Kolosov and N. I. Muskhelishvili. The use of complex variable theory methods for solving elasticity problems requires a good knowledge of the complex variable theory technique and, as a rule, is used for areas faceted with smooth contours [1; 2]. This article proposes a method for solving the same problems using conservation laws. Apparently, for the first-time conservation laws for elasticity equations were calculated in [3; 4]. These calculations were based on the theorem of E. Noether. The found laws did not find any practical application and were of purely academic interest [5]. In [6–10] it is shown that conservation laws can be successfully used in solving boundary value problems of the theory of plasticity and elastic-plasticity. G.P. Cherepanov [11; 12] noticed that the complex variable theory methods used in solving plane problems actually go back to conservation laws. In the presented work, an endless series of new conservation laws was found. These conservation laws made it possible to obtain formulas with the help of which one can effectively reduce the boundary value problems to quadratures. The latter are curvilinear integrals over a closed contour and can be easily solved numerically for given boundary conditions in displacements.

• 1.    Formulation of the problem. Consider the equations of two-dimensional elasticity in the stationary case

F1 = (X + 2Ц) uxx + (X + ц) vxy + Ц uyy = 0,

F 2 = ^{( X +} 2 ц ) v yy + ( Х + ц ) u xy + Ц v xx = 0, (1) where X , ц are constants called Lame parameters, u , v are components of the deformation vector, the indices below indicate the derivatives with respect to the corresponding components.

Let us find conservation laws of a special form for (1). The conservation law for the system of equations (1) is an expression of the form

A x ( x , У , u , v ) + B_y ( x , y , u , v ) to F 1 + и ² F 2 , (2) where и i some functions of x , y , which are simultaneously not identically equal to zero. The magnitudes A , B will be called the components of the conserved current. We are looking for the components of the conserved current in the form:

A = a 1 ( X + 2 ц ) u_x + p ¹ u_y +Y 1 M v_x + 5 ¹ v_y ,

^B = ^a 2 ^ux +P 2 ^uy +Y ^{2 v}x +3 2 ^vy , ⁽³⁾

where y ‘ , 3 ‘ are some functions depended on x , y only.

Substitute expressions (3) into (2). As a result, one can obtain a polynomial of the first degree in the variables u_x , u_y , u_xx , u_xy , u_yy , v_x , v_y , v_xx , v_xy , v_yy . Equating the coefficients at the derivatives in this polynomial to zero and taking into account that a ⁱ , P ⁱ , y i , 3 ⁱ are functions only from x , y , one can get after exclusion и ⁱ , i = 1, 2

a ² = -p 1 + ( X + ц )/, в ² =ца 1 ,

Y ² =-3 1 + ( X + ц ) a ¹, 3 ² = ( X + 2 ц ) Y ¹, ^{( X} + ² ц ^{) a} X + ^a y = 0, цY X +Y y = 0, e x +p y = 0, 3 x +3 2 = 0.

This implies:

(X + 2ц)aX -py + (X + ц)Y1y = 0, m — 31y + (X + ц)a1y = 0, px + May = 0, (X + 2ц^ +3x = 0.

Substituting (6) into (5) one can obtain

^{( X +} 2 M ) a 1x ^{+ ( X +} M ) Y xy ⁺ M ^a yy = 0, ( X + 2 M ) Y 1 yy + ( X + M ) a xy + _W xx = 0.

It is obvious that (7), up to notation, coincides with system (1). Therefore, a ¹, Y 1 is an arbitrary smooth solution to the system of equations (1). From (3) taking into account (4), one obtains the conserved current

A = ^a 1 ^{( X} + 2 ц ⁾⁽ u x ^{+ v}y ) ⁺ Y ¹ ц ⁽ v x ^{- u}y ),

B = Y 1 ( X + 2 ц )( u_x + v y ) + a ¹ ц ( v_x - u y ),

where a ¹, Y 1 is arbitrary solution of the system of equations (1).

2.    Construction of solutions to the boundary value problem. It is known that u = G1--1—(xG1 + yG2 + G 0)x = 4(1 -v)

= G ¹ -и ( xG ¹ + yG ² + G ⁰) _x ,            (9)

v = G2 -и(xG1 + yG2 + G0)y, is the solution to the system of equations (1). Here Gi are arbitrary harmonic functions, v is Poisson's ratio, which can be expressed through the Lamé parameters. From (9) one gets ux + vy = (Gx + Gy)(1 -и /2),

^- u y ⁺ vx = ^{- G}y ^{+ G}x .

Let x0, y0 be an arbitrary point of a domain Q with a smooth boundary   Г.   Let us assume

G1 = ln((x-x0)2 + (y-y0)2) = lnr2, G2 = 0. Then one obtains ux + vy = 2 x 2x0 и, - uy + vx = 2 y-2y°.(11)

rr

From (2), by Green's formula, follows jj (Ax + By) dxdy = ф - Ady + Bdx =

Q

= |-(a' (X + 2ц)(ux + vy) + y'm(vx -uy))dy +

Г

+ ( y ¹ ( X + 2 ц ) ( u_x + v_y ) + a 1 p ( v_x - u_y ) ) dx = 0.

Now let the contour Г 1 consist of two contours: Г and a circle of radius e : ( x - x ₀) 2 + ( y - y 0 ) 2 = £ 2 .

Assuming in (12) instead of Г the contour Г1 and taking into account relation (11), one can obtain

• -f a 1 ( X + 2 ц )2 x x ⁰ и-у 1 ц 2 y ^{У 0} I dy +

Г 1 I                  r ²                 r ² J

+ ^y 1 ( X + 2 ц ) 2 x 2 x ⁰ й+a ¹ ц ( v_x - u_y )J dx =    (13)

= f ⁺ f = 0.

Г    ⁽ x ^- x 0 ^{)2 + ( y - y} o )² = ^e 2

Let's represent the equation of a circle in parametric form:

x - x ₀ =e cos ф , y - y ₀ =e sin ф , 0 <ф< 2 п .

Then the last of the integrals in formula (13) will be written as follows:

⁽ x ^- X 0 ) ^{+ ( y - y} 0 ) = ^£

• 2 n

= {-(a1 (X + 2ц) 2йcos ф-y1ц2sin ф) cos ф d ф+ (14)

+ (y1(X + 2ц)2cos ф + a1ц2sin ф) sin ф d ф == 2ла' (x0,y0)(ц- (X + 2ц)ю.

In formula (14), the mean value theorem was used. Finally, from (13) one obtains

2 п ( ц- ( X + 2 ц ) и Г

• -fa ¹ ( X + 2 ц ) 2 й| x x ⁰ ]- 2 у 1 ц| y ^{У 0} ]f dy +

I I r J I r JJ (15) + fy 1 ( X + 2 ц ) 2 й| x 2 x ⁰ ]- 2 a ¹ цf y 2 y ⁰ ]] dx =

I I r2 J I r2 J J a1 (x0, у0).

Assuming now in equations (10) G ² = ln ( ( x - x ₀) ² + ( y - y ₀) ² ) = ln r ² , G ¹ = 0. In this case, similar reasoning and calculations give a formula ^for Y ¹⁽ x 0 , У 0 ).

One obtains

-1 ----------------x 2п(ц + (X + 2ц)м x ffa1 (X + 2ц) 2й| У У 0 ] + 2у1ц| x x0   dy +

Г I <  ^{r J} <  ^r X (16)

+ fY ¹ ( X + 2 ц ) 2 й[ ^У 2 ^{У 0} 1 + 2 a ¹ цf x 2 x ⁰ ]] dx =

I I r ² J I r ² J J

= Y 1 ( ^x 0 , У 0 ) .

Formulas (15), (16) make it possible to calculate a1, y1 at any point (x0, y0) in the area if the values of these functions are known on the boundary of Г. And since a1, y1 are a solution to equations (1), one can obtain a method for constructing solutions to system (1) according to their boundary conditions. Namely, let the following boundary value problem be set: there is an area Q with a boundary Г. On Г, the components of the displacement vector are given u |Г = uo(x,У), v |Г = vo(x,У). (17)

Then, based on formulas (15), (16), the solution to problem (17) for the system of equations (1) can be written in the form

- 1

2 п ( ц - ( X + 2 ц ) й

f

Г

• -1 u ₀ ( X + 2 ц ) 2 J x 1 - 2 v ₀ цf У ^- У 0 П dy +

I               I r ² J I r ² JJ (18)

, f             ( x - x o 1 о ( У - У о 1L

+ 1 v ₀ ( X + 2 ц ) 2 й| —2— I- 2 u ₀ ц| — 2— I I dx =

= ^u 0 ( ^x о , У о ) .

-1 ----------------x 2п(ц + (X + 2ц)й x f I u0 (X + 2ц)2й| У 2У0 | + 2vоц| x 2x0 11dy +

Г I V r   J I r   JJ (19)

+I v 0 ( X + 2 ц ) 2 й| ^У 2 ^{У 0} | + 2 u _о ц| x 2 x ⁰ 11 dx =

V               V r ² J V r ² JJ

= ^v 0 ( ^x о , У о ) .

Conclusion. In the article, conservation laws that depend on derivatives of the first order and linear on them are found. It is shown that two coefficients at the derivatives are solutions of the elasticity equations in the plane case. This made it possible to reduce the solution of boundary value problems in displacements to the calculation of contour integrals along the boundary of the area. The proposed technique can be easily generalized to multiply – connected domains [13–15], as well as to the three-dimensional case.