Conservation laws and solutions of the first boundary value prob-lem for equations of two-dimensional elasticity theory

Linear equations of elasticity theory from the group point of view have been studied for quite a long time [1; 2]. At first, a group of point transformations was found and all invariant solutions were listed [2]. Next, the group bundle of the Lamé equations was performed [3]. Although the group bundle technique has been known for a long time [1], it is not fulfilled for many systems of equations. In this regard, the equations of elasticity theory are a pleasant exception. Group stratification allowed to better understand why complex variable methods are so widely used in thtwo-dimensional elasticity theory. This is because the resolving system for the two-dimensional equations of elasticity theory is the system of Cauchy-Riemann equations. In [4; 5] conservation laws were used for the first time to solve boundary value problems, in particular, plasticity equations. Conservation laws for the plane theory of elasticity were constructed in [6], but they were not used to solve boundary value problems. In the proposed paper, new conservation laws are constructed for resolving and automorphic systems. On their basis, the first boundary value problem for two-dimensional elasticity equations is solved.

Problem statement

Let the following relation of stress tensors and strain tensor be proposed: o 11 = ( X + 2 ц ) е 1 1 + Xe ₂₂, g ₁₂ = 2 це ₁₂

g22 = (X + 2ц)е22 + Xs11, where Gj are the components of the stress tensor; eij - are the components of the strain tensor; X > 0,ц > 0 - are the Lame constants, i.e. (1) is the classical Hooke law for the isotropic homogeneous case.

Substituting (1) into the equilibrium equations, in the absence of mass forces, we obtain

^{( X +} 2 Ц ) u xx ^{+ X} v xy ⁺ Ц ^{( U} yy ⁺ v xy ) = ⁰, ц ⁽ u xy ⁺ v xy ) ^{+ ( X +} 2 Ц ) v yy ^{+ X} u xy = 0,

where u , v are the components of the deformation vector, the indices are at the bottom, unless otherwise indicated, mean derivatives with respect to the corresponding variables.

It is known that the system of equations (2) is of elliptic type. This determines the type of conservation laws and the solution of boundary value problems. Group properties of differential equations are described in [1]. The group properties of the elasticity equations are studied in [2]. In works [7; 8] the group properties of three-dimensional equations of the linear theory of elasticity and asymmetric equations of elasticity in the dynamic case are studied. There, in particular, it is shown that system (2) admits an infinite group of point transformations generated by the operators:

X = h'du + h2dv ,(3)

где h ¹, h ²is the arbitrary solution of the Cauchy–Riemann equations:

hX + hy = 0, h - hX = 0.(4)

Let’s make a group bundle of the system of equations (2) according to the method [1] on the subalgebra generated by (3). To do this, we continue the operators (3) to the first derivatives. We have

X = X + hXdux + h2dvy + hyduy + h^dvx ,(5)

The differential invariants for (5), taking into account (4), have the form

11 = X, I2 = У, I3 = Ux + vy , I4 = uy - vx ■(6)

Then the automorphic system of equations has the form ux+vy=θ(x,y), uy-vx=ω(x,y).                                (7)

Let’s recall some properties of automorphic systems. Any solution of an automorphic system can be obtained from one solution of this system with the help of transformations generated by the operator (3).

Substituting (7) into (2) we obtain the resolving system

F ₁ = ( λ+ 2 µ ) θ _x -µω _y = 0, F ₂ = ( λ+ 2 µ ) θ _y +µω _x = 0,                   (8)

Repeating almost verbatim the reasoning from [7], it can be argued that system (8) is equivalent to the system of equations (2).

Therefore, having constructed the solution of the system (8), we obtain the solution to system (2).

Let the following boundary value problem be posed for the system (8):

θ | _L =θ ₀( x , y ), ω | _L =ω ₀( x , y ),                                      (9)

where L is some smooth closed curve, θ ₀( x , y ), ω ₀( x , y ) - are known smooth functions.

To solve this problem, we construct conservation laws for the system of equations (8).

Conservation laws

Due to the linearity of system (8), it will have an infinite number of conservation laws. Only those conservation laws will be found in the paper that allow solving the boundary problem (9).

Definition. The conservation law for the system of equations (8) is an expression of the form

A_x ( x , y , θ , ω ) + B_y ( x , y , θ , ω ) =α F ₁ +β F ₂ = 0,

where α , β are some functions which are not identically equal to zero at the same time. A , B are called the conserved current components.

More detailed information on the construction of conservation laws for arbitrary systems of differential equations can be found in [8–10]. Note that for the first time the conservation laws for the equations of the linear theory of elasticity were found in [11; 12], but they could not be used to solve specific boundary value problems.

Let us assume that the components of the conserved current have the form A = a ¹ θ + a ² ω , B = b ¹ θ + b ² ω , where a ¹, a ², b ¹, b ² are some functions of x,y.

Substituting (11) into (10), after simple transformations, we obtain a ¹ = α ( λ + 2 µ ), a ² = βµ , b ¹ = β ( λ+ 2 µ ), a ² = -αµ , a ¹ x + b ¹ y = 0, a x ² + b y ² = 0.

Hence we have

α _x +β _y = 0, α _y -β _x = 0.

From (10) it follows

∫∫ ( A_x + B_y ) dxdy = ∫ - Ady + Bdx .

SL

Solution of the first boundary value problem

Let (x0,y0) ∈ S to be such a point at which the components of the conserved current have singular- ities, then from (14) it follows

∫ - Ady + Bdx = - ∫ - Ady + Bdx . ,

L

ε

where ε : ( x - x ₀)² + ( y - y ₀)² = ε ² is a circle of radius ε around the point ( x ₀, y ₀) ∈ S . Let’s calculate the integral in the right side of (15) for different solutions of the Cauchy-Riemann equations. As solutions, we choose those that have a singularity at the point ( x ₀, y ₀) ∈ S . Let

_α=       x - x 0       _{, β=}       y - y 0       _,

( x - x ₀) + ( y - y ₀)       ( x - x ₀) + ( y - y ₀)

then from the right side of (15) we have

∫ - Ady + Bdx = ∫ - ( α ( λ+ 2 µ ) θ+βµω ) dy + ( αµω + β ( λ + 2 µ ) θ ) dx .

We substitute (16) into (17) and make a change of variables according to the formulas x - x ₀ = ε cos ϕ , y - y ₀ = ε sin ϕ , we obtain

2 π

∫ - Ady + Bdx = ∫ [ - (( λ + 2 µ ) θ + µω ) + 2sin ϕ cos ϕµω )] d ϕ =

ε 0

=- 2 π [( λ+ 2 µ ) θ ( x ₀, y ₀) -µω ( x ₀, y ₀)].

In the formula (18) we aimed ε→ 0 and used the mean value theorem.

Now let's do the same calculations by setting α=-       ^{y - y 0}, β=       ^{x - x 0}.

( x - x ₀) + ( y - y ₀)       ( x - x ₀) + ( y - y ₀)

As the result we obtain

∫ - Ady + Bdx = - 2 πµω ( x ₀, y ₀). ε

Formulas (18) and (19) allow, taking into consideration the boundary conditions (9) and equality (15), to determine the values of the functions θ и ω at an arbitrary point ( x ₀, y ₀) ∈ S . They look like this: 2 π [( λ+ 2 µ ) θ ( x 0 , y 0 ) -µω ( x 0 , y 0 )] = ∫ - ^{( λ+ 2 µ} ₂ ^{)( x - x 0 ) θ 0} ₂ dy + ^{µ ( y} ₂ ^{- y 0) ω 0}   ₂ dx ,

_L  ( x - x ₀) + ( y - y ₀)      ( x - x ₀) + ( y - y ₀)

2πµω(x0,y0)]=∫ (λ+2µ2)(y-y0)θ02dy+    µ(x2-x0)ω0

_L ( x - x ₀) + ( y - y ₀)      ( x - x ₀) + ( y - y ₀)

Now, after restoring the solutions of the resolving system, we find solutions of the automorphic system, i.e., solutions of the original system of equations (2). We have

F3 =ux+vy-θ(x,y)=0, F4 =uy-vx-ω(x,y)=0.(20)

Here, in the right side, there are the known functions which were found in the previous paragraph. Let us find the conservation laws of the equations (20) in the following form:

A = a3θ+a4ω+c1, B =b3θ+b4ω+c2, where a,a,b ,b ,c ,c are some functions from x, y.

We have

A_x ( x , y , u , v ) + B_y ( x , y , u , v ) = α F ₃ +β F ₄ = 0, .

Splitting the system of equations (22), we obtain a3 =α, a4 = -β, b3 =β, b4 =α, a3+b3=0, a4 +b4 = 0, c1 +c2 =-αθ-βω. xy  xy  xy

From here we get

α _x +β _y = 0, α _y -β _x = 0.

Let the following boundary value task be posed for the system (2):

u | _L = u ₀( x , y ), v | _L = v ₀( x , y ), .

Let consider the conservation law in the form

∫ - Ady + Bdx = - ∫ - Ady + Bdx .

L ε

Let the solution of equations (24) have the form

_α=        x - x _{0        , β=}        y - y _{0        , .}

( x - x ₀)² + ( y - y ₀)²( x - x ₀)² + ( y - y ₀)²

We substitute (27) into the right side of (26), we get

∫-Ady+Bdx = ∫-(αu -βv+c1)dy + (βu + αv+c2)dx =

ε

ε

= ∫ - ( α cos φ-β sin φ+ c ¹) dy - ( β sin φ+α cos φ+ c ²) dx =- 2 π u ( x ₀, y ₀) . ε

Let the solution of equations (24) have the form

α=-       ^{y - y 0}      , β=       ^{x - x 0}      ,.

( x - x ₀) + ( y - y ₀)       ( x - x ₀) + ( y - y ₀)

We substitute (29) into the right side of (26), we get

∫ - Ady + Bdx = ∫ - ( α u -β v + c ¹) dy + ( β u +α v + c ²) dx =

ε

ε

= ∫ - ( - u sin φ- v cos φ+ c ¹) dy - ( u cos φ- v sin φ+ c ²) dx = - 2 π v ( x ₀, y ₀) . ε

As the result, we obtain formulas for calculating the components of the deformation vector 2 π u ( x ₀, y ₀) = ∫ - Ady + Bdx ,   2 π v ( x ₀, y ₀) = ∫ - Ady + Bdx ,

LL where c1 = ∫ αθdx ,c2 = ∫ βωdx.

Conclusion

In the article, new infinite series of conservation laws are obtained for the resolving system of equations, as well as for an automorphic system, constructed for two-dimensional elasticity equations. These laws made it possible to construct the analytical solution of the boundary value problem for the equations of the two-dimensional theory of elasticity in the stationary case. In the paper the solution of boundary value problems with the help of conservation laws, started in [13–15], is continued.