Savostyanova I. L. 3-dimensional solutions from two variables

The notion of “3-dimensional solutions from two variables” has been added to the title of the article. Mechanics know the plane deformed state – this is the case when in a two-dimensional Cartesian coordinate system two components of the strain rate vector and the hydrostatic pressure depend on x , y .The plane stress state is also known – this is when the components of the stress tensor a_z , t_xz , т are equal to zero, and the components ^ , s_y , t^ do not depend on z .

In our case, all components of the stress tensor do not depend on z , just such a case we called the spatial two-dimensional state.

The system of 3-dimensional equations of plasticity in the Cartesian coordinate system xx = x, x2 = y, x3 = z in the stationary case has the form d jsij = d iP, sij = X(d u +d iui)/2, d iuj = 0, sjsj = 2 k2, i, j =1,2,3                (1)

Here s ij , s ij are components of the tensor and deviator of the stress tensor; u 1 = u , u ₂ = v , u ₃ = w - components of the strain rate vector; X - non-negative function; k_s - constant plasticity; p - the hydrostatic pressure, summation is carried out over repeated indices.

Eliminating the stress tensor deviator components from the system of equations (1), we obtain the following nonlinear system of equations

• ⁵ i P = ^ j — ^Г'^ j , e j e ij = A 2 , ^ i u i = 0.               (2)

It is known that the system of equations (2) is of elliptic type. Let us describe the known solutions of this system.

The solutions of this system were constructed by R. Hill in 1948 [1], W. Prager in 1954 [2], D. D. Ivlev in 1960 [3; 4], MA Zadoyan in 1964 [5–8], as well as by the authors of this article [9–13]. We also note a number of exact solutions constructed by B. D. Annin [14] for the plasticity equations in the spatial case with the Tresk yield condition.

Symmetries of system (2)

The group of point symmetries of the system of equations (2) is generated by the following operators

X =d ,, Y =d_u ., N = x d., M = u,d_u ., i = 1,2,3. ix_i  iu_i     ix_i      iu_i

T 1 = x 2 ^d u ₃ ^{- x} 3 ^d u ₂ , T 2 = x 3 ^d u 1 ^- x 1 ^d u ₃ , T 3 = x 1 ^d u ₂ ^- x 2 ^d u 1 ,

Z 1 = ^X 2 ^d x ₃ ^{- X} 3 ^d x ₂ ^{+ u} 2 ^d u ₃ ^{- u} 3 ^d u ₂ , Z 2 = ^x 3 ^d x 1 ^{- x} 1 ^d x ₃ ^{+ u} 3 ^d u 1 ^{- u} 1 ^d u ₃ ,

Z 3 = x 1 ^d x ₂ - x 2 ^d x 1 ^{+ u} 1 ^d u ₂ ^{- u} 2 ^d u 1 , S = ^d p .

The operators Xi,Yi,N,M correspond to the following continuous transformations x'i = xi + ai, u 'i = ui + bi, x'i = xi exp a,  u'i = ui exp b, i = 1,2,3.

These are translations along the coordinates and components of the strain rate vector, as well as tension.

Transformations T show that the system of equations (2) does not change under rigid displacements u '2 = u2 - c1 x3, u '3 = u3 + c1 x2, u '1 = u1 + с2x3, u '3 = u3 — с2x1, u '2 = u2 + c3x1, u '1 = u1 - с3x2.

The groups generated by the operators Zi, are rotations around the coordinate axes x '2 = x2 cos ф1 + x3 sin ф1, x '3 = -x2 sin ф1 + x3 cos ф1, x'3 = x3 cosф2 + x1 sinф2, x'1 = -x3 sinф2 + x1 cosф2, x '1 = x1 cos ф3 + x2 sin ф3, x '2 = -x1 sin ф3 + x2 cos ф3.

The last transformation describes the invariance of the hydrostatic pressure with respect to displacements p ' = p + d.

In all these formulas a i , b i , C i , ф i , d are group parameters. It is usually assumed that they vary continuously in a neighbourhood of zero.

The investigations carried out have shown that all the solutions constructed by R. Hill, V. Prager, D. D. Ivlev, and M. A. Zadoyan are invariant solutions with respect to some one-dimensional subgroups of point transformations generated by operators (3). The invariance here means that the solutions do not change under some transformations generated by the symmetry group (3). Thus, R. Hill's solution is invariant under the subalgebra generated by the operator 2 C₀S + X_{ + a Y + P T , D. D. Ivlev's solution is invariant under the subalgebra 2 C 0 S + X 1 + a Y , Prager's solution is invariant under the subalgebra aS + X + T + a T , Zadoyan's solution is invariant under the same subalgebra. What does this fact mean? It says that in fact all these solutions are "two-dimensional", that is, in a suitable coordinate system they can be written as functions of only two independent variables. The same can be said about the solutions constructed by the authors of this work. Then the question arises: what is to be understood by a 3-dimensional solution? Based on the solutions presented here, the answer is as follows: 3-dimensional solutions are solutions that have three components of the velocity vector, pressure, which actually depend on two variables in a suitable coordinate system. These solutions are invariant solutions of rank 2. In this case, the problem of finding 3-dimensional solutions can be formally stated as follows: construction of new invariant solutions of rank 2 for 3-dimensional equations of ideal plasticity. The form of such solutions can be easily enumerated if we enumerate all the different, up to inner automorphisms, onedimensional subalgebras of the algebra (3).

There are several subalgebras on which invariant rank 2 solutions can be constructed. Let's list them.

X 3 + y S , X 1 + Z 1 + y S , a M + N + y S , N + Y 1 + Y S ,

Z1 + aN + Y + YS, Z1 + aN + pM + yS,

X 1 + a Z 1 + M + y S ,

X 1 + Y + a T 1 + y S , X 1 + Z 1 + T 1 + Y S , M + N + T 1 + Y S , Z 1 + Y +a T 1 +Y S , X 1 +a X ₂ + T ₂ +p T ₃ +y S , X 1 + Z 1 + Y 1 + a T 1 + Y S .

Here α , β , γ are arbitrary constants, different values of these constants correspond to dissimilar subalgebras.

In this paper, we consider only solutions invariant with respect to the subalgebra X +γ S .

Note. Other solutions based on subalgebras (4), as well as the form of all invariant solutions that can be constructed for the system of equations (1), can be found in [11].

Solutions invariant with respect to this subalgebra should be sought in the form u = u(x, y), v = v(x, y), w= w(x, y), p = p(x, y) +γz.                       (5)

Substituting relations (5) into the system of equations (1), we obtain

∂ _x σ _x + ∂ _y τ _xy = 0, ∂ _x τ _xy + ∂ _y σ _y = 0, ∂ _x τ _xz +∂ _y τ _yz =γ , ∂ _x u +∂ _y v = 0,

( σ _x -σ _y )² + ( σ x -σ z )² + ( σ _y -σ _z )² + 6( τ ² _xy +τ ² _xz +τ ² _yz ) = 6 k _s ²,

σ _x - p =λ∂ _x u , σ _y - p =λ∂ _y v , σ _z - p = 0, 2 τ _xy = λ ( ∂ _x v + ∂ _y u ),

2 τ _xz = λ∂ _xw , 2 τ _yz =λ∂ _y w .

We make the change of variables in (6) by the following formulas

σ _x = k ( а/ 3cos ω+ sin ω cos2 ϕ ), σ = k ( у/ 3cos ω - sin ω cos2 ϕ ), τ _xy = k sin ω sin2 ϕ .         (7)

Here k = δ k , 0 < δ <  1 is some constant.

Substituting these relations into (6), we obtain

( - cos ω+ у/ 3sin ω cos2 ϕ ) ∂ _x ω + у/ 3sin ω sin2 ϕ∂ ω - 2sin ω∂ _y ϕ = 0,

(cos ω + у/ 3sin ω cos2 ϕ ) ∂ ω + 3sin ω sin2 ϕ∂ _x ω + 2sin ω∂ _x ϕ = 0,

∂ τ +∂ τ =γ , ∂ u +∂ v = 0, τ ² +τ ² = k ² - k ² = K ²,                 (8)

x xz   y yz       xy        xz   yz   s

σ _x - p =λ∂ _x u , σ _y - p =λ∂ _y v , σ _z - p = 0, 2 τ _xy = λ ( ∂ _x v + ∂ _y u ),

2 τ _xz = λ∂ _xw , 2 τ _yz =λ∂ _y w .

From (8) we see that the original system split into two subsystems: the first two equations essentially coincide with the equations describing the plane stress state

(-cosω+ у/3sin ωcos2ϕ)∂ ω +у/3sinωsin2ϕ∂ ω -2sinω∂yϕ = 0,(9)

(cos ω + у/3sinωcos2ϕ)∂ ω +у/3sinωsin2ϕ∂ ω +2sinω∂xϕ = 0, and equations reminiscent of the equations describing the plastic torsion of a rod, at γ=0 and different yield point

∂ τ +∂ τ  =γ, τ2 +τ2 =k2 -k2 =K2.(10)

x xz     y yz         xz     yzs

Solving equations (9) and (10), it is possible to find the components of the stress tensor, while

σ _z = p = 1/2( σ _x + σ _y ). The only problem is to determine the constant k .

To determine the components of the velocity vector, we obtain the following equations

∂ v    ∂ v+∂ v∂

∂xu   =    y = xy, ∂x(  ∂xw  )+ ∂y(   y)

2σx-σy2σy-σx     6τxy          wx2+wy2         w2+ w2

xy

Note that the last equation for γ ≠ 0 has not yet been sufficiently studied, it has not even been included in the handbook [15].

As a result, it turned out that the solution of the system of equations (6) is actually a superposition of the plane stress state and plastic torsion around the z axis.

Let us give some other solutions of equations (6). For this, we find the group of point symmetries of the system of equations (6).

This group is generated by the operators

X 1 =∂ x , X 2 =∂ y , Y i =∂ u_i , N = x ∂ x + y ∂ y , M = u i ∂ u_i , i = 1,2,3.

T = x 1 ∂ u ₂

^{- x} 2 ^∂ u 1^{, Z} 3 ^{= x} 1 ^∂ x 2 ^{- x} 2 ^∂ x 1 ^{+ u} 1 ^∂ u 2

-

u ₂ ∂ _{u 1} , S =∂ _p .

We are looking for a solution that is invariant under the subalgebra generated by the operators X ₁ +α T + Y ₁ +β Y ₃ +γ S .

The solution should be sought in the form u = x+αxy+U(y),v = -α/2x2 + V(y), w= βx+ W(y), p = γx+P(y).

From the incompressibility equation we obtain

v = -α /2(x + y ) - y + C1.

Substituting (12) into (6) and we obtain the system of ordinary differential equations d (λU')=γ, d (λW')=0, 6ks2λ-2 =2(1+αy)2+6((U')2+β2+(W')2).

dy          dy

From (13) we have

λ U ' =γ y + C ₂, λ W ' = C ₃.

Here C , C , C – arbitrary constants.

We obtain

U '/ W ' = ( γ y + C ₂)/ C ₃,

6 k_s ² λ^{- 2} = 2(1 +α y )² + 6( β ² + (1 + ( C ₄ +ν y )²)( W ')²), ν=γ / C ₃, C ₄ = C ₂/ C ₃.

The system of equations (13) can be reduced to quadratures, which are expressed through elliptic integrals.

The constructed solution can be used to describe the plastic flow of a layer compressed by rigid plates orthogonal to the oz axis.

Let us write system (6) in a cylindrical coordinate system r , θ , z .

∂rσr + r-1∂θσrθ +(σr -σθ)/r=0, ∂rσrθ + r-1∂θσθ + 2σrθ /r=0,

∂ _r σ _rz + r ^{- 1} ∂_θσ_{θ z} +σ _rz / r = 0, σ _r - p = λ∂ _r u , σ_θ- p =λ ( u / r +∂_θ v / r ), σ _z - p = 0, 2 σ _{r θ} = λ ( ∂_θ u / r + r ∂ _r ( v / r )), 2 σ _{z θ} = λ r ^{- 1} ∂_θ w , 2 σ _zr = λ∂ _r w ,

( σ _r -σ_θ )² + ( σ _r -σ _z )² + ( σ_θ-σ _z )² + 6( σ _r ² _θ+σ ² _{θ z} +σ _r ² _z ) = 6 k_s ², ∂ r u + u / r +∂_θ v / r = 0.

We are looking for an invariant solution on a subalgebra Z ₃ + M , it has the form u = U ( θ ), v = V ( θ ), w = W ( θ ), p = P ( θ ) +α ln r .

Substituting (15) into (14), we obtain the system of ordinary differential equations d _θσ _r + ( σ _r -σ_θ ) =α , d _θσ_θ + 2 σ _{r θ} = 0, d _θσ _{r θ} + σ _rz = 0, σ _r - p = 0,

σ_θ - p = λ ( U / r + V '/ r ), σ _z - p = 0, 2 σ _{r θ} = λ ( U '/ r + r ∂ _r ( V / r )), 2 σ _{z θ} = λ r ^{- 1} W ', 2 σ _zr = λ / r ,

( σ _r -σ_θ )² + ( σ _r -σ _z )² + ( σ_θ-σ _z )² + 6( σ _r ² _θ+σ_θ ² _z +σ _r ² _z ) = 6 k _s ², U + V ' = 0.

Here the prime means the derivative with respect to θ .

From this we obtain the system of ordinary differential equations

(17 )

( X (U '- V ))' = a , X (U '- V ) = P ,  ( X W ) = 0,  6 kfr² = (U '- V )² + (W ')² + 1.

System (17) is solved completely similarly to system (13).

The found solution can be used to describe plastic flow in a converging flat channel with rigid and rough walls.

Other solutions of system of equations (1) can be found in [11].

Conclusion

In this work, a class of equations has been studied, which is called the equations describing the 3dimensional deformed state. For these equations, a group of point symmetries is found, admitted by them in the sense of Lie. It is shown that a two-dimensional stress state is a superposition of a plane stress state and torsion around the z axis. Several invariant solutions of these equations are constructed.