MODELING OF PLASTIC FLOW BETWEEN RIGID PLATES APPROACHING TO A CONSTANT ACCELERATION


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In this paper we study equations describing a slow plastic flow of material. In this case the material is in the flat state of stress. The paper presents the equations that can be used to simulate slow plastic material flows compressed between rigid plates, converging with constant acceleration. In the above equations we neglect convective terms, which greatly simplifies all calculations. The Lie algebra of point symmetries admitted by these equations is calculated for reduced equations. It has dimension eight. The optimal system of one-dimensional subalgebras is constructed for this algebra. It allows to give a view of all the different invariant solutions of rank two. That means such solutions depend only on two independent variables. To demonstrate this we offer a table of switches of all basis operators, as well as a table of all internal automorphisms functioning. One of the solutions, which simulates the slow plastic flow of the mate- rial compressed between rigid plates, converging with constant acceleration, built in. Among the most popular solu- tions in the flat theory of ideal plasticity is the Prandtl’s solution, which describes the compression of a plastic layer between rigid plates. In this case, the plates approach at a constant speed. The popularity of the solution is explained by its simplicity, as well as the fact that it can be used to describe various technological processes. The analogue of such a solution for the plane stress state cannot be constructed. In general, there are big problems with finding analytical solu- tions for the plane state of stress. It is caused by the fact that the equations describing this state are quite complex, even in spite of their linearization. In one of the previous works, one of the authors of the present article managed to find a solution that describes compression of a plastic layer between rigid plates which converge with constant acceleration. In this work the analogue of such a solution is found for the plane stress state. The authors hope that the suggested so- lution can also be used for the analysis of real technological processes.

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Introduction. The most popular decision in the flat theory of ideal plasticity is the one of Prandtl which de- éæ ¶u ¶v ö2 l = 3 3k êç 2 ¶x + ¶y ÷ æ ¶v ¶u ö2 + ç 2 ¶y + ¶x ÷ scribes compression of a plastic layer between rigid êëè ø è ø plates. At the same time plates approach at a constant speed. The popularity of the decision is explained by its simplicity and also by the fact that it can be used for the description of various technological processes as well as - æ 2 ¶u ç è ¶x + ¶v öæ 2 ¶v ¶ ÷çy øè ¶y + ¶u ö + ¶ ÷x ø 3 æ ¶v è 4 ç ¶x ¶u ö2 ù-2 ø û + ¶y ÷  , rocks [1-7]. The analog of such decision for flat tension state cannot be constructed. It is connected with the fact that the group of symmetries allowed by equations in case of flat deformation, differs from group of pure shear yield stress in case of flat tension [8-15]. In general, there is a big problem with analytical decisions for flat tension, which is caused by the fact that equations describing this state are rather complex, even despite linearization. In work [8] one of authors managed to find a solution de- scribing compression of a plastic layer between rigid plates which approach with continuous acceleration for the case of flat deformation. In this work the analog of such solution is also framed for flat tension state. Authors hope that the solution will be also used for the analysis of ¶u = ¶ l æ 2 ¶u + ¶v ö + ¶ l æ 2 ¶u + ¶v ö , (4) ¶t ¶x 3 ç ¶x ¶y ÷ ¶y 6 ç ¶y ¶x ÷ è ø è ø ¶t ¶x 6 ç ¶y ¶x ÷ ¶y 3 ç ¶y ¶x ÷ ¶v = ¶ l æ ¶u + ¶v ö + ¶ l æ 2 ¶v + ¶u ö . è ø è ø Group properties of the equations flat state of stress. We will find the group of point symmetries al- lowed by the system of flat tension equations. We will construct the optimum system of one-dimensional subal- gebras and also give a type of all invariant solutions of rank one. Theorem. Equations (4) allow the group of continuous transformations generated by operators real technological processes. We will consider the equations describing flat tension state flat state of stress X = ¶ , 0 ¶t X = ¶ , 1 ¶x X = ¶ , 2 ¶y ¶u ¶s ¶t ¶v ¶t ¶s Y = ¶ , Y = ¶ , T = x ¶ - y ¶ , = x + , = + y , (1) 1 ¶u 2 ¶v ¶v ¶u ¶t ¶x ¶y ¶t ¶x ¶y Z = y ¶ - x ¶ + v ¶ - u ¶ , (5) ¶x ¶y ¶u ¶v x y x y s2 + s2 - s s + 3t2 = 3k 2 , (2) ¶ ¶ ¶ ¶ ¶ ¶ ¶u ¶v ¶v + ¶u ¶x = ¶y = ¶x ¶y , (3) 2sx - sy 2sy - sx 6t where σx, σy, τ - components of stress tension tensor; u, v - velocity vector components; k - pure shear yield N = t ¶t + x ¶x + y ¶y , M = t ¶t + u ¶u + v ¶v . We will find the optimum system of one-dimensional subalgebras. For this purpose we will calculate commuta- tors of operators (5). They are presented in tab. 1. Automorphism corresponding to Xi, acting on operator Xj according to formula stress. System (1)-(3) is a system of five equations for five Ai ( X j ) = a2 + é ù + é é ùù+ X j a ë Xi , X j û 2! ë Xi , ë Xi , X j ûû unknown functions. + a3 é é é ùùù Applying the ratios (2)-(3), equations (1)-(3) may be written down only in terms of function u, v. We obtain 2s - s = l ¶u , 2s - s = l ¶v , x y ¶x y x ¶y è ø ç ÷ læ ¶v + ¶u ö = 6t , ¶y ¶x 3! ë Xi , ë Xi , ë Xi , X j ûûû +… Here a - some valid parameter. It is convenient to collect the influence of automor- phisms on operators (5) in tab. 2. Lemma. Operators X0, Xi, Yi, T generate the ideal in Lie algebra. Thus, automorphisms A0, …, A8 cannot change coeffi- cients under Z, M, N operators. Table 1 Commutator table X0 X1 X2 Y1 Y2 T Z N M X0 0 0 0 0 0 0 0 X0 X0 X1 0 0 0 0 0 Y2 -X2 X1 0 X2 0 0 0 0 0 -Y1 X1 X2 0 Y1 0 0 0 0 0 0 -Y2 0 Y1 Y2 0 0 0 0 0 0 Y1 0 Y2 T 0 -Y2 Y1 0 0 0 0 -T T Z 0 X2 -X1 Y2 -Y1 0 0 0 0 N -X0 -X1 -X2 0 0 T 0 0 0 M -X0 0 0 -Y1 -Y2 -T 0 0 0 Table 2 Table action of automorphisms X0 X1 X2 Y1 Y2 T Z N M A0 X0 X1 X2 Y1 Y2 T Z N - a 0X0 M - a 0X0 A1 X0 X1 X2 Y1 Y2 T - a 1Y2 Z + a 1X2 N - a 1X1 M A2 X0 X1 X2 Y1 Y2 T + a 2Y1 Z - a 2X1 N - a 2X2 M A3 X0 X1 X2 Y1 Y2 T Z + a 3Y2 N M - a 3Y1 A4 X0 X1 X2 Y1 Y2 T Z - a 4Y1 N M - a 4Y2 A5 X0 X1 + a 5Y2 X2 - a 5Y1 Y1 Y2 T Z N + a 5T M - a 5T A6 X0 X1 - a 6Y2 X2 + a 6X1 Y1 - a 6Y2 Y2 + a 6Y1 T Z N M A7 X0exp a 7 X1exp a 7 X2exp a 7 Y1 Y2 Texp(- a 7) Z N M A8 X0exp a 8 X1 X2 Y1exp a 8 Y2exp a 8 Texp a 8 Z N M In this respect it is necessary to consider one- dimensional subalgebras of four types D1 = Z + aM + bN + S , Suppose S1 = T + ai Xi + biYi , i = 1, 2 . We have D2 = M + aN + S , D3 = N + S , 2 A1 A2 S1 = T + a¢i Xi , A6 (T + a′i Xi ) = T + aX1 , Suppose where S = å(ai Xi + biYi ) + gT + a0 X 0 ; α, β, γ, αi, βi - i=1 S2 = Xi + aX 2 + biYi , A5 A6 S2 = X1 + aY1 . constants. Let us consider subalgebra D1 = Z + aM + bN + S . Under the influence of automorphisms A3, A4, A1, A2 we have A1, A2 , A3, A4 ( D1 ) = Z + a¢M + b¢N + g¢T + a′0 X 0 . The last disconjugate subalgebra from ideal has ap- pearance Y1. Now, taking into account that X0 is the centre of the ideal we will write down the final optimum system of one-dimensional subalgebras Here constants with strokes above are received as a re- sult of action of automorphisms in tab. 2. In this case there are only two types of disconjugate subalgebras Z + a (M - N ) + bX 0 + gT , Z + aM + bN , (M - N ) + aX 0 + gT , M + aN , M + aX1 , N + aY1 , Z + a(M - N ) + a0 X 0 + gT , Z + aM + bN , T + aX1 + bX 0 , X1 + aY1 + bX 0 , Y1 + aX 0 , X 0 . where α, β, α0, γ - arbitrary constants. We will consider D2 = M + αN + S. In this case there are three disconjugate subalgebras (M - N ) + a0 X 0 + gT , M + aN , M + aX1 . We will consider subalgebra D3 = N + S. After the automorphisms’ influence A0, A1, A2, A5, A6 we obtain not only disconjugate subalgebras N + b1Y1 . Remark 1. Dissimilar subalgebras correspond to vari- ous values of constants α, β, γ. Remark 2. With automorphisms A7, A8, as well as with external automorphisms of the system (4), the number of constants in optimum system can be reduced. 1. We will give a type of all invariant solutions of rank 2 which can be constructed on one-dimensional subalge- bras: Z + a (M - N ) + bX 0 + gT , if a ¹ 0 , then Now we have to make calculations with the ideal. u = r -1 f (x, h), v = r -1g (x, h) - 3g r2, r q The X0 operator forms the center of the ideal. Therefore it a can be excluded from consideration for now. x = bq - t, aq - ln r = h , if a = 0 , then ur = f (x, h), vq = g (x, h) - grq, x = bq - t, h = r . and are written as follows u = tu ( x, y ) , v = tv ( x, y ) . (6) Here and further r, q - polar coordinates; ur , vq - ve- Inserting (6) in (4) we obtain the system of two equalocity vector components in polar coordinates; f , g - tions in function u, v: some differentiable function of two variables. 2. Z + aM + bN, if b ¹ 0 , then u = ¶ l æ 2 ¶u + ¶v ö + ¶ l æ ¶u + ¶v ö, è ø è ø ç ÷ ç ÷ ¶x 3 ¶x ¶y ¶y 6 ¶y ¶x . (7) ln u - bq = f (x, h), ln v - bq = g (x, h), v = ¶ l æ ¶u + ¶v ö + ¶ l æ 2 ¶v + ¶u ö , r q ¶x 6 ç ¶y ¶x ÷ ¶y 3 ç ¶y ¶x ÷ x = (a + b) q - ln t, if b = 0 , then h = aq - ln r , where è ø è ø s = l æ 2 ¶u + ¶v ö , s = l æ ¶u + 2 ¶v ö , x 3 ç ¶x ¶y ÷ y 3 ç ¶x ¶y ÷ ur = f (x, h), vq = g (x, h) - grq, è ø è ø è ø l æ ¶u ¶v ö x = aq - ln t, h = aq - ln r . t = 6 ç ¶y + ¶x ÷ , 3. M - N + aX 0 + gT , éæ ¶u ¶v ö2 æ ¶v ¶u ö2 u = r -1 f (x, h), v = r -1g (x, h) - 3gr 2 , l = 3 3k êç 2 ¶x + ¶y ÷ + ç 2 ¶y + ¶x ÷ q êëè ø è ø -2 (8) x = a ln r - t, h = q . æ ¶u ¶v öæ ¶v ¶u ö 3 æ ¶v ¶u ö2 ù 4. M + aN , if a ¹ -1 , then u = t-(1+a) f (x, h) = t -(1+a) g (x, h), x = if a = -1 , then , h = t(1+a) x , - ç 2 ¶x + ¶y ÷ç 2 ¶y + ¶x ÷ + 4 ç ¶x + ¶y ÷  ,2 f = d kf ¢ , è øè ø è ø û dy d 2kg¢ u = tf ( x, y ), 5. M + aX1 , u = ax1 + f ( x, y ), v = tg ( x, y) . v = ax1 + g ( x, y ) . g = . dy By differentiating equation (8) at y, we have 6. N + aY1 , u = -a ln t + f (x t , y t ), v = g (x t , y t ). 2 f ¢ = d 2 dy2 kf ¢ g¢2 + 14 , f ¢2 . (9) 7. T + aX + bX , d 2 2kg¢ 1 0 2g¢ = . if a ¹ 0 , then dy2 g¢2 + 1 f ¢2 u = - xy a + f ( y,bx - at ), v = - 1 2a x2 + g ( y, bx - at ) . 4 Further, for simplicity we take k = 1 and enter new If a = 0,b ¹ 0 , then u = - yt b + f ( x, y ), v = - xt b + g ( x, y ) . variables f ¢ = 2w( y )cosh ( y ) , g¢ = w( y )sinh ( y ) . If a = 0, b = 0 , then there are no invariant solutions. The system (9) will be written as follows: 8. X1 + aY1 + bX 0, u = -ax + f (bx - t, y ), v = g (bx - t, y ) . 2wcosh ( y ) = (cosh ( y ))¢ , yy yy . (10) 9. Y1 + aX 0 , wsinh ( y ) = (2 sinh ( y ))¢ . 10. X 0 , u = -at + f ( x, y ), v = g ( x, y ) . Dividing the second equation of the system (10) by the first, we receive u = f ( x, y ), v = g ( x, y ) . sin (h) = 2(sin (h))¢ . Solution describing compression of the plastic layer between rigid plates approaching with continuous ac- 2 cos(h) (cos(h))¢ celeration. We will find the system of equations describ- ing plastic currents with continuous acceleration. These solutions are invariant to subalgebra M = t ¶ + u ¶ + v ¶ After simple transformations we receive the ordinary differential equation of the second order which coeffi- cients do not depend on y: h¢ (4 cos2 (h) + sin2 (h)) - 3h'2 cos(h)sin (h) = 0 . ¶t ¶u ¶v After standard replacement p = h', h" = pp', we obtain p¢ = 3cos(h)sin (h) . p 4 cos2 (h) + sin2 (h) By integrating the equation, we have Fig. 1 demonstrates dependence of function h on vari- able y. Now we calculate the components of the tension ten- sor. The graphs are provided in fig. 2-4. We have ln p = - 3 arctg 3 cos(h) + c . 3 t = kf ¢ g¢2 + 14 f ¢2 = 2k cos(h ( y )) , After the second integration, we have an implicit de- pendency h = h(y): h sy = 2k sin (h ( y )) , C ò exp( 0 3arctg 3 cos(h))dh = y . (11) sx = sy ± 2 . Fig. 1. Dependence of function h on variable y Рис. 1. График зависимости функции h от переменной y Fig. 2. Dependence of function τ on variable y Рис. 2. График зависимости τ от переменной y Fig. 3. Dependence of function σy on variable y Рис. 3. График зависимости σy от переменной y Fig. 4. Dependence of function σx on variable y Рис. 4. График зависимости σx от переменной y Conclusion. From the graphs and formulas it is clear that the solution provided can be used to describe com- pression of a plastic thin layer which is in conditions flat state of stress. Thickness of the layer does not exceed 0.06, however, the plates which compress the layer with continuous acceleration are characterized by constant pressure.
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作者简介

S. Senashov

Reshetnev Siberian State University of Science and Technology

Email: sen@sibsau.ru
31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation

E. Filyushina

Reshetnev Siberian State University of Science and Technology

31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation

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