Steady Flow of Incompressible Elastic-Plastic Medium in a Spherical
Matrix at Variable Loads
Marina V. POLONIK1,2, a, Egor E. ROGACHEV2, b
1
Far Eastern Federal University, 8 Suhanova Str., Vladivostok, 690950, Russian Federation
2
Institute for Automation and Control Processes of the Far Eastern Branch of the Russian
Academy of Sciences, 5 Radio Str., Vladivostok, 690041, Russian Federation
a
[email protected], [email protected]
Keywords: Elasticity, plasticity, large elastic-plastic deformations, stresses, rheology.
Abstract. In the framework of the model of large elastic-plastic deformations, the flow of the
material in a spherical matrix at varying loads is examined. Simulation is carried out under the
condition of steady elastic-plastic boundary. In order to find an exact solution the assumption of
an ideal smoothness of the walls and incompressibility of the material is accepted.
Introduction
Basic mathematical models in the calculation of the technologies of rolling and deep drawing of
material are either the theory of ideal plastic flow [1-3], or the theory of elastic-plastic processes
of A.A. Ilyushin [4, 5]. In the first case, as a rule, elastic deformations are neglected due to their
smallness, while remaining within the rigid-plastic analysis. This does not allow predicting the
elastic response of the material during unloading which is very important in the modern
technology. The deformation theory of plasticity, even at the possibility of taking into
consideration of large irreversible deformations [5], does not allow calculating elastic response
during unloading. Therefore, here we will use the mathematical model of large elastic-plastic
deformations [6-8], constructed as flow theory [9-13]. Within the framework of this model there
were obtained solutions of the problems of the theory of large elastic-plastic and elastic-viscousplastic deformations such as the formation of residual stresses in the vicinity of defects of
continuity [8-9, 14-17], on a linear flow [18] and viscous-metric flows [19], about the propagation
of elastic waves [20]. Namely within this model, one can construct an exact solution of the
problem of the theory of large elastic-plastic deformation about the steady flow of material in a
spherical diffuser under steady elastic-plastic boundary and variable loads. For the deep drawing
technologies such a situation is optimal and it is achieved by the development of technological
regimes. There is accepted an assumption of the ideal smoothness of the walls and
incompressibility of the material. Let us also note that the problem of steady flow of material in a
spherical diffuser with an elastic-plastic boundary and constant load was presented in [21].
Depending on model
A detailed description of the mathematical model of large elastic-plastic deformations is provided
in [6-7]. Here we mention only required dependencies. As in [7], we assume that the separation of
the strain tensor Almansi d ij to reversible (elastic) eij and irreversible (plastic) р ij components
follow the law:
d ij  eij  рij  1 / 2eik ekj  eik р kj  р jk ekj  eik р ks esj .
(1)
The result of the law of energy conservation and the hypothesis of independence of the free energy
from the irreversible deformations is the Murnaghan’s formula for determining the stresses:
 ij   р ij  W / d ik ( kj  2d kj ) .
(2)
In Eq. 2 р - additional hydrostatic pressure arising due to the condition of the medium
incompressibility    0  const . For the isotropic elastic-plastic medium, elastic potential
W  W ( L1 , L2 ) is given in the form
W  (a  ) L1  aL2  bL12  L1L2  L12 , L1  d kk , L2  d kmd mk .
(3)
Eq. 2 and Eq. 3 are true when in the medium рij  0 , wherein the condition from (1):
d ij  eij  1 / 2eik ekj  1 / 2(ui , j  u j ,i  u k ,i u k , j ) .
(4)
Continuum Motion is carried out in Euler coordinates; ui - Components of the displacement
vector;  - Shear modulus a,b, ,  - the elastic moduli of a higher order.
If the medium of irreversible deformation рij  0 invariants L1 and L2 should be replaced by
invariants of the tensor of reversible eij deformation: I1 and I 2 [14] that will provide passage to
the limit in the calculation of stresses from the formula
 ij   р ij  W / d ik ( kj  2ekj )
(5)
to formula Eq. 2 when tends to zero plastic strain р ij .
Statement of the problem
We believe that the motion of the medium is carried out in a spherical matrix    0 with
perfectly smooth surface. We apply a spherical coordinate system (r , , ) . We assume that the
motion of the medium is provided by the force on the borders of the matrix r  R and
r  s ( R  s)
 rr | r  R   P* (t ),   | r s  * (t ).
(6)
Furthermore, we assume that there is some material stationary surface r  R1 , dividing it into two
fields formed during deformation: elastic R1  r  R (I) and elastic-plastic s  r  R1 (II).
In the elastic region (I) the only nonzero component of the displacement vector ur  u for an
incompressible medium is determined by  /  0  1  u / r 1  u / r   1 the relationship:
u  r  (r 3  )1/ 3 ,   R03  R 3 .
(7)
The components of the strain tensor Almansi Eq. 1, taking into account Eq. 7 and   1   / r 3 :
d rr  1 / 2(1  r 4 (r 3  ) 4 3 )  1 / 2(1  4 3 ), d   1 / 2(1  r 2 (r 3  ) 2 3 )  1 / 2(1  2 3 ) .(8)
Using Eq. 8, according to the relationship Eq. 2 and Eq. 4, integrating the equilibrium equation
 rr ,r  2( rr    ) / r  0
(9)
we define the stress components
 rr  c2  F (),     rr  Ф().
(10)
In Eq. 10 the following notations are used:
F ()  b8  3  b9  2  b10  5 3  b11 1  b12  2 3  b13  1 3  b14 ln 1 3  b15 1 3  b16  2 3 
 b17   b19  2 ; Ф()  c3  4  c 4  8 3  c5  2  c6 ( 4 3   2 3 )  c7  2 3  c8  4 3  c9  2 ;
b8  1 / 6(   ), b9  3 / 2b8 , b10  2 / 5(a  b    9 / 5), b11    2, b12  a  b  1 / 2(5  9),
b14  3b11 , b13  2  4a  6b  9 / 4(   3), b15  2a  4b  7   18, b16  1 / 2b13 , b17  b11 ,
b18  1 / 4b15 , b19  1 / 2b11 , c3  3 / 4(  ), c 4  a  b  5 / 2  9 / 2, c5  3 / 4  9 / 4,
с9  3 / 2  3, с8  a  2b  7 / 2  9, c7  b    9 / 2, c6    2a  3b  9 / 4  27 / 4.
The integration constant с 2 is determined by the first boundary condition Eq. 6. Thus the stress in
R1  r  R are determined up to the unknown  :
 rr  F ()  F (1 )  P* (t ),    F ()  F (1 )  P* (t )  Ф(), 1  1   / R 3 .
(11)
In areas with irreversible deformations (II) stress are determined from the equilibrium equation
Eq. 9, taking into account the fulfillment of Tresca plasticity conditions [14] in this whole area
( rr    ) sr  R1  2k .
(12)
Then  rr  4k ln r  c1 ; k - elastic limit; c1 - the integration constant, which is determined from
the second boundary condition Eq. 6. The stress distribution around the plastic layer:
 rr  4k ln( r / s)  * (t ) ,   2k (1  2 ln( r / s))  * (t ) .
(13)
At the boundary of the elastic and plastic area r  R1 value  rr , calculated in accordance with
Eq. 11 and Eq. 13 must match. It results in the nonlinear equation
F (2 )  F (1 )  P* (t )  4k ln( R1 / s)  * (t )  0.
(14)
The second condition (10) and condition (12) is on the border r  R1 allow us to write:
Ф(2 )  2k , 2  1   / R12 .
(15)
Equation Eq. 15 allows us to determine 2 and therefore also  . Thus, equation Eq. 14 and Eq. 15
are related by the parameters P* (t ) , * (t ) , R , R1 , s . By varying these parameters, you can get the
necessary optimum mode of steady flow.
Numeric calculations
The calculations were carried out for two sets of the given matrix parameters (1 - R / R0  0.9987 ,
s / R0  0.01 , R1 / R0  0.0180 ; 2 - R / R0  0.9987 , s / R0  0.2 , R1 / R0  0.3600 ) and for the
changing loading forces P* (t ) /  and * (t ) /  . In the calculations there were used the values of
the material constants: a /   6.09 , b /   27.07 ,  /   135.33 ,  /   541 , k /   0.0203 ,
  25109 Па . All figures are given dimensionless quantities  /  , r / R0 , t ( rr /  - solid,
 /  - dashed line). Fig. 1 shows the stress distribution in the matrix under the given loading
forces
P* (t ) /   0.02(1  1 /(t 2  1)) with possible switching to steady loading. Fig. 2 -
P* (t ) /   0.02(1  cos(t )) - shows periodical loading. Fig. 3 - P* (t ) /   0.02(1  ln( 1  t )) - shows
infinitely increasing loading with a shift in the loading forces direction.
0.05
0.05
1
(t)
2
rr
*
0.04
0
rr
- 0.05
qq
s/R0 R1/R0
qq
s/R0
0.6
R1/R0 0.4
0.8
t3 0.03
t2
t1
r/R0 0.02
t1 t2
1
P* (t)
t
20
t3
60
Fig. 1. Steady loading P* (t ) /   0.02(1  1 /(t 2  1))
*(t)
2
0.05
0.075
1
rr
rr
0
t1 t3
0.05
0.025
- 0.05
qq
s/R0 R1/R0
qq
s/R0
0.6
R1/R00.4
0.8
1
t2
r/R0
0
t1
P*(t)
t3
t2 25
15
5
t
Fig. 2. Periodic loading P* (t ) /   0.02(1  cos(t ))
0.05
2
1
P (t)
0.06
0
rr
t1
0.04
qq
- 0.05
- 0.1
*
rr
0.02
t2
t3
r/R0
qq
s/R0 R1/R0
s/R0
R1/R0 0.4
0.6
0.8
1
(t)
0
*
t1
5 t2
10
15
t
t3
Fig. 3. Infinitely increasing loading with a shift in the loading forces direction
P* (t ) /   0.02(1  ln( 1  t ))
Thus, by changing the matrix parameters and forces applied onto the material, we can achieve the
necessary optimal mode.
Summary
Within the model of large elastic-plastic deformations there is found an exact solution to the
problem of flow of an incompressible elastic-plastic medium in a spherical diffuser with perfectly
smooth walls and fixed boundary at variable loading forces. By varying the parameters of the
matrix and the efforts attached to the molded material, we can achieve the desired optimal mode
of movement of the material.
Acknowledgments
We gratefully acknowledge partial financial support from the Russian Foundation for Basic
Research (RFBR) (Grant 14–01–31069).
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