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The prediction of macro-defects during the isothermal forging process by the rigid-viscoplastic finite-element method
Journal of Materials Processing Technology, 32 (1992) 599-608
599
Elsevier
The prediction of macro-defects during the isothermal forging process by the rigid-viscoplastic finite-element method Mingwang Fu a and Zijian L u o b ~Nanchang Institute of Aeronautic Technology, Nanchang, China bNorthwestern Polytechnic University, Xian, China (Received May 6, 1991; accepted in revised form September 17, 1991 )
Industrial Summary Macro-defects often occur during isothermal forging of workpieces of complicated shape. The occurrence of macro-defects is mainly attributable to an abnormal flow pattern of the metal. In order to reveal the cause of macro-defects during isothermal forging of a very large aluminium wheel for an aeroplane, the forging process is simulated by the rigid-viscoplastic finite-element method. The results so calculated show that the occurrence of a folding defect is caused by an abnormal flow pattern. Improvement in the die design and lubrication conditions are effective measures to avoid this kind of defect.
1. Introduction
Isothermal forging can produce complicated workpieces that are difficult to form by common forging technology. With regard to complicated workpieces, it is not easy to judge whether the flow patterns are normal or not. Macrodefects occurring during forging processes are caused mainly by abnormal flow patterns, thus it is necessary to understand the flow pattern during the process to be able to find the way to deal with the situation. A new development in the plastic-working theory to predict the occurrence of macro-defects is to use the finite-element method, thus providing a scientific basis for optimal plastic-working technology [ 1-5 ]. The rigid-viscoplastic finite-element method is an extension of the rigidplastic finite-element method and has become a powerful tool for the analysis of viscoplastic forming processes [ 1-3 ]. In the present study, an axisymmetric isothermal forging process for an aeroplane wheel has been simulated using this method. It is found that macro-defects (folding) occur due to an abnormal Correspondence to: M.W. Fu, Nanchang Institute of Aeronautic Technology, Nanchang, China.
0924-0136/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
600
flow pattern of the material. Some suggestions are made for the prevention of the occurrence of the folding defect.
2. Rigid-viscoplastic finite-element formulation According to the extremum principle, amongst all of the admissible velocity fields, the actual solution gives the minimum value of the following function:
O=f
dV- f FUdS+ I ,tiidV
Vo
SF
(1)
Vo
where E(.~) is the work function, Sr is the surface on which traction is prescribed, F is the traction vector, U is the velocity vector, 2 is the Lagrangean multiplier, and V0 is the volume of the deforming body. Suppose the deforming body is divided into m elements connected by n nodes. The velocity vector in eqn. (1) can be determined by the following equation: .U= _Nv
(2)
where v is the velocity vector at nodal point and _Nis the velocity interpolation matrix. The strain-rate vector is then derivable in the form:
=B_U
(3)
where B_is the strain rate-velocity matrix. The effective strain rate can be expressed as follows: i-=
(~TD_~)=
(uT_pv)I/2
(4)
where --2
2
D- =
~
2
D=
I
0
l
0 2 1
~ 1
n
m
for the axisymmetrical problem, _P= B_TD_B_. Equation (1) can be approximated by the summation of the functions of the element nodal velocities and Lagrangean multipliers: m
0J(v y)
(5)
j=l
where 0 j is the function of the j t h element, and vj and ~t~are the nodal velocities
601
and Lagrangean multiplier of the jth element, respectively. According to the stational condition, it is found that i=a 0v
J=~ -~-~7= 0
(6)
Solving eqns. (6) simultaneously, the velocity field and the mean stresses within the deforming body can be defined. Because eqn. (6) is a non-linear equation system, the Newton-Raphson method is employed as the solving process. Suppose the nth iteration nodal velocity vector and velocity increment vector for the nth iteration are v~ and Av~, respectively. The velocity vector for the (n+ 1 )th iteration can be determined as follows: ~n+l"~V~n'~-AV.n
(7)
The work function can be written as
o
o
where ~ii is the strain rate, a~i is the stress deviator tensor, 0 is the effective stress, and g-is the effective strain rate. Taking account of eqns. (7) and (8), expanding eqn. (5) into a series by the Newton-Raphson procedure and taking only the first two terms, the following linear equation system can be obtained:
•
f t (o/~-)P dV+ f R dV}
J= l ¢oj ["
Voj
~ f B_"r.CdV
j=l
Voj
Avn
~a ~ CTB dV Voj
J
(9)
Voi where
and 2,~ is the vector of the Lagrangean multiplier of each element and equals the hydrostatic stress of the relevant element.
6O2
After each iteration, the values of v are modified as Vn+l=Vn+flAv~
(10)
where fl is the coefficient to accelerate convergence, 0 < fl< 1. The convergence criterion is as follows:
llAvlllllvll ~
(11)
where IIv II is the Euclidean vector norm
llvll
(12)
= ( v T v ) 1/2
and g is a pre-determined small positive number, 10 -~ ~
3.1. Friction force The friction stress is determined by the following equation:
r= - m K [ (2/7~)tan(vr/a] VDI ]t
(13)
where m is the friction factor (0 < m < 1 ), K is the shear yield stress, vr is the magnitude of relative velocity between the die and the workpiece, a is a constant several orders of magnitude smaller than the die velocity, such as 10 -5 , ] VD [ is the absolute value of the die velocity, and t is the unit tangential vector.
3.2. Initial velocity field The initial velocity field can be determined by the following function [ 7 ]: (e2e-2Vo2j)- ~ (Fivi) +
Or= j=l
i=1
~tii dV
(14)
j= lyo j
where Fi and vi are the nodal force and velocity at the traction boundary.
3.3. Treatment of folding The folding is caused by the large frictional stress between the die and the workpiece. The metal flow along the contact surface is resisted by friction force, resulting in the lateral surface rotating upwards and folding occurring. The distorted mesh is illustrated in Fig. 1 to explain how to treat the folding phenomenon. Because of folding, side surface 3'4'5' becomes 3'4'5". The original intersection point 5' between the side surface of the workpiece and the die
603 ~,/i //i
lJxl
,/~/]/,-_/.///
8
l la ) ~
3
M
~ 3 '
l z (b)
LS~z
,
2
Fig. 1. The treatment of folding: (a) beforedeformation;(b) after folding. surface is substituted for 5", so after folding, 3'4' 5' cannot describe the real boundary: should an attempt be made for it to do so, this would lead to a great calculation error, and the calculation would not be able to be continued. In order to solve the problem, it is necessary to describe the boundary correctly. Side surface 3' 4' 5' is divided into n equidistance sections. After the nth step, point i can be determined by the following equations:
ri =N3ir3
"~'N4ir4 "t-Nsir5
Zi = N 3 i g 3
t-N4ig4
JFNsiZ5
(15)
where N3i, N4i, Nsi, r3, r4, rs, and za, z4, z5 are the values of the shape function and the coordinates of node 3', node 4' and node 5', respectively. At the (n + 1 )th step, the velocity of point i in the z-direction is V n+l ~N iz ~
3i
v n + l _1_ ~T . n + l . N 3z ~ ~ " 4i U 4z "r"
5i
v n+l 5z
(16)
where v3~, v4~, vs~ are the velocities of node 3', node 4' and node 5' at the (n + 1 )th step, respectively, when the following condition is satisfied for point i:
zi +vz~ Atn+l>~z5 +vs~
A t n+l
(17)
where At n+l is the time increment of the ( n + 1 )th step. Thus, point i becomes point 5", substituting point 5' and being the 5th node of element M. In order to avoid the distortion of element M, point 4' should be modified to the center of section 3' 5", becoming the 4th node. Similarly, the 6th node of element M is substituted for by point 6". Because the coordinates of the nodes are changed, the accumulated field variables of the corresponding nodes should be modified. For instance, the ( n + 1 )th step equivalent strain of point 5" is determined as follows: ~n,,+ 5 1 = e-T + A ~ 7 + 1
(18)
3.4. Mesh rezoning Simulating the large plastic deformation process by the finite-element method, it is necessary to redefine a new mesh system because of the serious distortion of the old mesh. The generation of the new mesh system consists of
604
1
2
3
Fig. 2. The determination of field variables at the new model.
generating a new mesh system and defining the field variables in the new mesh system: the latter can be generated by some available algorithm. The determination of the field variables in the new mesh system is based on its distribution in the old mesh system. In this paper, the authors adopted the following method to determine the field variables in the new mesh system. The accuracy of calculation of the field variables at the Gauss integration point inside the element is much greater than that at the element boundary and the field variables associated with the derivative of velocity may be discontinuous at the element boundary. In view of the above, the field variables at the element boundary are modified according to the field variables of the Gauss integration point. The quadrilateral element with eight nodes is divided into eight triangles (Fig. 2). The field variables of node K at the new mesh system can be determined according to the field variables at the nodes within triangle 128. After the field variables at all nodes, the new mesh has been determined, and the distribution of field variables in the new mesh defined. 4. Test material and experimental device
In order to simulate the isothermal forging process of aluminium alloy, pure lead is used as simulation material. The dimension of the sample is 19 X 23 mm. The constitutive equation as obtained experimentally is: # = Y(e-) [1+ (E-/r) n ]
t: LI _
2.Smm (a)
~
(19) punch
plate upper d ie
middle die b o t t o m die
I
I
(b)
Fig. 3. The dimensions of the sample (a) and the experimental device (b).
605 where Y(£) is the static yield stress, i-is the equivalent strain rate, and r is the parameter associated with the viscosity of materials: 6.380+53.9~ for ~<0.143 I]13.955+20.286 sin[n(~-0.143)/0.237]
..... ~te~=)15.758-2.685(e--0.310) ~,14.543 for ~>0.670
for 0.143<~<0.294 0.294
'0.512~+0.081 for C~<0.407 1.485-6.000~+9.400(2-4.600e 3 for 0.30 for ~>0.668
n=.
rn=
for
'
(20)
0.407<£~<0.668
- 1 4 . 1 4 0 ( + 6.426 for £~0.384 10.882-52.182£+86.718£2-46.509~ a for 0.781 for ~>0.621
0.384<~<0.621
(21)
(22)
where n is the strain-rate index. The dimensions of the sample are determined according to the law of similarity. The dimensions of the sample and the experimental device are shown in Fig. 3. The lubricant is machine oil, affording a friction factor of 0.22. 5. Pattern of metal flow determined by the finite-element method
In order to investigate the influences of strain rate and lubrication on the flow pattern, the following cases are simulated by the rigid-viscoplastic finiteelement method: (1) eo = 1 0 -3 s -1, m=0.22 (Co is the initial strain rate); (2) ~o = 3 × 10 -2 s -1, m=0.22; (3) eo = 3 × 10 -2 s -1, m=0.05. The flow patterns of the three cases are shown in Fig. 4. For the first case, all the material moves downwards at the initial stage. When the stroke of punch reaches 3.6 mm, the neutral layer appears. In Fig. 4, h represents the position of the neutral layer. As deformation progresses, the neutral layer moves down, the amount of material moving up becomes larger and its velocity becomes greater. When the stroke reaches 14 mm, the material coming from the upper exit contracts the shoulder of the punch and is forced to move downwards, leading to the disappearance of the neutral layer. When the lower cavity is almost filled with the material, the neutral layer occurs again, but it is located near the lower exit. Moreover, when the stroke is about 14 mm, the contour of punch forces the material at point A to move downwards and towards the left, at point B to move downwards and towards the right and at point C to move downwards and towards the left. The above flow pattern gives rise to the formation of folding at point B with the punch moving downwards (Fig. 4). The results of the finite-element simulation are consistent with the experimental results of Fig. 5.
606
/ 'i"" r
4
1
L &h=4.0mm
Ah=A.0mm
,I , , i w ,
Ah= 6.0mm
Ah=4.0mm
h
Ah= 6,0mm
Ah = 6,0ram
l:::
, :,:,:,
A h =14.0ram
Ah=14.(
A h = 15.3mm
Ah = 1 5 . 2 m m
Ah = 1 5 . 3 m m
(b)
(G)
(a)
mm
IAh : 1 4 . 0 m m
Fig. 4. T h e flow patterns of three cases, simulated by the rigid-viscoplastic finite-element method.
For the second case, the flow pattern is slightly different from that of the first case because of strain-rate sensitivity of the material. At first, because of the strain-rate sensitivity, strain-rate hardening is greater at the upper exit, resulting in the late occurrence of a neutral layer. Moreover, folding occurs and disappears repeatedly. When the neutral layer disappears, the material above
607
Fig. 5. The occurrenceof folding.
Fig. 6. Schematicdrawingof a suitablepunch for the avoidanceof the macro-defect. the neutral layer moves up. Vortex flow forms, which greatly increases the strain-rate hardening: however, this does not occur beneath the neutral layer. During the next loading step, moving-up of the material is difficult and the whole material moves downwards, the neutral layer disappearing. If the strainrate hardening at the lower exit can counteract the strain-rate hardening at the upper exit, the neutral layer occurs again. Therefore, whether the neutral layer occurs or not is mainly dependent on the strain-rate hardening at the upper exit and the lower exit. For the second case, because the neutral layer occurs and disappears alternatively, the amount of the material moving downwards is greater than that for the first case. For the third case, the improvement in lubrication makes the deformation homogeneous, which is beneficial for the avoiding of the occurrence of folding, according to the flow pattern predicted using the finite-element method. To sum up, a macro-defect such as folding is caused by an abnormal flow pattern. It is necessary to control the flow pattern for prevention of the occurrence of the macro-defect. The proper design of the die structure is the effective measure to avoid the occurrence of the macro-defect. For this purpose, a schematic drawing of a suitable punch is presented in Fig. 6. 6. C o n c l u s i o n s
(1) The occurrence of a macro-defect such as folding in the isothermal forging process can be predicted by means of the rigid-viscoplastic finiteelement method. (2) Macro-defects occurring during axisymmetric isothermal forging processes are caused mainly by an abnormal flow pattern.
608
(3)
I m p r o v e m e n t s in t h e die d e s i g n a n d in t h e l u b r i c a t i o n c o n d i t i o n s are decisive in t h e p r e v e n t i o n o f t h e o c c u r r e n c e s o f m a c r o - d e f e c t s .
References 1 S.I. Oh, N. Rebelo and S. Kobayashi, Finite element formulation for the analysis of plastic deformation of rate sensitive materials in metal forming, IUTAM Symposium on Metal Form ing Plasticity, Tutzing, 1978, p. 273. 2 P. Hartley, C.E.N. Sturgess and G.W. Rowe, Finite-element prediction of the influence of strain-rate and temperature variations on the properties of forged products, Proc. 8th NARMC, Rolla, MO, 1980, pp. 121-128. 3 S. Kobayashi, Thermoviscoplastic analysis of metal forming problems by the finite-element method, in: J.F.T. Pattman, O.C. Zienkiewicz, R.D. Wood and J.M. Alexander (Eds.), Numerical Analysis of Forming Process, Wiley, New York, 1984, pp. 45-69. 4 S.I. Oh, Finite-element analysis of metal forming processes with arbitrary shaped dies, Int. J. Mech. Sci., 24 (1982) 479. 5 S. Kobayashi, The role of the finite-element method in metal forming technology, Proc. 1st Int. Conf. Technology of Plasticity, Tokyo, 1984, Vol. 2, pp. 1035-1040. 6 T. Altan and S.I. Oh, CAD/CAM of tooling and process for plastic working, Proc. Ist Int. Conf. Technology of Plasticity, Tokyo, 1984, Vol. 1, pp. 531-544. 7 Bin Feng, The improvement of rigid-plastic f'mite-element method and its application on closed upsetting, Master's Thesis, Northwestern Polytechnic University, 1984 (in Chinese). 8 Fu Mingwang and Luo Zijian, A new finite mesh rezoning to simulate large deformation process, J. Nanchong Inst. Aeronaut. Technol., 1 (1989) 7-12 (in Chinese).
599
Elsevier
The prediction of macro-defects during the isothermal forging process by the rigid-viscoplastic finite-element method Mingwang Fu a and Zijian L u o b ~Nanchang Institute of Aeronautic Technology, Nanchang, China bNorthwestern Polytechnic University, Xian, China (Received May 6, 1991; accepted in revised form September 17, 1991 )
Industrial Summary Macro-defects often occur during isothermal forging of workpieces of complicated shape. The occurrence of macro-defects is mainly attributable to an abnormal flow pattern of the metal. In order to reveal the cause of macro-defects during isothermal forging of a very large aluminium wheel for an aeroplane, the forging process is simulated by the rigid-viscoplastic finite-element method. The results so calculated show that the occurrence of a folding defect is caused by an abnormal flow pattern. Improvement in the die design and lubrication conditions are effective measures to avoid this kind of defect.
1. Introduction
Isothermal forging can produce complicated workpieces that are difficult to form by common forging technology. With regard to complicated workpieces, it is not easy to judge whether the flow patterns are normal or not. Macrodefects occurring during forging processes are caused mainly by abnormal flow patterns, thus it is necessary to understand the flow pattern during the process to be able to find the way to deal with the situation. A new development in the plastic-working theory to predict the occurrence of macro-defects is to use the finite-element method, thus providing a scientific basis for optimal plastic-working technology [ 1-5 ]. The rigid-viscoplastic finite-element method is an extension of the rigidplastic finite-element method and has become a powerful tool for the analysis of viscoplastic forming processes [ 1-3 ]. In the present study, an axisymmetric isothermal forging process for an aeroplane wheel has been simulated using this method. It is found that macro-defects (folding) occur due to an abnormal Correspondence to: M.W. Fu, Nanchang Institute of Aeronautic Technology, Nanchang, China.
0924-0136/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
600
flow pattern of the material. Some suggestions are made for the prevention of the occurrence of the folding defect.
2. Rigid-viscoplastic finite-element formulation According to the extremum principle, amongst all of the admissible velocity fields, the actual solution gives the minimum value of the following function:
O=f
dV- f FUdS+ I ,tiidV
Vo
SF
(1)
Vo
where E(.~) is the work function, Sr is the surface on which traction is prescribed, F is the traction vector, U is the velocity vector, 2 is the Lagrangean multiplier, and V0 is the volume of the deforming body. Suppose the deforming body is divided into m elements connected by n nodes. The velocity vector in eqn. (1) can be determined by the following equation: .U= _Nv
(2)
where v is the velocity vector at nodal point and _Nis the velocity interpolation matrix. The strain-rate vector is then derivable in the form:
=B_U
(3)
where B_is the strain rate-velocity matrix. The effective strain rate can be expressed as follows: i-=
(~TD_~)=
(uT_pv)I/2
(4)
where --2
2
D- =
~
2
D=
I
0
l
0 2 1
~ 1
n
m
for the axisymmetrical problem, _P= B_TD_B_. Equation (1) can be approximated by the summation of the functions of the element nodal velocities and Lagrangean multipliers: m
0J(v y)
(5)
j=l
where 0 j is the function of the j t h element, and vj and ~t~are the nodal velocities
601
and Lagrangean multiplier of the jth element, respectively. According to the stational condition, it is found that i=a 0v
J=~ -~-~7= 0
(6)
Solving eqns. (6) simultaneously, the velocity field and the mean stresses within the deforming body can be defined. Because eqn. (6) is a non-linear equation system, the Newton-Raphson method is employed as the solving process. Suppose the nth iteration nodal velocity vector and velocity increment vector for the nth iteration are v~ and Av~, respectively. The velocity vector for the (n+ 1 )th iteration can be determined as follows: ~n+l"~V~n'~-AV.n
(7)
The work function can be written as
o
o
where ~ii is the strain rate, a~i is the stress deviator tensor, 0 is the effective stress, and g-is the effective strain rate. Taking account of eqns. (7) and (8), expanding eqn. (5) into a series by the Newton-Raphson procedure and taking only the first two terms, the following linear equation system can be obtained:
•
f t (o/~-)P dV+ f R dV}
J= l ¢oj ["
Voj
~ f B_"r.CdV
j=l
Voj
Avn
~a ~ CTB dV Voj
J
(9)
Voi where
and 2,~ is the vector of the Lagrangean multiplier of each element and equals the hydrostatic stress of the relevant element.
6O2
After each iteration, the values of v are modified as Vn+l=Vn+flAv~
(10)
where fl is the coefficient to accelerate convergence, 0 < fl< 1. The convergence criterion is as follows:
llAvlllllvll ~
(11)
where IIv II is the Euclidean vector norm
llvll
(12)
= ( v T v ) 1/2
and g is a pre-determined small positive number, 10 -~ ~
3.1. Friction force The friction stress is determined by the following equation:
r= - m K [ (2/7~)tan(vr/a] VDI ]t
(13)
where m is the friction factor (0 < m < 1 ), K is the shear yield stress, vr is the magnitude of relative velocity between the die and the workpiece, a is a constant several orders of magnitude smaller than the die velocity, such as 10 -5 , ] VD [ is the absolute value of the die velocity, and t is the unit tangential vector.
3.2. Initial velocity field The initial velocity field can be determined by the following function [ 7 ]: (e2e-2Vo2j)- ~ (Fivi) +
Or= j=l
i=1
~tii dV
(14)
j= lyo j
where Fi and vi are the nodal force and velocity at the traction boundary.
3.3. Treatment of folding The folding is caused by the large frictional stress between the die and the workpiece. The metal flow along the contact surface is resisted by friction force, resulting in the lateral surface rotating upwards and folding occurring. The distorted mesh is illustrated in Fig. 1 to explain how to treat the folding phenomenon. Because of folding, side surface 3'4'5' becomes 3'4'5". The original intersection point 5' between the side surface of the workpiece and the die
603 ~,/i //i
lJxl
,/~/]/,-_/.///
8
l la ) ~
3
M
~ 3 '
l z (b)
LS~z
,
2
Fig. 1. The treatment of folding: (a) beforedeformation;(b) after folding. surface is substituted for 5", so after folding, 3'4' 5' cannot describe the real boundary: should an attempt be made for it to do so, this would lead to a great calculation error, and the calculation would not be able to be continued. In order to solve the problem, it is necessary to describe the boundary correctly. Side surface 3' 4' 5' is divided into n equidistance sections. After the nth step, point i can be determined by the following equations:
ri =N3ir3
"~'N4ir4 "t-Nsir5
Zi = N 3 i g 3
t-N4ig4
JFNsiZ5
(15)
where N3i, N4i, Nsi, r3, r4, rs, and za, z4, z5 are the values of the shape function and the coordinates of node 3', node 4' and node 5', respectively. At the (n + 1 )th step, the velocity of point i in the z-direction is V n+l ~N iz ~
3i
v n + l _1_ ~T . n + l . N 3z ~ ~ " 4i U 4z "r"
5i
v n+l 5z
(16)
where v3~, v4~, vs~ are the velocities of node 3', node 4' and node 5' at the (n + 1 )th step, respectively, when the following condition is satisfied for point i:
zi +vz~ Atn+l>~z5 +vs~
A t n+l
(17)
where At n+l is the time increment of the ( n + 1 )th step. Thus, point i becomes point 5", substituting point 5' and being the 5th node of element M. In order to avoid the distortion of element M, point 4' should be modified to the center of section 3' 5", becoming the 4th node. Similarly, the 6th node of element M is substituted for by point 6". Because the coordinates of the nodes are changed, the accumulated field variables of the corresponding nodes should be modified. For instance, the ( n + 1 )th step equivalent strain of point 5" is determined as follows: ~n,,+ 5 1 = e-T + A ~ 7 + 1
(18)
3.4. Mesh rezoning Simulating the large plastic deformation process by the finite-element method, it is necessary to redefine a new mesh system because of the serious distortion of the old mesh. The generation of the new mesh system consists of
604
1
2
3
Fig. 2. The determination of field variables at the new model.
generating a new mesh system and defining the field variables in the new mesh system: the latter can be generated by some available algorithm. The determination of the field variables in the new mesh system is based on its distribution in the old mesh system. In this paper, the authors adopted the following method to determine the field variables in the new mesh system. The accuracy of calculation of the field variables at the Gauss integration point inside the element is much greater than that at the element boundary and the field variables associated with the derivative of velocity may be discontinuous at the element boundary. In view of the above, the field variables at the element boundary are modified according to the field variables of the Gauss integration point. The quadrilateral element with eight nodes is divided into eight triangles (Fig. 2). The field variables of node K at the new mesh system can be determined according to the field variables at the nodes within triangle 128. After the field variables at all nodes, the new mesh has been determined, and the distribution of field variables in the new mesh defined. 4. Test material and experimental device
In order to simulate the isothermal forging process of aluminium alloy, pure lead is used as simulation material. The dimension of the sample is 19 X 23 mm. The constitutive equation as obtained experimentally is: # = Y(e-) [1+ (E-/r) n ]
t: LI _
2.Smm (a)
~
(19) punch
plate upper d ie
middle die b o t t o m die
I
I
(b)
Fig. 3. The dimensions of the sample (a) and the experimental device (b).
605 where Y(£) is the static yield stress, i-is the equivalent strain rate, and r is the parameter associated with the viscosity of materials: 6.380+53.9~ for ~<0.143 I]13.955+20.286 sin[n(~-0.143)/0.237]
..... ~te~=)15.758-2.685(e--0.310) ~,14.543 for ~>0.670
for 0.143<~<0.294 0.294
'0.512~+0.081 for C~<0.407 1.485-6.000~+9.400(2-4.600e 3 for 0.30 for ~>0.668
n=.
rn=
for
'
(20)
0.407<£~<0.668
- 1 4 . 1 4 0 ( + 6.426 for £~0.384 10.882-52.182£+86.718£2-46.509~ a for 0.781 for ~>0.621
0.384<~<0.621
(21)
(22)
where n is the strain-rate index. The dimensions of the sample are determined according to the law of similarity. The dimensions of the sample and the experimental device are shown in Fig. 3. The lubricant is machine oil, affording a friction factor of 0.22. 5. Pattern of metal flow determined by the finite-element method
In order to investigate the influences of strain rate and lubrication on the flow pattern, the following cases are simulated by the rigid-viscoplastic finiteelement method: (1) eo = 1 0 -3 s -1, m=0.22 (Co is the initial strain rate); (2) ~o = 3 × 10 -2 s -1, m=0.22; (3) eo = 3 × 10 -2 s -1, m=0.05. The flow patterns of the three cases are shown in Fig. 4. For the first case, all the material moves downwards at the initial stage. When the stroke of punch reaches 3.6 mm, the neutral layer appears. In Fig. 4, h represents the position of the neutral layer. As deformation progresses, the neutral layer moves down, the amount of material moving up becomes larger and its velocity becomes greater. When the stroke reaches 14 mm, the material coming from the upper exit contracts the shoulder of the punch and is forced to move downwards, leading to the disappearance of the neutral layer. When the lower cavity is almost filled with the material, the neutral layer occurs again, but it is located near the lower exit. Moreover, when the stroke is about 14 mm, the contour of punch forces the material at point A to move downwards and towards the left, at point B to move downwards and towards the right and at point C to move downwards and towards the left. The above flow pattern gives rise to the formation of folding at point B with the punch moving downwards (Fig. 4). The results of the finite-element simulation are consistent with the experimental results of Fig. 5.
606
/ 'i"" r
4
1
L &h=4.0mm
Ah=A.0mm
,I , , i w ,
Ah= 6.0mm
Ah=4.0mm
h
Ah= 6,0mm
Ah = 6,0ram
l:::
, :,:,:,
A h =14.0ram
Ah=14.(
A h = 15.3mm
Ah = 1 5 . 2 m m
Ah = 1 5 . 3 m m
(b)
(G)
(a)
mm
IAh : 1 4 . 0 m m
Fig. 4. T h e flow patterns of three cases, simulated by the rigid-viscoplastic finite-element method.
For the second case, the flow pattern is slightly different from that of the first case because of strain-rate sensitivity of the material. At first, because of the strain-rate sensitivity, strain-rate hardening is greater at the upper exit, resulting in the late occurrence of a neutral layer. Moreover, folding occurs and disappears repeatedly. When the neutral layer disappears, the material above
607
Fig. 5. The occurrenceof folding.
Fig. 6. Schematicdrawingof a suitablepunch for the avoidanceof the macro-defect. the neutral layer moves up. Vortex flow forms, which greatly increases the strain-rate hardening: however, this does not occur beneath the neutral layer. During the next loading step, moving-up of the material is difficult and the whole material moves downwards, the neutral layer disappearing. If the strainrate hardening at the lower exit can counteract the strain-rate hardening at the upper exit, the neutral layer occurs again. Therefore, whether the neutral layer occurs or not is mainly dependent on the strain-rate hardening at the upper exit and the lower exit. For the second case, because the neutral layer occurs and disappears alternatively, the amount of the material moving downwards is greater than that for the first case. For the third case, the improvement in lubrication makes the deformation homogeneous, which is beneficial for the avoiding of the occurrence of folding, according to the flow pattern predicted using the finite-element method. To sum up, a macro-defect such as folding is caused by an abnormal flow pattern. It is necessary to control the flow pattern for prevention of the occurrence of the macro-defect. The proper design of the die structure is the effective measure to avoid the occurrence of the macro-defect. For this purpose, a schematic drawing of a suitable punch is presented in Fig. 6. 6. C o n c l u s i o n s
(1) The occurrence of a macro-defect such as folding in the isothermal forging process can be predicted by means of the rigid-viscoplastic finiteelement method. (2) Macro-defects occurring during axisymmetric isothermal forging processes are caused mainly by an abnormal flow pattern.
608
(3)
I m p r o v e m e n t s in t h e die d e s i g n a n d in t h e l u b r i c a t i o n c o n d i t i o n s are decisive in t h e p r e v e n t i o n o f t h e o c c u r r e n c e s o f m a c r o - d e f e c t s .
References 1 S.I. Oh, N. Rebelo and S. Kobayashi, Finite element formulation for the analysis of plastic deformation of rate sensitive materials in metal forming, IUTAM Symposium on Metal Form ing Plasticity, Tutzing, 1978, p. 273. 2 P. Hartley, C.E.N. Sturgess and G.W. Rowe, Finite-element prediction of the influence of strain-rate and temperature variations on the properties of forged products, Proc. 8th NARMC, Rolla, MO, 1980, pp. 121-128. 3 S. Kobayashi, Thermoviscoplastic analysis of metal forming problems by the finite-element method, in: J.F.T. Pattman, O.C. Zienkiewicz, R.D. Wood and J.M. Alexander (Eds.), Numerical Analysis of Forming Process, Wiley, New York, 1984, pp. 45-69. 4 S.I. Oh, Finite-element analysis of metal forming processes with arbitrary shaped dies, Int. J. Mech. Sci., 24 (1982) 479. 5 S. Kobayashi, The role of the finite-element method in metal forming technology, Proc. 1st Int. Conf. Technology of Plasticity, Tokyo, 1984, Vol. 2, pp. 1035-1040. 6 T. Altan and S.I. Oh, CAD/CAM of tooling and process for plastic working, Proc. Ist Int. Conf. Technology of Plasticity, Tokyo, 1984, Vol. 1, pp. 531-544. 7 Bin Feng, The improvement of rigid-plastic f'mite-element method and its application on closed upsetting, Master's Thesis, Northwestern Polytechnic University, 1984 (in Chinese). 8 Fu Mingwang and Luo Zijian, A new finite mesh rezoning to simulate large deformation process, J. Nanchong Inst. Aeronaut. Technol., 1 (1989) 7-12 (in Chinese).