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Finite element modelling of the high-velocity impact forging process by the explicit time integration method
Joumalof
Materials Processing Technology ELSEVIER
Journal of Materials Processing Technology 63 (1997) 718-723
Finite element modelling of the high-velocity impact forging process by the explicit time integration method Y. H. Yoo, D. Y. Yang Department ofMechanical Engineering, KAIST, Taejon 305-701, KOREA
Abstract The numerical simulations of the high-velocity impact forging processes are described in this paper. The explicit time integration finite element method was used to compute the deformation of the workpiece and the dies. In order to consider the effects of strain hardening, strain rate hardening and thermal softening, which are frequently observed in high-velocity deformation phenomena, the Johnson-Cook yield surface model was applied. Through the copper blow test simulation, the developed program was verified. Also, the developed program was applied to simulations of high-velocity multi-blow forging processes. Two types of workpiece configurations, cylinder and block, were used for high-velocity multi-blow forging simulations. As a result of multi-blow forging process simulation, it was found that the change of the blow efficiencies and the clash load generated by the blow operations could be efficiently analyzed. Keywords: Finite element modelling, Explicit time integration method, Simulation of copper blow test, High-velocity impact forging.
1. Introduction The high-velocity impact forging process using an energyrestricted type machine such as the hammer has long been a basic metal working method [I]. Hammers are the cheapest metal forming machines from the point of view of generating a large force. In order to avoid the breakage of hammer parts, the maximum clash load should be controlled within the limited value. During the actual forming, energy losses occur; only the remainder is transformed into useful work. In process design and die design of the high-velocity impact forging, the maximum clash load and the blow efficiency defined as the ratio of the useful work to the input kinetic energy are the very important factors. However, the accurate estimations of these values are extremely difficult because of complex physical phenomena during the impact operations. Several research studies on this problem using experimental and analytical methods have been reported[2-7]. Watermann [2] performed extensive experiments for hammer forging and proposed an empirical formula for estimation of the blow efficiency. Using the analytical method, Vajpayee et al. [5] analyzed changes of the blow efficiencies according to variation of process variables such as the input kinetic energy and the initial impact velocity. Through the iterative calculations considering the geometrical compatibility condition and the energy equilibrium, Haller [6] evaluated the maximum force generated in the impact forging operation. However, the existing research results are confined to two-dimensional simple geometry, such as the cylinder. Also, in these researches, the maximum clash load and the blow efficiency could not be treated simultaneously. 0924-0136/97/$15.00 © 1997 Elsevier Science 8.A All rights reserved PII 80924-0136(96)02713-6
In the present work, in order to simultaneously obtain the blow efficiency and the maximum clash load generated in highvelocity impact forging with a three-dimensional configuration, the explicit time integration finite element program, which is based on direct time integration of equation of motion, will be developed and applied to practical problems.
2. Method of simulation 2.1. Explicit finite element method The fundamentals of the elasto-plastic finite element method using explicit time integration is presented. The differential equation governing the motion of a material point at time t is on
(1)
'V
where 'cr iJ is the Cauchy stress tensor at time t, 'p is a density, t ai is an acceleration, 'b i is the body force per unit volume and a left superscript denotes a specific time. The variational equation is then given by the following:
J
J
J
J
tv
0v
's,
0v
'crijou1,JdV +
0p' a1ou1dV -
'f;ou1dS-
Ib10ujdV =0 (2)
where ou, is an arbitrary variation of the displacement field compatible with the boundary condition and '''fl is the given surface traction.
Y.H. Yoo, D.r. Yang I Journal of Materials Processing Technology 63 (1997) 718-723
Spatial discretization is performed with M eight node isoparametric hexahedral elements, then the finite element equation is obtained as follows:
(3)
where 'SI is the stress matrix, B u is the fmite element straindisplacement matrix [8], N IJ is the element shape function
,-
matrix and I A L , FK and 'B K denote the nodal acceleration vector, the nodal surface traction vector and the nodal body force vector, respectively. By mass lumping of the second term of the equation (3), the uncoupled finite element equation is obtained as follows: (I = 1,2,"',NDOF) where
'Ft"
(4)
is the nodal force resulting from the surface traction
and the body force, 'Ft" is the nodal force resulting from the stress divergence term, M 1 is the lumped nodal mass and NDOF is the total degree of freedom for the global system. In order to reduce a computation time for a volume integration, the one point Gaussian quadrature is used. In a reduced interation, zero energy modes called hourglass or keystone may be generated. To prevent these modes, the hourglass resisting force HI is considered as follows [9] : (5)
where k is a control constant, A and Jl are Lame constants, V is a volume, B is the strain-displacement matrix and ql is the magnitude of the hourglass velocity fields. The explicit time interation scheme is conditionally stable. At each time step cycle, a time increment ~'t is calculated by the Courant stability condition: ~'t S
L C
1;-
(6)
where the stability factor 1; is taken to be 0.85, L is the minimum characteristic length of the elements and C is the fastest wave velocity of the material. For an isotropic material, the elastic dilatational wave speed C is calculated by [10]
C=
~A+p2Jl
(7)
The central difference method [8] is used to integrate the nodal variables as follows:
AI
719
AI
'+2'VI ='-2'V1+'A1 '(~'t)
(8-a)
At each increment, the spherical and deviatoric strain rates are computed from the nodal velocities. Then, the deviatoric trial stress components are computed. In order to determine the plastic strain increment, the elastic predictor radial return method is used. Hydrostatic pressure is directly related to volumetric strain by the bulk modulus. In order to consider the effects of strain hardening, strain rate hardening and thermal softening, which are frequently observed in high-velocity deformation phenomena, the Johnson-Cook yield surface model [11] is applied. In the Johnson-Cook model, the von Mises flow stress, cr , is expressed as
where £ is the equivalent plastic strain,
£0
is a constant, Tmcll
and Tret are melting temperature and reference temperature, respectively. A, B, C, n and m are material constants. The expression in the first set of brackets represents the effect of strain hardening, The expressions in the second and third sets of brackets represent the effects of strain rate hardening and temperature softening, respectively. The temperature increase, ~T, by a adiabatic heating is expressed as
~T =
f -Lcredt o pC.
(10)
where C v is the constant-volume heat capacity of the material, and X defines the fraction of the plastic work which is used to increase the material temperature. A contact-searching scheme based on a master-slave algorithm [12] is used to treat the contact interface. In the onepass treatment, master nodes may penetrate into the slave surface. To minimize penetrations, the boundary that is modelled by a coarser mesh should be chosen as a master surface. But, this guide rule can be broken during severe deformation. To alleviate this problem. a symmetric treatment of the contact interfaces is introduced, in which contact treatment needs to be performed once more with the role of the slave and master surface interchanged. The kinematic contact condition is enforced by the penalty method [12]. In the penalty method, the normal contact force, 'f. is determined as
'f.
=cKL '1)
(11)
where c is a control constant, K is the bulk modulus, L is a characteristic length calculated from each contact element and '1) is a penetration depth. The frictional contact condition encountered in three dimensional metal forming processes is quite complex. In order
Y.H. Yoo, D.Y. Yang I Journal of Materials Processing Technology 63 (1997) 718-723
720
to model the frictional phenomena precisely, the non-classical Coulomb friction law [13] is implemented in the program. In this method, first, the trial frictional force at time t + 6.t, I +ftl y,T is calculated as
- velocity matrix, respectively, and K cxII is dermed as Bal ,B 4I • The equation (16) is solved iteratively using the NewtonRaphson method. The third integration part is linearized as follows:
(12) (17)
where Ef is the friction modulus, and t+ftl6.u I is the increment of tangential displacement during time increment. Second, the frictional force at time t + 6.t, .+ftl y., is calculated as follows: for a sticking node (I'+ftlyn $ Il'+"f.), '+ftlft = l+ftlftT
where (l3-a)
I
for a sliding node (I t+ftlftT > Il t+ftlf. ), t+ftlyT t+ftly =,,'H'f _ _,_ , ... • 1'+ftlftTI
(l3-b)
V(n.l)
and 6.V(.) are the nodal velocity obtained from the
previous iteration and its perturbation obtained from the current iteration, respectively. When the Newton-Raphson method is used, the generation of an initial velocity field is very important. In the present work, the method suggested by Oh [15] is used, where the linear viscous material relationship is assumed and the viscosity, Il, is expressed as follows :
2.2. Implicit finite element method considering the inertia effect
The essences of the rigid-plastic finite element method using implicit time integration is presented. In the present work, two basic assumptions are introduced. First, the material has a rigidplastic behavior. Second, the volume constancy of the material holds during deformation. The variational equation is then given by the following (14] :
(18) Neglecting the frictional effect, the finite element equation used for initial velocity generation is obtained as follows:
fJo+ lXtit'EH') oEdV +KLEvOEvdV +Jpa;ov;dV v
r(Ti +atiT Js,
I)
(14) ovidS
=0
where K, at. OV I are a penalty constant, an acceleration and an arbitrary variation of velocity, respectively. Using the Newmark method [8], the acceleration a l is expressed as
1 -vi l-y a =------a y tit Y VI
l
(19)
Because equation (19) is a linear equation, the solution of equation (19) is calculated without iteration.
p
l
(15)
where Y is a control constant, vi and ai are the velocity and acceleration obtained from the previous step, respectively. When spatial discretization is performed with M finite elements, the finite element equation is obtained as follows:
(16)
where Ball' Qcxare strain rate - velocity and volumetric strain rate
3. Simulation of copper blow test 3.1. Copper blow test
In order to assess available energy of a counter-blow hammer whose nominal capacity is 350,OOOJ, the copper blow test is performed. The two electrolytic OFHC (Oxygen Free High Conductivity) copper cylinder specimens of 80mm diameter X 120mm height are used for the test. In this test, the purity of the specimen must be over 99.9% so as to guarantee the uniform deformation of the test specimen. The counter-blow hammer used for the copper blow test is a hydraulic-coupling type machine and weights of the upper and lower dies are 40.2 ton and 43.2 ton, respectively. The two specimens are located in the center of the lower die, 400mm apart from each other, then one blow operation is performed under a working pressure of 7.5 bar and 1,400mm total stroke of both dies. To minimize the frictional effect between the specimens and
721
Y.H. Yoo, D.Y. Yang I Journal of Materials Processing Technology 63 (1997) 718-723
the dies, the molybdenum disulfide (MoS z) and a teflon sheet are used as lubricants at the same time. Under this lubrication condition, the ring compression test reveals that the values of the friction coefficient and friction factor are 0.052 and 0.09, respectively. The final height of the deformed specimen is 46.9mm, and no lateral crack in the deformed specimen is found. 3.2. Simulation ofcopper blow test
Both the explicit and the implicit method are used for simulation of the copper blow test. In order to compare theoretical results with experimental ones, simulation conditions are modelled cautiously within the limits of the possibility. The initial meshes are shown in Fig. 1. Because of symmetry, oneeighth of the full model is used for simulation. The calculation is started with the initial impact velocities of the dies. The initial impact velocity of the upper and lower die, Vi. is calculated from equation (20) considering the energy balance as follows:
(20) where Er is a total input energy, mt and mb are the masses of the upper and lower dies, respectively. The initial impact velocity of the upper and lower dies calculated from equation (20) is 2.897(m/s).
=
temperature softening properties, X= 0.9, C v 383. OJ I kgOC, Tmelt = 1083°C, T",f = 10°C, m 1.09 It is assumed that the material response of dies is elastic. The Young's modulus, Poisson's ratio and density of dies made by tool steel are 211.4 GPa, 0.293 and 7.83 g/cm3, respectively, but the density is scaled up by 100 times so the volume of dies is reduced with the same ratio, while the mass of dies is maintained to constant. Now, with the above mentioned simulation conditions, the copper blow test is simulated using the two different methods. First, the copper blow test is simulated using the developed explicit program with friction coefficient 0.052. The calculation is continued until the specimen and the dies of the forging equipment are separated completely. The deformed meshes obtained from the explicit method are shown in Fig. 2. The initial relative velocity of the dies (5.794 mls) is decreased during the blow operation and the fmal relative velocity of the dies is maintained in 0.84 mls. Hence, for the same time duration, the moved relative displacement between the upper and lower dies before blowing is much larger than the moved relative displacement between the upper and lower dies after blowing. The fmal height of the deformed specimen obtained by the explicit simulation is 49.2 mm.
:±t±±i±±±±t
(a) t= -lOms
-
(d) t
Cylinder Specimen
Block Specimen
Fig. 1. Mesh system for copper blow test simulation. The experimental data obtained from the Johnson-Cook's paper [11] are accepted as material properties of OFHC copper. The material constants for OFHC copper are summarized as follows: shear modulus, G = 48.3 GPa Poisson's ratio, v = 0.343 density, p = 8.96 g/cm3 strain hardening properties,A=90.0 MPa,B= 292.0 MPa,n =0.31 strain rate hardening properties, C = 0.025, Eo = 10-6 ~s-t
=20 ms
-J I I
=
I I ~
(b)t=Oms
(c)t=lOms
(e)t=30ms
(1) t =40 ms
Fig. 2. Deformed meshes obtained from copper blow test simulation using the explicit method at (a) t = -10 ms, (b) t = 0 ms, (c) t = 10 ms, (d) t = 20 ms, (e) t = 30 ms, (1) t = 40ms. Second, the developed implicit time integration rigid-plastic finite element program is applied to a simulation of the copper blow test with a friction factor of 0.09 so as to compare with the explicit results. The implicit calculation is continued until the height of the specimen becomes 49.2 mm which is the fmal height obtained by the explicit calculation. The comparison of fmal deformed configurations between the
Y.H. Yoo. D.Y. Yang/Journal of Materials Processing Technology 63 (1997) 718-723
722
experiment and the calculations is shown in Fig. 3. The calculated result reveals a good agreement in the [mal deformed configurations between the experiment and the simulations. Fig. 4 shows the equivalent plastic strain distributions obtained from the simulations. The explicit result is very similar to the implicit one in the equivalent plastic strain distribution. The time histories of the clash load and plastic deformation energy are shown in Fig. 5 and 6, respectively. The time history plots about
500000 . , - - - - - - - - - - - - - - - - - , Maximum (Implicit Analysis) : 334375J 400000
Maximum (Explicit Analysis) : 337456J
300000
200000 Explicit Analysis 100000
_ _ jL-0.052
Experimental Result
-Explicit Analysis
Implicit Analysis
Fig. 3. Comparison of final deformed configurations between experiment and calculations of copper blow test.
O.B
0.6
I
:/
,
o¥---'---,~---,-'----r--.-....~--r~--I 30 20 25 10 15 o 5 Contact Time (ms)
Fig. 6. Time history of plastic deformation energy obtained from copper blow test simulation. the clash load and plastic deformation energy have good agreements between the two methods. In the explicit simulation, the clash load is increased gradually to the maximum value 9, 182 kN at time 19.0 ms and then decreased to zero at time 22.3 IDS. The plastic deformation energy is increased gradually to the maximum value 337,456J and then maintained at the same level to the termination time 22.3 tns. The maximum load and the plastic deformation energy obtained from the implicit simulation. are 9.313 kN and 334.375J. respectively.
0.9
4. Application to multi-blow forging process
-f.65
Implicit Analysis
Explicit Analysis
Fig. 4. Comparison of equivalent plastic strain distribution between the explicit and the implicit methods in copper blow test simulation.
14000 Maximum (Implicit Analysis) : 9313kN 12000
.... ~
1co
Maximum (Explicit ADalysis) : 9182kN 10000 8000
..
6000
t3
4000
..J ~
ev Explicit Analysis 2000
_11-0.052
0 0
5
10
15
20
25
30
Contact Time (ms)
Fig. 5. Time history of clash load obtained from copper blow test simulation.
In this chapter, the high-velocity multi-blow forging process is simulated with the explicit program verified through the simulation of the copper blow test. The configurations of specimen used for the simulation of multi-blow forging processes are cylinder and block. The initial meshes are shown in Fig. 1. To inspect changes of the blow efficiencies and the clash load generated by blow operations. the sequentially operated four blows are simulated. The computational conditions for the simulation of multi-blow forging processes is the same as simulation of the copper blow test except for the dimensions of the block specimen. The block specimen of 100 rom long edge X 50 rom short edge X 120 rom height is used for the simulation in order that the contact area and volume of the specimens of both types become equal. The initial mesh and the [mal mesh obtained from each blow operation are shown in Fig. 7. The blow efficiencies obtained from simulation are presented in Table 1. When the number of blows is increased, the contact area between the specimen and the dies is increased and the blow efficiency is decreased and an amount of height reduction is decreased. The blow efficiencies of the block specimen are larger than the blow efficiencies of the cylinder specimen. especially in the fourth blow operation. It reflects the effect of the geometrical self-constraint The clash load history according to the accumulated contact time is shown in Fig. 8. When the number of blows is increased, the contact time between the specimen and dies is decreased and the maximum clash load generated from each blow is rapidly
Y.R. Yoo. D.Y. Yang / Journal of Materials Processing Technology 63 (1997) 718-723
increased. The maximum clash load generated in the block specimen is slightly larger than the cylinder specimen. However, the accumulated contact time of the block specimen is shorter than the cylinder specimen. Table 1 Blow efficiencies obtained from multi-blow forging simulation Blow number
Blowefficiency(%) cylinder
block
96.3 89.9 76.9 51.7
96.4 89.8
1 2 3 4
723
As a result of the multi-blow forging simulation, it was found that the changes of the blow efficiencies and the clash load generated by the blow operations could be efficiently analyzed.
5. Conclusion
In the present work, in order to simultaneously obtain the blow efficiency and the clash load generated in the high-velocity impact forging processes with three-dimensional geometry, the explicit time integration finite element program, which was based on direct time integration of equation of motion, was developed . The copper blow test was simulated using the developed program. The calculated result revealed a good agreement in the final deformed configurations between the experiment and the simulation. In order to compare with the explicit method, the implicit time integration rigid-plastic finite element program including inertia effect was also developed and then also applied to copper blow test simulation. As a result of the copper blow test simulation using the explicit program and the implicit program, it was found that the calculated results had good agreements in available plastic deformation energy, maximum load and equivalent plastic strain distribution. The developed explicit program was also applied to the multi-blow forging process simulation. As a result of the multi-blow forging process simulation, it was found that the changes of the blow efficiencies and the clash load generated by the blow operations could be efficiently analyzed.
77.5 59.8
Cylinder Specimen
References
Block Specimen Fig. 7. Deformed meshes of multi-blow forging.
10ססoo ...- - - - - - - - - - - - - - - - ,
80000
••••.•••••• Block Cylinder
60000
40000
2ססoo
o f........-.-...--,.....,.-.--rlI'--r~..,u>.--,r4-.l,-'........-I
o
5
ill
~
W
~
K. Lange, Handbook ofmetal forming, McGraw-Hill, 1985. H. D. Watermann, Industrie-Anzeiger, 77 (1963) 1727. A. J. Organ, Int. J . Mach. Tool Des. Res., 7 (1967) 325. C. E. N. Sturgess and M. G. Jones, Int. J. Mech. Sci., 13 (1971) 309. [5] S. Vajpayee, M. M. Sadek and S. A. Tobias, Int. J. Mach. ToolDes. Res., 19 (1979) 237. [6] H. W. Haller, Trans. ASME, J. Eng. Ind., 105 (1983) 270. [7] E. Benuzzi and F. Soavi, Trans. ASME, J. Eng. Ind., 107 (1985) 266. [8] K. J. Bathe, Finite element procedures in engineering analysis, Prentice Hall, Englewood Cliffs, 1982. [9]D. P. Flanagan and T. Belytschko, Internat. J. Numer. Methods Engrg., 17 (1981) 679. [10] W. Johnson, Impact strength of materials, Edward Arnold. London, 1972. [11] G. R. Johnson and W. H. Cook, Proc. of 7th Int. Symp. on Ballistics, (1983) 541. [12] J. O. Hallquist, G. L. Goudreau and D. J. Benson, J. Compo Meths. Appl. Mechs. Eng., 51 (1985) 101. [13] Z. H. Zhong, Finite element procedures for contact-impact problems, Oxford University Press, New York, 1993. [14] Y. J. Kim and D. Y. Yang, Int. J. Mech. Sci., 27 (1985) 487. [15] S. I. Oh, Int. J. Mech. Sci., 24 (1982) 479. [1] [2] [3] [4]
~
~
~
~
Accumulated Contact Time (ms)
Fig. 8. Time history of clash load obtained from multi-blow forging simulation.
Materials Processing Technology ELSEVIER
Journal of Materials Processing Technology 63 (1997) 718-723
Finite element modelling of the high-velocity impact forging process by the explicit time integration method Y. H. Yoo, D. Y. Yang Department ofMechanical Engineering, KAIST, Taejon 305-701, KOREA
Abstract The numerical simulations of the high-velocity impact forging processes are described in this paper. The explicit time integration finite element method was used to compute the deformation of the workpiece and the dies. In order to consider the effects of strain hardening, strain rate hardening and thermal softening, which are frequently observed in high-velocity deformation phenomena, the Johnson-Cook yield surface model was applied. Through the copper blow test simulation, the developed program was verified. Also, the developed program was applied to simulations of high-velocity multi-blow forging processes. Two types of workpiece configurations, cylinder and block, were used for high-velocity multi-blow forging simulations. As a result of multi-blow forging process simulation, it was found that the change of the blow efficiencies and the clash load generated by the blow operations could be efficiently analyzed. Keywords: Finite element modelling, Explicit time integration method, Simulation of copper blow test, High-velocity impact forging.
1. Introduction The high-velocity impact forging process using an energyrestricted type machine such as the hammer has long been a basic metal working method [I]. Hammers are the cheapest metal forming machines from the point of view of generating a large force. In order to avoid the breakage of hammer parts, the maximum clash load should be controlled within the limited value. During the actual forming, energy losses occur; only the remainder is transformed into useful work. In process design and die design of the high-velocity impact forging, the maximum clash load and the blow efficiency defined as the ratio of the useful work to the input kinetic energy are the very important factors. However, the accurate estimations of these values are extremely difficult because of complex physical phenomena during the impact operations. Several research studies on this problem using experimental and analytical methods have been reported[2-7]. Watermann [2] performed extensive experiments for hammer forging and proposed an empirical formula for estimation of the blow efficiency. Using the analytical method, Vajpayee et al. [5] analyzed changes of the blow efficiencies according to variation of process variables such as the input kinetic energy and the initial impact velocity. Through the iterative calculations considering the geometrical compatibility condition and the energy equilibrium, Haller [6] evaluated the maximum force generated in the impact forging operation. However, the existing research results are confined to two-dimensional simple geometry, such as the cylinder. Also, in these researches, the maximum clash load and the blow efficiency could not be treated simultaneously. 0924-0136/97/$15.00 © 1997 Elsevier Science 8.A All rights reserved PII 80924-0136(96)02713-6
In the present work, in order to simultaneously obtain the blow efficiency and the maximum clash load generated in highvelocity impact forging with a three-dimensional configuration, the explicit time integration finite element program, which is based on direct time integration of equation of motion, will be developed and applied to practical problems.
2. Method of simulation 2.1. Explicit finite element method The fundamentals of the elasto-plastic finite element method using explicit time integration is presented. The differential equation governing the motion of a material point at time t is on
(1)
'V
where 'cr iJ is the Cauchy stress tensor at time t, 'p is a density, t ai is an acceleration, 'b i is the body force per unit volume and a left superscript denotes a specific time. The variational equation is then given by the following:
J
J
J
J
tv
0v
's,
0v
'crijou1,JdV +
0p' a1ou1dV -
'f;ou1dS-
Ib10ujdV =0 (2)
where ou, is an arbitrary variation of the displacement field compatible with the boundary condition and '''fl is the given surface traction.
Y.H. Yoo, D.r. Yang I Journal of Materials Processing Technology 63 (1997) 718-723
Spatial discretization is performed with M eight node isoparametric hexahedral elements, then the finite element equation is obtained as follows:
(3)
where 'SI is the stress matrix, B u is the fmite element straindisplacement matrix [8], N IJ is the element shape function
,-
matrix and I A L , FK and 'B K denote the nodal acceleration vector, the nodal surface traction vector and the nodal body force vector, respectively. By mass lumping of the second term of the equation (3), the uncoupled finite element equation is obtained as follows: (I = 1,2,"',NDOF) where
'Ft"
(4)
is the nodal force resulting from the surface traction
and the body force, 'Ft" is the nodal force resulting from the stress divergence term, M 1 is the lumped nodal mass and NDOF is the total degree of freedom for the global system. In order to reduce a computation time for a volume integration, the one point Gaussian quadrature is used. In a reduced interation, zero energy modes called hourglass or keystone may be generated. To prevent these modes, the hourglass resisting force HI is considered as follows [9] : (5)
where k is a control constant, A and Jl are Lame constants, V is a volume, B is the strain-displacement matrix and ql is the magnitude of the hourglass velocity fields. The explicit time interation scheme is conditionally stable. At each time step cycle, a time increment ~'t is calculated by the Courant stability condition: ~'t S
L C
1;-
(6)
where the stability factor 1; is taken to be 0.85, L is the minimum characteristic length of the elements and C is the fastest wave velocity of the material. For an isotropic material, the elastic dilatational wave speed C is calculated by [10]
C=
~A+p2Jl
(7)
The central difference method [8] is used to integrate the nodal variables as follows:
AI
719
AI
'+2'VI ='-2'V1+'A1 '(~'t)
(8-a)
At each increment, the spherical and deviatoric strain rates are computed from the nodal velocities. Then, the deviatoric trial stress components are computed. In order to determine the plastic strain increment, the elastic predictor radial return method is used. Hydrostatic pressure is directly related to volumetric strain by the bulk modulus. In order to consider the effects of strain hardening, strain rate hardening and thermal softening, which are frequently observed in high-velocity deformation phenomena, the Johnson-Cook yield surface model [11] is applied. In the Johnson-Cook model, the von Mises flow stress, cr , is expressed as
where £ is the equivalent plastic strain,
£0
is a constant, Tmcll
and Tret are melting temperature and reference temperature, respectively. A, B, C, n and m are material constants. The expression in the first set of brackets represents the effect of strain hardening, The expressions in the second and third sets of brackets represent the effects of strain rate hardening and temperature softening, respectively. The temperature increase, ~T, by a adiabatic heating is expressed as
~T =
f -Lcredt o pC.
(10)
where C v is the constant-volume heat capacity of the material, and X defines the fraction of the plastic work which is used to increase the material temperature. A contact-searching scheme based on a master-slave algorithm [12] is used to treat the contact interface. In the onepass treatment, master nodes may penetrate into the slave surface. To minimize penetrations, the boundary that is modelled by a coarser mesh should be chosen as a master surface. But, this guide rule can be broken during severe deformation. To alleviate this problem. a symmetric treatment of the contact interfaces is introduced, in which contact treatment needs to be performed once more with the role of the slave and master surface interchanged. The kinematic contact condition is enforced by the penalty method [12]. In the penalty method, the normal contact force, 'f. is determined as
'f.
=cKL '1)
(11)
where c is a control constant, K is the bulk modulus, L is a characteristic length calculated from each contact element and '1) is a penetration depth. The frictional contact condition encountered in three dimensional metal forming processes is quite complex. In order
Y.H. Yoo, D.Y. Yang I Journal of Materials Processing Technology 63 (1997) 718-723
720
to model the frictional phenomena precisely, the non-classical Coulomb friction law [13] is implemented in the program. In this method, first, the trial frictional force at time t + 6.t, I +ftl y,T is calculated as
- velocity matrix, respectively, and K cxII is dermed as Bal ,B 4I • The equation (16) is solved iteratively using the NewtonRaphson method. The third integration part is linearized as follows:
(12) (17)
where Ef is the friction modulus, and t+ftl6.u I is the increment of tangential displacement during time increment. Second, the frictional force at time t + 6.t, .+ftl y., is calculated as follows: for a sticking node (I'+ftlyn $ Il'+"f.), '+ftlft = l+ftlftT
where (l3-a)
I
for a sliding node (I t+ftlftT > Il t+ftlf. ), t+ftlyT t+ftly =,,'H'f _ _,_ , ... • 1'+ftlftTI
(l3-b)
V(n.l)
and 6.V(.) are the nodal velocity obtained from the
previous iteration and its perturbation obtained from the current iteration, respectively. When the Newton-Raphson method is used, the generation of an initial velocity field is very important. In the present work, the method suggested by Oh [15] is used, where the linear viscous material relationship is assumed and the viscosity, Il, is expressed as follows :
2.2. Implicit finite element method considering the inertia effect
The essences of the rigid-plastic finite element method using implicit time integration is presented. In the present work, two basic assumptions are introduced. First, the material has a rigidplastic behavior. Second, the volume constancy of the material holds during deformation. The variational equation is then given by the following (14] :
(18) Neglecting the frictional effect, the finite element equation used for initial velocity generation is obtained as follows:
fJo+ lXtit'EH') oEdV +KLEvOEvdV +Jpa;ov;dV v
r(Ti +atiT Js,
I)
(14) ovidS
=0
where K, at. OV I are a penalty constant, an acceleration and an arbitrary variation of velocity, respectively. Using the Newmark method [8], the acceleration a l is expressed as
1 -vi l-y a =------a y tit Y VI
l
(19)
Because equation (19) is a linear equation, the solution of equation (19) is calculated without iteration.
p
l
(15)
where Y is a control constant, vi and ai are the velocity and acceleration obtained from the previous step, respectively. When spatial discretization is performed with M finite elements, the finite element equation is obtained as follows:
(16)
where Ball' Qcxare strain rate - velocity and volumetric strain rate
3. Simulation of copper blow test 3.1. Copper blow test
In order to assess available energy of a counter-blow hammer whose nominal capacity is 350,OOOJ, the copper blow test is performed. The two electrolytic OFHC (Oxygen Free High Conductivity) copper cylinder specimens of 80mm diameter X 120mm height are used for the test. In this test, the purity of the specimen must be over 99.9% so as to guarantee the uniform deformation of the test specimen. The counter-blow hammer used for the copper blow test is a hydraulic-coupling type machine and weights of the upper and lower dies are 40.2 ton and 43.2 ton, respectively. The two specimens are located in the center of the lower die, 400mm apart from each other, then one blow operation is performed under a working pressure of 7.5 bar and 1,400mm total stroke of both dies. To minimize the frictional effect between the specimens and
721
Y.H. Yoo, D.Y. Yang I Journal of Materials Processing Technology 63 (1997) 718-723
the dies, the molybdenum disulfide (MoS z) and a teflon sheet are used as lubricants at the same time. Under this lubrication condition, the ring compression test reveals that the values of the friction coefficient and friction factor are 0.052 and 0.09, respectively. The final height of the deformed specimen is 46.9mm, and no lateral crack in the deformed specimen is found. 3.2. Simulation ofcopper blow test
Both the explicit and the implicit method are used for simulation of the copper blow test. In order to compare theoretical results with experimental ones, simulation conditions are modelled cautiously within the limits of the possibility. The initial meshes are shown in Fig. 1. Because of symmetry, oneeighth of the full model is used for simulation. The calculation is started with the initial impact velocities of the dies. The initial impact velocity of the upper and lower die, Vi. is calculated from equation (20) considering the energy balance as follows:
(20) where Er is a total input energy, mt and mb are the masses of the upper and lower dies, respectively. The initial impact velocity of the upper and lower dies calculated from equation (20) is 2.897(m/s).
=
temperature softening properties, X= 0.9, C v 383. OJ I kgOC, Tmelt = 1083°C, T",f = 10°C, m 1.09 It is assumed that the material response of dies is elastic. The Young's modulus, Poisson's ratio and density of dies made by tool steel are 211.4 GPa, 0.293 and 7.83 g/cm3, respectively, but the density is scaled up by 100 times so the volume of dies is reduced with the same ratio, while the mass of dies is maintained to constant. Now, with the above mentioned simulation conditions, the copper blow test is simulated using the two different methods. First, the copper blow test is simulated using the developed explicit program with friction coefficient 0.052. The calculation is continued until the specimen and the dies of the forging equipment are separated completely. The deformed meshes obtained from the explicit method are shown in Fig. 2. The initial relative velocity of the dies (5.794 mls) is decreased during the blow operation and the fmal relative velocity of the dies is maintained in 0.84 mls. Hence, for the same time duration, the moved relative displacement between the upper and lower dies before blowing is much larger than the moved relative displacement between the upper and lower dies after blowing. The fmal height of the deformed specimen obtained by the explicit simulation is 49.2 mm.
:±t±±i±±±±t
(a) t= -lOms
-
(d) t
Cylinder Specimen
Block Specimen
Fig. 1. Mesh system for copper blow test simulation. The experimental data obtained from the Johnson-Cook's paper [11] are accepted as material properties of OFHC copper. The material constants for OFHC copper are summarized as follows: shear modulus, G = 48.3 GPa Poisson's ratio, v = 0.343 density, p = 8.96 g/cm3 strain hardening properties,A=90.0 MPa,B= 292.0 MPa,n =0.31 strain rate hardening properties, C = 0.025, Eo = 10-6 ~s-t
=20 ms
-J I I
=
I I ~
(b)t=Oms
(c)t=lOms
(e)t=30ms
(1) t =40 ms
Fig. 2. Deformed meshes obtained from copper blow test simulation using the explicit method at (a) t = -10 ms, (b) t = 0 ms, (c) t = 10 ms, (d) t = 20 ms, (e) t = 30 ms, (1) t = 40ms. Second, the developed implicit time integration rigid-plastic finite element program is applied to a simulation of the copper blow test with a friction factor of 0.09 so as to compare with the explicit results. The implicit calculation is continued until the height of the specimen becomes 49.2 mm which is the fmal height obtained by the explicit calculation. The comparison of fmal deformed configurations between the
Y.H. Yoo. D.Y. Yang/Journal of Materials Processing Technology 63 (1997) 718-723
722
experiment and the calculations is shown in Fig. 3. The calculated result reveals a good agreement in the [mal deformed configurations between the experiment and the simulations. Fig. 4 shows the equivalent plastic strain distributions obtained from the simulations. The explicit result is very similar to the implicit one in the equivalent plastic strain distribution. The time histories of the clash load and plastic deformation energy are shown in Fig. 5 and 6, respectively. The time history plots about
500000 . , - - - - - - - - - - - - - - - - - , Maximum (Implicit Analysis) : 334375J 400000
Maximum (Explicit Analysis) : 337456J
300000
200000 Explicit Analysis 100000
_ _ jL-0.052
Experimental Result
-Explicit Analysis
Implicit Analysis
Fig. 3. Comparison of final deformed configurations between experiment and calculations of copper blow test.
O.B
0.6
I
:/
,
o¥---'---,~---,-'----r--.-....~--r~--I 30 20 25 10 15 o 5 Contact Time (ms)
Fig. 6. Time history of plastic deformation energy obtained from copper blow test simulation. the clash load and plastic deformation energy have good agreements between the two methods. In the explicit simulation, the clash load is increased gradually to the maximum value 9, 182 kN at time 19.0 ms and then decreased to zero at time 22.3 IDS. The plastic deformation energy is increased gradually to the maximum value 337,456J and then maintained at the same level to the termination time 22.3 tns. The maximum load and the plastic deformation energy obtained from the implicit simulation. are 9.313 kN and 334.375J. respectively.
0.9
4. Application to multi-blow forging process
-f.65
Implicit Analysis
Explicit Analysis
Fig. 4. Comparison of equivalent plastic strain distribution between the explicit and the implicit methods in copper blow test simulation.
14000 Maximum (Implicit Analysis) : 9313kN 12000
.... ~
1co
Maximum (Explicit ADalysis) : 9182kN 10000 8000
..
6000
t3
4000
..J ~
ev Explicit Analysis 2000
_11-0.052
0 0
5
10
15
20
25
30
Contact Time (ms)
Fig. 5. Time history of clash load obtained from copper blow test simulation.
In this chapter, the high-velocity multi-blow forging process is simulated with the explicit program verified through the simulation of the copper blow test. The configurations of specimen used for the simulation of multi-blow forging processes are cylinder and block. The initial meshes are shown in Fig. 1. To inspect changes of the blow efficiencies and the clash load generated by blow operations. the sequentially operated four blows are simulated. The computational conditions for the simulation of multi-blow forging processes is the same as simulation of the copper blow test except for the dimensions of the block specimen. The block specimen of 100 rom long edge X 50 rom short edge X 120 rom height is used for the simulation in order that the contact area and volume of the specimens of both types become equal. The initial mesh and the [mal mesh obtained from each blow operation are shown in Fig. 7. The blow efficiencies obtained from simulation are presented in Table 1. When the number of blows is increased, the contact area between the specimen and the dies is increased and the blow efficiency is decreased and an amount of height reduction is decreased. The blow efficiencies of the block specimen are larger than the blow efficiencies of the cylinder specimen. especially in the fourth blow operation. It reflects the effect of the geometrical self-constraint The clash load history according to the accumulated contact time is shown in Fig. 8. When the number of blows is increased, the contact time between the specimen and dies is decreased and the maximum clash load generated from each blow is rapidly
Y.R. Yoo. D.Y. Yang / Journal of Materials Processing Technology 63 (1997) 718-723
increased. The maximum clash load generated in the block specimen is slightly larger than the cylinder specimen. However, the accumulated contact time of the block specimen is shorter than the cylinder specimen. Table 1 Blow efficiencies obtained from multi-blow forging simulation Blow number
Blowefficiency(%) cylinder
block
96.3 89.9 76.9 51.7
96.4 89.8
1 2 3 4
723
As a result of the multi-blow forging simulation, it was found that the changes of the blow efficiencies and the clash load generated by the blow operations could be efficiently analyzed.
5. Conclusion
In the present work, in order to simultaneously obtain the blow efficiency and the clash load generated in the high-velocity impact forging processes with three-dimensional geometry, the explicit time integration finite element program, which was based on direct time integration of equation of motion, was developed . The copper blow test was simulated using the developed program. The calculated result revealed a good agreement in the final deformed configurations between the experiment and the simulation. In order to compare with the explicit method, the implicit time integration rigid-plastic finite element program including inertia effect was also developed and then also applied to copper blow test simulation. As a result of the copper blow test simulation using the explicit program and the implicit program, it was found that the calculated results had good agreements in available plastic deformation energy, maximum load and equivalent plastic strain distribution. The developed explicit program was also applied to the multi-blow forging process simulation. As a result of the multi-blow forging process simulation, it was found that the changes of the blow efficiencies and the clash load generated by the blow operations could be efficiently analyzed.
77.5 59.8
Cylinder Specimen
References
Block Specimen Fig. 7. Deformed meshes of multi-blow forging.
10ססoo ...- - - - - - - - - - - - - - - - ,
80000
••••.•••••• Block Cylinder
60000
40000
2ססoo
o f........-.-...--,.....,.-.--rlI'--r~..,u>.--,r4-.l,-'........-I
o
5
ill
~
W
~
K. Lange, Handbook ofmetal forming, McGraw-Hill, 1985. H. D. Watermann, Industrie-Anzeiger, 77 (1963) 1727. A. J. Organ, Int. J . Mach. Tool Des. Res., 7 (1967) 325. C. E. N. Sturgess and M. G. Jones, Int. J. Mech. Sci., 13 (1971) 309. [5] S. Vajpayee, M. M. Sadek and S. A. Tobias, Int. J. Mach. ToolDes. Res., 19 (1979) 237. [6] H. W. Haller, Trans. ASME, J. Eng. Ind., 105 (1983) 270. [7] E. Benuzzi and F. Soavi, Trans. ASME, J. Eng. Ind., 107 (1985) 266. [8] K. J. Bathe, Finite element procedures in engineering analysis, Prentice Hall, Englewood Cliffs, 1982. [9]D. P. Flanagan and T. Belytschko, Internat. J. Numer. Methods Engrg., 17 (1981) 679. [10] W. Johnson, Impact strength of materials, Edward Arnold. London, 1972. [11] G. R. Johnson and W. H. Cook, Proc. of 7th Int. Symp. on Ballistics, (1983) 541. [12] J. O. Hallquist, G. L. Goudreau and D. J. Benson, J. Compo Meths. Appl. Mechs. Eng., 51 (1985) 101. [13] Z. H. Zhong, Finite element procedures for contact-impact problems, Oxford University Press, New York, 1993. [14] Y. J. Kim and D. Y. Yang, Int. J. Mech. Sci., 27 (1985) 487. [15] S. I. Oh, Int. J. Mech. Sci., 24 (1982) 479. [1] [2] [3] [4]
~
~
~
~
Accumulated Contact Time (ms)
Fig. 8. Time history of clash load obtained from multi-blow forging simulation.