ASM Metals HandBook Vol. 14 - Forming and Forging
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While it has not been stated what kind of tooling is being used to make this bend, the bending moment required can be
computed. To do this, the contribution of the product of stress, area, and distance from the neutral axis is integrated to
give the bending moment, M, about the neutral axis of the sheet:
(Eq 18)
While this integral can be evaluated for the assumptions stated so far, it can be simplified a great deal by neglecting the
elastic part of the material near the center of the sheet. For many practical ratios of sheet thickness to bend radius, this
does not introduce much error. Making this assumption, and using the values for plastic flow stress just calculated, the
following result is obtained:
(Eq 19)
At this point the springback may be computed by noting that the applied bending moment goes to zero upon unloading,
and therefore the internal moment must also go to zero. The (elastic) strains can be computed from the new geometry.
Letting r' be the new radius of curvature of the midplane after unloading:
(Eq 20)
Because the unloading is elastic, the appropriate material model for this part of the problem is Hooke's law, and solving
for the stress in the direction,
(Eq 21)
where E is Young's modulus and E' = E/(1 -
2
) is called the plane strain or bending modulus.
Fig. 7 Stresses and strains in bending.
The equilibrium condition here is that the change in the internal moment during unloading must be equal to the change in
the external moment. The change in the internal bending moment is obtained from the following integral, which may be
evaluated by using Eq 20 and 21:
(Eq 22)
Because the change in the external moment is equal to the original applied moment (Eq 19), the equilibrium equation is:
(Eq 23)
Solving for r', the radius of curvature of the midplane after springback:
(Eq 24)
This result is useful in predicting springback in simple bending. However, the assumptions made in deriving this
expression must be kept in mind when applying it, principally that the ratio of sheet thickness to bend radius must be
neither too small nor too large, and that the only load applied by the tooling is the bending moment. For instance, if the
sheet is bent while in tension, or if a significant compressive stress is applied across the thickness of the sheet (due to
contact with the die), the springback will be reduced. These conditions must be incorporated into the derivation if a
solution for either of these cases is desired.
There are other examples of closed-form analytical solutions of sheet bending processes. For example, R. Hill (Ref 32)
formulated the theory of plane strain bending for rigid-perfectly plastic materials. F. Proksa (Ref 33) used Hill's suggested
displacement equations in developing a theory for plane strain bending of rigid-linear hardening materials. Extensive
work has also been done by B.W. Shaffer and his collaborators (Ref 34, 35, 36).
If the process is more complicated than one of simple bending, or if a more accurate solution is desired, it may be
necessary to use a numerical approach to solve the equations. For example, in air bending, considerably more springback
will occur upon unloading than is predicted by Eq 24 because of the elastic unbending of the unsupported portion of the
sheet. An analysis of this process (Ref 37) considers the springback to be due to both the portion of the sheet in contact
with the punch and the free section of the sheet. These contributions are combined with the actual shape of the material
response curve to compute the springback, using an iterative-solution procedure.
Other numerical solutions of the bending process have also been developed. For example, K.H. Wolter (Ref 38) and H.
Verguts and R. Sowerby (Ref 39) took the deformation history of individual fibers into account. In a later study, further
improvements were made by P. Dadras and S.A. Majlessi (Ref 40, 41).
Drawing Analysis. The analysis of sheet drawing or stretching is different in some respects, as described earlier, and
an analysis of the cup-drawing process provides a good example. The following analysis is similar to work by R. Whitely
(Ref 42) and shows the effect of anisotropy on the limiting drawing ratio.
The geometry of cup drawing is clearly more complex than can be represented by radial drawing alone, but an analysis of
the flange, or radial drawing portion of the cup can yield useful results. To simplify the analysis, it is assumed that the
thickness change may be neglected, or
t
= 0 (see Fig. 8).
Fig. 8 Geometry definitions for drawing analysis.
The circumferential strain, , is related to the change in circumference at any radius by:
(Eq 25)
where r is the radius of any element of material and R is the initial radius of that element. Based on the assumption of
volume constancy with
t
= 0, the radial strain is given by
r
= - .
To relate the stresses to the strains, a rigid-perfectly plastic material model is assumed. To further simplify the analysis,
the Tresca yield criterion is chosen, so that the difference between largest and smallest principal stress is assumed to be a
constant at any point in the sheet. In the flange, shear stresses induced by friction are neglected, and the coordinate
stresses are taken to be the principal stresses. The radial stress,
r
, is therefore the (algebraically) largest principal stress,
and the compressive circumferential stress, , is the smallest. Therefore:
r
- = Y
f
(Eq 26)
where Y
f
is the flow strength of the material in the flange under the condition that
t
= 0.
For the equilibrium equation, consider a small element of material in the flange between the radii r and r + dr (see Fig. 9).
Because in practice the flange is slightly thicker at the outer edge than anywhere else, it is assumed that any blankholder
load is concentrated there. As a result, σ
t
= 0 throughout the flange. Balancing the forces acting on this element in the r
direction:
σ
r
|
r
trθ + 2σ tdr sin (θ/2)
= σ
r
|
r + dr
t(r + dr)θ
(Eq 27)
where
r
|
r
is equal to
r
evaluated at radius r. For small angles, sin (θ) ~θ, or:
r
|
r
trθ + tdrθ =
r
|
r + dr
t(r + dr)
(Eq 28)
Rearranging and dividing through by r, t, and θ:
(Eq 29)
Taking the limit as dr 0, the equilibrium equation at a point becomes:
(Eq 30)
Because of the assumption of a rigid-perfectly plastic material, the equilibrium equation turns out such that the stresses
may be computed without reference to the strain distribution. Integrating Eq 30:
r
(r) = C - Y
f
ln(r)
(Eq 31)
where C is an integration constant. Because
r
(R
0
) is known, where R
0
is the radius of the outer edge of the flange, C = Y
f
ln(R
0
) +
r
(R
0
). This gives:
(Eq 32)
For the case of no blankholder,
r
(R
0
) = 0, or:
(Eq 33)
Now if the effects of bending and friction over the draw ring are neglected, the radial stress that must be supported by the
cup wall to draw successfully may be computed from Eq 33:
(Eq 34)
where R
p
is the punch radius. The draw will fail if
r
- Y
w
, the flow stress in the wall. The flow in the wall is
characterized by = 0, so that for an anisotropic material, Y
w
will be different from Y
f
. Taking = 0 at the top of the
cup wall, the drawing limit is reached when:
(Eq 35)
This gives the limit drawing ratio as:
(Eq 36)
This equation predicts a limiting drawing ratio for an isotropic material to be ln(1), or approximately 2.72, but in fact this
is quite high. Actual values obtained experimentally are closer to 2.1 or 2.2 (Ref 43). The main reason for the discrepancy
is the fact that frictional forces and bending were neglected in the equilibrium equation.
Fig. 9 Flange element for equilibrium equation in drawing.
This analysis may be continued, relating the ratio of flow strengths to the R value of the material, and an efficiency factor
may be introduced to account for frictional losses and to give a better correlation with experimental results. The resulting
theory, while not perfect, does demonstrate the strong dependence of the limiting drawing ratio on material anisotropy.
For a more accurate analysis, a closed-form solution has not yet been obtained. However, a very good solution requiring a
computer program and an iterative solution procedure was developed by D.M. Woo (Ref 44, 45) following the same basic
steps outlined here, but with much less restrictive assumptions regarding material behavior, friction, and geometry of the
sheet. Using the Von Mises yield criterion and a rigidpower law hardening material model, the force equilibrium
equations derived above are integrated by approximating the variation of stress and strain (including thickness strain) to
be linear across each element. In this manner, the stress and strain distribution for the entire sheet was obtained for each
small advance of the punch into the material, with the incremental strains at each stage being added to find the total strain.
Many other researchers have since adopted and expanded the resulting finite-difference solution (Ref 46, 47, 48).
References cited in this section
25.
J.A. Schey, Introduction to Manufacturing Processes, McGraw-Hill, 1977, p 157-179
26.
K. Lange, Handbook of Metal Forming, McGraw-Hill, 1985, p 20.1-20.21
27.
E. Siebel and H. Beisswänger, Deep Drawing, Carl Hansen Verlag, 1955, (in German)
28.
B.W. Senior, Flange Wrinkling in Deep-Drawing Operations, J. Mech. Phys. Solids, Vol 4, 1956, p 235-246
29.
J.M. Alexander, An Appraisal of the Theory of Deep Drawing, Met. Rev., Vol 5, 1960, p 349-411
30.
E. Doege and N. Sommer, Optimization of Blankholder Force During Deep Drawing of Rectangular Parts,
Stahl
Eisen, Vol 103 (No. 3), 1983, p 139-141
31.
E.I. Odell, Analysis and Design of a Variable Position Minimum Pressure Blank-
Holder System for Axisymmetric
Deep Drawing, Int. J. Mach. Tool Des. Res., Vol 17, 1977, p 77-90
32.
R. Hill, The Mathematical Theory of Plasticity, Oxford University Press, 1950
33.
F. Proksa, Plastisches Biegen Von Blechen, Stahlbau, Vol 28 (No. 2), Feb 1959, p 29-36
34.
B.W. Shaffer and E.E. Ungar, Mechanics of the Sheet-Bending Process, J. Appl. Mech. (Trans. ASME),
Vol 27 (No.
1), March 1960, p 34-40
35.
B.W. Shaffer and R.N. House, Jr., Displacements in Wide Curved Bar Subjected to Pure Elastic-Plastic Bending,
J.
Appl. Mech., Vol 24 (No. 3), 1957, p 447-452
36.
B.W. Shaffer and R.N. House, Jr., The Significance of Zero Shear Stress in the Pure Ben
ding of a Wide Curved Bar,
J. Aeronaut. Sci., April 1957, p 307-308
37.
K.A. Stelson and D.C. Gossard, An Adaptive Pressbrake Control Using an Elastic-Plastic Material Model,
J. Eng.
Ind. (Trans. ASME), Vol 104, 1982, p 389-399
38.
K.H. Wolter, Freies Beigen Von Blechen, VDI-Forschungsheft, Vol 18 (No. 435, Ausgabe B), VDI-Verlag, 1952, p 1-
32
39.
H. Verguts and R. Sowerby, The Pure Plastic Bending of Laminated Sheet Metals, Int. J. Mech. Sci.,
Vol 17, 1975, p
31-51
40.
P. Dadras and S.A. Majlessi, Plastic Bending of Work Hardening Materials, J. Eng. Ind., Vol 104, 1982, p 224-230
41.
S.A. Majlessi and P. Dadras, Pure Plastic Bending of Sheet Laminates Under Plane Strain Condition,
Int. J. Mech.
Sci., Vol 25, 1983, p 1-14
42.
R. Whitely, The Importance of Directionality in Drawing Quality Sheet Steel, Trans. ASM, Vol 52, 1960, p 154-169
43.
W.F. Hosford and R.M. Caddell, Metal Forming: Mechanics and Metallurgy, Prentice-Hall, 1983, p 274-279
44.
D.M. Woo, Analysis of the Cup-Drawing Process, J. Mech. Eng. Sci., Vol 6, 1964, p 116-131
45.
D.M. Woo, On the Complete Solution of the Deep-Drawing Problem, Int. J. Mech. Sci., Vol 10, 1968, p 83-94
46.
E.I. Odell and W.E. Clausen, Numerical Solution of a Deep-Drawing Problem, J. Eng. Ind., Vol 99, 1977, p 206-209
47.
B. Kaftanoglu and A.E. Tekkaya, Complete Numerical Solution of the Axisymmetrical Deep-Drawing Process,
J.
Eng. Mater. Technol., Vol 103, 1981, p 326-332
48.
K.R. Vemuri, P.S. Raghupathi, and K. Hoang-Vu, A Computer-Aided Design System for Axi
symmetric Deep
Drawing Process, in Proceedings of the 14th North American Manufacturing Research Conference,
Vol 14, Society
of Manufacturing Engineers, 1986, p 365-371
Process Modeling and Simulation for Sheet Forming
D. Lee, S.A. Majlessi, and J. H. Vogel, Department of Mechanical Engineering, Rensselaer Polytechnic Institute
Finite-Element Analysis Methods
With recent advances in high-speed computers, there has been a marked progress in the development and application of
numerical methods for mechanics-related problems. Among these techniques, the finite-element analysis method has been
widely used in the treatment of diverse classes of problems in this field. In particular, material forming processes, because
of their complexity and large number of process variables, are ideally suited for the application of the finite-element
analysis technique. A systematic survey of the developments and applications of the technique in the field of material
processing has been given by A.S. Wifi (Ref 49). In another review paper, S. Kobayashi (Ref 50) has outlined the
important role of the finite-element method in metal forming modeling work.
Although early development of the finite-element method was primarily concerned with linear problems, the method has
been extended to nonlinear problems by the initial strain method (Ref 51, 52) and by the tangent modulus method (Ref
53, 54). The actual application of this technique to metal forming problems was made in the 1960s by P.V. Marcal et al.
(Ref 55), Y. Yamada et al. (Ref 56), and O.C. Zienkiewicz et al. (Ref 57); additional applications were made in later
years by employing the small-strain analysis method (Ref 58, 59, 60, 61, 62, 63, 64, 65).
Metal forming operations, however, inherently involve large strains and therefore different formulations were needed for
proper analyses of these processes. During the 1970s two basic approaches for treating large deformation were developed.
One of these was the rigid-plastic formulation (Ref 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77) where it is assumed that
the elastic strains are small compared to plastic strains, and therefore can be ignored. In these analyses, a material model
that obeyed the von Mises yield criterion and the associated flow rule were used. The incompressibility constraint was
treated by either a Lagrange multiplier (Ref 66) or a penalty function (Ref 78, 79).
The rigid-plastic formulation, however, does not account for elastic unloading; therefore, such phenomena as springback
and development of residual stresses cannot be treated by the model. Development of the second approach, elastic-plastic
large-deformation finite-element analyses, has been accomplished partially as a result (Ref 80, 81, 82, 83, 84, 85, 86, 87,
88, 89, 90, 91). The classic paper of H.D. Hibbitt et al. (Ref 80) introduced one of the first large-strain finite-element
formulations, in which a total Lagrangian formulation (TLF) was employed. In this formulation, the initial undeformed
configuration of the material is chosen to be the reference state. A.S. Wifi (Ref 81) developed a complete finite-element
program and analyzed the axisymmetric stretch-forming and deep-drawing processes, using the incremental TLF. N.M.
Wang and B. Budiansky also employed a TLF-based nonlinear membrane theory and simulated the axisymmetric
hemispherical punch stretching of sheet metals (Ref 91).
An alternative approach called the updated Lagrangian formulation (ULF), in which the current configuration of the
deforming material is considered to be the reference state, was also developed and extensively used (Ref 83, 92, 93).
Some of the key papers discussing the development of finite-element analysis using ULF for large elastic-plastic
deformation are those of R.M. McMeeking and J.R. Rice (Ref 92) and E.H. Lee (Ref 93). The formulation of initial
displacement matrix (the undeformed configuration of the material) is not required in the ULF method.
The basis of the finite-element analysis technique, using the variational approach, is to formulate a functional based on
the specific constitutive relations. This functional is usually a statement of the potential energy of the deforming
continuum. The finite-element equations are then formulated by the condition that the first variation of the functional
should vanish. E.P. Popov et al. (Ref 94) employed the shell theory and developed a finite-element program to analyze
sheet bending problems. Using this theory gives satisfactory results only when the deformation is moderate and the ratio
of thickness to bending radius is small. For analysis of large-strain bending processes, V. Cupka et al. (Ref 95) employed
the finite-element formulation developed by H.D. Hibbitt et al. (Ref 80) and solved a three-point bending problem with
counterpressure at the bottom of a sheet. S.-I. Oh and S. Kobayashi (Ref 96) developed two finite-element analysis
programs, using formulations suitable for rigid-plastic and elasto-plastic material behavior. It was shown that the results
of the two analyses agreed very well with each other. In a recent effort, A. Makinouchi (Ref 97) formulated an elaborate
constitutive equation assuming a varying Young's modulus, and developed a finite-element program using a ULF
approach. As an example, the essence of that work is outlined in Example 1 of the following section.
Example 1: Finite-Element Analysis of Sheet Bending and Hemming.
An elastic-plastic incremental finite-element analysis program was developed for sheet bending and hemming processes
of metals under plane strain conditions (Ref 97). A constitutive equation within the framework of finite deformation was
also developed. The derivation of the constitutive equation follows the decomposition of the deformation gradient F as
proposed by E.H. Lee (Ref 98):
F = V
e
F
p
(Eq 37)
where V
e
and F
p
are elastic and plastic deformation gradients, respectively. The velocity gradient L in the current
configuration, x, and the plastic velocity gradient L
p
in the plastically deformed configuration, α, are given by:
L = F
-1
2L
p
=
p
F
p-1
(Eq 38)
The rate of deformation tensor D, which is the symmetric part of L, is expressed as:
(Eq 39)
where T stands for transposed. A measure of elastic strain
e
is introduced with respect to the plastically deformed
configuration , and is defined as follows:
(Eq 40)
where u
e
is the elastic displacement from the plastically deformed configuration to the current configuration, x. Using the
above definitions, it can be shown that:
V
e
= I +
e
(Eq 41)
where I is the unity tensor. Substituting the material time derivative of Eq 41 into Eq 39, we obtain:
(Eq 42)
where D
p
is the plastic rate of deformation tensor, D
p
= (L
p
+ L )/2, and W
p
is the plastic spin tensor W
p
= (L
p
= L )/2.
Because the analysis is concerned with small elastic strains, the elastic strain rate
e
can be neglected compared to plastic
rate of deformation D
p
; thus Eq 42 reduces to:
D =
e
- W
pe
+
e
W
p
+ D
p
(Eq 43)
It is important to note that the first three terms in Eq 43 have the form of a Jaumann rate (Ref 92) of
e
. Therefore:
D = MATH OMITTED
e
+ D
p
(Eq 44)
where MATH OMITTED
e
is the Jaumann rate of
e
. The above relationship is the rate form of the elastic-plastic
decomposition for cases with small-elastic but finite-plastic deformation, and replaces the following well-known
relationship:
D = D
e
+ D
p
(Eq 45)
In one finite-element analysis program, R. Hill's variational principle (Ref 99) and the updated Lagrangian formulation
has been employed. Hill's variational principle is expressed in the form:
(Eq 46)
Constant-strain triangular elements were used in the calculations. In order to verify the theory, a number of experiments
using mild steel and high-strength steel sheet were carried out. The stress-plastic strain relation obtained from the tensile
tests is expressed by a Swift-type equation:
= C(
0
+
p
)
n
(Eq 47)
It has been shown that the evolution of the elastic modulus can also be presented by a similar relation:
E = E
0
(1.0 + 100
p
)
m
(Eq 48)
Figure 10 shows a comparison between computed and measured springback angle. Two computed results are shown for
each material. One is obtained by using a constant Young's modulus E
0
, and the other by using the variable Young's
modulus E. As shown in Fig. 10, a closer agreement with the experimental data is achieved when a variable elastic
modulus is used. Figure 11(a) illustrates the computed effective strain and the longitudinal stress profile at a 45° bent
angle. In Fig. 11(b), the distribution of longitudinal stress σ
ll
, normal stress σ
nn
, lateral stress σ
zz
, and shear stress σ
ln
are
depicted for the bent angle of 45°.
Fig. 10
Comparisons of the springback angle obtained experimentally and using two different computations. (a)
Low-carbon steel. (b) High-strength steel. E
0
, constant Young's modulus; E, variable Young's modulus.
Source:
Ref 97.
Fig. 11
Distribution of (a) effective strain and longitudinal stress and (b) stress components in the cross section
for a high-strength steel material model with bend angle of 45°. Punch diameter, 2.0 mm (0.08 in.); die spa
n,
6.4 mm (0.25 in.); sheet thickness, 0.8 mm (0.03 in.). Source: Ref 97.
References cited in this section
49.
A.S.
Wifi, Finite Element Correction Matrices in Metal Forming Analysis (With Application to Hydrostatic Bulging
of a Circular Sheet), Int. J. Mech. Sci., Vol 24, 1982, p 393-406
50.
S. Kobayashi, Metalworking Process Modeling and the Finite Element Method, in
Proceedings of the Ninth North
American Manufacturing Research Conference, Vol 9, Society of Manufacturing Engineers, 1981, p 16-21
51.
J.H. Argyris, Continua and Discontinua, in
Proceedings of the Conference on Matrix Methods in Structural
Mechanics, AFFDL-TR-66-80, Nov 1966, p 11-189
52.
R.H. Gallagher, J. Padlog, and P.P. Bijlard, Stress Analysis of Heated Complex Shapes, J. Am. Rocket Soc.,
Vol 32,
1962, p 700-707
53.
G.G. Pope, The Application of the Matrix Displacement Method in Plane Elasto-Plastic Problems, in
Proceedings of
the Conference on Matrix Methods in Structural Mechanics, AFFDL-TR-66-80, Nov 1966, p 635-654
54.
J.L. Swedlow and W.H. Yang, "Stiffness Analysis of Elastoplastic Plates," AFRPL-TR-66-
5, Graduate Aeronautical
Laboratories, California Institute of Technology, 1966
55.
P.V. Marcal and I.P. King, Elastic-Plastic Analysis of Two-
Dimensional Stress Systems by the Finite Element
Method, Int. J. Mech. Sci., Vol 9, 1967, p 143-155
56.
Y. Yamada, N. Yoshimura, and T. Sakurai, Plastic Stress-Strain Matrix and Its Application for the Solution of Elastic-
Plastic Problems by the Finite Element Method, Int. J. Mech. Sci., Vol 10, 1968, p 343-354
57.
O.C. Zienkiewicz, S. Valliappan, and I.P. King, Elasto-Plastic Solutions of Engineering P
roblems "Initial Stress"
Finite Element Approach, Int. J. Num. Meth. Eng., Vol 1, 1969, p 75-100
58.
C.H. Lee and S. Kobayashi, Elastoplastic Analysis of Plane-
Strain and Axisymmetric Flat Punch Indentation by the
Finite Element Method, Int. J. Mech. Sci., Vol 12, 1970, p 349-370
59.
C.H. Lee, S. Masaki, and S. Kobayashi, Analysis of Ball Indentation, Int. J. Mech. Sci., Vol 14, 1972, p 417-426
60.
C.H. Lee, H. Iwasaki, and S. Kobayashi, Calculation of Residual Stresses in Plastic Deformation Processes,
J. Eng.
Ind., Vol 95, Feb 1973, p 283-291
61.
Y. Yamada, Recent Advances in Matrix Displacement Method for Elastic-Plastic Problems in Japan, in
Recent
Advances in Matrix Methods of Structural Analysis and Design, R.H. Gallager et al., Ed., UAH, 1971, p 283
62.
M.M. Abou Ellail and M.M. Faraf, "Analysis of Direct Extrusion by the Finite Element Technique," Paper presented
at the First Cairo University Conference in Mechanical Design and Production, Cairo, Dec 1979
63.
M. Gotoh, A Finite Element Analysis of General Deformation of Sheet Metals, Int. J. Num. Meth. Eng.,
Vol 8, 1974,
p 731-741
64.
A. Nagamatsu, Influence of Friction on Elastic-Plastic Deformation in Metal Working by Compression, Theor.
Appl.
Fract. Mech., Vol 21, 1973, p 257
65.
K. Iwata, K. Osakada, and S. Fujino, Analysis of Hydrostatic Extrusion by the Finite Element Method, J. Eng. Ind.,
Vol 94, 1972, p 697-703
66.
C.H. Lee and S. Kobayashi, New Solutions to Rigid-Plastic Deformation Problems Using Matrix Method,
J. Eng.
Ind., Vol 95, 1973, p 865-873
67.
S. Kobayashi and C.H. Lee, Deformation Mechanics and Workability in Upsetting Solid Circular Cylinders, in
Proceedings of the First North American Manufacturing Research Conference,
Vol 1, Society of Manufacturing
Engineers, 1973, p 185
68.
S.N. Shah and S. Kobayashi, "The Matrix Method for the Analysis of Metal Forming Processes," Paper presented at
the Sagamore Army Material Research Conference, Requette Lake, NY, 1974
69.
S.N. Shah, C.H. Lee and S. Kobayashi, Compression of Tall Cir
cular Solid Cylinder Between Parallel Flat Dies, in
Proceedings of the International Conference on Production Engineering,
Japan Society of Precision Engineering,
Tokyo, 1974, p 295
70.
S.N. Shah and S. Kobayashi, Rigid Plastic Analysis of Cold Heading by the Matrix Method, in
Proceedings of the
15th International Machine Tool Design and Research Conference, Halsted Press, 1975, p 603-610
71.
S. Kobayashi, Rigid-Plastic Finite Element Analysis of Axisymmetric Metal Forming Processes, in
Numerical
Modelling of Manufacturing Processes, PVP-PB-025, American Society of Mechanical Engineers, 1977, p 49-65
72.
C.C. Chen and S. Kobayashi, Rigid-Plastic Finite Element Analysis of Ring Compression, in
Application of
Numerical Methods to Forming Processes, American
Society of Mechanical Engineers, Applied Mechanics Division,
Vol 28, 1978, p 163-174
73.
C.C. Chen and S. Kobayashi, Deformation Analysis of Multi-Pass Bar Drawing and Extrusion, in Annals CIRP,
Vol
27 (No. 1), 1978, p 151-155
74.
W. Klie, Plastic Deformation With Free Boundaries--A Finite Element Approach, in Metal Forming Plasticity,
H.
Lippmann, Ed., Springer-Verlag, 1979, p 260
75.
K. Roll, Calculation of Metal Forming Processes by Finite Element Method, in
Application of Numerical Methods to
Forming Processes, American Society of Mechanical Engineers, Applied Mechanics Division, Vol 28, 1978, p 67
76.
M. Lung and O. Mahrenholtz, Finite Element Procedure for Analysis of Metal Forming Processes, Trans.
Can. Soc.