ASM Metals HandBook Vol. 14 - Forming and Forging
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Mech. Eng., Vol 2 (No. 1), 1973, p 31-36
77.
E. Anderheggen, Finite Element Analysis Assuming Rigid-Ideal-Plastic Material Behavior,
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Stuttgart, 1977
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D.C. Zienkiewicz and P.H. Godbole, A Penalty Function Approach to Problems of Plastic Flow of Metals With Large
Surface Deformations, J. Strain Anal., Vol 10, 1975, p 180-183
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Displacement, Int. J. Solids Struct., Vol 6, 1970, p 1069-1086
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A.S. Wifi, An Incremental Complete Solution of the Stre
tch Forming and Deep Drawing of a Circular Blank Using a
Hemispherical Punch, Int. J. Mech. Sci., Vol 18, 1976, p 23-31
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A.S. Wifi and Y. Yamada, Elastic-
Plastic Large Strain Analysis of Metal Forming by the Finite Element Method,
Monthly J. Inst. Ind. Sci., Vol 28, 1976, p 25
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Y. Yamada, T. Hirakawa, and A.S. Wifi, Analysis of Large Deformation and Bifurcation in Plasticity Problems by
Finite Element Method, in Finite Elements in Nonlinear Mechanics, Tapir, 1978, p 393
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d T. Hirakawa, Analysis of Large Deformation and Stress in Metal Forming Processes by
the Finite Element Method, in Metal Forming Plasticity, Springer-Verlag, 1979, p 158
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inite Element and a
Microcomputer System Development," Paper presented at the International Conference on Computer Applications in
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Current Advances in Mechanical
Design and Production, First Cairo University Mechanical Design and Production Conference,
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Metwalli, Ed., Pergamon Press, 1980
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Numerical Modelling of Manufacturing Processes, PVP-PB-025
, American Society of Mechanical Engineers, 1977, p
19-33
89.
E.H. Lee, R.L. Mallett, and R.M. McMeeking, Stress and Deformation Analysis of Metal Forming Processes, in
Metal
Forming Plasticity, International Union of Theoretical and Applied Mechanics (IUTAM) Symposium, Springer-
Verlag, 1979, p 177-189
90.
K.A. Derbalian, E.H. Lee, R.L. Mallett, and R.M. McMeeking, Finite Element Analysis With Specially Fixed Mesh,
in Application of Numerical Methods to Forming Processes, American Society of Mechanical En
gineers, Applied
Mechanics Division, Vol 28, 1978, p 39-47
91.
N.M. Wang and B. Budiansky, Analysis of Sheet Metal Stamping by a Finite Element Method, J. Appl. Mech.,
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45, 1978, p 73-82
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E.H. Lee, The Basis for an Elastic-Plastic Code, Stanford University, SUDAM 76-1
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95.
V. Cupka, H. Miyamoto, T. Nakagawa, T. Miyoshi, and K. Suzuki, Analysis of Counter-
Pressure Bending by FEM, in
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an Society of Precision Engineering,
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96.
S.-I. Oh and S. Kobayashi, Finite Element Analysis of Plane-Strain Sheet Bending, Int. J. Mech. Sci.,
Vol 22, 1980, p
583-594
97.
A. Makinouchi, Elastic-Plastic Stress Analysis of Bending and Hemming of Sheet Metal, in
Computer Modeling of
Sheet Metal Forming Process, N.M. Wang and S.C. Tang, Ed., The Metallurgical Society, 1986, p 161-176
98.
E.H. Lee, Elastic-Plastic Deformation at Finite Strain, J. of Appl. Mech., Vol 36, 1969, p 1-6
99.
R. Hill, Some Basic Principles in the Mechanics of Solids Without a Natural Time, J. Mech. Phys. Solids,
Vol 7,
1968, p 209-225
Process Modeling and Simulation for Sheet Forming
D. Lee, S.A. Majlessi, and J. H. Vogel, Department of Mechanical Engineering, Rensselaer Polytechnic Institute
Sheet Stretching
Punch stretching can be characterized as a nonsteady metal-forming process because of the presence of the moving
boundary that separates the sheet in contact with the punch head from the unsupported region of the sheet. Because of its
complexity, a detailed analysis of the stretching process can only be made by numerical methods such as finite-difference
(Ref 100, 101, 102) and finite-element techniques (Ref 91, 103, 104, 105, 106).
N.M. Wang and B. Budiansky (Ref 91) assumed that the sheet metal is rate insensitive, elasto-plastic, and exhibits normal
anisotropy. Using a convected coordinate system, the variational principle was formulated and the resulting equations
from finite-element discretization were solved. S. Kobayashi and his collaborators have also analyzed the sheet stretching
process in rate-sensitive and rate-insensitive materials. The common assumption in those analyses is that the elastic
strains are small compared to plastic strains and therefore can be ignored. The following example is presented to illustrate
some of the important parts of this type of finite-element formulation.
Example 2: Finite-Element Analysis of Punch Stretching of Rate-Sensitive
Materials.
In Ref 105, a finite-element program has been developed to analyze the axisymmetric punch stretching of rate-sensitive
materials. A rigid-viscoplastic material model with normal anisotropy has been assumed. The finite-element model was
based on the following variational formulation:
(Eq 49)
where is the effective stress, d is the incremental effective logarithmic strain, and H is the slope of the stress-strain
curve. The second integral represents the variation of the work done by the traction F. In developing Eq 49, it was
assumed that the principal axes of true strain rates maintain the same direction in each element and that the ratio of the
principal strain rate components remain constant during each time increment. The following constitutive equations for a
rate-sensitive material with normal anisotropy have been employed:
(Eq 50)
where Y is the yield stress and γ and m are material parameters. Following R. Hill's theory of anisotropy (Ref 32),
effective stress and increment of effective strain are defined as:
(Eq 51)
and
(Eq 52)
The incremental-strain components are defined as follows:
(Eq 53)
and
(Eq 54)
The sheet geometry is divided into a number of conical frustums, and the functional is expressed within each element
in terms of the nodal values associated with the particular element. The condition that the first variation of the functional
vanishes provides the stiffness equations. These equations are nonlinear and are solved by the Newton-Raphson iterative
scheme.
For verification purposes, a number of aluminum-killed steel specimens were stretched, and the results were compared
with the predicted values of the finite-element program. Figure 12 shows the computed radial, ε , and hoop, ε , strain
distributions for two coefficients of friction (μ = 0.25 and μ= 0.4) at the punch-sheet interface. The agreement with the
measured strains is generally good. In order to examine the sensitivity of the material to strain rate, four separate
computations were carried out with different punch speeds. In one case, a rate-insensitive material model was used. The
computed load versus punch travel is depicted in Fig. 13.
Fig. 12 Comparison of computed and measured strain distribution at several punch depths for
two coefficients
of friction. (a) μ= 0.25. (b) μ= 0.4 Source: Ref 105.
Fig. 13 Punch load-displacement curves for three punch speeds in stretching of rate-
sensitive material and one
punch speed for rate-insensitive material. Source: Ref 105.
References cited in this section
32. R. Hill, The Mathematical Theory of Plasticity, Oxford University Press, 1950
91. N.M. Wang and B. Budiansky, Analysis of Sheet Metal Stamping by a Finite Element Method, J. Appl. Mech.,
Vol
45, 1978, p 73-82
100.
D.M. Woo, The Stretch-Forming Test, Engineer, Vol 220, 1965, p 876-880
101.
N.M. Wang and W.J. Gordon, "On the Stretch
ing of a Circular Sheet by a Hemispherical Punch," General Motors,
1968
102.
N.M. Wang, Large Plastic Deformation of a Circular Sheet Caused by Punch Stretching, J. Appl. Mech.,
Vol 37,
1970, p 431-440
103.
H.S. Mehta and S. Kobayashi, Finite Element Analysis and Experimental Investigation of Sheet Metal Stretching,
J.
Eng. Ind., Vol 95, 1973, p 874-880
104.
J.H. Kim, S.-I. Oh, and S. Kobayashi, Analysis of Stretching of Sheet Metals With Hemispherical Punch,
Int. J.
Mach. Tool Des. Res., Vol 18, 1978, p 209-226
105.
N. Rebelo and S. Kobayashi, Axisymmetric Punch Stretching of Strain-Rate Sensitive Sheet Metals, in
Proceedings
of the Eighth North American Manufacturing Research Conference,
Vol 8, Society of Manufacturing Engineers,
1980, p 235-238
106.
S. Kobayashi and J.H. Kim, Deformation Analysis of Axisymmetric Sheet Metal Forming Processes by the Rigid-
Plastic Finite Element Method, in Mechanics of Sheet Metal Forming,
D.P. Koistinen and N.M. Wang, Ed., Plenum
Press, 1978, p 341-365
Process Modeling and Simulation for Sheet Forming
D. Lee, S.A. Majlessi, and J. H. Vogel, Department of Mechanical Engineering, Rensselaer Polytechnic Institute
Deep Drawing
Several finite-element analysis programs using different formulations have been proposed to study the deep-drawing
process. The majority of these analyses employed a rigid-plastic material model and the incremental theory of plasticity to
analyze axisymmetric geometries. The process of nonaxisymmetric sheet metal forming has also been examined by
several authors. C.H. Toh and S. Kobayashi (Ref 107, 108, 109) used a rigid-plastic finite-element analysis method based
on the membrane theory to model deep drawing of nonaxisymmetric geometries. Y.J. Kim and D.Y. Yang (Ref 110) also
developed an incremental rigid-plastic finite-element analysis method to analyze a hydrostatic-bulging process of
rectangular sheet diaphragms.
As an alternative approach, S. Levy et al. applied the deformation theory of plasticity and developed a finite-element
analysis program to approximate a cup-drawing process of an elastic-plastic material. It was shown that considerable
savings in computational time was achieved by using the deformation theory (Ref 111). D. Lee and coworkers (Ref 10,
112, 113) further improved this approach to account for the anisotropic behavior of materials, and demonstrated that a
computer software system based on the above approach can be applied to simulate actual sheet metal forming processes.
A subsequent refinement of the analysis has also been made, with improved analytical and numerical methods (Ref 114).
The basic approach has also been used by R.W. Logan and W.F. Hosford (Ref 115) to analyze necking and wall
wrinkling in sheet metal stretching/drawing. They included a scheme for updating the reference state to the current
configuration in the analysis, and reported an excellent computational efficiency over the stepwise incremental
formulation.
Although a number of general-purpose finite-element codes have been developed lately for analyzing nonlinear problems
(Ref 116, 117), they are not suitable for modeling sheet metal forming processes. Moreover, the prohibitive length of
computation time has prevented the use of such programs as an interactive tool at the design stage. The main advantage of
the membrane-type elements and deformation theory of plasticity is the simplicity. With such a simplified method,
solutions may be obtained quickly, and yet they are within an acceptable range of engineering approximations.
Example 3: Using Final Part Configuration to Compute Size of the Undeformed
Blank.
In Ref 114, a number of isoparametric elements were specified on the prescribed final configuration of an axisymmetric
part, and the shape and size of the undeformed blank sheet were computed. It is shown that considerable savings in
computer time can be achieved by using deformation theory.
The computational scheme is composed of first defining the locations of the nodal points in the undeformed configuration
(trial nodal points), and then improving these locations by applying a Newton-Raphson iterative scheme. The principle of
minimum potential energy is employed in the following manner. The potential energy of deformation ψ is decomposed
into two parts:
ψ= ψ
x
+ ψ
u
(Eq 55)
The first term, ψ
x
, is the energy associated with the deformation of the trial nodes from the initial to the final
configurations, and is given by:
(Eq 56)
The volume integral in the above equation refers to the strain energy where
x
, is the strain due to deformation of the trial
nodes from their beginning to end positions. The surface integral represents the work of traction forces. For a given final
configuration, ψ
x
, holds a constant value once the trial nodal positions are assumed.
The second term in Eq 55, ψ
u
, is the increment of potential energy due to the small displacements, u
i
, of the trial nodal
points in the initial configuration, and is given by:
(Eq 57)
where ε
f
, denotes the final strain. It has been shown that the surface integral of Eq 57 represents the frictional work only.
Because ψ
u
is a function of the small increments, u
i
, the correct values of u
i
can be determined by minimization of ψ
u
:
(Eq 58)
Because of the nonlinear nature of the problem, the correct values of the initial nodal positions have to be determined
through several iterations.
In Fig. 14, the computed strain distributions in radial and hoop directions are shown for a hypothetical cup model. In
order to verify the model, several experiments have been conducted for axisymmetric cups made of drawing quality steel
sheet. As shown in Fig. 15, the measured strain values agree well with the computed results, although there are some
discrepancies in the punch profile region where the description of friction is not well defined.
Fig. 14 Strain distributions at the end of each step of a four-step forming process for a steel cup.
Fig. 15
Comparison of the computed strains along the final generator with experimental data of 11.10 mm
(0.437 in.) deep cup made without lubrication. Inset shows the cup geometry. Source: Ref 114.
References cited in this section
10. D. Lee, Computer-Aided Control of Sheet Metal Forming Processes, J. Met., Vol 34, 1982, p 20-29
107.
C.H. Toh, "Process Modeling of Sheet Metal Forming of General Shapes by the Finite-
Element Method Based on
Large Strain Formulation," Ph.D. thesis, University of California, Berkeley, Sept 1983
108.
C.H. Toh and S. Kobayashi, Finite-
Element Process Modeling of Sheet Metal Forming of General Shapes, in
Fundamentals of Metal Forming Technique--States and Trends, Proceedings of International Symposium, Springer-
Verlag, Stuttgart, 1983, p 39-56
109.
S. Kobayashi, Deformation Analysis and Blank Design in Square Cup Drawing, in
Advanced Systems for
Manufacturing, 12th Conference on Production Research and Technology,
National Science Foundation, 1985, p
258-294
110.
Y.J. Kim and D.Y. Yang, A Rigid-
Plastic Finite Element Formulation Considering the Effect of Geometric Change
and Its Application to Hydrostatic Bulging, Int. J. Mech. Sci., Vol 27, 1985, p 453-463
111.
S. Levy, C.F. Shih, J.P.D. Wilkinson, P. Stine, and R.C. McWils
on, Analysis of Sheet Metal Forming to
Axisymmetric Shapes, in Formability Topics--Metallic Materials,
STP 647, B.A. Niemeier, A.K. Schmieder, and
J.R. Newby, Ed., American Society for Testing and Materials, 1978, p 238-260
112.
D. Lee and C.M. Forth, Development of a Computer-
Aided Analysis Method for Sheet Material Forming Processes,
in Innovations in Materials Processing,
Proceedings of the 30th Sagamore Conference, G. Bruggeman and V. Weiss,
Ed., Plenum Press, 1985, p 161-185
113.
D. Lee and P.A. Stine, "Computer-
Aided Prediction of Sheet Metal Manufacturing Processes and Its Experimental
Verification," Paper presented at the Fourth International Conference on Mechanical Behavior of Materials,
Stockholm, Aug 1983
114.
S.A. Majlessi and D. Lee, Further Development of Sheet Metal Forming Analysis Method, J. Eng. Ind.,
1987, in
press
115.
R.W. Logan and W.F. Hosford, Wall Wrinkling During Stretch Draw: Simulation and Experiment, in
Computer
Modeling of Sheet Metal-Forming Process, N.M. Wang and S.C. Tang, Ed., 1986, p 225-242
116.
S.-I. Oh, G.D. Lahoti, and T. Altan, ALPID--A General Purpose FEM Program for Metal Forming, in
Proceedings
of the Ninth North American Manufacturing Research Conference,
Vol 9, Society of Manufacturing Engineers,
1981, p 83-88
117.
K. Bathe, ADINA, A Finite Element Program for Automatic Dynamic Incremental Nonlinear Analysis,
Massachusetts Institute of Technology, Dec 1978
Process Modeling and Simulation for Sheet Forming
D. Lee, S.A. Majlessi, and J. H. Vogel, Department of Mechanical Engineering, Rensselaer Polytechnic Institute
Case Studies
This section will examine a total of three cases in order to illustrate different possibilities that occur under these sheet
forming conditions. All the cases represent actual forming for which experiments have been conducted to verify the
predictions.
In the cup forming, the frictional boundary condition was changed while the geometry was kept the same. In the vent-
forming case, a portion of the three-dimensional part was approximated as an axisymmetric geometry. The box forming
process was analyzed using three-dimensional sheet forming analysis.
Example 4: Cup Forming.
A photograph of the formed cup and the corresponding finite-element representation of the cup geometry are shown in
Fig. 16. Details of the imposed frictional boundary conditions are that the measured coefficient of friction (0.14) was used
in cup 1, and an exceedingly low coefficient of friction (0.02) was applied to cup 2 as a hypothetical case. Appropriate
normal loads were also specified, and the relative motion of the sheet metal with respect to the die surface was accounted
for by the sign of friction coefficient. Selection of specific nodal points for the application of the normal force has been
found to be not critical; the normal force may be distributed over the appropriate nodal points.
Fig. 16 (a) Photograph of the formed part. (b) A finite-element model for cup.
Given the material properties, the geometry file, and the loading boundary conditions, the next step in the flow diagram
(Fig. 3) is to compute the strain distributions when the cups have been formed. A parallel analysis starting from the
material data base yields a computed limiting strain FLD. The required inputs were strain rate sensitivity m, strain
hardening coefficient n, offset strain ε
0
, anisotropy index M
33
, and the initial inhomogeneity index, η. Superimposing the
safe/marginal shop FLD on the computed limiting FLD gives all the information that is necessary to establish a base for
FLD criteria.
All the necessary information to make a decision is in place; the strain distributions for the particular formed part will be
compared against the failure criteria, that is, the composite FLD, as shown in Fig. 17. The computed radial (major) and
hoop (minor) engineering strains at each nodal point are identified in the composite FLD by the corresponding nodal
point numbers. The resulting display gives the information necessary to proceed with the next step: Strain levels in some
of the nodal points in cup 1 exceed the safe/marginal boundary, while with cup 2 it is entirely safe to proceed.