Deep drawing of sheet metal is an important manufacturing technique. In the deep drawing process,
a “blank” of sheet metal is clamped by a blank holder against a die. A punch is then moved
against the blank, which is drawn into the die. Unlike the operation described in the
hemispherical punch stretching example (Stretching of a thin sheet with a hemispherical punch), the
blank is not assumed to be fixed between the die and the blank holder; rather, the blank is
drawn from between these two tools. The ratio of drawing versus stretching is controlled by
the force on the blank holder and the friction conditions at the interface between the blank
and the blank holder and the die. Higher force or friction at the blank/die/blank holder
interface limits the slip at the interface and increases the radial stretching of the blank.
In certain cases drawbeads, shown in Figure 1, are used to restrain the slip at this interface even further.
To obtain a successful deep drawing process, it is essential to control the
slip between the blank and its holder and die. If the slip is restrained too
much, the material will undergo severe stretching, thus potentially causing
necking and rupture. If the blank can slide too easily, the material will be
drawn in completely and high compressive circumferential stresses will develop,
causing wrinkling in the product. For simple shapes like the cylindrical cup
here, a wide range of interface conditions will give satisfactory results. But
for more complex, three-dimensional shapes, the interface conditions need to be
controlled within a narrow range to obtain a good product.
During the drawing process the response is determined primarily by the
membrane behavior of the sheet. For axisymmetric problems in particular, the
bending stiffness of the metal yields only a small correction to the pure
membrane solution, as discussed by Wang and Tang (1988). In contrast, the
interaction between the die, the blank, and the blank holder is critical. Thus,
thickness changes in the sheet material must be modeled accurately in a finite
element simulation, since they will have a significant influence on the contact
and friction stresses at the interface. In these circumstances the most
suitable elements in
Abaqus
are the 4-node reduced-integration axisymmetric quadrilateral, CAX4R; the first-order axisymmetric shell element, SAX1; the first-order axisymmetric membrane element, MAX1; the first-order finite-strain quadrilateral shell element, S4R; the fully integrated general-purpose finite-membrane-strain
shell element, S4; and the 8-node continuum shell element, SC8R.
Membrane effects and thickness changes are modeled properly with CAX4R. However, the bending stiffness of the element is low. The
element does not exhibit “locking” due to incompressibility or parasitic shear.
It is also very cost-effective. For shells and membranes the thickness change
is calculated from the assumption of incompressible deformation of the
material.
Geometry and model
The geometry of the problem is shown in
Figure 2.
The circular blank being drawn has an initial radius of 100 mm and an initial
thickness of 0.82 mm. The punch has a radius of 50 mm and is rounded off at the
corner with a radius of 13 mm. The die has an internal radius of 51.25 mm and
is rounded off at the corner with a radius of 5 mm. The blank holder has an
internal radius of 56.25 mm.
The blank is modeled using 40 elements of type CAX4R or 31 elements of type SAX1, MAX1, S4R, S4, or SC8R. An 11.25° wedge of the circular blank is used in the
three-dimensional S4R and S4 models. These meshes are rather coarse for this analysis.
However, since the primary interest in this problem is to study the membrane
effects, the analysis will still give a fair indication of the stresses and
strains occurring in the process.
The contact between the blank and the rigid punch, the rigid die, and the rigid blank holder is
modeled with a contact pair in most cases. The top and bottom surfaces of the blank are
defined as surfaces in the model. The rigid punch, the die, and the blank holder are modeled
as analytical rigid surfaces. The mechanical interaction between the contact surfaces is
assumed to be frictional contact. Therefore, friction is used in conjunction with the
various contact property definitions to specify coefficients of friction.
At the start of the analysis for the CAX4R model, the blank is positioned precisely on top of the die and
the blank holder is precisely in touch with the top surface of the blank. The
punch is positioned 0.18 mm above the top surface of the blank.
In the case of shells and membranes, the positioning of the blank depends on the contact
formulation used. Node-to-surface and surface-to-surface contact formulations are available
in Abaqus/Standard. For the node-to-surface formulation, the shell/membrane thickness is modeled using an
exponential pressure-overclosure relationship (Contact Pressure-Overclosure Relationships). The blank
holder is positioned a fixed distance above the blank. This fixed distance is the distance
at which the contact pressure is set to zero using an exponential pressure-overclosure
relationship. However, the surface-to-surface contact formulation (which applies to both
contact pairs and general contact) automatically takes thickness into account, and the need
for specifying pressure overclosure relations is eliminated. Examples of the
surface-to-surface contact formulation with shell elements are provided in this problem.
Material properties
The material (aluminum-killed steel) is assumed to satisfy the
Ramberg-Osgood relation between true stress and logarithmic strain:
The reference stress value, K, is 513 MPa; and the
work-hardening exponent, n, is 0.223. The Young's modulus
is 211 GPa, and the Poisson's ratio is 0.3. An initial yield stress of 91.3 MPa
is obtained with these data. The stress-strain curve is defined in piecewise
linear segments in the metal plasticity specification, up to a total
(logarithmic) strain level of 107%.
The coefficient of friction between the interface and the punch is taken to
be 0.25; and that between the die and the blank holder is taken as 0.1, the
latter value simulating a certain degree of lubrication between the surfaces.
The stiffness method of sticking friction is used in these analyses. The
numerics of this method make it necessary to choose an acceptable measure of
relative elastic slip between mating surfaces when sticking should actually be
occurring. The basis for the choice is as follows. Small values of elastic slip
best simulate the actual behavior but also result in a slower convergence of
the solution. Permission of large relative elastic displacements between the
contacting surfaces can cause higher strains at the center of the blank. In
these runs we let
Abaqus
choose the allowable elastic slip, which is done by determining a
characteristic interface element length over the entire mesh and multiplying by
a small fraction to get an allowable elastic slip measure. This method
typically gives a fairly small amount of elastic slip.
Although the material in this process is fully isotropic, the local
coordinate system is used with the CAX4R elements to define a local orientation that is coincident
initially with the global directions. The reason for using this option is to
obtain the stress and strain output in more natural coordinates: if the local
coordinate system is used in a geometrically nonlinear analysis, stress and
strain components are given in a corotational framework. Hence, in our case
throughout the motion, S11 will be the stress in the
r–z plane in the direction of the
middle surface of the cup. S22 will be the stress in the thickness direction, S33 will be the hoop stress, and S12 will be the transverse shear stress, which makes interpreting
the results considerably easier. This orientation definition is not necessary
with the SAX1 or MAX1 elements since the output for shell and membrane elements is
already given in the local shell system. For the SAX1 and MAX1 model, S11 is the stress in the meridional direction and S22 is the circumferential (hoop) stress. An orientation definition
would normally be needed for the S4R and S4 models but can be avoided by defining the wedge in such a manner
that the single integration point of each element lies along the global
x-axis. Such a model definition, along with appropriate
kinematic boundary conditions, keeps the local stress output definitions for
the shells as S11 being the stress in the meridional plane and S22 the hoop stress. There should be no in-plane shear, S12, in this problem. A transformation is used in the S4R and S4 models to impose boundary constraints in a cylindrical system.
Loading
The entire analysis is carried out in five steps. In the first step the
blank holder is pushed onto the blank with a prescribed displacement to
establish contact. In the shell models this displacement roughly corresponds to
zero clearance across the interface.
In the second step the boundary condition is removed and replaced by the
applied force of 100 kN on the blank holder. This force is kept constant during
Steps 2 and 3. This technique of simulating the clamping process is used to
avoid potential problems with rigid body modes of the blank holder, since there
is no firm contact between the blank holder, the blank, and the die at the
start of the process. The two-step procedure creates contact before the blank
holder is allowed to move freely.
In the third step the punch is moved toward the blank through a total
distance of 60 mm. This step models the actual drawing process. During this
step the time incrementation parameters are set to improve efficiency for
severely discontinuous behavior associated with frictional contact.
The last two steps are used to simulate springback. In the fourth step all
the nodes in the model are fixed in their current positions and the contact
pairs are removed from the model. This is the most reliable method for
releasing contact conditions. In the fifth, and final, step the regular set of
boundary conditions is reinstated and the springback is allowed to take place.
This part of the analysis with the CAX4R elements is included to demonstrate the feasibility of the
unloading procedure only and is not expected to produce realistic results,
since the reduced-integration elements have a purely elastic bending behavior.
The springback is modeled with more accuracy in the shell element models.
Results and discussion
Figure 3
shows deformed shapes that are predicted at various stages of the drawing
process for the CAX4R model. The profiles show that the metal initially bends and
stretches and is then drawn in over the surface of the die. The distributions
of radial and circumferential strain for all three models and thickness strain
for the CAX4R model are shown in
Figure 4.
The thickness for the shell or membrane models can be monitored with output
variable STH (current shell or membrane thickness). The thickness does not
change very much: the change ranges from approximately −12% in the cylindrical
part to approximately +16% at the edge of the formed cup. Relatively small
thickness changes are usually desired in deep drawing processes and are
achieved because the radial tensile strain and the circumferential compressive
strain balance each other.
The drawing force as a function of punch displacement for various element types is shown in Figure 5, where the curves are seen to match closely. Similarly, the drawing force as a function
of punch displacement with S4R elements using
the node-to-surface and surface-to-surface contact pair formulations is shown in Figure 6. The differences in the reaction force history are due to consideration of the blank
thickness explicitly in the surface-to-surface contact formulation as compared to the
node-to-surface contact formulation where a pressure-overclosure relationship is specified.
In all of the cases, oscillations in the force history are seen. These oscillations are a
result of the rather coarse mesh—each oscillation represents an element being drawn over the
corner of the die. Compared to the shell models, the membrane model predicts a smaller punch
force for a given punch displacement. Thus, toward the end of the analysis the results for
punch force versus displacement for the MAX1
model are closer to those for the CAX4R model.
The deformed shape after complete unloading is shown in
Figure 7,
superimposed on the deformed shape under complete loading. The analysis shows
the lip of the cup springing back strongly after the blank holder is removed
for the CAX4R model. No springback is evident in the shell models. As was noted
before, this springback in the CAX4R model is not physically realistic: in the first-order
reduced-integration elements an elastic “hourglass control” stiffness is
associated with the “bending” mode, since this mode is identical to the
“hourglass” mode exhibited by this element in continuum situations. In reality
the bending of the element is an elastic-plastic process, so that the
springback is likely to be much less. A better simulation of this aspect would
be achieved by using several elements through the thickness of the blank, which
would also increase the cost of the analysis. The springback results for the
shell models do not exhibit this problem and are clearly more representative of
the actual elastic-plastic process.
Model using the incompatible mode element, CAX4I, as an alternative to the CAX4R element. In contrast to the reduced-integration, linear
isoparametric elements such as the CAX4R element, the incompatible mode elements have excellent bending
properties even with one layer of elements through the thickness (see
Geometrically nonlinear analysis of a cantilever beam)
and have no hourglassing problems. However, they are computationally more
expensive.
Wang, N. M., and S.
C. Tang, “Analysis of Bending Effects in Sheet Forming
Operations,” International Journal for
Numerical Methods in Engineering, vol. 25, pp. 253–267, January
1988.
Figures
Figure 1. A typical drawbead used to limit slip between the blank and
die. Figure 2. Geometry and mesh for the deep drawing problem. Figure 3. Deformed shapes at various stages of the analysis. Figure 4. Strain distribution at the end of the deep drawing step. Figure 5. Punch force versus punch displacement using the node-to-surface
contact formulation. Figure 6. Comparison of punch force versus punch displacement with S4R elements
for different contact formulations. Figure 7. Deformed shape after unloading.
