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## 14. Slab Analysis of Bulk Forming Processes

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### ME 612 Metal Forming and Theory of Plasticity 14. Slab Analysis of Bulk Forming Processes Assoc.Prof.Dr. Ahmet Zafer enalp e-mail: azsenalp_at_gmail.com – PowerPoint PPT presentation

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Title: 14. Slab Analysis of Bulk Forming Processes

1
14. Slab Analysis of Bulk Forming Processes

ME 612 Metal Forming and Theory of Plasticity
• Assoc.Prof.Dr. Ahmet Zafer Senalp e-mail
azsenalp_at_gmail.com
• Mechanical Engineering Department
• Gebze Technical University

2

14. Slab Analysis of Bulk Forming Processes
• This method entails a force balance on a slab of
metal of differential thickness. This produces a
differential equation where variations are
considered in one direction only. Using pertinent
boundary conditions, an integration of this
equation then provides a solution. The
assumptions involved are the following
• 1. Friction does not influence the orientation of
the principal axes. In the Figures below we
assume that x,y,z are fixed principal axes within
the deformation zone.
• 2. Plane sections remain plane, thus the
deformation is homogeneous in regard to the
determination of induced strain.
• 3. The principal stresses do not vary on the yz
plane (see Figure below)

3

14.1. Plane Strain Drawing
14. Slab Analysis of Bulk Forming Processes

Figure 14.1. An example of sheet drawing showing
a slab. The axes X, Y, Z are assumed to be
principal stress axes.
4

14.1. Plane Strain Drawing
14. Slab Analysis of Bulk Forming Processes
• The drawing stress is defined as follows
• Using incompressibility
• and plane strain conditions,
• results in the following
• We define the homogeneous strain as

(14.1)
(14.2)
(14.3)
(14.4)
(14.5)
5

14.1. Plane Strain Drawing
14. Slab Analysis of Bulk Forming Processes

Figure 14.2. Sheet Drawing. Free body diagrams to
calculate and
6

14.1. Plane Strain Drawing
14. Slab Analysis of Bulk Forming Processes
• Taking in the left free body
diagram,
• And since
• In general, ltlt 1 so
• indicating that is compressive.
• Using the flow rule for, we have
that or finally using equation
(14.7) and the definition of the deviatoric
stress,
• In terms of principal stresses, since is
obviously tensile, then
• Substitution in the von-Mises criterion, gives
the following relation between the stress
components to initiate and sustain plastic
deformation (plane strain)

(14.9)
(14.7)
(14.8)
(14.9)
7

14.1. Plane Strain Drawing
14. Slab Analysis of Bulk Forming Processes
• or
• This equation is valid in any location inside the
deformation zone but we here assume a
nonhardening material (Y - constant).
• Considering equilibrium of forces in the X
direction
• or after neglecting higher order terms,
• and using dt/2 ds sina we finally have
• Let us define B as follows
• The equilibrium equation is now simplified as
follows

(14.10)
(14.11)
(14.12)
(14.13)
(14.14)
(14.15)
8

14.1. Plane Strain Drawing
14. Slab Analysis of Bulk Forming Processes
• Substituting p from equation (14.10) in the
equation above gives
• The solution is based on the following
assumptions.
• An average constant value of µ describes the full
contact region.
• The metal does not work harden, or a mean value
of yield stress strength adequately describes any
work-hardening effects in either case,Y is
treated as a constant.
• The semi-die angle a is a constant.
• Direct integration using the conditions that
when and when
• gives

(14.16)
(14.17)
9

14.1. Plane Strain Drawing
14. Slab Analysis of Bulk Forming Processes
• or using homogeneous strain,
• olur.
• Note Consider (i.e. no
friction). Then using a Taylor series expansion
of the exponential term in equation (14.18) leads
to
• which as gives
• .

(14.18)
(14.19)
(14.20)
10

14.1. Plane Strain Drawing
14. Slab Analysis of Bulk Forming Processes
• which is the answer also provided by the ideal
work method!! From this it can be realized that
the slab analysis method simply extends the
information provided by the ideal-work method to
include frictional effects.
• Example
•
• A sheet of metal having an initial thickness of
0.100 in. and width of 12 in. is to be drawn
through straight-sided dies having an included
angle of 300. If the average of the yield stress
is ksi and an average value for the
coefficient of friction is 0.08, calculate the
force needed to complete this operation for a
reduction of 10.
• Solution
• From where a is the
semi-die angle,

11

14.1. Plane Strain Drawing
14. Slab Analysis of Bulk Forming Processes
• The drawing force , so

(14.21)
(14.22)
(14.23)
(14.24)
(14.25)
ksi
lbf
(14.26)
12

14.2. Wire or Rod Drawing
14. Slab Analysis of Bulk Forming Processes
• In terms of principal stresses,
• where z is the main axis and the directions 2 2
and 3 3 are the hoop and radial directions.
Substitution of the above equation in the
von-Mises criterion, gives the following relation
between the stress components to initiate and
sustain plastic deformation (axisymmetric
problems)
• or
• For a wire or rod of circular cross section the
basic governing equation is
• so that,
• Integrating gives

(14.27)
(14.28)
(14.29)
where
(14.30)
13

14.2. Wire or Rod Drawing
14. Slab Analysis of Bulk Forming Processes
• where

(14.31)
(14.32)
Figure 14.3. Slab analysis for rod drawing.
14

14.3. Plane Strain Extrusion
14. Slab Analysis of Bulk Forming Processes
• We here list only the equations that are
different from those of plane strain drawing

Figure 14.4. Plane strain and axisymmetric
extrusion.
15

14.3. Plane Strain Extrusion
14. Slab Analysis of Bulk Forming Processes
• Axisymmetric extrusion
• Notes
• The equations above are for a non-hardening
material. In the case of a hardening material,
one can use the above equations (as an
approximation) by taking Y to be the mean yield
stress over the range of strain induced by the
shape change.
• It should be noted that these analyses become
unrealistic at high die angles and low
reductions. Assuming that P is a principal stress
is reasonable only if a is small and friction is
low.
• All slab analysis calculations do not account for
redundant (non-homogeneous) deformation.

(14.33)
(14.34)