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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)
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