ME 612
Metal Forming and Theory of Plasticity
13. The Ideal Work Method for the
Analysis of Forming Processes
Assoc.Prof.Dr. Ahmet Zafer Şenalp
e-mail: [email protected]
Mechanical Engineering Department
Gebze Technical University
13. The Ideal Work Method for the
Analysis of Forming Processes
In general the prediction of external forces needed to cause metal flow is needed.
Such prediction is difficult due to uncertainties introduced from frictional effects and
non-homogeneous deformation as well as from not knowing the true manner of strain
hardening.
Each solution method is based on several assumptions. The easiest method is the
ideal work method. The work required for deforming the workpiece is equated to the
external work. The process is considered ideal in the sense that the external work is
completely utilized to cause deformation only.
Friction and non-homogeneous deformation are neglected.
Dr. Ahmet Zafer Şenalp
ME 612
Mechanical Engineering Department,
GTU
2
13.1. Axisymmetric Extrusion and Drawing
13. The Ideal Work Method for the
Analysis of Forming Processes
Figure 13.1 Illustration of direct or forward extrusion assuming ideal deformation.
Dr. Ahmet Zafer Şenalp
ME 612
Mechanical Engineering Department,
GTU
3
13.1. Axisymmetric Extrusion and Drawing
13. The Ideal Work Method for the
Analysis of Forming Processes
Let us consider axisymmetric extrusion (Fig 13.1) where the diametral area is reduced
from A0 to Af . The ideal work is
wi 
εf
 σdε
(13.1)
0
Here
   axial  ln
f
A0
1
 ln
Af
1 r
and r is the percent area reduction:
r
A0  Af
100%
A0
The final axial strain is usually called the homogeneous strain and denoted as  h
1
 axial f   h  ln
1 r
n
Assuming   K we finally can write:
f
n 1
n 1
K f
K h
w i    d 

n 1
n 1
0
Dr. Ahmet Zafer Şenalp
ME 612
(13.2)
Mechanical Engineering Department,
GTU
4
13.1. Axisymmetric Extrusion and Drawing
Note that if there is no hardening (n = 0 and 
w i  Y f  Y h
13. The Ideal Work Method for the
Analysis of Forming Processes
 Y ),
The external work (actual work) applied; W
W  Fe 
(13.3)
or per unit volume:
w 
W
F 
 e
 Pe
A 0  A 0 
(13.4)
Where Pe is the applied extrusion pressure. For an ideal process:
w  w i
f
n 1
n 1
K f
K h
Pe    d 

n 1
n 1
0
(13.5)
In reality:
f
n 1
n 1
K f
K h
Pe    d 

n 1
n 1
0
Dr. Ahmet Zafer Şenalp
ME 612
(13.6)
Mechanical Engineering Department,
GTU
5
13.1. Axisymmetric Extrusion and Drawing
13. The Ideal Work Method for the
Analysis of Forming Processes
Similar results can be obtained for rod or wire drawing (Figure 13.2).
The external work/volume in drawing is
wa 
Fd
 d
Af
and so in general we have:
f
n 1
n 1
K f
K h
 d    d 

n 1
n 1
0
(13.7)
Where  d is the applied drawing stress.
Figure 13.2. Illustration of rod or wire drawing.
Dr. Ahmet Zafer Şenalp
ME 612
Mechanical Engineering Department,
GTU
6
13.2. Friction, Redundant Work and Efficiency
13. The Ideal Work Method for the
Analysis of Forming Processes
The actual work:
wa  wi  wf  w r
w f and w r are usually combined. We define the mechanical efficiency

wi
w
 as follows:
(13.8)
is a function of the die, lubrication, reduction rate, etc; ,
The efficiency 
0.5    0.65
Usually
Figure 13.3. Comparison of ideal and actual deformation to illustrate the meaning of redundant
deformation.
Dr. Ahmet Zafer Şenalp
ME 612
Mechanical Engineering Department,
GTU
7
13.2. Friction, Redundant Work and Efficiency
13. The Ideal Work Method for the
Analysis of Forming Processes
Generalizing the formulas given above for the extrusion pressure and drawing stress,
we can write the following:
f
Pe 
  d
0

n 1
n 1
K f
K h


(n  1) (n  1)
(13.9)
f
d 
  d
0

n 1
n 1
K f
K h


( n  1) (n  1)
Dr. Ahmet Zafer Şenalp
ME 612
(13.10)
Mechanical Engineering Department,
GTU
8
13.2. Friction, Redundant Work and Efficiency
13. The Ideal Work Method for the
Analysis of Forming Processes
Figure 13.4. The stress-strain behavior is depicted in (c), the metal obeying   K
 1 is to be considered as the true stress needed to reduceD 0 toD f ( 1 is the
corresponding true strain).

Dr. Ahmet Zafer Şenalp
ME 612
Mechanical Engineering Department,
GTU
9
n
13. The Ideal Work Method for the
Analysis of Forming Processes
Example:
As shown in Fig 13.4.(a) A round rod of initial diameter, D 0can be reduced to diameter
D f by pulling through a conical die with a necessary load,Fd as shown in sketch 13.4(a).
A similar result can occur by applying a uniaxial tensile load, as shown in sketch
13.4(b). Using the ideal-work method for both the drawing and tensile operations,
compare the load Fd with the load F1 (or the “drawing stress”  d with the tensile stress
 1 ) needed to produce equivalent reductions.
For drawing
we showed that:
n 1
d 
K h
(n  1)
(13.11)
For tension:
 t  K h n
(13.12)
From the two equations above:
d

 h
 t n 1
(13.13)
Dr. Ahmet Zafer Şenalp
ME 612
Mechanical Engineering Department,
GTU
10
13. The Ideal Work Method for the
Analysis of Forming Processes
Example:
But,  h  n (strain at ultimate load – max strain to avoid necking). So finally:
d

n
 h 
1
 t n 1 n 1
Also,
n 2
Fd   d D f
4
n 2
Ft   t D f
4
Then,
 d Fd
 1
 t Ft
Dr. Ahmet Zafer Şenalp
ME 612
(13.14)
(13.15)
Mechanical Engineering Department,
GTU
11
13.3. Maximum Drawing Reduction
in Axisymmetric Drawing
13. The Ideal Work Method for the
Analysis of Forming Processes
Figure 13.5. The tensile stress-strain curve and the drawing stress-strain behavior for two levels of
deformation efficiency. The intersection points,  * , are the limit strains in drawing.
Dr. Ahmet Zafer Şenalp
ME 612
Mechanical Engineering Department,
GTU
12
13.3. Maximum Drawing Reduction
in Axisymmetric Drawing
13. The Ideal Work Method for the
Analysis of Forming Processes
With greater reduction the drawing stress;  d increases. Its value can’t be higher
than the yield stress of the material at the exit. From the previous analysis
n 1
K h
d 
(n  1)
(13.16)
The maximum possible value of  d is K fn* , where we denote as
1
 f *   h*  ln
the final axial strain corresponding to maximum reduction.
1  rmax
From the above equations
K h 
n
K h *
n 1
( n  1)
(13.17)
From here
 h*   (n  1)
with
A
A
 h*  ln 0  0  e n 1
Af *
Af *
Dr. Ahmet Zafer Şenalp
ME 612
Mechanical Engineering Department,
GTU
13
13.3. Maximum Drawing Reduction
in Axisymmetric Drawing
13. The Ideal Work Method for the
Analysis of Forming Processes
and maximum reduction per pass:
rmax  1 
Af *
A0
 1  e  ( n 1)
(13.18)
For   1 (perfect drawing) the maximum reduction is given as rmax  1  e  n 1
and for n=0 (perfectly plastic material – no hardening) we have that: rmax  1  e 1  63%
Dr. Ahmet Zafer Şenalp
ME 612
Mechanical Engineering Department,
GTU
14
13.3. Maximum Drawing Reduction
in Axisymmetric Drawing
13. The Ideal Work Method for the
Analysis of Forming Processes
Figure 13.6. Influence of semi-die angle on the actual work; w a during drawing where the individual
contributions of ideal , w i frictional, w f and redundant work w r are shown
Dr. Ahmet Zafer Şenalp
ME 612
Mechanical Engineering Department,
GTU
15
13.3. Maximum Drawing Reduction
in Axisymmetric Drawing
13. The Ideal Work Method for the
Analysis of Forming Processes
Figure 13.7. Effect of semi-die angle on drawing efficiency for various reductions; note the change in
*
the optimal die angle, 
Dr. Ahmet Zafer Şenalp
ME 612
Mechanical Engineering Department,
GTU
16
13.4. Plane Strain Extrusion And Drawing
13. The Ideal Work Method for the
Analysis of Forming Processes
Figure 13.8. Plane strain drawing.
The calculations and previous definitions are applicable to plane strain problems with
only minor modifications. The differences arise from the new form of the yield
condition and the new expression for the equivalent strain. They are as follows:
2
Yield condition:  x  p 
Y.S where Y.S. is the yield stress of the material at any
3
location in the deformation zone.
Dr. Ahmet Zafer Şenalp
ME 612
Mechanical Engineering Department,
GTU
17
13.4. Plane Strain Extrusion And Drawing
13. The Ideal Work Method for the
Analysis of Forming Processes
Equivalent strain:
2

x
3
The above changes will modify the final results as follows:
Plane strain extrusion:
Extrusion Pressure:
Pe 
F
wt 0
Pe  w  
wi
r
f
 d

(13.19)
0
f 
where,
 h  ln


1
1
1 r
2
h
3
with the homogeneous strain
t0  tf
t0
Dr. Ahmet Zafer Şenalp
ME 612
Mechanical Engineering Department,
GTU
18
13.4. Plane Strain Extrusion And Drawing
For
Pe 
For
 Y
Y f

Y

13. The Ideal Work Method for the
Analysis of Forming Processes
(rigid plastic material):
2
h
3

  K n (power law hardening):
 2

K
h 

n 1
K f
3 
Pe 
 
 (n  1)
 (n  1)
n 1
Plane strain drawing:
Drawing Stress:
F
d 
wt f
 d  w 
wi


1

Dr. Ahmet Zafer Şenalp
ME 612
f
  d
(13.20)
0
Mechanical Engineering Department,
GTU
19
13.4. Plane Strain Extrusion And Drawing
f 
where,
 h  ln
For
2
 h with the homogeneous strain (x-strain)
3
1
1 r
r
t0  tf
t0
  Y (rigid plastic material):
d 
For
13. The Ideal Work Method for the
Analysis of Forming Processes
Y f

Y

2
h
3

  K n (power law hardening):
 2

K


n 1
h
K f
3

d 
 
 (n  1)
 (n  1)
Dr. Ahmet Zafer Şenalp
ME 612
n 1
Mechanical Engineering Department,
GTU
20
13.4. Plane Strain Extrusion And Drawing
13. The Ideal Work Method for the
Analysis of Forming Processes
For max reduction:
n
2
2  2

 d  (yield stress at exit)  K  h 
3  3 
3
(13.21)
from which we finally conclude that:
rmax  1  e  ( n 1)
(13.22)
Note that the max reduction is the same for both plane strain and axially symmetric
problems.
Dr. Ahmet Zafer Şenalp
ME 612
Mechanical Engineering Department,
GTU
21