13. The Ideal Work Method for the Analysis of Forming Processes e-mail: Assoc.Prof.Dr. Ahmet Zafer Şenalp e-mail: [email protected]@gmail.com.

13. The Ideal Work Method for the13. The Ideal Work Method for theAnalysis of Forming ProcessesAnalysis of Forming Processes

Assoc.Prof.Dr. Ahmet Zafer Şenalpe-mail: e-mail: [email protected]@gmail.com

Mechanical Engineering DepartmentGebze Technical University

ME 612ME 612 Metal Forming and Theory of Plasticity Metal Forming and Theory of Plasticity

mailto:[email protected]

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

2Mechanical Engineering Department, GTU

13. The Ideal Work Method for the13. The Ideal Work Method for the

Analysis of Forming ProcessesAnalysis of Forming Processes




Analysis of Forming ProcessesAnalysis of Forming Processes13.1. 13.1. Axisymmetric Extrusion and DrawingAxisymmetric Extrusion and Drawing

Figure 13.1 Illustration of direct or forward extrusion assuming ideal deformation.

Let us consider axisymmetric extrusion (Fig 13.1) where the diametral area is reduced from A0 to Af . The ideal work is

Here

and r is the percent area reduction:

The final axial strain is usually called the homogeneous strain and denoted as

Assuming we finally can write:





(13.1)

(13.2)

fε

0

i εdσw

r1

1ln

A

Aln

f

0axialf

%100A

AAr

0

f0

h

r1

1lnhaxialf

nK

1n

K

1n

Kdw

1nh

1nf

0

i

f

Note that if there is no hardening (n = 0 and ),

The external work (actual work) applied; or per unit volume:

Where Pe is the applied extrusion pressure. For an ideal process:

In reality:





(13.3)

Yhfi YYw

W eFW

e0

e

0

PA

F

A

Ww

(13.4)

iww

1n

K

1n

KdP

1nh

1nf

0

e

f

(13.5)

1n

K

1n

KdP

1nh

1nf

0

e

f

(13.6)

Similar results can be obtained for rod or wire drawing (Figure 13.2). The external work/volume in drawing is

and so in general we have:

Where is the applied drawing stress.





(13.7)

df

da A

Fw

1n

K

1n

Kd

1nh

1nf

0

d

f

d

Figure 13.2. Illustration of rod or wire drawing.

The actual work:

and are usually combined. We define the mechanical efficiency as follows:

The efficiency is a function of the die, lubrication, reduction rate, etc; , Usually




Analysis of Forming ProcessesAnalysis of Forming Processes13.2. 13.2. Friction, Redundant Work and EfficiencyFriction, Redundant Work and Efficiency

(13.8)

rfia wwww fw

rw

w

w i

65.05.0

Figure 13.3. Comparison of ideal and actual deformation to illustrate the meaning of redundant deformation.

Generalizing the formulas given above for the extrusion pressure and drawing stress, we can write the following:





(13.9)

)1n(

K

)1n(

Kd

P1n

h1n

f0e

f

)1n(

K

)1n(

Kd 1n

h1n

f0d

f

(13.10)

Figure 13.4. The stress-strain behavior is depicted in (c), the metal obeying is to be considered as the true stress needed to reduce to ( is the corresponding true strain).





nK 1 0D fD 1

As shown in Fig 13.4.(a) A round rod of initial diameter, can be reduced to diameter by pulling through a conical die with a necessary load, 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” with the tensile stress

) needed to produce equivalent reductions.For drawing we showed that:

For tension:

From the two equations above:




Analysis of Forming ProcessesAnalysis of Forming ProcessesExample:Example:

0D

dFfD

d

1

)1n(

K 1nh

d

(13.11)

nht K (13.12)

1nh

t

d

(13.13)

But, (strain at ultimate load – max strain to avoid necking). So finally:

Also,

Then,




Analysis of Forming ProcessesAnalysis of Forming ProcessesExample:Example:

(13.14)

(13.15)

nh

11n

n

1nh

t

d

2fdd D

4

nF

2ftt D

4

nF

1F

F

t

d

t

d




Analysis of Forming ProcessesAnalysis of Forming Processes13.3. 13.3. Maximum Drawing Reduction Maximum Drawing Reduction iin Axisymmetric Drawingn Axisymmetric Drawing

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.*

With greater reduction the drawing stress; increases. Its value can’t be higher than the yield stress of the material at the exit. From the previous analysis

The maximum possible value of is , where we denote as the final axial strain corresponding to maximum reduction.

From the above equations

From here

with





d

)1n(

K 1nh

d

(13.16)

d n*fK

max*h*f r1

1ln

)1n(

KK

1n

hnh

*

(13.17)

)1n(*h

1n

*f

0

*f

0*h e

A

A

A

Aln

and maximum reduction per pass:

For (perfect drawing) the maximum reduction is given asand for n=0 (perfectly plastic material – no hardening) we have that:





(13.18))1n(

0

fmax e1

A

A1r

*

1%63e1r 1

max

1nmax e1r





Figure 13.6. Influence of semi-die angle on the actual work; during drawing where the individual contributions of ideal , frictional, and redundant work are shown

awiw fw rw





Figure 13.7. Effect of semi-die angle on drawing efficiency for various reductions; note the change in the optimal die angle, *

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:

Yield condition: where Y.S. is the yield stress of the material at any location in the deformation zone.




Analysis of Forming ProcessesAnalysis of Forming Processes13.4. 13.4. Plane StraPlane Straiin Extrusn Extrusiion And Drawon And Drawiingng

Figure 13.8. Plane strain drawing.

S.Y3

2px

Equivalent strain:

The above changes will modify the final results as follows:Plane strain extrusion:Extrusion Pressure:

where, with the homogeneous strain





x3

2

0e wt

FP

f

0

ie d

1wwP

(13.19)

hf3

2

r1

1lnh

0

f0

t

ttr

For (rigid plastic material):

For (power law hardening):

Plane strain drawing:Drawing Stress:





(13.20)

Y

h

f

e3

2Y

YP

nK

)1n(3

2K

)1n(

KP

1n

h1nf

e

fd wt

F

f

0

id d

1ww

where, with the homogeneous strain (x-strain)

For (rigid plastic material):

For (power law hardening):





hf3

2

r1

1lnh

0

f0

t

ttr

Y

hf

d3

2Y

Y

nK

)1n(3

2K

)1n(

K

1n

h1nf

d

For max reduction:

from which we finally conclude that:

Note that the max reduction is the same for both plane strain and axially symmetric problems.





(13.21)3

2d (yield stress at exit)

n

h3

2K

3

2

)1n(max e1r (13.22)

13. The Ideal Work Method for the Analysis of Forming Processes e-mail: Assoc.Prof.Dr. Ahmet Zafer Şenalp e-mail: [email protected]@gmail.com.

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