Effect of Temperature: At elevated temperatures, the rate of strain hardening (represented by the

strain hardening exponent) falls rapidly in most metals. The flow stress

and tensile strength, measured at constant strain and strain-rate, also drop

with increasing temperature.

FIGURE: Typical effects of

temperature on stress-strain curves.

Temperature affects the modulus of

elasticity, yield stress, ultimate

tensile strength, and toughness of

materials.

Decreasing the strain rate has the same effect on the flow stress as raising

the temperature but at much lower rates.

FIGURE: The effect of strain rate on the ultimate tensile strength of

aluminum. Note that as temperature increases, the slope increases thus,

tensile strength becomes more sensitive to strain rate as temperature

increases.

Hot Working:

Hot working takes advantage of the decrease in flow stress at high

temperature to lower tool forces, equipment size, and power

requirements.

Hot working is defined as working above the recrystallization

temperature so that the work metal recrystalizes as it deforms. The

resultant product is in an annealed state.

However, hot working has some undesirable effects:

1. Lubrication is more difficult. Many hot working processes are done

without lubrication.

2. The work metal tends to oxidize forming a scale layer that causes loss

of metal and roughened surfaces. Processing under inert atmosphere is

prohibitively expensive and is applied only in case of reactive metals

such as titanium.

3. Tool life is shortened because of heating. Sometimes scale breakers

are employed and rolls are cooled by water spray to minimize tool

damage.

4. Poor surface finish and loss of precise gauge control.

5. Lack of work hardening is undesirable where the strength level of a

cold worked product is needed.

Because of these limitations, it is common to hot roll steel to about 6 mm

thickness. The product is then pickled to remove scale, and further rolling

is done cold to ensure good surface finish, and optimum mechanical

properties.

Temperature rise during deformation:

The heat generated by mechanical work raises the temperature of the

metal during plastic deformation. As shown earlier,

w d

Only a small fraction of this energy is stored. This fraction drops from

5% initially to 1 or 2% at high strains. The rest is released as heat. If

deformation is adiabatic, i.e., no heat transfer to the surroundings, the

temperature is given by:

CC

dT a

Where a is the average value of over the strain interval 0 to , is

the density, C is the mass heat capacity, and is the fraction energy

stored as heat ( about 0.98).

Any temperature increase causes the flow stress to drop leading to

thermal softening, especially at high strain-rates where heat cannot be

transferred to surroundings.

Deformation work

One concern regarding forming operations is the prediction of externally

applied loads needed to cause the metal to deform (flow) to the required

shape.

Uncertainties: frictional effects, non-homogeneous deformation, and true

manner by which strain hardening occurs during complex deformations.

Due to these uncertainties, the exact values of force requirements are

seldom predictable.

Ideal work of deformation:

The process is assumed to be ideal in the sense that external work is

completely utilized to cause deformation only. Effects of friction and

non-homogeneous deformation are ignored. Assuming power-law for

strain- hardening materials:

dkdW n

i

Where )/ln(ln 00

AAl

l

.

1

1

n

KW

n

i

If other forms of work hardening or idealized relations are more

appropriate, they would be used.

E.g., when a mean flow stress σm is sometimes used over the range of

homogeneous strain:

miW , and by comparing with the previous equation:

11

nn

k n

m

Effect of friction, redundant work, and mechanical efficiency:

Frictional work per unit volume, fW is consumed at the interface between

the deforming metal and the tool faces that constrain the metal.

Redundant work rW is due to internal distortion in excess of that needed

to produce the required shape.

FIGURE: Deformation of grid patterns in a workpiece to compare ideal

deformation and inhomogeneous deformation with additional shearing,

which requires higher work (redundant)

Then the actual work is the sum of the three terms:

rfia WWWW

In general, it is difficult in practice to separate fW from rW since they are

not mutually independent. The deformation efficiency is then defined

as:

a

i

W

W

Often in practice, varies between 0.5 and 0.65.

Analysis of forming processes:

Several techniques have been developed for modeling of forming

processes for the purpose of understanding the pattern of metal flow,

calculation of load and energy requirements, and proper design of the

tooling required to carry out the process.

These techniques are classified as follows:

1. Slab analysis: or free body-equilibrium approach. This is the

simplest and most widely applied method.

2. Slip line field method: based on a presumed deformation field that is

geometrically consistent with the shape change.

3. Upper bound analysis: predicts a load that is at least equal to or greater

than the exact load needed to cause plastic flow.

4. Finite element analysis: based on numerical approach, and accordingly

gives near actual results.

Slab Analysis:

The method is based on carrying out a force balance on a slab of metal of

differential thickness in the deforming body. The resulting differential

equation is solved using pertinent boundary conditions. The technique

assumes homogeneous deformation, i.e. no internal distortion.

Direct compression in plane strain

Ideal deformation of a billet with circular or rectangular cross section

between flat platens, leads to a reduction in height and uniform increase

in area. Considering volume constancy, the area corresponding to any

height is calculated as:

(9.1)

However, lateral friction at the billet-platen interfaces leads to restriction

of material flow on these surfaces causing a barrel shape of the lateral

sides, and a consequent increase in the pressure required to carry out the

upsetting process as shown in Fig... The pressure ascends symmetrically

from the edges of the billet, reaching maximum value at the center,

forming what is known as the friction hill. The average pressure, and

accordingly the force required to carry out the upsetting process will

therefore increase due to friction.

FIGURE: (a) Ideal deformation of a solid cylindrical specimen

compressed between flat frictionless dies (upsetting). (b) Deformation in

upsetting with friction at the die-workpiece interfaces.

FIGURE: Grain flow lines in upsetting a solid steel cylinder at elevated

temperatures. Note the highly inhomogenous deformation and barreling.

The different shape of the bottle, section of the specimen (as compared

with the top) results from the hot specimen resting on the lower, cool die

before deformation proceeded. The bottom surface was chilled; thus it

exhibits greater strength and hence deforms less than the top surface.

The following Fig. represents a billet with rectangular section under

compression through platens. The billet width is a, and height is h. The

figure shows also the stresses on a vertical element in that section. As

discussed earlier, plane strain condition assumes that the thickness of the

section in the third direction (b) is much larger than the side (a), such that

there is no strain in that thickness, i.e. ( h<a<<b). This condition

resembles actual open-die forging of long shafts or blooms between flat

platens.

h

dx

a/2

x

y

σy

σy = P

µσy

µσy

σxσx + dσx

σy

σy

σxσx + dσx

dx

b

σx + σy

2σz=

Fig. .. Stresses on an element in plane-strain upsetting between flat dies

Considering a force balance on the element (slab) in the x direction:

0)()(2 hddxph xxx

xhdpdx 2

x & y (taken as –p) are principal stresses. For plane strain

deformation, the flow stress (σf) is calculated from the Eq. (σ1 – σ3 = σf) ,

and considering the maximum difference between the two yield criteria

as stated earlier, then; ffx p 15.1)( ' ,

Where '

f is the flow yield stress under plane strain = 1.15 σf.

Considering the flow stress to be constant (perfect plastic material),

differentiation of the above equation gives: dσx = - dp

hdppdx 2 or dxhp

dp 2

At 2/ax , 0x , and '

fp , so the solution is:

)29(2

2exp

'

x

a

h

p

f

The maximum value of p occurs at the center line, with a maximum

value: )3.9(exp

max

'

h

ap

f

A plot of '/ fp versus x gives the friction hill as shown in Fig..a.

a

h

a/2

x

h

0

1

a) Sliding friction

a/2

xb) Sticking friction

0

1

p/σ'f

σx/σ'f1+

e

µha

2h

Fig. .. Pressure distribution and horizontal stress distribution in plane

strain upsetting with sliding (a) and sticking (b) friction

To calculate σx : )4.9(1)

2(

2

''

xa

hffx ep

Average pressure

The mean or average pressure is of great interest as it can be used to

calculate the applied force on the contact area.

dxea

pdxa

px

a

h

a

f

a

a

)2

(22

0

'2

0

22

)5.9(2

1'

h

ap fa

Then the upsetting force is given by:

Sticking friction at the interface

To avoid shearing of the workpiece at the billet-platen interface, fp ,

as explained earlier, where f is the shear yield strength of the billet

material (equal to half the flow stress). Since, 12

f

p

, thus 5.0 if

sliding friction is to take place. If the limit at which sliding friction is

exceeded, sticking takes place. To calculate the pressure distribution

under sticking, the frictional forces indicated previously as p is replaced

by the shear yield strength '5.0 ff in the previous analysis, which

leads to:

)6.9(2/

1' h

xap

f

This equation represents a linear variation of p with x, i.e. linear friction

hill. The maximum value which occurs at the centerline is:

)7.9()2

1('

maxh

ap f

A plot of the pressure distribution under sticking friction is shown in Fig.

..b. for comparison with the case of sliding friction. The average pressure

with sticking friction is:

)8.9()4

1('

h

aP fa

Since pressure is maximum at the center, then sticking starts primarily at

the center of the interface area and spreads outwards. However, there

could be a case where the outer parts may still be under sliding friction.

The point of intersection x1 between the exponential relation represented

by Eq. 9.2, and the linear relation represented by Eq. 9.6, is the point

separating the sticking and the sliding area, where:

)9.9()2

1ln(

221

hax

Pressure distributions

at rectangular

sections: FIGURE: Normal stress

(pressure) distribution in

the compression of a rectangular workpiece with sliding friction under

conditions of plane stress, using the distortion-energy criterion. Note that the

stress at the corners is equal to the uniaxial yield stress, Y.

FIGURE: Increase in contact area of a rectangular specimen (viewed

from the top) compressed between flat dies with friction. Note that the

length of the specimen has increased proportionately less than its width.

Forging of a solid cylindrical workpiece:

The upsetting of a cylindrical workpienc is modeled using the same

analysis followed with a rectangular specimen in the previous section.

However, cylindrical coordinates are applied as shown in Fig. 9.16, using

stress components σr in the radial direction, σө in the tangential direction,

and σz in the axial direction.

r = d2

r dr

h

dθ

dθ

2

σr

σθ

µσz

dθ

σr + dσr

σz

σz

σθ

dθ

2

µσz

Fig. Stresses on an element during upsetting of a cylindrical specimen

Following a similar approach for the case of plane strain, a similar

equation is obtained:

0)()(2

2Pr2

ddrrhd

hdrddrdhrd rrr

0Pr2 rr hrddrhdrhdr

For axisymmetric flow, r , then; r

For yielding , frz , back substitution gives the pressure

distribution:

)10.9()

2(

2x

d

h

f

ep

Notice the similarity with Eq. 9.2, and that the flow stress is used in Eq.

9.10 directly without multiplying by 1.15. This is indicative that in the

case of axial symmetry, the two yield criteria coincide.

The average pressure: )11.9()3

21(

h

rp fa

The upsetting force is: )4/( 2dpF a

The value of the coefficient of friction can be estimated to be 0.05 to

0.1 for cold forging, and 0.1 to 0.2 for hot forging.

Under sticking friction the stress distribution again is linear:

FIGURE: Ratio of average die pressure to yield stress as a function of

friction and aspect ratio of the specimen: (a) plane-strain compression;

and (b) compression of a solid cylindrical specimen.

Notice the similarity of equations for both plane strain and cylindrical

billets, and same trends of the results representing change of the upsetting

)12.9())2/(

1(h

xdp f

pressure with the width/height (aspect) ratio at both conditions as shown

in Fig. 9.17. It should be observed that:

higher aspect ratios lead to higher pressures at the same frictional

conditions.

that the pressures are higher for a plane-strain specimen compared

to a cylindrical specimen with the same aspect ratio, and the same

frictional conditions.

Impression-die forging:

FIGURE: Schematic illustration of stages in impression-die forging.

Note the formation of flash, or excess material that is subsequently

trimmed off.

Accurate calculation of forces in impression-die forging is difficult. To

simplify force calculation, a pressure-multiplying factor Kp is

recommended:

Where F is the forging load, A is the projected area of the forging

(including the flash), and σf is the flow stress of the material.

Typical Kp ranges are shown in the following table:

Forging Shape Kp Kh

Simple shapes, without flash Simple shapes, with flash Complex shapes, with flash

3-5 5-8 8-12

2 – 2.5 3 4

The capacity of the press to be used should have a rated maximum force

well above the estimated load.

During forging under hot working conditions, the strain rate affects the

stresses required to carry out the process. The strain rate is calculated as

given:

1

0

11

0. 1)ln(

h

v

dt

dh

hh

h

dt

d

dt

d

Where v is the relative velocity between the platens.

To obtain an approximate estimate of the strain rate in impression-die

forging:

V

Av

h

v

m

m .

Where V is the volume of the metal, and A is the projected area.

Similarly, an average strain may be estimated as:

)ln()ln( 0

)

0

V

Ah

h

h

m

m

As hammers are rated according to their energy, the energy required for

forging is calculated approximately from the equation:

hmf VKE

Where E is the forging energy, and Kh is the multiplying factor for

hammers obtained from the previous table. The rated hammer capacity

should be greater than the estimated energy.

In hot forging forces are much reduced with heated dies, but these

preheated dies require very slow forging speeds.

Ring compression test for determination of µ during forming:

The ring-compression test is an experimental method used for the

determination of frictional conditions in bulk metal forming. The concept

of the test is the increasing or decreasing of the inner diameter of a short

ring specimen when it is compressed between two flat, parallel platens. If

the friction is low (good lubrication) the internal diameter increases;

while if the friction is high (poor lubrication) the internal diameter is

decreases as shown in Figure

Low Friction(Good Lubrication) High Friction(Poor Lubrication)