Cold and Hot Forging: Fundamentals and Applications / Edited by Taylan Altan, Gracious Ngaile, Gangshu Shen

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12 / Cold and Hot Forging: Fundamentals and Applications

Fig. 2.5

Forward and backward extrusion processes. (a) Common cold extrusion processes (P, punch; W, workpiece; C, container;

E, ejector). [Feldman, 1977]. (b) Example of a component produced using forward rod and backward extrusion. Left to

right: sheared blank, simultaneous forward rod and backward cup extrusion, forward extrusion, backward cup extrusion, simultaneous

upsetting of ﬂange and coining of shoulder. [Sagemuller, 1968]

Fig. 2.6 Radial forging of a shaft Fig. 2.7 Hobbing. (a) In container. (b) Without restriction

Materials. Titanium alloys, aluminum alloys.

Process Variations. Closed-die forging with

or without ﬂash, P/M forging.

Application. Net- and near-net shape forg-

ings for the aircraft industry.

2.3.9 Open-Die Forging (Fig. 2.9)

Deﬁnition. Open-die forging is a hot forging

process in which metal is shaped by hammering

or pressing between ﬂat or simple contoured

dies.

Equipment. Hydraulic presses, hammers.

Materials. Carbon and alloy steels, aluminum

alloys, copper alloys, titanium alloys, all forge-

able materials.

Process Variations. Slab forging, shaft forg-

ing, mandrel forging, ring forging, upsetting be-

tween ﬂat or curved dies, drawing out.

Application. Forging ingots, large and bulky

forgings, preforms for ﬁnished forgings.

2.3.10 Orbital Forging (Fig. 2.10)

Deﬁnition. Orbital forging is the process of

forging shaped parts by incrementally forging

Forging Processes: Variables and Descriptions / 13

Fig. 2.8

Isothermal forging with dies and workpiece at ap-

proximately the same temperature

Fig. 2.9 Open-die forging

Fig. 2.10 Stages in orbital forging

Fig. 2.11 Powder metal (P/M) forging

Fig. 2.12 Upset forging

Application. Bevel gears, claw clutch parts,

wheel disks with hubs, bearing rings, rings of

various contours, bearing-end covers.

2.3.11 Powder Metal

(P/M) Forging (Fig.2.11)

Deﬁnition. P/M forging is the process of

closed-die forging (hot or cold) of sintered pow-

der metal preforms.

Equipment. Hydraulic and mechanical

presses.

Materials. Carbon and alloy steels, stainless

steels, cobalt-base alloys, aluminum alloys, ti-

tanium alloys, nickel-base alloys.

Process Variations. Closed-die forging with-

out ﬂash, closed-die forging with ﬂash.

Application. Forgings and ﬁnished parts for

automobiles, trucks, and off-highway equip-

ment.

2.3.12 Upsetting or Heading (Fig. 2.12)

Deﬁnition. Upsetting is the process of forg-

ing metal (hot or cold) so that the cross-sectional

area of a portion, or all, of the stock is increased.

Equipment. Hydraulic, mechanical presses,

screw presses; hammers, upsetting machines.

Materials. Carbon and alloy steels, stainless

steels, all forgeable materials.

Process Variations. Electro-upsetting, upset

forging, open-die forging.

(hot or cold) a slug between an orbiting upper

die and a nonrotating lower die. The lower die

is raised axially toward the upper die, which is

ﬁxed axially but whose axis makes orbital, spi-

ral, planetary, or straight-line motions.

Equipment. Orbital forging presses.

Materials. Carbon and low-alloy steels, alu-

minum alloys and brasses, stainless steels, all

forgeable materials.

Process Variations. This process is also

called rotary forging, swing forging, or rocking

die forging. In some cases, the lower die may

also rotate.

14 / Cold and Hot Forging: Fundamentals and Applications

Fig. 2.15 Ironing operation

Fig. 2.14 Coining operation

Fig. 2.13 Nosing of a shell

Application. Finished forgings, including

nuts, bolts; ﬂanged shafts, preforms for ﬁnished

forgings.

2.3.13 Nosing (Fig. 2.13)

Deﬁnition. Nosing is a hot or cold forging

process in which the open end of a shell or tu-

bular component is closed by axial pressing with

a shaped die.

Equipment. Mechanical and hydraulic

presses, hammers.

Materials. Carbon and alloy steels, aluminum

alloys, titanium alloys.

Process Variations. Tube sinking, tube ex-

panding.

Applications. Forging of open ends of am-

munition shells; forging of gas pressure contain-

ers.

2.3.14 Coining (Fig. 2.14)

Deﬁnition. In sheet metal working, coining

is used to form indentations and raised sections

in the part. During the process, metal is inten-

tionally thinned or thickened to achieve the re-

quired indentations or raised sections. It is

widely used for lettering on sheet metal or com-

ponents such as coins. Bottoming is a type of

coining process where bottoming pressure

causes reduction in thickness at the bending

area.

Equipment. Presses and hammers.

Materials. Carbon and alloy steels, stainless

steels, heat-resistant alloys, aluminum alloys,

copper alloys, silver and gold alloys.

Process Variations. Coining without ﬂash,

coining with ﬂash, coining in closed die, sizing.

Applications. Metallic coins; decorative

items, such as patterned tableware, medallions

and metal buttons; sizing of automobile and air-

craft engine components.

2.3.15 Ironing (Fig. 2.15)

Deﬁnition. Ironing is the process of smooth-

ing and thinning the wall of a shell or cup (cold

or hot) by forcing the shell through a die with a

punch.

Equipment. Mechanical presses and hydrau-

lic presses.

Materials. Carbon and alloy steels, aluminum

and aluminum alloys, titanium alloys.

Applications. Shells and cups for various

uses.

REFERENCES

[Altan et al., 1983]: Altan, T., Oh, S.-I., Gegel,

H.L., Metal Forming Fundamentals and Ap-

plications, ASM International, 1983.

[Feldman, 1977]: Feldman, H.D., Cold Extru-

sion of Steel, Merkblatt 201, Du¨sseldorf, 1977

(in German).

[Sagemuller, 1968]: Sagemuller, Fr., “Cold Im-

pact Extrusion of Large Formed Parts,” Wire,

No. 95, June 1968, p 2.

SELECTED REFERENCES

[Altan, 2002]: Altan, T., “The Greenﬁeld Coa-

lition Modules,” Engineering Research Cen-

Forging Processes: Variables and Descriptions / 15

ter for Net Shape Manufacturing, The Ohio

State University, 2002.

[ASM, 1989]: Production to Near Net Shape

Source Book, American Society for Metals

1989, p 33–80.

[ASM Handbook]: Forming and Forging, Vo l

14, ASM Handbook, ASM International,

1988, p 6.

[Kalpakjian, 1984]: Kalpakjian, S., Manufac-

turing Processes for Engineering Materials,

Addison-Wesley, 1984, p 381–409.

[Lange et al., 1985]: Lange, K., et al., Hand-

book of Metal Forming, McGraw-Hill, 1985,

p 2.3, 9.19.

[Lindberg, 1990]: Lindberg, Processes and Ma-

terials of Manufacture, 4th ed., Allyn and Ba-

con, 1990, p 589–601.

[Niebel et al., 1989]: Niebel, B.W.,

Draper, A.B., Wysk, R.A., Modern Manu-

facturing Process Engineering, 1989, p 403–

425.

[Schuler Handbook, 1998]: Schuler, Metal

Forging Handbook, Springer, Goppingen,

Germany, 1998.

[SME Handbook, 1989]: Tool and Manufac-

turers Engineering Handbook, Desk Edition

(1989), 4th ed., Society of Manufacturing En-

gineers, 1989, p 15-8.

CHAPTER 3

Plastic Deformation:

Strain and Strain Rate

Manas Shirgaokar

Gracious Ngaile

3.1 Introduction

The purpose of applying the plasticity theory

in metal forming is to investigate the mechanics

of plastic deformation in metal forming pro-

cesses. Such investigation allows the analysis

and prediction of (a) metal ﬂow (velocities,

strain rates, and strains), (b) temperatures and

heat transfer, (c) local variation in material

strength or ﬂow stress, and (d) stresses, forming

load, pressure, and energy. Thus, the mechanics

of deformation provide the means for determin-

ing how the metal ﬂows, how the desired ge-

ometry can be obtained by plastic forming, and

what are the expected mechanical properties of

the part produced by forming.

In order to arrive at a manageable mathemat-

ical description of the metal deformation, sev-

eral simplifying (but reasonable) assumptions

are made:

●

Elastic deformations are neglected. How-

ever, when necessary, elastic recovery (for

example, in the case of springback in bend-

ing) and elastic deﬂection of the tooling (in

the case of precision forming to very close

tolerances) must be considered.

●

The deforming material is considered to be

in continuum (metallurgical aspects such as

grains, grain boundaries, and dislocations are

not considered).

●

Uniaxial tensile or compression test data are

correlated with ﬂow stress in multiaxial de-

formation conditions.

●

Anisotropy and Bauschinger effects are ne-

glected.

●

Volume remains constant.

●

Friction is expressed by a simpliﬁed expres-

sion such as Coulomb’s law or by a constant

shear stress. This is discussed later.

3.2 Stress Tensor

Consider a general case, where each face of a

cube is subjected to the three forces F

, and

(Fig. 3.1). Each of these forces can be re-

solved into the three components along the three

coordinate axes. In order to determine the

stresses along these axes, the force components

are divided by the area of the face upon which

they act, thus giving a total of nine stress com-

ponents, which deﬁne the total state of stress on

this cuboidal element.

This collection of stresses is referred to as the

stress tensor (Fig. 3.1) designated as r

and is

expressed as:

rrr

xx yx zx

r ⳱ rrr

ij xy yy zy

冷冷

rrr

xz yz zz

A normal stress is indicated by two identical

subscripts, e.g., r

, while a differing pair indi-

cates a shear stress. This notation can be sim-

pliﬁed by denoting the normal stresses by a sin-

gle subscript and shear stresses by the symbol s.

Thus one will have r

⬅ r

and r

⬅ s

Cold and Hot Forging Fundamentals and Applications

Taylan Altan, Gracious Ngaile, Gangshu Shen, editors, p17-23

DOI:10.1361/chff2005p017

www.asminternational.org

18 / Cold and Hot Forging: Fundamentals and Applications

Fig. 3.1 Forces and the stress components as a result of the forces. [Hosford & Caddell, 1983]

In case of equilibrium, r

⳱ r

, thus im-

plying the absence of rotational effects around

any axis. The nine stress components then re-

duce to six independent components. A sign

convention is required to maintain consistency

throughout the use of these symbols and prin-

ciples. The stresses shown in Fig. 3.1 are con-

sidered to be positive, thus implying that posi-

tive normal stresses are tensile and negative ones

are compressive. The shear stresses acting along

the directions shown in Fig. 3.1 are considered

to be positive. The double sufﬁx has the follow-

ing physical meaning [Hosford & Caddell,

1983]:

●

Sufﬁx i denotes the normal to the plane on

which a component acts, whereas the sufﬁx

j denotes the direction along which the com-

ponent force acts. Thus r

arises from a

force acting in the positive y direction on a

plane whose normal is in the positive y di-

rection. If it acted in the negative y direction

then this force would be compressive instead

of tensile.

●

A positive component is deﬁned by a com-

bination of sufﬁxes where either both i and

j are positive or both are negative.

●

A negative component is deﬁned by a com-

bination of sufﬁxes in which either one of i

or j is negative.

3.3 Properties of the Stress Tensor

For a general stress state, there is a set of co-

ordinate axes (1, 2, and 3) along which the shear

stresses vanish. The normal stresses along these

axes, viz. r

, r

, and r

, are called the principal

stresses. Consider a small uniformly stressed

block on which the full stress tensor is acting in

equilibrium and assume that a small corner is

cut away (Fig. 3.2). Let the stress acting normal

to the triangular plane of section be a principal

stress, i.e., let the plane of section be a principal

plane.

The magnitudes of the principal stresses are

determined from the following cubic equation

developed from a series of force balances:

r ⳮ I r ⳮ I r ⳮ I ⳱ 0 (Eq 3.1)

i1i2i3

where

I ⳱ r Ⳮ r Ⳮ r

1xxyyzz

I ⳱ⳮrr ⳮ rr ⳮ rr Ⳮ r

2 xxyyyyzzzzxxxy

Ⳮ r Ⳮ r

yz zx

I ⳱ rrr Ⳮ 2rrr ⳮ rr

3 xxyyzz xyyzzx xxyz

ⳮ rr ⳮ rr

yy zx zx xy

In terms of principal stresses the above equa-

tions become:

I ⳱ r Ⳮ r Ⳮ r (Eq 3.1a)

1123

I ⳱ⳮ(rr Ⳮ rr Ⳮ rr) (Eq 3.1b)

2 122331

I ⳱ rrr (Eq 3.1c)

3123

The coefﬁcients I

, and I

are independent

of the coordinate system chosen and are hence

Plastic Deformation: Strain and Strain Rate / 19

Fig. 3.4

Cut at an arbitrary angle h in the x-y plane. [Hosford

& Caddell, 1983]

Fig. 3.3 Stresses in the x-y plane. [Hosford & Caddell, 1983]

Fig. 3.2

Equilibrium in a three-dimensional stress state.

[Backofen, 1972]

Fig. 3.5

Metal ﬂow in certain forming processes. (a) Non-

steady-state upset forging. (b) Steady-state extrusion.

[Lange, 1972]

called invariants. Consequently, the principal

stresses for a given stress state are unique. The

three principal stresses can only be determined

by ﬁnding the three roots of the cubic equation.

The invariants are necessary in determining the

onset of yielding.

3.4 Plane Stress or

Biaxial Stress Condition

Consider Fig. 3.1 with the nine stress com-

ponents and assume that any one of the three

reference planes (x, y, z) vanishes (Fig. 3.3). As-

suming that the z plane vanishes, one has r

⳱

⳱ s

and a biaxial state of stress exists. To

study the variation of the normal and shear stress

components in the x-y plane, a cut is made at

some arbitrary angle h as shown in Fig. 3.4, and

the stresses on this plane are denoted by r

and

r Ⳮ rrⳮ r

xyxy

r ⳱Ⳮcos 2h

Ⳮ s sin 2h (Eq 3.2)

r ⳮ r

s ⳱ⳮ sin 2h Ⳮ s cos 2h (Eq 3.3)

h xy

冢冣

The two principal stresses in the x-y plane are

the values of r

on planes where the shear stress

⳱ 0. Thus, under this condition,

tan 2h ⳱ (Eq 3.4)

r ⳮ r

20 / Cold and Hot Forging: Fundamentals and Applications

Fig. 3.6 Displacement in the x-y plane. [Altan et al., 1983]

Thus, with the values of sin 2h and cos 2h the

equation becomes

r , r ⳱ (r Ⳮ r )

12 x y

2 2 1/2

Ⳳ [(r ⳮ r ) Ⳮ 4s ] (Eq 3.5)

xy xy

To ﬁnd the planes where the shear stress s

maximum, differentiate Eq 3.3 with respect to h

and equate to zero. Thus,

2 2 1/2

s ⳱ [(r ⳮ r ) Ⳮ 4s ] (Eq 3.6)

max x y xy

3.5 Local Deformations and the

Velocity Field

The local displacement of the volume ele-

ments is described by the velocity ﬁeld, e.g., ve-

locities, strain rates, and strains (Fig. 3.5). To

simplify analysis, it is often assumed that the

velocity ﬁeld is independent of the material

properties. Obviously, this is not correct.

3.6 Strains

In order to investigate metal ﬂow quantita-

tively, it is necessary to deﬁne the strains (or

deformations), strain rates (deformation rates),

and velocities (displacements per unit time).

Figure 3.6 illustrates the deformation of an in-

ﬁnitesimal rectangular block, abcd, into a par-

allelogram, a⬘b⬘c⬘d⬘, after a small amount of

plastic deformation. Although this illustration is

in two dimensions, the principles apply also to

three-dimensional cases.

The coordinates of a point are initially x and

y (and z in three dimensions). After a small de-

formation, the same point has the coordinates x⬘

and y⬘ (and z⬘ in 3-D). By neglecting the higher-

order components, one can determine the mag-

nitude of the displacement of point b, u

,asa

function of the displacement of point a. This

value, u

, is different from the displacement of

point a, u

, about the variation of the function

over the length dx, i.e.,

⳵u

u ⳱ u Ⳮ dx (Eq 3.7)

bx x

⳵x

Note that u

also depends on y and z.

The relative elongation of length ab (which is

originally equal to dx), or the strain in the x di-

rection, e

, is now:

(u ⳮ u)

bx x

e ⳱

⳵u ⳵u

e ⳱ u Ⳮ dx ⳮ udx⳱ (Eq 3.8a)

xx x

冢冣

冫

⳵u ⳵x

Similarly, in the y and z directions,

⳵u ⳵u

e ⳱ ; e ⳱ (Eq 3.8b)

⳵y ⳵z

The angular variations due to the small de-

formation considered in Fig. 3.6 are inﬁnitesi-

mally small. Therefore, tan ␣

⳱ ␣

and tan

␣

⳱ ␣

. Thus:

␣ ⳱ (u ⳮ u )/(u Ⳮ dx ⳮ u ) (Eq 3.9)

xy by y bx x

The expression for u

is given in Eq 3.7, and

that for u

can be obtained similarly as:

⳵u

u ⳱ u Ⳮ dy (Eq 3.10)

by y

⳵y

Using Eq 3.7 and 3.10, and considering that

⳱ ⳵u

/⳵

is considerably smaller than 1, Eq

3.9 leads to:

⳵u

␣ ⳱ (Eq 3.11a)

⳵x

Plastic Deformation: Strain and Strain Rate / 21

and similarly,

⳵u

␣ ⳱ (Eq 3.11b)

⳵y

Thus, the total angular deformation in the xy

plane, or the shear strain, c

, is:

⳵u ⳵u

c ⳱ ␣ Ⳮ ␣ ⳱Ⳮ (Eq 3.12a)

xy xy yx

⳵x ⳵y

Similarly:

⳵u ⳵u

c ⳱Ⳮ (Eq 3.12b)

⳵z ⳵y

and

⳵u ⳵u

c ⳱Ⳮ (Eq 3.12c)

⳵x ⳵z

3.7 Velocities and Strain Rates

The distribution of velocity components (v

) within a deforming material describes the

metal ﬂow in that material. The velocity is the

variation of the displacement in time or in the x,

y, and z directions [Backofen, 1972, and Rowe,

1977].

⳵u ⳵u ⳵u

xyz

v ⳱ ;v ⳱ ;v ⳱ (Eq 3.13)

xyz

⳵t ⳵t ⳵t

The strain rates, i.e., the variations in strain

with time, are:

⳵e ⳵⳵(u ) ⳵⳵u ⳵v

xx xx

˙e ⳱⳱ ⳱ ⳱

冢冣

⳵t ⳵t ⳵x ⳵x ⳵t ⳵x

Similarly,

⳵v ⳵v ⳵v

xyz

˙e ⳱ ;˙e ⳱ ;˙e ⳱ (Eq 3.14a)

xyz

⳵x ⳵y ⳵z

⳵v ⳵v

˙c ⳱Ⳮ (Eq 3.14b)

⳵y ⳵x

⳵v ⳵v

˙c ⳱Ⳮ (Eq 3.14c)

⳵z ⳵y

⳵v ⳵v

˙c ⳱Ⳮ (Eq 3.14d)

⳵z ⳵x

The state of deformation in a plastically de-

forming metal is fully described by the displace-

ments, u, velocities, v, strains, e, and strain rates,

(in an x, y, z coordinate system). It is possible˙e

to express the same values in an x⬘,y⬘,z⬘ sys-

tem, provided that the angle of rotation from x,

y,ztox⬘,y⬘,z⬘ is known. Thus, in every small

element within the plastically deforming body,

it is possible to orient the coordinate system such

that the element is not subjected to shear but

only to compression or tension. In this case, the

strains c

, c

all equal zero, and the ele-

ment deforms along the principal axes of defor-

mation.

In uniaxial tension and compression tests (no

necking, no bulging), deformation is also in the

directions of the principal axes.

The assumption of volume constancy made

earlier neglects the elastic strains. This assump-

tion is reasonable in most forming processes

where the amount of plastic strain is much larger

than the amount of elastic strain. This assump-

tion can also be expressed, for the deformation

along the principal axes, as follows:

e Ⳮ e Ⳮ e ⳱ 0 (Eq 3.15)

xyz

and

˙e Ⳮ ˙e Ⳮ ˙e ⳱ 0 (Eq 3.16)

xyz

3.8 Homogeneous Deformation

Figure 3.7 considers “frictionless” upset forg-

ing of a rectangular block. The upper die is mov-

ing downward at velocity V

. The coordinate

axes x, y, and z have their origins on the lower

platen, at the center of the lower rectangular sur-

face.

The initial and ﬁnal dimensions of the block

are designated by the subscripts 0 and 1, respec-

tively. The instantaneous height of the block

during deformation is h. The velocity compo-

nents v

, and v

, describing the motion of

each particle within the deforming block, can be

expressed as the linear function of the coordi-

nates x, y, and z as follows:

Vx Vy Vz

DD D

v ⳱ ;v ⳱ ;v ⳱ⳮ (Eq 3.17)

xyz

2h 2h h

In order to demonstrate that the velocity ﬁeld

described by Eq 3.17 is acceptable, it is neces-

22 / Cold and Hot Forging: Fundamentals and Applications

Fig. 3.7

Homogeneous deformation in frictionless upset

forging

sary to prove that these velocities satisfy (a) the

volume constancy and (b) the boundary condi-

tions [Johnson et al., 1975].

Satisfaction of the boundary conditions can be

shown by considering the initial shape on the

block before deformation (Fig. 3.7). At the ori-

gin of the coordinates, all the velocities must be

equal to zero. This condition is satisﬁed because,

at the origin, for x ⳱ y ⳱ z ⳱ 0, one has, from

Eq 3.17, v

⳱ v

⳱ 0. At the boundaries:

At x ⳱ l

/2; velocity in the x direction:

v ⳱ V l /4h (Eq 3.18a)

xo D o o

At y ⳱ w

/2; velocity in the y direction:

v ⳱ V w /4h (Eq 3.18b)

yo D o o

At z ⳱ h

; velocity in the z direction:

v ⳱ⳮV (Eq 3.18c)

zo D

It can be shown easily that the volume con-

stancy is also satisﬁed. At the start of deforma-

tion, the upper volume rate or the volume per

unit time displaced by the motion of the upper

die is:

Volume rate ⳱ V w h (Eq 3.19)

Doo

The volumes per unit time moved toward the

sides of the rectangular block are:

2v h w Ⳮ 2v l h (Eq 3.20)

xo o o yo o o

Using the values of v

and v

given by the

Eq 3.18(a) and 3.18(b), Eq 3.20 gives:

Volume rate ⳱ 2h (w V l Ⳮ l V w )/4h

ooDo oDo o

(Eq 3.21a)

Volume rate ⳱ V w h (Eq 3.21b)

Doo

The quantities given by Eq 3.19 and 3.21 are

equal; i.e., the volume constancy condition is

satisﬁed. The strain rates can now be obtained

from the velocity components given by Eq 3.17.

⳵vV

˙e ⳱⳱ (Eq 3.22a)

⳵x2h

Similarly:

˙e ⳱ ;˙e ⳱ⳮ (Eq 3.22b)

2h h

It can be easily seen that:

˙c ⳱ ˙c ⳱ ˙c ⳱ 0

xy xz yz

In homogeneous deformation, the shear strain

rates are equal to zero. The strains can be ob-

tained by integration with respect to time, t.

In the height direction:

e ⳱ ˙e dt ⳱ⳮdt (Eq 3.23)

冮冮

For small displacements, dh ⳱ⳮV

dt. Thus,

Eq 3.23 gives:

dh h

e ⳱ e ⳱⳱ln (Eq 3.24a)

冮

Other strains can be obtained similarly:

e ⳱ e ⳱ ln ; e ⳱ e ⳱ ln (Eq 3.24b)

1x by

Volume constancy in terms of strains can be

veriﬁed from: