Fundamentals of Metal Forming

TALAT Lecture 3300

Fundamentals of Metal Forming
18 pages, 27 figures

Basic Level

prepared by Klaus Siegert and Eckart Dannenmann, Institut für Umformtechnik,
Universität Stuttgart

Objectives:
− a brief review of the fundamental terms and laws governing metal forming at
room temperature as well as at high temperatures. This lecture is a necessary
prerequisite to understand the more specific treatment of metal forming
subjects such as forging, impact extrusion and sheet metal forming in the
subsequent TALAT Lectures 3400 to 3800.

Prerequisites:
− General background in production engineering, machine tools

Date of Issue: 1996
 EAA - European Aluminium Association

3300 Fundamentals of Metal Forming

Table of Contents

3300 Fundamentals of Metal Forming ............................................................2
3301 Introduction................................................................................................... 3
3301.01 Definition of Forming ..............................................................................3
3302 Terms for Classifying Forming Processes .................................................. 4
3302.01 Classification by State of Stress (Figure 3302.01.01)..............................4
3302.02 Classification by Type of Raw Material (Figure 3302.02.01) .................4
3302.03 Classification by Forming Temperature...................................................5
3302.04 Classification by Methods of Induction of Forces into the Work-Piece..5
3303 Characteristic Values and Basic Laws of Metal Forming......................... 6
3303.01 Flow Stress...............................................................................................6
3303.02 Plastic Strain, Rate and Acceleration.......................................................7
Logarithmic (True) Plastic Strain....................................................................... 7
Logarithmic Strain in Upsetting ......................................................................... 7
Law of Volume Constancy .................................................................................. 8
Plastic Strain Rate .............................................................................................. 8
Plastic Strain Acceleration ................................................................................. 9
3303.03 Plastic Flow under Combined Stresses ....................................................9
Criteria for Plastic Flow..................................................................................... 9
Maximum Shear Stress Hypothesis................................................................... 10
Mises Flow Criterion ........................................................................................ 11
Yield Criteria for Plane Stress (Yield Locus) ................................................... 12
3303.04 Law of Plastic Flow ...............................................................................12
3303.05 Flow Curves .............................................................................................13
General Definition of the Flow Curve .............................................................. 13
Flow Curves at Room Temperature.................................................................. 13
Flow Curves at Elevated Temperatures............................................................ 14
3303.06 Average Flow Stress ..............................................................................16
3303.07 Energy Considerations ...........................................................................16
Forming Energy ................................................................................................ 16
Heat Development during Forming .................................................................. 17
Literature/References ............................................................................................. 18
List of Figures.......................................................................................................... 18

TALAT 3300 2

3301 Introduction

3301.01 Definition of Forming

The general description given in Figure 3301.01.01 defines metal forming by plastic
deformation and distinguishes it from other forming and shaping processes like casting
and machining.

Definition of Forming

Forming is a fabrication process for solid substances by controlled
plastic deformation in order to obtain alterations of:

- the form,
- the material properties and/or
- the surface properties,

whereby the mass and material continuum remain unchanged.

K.Siegert

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Training in Aluminium Application Technologies
Definition of Forming 3301.01.01

TALAT 3300 3

3302 Terms for Classifying Forming Processes

• Classification by State of Stress
• Classification by Type of Raw Material
• Classification by Forming Temperature
• Classification by Methods of Induction of Forces into the Work-Piece

3302.01 Classification by State of Stress (Figure 3302.01.01)

Terms for classifying the forming process:
Classification by State of Stress
Rolling, Open-Die Forming,
Pressure Forming DIN 8583 Die Forming, Indenting,
Pressing Through

Tension-Compression Drawing, Deep Drawing,
Forming DIN 8584 Collar Forming, Compressing,
Upset Bulging

Tension Forming DIN 8585 Stretch Reducing, Bulge Forming,
Stretch Forming

Bending with Linear Tool Movement,
Bend Forming DIN 8586
Bending with Rotating Tool Movement

Shear Forming DIN 8587 Shear Displacement, Twisting

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Classifying the Forming Process by State of Stress 3302.01.01

3302.02 Classification by Type of Raw Material (Figure 3302.02.01)

Type of Raw Material
Sheet Forming

The raw material consists of flat parts of constant thickness.

The parts produced have a three-dimensional form with
approximately constant wall thickness ( ≈ raw material thickness).

Bulk Forming
Raw parts are three-dimensional.

The parts produced are also three-dimensional but often have
very different wall thicknesses and/ or cross-sections.

alu Classifying the Forming Process by
3302.02.01
Training in Aluminium Application Technologies Types of Raw Materials

TALAT 3300 4

3302.03 Classification by Forming Temperature
(Figure 3302.03.01)

Forming Temperature

Cold Forming
Process in which the work-piece is not heated before forming,
but instead formed at room temperature.
ϑ = ϑRoom (ca. 20° C )

Warm Forming
Process in which the work-piece is heated to an elevated temperature
higher than room temperature before forming.
ϑ > ϑ Room (ca. 20° C )

Training in Aluminium Application Technologies Forming Temperature 3302.03.01

3302.04 Classification by Methods of Induction of Forces into the Work-Piece
(Figure 3302.04.01)

Methods of Applying Force
Process with Indirect Force Application Process with Direct Force Application
(Deep Drawing) (Upsetting) F
FN FSt

Forming Zone

Bending Zone

Force Transmission Zone

Force Induction Zone

K. Siegert F
Training in Aluminium Application Technologies Methods of Applying Force 3302.04.01

TALAT 3300 5

3303 Characteristic Values and Basic Laws of Metal Forming
• Flow Stress
• Plastic Strain, Rate and Acceleration
• Plastic Flow under Combined Stresses
• Law of Plastic Flow
• Flow Curves
• Average Flow Stress
• Energy Considerations

3303.01 Flow Stress
(Figure 3303.01.01)

Characteristic Values, Basic Laws
Flow Stress
F
A
The flow stress (resistance to plastic deformation) of a
material is the stress required to initiate or continue
plastic deformation under uniaxial state of stress. A0

F
Flow stress kf =
(true stress) A l0

By comparison F l
σ=
(technical stress) A0

Conversion:
ε = (l − l0 ) / l0
with
one obtains k f = σ (ε l + 1)

F

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Definition of Flow Stress 3303.01.01

TALAT 3300 6

3303.02 Plastic Strain, Rate and Acceleration

Logarithmic (True) Plastic Strain
(Figure 3303.02.01)

Logarithmic (True) Strain
The logarithmic strain ϕ (true strain; degree of deformation)
is a measure of the permanent (plastic) deformation F
dl
logarithmic (or true) strain dϕ =

dl
2
l
l1
dl l
ϕl = ∫ = ln l1 − ln l0 = ln 1 .
l0
l l0
A0
dl
As opposed to this, elongation dε = A
l0
(relative deformation)

l1

l0
l
A1
dl l − l l
l1

ε = ∫ = 1 0 = 1 − 1.
l
l0 0
l0 l0

Conversion:  l1 
with   = (ε + 1)
 l0 

dl
2
F
one obtains ϕ l = ln(ε + 1)

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Logarithmic (True) Strain 3303.02.01

Logarithmic Strain in Upsetting
(Figure 3303.02.02)

Logarithmic Strain in Upsetting

 h1 
ϕ h = ln 
 h0 
Cuboid
b 
ϕ b = ln 1 
h0

h1

 b0 
l 
b0
l0 l1 ϕ l = ln 1 
b1  l0 

r0
r1
 h1 
Circular ϕ h = ln 
h0

 h0 
h1

Cylinder
r 
ϕ r = ln 1  = ϕ t
 r0 

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Logarithmic Strain in Upsetting 3303.02.02

TALAT 3300 7

Law of Volume Constancy
(Figure 3303.02.03)

Law of Volume Constancy

h0

h1
l1
l0
b0 b1

Assuming that during forming the volume of a cuboid remains constant,
the following equation is valid for a homogeneous compression:
V = h1 ⋅ b1 ⋅ l1 = h 0 ⋅ b 0 ⋅ l 0

so that:  h1   b1   l 1 
  ⋅  ⋅  = 1
 h 0   b0   l 0 
Taking the logarithms h  b  l 
of both sides: ln 1  + ln 1  + ln 1  = 0
 h0   b0   l0 
or: ϕh +ϕh +ϕh = 0 or ∑ϕ = 0
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Law of Volume Constancy 3303.02.03

Plastic Strain Rate
(Figure 3303.02.04)

Plastic Strain Rate
(True strain rate)
·
The true strain rate or logarithmic strain rate ϕ is the derivative of the
logarithmic strain (true strain) ϕ with time t:
dϕ
ϕ=
!
dt
From the law of volume constancy one obtains:
. . .
ϕh + ϕb + ϕl = 0
.
Σϕ = 0
·
One has to differentiate between the strain rate ϕ and the
tool speed vwz. For a homogeneous compression where
h = instantaneous height of the material being compressed,
the following relation is valid:
vwz
ϕ=
!
h
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Plastic Strain Rate (True Strain Rate) 3303.02.04

TALAT 3300 8

Plastic Strain Acceleration
(Figure 3303.02.05)

Plastic Strain Acceleration

""

The plastic strain acceleration ϕ is the derivative
"

of the plastic strain rate ϕ with time t:

&
dϕ
&
&
ϕ=
dt
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Forming Strain Acceleration 3303.02.05

3303.03 Plastic Flow under Combined Stresses

Criteria for Plastic Flow
(Figure 3303.03.01)

Criteria for Plastic Flow
Criteria for plastic flow describe the requirements
which must be fulfilled in order for plastic deformation to
occur under a multiaxial state of stress.

The requirements for flow are met when an equivalent
stress, σv , derived from the multiaxial stress state,
equals the flow stress kf:

σv = kf

In the forming technology, two types of flow hypothesis
are used to derive the comparative stress σv :

-the shear stress hypothesis according to TRESCA
-the forming energy hypothesis according to v. MISES

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Criteria for Plastic Flow 3303.03.01

TALAT 3300 9

Maximum Shear Stress Hypothesis
(Figure 3303.03.02, Figure 3303.03.03, Figure 3303.03.04)

Shear Stress Hypothesis (1)
Flow occurs when the maximum shear stress τmax reaches a value
characteristic for the material, the so-called shear yield stress k:

τmax = k .

Based on the stresses ( σ1 > σ2 > σ3 ) in the MOHR stress circle,
the following equations can be derived:
τ

1 τmax
τmax = ( σ1 − σ3 ) = k
2 (=k)

σ2 σ
σ3 σ1 = σmax
1
( σmax − σmin ) = k = σmin
τmax =
2

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Shear Stress Hypothesis (1) 3303.03.02


For a uniaxial state of tensile stress ( σ1 = σmax , σ2 = σ3 = σmin = 0 ),
the following relationship is valid:
τ
σmax - σmin = σ1 = 2k

and ( definition of flow stress ) τmax= k

σ2 = σ3 = 0 σ1 σ
σ1 = kf .

Considering the flow conditions ( σv = kf ), one
obtains:
σv = kf = σmax - σmin.

Thus, flow starts when the difference between the largest and
smallest principal normal stresses equals the flow stress.

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TALAT 3300 10


According to the shear stress theory, the
comparative deformation strain is equal to the
largest value of the logarithmic deformation strain:

ϕ g = ( ϕ 1 , ϕ 2 , ϕ 3 ) max .

ϕg is the principal logarithmic (or true) strain.

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Mises Flow Criterion
(Figure 3303.03.05)

Distortion Energy (von-Mises) Theory
Flow sets in when the elastic distortion energy for changing the form exceeds a criterical value.

If σ1 > σ2 > σ3, then the comparative stress is given by

1
σ v = kf = (σ 1 − σ 2 ) + (σ 2 − σ 3 ) + (σ 3 − σ 1 ) .
2 2 2

2
Using the average stress
1
σm = (σ 1 + σ 2 + σ 3 )
3
one obtains
3
σ v = kf = (σ 1 − σ m )2 + (σ 2 − σ m )2 + (σ 3 − σ m ) .
2

2
Then, according to the distortion energy theory, the comparative deformation strain becomes

ϕv =
3
(ϕ 1 + ϕ 2 2 + ϕ 32 ).
2 2

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Distortion Energy Hypothesis (v. Mises) 3303.03.05

TALAT 3300 11

Yield Criteria for Plane Stress (Yield Locus)
(Figure 3303.03.06)

Yield Criteria for Plane Stress

σ3 The comparision of yield criteria for a
σ2 = 0 plane state of stresses (σ2 = 0) indicates
the stress combinations σ1 , σ3 at which
flow sets in.
kf
The curves depicting the yield criteria can
-kf thus be considered as an illustration of the
−σ1 σ1 flow theory in the σ1 , σ3 plane state of
kf
stresses.
-kf Distortion energy
theory (v. Mises)
The two flow theories sometimes deliver
somewhat different stress values for the
−σ3 start of flow. This difference is, however,
small and does not exceed 15%.
Maximum shear stress theory
(Tresca)

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Yield Criteria for Plane Stress 3303.03.06

3303.04 Law of Plastic Flow
(Figure 3303.04.01)

Law of Plastic Flow
The law of plastic flow describes the correlation between stress
and the resulting deformation strain for the plastic state.
Assuming that the ratios of the deformation strains among each other
remain constant during the process, the correlation between the
deformation strain ϕ and the principal stress σ which causes the
deformation can be given by:
1
σm = (σ 1 + σ 2 + σ 3 )
3
ϕ 1 : ϕ 2 : ϕ 3 = (σ 1 − σ m ) : (σ 2 − σ m ) : (σ 3 − σ m )
Important consequences:
The logarithmic deformation strain attains the value zero
when the corresponding principal stress equals σm.
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Law of Plastic Flow 3303.04.01

TALAT 3300 12

3303.05 Flow Curves

General Definition of the Flow Curve
(Figure 3303.05.01)

Flow Curve

The flow stress kf of a material depends on
- the logarithmic principal deformation strain ϕg
"
- the logarithmic principal deformation strain rate ϕg
- the temperature ϑ

and, during the high speed forming, also on
""
- the principal deformation strain acceleration ϕg
" ""
i.e. kf = f(ϕg, ϕg, ϑ, (ϕg)).

The flow curve is the illustration of the flow stress kf as a
" ""
function of ϕg, ϕg, ϑ and ϕg.

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Flow Curve 3303.05.01

Flow Curves at Room Temperature
(Figure 3303.05.02, Figure 3303.05.03)

Flow Curves at Room Temperature (1)
In the range of cold forming (forming temperature ϑ
is considerably lower than the recrystallisation
temperature ϑRekr ), the flow stress kf for most metallic
materials depends only on the logarithmic principal
deformation strain ϕg:
k f = f (ϕg )
kf

This dependence can be replaced for a number of
materials by the approximate relationship

k f = a ·ϕ ng
k f = a ·ϕ ng
(valid for ϕg ≠ 0, i.e. for kf ≥ Rp0.2 respectively
approximation for kf ≥ ReH)

(a and n are material constants;
ϕg
n is called strain hardening exponent)

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Flow Curves at Room Temperature (1) 3303.05.02

TALAT 3300 13

Flow Curves at Room Temperature (2)
By taking the logarithm of the equation kf = a·ϕgn

log k f = n log ϕ g + log a
· ·
The illustration of a flow curve of the form kf = a· ϕgn
log k f in a coordinate system with axes in a logarithmic
scale is a straight line whose gradient is the
log k f = n log ϕ g + log a
· · coefficient n.
The coefficient n is a measure of the amount by which
∆ log k f
α a material hardens with increasing deformation strain
ϕg and is therefore known as the strain hardening
∆ log ϕ g coefficient.

∆ log k f Typical values of n for an aluminium-killed steel
n = tan α = are in the range of
∆ log ϕ g
0.2 ≤ n ≤ 0.25
log ϕ g
and for aluminium sheet alloys (AlMg0,4Si1,2 ka,
AlMg,5Mn w) in the range of
0.2 ≤ n ≤ 0.3

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Flow Curves at Room Temperature (2) 3303.05.03

Flow Curves at Elevated Temperatures
(Figure 3303.05.04, Figure 3303.05.05 and Figure 3303.05.06)

Flow Curves at Elevated Temperature (1)
180 Material Al 99.5
N / mm² " 20°C During forming at elevated temperatures
ϕg = 4 s
-1

140 120°C (warm forming), the flow stress kf depends on
the logarithmic principal deformation strain ϕg ,
120 240°C the forming temperature ϑ and the
Flow Stress kf

100 logarithmic principal deformation strain rate
"
(deformation strain rate) ϕg:
80
60 360°C "
kf = f(ϕg , ϑ , ϕg)
40 480°C
As a rule, the flow stress kf decreases with
20
increasing temperature ϑ for a given
logarithmic principal deformation strain ϕg,
0 0.2 0.4 0.6 0.8 1.0 1.2
Logarithmic Principal
Deformation Strain ϕg
Source: Bühler a.o

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Flow Curves at Elevated Temperature (1) 3303.05.04

TALAT 3300 14


100
N / mm²
" - With increasing temperature ϑ , the influence
ϕg = 63 s
-1
of the logarithmic principal deformation strain
80 ϕg on the flow stress kf decreases.
Flow Stress kf

"
ϕg = 4 s
-1
60 "
"
- The influence of the deformation strain rate ϕg
ϕg = 0.25 s
-1
on the flow stress kf increases with increasing
40 temperature ϑ. For "a given logarithmic principal
deformation strain ϕg and a given forming
.
20 Material Al 99.5 temperature ϑ, the flow stress kf increases with
"
Temperature ϑ = 360°C increasing deformation strain rate ϕg.

0 0.2 0.4 0.6 0.8 1.0 1.2
Logarithmic Principal
Deformation Strain ϕg
Source: Bühler a.o.

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N / mm²
200
20°C 120°C For a given logarithmic principal deformation
"
strain ϕg and a given forming temperature ϑ,
100 240°C the relationship between the flow stress kf and
Flow Stress kf

80 the deformation strain rate ϕg is given
60 approximately by the equation
360°C
40 k f = b ⋅ϕ g m
!
C ϕg = 1
480°
20 Material Al 99.5 The exponent m is a measure of the hardening
0.25 40 s-1 63 of the material which depends on the
Logarithmic Principal deformation strain rate.
"
Deformation Strain Rate ϕg It increases with increasing temperature.

Source: Bühler a.o

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TALAT 3300 15

3303.06 Average Flow Stress
(Figure 3303.06.01)

Average Flow Stress

The calculation of stresses, energies and forces is
simplified, if one uses an equivalent constant flow
kf stress instead of the actual flow stress kfm. For a
forming process in the range 0 ≤ ϕg ≤ ϕg1 , it is
defined as
kf1 kf (ϕg) ϕ g1

∫k ( ϕ ) dϕ
1
kfm k fm =
kf0 ϕ g1 ϕ g1 f g

∫k ( ϕ ) dϕ
0

f g
0
As an approximation, one can use
kf 0 + kf 1
k fm =
2
0 ϕg1 ϕg kf0 - flow stress at start of process (ϕg = 0)
kf1 - flow stress at end of process (ϕg = ϕg1)

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Average Flow Stress 3303.06.01

3303.07 Energy Considerations

Forming Energy
(Figure 3303.07.01)

Forming Energy

Ideal forming energy Total energy (effective forming
energy)
The ideal forming energy W id is that part of
mechanical energy which has to be used for The total forming energy Wges includes the
the forming process without external friction ideal forming energy Wid as well as the
and internal displacements. energy required to overcome the external
friction (WR) and the internal displacement
The following relation is valid for the forming (WSch), so that
energy Wid:
Wges = Wid + WR + WSch
Wid = V ∫ k f ( ϕ g ) dϕ
where V is the formed volume.

Using the average flow stress kfm, one obtains

Wid = V k fm ϕ g

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Forming Energy 3303.07.01

TALAT 3300 16

Heat Development during Forming
(Figure 3303.07.02)

Heat Developed during Forming

Forming processes are irreversible processes. The major part of
the energy applied for the forming process is converted into heat,
causing the work-piece to warm up. Assuming that the total forming
energy Wges is converted into heat and that no heat is lost to the
environment (tool), the rise in temperature is given by the equation
Wges
∆ϑ =
V * ρ * cp

(V formed volume, ρ density and cp specific heat of the work-piece
material)

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Heat Developed during Forming 3303.07.02

TALAT 3300 17

Literature/References

K. Lange (Editor): Umformtechnik, Springer-Verlag, Berlin, Heidelberg, New York,
Tokyo, 1984

List of Figures

Figure No. Figure Title (Overhead)

3301.01.01 Definition of Forming

3302.01.01 Classifying the Forming Process by State of Stress
3302.02.01 Classifying the Forming Process by Types of Raw Materials
3302.03.01 Classifying the Forming Process by Forming Temperature
3302.04.01 Classifying the Forming Process by Methods of Applying Force

3303.01.01 Characteristic Values, Basic Laws: Flow Stress
3303.02.01 Logarithmic (True) Strain
3303.02.02 Logarithmic Strain in Upsetting
3303.02.03 Law of Volume Constancy
3303.02.04 Plastic Strain Rate (True Strain Rate)
3303.02.05 Forming Strain Acceleration
3303.03.01 Criteria for Plastic Flow
3303.03.02 Shear Stress Hypothesis (1)
3303.03.05 Distortion Energy Hypothesis (v. Mises)
3303.03.06 Yield Criteria for Plane Stress
3303.04.01 Law of Plastic Flow
3303.05.01 Flow Curve
3303.05.02 Flow Curves at Room Temperature (1)
3303.05.03 Flow Curves at Room Temperature (2)
3303.05.04 Flow Curves at Elevated Temperature (1)
3303.06.01 Average Flow Stress
3303.07.01 Forming Energy
3303.07.02 Heat Developed during Forming

TALAT 3300 18

Fundamentals of Metal Forming

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