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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TALAT 3300
Definition of Forming
3
3301.01.01
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
Pressure Forming
DIN 8583
Rolling, Open-Die Forming,
Die Forming, Indenting,
Pressing Through
Tension-Compression
Forming
DIN 8584
Drawing, Deep Drawing,
Collar Forming, Compressing,
Upset Bulging
Tension Forming
DIN 8585
Stretch Reducing, Bulge Forming,
Stretch Forming
Bend Forming
DIN 8586
Bending with Linear Tool Movement,
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
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3302.02
Classification by Type of Raw Material (Figure 3302.02.01)
Terms for classifying the forming process:
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.
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Classifying the Forming Process by
Types of Raw Materials
4
3302.02.01
3302.03
Classification by Forming Temperature
(Figure 3302.03.01)
Terms for classifying the forming process:
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 )
Classifying the Forming Process by
Forming Temperature
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3302.04
3302.03.01
Classification by Methods of Induction of Forces into the Work-Piece
(Figure 3302.04.01)
Terms for classifying the forming process:
Methods of Applying Force
Process with Indirect Force Application
(Deep Drawing)
FN
Process with Direct Force Application
(Upsetting)
F
FSt
Forming Zone
Bending Zone
Force Transmission Zone
Force Induction Zone
F
K. Siegert
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TALAT 3300
Classifying the Forming Process by
Methods of Applying Force
5
3302.04.01
3303 Characteristic Values and Basic Laws of Metal Forming
•
•
•
•
•
•
•
3303.01
Flow Stress
Plastic Strain, Rate and Acceleration
Plastic Flow under Combined Stresses
Law of Plastic Flow
Flow Curves
Average Flow Stress
Energy Considerations
Flow Stress
(Figure 3303.01.01)
Characteristic Values, Basic Laws
Flow Stress
kf =
F
A
By comparison
(technical stress)
σ=
F
A0
A0
l0
Flow stress
(true stress)
F
Conversion:
with
one obtains
l
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.
A
ε = (l − l0 ) / l0
k f = σ (ε l + 1)
F
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Definition of Flow Stress
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TALAT 3300
6
3303.01.01
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
A0
dl
l0
A
l0
As opposed to this, elongation
(relative deformation)
dε =
l
l0
dl
l
= ln l1 − ln l0 = ln 1 .
l
l0
l1
l1
ϕl = ∫
dl
2
dl
logarithmic (or true) strain dϕ =
l
A1
dl l − l l
ε = ∫ = 1 0 = 1 − 1.
l
l0
l0
l0 0
Conversion:
with
 l1 
  = (ε + 1)
 l0 
one obtains
ϕ l = ln(ε + 1)
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dl
2
l1
F
Logarithmic (True) Strain
3303.02.01
Training in Aluminium Application Technologies
Logarithmic Strain in Upsetting
(Figure 3303.02.02)
Logarithmic Strain in Upsetting
 h1 

 h0 
ϕ h = ln
b 
 b0 
ϕ b = ln 1 
h1
h0
Cuboid
b0
l0
l1
b1
l 
 l0 
ϕ l = ln 1 
r0
r1
 h1 

 h0 
ϕ h = ln
h1
h0
Circular
Cylinder
r 
 r0 
ϕ r = ln 1  = ϕ t
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Logarithmic Strain in Upsetting
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TALAT 3300
7
3303.02.02
Law of Volume Constancy
(Figure 3303.02.03)
h1
h0
Law of Volume Constancy
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 
ln 1  + ln 1  + ln 1  = 0
of both sides:
 h0 
 b0 
 l0 
ϕh +ϕh +ϕh = 0
or:
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or
∑ϕ = 0
Law of Volume Constancy
3303.02.03
Training in Aluminium Application Technologies
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:
ϕ!=
dϕ
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:
ϕ!=
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TALAT 3300
vwz
h
Plastic Strain Rate (True Strain Rate)
8
3303.02.04
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:
"
&=
ϕ&
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dϕ&
dt
Forming Strain Acceleration
3303.02.05
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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
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TALAT 3300
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3303.03.01
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
τmax =
τmax =
1
( σ1 − σ3 ) = k
2
τmax
(=k)
1
( σmax − σmin ) = k
2
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σ3
= σmin
σ2
σ1 = σmax
σ
3303.03.02
Shear Stress Hypothesis (1)
Shear Stress Hypothesis (2)
For a uniaxial state of tensile stress ( σ1 = σmax , σ2 = σ3 = σmin = 0 ),
the following relationship is valid:
τ
σmax - σmin = σ1 = 2k
τmax= k
and ( definition of flow stress )
σ2 = σ3 = 0
σ1 = kf .
σ1
σ
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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Shear Stress Hypothesis (2)
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TALAT 3300
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3303.03.03
Shear Stress Hypothesis (3)
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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Shear Stress Hypothesis (3)
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3303.03.04
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
σ v = kf =
1
2
2
2
(σ 1 − σ 2 ) + (σ 2 − σ 3 ) + (σ 3 − σ 1 ) .
2
Using the average stress
σm =
1
(σ 1 + σ 2 + σ 3 )
3
one obtains
σ v = kf =
3
2
(σ 1 − σ m )2 + (σ 2 − σ m )2 + (σ 3 − σ m ) .
2
Then, according to the distortion energy theory, the comparative deformation strain becomes
ϕv =
2 2
(ϕ 1 + ϕ 2 2 + ϕ 32 ).
3
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Distortion Energy Hypothesis (v. Mises)
11
3303.03.05
Yield Criteria for Plane Stress (Yield Locus)
(Figure 3303.03.06)
Yield Criteria for Plane Stress
σ3
The comparision of yield criteria for a
plane state of stresses (σ2 = 0) indicates
the stress combinations σ1 , σ3 at which
flow sets in.
σ2 = 0
kf
−σ1
-kf
σ1
kf
-kf
Distortion energy
theory (v. Mises)
−σ3
Maximum shear stress theory
(Tresca)
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The curves depicting the yield criteria can
thus be considered as an illustration of the
flow theory in the σ1 , σ3 plane state of
stresses.
The two flow theories sometimes deliver
somewhat different stress values for the
start of flow. This difference is, however,
small and does not exceed 15%.
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
(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 )
σm =
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
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TALAT 3300
12
3303.04.01
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
"
""
kf = f(ϕg, ϕg, ϑ, (ϕg)).
i.e.
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
Training in Aluminium Application Technologies
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
(valid for ϕg ≠ 0, i.e. for kf ≥ Rp0.2 respectively
approximation for kf ≥ ReH)
(a and n are material constants;
n is called strain hardening exponent)
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ϕg
Flow Curves at Room Temperature (1)
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TALAT 3300
k f = a ·ϕ ng
13
3303.05.02
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
in a coordinate system with axes in a logarithmic
scale is a straight line whose gradient is the
coefficient n.
log k f
log k f = n log
· ϕ g + log
· a
∆ log k f
α
∆ log ϕ g
n = tan α =
∆ log k f
∆ log ϕ g
The coefficient n is a measure of the amount by which
a material hardens with increasing deformation strain
ϕg and is therefore known as the strain hardening
coefficient.
Typical values of n for an aluminium-killed steel
are in the range of
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)
Flow Stress kf
180 Material Al 99.5
N / mm² "
-1
140
120
ϕg = 4 s
240°C
100
80
60
During forming at elevated temperatures
(warm forming), the flow stress kf depends on
the logarithmic principal deformation strain ϕg ,
the forming temperature ϑ and the
logarithmic principal deformation
strain rate
"
(deformation strain rate) ϕg:
360°C
"
kf = f(ϕg , ϑ , ϕg)
480°C
40
20
0
20°C
120°C
0.2 0.4 0.6 0.8 1.0 1.2
As a rule, the flow stress kf decreases with
increasing temperature ϑ for a given
logarithmic principal deformation strain ϕg,
Logarithmic Principal
Deformation Strain ϕg
Source: Bühler a.o
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Flow Curves at Elevated Temperature (1)
14
3303.05.04
Flow Curves at Elevated Temperature (2)
Flow Stress kf
100
N / mm²
80
"
ϕg = 63 s
"
60
ϕg = 4 s
20
- With increasing temperature ϑ , the influence
of the logarithmic principal deformation strain
ϕg on the flow stress kf decreases.
-1
"
- The influence of the deformation strain rate ϕg
on the flow stress kf increases with increasing
ϕg = 0.25 s
temperature ϑ. For "a given logarithmic principal
deformation strain ϕg and a given
. forming
temperature ϑ, the flow stress kf increases
with
Material Al 99.5
"
increasing
deformation
strain
rate
.
ϕg
Temperature ϑ = 360°C
"
40
-1
-1
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 (2)
3303.05.05
Training in Aluminium Application Technologies
Flow Curves at Elevated Temperature (3)
Flow Stress kf
N / mm²
200
20°C 120°C
100 240°C
80
60
360°C
40
ϕg = 1
C
480°
Material Al 99.5
20
40
s-1 63
0.25
Logarithmic Principal
"
Deformation Strain Rate ϕg
For a given
logarithmic principal deformation
"
strain ϕg and a given forming temperature ϑ,
the relationship between the flow stress kf and
the deformation strain rate ϕg is given
approximately by the equation
m
k f = b ⋅ ϕ!
g
The exponent m is a measure of the hardening
of the material which depends on the
deformation strain rate.
It increases with increasing temperature.
Source: Bühler a.o
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Flow Curves at Elevated Temperature (3)
15
3303.05.06
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
stress instead of the actual flow stress kfm. For a
forming process in the range 0 ≤ ϕg ≤ ϕg1 , it is
defined as
kf
kf (ϕg)
kf1
kfm
kf0
ϕ g1
∫k ( ϕ
f
0
g
k fm =
) dϕ
ϕg1
∫k ( ϕ
f
g
0
) dϕ
kf 0 + kf 1
2
kf0 - flow stress at start of process (ϕg = 0)
kf1 - flow stress at end of process (ϕg = ϕg1)
ϕg
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ϕ g1
ϕ g1
As an approximation, one can use
k fm =
0
1
Average Flow Stress
3303.06.01
Training in Aluminium Application Technologies
3303.07
Energy Considerations
Forming Energy
(Figure 3303.07.01)
Forming Energy
Total energy (effective forming
energy)
Ideal forming energy
The ideal forming energy W id is that part of
mechanical energy which has to be used for
the forming process without external friction
and internal displacements.
The total forming energy Wges includes the
ideal forming energy Wid as well as the
energy required to overcome the external
friction (WR) and the internal displacement
(WSch), so that
The following relation is valid for the forming
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
16
3303.07.01
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
∆ϑ =
Wges
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
17
3303.07.02
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
3302.02.01
3302.03.01
3302.04.01
Classifying the Forming Process by State of Stress
Classifying the Forming Process by Types of Raw Materials
Classifying the Forming Process by Forming Temperature
Classifying the Forming Process by Methods of Applying Force
3303.01.01
3303.02.01
3303.02.02
3303.02.03
3303.02.04
3303.02.05
3303.03.01
3303.03.02
3303.03.03
3303.03.04
3303.03.05
3303.03.06
3303.04.01
3303.05.01
3303.05.02
3303.05.03
3303.05.04
3303.05.05
3303.05.06
3303.06.01
3303.07.01
3303.07.02
Characteristic Values, Basic Laws: Flow Stress
Logarithmic (True) Strain
Logarithmic Strain in Upsetting
Law of Volume Constancy
Plastic Strain Rate (True Strain Rate)
Forming Strain Acceleration
Criteria for Plastic Flow
Shear Stress Hypothesis (1)
Shear Stress Hypothesis (2)
Shear Stress Hypothesis (3)
Distortion Energy Hypothesis (v. Mises)
Yield Criteria for Plane Stress
Law of Plastic Flow
Flow Curve
Flow Curves at Room Temperature (1)
Flow Curves at Room Temperature (2)
Flow Curves at Elevated Temperature (1)
Flow Curves at Elevated Temperature (2)
Flow Curves at Elevated Temperature (3)
Average Flow Stress
Forming Energy
Heat Developed during Forming
TALAT 3300
18