TALAT lecture 3300: fundamentals of metal forming - CORE-Materials

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. 

K.Siegert 

alu 

Training in Aluminium Application Technologies 

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. 


TALAT 3300 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) 

alu 


Terms for classifying the forming process: 

Classification by State of Stress 

Pressure Forming 

Tension-Compression 

Forming 

Tension Forming 

Bend Forming 

Shear Forming 

DIN 8583 

DIN 8584 

DIN 8585 

DIN 8586 

DIN 8587 

Classifying the Forming Process by State of Stress 


Rolling, Open-Die Forming, 

Die Forming, Indenting, 

Pressing Through 

Drawing, Deep Drawing, 

Collar Forming, Compressing, 

Upset Bulging 

Stretch Reducing, Bulge Forming, 

Stretch Forming 

Bending with Linear Tool Movement, 

Bending with Rotating Tool Movement 

Shear Displacement, Twisting 

3302.01.01 

3302.02 Classification by Type of Raw Material (Figure 3302.02.01) 

alu 



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. 

Classifying the Forming Process by 

Types of Raw Materials 

3302.02.01

3302.03 Classification by Forming Temperature 

(Figure 3302.03.01) 

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Forming Temperature 

Cold Forming 

Process in which the work-piece is not heated before forming, 

but instead formed at room temperature. 

Warm Forming 

ϑ = ϑRoom (ca. 20° C ) 

Process in which the work-piece is heated to an elevated temperature 

higher than room temperature before forming. 

ϑ > ϑ Room (ca. 20° C ) 


Forming Temperature 


3302.03.01 

3302.04 Classification by Methods of Induction of Forces into the Work-Piece 

(Figure 3302.04.01) 

alu 



Methods of Applying Force 

Process with Indirect Force Application 

(Deep Drawing) 

K. Siegert 

FN 

Forming Zone 

F St 

Bending Zone 

Force Transmission Zone 

Force Induction Zone 


Methods of Applying Force 

Process with Direct Force Application 

(Upsetting) F 

F 

3302.04.01

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) 

alu 


Characteristic Values, Basic Laws 

Flow Stress 

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. 

Flow stress 

(true stress) 

By comparison 

(technical stress) 

Conversion: 

with 

one obtains 

F 

k f 

A 

= 

σ = F 

A 0 

ε = ( l−l )/ l 0 0 

= σε ( + 1) 

k f l 


A 

A 0 

F 

l 0 

l 

Definition of Flow Stress 3303.01.01 

F

3303.02 Plastic Strain, Rate and Acceleration 

Logarithmic (True) Plastic Strain 

(Figure 3303.02.01) 

logarithmic (or true) strain dϕ 

= 

l 

∫ 

l0 

ε = = − ∫ = − 

dl 

1 

l1 l0 

l1 

1. 

l l l 

alu 


l 

l 0 0 

⎛ l ⎞ 1 

⎜ ⎟ = ( ε + 1) 

⎝ l ⎠ 

0 

0 

ϕ l = ln( ε + 1) 

Logarithmic (True) Strain 

The logarithmic strain ϕ (true strain; degree of deformation) 

is a measure of the permanent (plastic) deformation 


0 

dl 

l 

1 

dl 

l1 

ϕ l = = lnl1− lnl0 = ln . 

l 

l 

As opposed to this, elongation 

(relative deformation) 

Conversion: 

with 

one obtains 

Logarithmic Strain in Upsetting 

(Figure 3303.02.02) 

Cuboid 

Circular 

Cylinder 

h 0 

alu 


0 

dl 

dε 

= 

l 

0 

Logarithmic (True) Strain 

l 1 

dl 

2 

l 

dl 

2 

Logarithmic Strain in Upsetting 

h0 

b 0 

l 0 

r 0 

h1 

h 1 

b 1 

r 1 

l 0 

F 

h 

ϕ h = 

h 

b 

ϕ b 

b 

l 

ϕ l 

l 

⎛ ⎞ 

⎜ ⎟ 

⎝ ⎠ 

= ⎛ ⎞ 

⎜ ⎟ 

⎝ ⎠ 

= ⎛ 

1 ln 

0 

1 ln 

0 

⎞ 1 ln⎜ 

⎟ 

⎝ ⎠ 

F 

3303.02.01 

h 

ϕ h = 

h 

r 

ϕr ϕt 

r 

⎛ ⎞ 

⎜ ⎟ 

⎝ ⎠ 

= ⎛ 

1 ln 

0 

⎞ 1 ln⎜ 

⎟ = 

⎝ ⎠ 

Logarithmic Strain in Upsetting 3303.02.02 

l 1 

0 

0 

A 0 

A 

A 1

Law of Volume Constancy 

(Figure 3303.02.03) 

Taking the logarithms 

of both sides: 

or: 

alu 


h 0 

Law of Volume Constancy 

⎛ h ⎞ 1 b1 

l1 

ln⎜ ⎟ + ln ln 0 

⎝ h ⎠ b l 

⎛ ⎞ 

⎜ ⎟ + 

⎝ ⎠ 

⎛ 

V=h1⋅b1⋅ l1 = h0⋅b0⋅l0 ⎛ h ⎞ 1 b1 

l1 

⎜ ⎟ ⋅ 1 

⎝ h0 

⎠ b0 

l0 

⎞ 

⎜ ⎟ = 

⎝ ⎠ 

⎛ ⎞ 

⎜ ⎟ ⋅ 

⎝ ⎠ 

⎛ ⎞ 

⎜ ⎟ = 

⎝ ⎠ 


b 0 

0 

l 0 

h 1 

Assuming that during forming the volume of a cuboid remains constant, 

the following equation is valid for a homogeneous compression: 

so that: 

Plastic Strain Rate 

(Figure 3303.02.04) 

0 

0 

b 1 

ϕ + ϕ + ϕ = 0 or ϕ = 0 

h h h 

∑ 

Law of Volume Constancy 3303.02.03 

· 

The true strain rate or logarithmic strain rate ϕ is the derivative of the 

logarithmic strain (true strain) ϕ with time t: 

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: 

alu 


Plastic Strain Rate 

(True strain rate) 

ϕ 

ϕ = d 

dt 

ϕ = vwz 

h 

Plastic Strain Rate (True Strain Rate) 

l 1 

3303.02.04

Plastic Strain Acceleration 

(Figure 3303.02.05) 

 

The plastic strain acceleration ϕ is the derivative 

 

of the plastic strain rate ϕ with time t: 

alu 


Plastic Strain Acceleration 

& & ϕ 

ϕ = d 

dt 


Forming Strain Acceleration 3303.02.05 

3303.03 Plastic Flow under Combined Stresses 

Criteria for Plastic Flow 

(Figure 3303.03.01) 

alu 


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 = k f 

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 

Criteria for Plastic Flow 3303.03.01

Maximum Shear Stress Hypothesis 

(Figure 3303.03.02, Figure 3303.03.03, Figure 3303.03.04) 

Flow occurs when the maximum shear stress τmax reaches a value 

characteristic for the material, the so-called shear yield stress k: 

alu 


τmax = k . 

Based on the stresses ( σ1 > σ2 > σ3 ) in the MOHR stress circle, 

the following equations can be derived: 

τ 

τmax = 

1 

2 ( σ1 − σ3 ) = k 

1 

τmax = ( σmax − σmin ) = k 

2 

alu 


Shear Stress Hypothesis (1) 

σ3 

= σmin 


σ2 

τmax 

(=k) 



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 ) 

σ1 = k f . 

Considering the flow conditions ( σv = k f ), one 

obtains: 

σv = k f = σmax - σmin. 

σ2 = σ3 = 0 

Thus, flow starts when the difference between the largest and 

smallest principal normal stresses equals the flow stress. 

τmax= k 

σ1 = σmax 

σ1 

σ 

3303.03.02 

Shear Stress Hypothesis (2) 3303.03.03 

σ

alu 


Mises Flow Criterion 

(Figure 3303.03.05) 

alu 



According to the shear stress theory, the 

comparative deformation strain is equal to the 

largest value of the logarithmic deformation strain: 

( ) 

ϕ ϕ , ϕ , ϕ . 

g = 1 2 3 

ϕg is the principal logarithmic (or true) strain. 


Distortion Energy (von-Mises) Theory 


max 

3303.03.04 

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 σ σ σ σ σ σ 

f k 

1 

= = 1− 2 + 2 − 3 + 3 − 1 

2 

Using the average stress 

1 

σ m = ( σ1+ σ 2 + σ 3) 

3 

one obtains 

2 

2 

( ) ( ) ( ) 

3 

σ = k = σ − σ + σ − σ + σ −σ 

v f 1 m 2 m 3 m 

2 

2 

2 

( ) ( ) ( ) 

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

2 2 2 2 

ϕv = ( ϕ1+ ϕ2+ ϕ3) 

. 

3 

2 

2 

. 

. 

Distortion Energy Hypothesis (v. Mises) 3303.03.05

Yield Criteria for Plane Stress (Yield Locus) 

(Figure 3303.03.06) 

−σ1 

-k f 

k f 

Maximum shear stress theory 

(Tresca) 

alu 


σ3 

-k f 

−σ3 

Yield Criteria for Plane Stress 

Distortion energy 

theory (v. Mises) 


k f 

σ2 = 0 

σ1 

3303.04 Law of Plastic Flow 

(Figure 3303.04.01) 

alu 


The comparision of yield criteria for a 

plane state of stresses (σ2 = 0) indicates 

the stress combinations σ1 , σ3 at which 

flow sets in. 

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 

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 = ( σ + σ m 

1 2 

3 

+ σ 3) 

ϕ : ϕ : ϕ = σ −σ : σ −σ : σ −σ 

( ) ( ) ( ) 

1 2 3 1 m 2 m 3 m 

Important consequences: 

The logarithmic deformation strain attains the value zero 

when the corresponding principal stress equals σm. 

Law of Plastic Flow 3303.04.01

3303.05 Flow Curves 

General Definition of the Flow Curve 

(Figure 3303.05.01) 

alu 


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. k f = f(ϕg, ϕg, ϑ, (ϕg)). 

The flow curve is the illustration of the flow stress k f as a 

 

function of ϕg, ϕg, ϑ and ϕg. 

Flow Curves at Room Temperature 

(Figure 3303.05.02, Figure 3303.05.03) 

alu 



Flow Curve 3303.05.01 

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 k f for most metallic 

materials depends only on the logarithmic principal 

deformation strain ϕg: 

k = f ( ) 

f ϕg 

This dependence can be replaced for a number of 

materials by the approximate relationship 

n k = a· 

ϕ 

f g 

(valid for ϕg ≠ 0, i.e. for k f ≥ R p0.2 respectively 

approximation for k f ≥ R eH) 

(a and n are material constants; 

n is called strain hardening exponent) 

Flow Curves at Room Temperature (1) 3303.05.02 

k f 

 

n k = a· 

ϕ 

f g 

ϕg

log k f 

log kf = n log · ϕ g + log · a 

∆ log k f 

n = tanα 

= 

∆ logϕ 

alu 


Flow Curves at Room Temperature (2) 

α 

∆ logϕ g 

g 

logϕ g 

∆ log k f 


By taking the logarithm of the equation kf = a · ϕg n 

log k f = n log · ϕ g + log · a 

The illustration of a flow curve of the form kf = a · ϕg n 

in a coordinate system with axes in a logarithmic 

scale is a straight line whose gradient is the 

coefficient n. 

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 

and for aluminium sheet alloys (AlMg0,4Si1,2 ka, 

AlMg,5Mn w) in the range of 

0.2 ≤ n ≤ 0.3 

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) 

Material Al 99.5 

ϕg = 4 s-1 180 

N / mm² 

140 

120 

100 

80 

60 

40 

20 

Flow Stress kf 

Source: Bühler a.o 

Flow Curves at Elevated Temperature (1) 

alu 


20°C 

120°C 

240°C 

360°C 

480°C 

0 0.2 0.4 0.6 0.8 1.0 1.2 

Logarithmic Principal 

Deformation Strain ϕg 

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: 

k f = f(ϕg , ϑ , ϕg) 

As a rule, the flow stress k f decreases with 

increasing temperature ϑ for a given 

logarithmic principal deformation strain ϕg, 

Flow Curves at Elevated Temperature (1) 3303.05.04

100 

N / mm² 

80 

Flow Stress k f 

60 

40 

20 

Source: Bühler a.o. 


alu 


N / mm² 

200 

Flow Stress kf 

100 

80 

60 

Source: Bühler a.o 

ϕg = 63 s -1 

 

 

 

ϕg = 4 s -1 

ϕg = 0.25 s -1 


Temperature ϑ = 360°C 

0 0.2 0.4 0.6 0.8 1.0 1.2 


Deformation Strain ϕg 


- With increasing temperature ϑ , the influence 

of the logarithmic principal deformation strain 

ϕg on the flow stress k f decreases. 

 

- The influence of the deformation strain rate ϕg 

on the flow stress kf increases with increasing 

temperature ϑ. For a given logarithmic principal 

 

deformation strain ϕg and a given . forming 

temperature ϑ, the flow stress kf increases with 

 

increasing deformation strain rate ϕg. 

Flow Curves at Elevated Temperature (2) 3303.05.05 


20°C 120°C 

240°C 

360°C 

ϕg = 1 


0.25 40 s 63 


Deformation Strain Rate ϕg 

-1 

40 

20 

 

480°C 

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 

k = b⋅ 

ϕ 

m 

f 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. 

alu 

Training in Aluminium Application Technologies Flow Curves at Elevated Temperature (3) 3303.05.06

3303.06 Average Flow Stress 

(Figure 3303.06.01) 

k f 

k f1 

k fm 

k f0 

0 ϕg1 ϕg 

alu 


ϕ 

g1 

∫ 

0 

( ) 

k f (ϕg) 

k ϕ dϕ 

f g 

3303.07 Energy Considerations 

Forming Energy 

(Figure 3303.07.01) 

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 following relation is valid for the forming 

energy W id: 

alu 


( ) 

W = V k d 

id ∫ ϕ ϕ 

f g 

where V is the formed volume. 

Using the average flow stress k fm, one obtains 

W = V k ϕ 

id fm g 

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 k fm. For a 

forming process in the range 0 ≤ ϕg ≤ ϕg1 , it is 

defined as 

ϕg1 

1 

k = k ( ) d 

fm ∫ ϕ ϕ 

f g 

ϕ 

g1 

0 

As an approximation, one can use 

k 

fm 

k + k 

= 

2 

f0 f1 

kf0 - flow stress at start of process (ϕg = 0) 

k f1 - flow stress at end of process (ϕg = ϕg1) 

Average Flow Stress 3303.06.01 



Total energy (effective forming 

energy) 

The total forming energy W ges includes the 

ideal forming energy W id as well as the 

energy required to overcome the external 

friction (W R) and the internal displacement 

(W Sch), so that 

W = W + W + 

W 

ges id R Sch 

3303.07.01

Heat Development during Forming 

(Figure 3303.07.02) 

alu 


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* * 

c 

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

material) 

Heat Developed during Forming 


ρ 

p 

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 


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

TALAT lecture 3300: fundamentals of metal forming - CORE-Materials

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