1

Fundamentals of Metal Forming

1. Overview2. Material Behavior

3. Temperature Effects4. Strain Rate Effect

5. Friction and Lubrication

2

Introduction• Metal Forming uses plastic deformation to

change the shape of metal workpieces– Materials (metals) – plastic deformation– External loads – Typically compressive

• Sometimes Stretch the metal (tensile), bend the metal (tensile and compressive), shear stresses

– Shape - die and tools• Classification

– Bulk Deformation – Sheet Metal Forming – High surface area-to-volume

ratio • Parts are called stampings• Usual tooling: punch and die

3

Introduction• Bulk Deformation

Process– Rolling– Forging– Extrusion– Drawing

• Sheet metal forming– Bending– Drawing– Shearing– (Stamping)

+

+

v

Fv

F

F

RAMF

vF

v

4

Forging Sequence of Part and Die

5

Material Properties in Forming

• Desirable material properties: – Yield strength ?– Ductility ?– Second phase or Inclusion ?

• These properties are affected by temperature: – When work temperature is raised, ductility increases

and yield strength decreases while loosing on surface finish and dimension accuracy.

• Other factors: – Strain rate and friction

6

Theories of Failure• The limit of the stress state on a material

– Ductile Materials - Yielding– Brittle Materials - Fracture

• In a tensile test, Yield or Failure Strength of a material.

• In a multiaxial state of stress, how do we use Yield or Failure Strength?

7

Yielding: Ductile MaterialsA. Maximum Shear Stress Theory

(Tresca Criterion)

2maxyS

=τ

y

y

y

S

S

S

=−

=

=

21

2

1

σσ

σ

σFor Plane Stress:σ1, σ2 have same signs.

σ1, σ2 have opposite signs.

In a tension specimen: τmax

Sy

(σ3=0)

The diameter of the Mohr circle = Sy

8

Ductile Materials

[ ] ( )[ ]31232123

22

21332211 2

21

21 σσσσσσνσσσεσεσεσ ++−++=++=

Eu

For the three principal stresses;

in an uniaxial tension test. 2

31

yd SE

u ν+=

For plane stress; [ ]2221

213

1 σσσσν+−

+=

Eud

22221

21 yS=+− σσσσ

σ2

σ1

A

B

After taken out the hydrostatic stress (σave=(σ1+σ2+σ3)/3)

( ) ( ) ( )aveaveave σσσσσσ −−− 321 ,,Now substitute σ1, σ2 and σ3 with

B. Maximum Distortion Energy Theory (von Mises Criterion)

[ ] 22221

21 3

13

1yS

EEνσσσσν +

=+−+

In a biaxial case, the same amount of distortion energy

9

Plasticity• Flow theory (Classical theory)

– The current strain rates depend on the stress.• Deformation theory (Hencky theory)

– The total strain is related to the stress.– Ideal for nonlinear elasticity– Still work for monotonically increasing stresses

everywhere in a body• Pressure-independent – Hydrostatic pressure

does not affect dislocation motion.• Bauschinger effect

– The different behaviors in tension and compression

Stress Representation

10

σx

σz

σy

τyzτxz

τxy

ij

zyzxz

yzyxy

xzxyx

σσσσσσσσσσ

στττστττσ

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

333231

232221

131211

Tensor Notation

General 3-D Stress Plane Stress

σx

σyτxy

σx τ xy 0

τ xy σy 0

0 0 0

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

= σij

z

Stress Transformation (Plane Stress)

11

σx' =σx +σy

2+

σx −σy

2cos2θ +τ xy sin2θ

τ x'y ' = −σx −σy

2sin2θ +τ xy cos2θ

σy' =σx +σy

2−

σx −σy

2cos2θ −τ xy sin2θ

σx' +σy ' =σx +σy

2+

σx −σy

2cos2θ +τ xy sin2θ +

σx +σy

2−

σx −σy

2cos2θ −τ xy sin2θ = σx +σy

−σx 'σy' +τ x 'y'2 = −

σx +σy

2+

σx −σy

2cos2θ +τ xy sin2θ

⎛

⎝ ⎜

⎞

⎠ ⎟

σx +σy

2−

σx −σy

2cos2θ −τ xy sin2θ

⎛

⎝ ⎜

⎞

⎠ ⎟ + −

σx −σy

2sin2θ +τ xy cos2θ

⎛

⎝ ⎜

⎞

⎠ ⎟

2

= −σx +σy

2

⎛

⎝ ⎜

⎞

⎠ ⎟

2

+σx −σy

2cos2θ +τ xy sin2θ

⎛

⎝ ⎜

⎞

⎠ ⎟

2

+ −σx −σy

2sin2θ +τ xy cos2θ

⎛

⎝ ⎜

⎞

⎠ ⎟

2

=σx +σy

2

⎛

⎝ ⎜

⎞

⎠ ⎟

2

−σx −σy

2cos2θ +τ xy sin2θ

⎛

⎝ ⎜

⎞

⎠ ⎟

2

+ −σx −σy

2sin2θ +τ xy cos2θ

⎛

⎝ ⎜

⎞

⎠ ⎟

2

= −σxσy +τ xy2

12

Yield function in 3Dσij[ ]= σ'ij +σmδ ij[ ]

where σ'ij = DeviatoricStress Tensor,

σm =1

3σ1 +σ2 +σ3( )= Hydrostatic Component of Stress

δ ij =1 for i = j

0 for i ≠ j

( )( )

321

2223

132321

2222

3211

322

13

2

where0

σσσ

τστστστττσσσ

σσσσσσ

τττσσσσσσ

σσσσσσσσσ

=

−−−+=

++−=

+++++−=

++=++==−−−

xyzxzyyzxxzyzxyzyx

zxzyyx

zyx

I

I

IIII

zxyzxy

( ) ( ) ( ) ( ) CJJfIIIfff Iij ==== 32321 ,,,σσ

Stress Tensor :

Three Stress Invariants

Yield function:

( )zyxJ

JJ

JJJ

'''''''''

0 where0'''

3

1323212

1

322

13

σσσσσσσσσ

σσσ

=++−=

==−−−

Three Deviatoric Stress Invariants

13

Example: Stresses in 3D

[ ]

( )( ) ( ) ( )( )( )( )

( )72

and54)182412( ,13

MPa6andMPa4 MPa,307254130634

4445

0401

041115

MPa401041115

3213

1323212

3211

321

23

2

3

2

1

==−=++−=++−=

=++====

=−+−

=−−−==−−−−−−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−

−−−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−−=

σσσσσσσσσ

σσσσσσ

σσσ

σσσσσσσ

σσ

σ

σ

III

nnn

ij

To find Principal Stresses:

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

zyzxz

zyyxy

zxyxx

ij

στττστττσ

σσσσσσσσσ

σ

332313

322212

312111

x

y

z

14

Example: Continue

[ ] [ ]

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−

−−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−−=

+=

33.001033.011167.0

33.400033.400033.4

401041115

'

'

ij

ijmijij

σ

δσσσ

[ ]

MPa6&4,3

MPa401041115

31 =

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−−=

−σ

σ ij

( ) ( ) MPa33.464331445

31

=++=++=mσ

74.0;33.2;00''33.274.0

0''31

920)'33.0(

0)'33.0(2)'67.0()'33.0(

'33.0010'33.0111'67.0

321

3

2

2

=−===−+

=⎟⎠⎞

⎜⎝⎛ −++

=++−+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−−

−−−

JJJσσ

σσσ

σσσ

σσ

σ

Deviatoric Stress

15

Presentation of Yield Surface

( )CS

Cf

y =

=−= 313,2,1 σσσ

( ) ( )

3

;461, 222

232

yS

CJJJf

=

=++==

σ

σσσ

0,, 321 ==−= σσσσσ

Max. Shear Stress Theory: ( )( ) CkJkJkJJJJf

Cf

=−+−−=

=−=6

242

222

33232

313,2,1

649636274,

σσσ

( )

( ) ( ) ( ) ( )[ ] CJf

CkJJJf

=−+−+−==

===

213

232

22123,2,1

2232

61

,

σσσσσσσ

Maximum Distortion Energy Theory:

Based on uniaxial yielding: 0, 321 === σσσσ

( ) ( )

33

;61,

22

22232

ySC

CJJJf

==

=+==

σ

σσ

Maximum Distortion Energy TheoryMax. Shear Stress Theory

For shear:

2

;2)( 3,2,1

y

y

S

SCf

=

===

σ

σσ

Sy

Sy

2yS

2yS

16

Plasticity

( )

( )

( ) 0,

0,

0,

=∂∂

==

<∂∂

==

>∂∂

==

ijij

ij

ijij

ij

ijij

ij

fdfCf

fdfCf

fdfCf

σσ

σ

σσ

σ

σσ

σLoading:

Unloading:

Neutral:

1

2

3

Isotropic hardening

SY

SY

S<SY

Bauschinger effect

S`Y

Plastic State:

Elastic State:

Impossible:

( )( )( ) 0

0

0

>

<

=

ij

ij

ij

f

f

f

σ

σ

σ

17

Strain HardeningIsotropic Hardening

Kinematics Hardening

Combined Hardening General Hardening

ijij

fddσ

λε∂∂

=Normality Rule:

Convexity

18

2. Behavior in Metal Forming• The stress-strain relationship beyond elastic range

assuming no unloading at anytime and anywhere.

• Flow stress – The instantaneous value of stress required to continue deforming the material.

• Average Flow Stress

• For any metal, K and n in the flow curve depend on temperature

nKεσ =

nf KY ε=

nKdKd

Yn

f

f

n

ff

ff

+===

∫∫1

00 εε

εε

ε

εσεε

fYYs

εf

fY

19

3. Temperature in Metal Forming• Cold Working – Performed at room

temperature or slightly above – Near net shape or net shape– Better accuracy, closer tolerances & surface finish– Strain hardening increases strength and hardness– Grain flow during deformation can cause desirable

directional properties in product– No heating of work required– Higher forces and power required– Surfaces must be free of scale and dirt– Ductility and strain hardening limit the amount of

forming

20

Warm Working• Performed at above room temperature but below

recrystallization temperature• 0.3Tm, where Tm = melting point (absolute

temperature)– Lower forces and power than in cold working– More intricate work geometries possible– Need for annealing may be reduced or eliminated

• Isothermal Forming – eliminate surface cooling especially highly alloyed steels and Ti and Ni alloys

21

Hot Working• Deformation process at temperatures above

recrystallization temperature (0.5Tm) • A perfectly plastic material - Strain hardening

exponent is zero (theoretically)– Lower forces and power required– Metals become ductile– Strength properties are generally isotropic– No work hardening of part– part can be subsequently cold formed– Lower dimensional accuracy– Higher total energy required– Poorer surface finish including oxidation (scale), – Shorter tool life

22

4. Effect of Strain Rate

• Sensitive to strain-rate at elevated temperatures• Strain rate: • Relationship:• A more complete relationship:• Evaluation of strain rate is complicated by

– Workpart geometry– Variations in strain rate on the part

• Strain rate can reach 1000 s-1 or more for some metal forming operations

hv

=ε&m

f CY ε&=mn

f AY εε &=

23

Effect of temperature on flow stress

Increasing temperature decreases C & increases mAt room temperature, effect of strain rate is almost negligible

ε&

Yf

C

1.0 2.0 3.0

Yf

1 100.1 100 ε&ε&

Yf

C

1 100.1 100

m

Room Temperature

400°C

800°C

1200°C

24

5. Friction and Lubrication• Friction – retard metal flow and increase power and wear

• Lubrication – reduce friction & heat, improve surface finish– Choosing a Lubricant – Type of operation, reactivity, work

materials, cost and ease of applications• Cold working – mineral oil, fats, fatty oils, water-based emulsions,

soaps and coating• Hot working – mineral oil, graphite and glass

Categories Temp. Range Strain-rateSensitivity exponent

Coefficient of Friction

Cold Working <0.3Tm 0<m<0.05 0.1

Warm Working 0.3Tm-0.5Tm 0.05<m<0.1 0.2

Hot Working 0.5Tm-0.75Tm 0.05<m<0.4 0.4-0.5