1 Plastic Deformation 2013 (1)

7/27/2019 1 Plastic Deformation 2013 (1)

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1

1: Plastic deformation

Stefan Jonsson

2013

Technological and true definitions

Le

ln

L

L

L

dLL

LdL

d

0A

Ps

0

A

P

L0 + L = L

P P

A0

A


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2

Elastic and plastic volumechange

000 LA

P P zzyyxxel

V

V

0

000 LAALV

Volume is unaffected by plastic deformation

because no atoms are added/removed.

True & technological relations

0 LLLL

000 LLL

)1(0

0

0000esL

LLsLA

LP

LAA

ALP

A

P

e

s

.

variables with the wanted ones.

Remember V = constant


3/24

3

True and technological curves

400

450

150

200

250

300

350

s

,,

[MPa]

true valuesSSAB, Tunnplt

es

< e

> s

0

50

100

0 2 4 6 8 10 12 14 16 18 20 22

e , , [%]

tech valuesDomex DD 200, Rolling directionstrain rate: 5E-03 (1/s)

Addition of strains

toteLLLL

eee

0

03

2

23

1

12

0

01321

tot

L

L

L

L

L

L

L

LL

dL

L

dL

L

dL

L

dL

3

0

3

2

2

1

1

0

321

Only true strains can be added linearly


4/24

4

Elastic and plastic deformation

petot

pe

y

y

p e

Elastic and plastic deformation

y

Elastic deformation,

full recoverable

Elastic & plastic

deformation,

partly recoverable

p eAs long as there is a stress,

there is an elastic deformation!


5/24

5

Necking

P2 P2

P1 P1

A B t0

t1

t2

xy

z

m03:i

n

A

B

t1 t2

1

t0

Deformation is

homogeneous until

onset of necking

y

1

n

A

B

t1

t2

t0

When a neck is

formed, deformation

becomes highly

localized

Instability criterion for straining

ddAdA

ddAddA

dAdA

dP

P=[(y)]A[(y)] V=A()L() ?

dydddyddyddydydy

0

AddA

Ld

dL

Ad

dA

Ld

dL

Ad

dA

Ld

dV

AdA

For homogeneous

0

dy

dA

d

d

dy

dP

0

dy

d0

d

d

Hom. Deform. Necking


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6

Macro- & microscopic deform.

P P

deformation

is only an

average over

the sample

volume.

Much

information is

hidden.

The Bauschinger effect

soft

, ||

f

b

easier

permanent softening

f


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7

Definitions

o po n s

on the

stress

s

Yield stressProportionality

(Ultimate)

Tensile strength

curvee

0.2%homogeneousdeformation

necking fracture

Lders strain

Propagating y

er s an s

Inhomogeneous

deformation Upper & lower

yield stress

u

L

Continues with

normal plastic

deformationL

l


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8

Strain rate sensitivity

)( fm

logln

m

)(logloglog fm

ogn Increasedstrain rate

Elongation v.s. m-value

011

md

m

f

/

)(

1

Deformation

continues outside

the neck

Grain boundary

sliding gives high

m-values


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9

Superplasticity

Grain boundary sliding

Limited to a range in T and strain rate

Total strain energy

ddLPdLPdW

200

250

300

350

400

450

,

[MPa]

2

1

dWV

0

50

100

150

0 5 10 15 20

, [%]

Per volume unit!!


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10

Elastic strain energy

Eheigthwidth

W elV

2

250

300

350

400

450

,[MPa]

0

50

100

150

0 5 10 15 20

, [%]

Plastic strain energy

400

450

400

450

0

50

100

150

200

250

300

350

0 5 10 15 20

, [%]

,

[MPa]

0

50

100

150

200

250

300

350

0 5 10 15 20

, [%]

,

[MPa]

pl

V

pl

V

el

V

tot

V WWWW


11/24

11

Forcesonabody

2

3Face 3

1

Face 2

Face 1

F

FN

F

F2

T

F = FT + FN = F1 + F2 + F3

Forcesoneach surface produces 1normalstress

and2perpendicular shear stresses

Stress components

2

333

333231

232221

131211

ij

1

11

22

Firstindex: Forcedirection

Secondindex: Faceindex

33

2322

131211

ij

ij = ji32 32

23

23Symm.

6independentelementsofthestresstensor


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12

Normalstrains,iiMechanical equilibrium,i.e.no

accelerationsarecreated by

11

3

11

22

applied stresses.

11

22 2233

1 2

33

03322110

VV

33

0332211

Normalplasticstrain shall not

produce volume change.

Positiveshear stress;12=21

2

3

12-plane

Neutral direction

1

Mechanical equilibrium,i.e.no

rotationsarecreated byapplied

shear stresspairShear deformationcannot

produce volume change.


13/24

13

Negativeshear stress;12=21

2

3

12-plane

Neutral direction

1


2

3Neutral direction

1

13-plane


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14


2

3Neutral direction

1

13-plane


2

3

Neutral direction

1

23-plane


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15


2

3

Neutral direction

1

23-plane

Shear strains;ij=ji

1 2

3

NOTE!ij=ji


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16

Definitionofstrain tensorDisplacements u1,u2andu3are

found bymoving inx1,x2andx3

i

j

j

iij

x

u

x

u

2

1

Thetensor comp.isacombination

oftwo dis lacements u andu

Thetensor comp.isthe

displacement,uifound

bymoving inxi

i=j

i

1 2

3

found bymoving inxjandxi

ji

j

i

i

j

i

j

j

iij

x

u

x

u

x

u

x

u

2

1

2

1

Engineering shear, y

Pure shear ySimple shear

x2

1

x

1

Produced bydislocation slip

21 jiijij

ijjiiji

j

j

iij

x

u

x

u

2

1

2

1

2

1

This is only half of the

engineering shear!!


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17

Strain components, ij

0

02

1

00

00

3231

2321

33

22

332313

232212

ij

5 Independent strains: 2 3

Symm. Symm. Symm.

AngularVolumetric

03322110

VV

5 independent slip systems must be activated to produce a general deformation.

Fragmentation of grains reduces this number locally.

e orma one orma on

Stress states

y

x

h

tanh

cos

yy

xx

xy

xy

2cos2sin2

cossin2sincos 22

xy

xxyy

xyyyxx


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18

Mohrs circle for stresses (xx ,xy)

R (, )

(1 ,0)(2 ,0)

( ,-x )

22

2sinxy

R

R

R

yyxx

yyxx

)(2

1

)(2

1

2

1

2

2

2

22cos

xy

yyxx

yyxx

R

R

Rotationbetween principaldirections

z

z

=90 =45 0

y

y

z

y

2=180 2=90 20


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19

45 direction isbestforone principal

stress

zz zz zz

=45

zz zz zz

2=90

2

zz

General stress state

31

123

z

zz

yyxx

yzzy

zx

yx xy

xz

2max

x


20/24

20

123

Hydrostatic

pressure

hydrostatic

pressure

123

Tri-axial stress state

)(2

131max

3

123

1

2

3


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21

Spherical stress state

3=2=1

Cylindrical stress state

13=2


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22

Planar stress state

13= 2Free surface

Membranes

12=3

23

1=

Slipsystemsinfcc

110111typeofsSlipsystem12

110typeofvectorsBurgers3

111typeofslipplanes4

0bn


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23

Slipsystemsinbcc

110typeofslipplanes6 112typeofslipplanes12

111110typeofsSlipsystem12 ypeovec orsurgers

111112typeofsSlipsystem12ypeovec orsurgers

Totally 24 slip systems are easily activated.

Critically resolved shear stresses, CRSS, are

different for {110} and {112}

SlipsystemsinhcpEasily

activated

0211(0001)typeofsSlipsystem3


0001(slipplanebasal1 )

02110011typeofsSlipsystem3


0011slipplanesprismatic3

6 easily activated slip systems, different CRSS


24/24

Schmids formula,CRSS,c

0P0P

cos0PPb

b1

b2

b3

cos0A

coscos

cos/

cos

0

0

A

Pgb

dA=ab/cos

=dA0cos

a/cos

cos0PPn

c coscos

A0a b

dA0=ab

Taylors formula (inverted Schmid)

c coscos dddW

cy Single

cc m

coscos

1

cy m

m

ddd

y

c

m

Averaging

Pol cr stals

06.3

06.3

8.2

hcp

fcc

bcc

m

m

mRandom

orientations

bcc deforms best

hcp deforms worst

Documents

1 Plastic Deformation 2013 (1)