Analysis of Multiaxial Elastic-Plastic Stressand Strain States at Notches Induced by

Non-Proportional Loading Paths

June 2nd – 6th, 2014,Aalto University,

Espoo, Finland

Prof. Grzegorz GlinkaUniversity of Waterloo, Canada

© 2010 Grzegorz Glinka. All rights reserved. 1

Multiaxial Elastic-Plastic Notch Tip Stressesand Strains Induced by Non-Proportional

Cyclic Loading Histories ??!!

Smooth laboratory specimens used for thedetermination of the material stress-strain curve

Stress and strain state in a specimen used for the determination of material properties

ij

xx

yy

zz

0 00 00 0

ij yy

0 0 00 00 0 0

6-8 mm

yy

x

y

z

yy

yy

yy

yy

xx zz yy

zz

xx

yy

yy0

E= yy/ yy= - xx/ yy= - zz/ yy

4

The stress state at a point in a body

kl

ij ijij A

kl k l

F FA x x0

lim

F1 F2

F3

F4

F2

xy

z

F1

Fz

y

zFz

Fz

FyFx

xxy

z

F1

Fz

Fy

Fz

FyFx

Fzz

Fzx Fzy

xx

x

z

yxx

xz

xy yy

yz

yx

zyzx

zz

5

11 ;

11 ;

11 ;

xyxx xx yy zz xy

yzyy yy zz xx yz

zxzz zz xx yy yz

E E

E E

E E

Strains in terms of Stresses in the elasticstate – The Hooke law

Valid when:

eq yield

eq xx yy yy zz zz xx xy yz zx

0 0.2

2 2 2 2 2 21 62

22

ABC

t

t

23

0

0A

B

C

23

220

ABC

t

t

23

0

0

22

A

B

C

23

22

0

Non-proportional loading path Proportional loading path

F2R

t

x2

x3

F

T

T22

33

22

33

x2

x3

23

Fluctuations and complexity of the stress state at thenotch tip

© 2008 GrzegorzGlinka. All rights reserved. 6

7

yyy

nyy

yyy e pyy E K

1

eyy

yy

pyyyy

yy H EE 011

Stre

ss

Strain yy

yy

pyy eyy

E

0

H

0

Stre

ss

Strain yy

yy

pyy eyy

E E

0

Various mathematical approximations of uniaxial material stress-strain curves in the elastic-plastic stress state

pyyyyyy

eyy yyEf

Ramberg-Osgood material curve

Bi-linear material curve

General form of the material curve

8

A key feature of the deformation theory of plasticity (proportional loading) is its predictionthat a single curve relates the equivalent stress, eq, and the equivalent plastic strain, peq,for all stress states.

, ( )p eq eqfThe function f( eq) depends on the material and it is the same as the uni-axial stress-strainrelationship determined from uni-axial tension tests, i.e.

1

, ,

nyy yy

yy e yy p yy E K

Uniaxial Ramberg-Osgood stress-strain curve

Plastic strain in theuni-axial stress state

Equivalent plastic strain inthe multi-axial stress state

1/ 1/

, ,;n n

yy eqp yy p eqK K

, ( )p yy yyf , ( )p eq eqfThe function f( ) can be determined experimentally from an uni-axial tension test (see examples);

9

Strains vs. Stresses in Hencky’s Total Deformation Theoryof Plasticity

1 1 1( ) ( )2

1 1 1( ) ( )2

1 1 1( ) ( )2

2(1 ) 3

2(1 ) 3

2(1 )

xx xx yy zz xx yy zzp

yy yy zz xx yy zz xxp

zz zz xx yy zz xx yyp

xy xy xyp

yz yz yzp

zx

E E

E E

E E

E E

E E

E3

zx zxpE

These are the Hencky equations of the total deformation theory of plasticity!!

eq eqp

peq peq

Ef

where:

10

Proportional Multiaxial LoadingStress-Strain Equations of Total Deformation Plasticity

i

peq

ij ij kk ijeq

j1+ 3- +

E E 2S

11 22 33

p p peq ij ijeq ij ij

ijij ij kk kk

3 2= S S =2 3

1S = - = + +3

Where:

2222 22f

E

Uniaxial

peq eqf

Multiaxial

11

Stresses in a prismatic notched body

k

n

ne tp a

FSA

andK

A, B, C

22

22

22

33

D

E

11

max

peak

n

22

11

33

3

2

A D B

C

F

F

1E

12

1

max

peak

n

22

11

33

3

2

A D B

C

F

F

Stresses in axisymmetric notched body

A, B

22

22

22

22

33

11

C

D

n

peak t n

FSA

andK

22a

11a= 0

32a

33a

31a 13

a= 0

12a= 0

21a= 0

23a

1

2

3

22

0 0 00 0 ;0 0 0

a aij

Stress State at The Notch Tip

22

33

0 0 00 00 0

a aij

a

Component in plane stress-uniaxial stress state at the notch

Component in plane strain-biaxial stress state at the notch

Neuber’s Rule The ESED Method

22e

22a

22e

22a

0

B

A

22e

22a

22e

22a

0

B

A A

22 22

2

22 22e e a atnK

E

22

22 2222

2

22

02 2

a

ae

n te

aKd

E22

22 22

aa af

E22

22 22

aa af

E© 2010 Grzegorz Glinka. All rights reserved.14

The Neuber Rule in the Uniaxial stress State(notch tip in plane stress state)

The strain-stress constitutive equations and the Neuber rule for a notch tip inplane stress ( 11 33 = 0) state can be written as:

2 2 2 211

2222

2222

22 22 2 2 22

......................( )

.............................( )

......................( )

....................................( )

( )2

( )

( )2

a aa

aa a

eq

a aeqa

ae ae

a

b

c

d

fE

fE

fE

a aeq 22Where:

aeqa a a a aaeq

aeqa a a a aaeq

aeqa a a aaeq

a ae e

fa

E

fb

E

fc

E

11 22 33 22 33

22 22 33 22 33

33 22 3

22 2

3 22

2 22 22

( )........( )

2

( )1 2 .......( )2

( )10 2 ........( )2

................................... d..................( )

The Neuber Rule for the notch tip in Plane StrainThe constitutive strain-stress equations and the Neuber rule for a notch tip inplane strain ( 11= 0 and 33 = 0) state can be written as:

where: 22 22 22 33

2 2a a a a aeq = - +

22a

11a

32a=0

33a

31a 13

a= 0

12a= 0

21a= 0

23a

11

22

33

0 00 00 0

a

a aij

a

Stress state away from the notch tip (in the symmetry plane)

1

2

3

11 11 22 33 11 22 33

22 22 33 11 22 33 11

33 33

( )1 1( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 2

( )1 1( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 2

1( )

aeqa a a a a a a

aeq

aeqa a a a a a a

aeq

a a

f xx x x x x x x

E x

f xx x x x x x x

E x

xE 11 22 33 11 22

11 11

22 22

33 33

11 11

22 22

33 33

( ) 1( ) ( ) ( ) ( ) ( )

( ) ( )

( ) (

( )( ) 2

( ) ( )

( ) ( )

( ) () ) )

)

( (

aeqa a a a a

a

e e

e e

e

e

e

q

a a

a a

a a

f xx x x x x x

x

x x

x x

x

x

x x

x x x

x

Equations of the multiaxial Neuber rule and Hencky’s plasticityequations (General tri-axial stress state)

1'

2 2 2

11 22 22 33 33 11

( )( )

'

1( ) ( ) ( ) ( ) ( ) ( ) ( )

:

2

naeqa

eq

a a a a a a aeq

xf x

K

x x

wher

x

e

an x x x xd

11 11 22 33 11 22 33

22 22 33 11 22 33 11

33

( )1 1( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 2

( )1 1( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 2

10 ( )

aeqa a a a a a a

aeq

aeqa a a a a a a

aeq

a

f xx x x x x x x

E x

f xx x x x x x x

E x

xE 11 22 33 11 22

11 111 1 1

22

1

22 22 22

( ) 1( ) ( ) ( ) ( ) ( )( ) 2

( ) ( )

( ) (

( ) ( )

( ) ( ) )

e

aeq

e

e

a a a a aa

ae

eq

a a

a

f xx x x x x

x

x

x xx

x xx x

Equations of the multiaxial Neuber’s rule andHencky’s equations of plasticity (Plane strain, 33=0)

1'

2 2 2

11 22 22 33 33 11

( )( )

'

1( ) ( ) ( ) ( ) ( ) ( ) ( )

:

2

naeqa

eq

a a a a a a aeq

xf x

K

x x

wher

x

e

an x x x xd

a

a2R

t

x2

x3

a

PT

0

00

0

94400 , 0.3,4720 , 550

forE

forE H

E MPaH MPa MPa

33

22

3.89 2.19

0.27 0.27

P Te

ne

n

K K

Cylindrical Notched Specimen under ProportionalTension and Torsion Load

Input: Pseudo-Elastic Stress and Strain components atthe notch tip (proportional loading)

n - nominal stress induced by the axial load, n = P/An = 0.27 n-nominal shear stress induced by the torque

The notch tip hypothetical (elastic) stress components

11e= 0; 22

en·KP; 33

e= 0.27 n·KP

12e= 0; 23

e= 0.27· n·KT; 31e= 0.27 n·KT

e ek

e2

3

0 0 00 00 0

e e e ee e

e e e ee e

2222 33 22 33

2 23

2222 33 22 33

3 23

2 2

2 2

Pseudo-Elastic Principal Stresses at the notch tip

e eij

e22

33

0 0 00 00 0

e

e eij

e

11

22

33

0 00 00 0

N Nij

N22

33

0 0 00 00 0

N

N Nij

N

11

22

33

0 00 00 0

Input: Pseudo-Elastic Stress and Strain components(Principal Stresses)

Output: Actual Elastic-Plastic Stresses and Strains(Principal Stresses)

Hooke’s law

Hencky’s Eqns

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 400 800

Notc

hTi

pSt

rain

,

2FE

M

2N

2E

3N

3FE

MNominal Stress, Sn= n

2+3 n2 (MPa)

0

100

200

300

400

500

600

700

800

0 100 200 300 400 500 600 700 800

Notc

hTi

pSt

ress

(MPa

)FEM

FEM

Nominal Stress, Sn= n2+3 n

2 (MPa)

Plastic deformation induced by non-proportionalloading paths

When the stress component ratios change during the loading process (i.e. if theload path is non-proportional the resulting elastic-plastic strains at the load pathend are not the same - in spite of the fact that the final loads or correspondingnotch tip hypothetical “elastic” stress states are the same.The three various load paths below will produce at the end three differentstress and strain states. In order to determine the final stress-strain stateat the notch tip the load path needs to be followed up incrementally asapplied!Therefore the constitutive stress-strain relationships must be written interms of increments because the current stress-strain increments dependon the previous stress/load path.!

a

a2R

t

x2

x3

a

PT

T

P0

Proportionalload path

11 11 22 33 11 22 33

22 22 33 11 22 33 11

33 33 11 22 33 11 22

12

1 12

1 12

1 12

1

aeqa a a a a a aaeq

aeqa a a a a a aaeq

aeqa a a a a a aaeq

a

E H

E H

E H

E 12 12

23 23 23

13 13 13

32

1 32

1 32

aeqa aaeq

aeqa a aaeq

aeqa a aaeq

H

E H

E H

Incremental Constitutive Stress-Strain Equations of theIncremental Theory of Plasticity

, ,

,

1

1

ap a p a

a a a

ap a

a

a

d fdd d H

or HH d f

d

a

p,a

,

,

p a a

p a aeq eq

f

f

a

p,a

The Generalized Constitutive ‘Stress-Plastic Strain’ Material Curve

Multiple Plasticity Surfaces of the Mroz-Garud Model

,3 32 2a a

eq

a p aeq

q

eq

e

dH

In order to make the numerical analysis easier and more general the constitutive stress-strain curveis approximated by a series of linear segments. It enables to model any stress-strain curve and notonly those approximated by the Ramberg-Osgood expression! Each segment is defined by its ownmodulus ‘H’!

As soon as the linear piece with the current stress state is identified (based on the current value ofeq) the elastic-plastic strain increments can be determined from the incremental constitutive

equations.

0

f1

f2

f3

0 a,peq

aeq

1

2

3

H

ys

ys

O1

O2

O1,

B1

B2

P

ij

f 1

f1‘

f2

n

n

In order to account for the memory effect the plasticity surface associated with thecurrent state (and also current linear segment) is translated in the stress spaceaccording to certain rules. The Mroz-Garud model requires such a translation of thecurrent plasticity surface , enforced by the application of stress increments ij, thatPoints B1 and B2, having the same normal directional vector ‘n’, must meet when thestress state reaches’ the end of the current linear piece of the stress-strain curve.

The essence of the plasticity model:The multiaxial plasticity model is required for two main reasons:

a) To identify the current ‘plastic modulus ‘H’ for the determination ofthe relation of plastic strain and associated stress increment, i.e.

/H.

b) To model the material memory effect, i.e. to select appropriatemodulus ‘H’ depending on the previous stress history.

c) The current modulus is ‘H’ defined by the active largest plasticitysurface.

d) When two plasticity surfaces touch each other (at the end ofprevious linear piece of the stress-strain curve) the larger surfacedefines the current plastic modulus ‘H’ and the current linear pieceof the stress-strain curve.

31

time

max

min0

stre

ss

Translation of plasticity surfaces due to uni-axial cyclic stress state

max

0

ys

0

min

ys

32

0

1,max

2,m

ax

A

0

1,min

2,m

in

B

time1,

max

0

stre

ss

1,min

1

2

2,m

ax

A

B

Translation of plasticity surfaces under bi-axial cyclic load

33

Translation of Plasticity Surfaces of the Mroz-Garud Modelfor under arbitrary Non-proportional Loading Path

0, 0300, 0.0017400, 0.0027450, 0.0033500, 0.0043550, 0.0058610, 0.0075650, 0.0101700’ 0.0159750, 0.023800, 0.0311

Material input file (the stress-strain curve)

Material stress-strain curve

0

100

200

300

400

500

600

700

800

900

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

Strain

Str

ess

Linear pieces

"Elastic" Stress History

0

100

200

300

400

500

600

700

0 50 100 150 200 250

Time

Stre

ss

Input Stress History (smooth material element)

-200

-100

0

100

200

300

400

500

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007

Strain

Stre

ss

SIG(1,1)SIG(1,2)

Resulting Elastic-Plastic Stress StrainResponse (smooth element)

Notched Bodies:

The cyclic plasticity model and theNeuber/ESED rule combined

11 12 13

21 22 23

32 32 33

a a a

a a a aij

a a a

Stress-Strain States at the Notch Tip - GENERAL

Elastic input

11 12 13

21 22 23

31 32 33

a a a

a a a aij

a a a

Elastic-plastic output

11 12 13

21 22 23

31 32 33

e e e

e e e eij

e e e

11 12 13

21 22 23

32 32 33

e e e

e e e eij

e e e

Incremental Neuber’s Rule Incremental ESED Method

0

ija

ija

ije

ije

ija

ija

ije

ija

a ae e ae e a

0

ije

ija

ije

ija

ije

ija

ije

ija

e e a a

11 11 11 11

12 12 12 12

13 13 13 13

22 22 22 22

23 23 23 23

11 11 11 11

12 12 12 12

13 13 13 13

22 22 22 22

23 23 23 23

33

a a a a

a a a a

a a a

e e e e

e e e e

e e e e

e e e

a

e

e e e e

a a a a

a a a

e

a

33 33 33 33 33 33 33ae a ae e a

General Incremental Multiaxial Neuber’s Rule

11 11 22 33 11 22 33

22 22 33 11 22 33 11

33 33 11 22 33 11 22

12

1 12

1 12

1 12

1

aeqa a a a a a aaeq

aeqa a a a a a aaeq

aeqa a a a a a aaeq

a

E H

E H

E H

E 12 12

23 23 23

13 13 13

32

1 32

1 32

aeqa aaeq

aeqa a aaeq

aeqa a aaeq

H

E H

E H

Incremental Constitutive Stress-Strain Equations

11 11

12 12

13 13

22 22

23 23

33

11 11

12 12

13 13

22 22

23 23

3333 33

e e

e e

e e

e e

a a

a a

a a

a a

e e

a

a

e e a

a

General Incremental Multiaxial ESED Rule

11 11 22 33 11 22 33

22 22 33 11 22 33 11

33 33 11 22 33 11 22

12

1 12

1 12

1 12

1

aeqa a a a a a aaeq

aeqa a a a a a aaeq

aeqa a a a a a aaeq

a

E H

E H

E H

E 12 12

23 23 23

13 13 13

32

1 32

1 32

aeqa aaeq

aeqa a aaeq

aeqa a aaeq

H

E H

E H

Incremental Constitutive Stress-Strain Equations

Notched Bodies:

Non-Proportional MonotonicLoading Path

(Mutiaxial Neuber/ESED method vs. FEM)

a

a2R

t

x2

x3

a

PT

0

00

0

94400 , 0.3,4720 , 550

forE

forE H

E MPaH MPa MPa

33

22

3.89 2.19

0.27 0.27

P Te

ne

n

K K

Material Curve ~SAE 1045

0

100

200

300

400

500

600

700

0 0.01 0.02 0.03 0.04

Strain

Stre

ss(M

Pa)

0

20

40

60

80

100

120

140

160

180

200

0 50 100 150 200 250 300 350 400

Axial notch tip stress, 22e (MPa)

Shea

rnot

chtip

stre

ss,

23e

(MPa

)

initiation of plasticyielding at the notch tip

Torsion-Tension Non-Proportional Loading Path(torsion followed by tension)

The loading path

0

0.0005

0.001

0.0015

0.002

0 0.0005 0.001 0.0015 0.002

Axial notch tip stress, 22a

Shea

rmot

chtip

stra

in,

23a

NeuberFEMESED

Shear vs. normal elastic-plastic notch tip strains

0

20

40

60

80

100

120

140

160

0 50 100 150 200 250 300Axial notch tip stress, 22

a (MPa)

Shea

rnot

chtip

stre

ss,

23a

(MPa

)

NeuberFEMESED

The shear vs. normal notch tip stresses

Notched Bodies:

Non-Proportional Cyclic Loading Paths(Mutiaxial Neuber/ESED method vs. Experiments)

Experimental data:Courtesy of D. Socie and P. Kurath,

University of Illinois, USA

a

a2R

t

x2

x3

a

PT

0

00

0

94400 , 0.3,4720 , 550

forE

forE H

E MPaH MPa MPa

33

22

3.89 2.19

0.27 0.27

P Te

ne

n

K K

Cylindrical Notched Specimen under Tension andTorsion Loading

(Non-Proportional Cyclic Loading Paths)

Material - 1070 Steel

0

100

200

300

400

500

600

700

0 0.002 0.004 0.006 0.008 0.01

Strain

Stre

ss(M

Pa)

E=210000 MPa, =0.3

K’=1736 MPa, n’=0.199

Rectangular Stress Path

-250

-200

-150

-100

-50

0

50

100

150

200

250

-500 -400 -300 -200 -100 0 100 200 300 400 500

Axial stress, 22e (MPa)

Shea

rstr

ess,

23e

(MPa

)

Hypothetical (elastic) shear vs. normal notch tipInput stresses

-0.0025

-0.002

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

-0.003 -0.002 -0.001 0 0.001 0.002 0.003

Axial notch tip strain, 23a

Shea

rnot

chtip

stra

in,

a

Calc.Exper.Exper.Exper.Input

Shear vs. normal elastic-plastic notch tip strains(calculated vs. measured)

-250

-200

-150

-100

-50

0

50

100

150

200

250

-400 -200 0 200 400

Normal stress, 22 [MPa]

Shea

rstr

ess,

23[M

Pa]

Calculated shear vs. normal notch tip stresses

-500

-400

-300

-200

-100

0

100

200

300

400

500

-0.003 -0.002 -0.001 0.000 0.001 0.002 0.003

Strain 22

Stre

ss22

[MPa

]

23 - 23

-250

-200

-150

-100

-50

0

50

100

150

200

250

-0.002 -0.001 0.000 0.001 0.002

Strain 23

Stre

ss23

[MPa

]

Elastic-Plastic Stress-Strain paths at the notch tip

Shear vs. normal notch tip hypothetical-elasticInput Stresses

Unequal-Frequency Stress Path

-250

-200

-150

-100

-50

0

50

100

150

200

250

-600 -400 -200 0 200 400 600

Axial Stress, 22e (MPa)

Shea

rStr

ess,

23e

(MPa

)

-0.0025

-0.002

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

-0.003 -0.002 -0.001 0 0.001 0.002 0.003

Axial notch tip strain, 22a

Shea

rnot

chtip

stra

in,

23a

Calc.Exper.

Shear vs. normal elastic-plastic notch tip strains(calculated vs. measured)

Elastic Stress Path,

-150

-100

-50

0

50

100

150

0 50 100 150 200 250

Normal Axial Stress, MPa

Shea

rStr

ess,

MPa

Axial stress-Axial strain,

-250

-200

-150

-100

-50

0

50

100

150

200

0 0.0005 0.001 0.0015 0.002

Axial strain

Axi

alst

ress

[MPa

]

ABAQUSNEUBER

Shear stress-Shear strain

-150

-100

-50

0

50

100

150

-0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015

Shear strain

Shea

rStre

ss,M

Pa

ABAQUSNEUBER

Material stress-strain curve

0

100

200

300

400

500

600

0 0.005 0.01 0.015 0.02 0.025 0.03

Strain

Stre

ss

FEM and Neuber elastic-plastic notch tip stress-strain components induced byasymmetric non-proportional stress-path

Cyclic Plasticity, Notches??

Enough for now!!!