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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??
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