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Constitutive
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Constitutive models
Part 2
Elastoplastic
Elastoplastic material models
• Elastoplastic materials are assumed to behave elastically up to a certain stress limit after which combined elastic and plastic behaviour occurs.
• Plasticity is path dependent – the changes in the material structure are irreversible
Stress-strain curve of a hypothetical materialIdealized results of one-dimensional tension test
Engineering stress
Engineering strain
Yield pointYield stress
areainitialforce /
l
l0
Johnson’s limit … 50% of Young modulus value
Real life 1D tensile test, cyclic loading
Hysteresis loops moveto the right - racheting
Where is the yield point?
Conventional yield point
Lin. elast. limit
Mild carbon steel before and after heat treatment
Conventional yield point … 0.2%
The plasticity theory covers the following fundamental points
• Yield criteria to define specific stress combinations that will initiate the non-elastic response – to define initial yield surface
• Flow rule to relate the plastic strain increments to the current stress level and stress increments
• Hardening rule to define the evolution of the yield surface. This depends on stress, strain and other parameters
Yield surface, function• Yield surface, defined in stress space separates stress states
that give rise to elastic and plastic (irrecoverable) states• For initially isotropic materials yield function depends on
the yield stress limit and on invariant combinations of stress components
• As a simple example Von Mises …• Yield function, say F, is designed in such a way that
plasticity analyticalfor leinadmissib outside,0
surface theon0
surface the withinstate stress0
F
F
F
0yieldeffective F
0...),,( P KF ijij
Three kinematic conditions are to be distinguished
• Small displacements, small strains– material nonlinearity only (MNO)
• Large displacements and rotations, small strains– TL formulation, MNO analysis
– 2PK stress and GL strain substituted for engineering stress and strain
• Large displacements and rotations, large strains– TL or UL formulation
– Complicated constitutive models
Rheology models for plasticity
Ideal or perfect plasticity, no hardening
Loading, unloading, reloading and cyclic loading in 1D
stre
ss
strain
+ n
ew
yie
ld
stre
ss 1
- n
ew
yie
ld s
tre
ss 1
initi
al y
ield
str
ess
Isotropic hardening
ne
w y
ield
str
ess
2 loading
unloading
reloading
Isotropic hardening in principal stress space
321Y31 ,0)(
tensionin stress yield1D and stresses principalby expressedTresca
F
02])()()[(
tensionin stress yield1D and stresses principalby expressed Misesvon2Y
213
232
221 F
- plane
arccos (2/sqrt(3))
stre
ss
strain
+ n
ew
yie
ld
stre
ss 1
initi
al y
ield
str
ess
Kinematic hardening
loading
unloading
reloading
Loading, unloading, reloading and cyclic loading in 1D
Kinematic hardening in principal stress space
constant ..., where,0)( takewe
hardening) isotropic of case in (as0)( of insteadP ccF
F
ijijijij
ij
Von Mises yield condition, four hardening models
1. Perfect plasticity – no hardening 2. Isotropic hardening
3. Kinematic hardening 4. Isotropic-kinematic
Different types of yield functions
),,(
have could weall,at generalnot is whichGenerally,
invariant. anusually , of functionscalar a is)( e wher
hardening isotropic),(
waydifferent a in ofcomponent every on depends hardening
hardening isotropic-non),(
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hardening kinematic)(
. strain plasticpermanent theon depends h whic
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... nsdislocatio of nition Defi
ns.dislocatio of motionby caused isflow material tic Plas
region. plasticity thearound
exists whichstructure material healthy'' by the stabilized isit practiceIt
forever. so do toinclided is andflow tostarts material hardening, no means
plasticityperfect )(
P
PP
P
P
P
KFF
KK
KFF
FF
cc
FF
FF
ijij
ijij
ij
ij
ijij
ijij
ijij
ij
ij
Plasticity models – physical relevance
• Von Mises- no need to analyze the state of stress- a smooth yield sufrace- good agreement with experiments
• Tresca- simple relations for decisions (advantage for hand calculations)- yield surface is not smooth (disadvantage for programming, the normal to yield surface at corners is not uniquely defined)
• Drucker Pragera more general model
1D example, bilinear characteristics
PET ddd
d
plasticelastic
strain
stress
total
PT
ddd
EEE
EPT ddd Etan
Ttan E
Strain hardening parameter
Y
HEE
E
EE
EEE
/1 T
T
T
TP
… means total or elastoplastic… elastic modulus
… tangent modulus
Strain hardening parameter again
Elastic strains removed
Initial yield
Upon unloading and reloading the effective stress must exceed
Geometrical meaning of the strain hardening parameter is the slope of the stress vs. plastic strain plot
How to remove elastic part
T
TP EE
EEE
1D example, bar (rod) elementelastic and tangent stiffness
L
A
F F
Y
Y L
EAFk
E
L
AE
L
AFk
PE
PP
TT dd
d
d
d
d
d
P
P
P
PPT 1
/d/d
/Ed
EE
E
L
EA
EEL
AEk
Elastic stiffness
Tangent stiffness
Results of 1D experiments must be correlated to theories capable to describe full 3D behaviour of
materials
• Incremental theories relate stress increments to strain increments
• Deformation theories relate total stress to total strain
Relations for incremental theoriesisotropic hardening example 1/9
tt d
dlim:rates and increments between Relation
0
surface yield back to go0
0 that meansit - neutral0and0
ticelastoplas0and0
elastic0and0
elastic0if
and on dependsn deformatio ofincrement
0),( is surface yield Let the
Peff
eff
eff
eff
P
F
F
F
F
F
F
F
ij
ijij
Parameter only
Relations for incremental theoriesisotropic hardening example 2/9
Eq. (i) … increment of plastic deformation has a direction normal to F while its magnitude (length of vector) is not yet known
defines outer normal to Fin six dimensional stress space
0d so ns,deformatio plastic during zero bemust which
ddd
aldifferenti totala as expressed becan
}{
andscalar unknown far so is where
1947) (Drucker, form in the assumed is rule Flow
PP
PP
T
3111
P
F
FFFFF
F
FF
F
ijij
ijij
ijij
ijij
ijij
q
q
Relations for incremental theoriesisotropic hardening example 3/9
elastic total plastic deformations
matrix of elastic moduli
(iii) eq. )(
are increments stress
(ii) eq.0dd
form thein expressed be can 0d condition the
}{ Denoting
PE
PTTPTT
TP31
P11
εεEεEσ
εpqεpq
p
F
FF
Relations for incremental theoriesisotropic hardening example 4/9
qEqqp
EqTT
T
get we(iii) increments stressfor and (ii) 0d
(i), ruleflow for relations theCombining
F
Dot product and quadratic form … scalar
Row vector
Column vector
Lambda is the scalar quantity determining the magnitude of plastic strain increment in the flow rule
Still to be determined
Relations for incremental theoriesisotropic hardening example 5/9
PPE with)(
writecan weincrement stress for the Now,
εεεEεEσ q
determined be tohas still where
)(
with
form theinincrement strain totalof functiona as
increment stress get the wefor ngSubstituti
TT
TEP
EP
p
Eqqqp
EqEqEE
εEσ
equal to zero for perfect plasticity
diadic product
Relations for incremental theoriesisotropic hardening example 6/9
ijij
tt
ijij
tt
t
t
ijt
ijij
ijij
t
WF
AW
W
W
F
WWf
fF
ssJ
JF
FF
P
P
Y32
Y
PY
Y32
Pij
P
PY
YPij
Pij
PPY
PYP
21
D2
2Y3
1D2
TP31
P11
and using
F
rule Chain
increments plasticby donework d),(
at suggest th sExperiment
)( need we evaluate to
invariant deviatoric second theis where
0 condition yield Mises vonAssume
}{ of ionDeterminat
p
A new constant defined
At time t
Relations for incremental theoriesisotropic hardening example 7/9
PE
PETE
Y0
Yt
P0 Pt
PW
T312211
T
TP
Y
P
YPY
Y
P
2Y
02YP
P
PPY
0Y
t
PY
0Y2
1P
}{finaly so
3
2
3
2
3
2
)(2
1
)(stics characteribilinear 1D
)( done work elastic the1D in
A
EE
EEE
EA
E
W
EW
E
W
tt
t
t
t
t
tt
p
W
Relations for incremental theoriesisotropic hardening example 8/9
εEσ
bbEE
bqEqqEqbqp
p
q
s
σε
EP
TEP
TTT
T312312332211
T
T
T312312332211
T312312m33m22m11
T312312332211
33221131
m
Y
,,
}{
3
2
}222{
}{}{
)(
follows as compute can we and and given For Summary.
ca
ca
A
EE
EEA
ssssss
ssssss
ijij
J2 theory, perfect plasticity 1/6 alternative notation … example of numerical treatment
)2,2,2,1,1,1(diag][ with},]{[}{or
)222(
deviator stress ofinvariant second
}{}{
deviator stress
stress mean)(
}{}{
}{}{
law sHooke'}...]{[}{
T21
2
22222221
2D2
Tmmm
31
m
T
T
MsMsJ
ssssssJJ
s
E
zxyzxyzzyyxx
zxyzxyzzyyxx
zzyyxx
zxyzxyzzyyxx
zxyzxyzzyyxx
J2 theory, numerical treatment …2/6
Yeff
T2eff
TT
behaviour plasticperfectly for criterion yield
2/}]{[}{33 stress effective Misesvon
)1/(with},{2}]{][[ also and
0since},]{[}{}]{[}{
thatprove can one
sMsJ
EGsGsME
ssssMsMs zzyyxx
J2 theory, numerical treatment …3/6
endif
0else
,0then if
by expressed be can region elastic in ndeformatio plastic no
increment ,derivative timeits...}{2}]{[}]{[}]{[}{
law sHooke'...}]{[}]{[}{
parameter unknownfar so is...}]{[}{
}{
hypothesis Reuss- Prandtl toaccording ruleFlow
Yeff
P
PE
T
sGEEE
EE
sMF
Six nonlinear differential equations + one algebraic constraint (inequality)There is exact analytical solution to this. In practice we proceed numerically
J2 theory, numerical treatment …4/6
T2Y
EPEP
2Y
T
2Y
T
2Y
2eff2
T
TT
TT
eff
T
Teff
eff
Yeff
3with
finally
2
3
4
3
get we
3/43/442
that realizing and
2
for ngSubstituti
0 also and 0
02
3
condition plasticity atingDifferenti
ssEEεEσ
εsεMEs
Mss
MssεMEs
σ
σMsMss
Msss
s
G
G
GGJGG
G
System of six nonlineardifferential equationsto be integrated
J2 theory, numerical treatment …5/6predictor-corrector method, first part: predictor
1. known stress
2. test stress (elastic shot)
3a. elastic part of increment
T)1( s r
Tsr
3b. plastic part of increment
Tc.4 sss rt
tσ
)2/()1(3.5 2Y
Tc εs r
εEσσσσ tt TT
cT' 2.6 sσσ Gtt
J2 theory, numerical treatment …6/6predictor-corrector method, second part: corrector
''
''
'eff
Y
Y'
eff
Y'
eff
Yeff
'
)1(
have weionsconsiderat plasticity intoenter not does
tensor stress theofpart spherical thesince and
)1(
)(
)(
)(
)(
a way that such in findFor
Correction
tttttt
tttttt
tt
tt
tt
tt
tttt
sσσ
sss
s
s
s
s
ss
Secant stiffness method and the method of radial return