Download - FY Lecture1
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Constitut ive modeling of large-strain
cyclic plasticityfor anisotropic metals
Fusahito YoshidaDepartment of Mechanical Science and Engineering
Hiroshima University, JAPAN
1: Basic framework of modeling
2: Models of orthotropic anisotropy
3: Cyclic plastici ty Kinematic hardening model
4: Applications to sheet metal forming and some
topics on material modeling
Lecture 1: Contents
Introduction:
purpose of constitutive modeling,
Stress and strain
Yielding of isotropic solids
Plastic potential and associated flow rule
Isotropic/kinematic hardening models
Isotropic hardening law
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Material behavior under uniaxial tension
(Tensile strength)
(yield strength)
yield point
stress
strain
necking
Void nucleation
Void growth/coalescence
Fracture
Necking occurs at a nominal stress peak, and it develops rapidly with increasing
strain. The specimen fractures as a consequence of void nucleation, growth and
coalescence.
Ductile fracture
Stress-strain curves of various metals
Experiment
Models
SNCM439
S35C
SUS304
BsBM1
A1100
Upper yield point
Elastic region
Lu ders bands
Lower yield point
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What material behaviors are our interests
in plasticity modeling?
Anisotropy (r-value, stress directionality)
Cyclic plasticity (the Bauschinger effect,
cyclic hardening, ratcheting, ,etc.)
Damage (evolution of voids, )
Rate-dependent behavior (viscoplasticity,
creep) Thermo-mechanical coupling
. etc.
98 TS
9 TS
Modeling ofAnisotropy andHardening ( - responses)
includingthe Bauschinger effect
Earing in cylindrical cup
deep drawing Springback
Barlat Yld2000-2d
Yoshida-Uemori
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Predictions of cracking and wrinkle
By PAM-STAMP 2G
(Yoshida-Uemori model)
Stamped panel
FE simulationBy JSTAMP
Cracking
Cracking
Sheet thinning
Photo
3D measurement
FE simulation
Deformation of solids
X
Current (t= t) configuration
Reference (t= 0) configuration
F : Deformation gradient
L : Velocity gradient
D : Rate of deformation
(stretching) tensor
W: Spin tensor
E: Lagrangian strain tensor
( ) ( )1 1
,2 2
/
iij
j
kkm
m
T T
T
e p
xd d , F X
vd d L
x
-
d dt
= =
= =
= + =
=
x F X
v L x,
D L L W L L
E F D F D
D = D + D
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Stress (1)
( ) ( )
1 2 3
[ ]
0, , ,
xx xy xz
ij yx yy yz
zx zy zz
p p
ij ij
= =
= =
Deviatoric stress and its Invariants
Cauchy stress, principal stress
( ) ( )1 2 31 1
3 3
ij ij m ij
m xx yy zz
s
=
= + + = + +
=hydrostatic stess (or mean stress)
Stress (2)
1 2 3
' ' '
1 2 3
1 13 , ,
2 3
1 10, ,
2 3
ii m ij ij ij jk ki
ii ij ij ij jk ki
J J J
J s J s s J s s s
= = = =
= = = =
Stress invariants
Jaumann rate (objective rate)
, ij ij im mj im mjW W W W = + = +& &
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Initial Yielding of Isotropic Solids
( ) ( ) ( )' '1 2 3 1 2 3 2 3, , , , ,f f f J J J f J J = = =
Since the yielding is not affected by the hydrostatic
stress component (i.e., incompressible), initial yielding
of an isotropic solid is expressed by the function (yield
function):
For example,
von Mises
Drucker
( ) ( )'3 '2 62 3 027D ijf s J J = = =
( ) ' 223
2M ij of s J = = =
Yield locus
Thin-walled tube in axial loads
& internal/outernal pressure
Yield locus is a description of yield criterion in stress space.
Thin-walled tube in axial loads
& torsion
von Misesvon Mises Tresca
Tresca
Crystal plasticity theory (FCC)
Carbon steel (S25C)
Stainless steel (SUS304)
Brass (BsBM1)
Aluminum alloy (A2017)
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Physical background:Plastic deformations in a crystal
{1 1 1}
[1 0 1]
atom( )
slip plane
grain
grain
boundary
1m100m
A few
Slip occurs most readily in specific directions (slip directions) on
certain crystallographic planes (slip planes) .
Why is the yielding not affected by hydrostatic stress?
Schmids law: Slip of a crystal occurs when the
resolved shear stress reaches its critical value, CRSS.
( ) cos cosR =
Resolved shear stress
Schmid factor
Yield cri terion for a crystal
( )R
crk = Critical resolved shear stress
(CRSS)Slip direction Slip plane
Keywords: Slip system = Slip plane and slip direction
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Resolved shear stress is not changed by the
hydrostatic stress (pressure):
At the atmosphere Under hydrostatic
pressure
a b =
Plastic potential & associated flow rule
Unloading
Neutral loading
LoadingInitial yield locus F
Subsequent yield locus f
Loading
Unloading
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Druckers postulate on stable stress-strain response
( ) ( )2
e p e pd d d d d E d d d = + = +
0, 0p pij ij
d d d d
(a) Stable (b) Unstable (c) Multiaxial stress state
( ) ( ) ( )
( )
( ) ( )
* * *
*
* *
0
0 0
e p
p
p p
ij ij ij ij ij ij
d d d
d
d d
= +
=
Stress cycle
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Principle of maximum plastic work
Convexity of yield locus
Normality rule for plastic
strain rate vector
( ) ( )* *0 or 0p pij ij ij ij ij ijd s s d
or
p
ij
ij
p
ij
ij
fd d
fd d
s
=
=
Associated flow rule
Yield locus of mild steel
Kuwabara
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Kinematic hardening model
Subsequent yield function
( ) ( )
( ) ( )
2
' 2
0
or 0
ij ij ij
ij ij ij
f Y
f s s Y
= =
= =
( )'3
2
p
ij ij ij
dd s
Y
=
backstress)Associated flow rule
Combined hardening model
Subsequent yield function
( ) ( )
( ) ( )
2
' 2
0
or 0
ij ij ij o
ij ij ij o
f
f s s
= =
= =
Associated flow rule
( )'3
2
p
ij ij ij
o
dd s
=
Appropriate evolution equations for isotropic hardening
and kinematic hardening is of vital importance.
ijY
oij
O
p
ijd
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Stress-Strain Response of a High Strength
Steel Sheet of 590 MPa Grade
Isotropic hardening (IH) model
Permanent
stress offsetTransient Bauschingereffect
Early
re-yielding
Hardening law
= description of
expansion of yield locus
(isotropic hardening)
movement of the center
of yield locus (kinematic
hardening: evolution of the
back stress)
ijY
oij
O
p
ijd
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Isotropic Hardening Law by means of
Effective Stress and Effective Plastic Strain
( ) ( ) 0ijF Y s Y = = =s
( ) ( )ij
s = =s
( ) ( ) ( ) ( ) 0ijf Y R s Y R = + = + =s
For initial yielding:
For the subsequent yielding:
Isotropic hardening stress
Effective stress:
For von Mises material:
'
2
33
2 i j i j
J s s= =
Effective Plastic Strain Increment d
p p p
ij ij ij ijdw d s d d = = =
2
3
p p
ij ijd d d =
Work conjugate formulation:
When using von Mises effective stress:
Effective plastic strain:
d dt = = &
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Linear hardening
Ludwik
Swift
Voce
Isotropic Hardening Laws
nY C = +
( )0n
C = +
For example,
( ) ( )Y R = = +
'Y H = +
[ ]1 exp( )SatY R = +
Uniaxial tension stress-strain curves
( )
( )
( )
p n
p n
p n
o
C
Y C
C
=
= +
= +
Ludwik
Swift
Perfectly plastic solid Linearly hardening
plastic solid
Non-linearly hardening
plastic solids
Power-law hardening