elasto-viscoplasticity - mms2

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ATHENS
Course MP06
Nonlinear Computational Mechanics
March 16 to 20, 2009
ELASTO-VISCOPLASTICITY
Jean-Louis Chaboche
ONERA, 29 av. de la Division Leclerc
92320 Châtillon, France
[email protected]
ATHENS – Course MP06 – 16 – 20 March 2009
Elasto-viscoplasticity
Classical elasto-viscoplasticity
VARIOUS VISCOSITY AND CREEP EFFECTS
NOTION OF VISCOPLASTIC POTENTIAL
EXAMPLES OF SECONDARY CREEP AND
MULTIPLICATIVE HARDENING RULE
ELASTO-VISCOPLASTICITY BASED ON
NON-LINEAR KINEMATIC HARDENING
STATIC RECOVERY EFFECTS
IDENTIFICATION OF UNIFIED VISCOPLASTIC
CONSTITUTIVE EQUATIONS
Viscosity effects
σ
ε =∞
σ
ε
ε=
ε
ε
Strain rate effects on the monotonic and cyclic behaviour
σ
∆σ
ε= &
−
&
−
&
&
'
−
−
−
&
(
(
' !
!
∆ε % "#$
ε
) *
Two strain rates test
+
', &,
ε =
−
−
ε =
−'
−
-
!
Modelling of « creep-forming »
process : Metal forming by
simultaneous creep and
heat treatment
conical panel
Test
model
mold
Final state, T 7351 - 173°C
'
Creep tests
ε
constant stress and temperature
ε
σ =!
+
,
" . $/
+
ε = ," σ $
β
0β 0
Secondary creep – Norton’s law (1930)
σ
ε =
Λ
21
1
σ
ε1
Strain-hardening and time-hardening assumptions
7
3 +
4
56
σ
σ
σ
+
σ
ε
+
.
Elastoplasticity and viscoplasticity : equipotential surfaces
7
Ω =∞
Ω=
"
"*
$
$
Strain rate effect and equipotentials
σ
σ
ε =∞
8=
σ7
ε=
Ω=Ω∗
7
=
ε
ε
ε=
σ
ε =∞
Strain rate – stress relationship at fixed hardening
(σ )
+
σ
6
6
σ 9
9
σ7 9 7
7
(ε )
Viscoplasticity equations – Normality rule
potential :
Ω = Ω (σ -
- :)
hardening
variables
σ
Ω 9Ω∗
normality :
Ω9
:;<
σ7
σ
∂Ω
∂σ
σ
σ
∂Ω
ε =
∂σ
Odqvist law
secondary creep
?
/
ε
=
∂Ω
∂σ
=
Λ
Ω=
+
∂Ω ∂σ
∂σ
4
4
∂σ
=
σ 4
Λ
σ
+
4
σ
Λ
σ 4
=
Λ
5 +
3
+/
4
2
/
,
=
= 4
7
σ 4
=
Λ
−
∆
<5
> *+
'
Rabotnov – Lemaitre equation (1969)
,
;
/
ε = ," σ $
ε =
β
β
β−
," σ $
ε=
+
+
B
;
=
+
σ
4
A
σ
A
−
+
2
ε
$ ;
ε =
+
@
−β
6
6 /
∂Ω
=
∂σ
σ
4
B
σ
Λ
β
"
A
Ω=
+
(
," σ $) β ε
β
; 2
ε =
7
primary creep
4
=A
7
+
+
−β
Modelling of creep tests
C2
ε "#$
!
.B
σ
ε "#$
E
9
σ 9
E
.B
σ 9
σ 9
σ 9
σ 9
σ 9
6 "+
σ 9
7 $
2
D
A
+
"
7 $
Viscoplasticity with Non-Linear Kinematic hardening
Ω=
potential :
normality :
multiplier :
ε
=
∂Ω
∂σ
A
+
F (σ −
)− : − <
+
!
)
. '
A
−
=
F (σ −
G
)
=
∂Ω
F (σ − ) − : − <
=
=
∂σ 7
A
Non-Linear Kinematic hardening :
=
=
ε /ε
!ε −@
isotropic hardening superposition :
• on the yield stress :
:
• on the hardening modulus : @ 9 @":$
• on the « drag stress » :
A 9 A ":$
.
Microcreeps after creep and unloading
(« dip-test » technique )
8 ! ?
!
−' −
σ *θ "BE $
ε ** = &
ε *θ = &
−' −
'
'
'
σ ** " BE $
DB, >
H
Modelling of creep tests
C2
ε "#$
!
.B
σ
ε "#$
E
9
σ 9
E
.B
σ 9
σ 9
σ 9
σ 9
σ 9
6 "+
σ 9
7 $
2
D
A
+
"
7 $
Modelling of relaxation tests
σ " BE $
C2
!
6
7
+
+
7
+
"$
Strain recovery
σ
σ
σ7
+
<
3
ε
ε
ε
<
=
!ε −@
4
ε
Static or thermal recovery
5
+
7
/
+)
σ
J
σ
@
> !
6 +
J
>
@
+
!
,
,
&
+
B
I
ε
B
ε
I
7
'
Static or thermal recovery
A
+
"
/
"E
$=
+
7
$
! ε −@
=
−
"
+
$
7
/
: = ) (K − : )
5
+
7
+
C
/
B
7
/
"&$
− ":$
"&$
5
Static recovery on kinematic hardening
/
ε =
)
6
σ−
)
−<
A
7
ε
/
σ
Static recovery effect in the two-level creep test
7
6
σ
σ
ε
σ
σ
σ
σ
<σ
+
.
Identification steps
E
/
+
/
7
+
7
7
?
/
<
+
6
σ
−'
+
<
)
+
/
- <- A
<ε <
−
−
Plasticity – Viscoplasticity adjustment
σ
ε1 ≅ ε1 = !
σ
σ 7 (ε
<
)
<1
<1
<
<
<1
ε
ε
<
ε =
( )
< = <1 + σ 7 ε 1
7
Intrinsic dependancy between threshold and exponent
2 9
6
6
ε =
2
σ −σ
A
σ
D(
(
!
2
σ
σ
σ
9
9 ' BE
BE
ε
σ =
)
+ :+<
σ
σ
Intrinsic dependancy between threshold and exponent
σ−σ
σ 9
B
/
D(
(
!
σ 9 ' B
/
L
M
2
σ
σ
ε
9
9 ' BE
BE
σ
Identification steps
E
/
+
/
7
+
7
<
)
+
<
+
7
?
/
σ
−
+
(
7
6
/
- <- A
/
−
"+
5 +
N +
$
<ε <
<ε <
−
−
−
−
Viscosity with two slopes
(E
σ7 = σ −σ
=
=
A =
A =
(
ε =
(
σ7
A
+
!
σ7
A
7
ε
" $
'
Zener – Hollomon master curve
=ε1
"5 $
σ
A"5 $
= θ "5 $ P
&/ P (&&&) =
σ
σ 1 "5 $
5
P
5
5
σ
σ
8
- O < - ..
(&&&)
σ

elasto-viscoplasticity - mms2

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