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BRITTLE FRACTURE & BRITTLE to DUCTILE FRACTURE TRANSIT ION MICROMECHANISMS
André PINEAUCentre des Matériaux - Ecole des Mines de Paris. ParisTech
UMR CNRS 7633andre.pineau@ensmp.fr
I. INTRODUCTION : Various Scales / History of Fracture Mechaniscs
II. BRITTLE CLEAVAGE FRACTURE – THEORYII.1. Main characteristics of Brittle Cleavage FractureII.2. Weibull – Beremin TheoryII.3. Multiple barrier modelsII.4. Effect of Deformation – Second Order approximation
III. DUCTILE FRACTURE – THEORY (Introduction)III.1. Cavity initiation, growth & coalescence. GTN modelIII.2. Computational strategies to simulate ductile crack growth
IV. DUCTILE TO BRITTLE TRANSITION IN STEELSIV-1. Qualitative explanations for the existence of DBTIV-2. Fracture toughness transitionIV-3. Impact Charpy test
V. CONCLUSIONS
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
2
BRITTLE FRACTURE & BRITTLE to DUCTILE FRACTURE TRANSIT ION MICROMECHANISMS
André PINEAUCentre des Matériaux - Ecole des Mines de Paris. ParisTech
UMR CNRS 7633andre.pineau@ensmp.fr
I. INTRODUCTION : Various Scales / History of Fracture Mechaniscs
II. BRITTLE CLEAVAGE FRACTURE – THEORYII.1. Main characteristics of Brittle Cleavage FractureII.2. Weibull – Beremin TheoryII.3. Multiple barrier modelsII.4. Effect of Deformation – Second Order approximation
III. DUCTILE FRACTURE – THEORY (Introduction)III.1. Cavity initiation, growth & coalescence. GTN modelIII.2. Computational strategies to simulate ductile crack growth
IV. DUCTILE TO BRITTLE TRANSITION IN STEELSIV-1. Qualitative explanations for the existence of DBTIV-2. Fracture toughness transitionIV-3. Impact Charpy test
V. CONCLUSIONS
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
3
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
1
10
100(m)
(µm)(nm)
1
10
100
1000
10
100
1000
First Talk
Second Talk
A. Pineau, IJF, 2008
Y. Bréchet, Mat. Nuc., 2008Pareige, JNM, 2007
2 ¼ Cr – E. Bouyne, 1998
4
Standard Fracture Toughness specimens and wide plate components linked with the equivalent CTOD ratio β
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
2c
a
2a
σσσσ∞∞∞∞
t
ca
σσσσ∞∞∞∞c a
a
a
σσσσ∞∞∞∞
t
σσσσ∞∞∞∞
σσσσ∞∞∞∞
t
σσσσ∞∞∞∞σσσσ∞∞∞∞
t
σσσσ∞∞∞∞
Standard fracture toughness specimens
Equivalent CTOD ratio β
Surface crack
Surface crack
Through-thickness crack
Through-thickness crack
Center surface crack panel (CSCP) Center through-thickness crack panel (CTCP)
Edge surface crack panel (ESCP) Edge through-thickness crack panel (ETCP)
B = t
W a0= BH H
S = 4W
1.25W
0.6W
a0 = BW
B = t
0.6W
Structural components
(0.45 < a0 / W < 0.55)
3-point bendCompact
5
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
1
10
100(m)
(µm)(nm)
1
10
100
1000
10
100
1000
Bouchard,(NH),1978, R.M. Pelloux
Fe-3%Si- Qiao, Argon, 2003
A. Lambert-Perlade, 2001
A. Lambert-Perlade, 2001
6
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
1
10
100(m)
(µm)(nm)
1
10
100
1000
10
100
1000
2IC m/J62G −=
2GBIC
GBIC
m/J20020G
mMPa62k
−=
−=
23IC
IC
m/J10100G
mMPa10050K
×=
−=
X 10X 1000
23IC
IC
m/J1082G
mMPa4020K
×−=
−=
X 0.10
X 1000
7
( )mma∆
J(KJ/m2)
�
�
��
��
�
DUCTILE TO BRITTLE TRANSITION - DEFINITION
� T1
T2 > T1
OR DECREASING STRAIN RATE
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
8
WARM-PRESTRESS EFFECT
*
LUCF
*LCF
T1TEMPERATURE
FRACTURE
TOUGHNESS
( )mMPa
T2
See also D.J. Smith, CSI 2003, pp. 289-345
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
9
Crack
εεεεσσσσ,
x
GLOBAL APPROACH
Fieldεεεε−−−−σσσσSingle parameterK, J, CTOD
K-T; J-QTwo parameters
LOCAL APPROACH
Crack Tip
Fieldεεεε−−−−σσσσ+ Physically-based
Fracture Criteria
COMPUTATIONAL FRACTURE MECHANICS
1960 1970 1980 1990 2000
LEFM
EPFM
1981ICF5
2010
LAF
1986
1st Sem.1992AIME
Sem.
19962nd Sem.
20063st Sem.
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
10
I - EXPERIMENTS
• Mechanical Tests
• Microstructural Observations
• Micromechanisms
II - MODELS
• Constitutive Equations
• Damage Modelling
III - IDENTIFICATION
• Tuning of Models Parameters
IV - SIMULATION
• Laboratory Tests
V - SIMULATION
• Industrial Components
• Complex Loading Conditions
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
11
MECHANICAL TESTS OBSERVATIONS
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
12
(((( )))) (((( ))))(((( )))) (((( ))))(((( ))))(((( ))))
2,1i;K
R11F1
p
1
RFp
pbexp1Qpbexp1QRT,pR
EQUATIONSVECONSTITUTI
in
ii
21
22110
====−−−−σσσσ====εεεεεεεε
++++εεεε
========
−−−−σσσσ====
−−−−−−−−++++−−−−−−−−++++====
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅
elsmodCoupled
elsmodUncoupled
RUPTUREDUCTILE
dVV1
exp1P
STRESSWEIBULL
FRACTURECLEAVAGE
MODELLINGDAMAGE
m/1mI
uw
m
u
wR
−−−−−−−−••••
σσσσ====σσσσ
σσσσσσσσ−−−−−−−−====
••••
∫∫∫∫
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
13
CLEAVAGE FRACTURE
Probabilistic Model
Weibull Stress
DUCTILE RUPTURE
Coupled Models
• Gurson-Tvergaard-NeedlemanPotential with an acceleratingfunction
• Rousselier Potential
• Statistical models in DuplexStainless Steels
Charlotte Bouchet
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
14
NS (AE)
CT
CHARPY V NOTCH
SIMULATION
B. Tanguy & J. Besson
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
15
SIMULATIONPRESSURE VESSEL
ALOHA, 1988
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
16
BRITTLE FRACTURE & BRITTLE to DUCTILE FRACTURE TRANSIT ION MICROMECHANISMS
André PINEAUCentre des Matériaux - Ecole des Mines de Paris. ParisTech
UMR CNRS 7633andre.pineau@ensmp.fr
I. INTRODUCTION : Various Scales / History of Fracture Mechaniscs
II. BRITTLE CLEAVAGE FRACTURE – THEORYII.1. Main characteristics of Brittle Cleavage FractureII.2. Weibull – Beremin TheoryII.3. Multiple barrier modelsII.4. Effect of Deformation – Second Order approximation
III. DUCTILE FRACTURE – THEORY (Introduction)III.1. Cavity initiation, growth & coalescence. GTN modelIII.2. Computational strategies to simulate ductile crack growth
IV. DUCTILE TO BRITTLE TRANSITION IN STEELSIV-1. Qualitative explanations for the existence of DBTIV-2. Fracture toughness transitionIV-3. Impact Charpy test
V. CONCLUSIONS
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
17
BRITTLE FRACTUREMAIN CHARACTERISTICS
• SCATTER
• SIZE EFFECT
• GEOMETRICAL DEPENDENCE
• LOADING RATE EFFECT
• EFFECT OF METALLURGICAL
VARIABLES & INHOMOGENEITIES
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
18
0
50
100
150
200
250
300
350
400
-200 -150 -100 -50 0 50
KIC (NS)
KIC (S)
KJ (NS)
KJ-Kcpm(S)
Kcpm (NS)
DIDR
Code RCC-M
Fra
ctur
e T
ough
ness
(MP
am1/
2)
T - RTNDT (°C)
C. Naudin, A. Pineau and J.M. Frund (2001), 10th Conf. on environmental degradation
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
19
Size Effect
I. Iwadate et al., Nucl. Eng. Design (1985), Vol. 85, pp. 89-99
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
20
Short
Crack
Effect
CONSTRAINT EFFECT- W.A. Sorem et al. Int. J. Fracture, (1991), 47, 105-126- D. Tigges et al. ECF 10, (1994), 1, 637-646
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
21
E36 Steel (-50°C)
0
50
100
150
200
250
300
0,00 0,20 0,40 0,60 0,80
a/W
J IC
(KJ/
m²)
SENB 0.03 < a/W < 0.77CCP 0.63 < a/W < 0.77
Sumpter 1993
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
CONSTRAINT EFFECT
22
0
50
100
150
200
250
300
350
-220 -200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20
TEMPERATURE (°C)
FR
AC
TU
RE
TO
UG
HN
ES
S (
MP
am1/
2 )
Pr=90%
Pr=10% Pr=90%
Pr=10%
.K
MPam1/2/s
4102
2
. •o Theoretical curves for failure
Probabilities of 10% and 90% (Beremin model)
Henry et al., 1985A 508 Steel
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
23
Weld structure Correspondingthermal cycles
CorrespondingHAZ microstructures
Run #1
Run #2
Run #3
A
BC
D
Coarse gained HAZ(CG HAZ)
Intercritically reheatedCG HAZ
Supercritically reheatedCG HAZ(fine grains)
~ CG HAZ
A
B
C
D
METALLURGICAL
INHOMOGENEITIES
A. Lambert-Perlade et al., Metall. And Mater. Trans. A (2004), Vol. 35A, pp. 1039-1053
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
24
10µm
10µm10µm
30 µmtransverse direction
short transversedirection
E 450BASE METAL CGHAZ-25
ICCGHAZ-25 CGHAZ-120
A. Lambert-Perlade et al., Metall and Mater. Trans. A, (2004), Vol. 35A, pp. 1039-1053
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
25
martensite
austenite
5 µm
1 µm
martensite
austenite
1 µm
martensite
austenite
MARTENSITE - AUSTENITECONSTITUENTS
E 450 STEEL
A. Lambert-Perlade et al., Metall. and Mater. Trans. A, (2004), Vol. 35 A, pp. 1039-1053
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
26
0
50
100
150
200
250
300
-200 -150 -100 -50
Base metal
KJc (MPa.m1/2)
T (°C)
base metal
0
50
100
150
200
250
300
-200 -150 -100 -50 0 50
CGHAZ-25
KJc (MPa.m1/2)
T (°C)
CG HAZ 1
0
50
100
150
200
250
300
-200 -150 -100 -50 0 50
ICCGHAZ-25
KJc (MPa.m1/2)
T (°C)
IC CG HAZ 1
0
50
100
150
200
250
300
-200 -150 -100 -50 0 50
CGHAZ-120
KJc (MPa.m1/2)
T (°C)
CG HAZ 2
ductile stable crack propagation over > 0.2 mmA. Lambert-Perlade et al., Metall. Mater. Trans. A, (2004), Vol. 35A, pp. 1039-1053
T0 = - 130°C T0 = - 45°C
T0 = - 12°C T0 = - 6°C
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
27
BRITTLE FRACTURE & BRITTLE to DUCTILE FRACTURE TRANSIT ION MICROMECHANISMS
André PINEAUCentre des Matériaux - Ecole des Mines de Paris. ParisTech
UMR CNRS 7633andre.pineau@ensmp.fr
I. INTRODUCTION : Various Scales / History of Fracture Mechaniscs
II. BRITTLE CLEAVAGE FRACTURE – THEORYII.1. Main characteristics of Brittle Cleavage FractureII.2. Weibull – Beremin TheoryII.3. Multiple barrier modelsII.4. Effect of Deformation – Second Order approximation
III. DUCTILE FRACTURE – THEORY (Introduction)III.1. Cavity initiation, growth & coalescence. GTN modelIII.2. Computational strategies to simulate ductile crack growth
IV. DUCTILE TO BRITTLE TRANSITION IN STEELSIV-1. Qualitative explanations for the existence of DBTIV-2. Fracture toughness transitionIV-3. Impact Charpy test
V. CONCLUSIONS
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
28
V
V0
WEAKEST LINK THEORY. WEIBULL-BEREMIN MODEL
(((( )))) (((( )))) (((( ))))
(((( ))))
σσσσ∫∫∫∫−−−−−−−−====
σσσσαααα====→→→→
∫∫∫∫====σσσσ→→→→ ∞∞∞∞
0PZR
2c
00l000
VdV
pexp1P:V
lGriffith
dllppdllp:Vc
(((( ))))
22m
VdV
;exp1P
VdV
exp1P
l/lp
LawPower
m/1
0
m1PZw
m
u
wR
0
m
u
1PZR
00
−−−−ββββ====
σσσσ====σσσσ
σσσσσσσσ−−−−−−−−====
σσσσσσσσ−−−−−−−−====
γγγγ====
∫∫∫∫
∫∫∫∫
ββββ
(((( ))))
σσσσσσσσ−−−−σσσσ−−−−−−−−−−−−====
−−−−−−−−====>>>>
∫∫∫∫0
n
20
2u
21
decohR
n
0
u
VdV
/1
/1/1expexp1P
lll
explSizep
LawlExponentia
See also Kroon & Faleskog in Inter. Journal of Fract. Vol. 118, (2002), pp.99-118 & Eng. Fract. Mechanics,
Vol. 71, (2004), pp. 57-79.
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
29
BEREMIN MODEL (83) (S.S.Y.)
CRACK
HRR Field0
σ
( ) ( )02
0 /J/x;/K/x σσ
HRR ( )
σσ=
σσ=σ
002
00YY /J
xg
/K
xf
B : Specimen Thickness
Cm (n) : constant
σ
σ−−=
−
mu0
mCB4m
04IC
RV
Kexp1P
−= ν22IC
IC 1E
KJ
( )2 2 4
022
0
1 exp1
mIC m
Rmu
J E BCP
V
σν σ
− = − −
−
yyσ
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
30
A533 SteelServer & Wullaert
0
20
40
60
80
100
120
140
160
0 50 100 150 200 250 300TEMPERATURE (K)
FR
AC
TU
RE
TO
UG
HN
ES
S, K
IC
(MP
am1/
2 )
Experimental Results 1T to 11T
Theory, ICF5, Cannes, 1981m = 21 ; σσσσu = 2530 MPa ; Vo = 60 µm 3
PR = 0.90
0.50
PR = 0.10
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
31
J∆ LARGE SCALE YIELDING
T1 T2 T3
2
Ln (J/σσσσ0)
Ln (
σσ σσ w/ σσ σσ
u)m
= L
n [L
n (1
/1-P
R)]
T1< T2< T3
SMALL SCALE YIELDING (SSY)
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
32
BEREMIN MODEL & MASTER CURVE
σσσσσσσσ−−−−−−−−====
−−−−
muu
m4m4
ICR V
CBKexp1P
MODELBEREMIN
(((( )))) (((( ))))[[[[ ]]]](((( ))))
(((( ))))mm25Bfor
mMPa100medianKT
CinT;mMPainK
TT019.0exp7030mediumK
63.0PK
mMPa20minK
minKKminKK
BB
exp1P
CURVEMASTER
JCo
oJC
Ro
I
4
Io
IIC
oR
========→→→→
°°°°
−−−−++++====−−−−====⇒⇒⇒⇒−−−−
≈≈≈≈−−−−
−−−−−−−−−−−−−−−−====
(((( )))) (((( ))))(((( )))) CstT/mediumKwhen
minKK,Kif
DEPENDENCE ETEMPERATUR
SAME THE PREDICT MODELS BOTH
4/m1oJC
IJCIC
====σσσσ−−−−
>>>>>>>>−−−−
−−−−
A. Lambert-Perlade, PhD Thesis, 2001
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
33
COMPARISON OF BEREMIN ( ) & MASTER CURVE ( ) (PR = 0.10 - 0.90)
m = 27σσσσu = 2158 MPa
m = 20σσσσu = 2351 MPa
m = 20σσσσu = 2670 MPa
m = 20σσσσu = 2085 MPa
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
34
BEREMIN MODEL - LIMITATIONS
(1) � THRESHOLD WEIBULL STRESS ?
(2) � STRAIN and/or TEMPERATURE DEPENDENCE OF MODEL PARAMETERS
(3) � EXISTENCE OF DIFFERENT MICROSTRUCTURAL BARRIERS FOR A PROPAGATING CRACKMULTIPLE BARRIER MODELS
(((( ))))?andm uσσσσ
⇒⇒⇒⇒
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
35
BRITTLE FRACTURE & BRITTLE to DUCTILE FRACTURE TRANSIT ION MICROMECHANISMS
André PINEAUCentre des Matériaux - Ecole des Mines de Paris. ParisTech
UMR CNRS 7633andre.pineau@ensmp.fr
I. INTRODUCTION : Various Scales / History of Fracture Mechaniscs
II. BRITTLE CLEAVAGE FRACTURE – THEORYII.1. Main characteristics of Brittle Cleavage FractureII.2. Weibull – Beremin TheoryII.3. Multiple barrier modelsII.4. Effect of Deformation – Second Order approximation
III. DUCTILE FRACTURE – THEORY (Introduction)III.1. Cavity initiation, growth & coalescence. GTN modelIII.2. Computational strategies to simulate ductile crack growth
IV. DUCTILE TO BRITTLE TRANSITION IN STEELSIV-1. Qualitative explanations for the existence of DBTIV-2. Fracture toughness transitionIV-3. Impact Charpy test
V. CONCLUSIONS
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
36
MULTIPLE BARRIER MODELS
20 µm
53°40° 58°
18°23° 52°
A. Lambert-Perlade & A.-F. Gourgues
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
37
A DOUBLE BARRIER MODEL FOR CLEAVAGE FRACTURE
A. Martin-Meizoso et al., Acta Metall. Mater. (1994), Vol. 42, pp. 2057-2068.
• STEP 1 : Fracture Probability of a M-A constituent / Critical Stress Criterion.
• STEP 2 : Probability of propagating a crack at the MA/Matrix Interface.
• STEP 3 : Probability of crossing a packet boundary.
dVCCFxNDCCFxNexp1P*
ccv
**g
gvPZR
⟩+
⟨⟨∫−−=
Fg & F c Determined by Metallography2
1
g/cIa
21
2
g/c* K
1
EC
σβ=
σ
ν−
γπ=
2
1
g/gIa
21
2
g/g* K
1
ED
σβ=
σ
ν−
γπ=C D
2 31g/c
IaK
g/gIaK
σ
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
38
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
39
CRACKS AT GRAIN BOUNDARIES (1)
Qiao, Argon, Mat. Letters, vol.54, 2008, pp.3156-3160
See also Qiao , Argon, Mech. Materials, vol.35, 2003, pp.129-154 and pp.313-331
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
40
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
41
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
42
CRACKS AT GRAIN BOUNDARIES (2)Qiao, Argon, Mat.
Letters, vol.54, 2008,
pp.3156-3160
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
43
BRITTLE FRACTURE & BRITTLE to DUCTILE FRACTURE TRANSIT ION MICROMECHANISMS
André PINEAUCentre des Matériaux - Ecole des Mines de Paris. ParisTech
UMR CNRS 7633andre.pineau@ensmp.fr
I. INTRODUCTION : Various Scales / History of Fracture Mechaniscs
II. BRITTLE CLEAVAGE FRACTURE – THEORYII.1. Main characteristics of Brittle Cleavage FractureII.2. Weibull – Beremin TheoryII.3. Multiple barrier modelsII.4. Effect of Deformation – Second Order approximation
III. DUCTILE FRACTURE – THEORY (Introduction)III.1. Cavity initiation, growth & coalescence. GTN modelIII.2. Computational strategies to simulate ductile crack growth
IV. DUCTILE TO BRITTLE TRANSITION IN STEELSIV-1. Qualitative explanations for the existence of DBTIV-2. Fracture toughness transitionIV-3. Impact Charpy test
V. CONCLUSIONS
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
44
A SIMPLIFIED THEORY
BEREMIN MODEL (1983)+
S. BORDET (2005)
Second Order Approximation
• The cleavage stress depends on plastic strain
• This dependence remains deterministic
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
45
J.-J.Gilman, (1958)
ZINC SINGLE CRYSTALS
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
46
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
47
S. Kotrechko, Theoretical & Applied Fracture Mechanics, 40, (2003), 271-277
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
48
EFFECT OF PLASTIC STRAIN
• Strong Effect in Zn single crystals
• CLEAVAGE INITIATED FROM PARTICLESNucleation TiN in IN718 (F. Alexandre, 2005)
Bordet et al. (2005) (EFM, Part I, 435-452; Part II, 453-474)
( ) ( )1981min,Bereexp1P
Na
uN
0eq1nucl
σ
σ−σλ+σ−−=
( ) nuclt
0 propcleav PPtP δ= ∫
( )
εε
σσ−−=⇒εσ−=δ
0eq
eq
0Y
0nucleq0nucl0nucl exp1PdP1NP
[ ]
σσ−−=δ−−= ∫ ∫
m
*u
*w
PZ
t
0 nuclpropR exp1dVNPexp1P
( ) ( ) ( )( )u
PZ 00eq
eqnucl
mth
m1
0Y
0m*
w V
dVdtP1eq
∫ ∫
εε
−σ−σσσ=σ ε
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
49
A SOPHISTICATED THEORY TO BE DEVELOPED !
Third Order Approximation
• The local stresses are no longer uniform
- Random distribution due to elastic anisotropy(S. Kotrechko + S. Pommier)
- Crystalline plasticity(Ecole des Mines)
• "Self organisation" of the local stresses : Arching type effect(S. Pommier)
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
50
INHOMOGENEITY IN LOCAL STRESS DISTRIBUTION (EL ASTICITY )
• S. KOTRECHKO (Theoretical and Applied Fracture Mechanics, 47, (2007), 171-181)
• See also: S. POMMIER (Fatigue & Fracture of Engineering Materials & Structures, 25, (2002), 331-348.
( ) ( )[ ]( ) ( )[ ]( ) ( )[ ]21II2313I
221II
23I33
31II3212I2
31II22I22
32II3121I2
32II21I11
µµ2DDD
µµ2DDD
µµ2DDD
ΣΣ+ΣΣ+ΣΣ+Σ+Σ+Σ=σ
ΣΣ+ΣΣ+ΣΣ+Σ+Σ+Σ=σ
ΣΣ+ΣΣ+ΣΣ+Σ+Σ+Σ=σ
( )
[ ][ ]113322
1111
2IIII
2I
2I
IIIIII
ii
24.024.0and
40.160.0.dev.St3forTensionUniaxial
10x66.0µD;10x72.0µ;10x7.1D
grainsofondistributiRandomFe:Ex
anisotropyelastictheoffunctionstscoefficien:µ,µ,D,D
stresseslocaltheofVariance:D
Σ−Σ−σσΣ−Σσ
====
σ
−−−
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
51
Analogy with Granular Materials[Dantu, Radjai, Roux, Evesque …]
F
Distribution of maximum principal stress (FEM)
Characteristic length > grain = arch dimension
sand = water + grains
S. Pommier, FFEMS, (2002), 25, 331
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
52
LOCAL STRESS HETEROGENEITY : 3D
Max. principalstress
Tensile loading Copper
S. Pommier
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
53
Distribution calculated from the results of 70 calculations
Zirconium Titanium Zinc Copper Iron Aluminum
Minimum value (MPa) Maximum value (MPa) Mean value (MPa)
926 1046 976
1058 1225 1141
792 1269 1036
679 1283 1018
1379 2107 1799
640 726 681
Standard Deviation (MPa) 27 35 123 121 144 16
3 times the standard deviation in percentage of the mean value
8.5% 9.25% 35% 35% 24% 7%
S. Pommier (FFEMS, 25, (2002, 331-348)
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
54
F. Barbe, S. Forest, G. Cailletaud, Int. J. Plasticity, 17, (2001), 537-563
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
55
F. Barbe, S. Forest, G. Cailletaud, Int. J. Plasticity, 17, (2001), 537-563
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
56
BAINITIC STEEL N. Osipov
G. Cailletaud
A.F. Gourgues
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
57
BAINITIC STEEL
N. Osipov
G. Cailletaud
A.F. Gourgues
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
58
( )σ>σlocP
( )if
ifp σ=σ
i = 1 i = 2
( )locσ
1
0
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
( )ifσApplied local stress ; Local fracture stress
Pro
babi
lity
Den
sity
( )ifp
Cum
ulat
ive
Pro
babi
lity,
( )σ>σlocP
N. Osipov
G. Cailletaud
A.F. Gourgueslocσ
ifσ
( ) ( ) σσσσσ dpPP if
ifo locR =×⟩= ∫
∞
59
STATISTICAL ASPECTS OF CLEAVAGE FRACTURE - CONCLUSIONS
• Various sources of scatter for cleavage fracture
• This scatter has strong practical implications. Threshold stress ? Size effect
• Modelling. Three approximations :
- Engineering purposes. 1st Order Approximation will remain largely usedHowever attempts made to consider that in the master curve approach, T0 is statistically distributed
- Most urgent problem : Strain and Temperature dependence of
• Many metallurgical means to develop microstructure with improved fracturetoughness
A 508 steel / 2 1/4 - 1 Mo steel
uσ
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
60
BRITTLE FRACTURE & BRITTLE to DUCTILE FRACTURE TRANSIT ION MICROMECHANISMS
André PINEAUCentre des Matériaux - Ecole des Mines de Paris. ParisTech
UMR CNRS 7633andre.pineau@ensmp.fr
I. INTRODUCTION : Various Scales / History of Fracture Mechaniscs
II. BRITTLE CLEAVAGE FRACTURE – THEORYII.1. Main characteristics of Brittle Cleavage FractureII.2. Weibull – Beremin TheoryII.3. Multiple barrier modelsII.4. Effect of Deformation – Second Order approximation
III. DUCTILE FRACTURE – THEORY (Introduction)III.1. Cavity initiation, growth & coalescence. GTN modelIII.2. Computational strategies to simulate ductile crack growth
IV. DUCTILE TO BRITTLE TRANSITION IN STEELSIV-1. Qualitative explanations for the existence of DBTIV-2. Fracture toughness transitionIV-3. Impact Charpy test
V. CONCLUSIONS
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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L. Devillers-Guerville, 1998DUPLEX STAINLESS STEEL
Cavity nucleation & growth from embrittled ferrite
DUCTILE FRACTURE MICROMECHANISMS
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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CAVITY COALESCENCE
DUCTILE FRACTURE MICROMECHANISMS
Steel C-Mn Aluminium 6156
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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GTN Model (Growth)
• Isotropic hardening + Elasticity
• Comparison with calculations on elementary cells (cavity + matrix)See Koplik & Needleman, 1988
• "New" Yield Surface
Most often :
But new calculations(Faleskog et al., 1998) showed thatq1 and q2 dependon and work hardening exponent, n
( )pyσ
221
y
kk212
y
2eq fq1
2
qcoshfq2 −−
σσ+
σσ
=φ
1q,2/3q 21 ==
E/yσ
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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GTN Model (Cavity Nucleation)
( ) ⋅⋅⋅ +ε−= pAtrf1f np
Growth Nucleation
• Phenomelogical Approach- Fitting of An on macroscopic tests- See e.g. (Chu & Needleman, 1980)
• Experimental Approach- See e.g. Duplex Stainless Steels (Devillers-Guerville et al., 1997)
( )
ε−−π
= 2n
2n
n
nn
s2
pexp
s2
fA
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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GTN Model (Coalescence)
Definition of an effective cavity volume fraction, f*
0fq12
qcosh*fq2 2*2
1y
kk212
y
2eq =−−
σσ+
σσ
=φ
=*fwhere ( ) ccc
c
ffiffff
ffiff
>−δ+<
δ*f
cf
Not necessary to introduce accelerating factor, when statistical cavity distribution is taken into account(Devillers-Guerville et al., 1997; Decamp et al., 1997)
δ
cf
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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GTN MODEL - EXTENSIONS
• Plastic Anisotropy (Bron - Besson, 2004)
• Cavity Shape (Gologanu - Leblond, 1993, 1997)
• Thomason, (1985, 1990)
• Pardoen and Hutchinson (2000)
• Benzerga (2000)
COALESCENCE THOMASON MODEL
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
67
BRITTLE FRACTURE & BRITTLE to DUCTILE FRACTURE TRANSIT ION MICROMECHANISMS
André PINEAUCentre des Matériaux - Ecole des Mines de Paris. ParisTech
UMR CNRS 7633andre.pineau@ensmp.fr
I. INTRODUCTION : Various Scales / History of Fracture Mechaniscs
II. BRITTLE CLEAVAGE FRACTURE – THEORYII.1. Main characteristics of Brittle Cleavage FractureII.2. Weibull – Beremin TheoryII.3. Multiple barrier modelsII.4. Effect of Deformation – Second Order approximation
III. DUCTILE FRACTURE – THEORY (Introduction)III.1. Cavity initiation, growth & coalescence. GTN modelIII.2. Computational strategies to simulate ductile crack growth
IV. DUCTILE TO BRITTLE TRANSITION IN STEELSIV-1. Qualitative explanations for the existence of DBTIV-2. Fracture toughness transitionIV-3. Impact Charpy test
V. CONCLUSIONS
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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COMPUTATIONAL STRATEGIES TO SIMULATE DUCTILE CRACK GROWTH
STRATEGY 1 • Use of the void Nucleation, Growth and Coalescence laws
• Integrate them using solutions of elastoplastic crack problems
• Neglect coupling effects between mechanical fields and DamageSee e.g. d'Escatha and Devaux, 1979
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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COMPUTATIONAL STRATEGIES TO SIMULATE DUCTILE CRACK GROWTH
STRATEGY 2 • Explicit modeling of the cavities using refined finite element mesh
• Requires a criterion for void coalescence
• Limitation to small crack extension and simple void geometrySee e.g. Aravas, McMeeking, 1985. Tvergaard, Hutchinson, 2002
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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COMPUTATIONAL STRATEGIES TO SIMULATE DUCTILE CRACK GROWTH
STRATEGY 2STRATEGY 2STRATEGY 2
STRATEGY 3 • Pursued mainly by groups in France, Germany, U.K. and U.S.
• Constitutive model (Gurson or Rousselier) with damage-induced softening
• Averaging over a critical distance
• Introduction of an internal length (e.g. Mudry et al., 1989. Rivalin et al., 2001)
COMPUTATIONAL CELLS
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
71
COMPUTATIONAL STRATEGIES TO SIMULATE DUCTILE CRACK GROWTH
STRATEGY 2STRATEGY 2STRATEGY 2
STRATEGY 4 • Introduction of the Cohesive Zone Model
• CZM parameters depend on constraint effects
• Crack path must be prescribed in advanceSee e.g. Tvergaard and Hutchinson, 1992. Brocks et al., 2003
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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SIMULATION OF DUCTILE CRACK GROWTH RESISTANCE CURVES
• Considerable success of the so-called "local approach" (Pineau, CSI, 2003)or "top-down" approach to ductile fracture (Hutchinson and Evans, 2000)
• Assessment of the fracture integrity of structural components under large scale yielding conditions
• Main interest : microstructural parameters(void volume fraction, void spacing,etc…) must be set such that the model reproduces experiments on specific specimens
• 3D aspects of crack initiation and growth have been simulatedComparison with experiments scarce (Gao et al., 1998; Rivalin et al., 2001)
Very much remains to be done : flat/slant fracture in these sheets
SUMMARY
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
73
BRITTLE FRACTURE & BRITTLE to DUCTILE FRACTURE TRANSIT ION MICROMECHANISMS
André PINEAUCentre des Matériaux - Ecole des Mines de Paris. ParisTech
UMR CNRS 7633andre.pineau@ensmp.fr
I. INTRODUCTION : Various Scales / History of Fracture Mechaniscs
II. BRITTLE CLEAVAGE FRACTURE – THEORYII.1. Main characteristics of Brittle Cleavage FractureII.2. Weibull – Beremin TheoryII.3. Multiple barrier modelsII.4. Effect of Deformation – Second Order approximation
III. DUCTILE FRACTURE – THEORY (Introduction)III.1. Cavity initiation, growth & coalescence. GTN modelIII.2. Computational strategies to simulate ductile crack growth
IV. DUCTILE TO BRITTLE TRANSITION IN STEELSIV-1. Qualitative explanations for the existence of DBTIV-2. Fracture toughness transitionIV-3. Impact Charpy test
V. CONCLUSIONS
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
74
WHY DOES A TRANSITION EXIST ?
1 - PROBABILISTIC ASPECT
Volume of sampled material increases with crack growth
Sketch
2 - MESO-MECHANICAL ASPECT
Maximum stress increases with crack growth
B. Tanguy
3 - MICRO-MECHANICAL ASPECT
Local stresses amplified by cavity growth
(Petti & Dodds, IJSS (2005), p. 3655) J.C. Lautridou
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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PROPAGATING CRACK
CRACK CRACK
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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WHY DOES A TRANSITION EXIST ?
1 - PROBABILISTIC ASPECT
Volume of sampled material increases with crack growth
Sketch
2 - MESO-MECHANICAL ASPECT
Maximum stress increases with crack growth
B. Tanguy
3 - MICRO-MECHANICAL ASPECT
Local stresses amplified by cavity growth
(Petti & Dodds, IJSS (2005), p. 3655) J.C. Lautridou
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
77
B. Tanguy, 2001
MAXIMUM STRESS AHEAD OF A PROPAGATING CRACK
CHARPY V SPECIMEN
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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WHY DOES A TRANSITION EXIST ?
1 - PROBABILISTIC ASPECT
Volume of sampled material increases with crack growth
Sketch
2 - MESO-MECHANICAL ASPECT
Maximum stress increases with crack growth
B. Tanguy
3 - MICRO-MECHANICAL ASPECT
Local stresses amplified by cavity growth
(Petti & Dodds, IJSS (2005), p. 3655) J.C. Lautridou
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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Steel 16 MND5
J.C. Lautridou , 1980
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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SAMPLING EFFECT DUE TO CRACK GROWTH
( )22IK
• WALLIN (1989)- Assumes that the "effective" volume continues to grow as
( )( ) ( )( )
( )thicknessspecimenB;stresscleavage
SSYundercalculatedeasilyKforvaluegnormalizinK;afK
daafBK
2
K
af
P1
1Ln
f
0I
a
0
240
2f
40
4
R
==σ=∆=
∆σ+∆=− ∫
∆
• BRUCKNER & MUNZ (1984)- Assumes that the size of the active volume is constant during crack extension
( )( )
volumeactivetheofsizethedescribingttancons:W;initiationcrackatKofvalueK
daafWK
1
K
K
P1
1Ln
ii
4a
0i40
4
0
i
R
=
∆+
=
− ∫∆
���� In both cases, or R curve must be known( )aK ∆
( ) sizezoneprocesscleavage/K/xwhereK
a21
K
K
P1
1Ln
)1993(WallinbyproposedcorrectionDCGtheofressionexpSimplified
2fi2
i
2f
0
i
4/1
R
σ≈β
βσ∆+=
−
⇒
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
81
ELIMINATION OF ELIGIBLE NUCLEATION SITES
( )( )
elementiththeofVvolumethein
particleafrommicrovoidanucleatetoobabilityPrP
V
VP1
i
ivoid
m/1
0
ini
Iivoid
n1iw
=
σ−Σ=σ =
See e.g. Koers et al. (1995)
• Very few results in the literature
• This effect is usually neglected
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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MnS
CLEAVAGE
A 508 Steel Notched Specimen T = - 150°C
CLEAVAGE INITIATED FROM MnS INCLUSIONSTanguy, 2007
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
83
BRITTLE FRACTURE & BRITTLE to DUCTILE FRACTURE TRANSIT ION MICROMECHANISMS
André PINEAUCentre des Matériaux - Ecole des Mines de Paris. ParisTech
UMR CNRS 7633andre.pineau@ensmp.fr
I. INTRODUCTION : Various Scales / History of Fracture Mechaniscs
II. BRITTLE CLEAVAGE FRACTURE – THEORYII.1. Main characteristics of Brittle Cleavage FractureII.2. Weibull – Beremin TheoryII.3. Multiple barrier modelsII.4. Effect of Deformation – Second Order approximation
III. DUCTILE FRACTURE – THEORY (Introduction)III.1. Cavity initiation, growth & coalescence. GTN modelIII.2. Computational strategies to simulate ductile crack growth
IV. DUCTILE TO BRITTLE TRANSITION IN STEELSIV-1. Qualitative explanations for the existence of DBTIV-2. Fracture toughness transitionIV-3. Impact Charpy test
V. CONCLUSIONS
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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A SIMPLIFIED APPROACH TO DB TRANSITION
• Based on the assumption that the stress-strain field is not modifiedby crack extension except the loading parameter, J
See e.g. Amar, Pineau, 1987; Koers et al., 1995
• Strong limitations to this approach
Does not account for modifications in the stress field of a propagating crack
Does not rely on a possible change in micromechanisms
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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Amar & Pineau, 1987
See also Koers et al., 1995. Fe 510 Nb
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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MORE ADVANCED MODELS FOR DB TRANSITION
OBJECTIVES
(i) • Stress Profiles ahead of a growing crack
Strongly dependent on geometry (bend vs tension)
(ii) • Introduction of damage ahead of the crack tip produces a macroscopic reduction
of the "local" stresses (GTN model). At metallurgical scales ductile damage
amplify local stress(See Petti & Dodds, 2005)
(iii) • Sampling effect due to crack growth
(iv) • Elimination of eligible particles (or nucleation sites)
(v) • Effect of plastic strain on cleavage stress
(See above, Beremin, 1983, Bordet et al., 2005)
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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Toyoda et al., 1991; Ruggieri & Dodds, 1996HSLA Steel
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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Ruggieri & Dodds, 1996HSLA Steel
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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Gao et al., 1999
GTN Model f0 = 0.0045, fc = 0.20, D = 0.30 mm, m = 11.86, MPa2490u =σ
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
90
BRITTLE FRACTURE & BRITTLE to DUCTILE FRACTURE TRANSIT ION MICROMECHANISMS
André PINEAUCentre des Matériaux - Ecole des Mines de Paris. ParisTech
UMR CNRS 7633andre.pineau@ensmp.fr
I. INTRODUCTION : Various Scales / History of Fracture Mechaniscs
II. BRITTLE CLEAVAGE FRACTURE – THEORYII.1. Main characteristics of Brittle Cleavage FractureII.2. Weibull – Beremin TheoryII.3. Multiple barrier modelsII.4. Effect of Deformation – Second Order approximation
III. DUCTILE FRACTURE – THEORY (Introduction)III.1. Cavity initiation, growth & coalescence. GTN modelIII.2. Computational strategies to simulate ductile crack growth
IV. DUCTILE TO BRITTLE TRANSITION IN STEELSIV-1. Qualitative explanations for the existence of DBTIV-2. Fracture toughness transitionIV-3. Impact Charpy test
V. CONCLUSIONS
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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APPLICATION OF THE LOCAL APPROACH TO DUCTILE-BRITTLE TRANSITION IN FERRITIC STEELS
MODELING OF CHARPY IMPACT TEST
• MATERIAL : PWR A 508 Steel
• NUMERICAL MODELING- 3D- Viscoplastic Constitutive Laws- GTN & ROUSSELIER Models for Ductile Fracture- BEREMIN Model for Cleavage Fracture
• COMPARISON EXPERIMENTS & NUMERICAL SIMULATIONS- Interrupted tests to measure ductile crack growth
See : B. Tanguy et al., Eng. Fracture Mechanics (2005), Vol. 72. Part I : pp. 49-72. Part II : pp. 413-434.
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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CHARPY TEST SIMULATIONB. Tanguy
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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Fracture at –60°C under dynamic
conditions :
a) Experimental fracture surface,
b) Simulation of ductile crack
propagation
(contour plots indicate )Iσ
A 508 Steel• No Inertial Effect
• Strain Rate Effect (Viscoplasticity)
• Ductile Damage (Rousselier & Gurson)
• Adiabatic Heating
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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A 508 STEEL
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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MODELING CHARPY-V TEST - CONCLUSIONS
1 - Accurate Description of Load-Displacement Curves
and Ductile Crack Growth Provided that 3D Modeling is Used
2 - Good Description of Tunnelling Effect for Ductile Crack Growth
3 - Necessity of Using a Viscoplastic Constitutive Equation and to account
for Adiabatic Heating
4 - Temperature Increase as Large as 150°C
5 - Easy Prediction of Index Temperatures TK28 and TK68
6 - Above 100 Joules, appears to be an Increasing Function of Temperatureuσ≈≈≈≈
CHARPY TESTS TUNING DAMAGE MODEL PARAME TERS FRACTURE TOUGHNESS PREDICTION
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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CONCLUSIONS and ADDITIONAL COMMENTS
• Illustration of the benefit of the micromechanical modeling of fracture
• More complex but rewarding : size effect; non-isothermal loading; mixed mode etc…
• Many research areas to be explored :
(1) Cleavage Stress or See Kotrechko et al.
(2) Ductile Fracture Models (too) sophisticated ?
(3) New experimental techniques (X-ray 3D Tomography) : Nucleation of cavities
(4) Existing models based on CSM. How far ? Crystallographic aspects
(5) Development of simulation tools : more complex situations (Welds etc…)
(6) Extension of ductile cracks over long distances remains a real challenge
( )?T,fu ε=σ
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
97
BRITTLE FRACTURE & BRITTLE to DUCTILE FRACTURE TRANSIT ION MICROMECHANISMS
André PINEAUCentre des Matériaux - Ecole des Mines de Paris. ParisTech
UMR CNRS 7633andre.pineau@ensmp.fr
I. INTRODUCTION : Various Scales / History of Fracture Mechaniscs
II. BRITTLE CLEAVAGE FRACTURE – THEORYII.1. Main characteristics of Brittle Cleavage FractureII.2. Weibull – Beremin TheoryII.3. Multiple barrier modelsII.4. Effect of Deformation – Second Order approximation
III. DUCTILE FRACTURE – THEORY (Introduction)III.1. Cavity initiation, growth & coalescence. GTN modelIII.2. Computational strategies to simulate ductile crack growth
IV. DUCTILE TO BRITTLE TRANSITION IN STEELSIV-1. Qualitative explanations for the existence of DBTIV-2. Fracture toughness transitionIV-3. Impact Charpy test
V. CONCLUSIONS
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
98
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
CONCLUSIONS
• Illustration of the benefit of micromechanical modelling of fracture
• LAF more complex but rewarding : size effect , non-isothermal loading , etc…
• Many research areas remain to be explored :
- Cleavage stress σu (temperature)
- Ductile fracture models ( too) sophisticated
- Shear dominated ductile fracture
- New experimental techniques ( Tomography )- Cavity nucleation
- Existing models based on C.M.- How far ? Crystallographic aspects?
- Development of simulation tools
- Extension of ductile cracks over long distances remainsa real challenge
99
THANK YOU FOR YOUR ATTENTION
For further information
andre.pineau@ensmp.fr
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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LIMITATION (1) : THRESHOLD WEIBULL STRESS
Actually, existence of a threshold stress in Beremin model since plastic deformation over a critical distance (Xc) is a prerequisite to initiate cleavage fracture
cominI X3~K ππππσσσσ
Increasing Temperature
Ln
KIC
(MP
am1/
2 )
for
PR
= C
ons t
ant
(((( ))))TLn oσσσσ
Centre des Matériaux - Ecole des Mines de Paris ICF Moscow 7-12 July 2007 A. Pineau
101
Experimental Results (Soehmaker - Rolfe.1971)
(-177°)
(-144°)
(-196°)
(-128°)
(-137°)
(-196°)(-147°)
(- 84°)
(- 73°)
(-104°)
(- 36°)
(-125°)
(-116°)
(- 83°)
2.0
2.5
3.0
3.5
4.0
4.5
6 6.25 6.5 6.75 7Ln σσσσ0 (σσσσ0 in MPa)
Ln K
I C (
KI C
in M
Pa
m1/
2)
ABS - C Mild Steel(σσσσ y at R.T. = 280 MPa)
( )Cin °
−•
−−•
−−•
≈
≈
≈
θεεε
1
11
15
sec20
sec10
sec10.5
Centre des Matériaux - Ecole des Mines de Paris ICF Moscow 7-12 July 2007 A. Pineau
102
0
50
100
150
200
-200 -150 -100 -50 0 50
ICCGHAZ-25
Results of the “double barrier” model with TPB specimens.Open circles denote microcrack events, solid circles are for final fracture of the specimens.Solid (resp. dotted) lines represent 10%, 50% and 90% probabilities for the specimen to fracture (resp. for a cleavage microcrack to propagate across a particle/matrix boundary) as given by the DB model.
TEMPERATURE (°C)
TO
UG
HN
ESS
(M
Pa √√ √√
m)
( )
( )25CGHAZ
mMPa28K
25ICCGHAZ
mMPa18K
mMPa8.7K
MPa2112
g/gIC
g/gIC
g/cIC
CAM
−=
−=
=
=σ −
E 450
Centre des Matériaux - Ecole des Mines de Paris ICF Moscow 7-12 July 2007 A. Pineau
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Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
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Standard Fracture Toughness specimens and wide plate components linked with the equivalent CTOD ratio β
Summer School on Fracture, CARGESE, 07-19 June 2010 A. Pineau
2c
a
2a
σσσσ∞∞∞∞
t
ca
σσσσ∞∞∞∞c a
a
a
σσσσ∞∞∞∞
t
σσσσ∞∞∞∞
σσσσ∞∞∞∞
t
σσσσ∞∞∞∞σσσσ∞∞∞∞
t
σσσσ∞∞∞∞
Standard fracture toughness specimens
Equivalent CTOD ratio β
Surface crack
Surface crack
Through-thickness crack
Through-thickness crack
Center surface crack panel (CSCP) Center through-thickness crack panel (CTCP)
Edge surface crack panel (ESCP) Edge through-thickness crack panel (ETCP)
B = t
W a0= BH H
S = 4W
1.25W
0.6W
a0 = BW
B = t
0.6W
Structural components
(0.45 < a0 / W < 0.55)
3-point bendCompact
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