lecture 4: fracture of metals and polymers ii. crack
TRANSCRIPT
Institute of Mechanics, Materials and Civil Engineer ingEcole Polytechnique de Louvain& Research Center on Architectured and Composite Materials (ARCOMAT)
Lecture 4: Fracture of metals and polymers II. Crack propagation
10 mars 2016Institut de Mathématiques, Quartier Polytech 1, Allée de la Découverte12, Bâtiment B37, 4000 Liège - Auditoire 02
Chaire Francqui 2016, ULg
Thomas Pardoen
Outline
1. Experimental characterization of ductile tearinga. The traditional JR curve approach & cob. The essential work of fracturec. A few words on the mechanisms
2. Modelling strategies
3. Example 1 : Thick components - prediction of JR curves with Gurson type model
4. Example 2 : Thin sheet fracture
5. Example 3 : Application of « stress at a distance » model in adhesive joints
6. Example 4 : Fracture in Al with PF
y
x
z
rθ
σx
y στxy
zσ
σrθσ
xy
z
Mode I Mode II
x
z
y x
z
Mode III
y
( ) ( ) ( )∑∞
=
−+=
,...3,2
12
1 ,,2 n
n
nijij fraKCfr
K θθπ
σ II
IIII dimensionsother nij
2a
σ
σ ∞
∞
KI = σ ∞ πa
1.5
1.
0.5
0.0.0001 0.001 0.01 0.1 1. 10.
r/a
σ yyapprox./σ yyexact
1st term only
1st and 2nd terms
1st, 2nd and 3nd terms
exact
( ) ( ) ( )θπ
θπ
θπ
σ III1
IIIII1
III1
I
r2r2r2ijijij f
Kf
Kf
Kij ++=
aYK ∞= σIIIor IIor I
Back to linear elastic staticcrack analysis
JR curves
2
strain plane
3
1
=
0
I
σπK
rY
2
stress plane 1
=
0
I
σπK
rY
log σyslope -1/2
log r/L Finite strain zone
Plastic zone
K-dominated zone
K-field validity
Small Scale Yielding
Validity of LEFM
JR curves
F
F
hB
a
h
A
CF
∂∂
∂∂
2
2
=A
-=P
G
32
2212
hEB
aF=G
−++=
ν1*
1 222 III
III
KKK
EG
Energetic approach (energy release rate)Linear elastic
P is the potential energy of the systemC is the compliance of the structureA is the crack area
JR curves
G ≥ Gc (J/m2) or KI ≥ KIc (MPa.m1/2)
K ≥ KR(∆a) (or G ≥ GR(∆a))
∆a
R
K Ic
K (∆a)
∆a
K ss
∆aRef
K eng
(b)
a
RG , G
ainitial
G G R
Stable
Unstable
Fracture resistance curves(linear elastic)
Cracking initiation (fracture toughness) :
Crack propagation
Stability
JR curves
( )
∫
−=
−=
∂∂
σ
∂∂
dsx
unnW
AJ
jijixV
2J/m P
( ) ∫−=
F
FduawB
J0
η
F
Ψ
uw-a
S
B
wa
( ) duFawB
JF
∫−=
0
2SENBdeep
Non linear fracture mechanics: J integral
valid for radial loadingsWv is the strain energy density
JR curves
J.R. Rice, 1968
0
1
2
3
4
5
6
0 2 4 6 8 10
σyy
/σ0
rσ0/J
Material with N=10 αε0=0.001
εplan
σplan
HRR fields
J2 deformation theory(non linear elastic response)
See HHR tables by Fong Shih, 1983 (Brown University )
JR curves
Hutchinson, JMPS1968; Rice and Rosengren, JMPS 196 8
N
=
00 σσα
εε
( )NrI
Jij
N
Nij ,~1
1
000 θσ
εασσσ
+
=
( )NrI
J N
N
,~1
1
000 θσ
εασσσ
+
=
( )NrI
Jij
N
N
Nij ,~1
000 θε
εασαεε
+
=
JR curve and definition of fracture toughness
JR curves
From B. Tanguy, CEA
Fatigue precracking of CT or SENB specimens,
then load with partial unloading sequences
JR curve and definition of fracture toughness
JR curves
From B. Tanguy, CEA
JR curve and definition of fracture toughness
Ji
∆a0.2 mm
J
loss of constraint
Steady state regime
active plasticzone
crack wake
JIc ΓSS
Γp
JR curves
J = Γ0 + Γp
Ji
y
x
δ
Initial crackr
u
uy
x
45°
( )NurI
Jru i
N
N
Ni ,~1
000 θ
εασαε
+
=
Fracture toughness can also bedefined as critical CTOD δc
JR curves
( )0
0,σ
αεδ JNd=
Crack tip opening displacement
Within the finite strain zone
3 41 5 62
0.05-0.1
εp
0.577
Stress triaxiality
loss of constraint
increasingn
constraint changes
r/δFracture process zone
this is where damage develops !
JR curves
σyy/σ0
3
Evolution of stress fields duringcrack propagation
JR curves
Slight stress elevation during crack propagation
3 41 5 62
HRR ~ 0.6
Stress triaxiality
increasingsheet thickness
r/δ
Fracture process zone = necking zone
0.6
1.
mid-planesurface
Additional complexitycoming from 3D effects
JR curves
log σy
slope -1/(N+1)
slope -1/2
log r/L
log σ y
slope -1/(N+1)
log r/L
log σ y
slope -1/(N+1)
log r/L
slope -1/2
Finite strain zoneHRR validity
J-dominated zone
Plastic zone
K-dominated zone
K-field validity
Finite strain zone
HRR validity
J-dominated zone
Plastic zone
Finite strain zonePlastic zone
Small Scale Yielding Large Scale Yielding
No single parameter approach
slope -1/2
(a) (b)
The big picture on the validity
*
2
E
KJ =≡G only J valid
JR curves
Outline
1. Experimental characterization of ductile tearinga. The traditional JR curve approachb. The essential work of fracturec. A few words on the mechanisms
2. Modelling strategies
3. Example 1 : Thick components - prediction of JR curves with Gurson type model
4. Example 2 : Thin sheet fracture
5. Example 3 : Application of « stress at a distance » model in adhesive joints
6. Example 4 : Fracture in Al with PF
pef wlww α0+=
00tl
Ww f
f =
DENT geometry
pepef wltwltWWW 20000 α+=+=
Essential Work of Fracture (J)
Plastic Work (J)
Total work of Fracture (J)
Thickness samples (m)Ligament length (m)
Essential specific Work of Fracture (J/m²)
Plastic work density (J/m 3)
(Double EdgeNotch Tension)
The essential of fracture method
Essential work of fracture
Cotterell and Reddel, IJF, 1977
l0
Essential work of fracture
Example EWF method
0
5 104
1 105
1.5 105
2 105
2.5 105
3 105
3.5 105
4 105
0 5 10 15 20 25
t0 = 1 mm
t0 = 2 mm
t0 = 3 mm
t0 = 4 mm
t0 = 5 mm
t0 = 6 mm
wf (J/m2)
l0 (mm)
Al 6082 O temper– 6 thicknesses
Pardoen, Marchal, Delannay, JMPS 1999
Essential work of fracture
EWF method widely applied to polymers
Outline
1. Experimental characterization of ductile tearinga. The traditional JR curve approachb. The essential work of fracturec. A few words on the mechanisms
2. Modelling strategies
3. Example 1 : Thick components - prediction of JR curves with Gurson type model
4. Example 2 : Thin sheet fracture
5. Example 3 : Application of « stress at a distance » model in adhesive joints
6. Example 4 : Fracture in Al with PF
Classical ductile crack propagation mechanisms
Initial sharp crack (e.g. fatigue precrack)
Crack tip blunting
Physical initiation of cracking
Fracture mechanisms at crack tip
Fracture mechanisms at crack tip
Crack tip just after initiation in Cu from Pardoen & De lannay, EFM 2000
Fatigue zoneBlunting zone =
stretch zone width(≈ δc/2) Ductile tearing
Fracture mechanisms at crack tip
Fracture surface near the original crack tip
Propagating crack remains sharp (opening on the order of voidsize and not much larger like at cracking initiation)
Crack tip opening angle remains relatively constant af ter the initiation transient
CTOA
Fracture mechanisms at crack tip
Classical ductile crack propagation mechanisms
Additional complexity
Mechanisms at intermediate scale change with plate thickness
and also with loading rate, environment and temperature …
Fracture mechanisms at crack tip
Recent result : localization leading to slant fracture can occur before damage
Recent progress in the characterization of slant fracture
Fracture mechanisms at crack tip
Morgeneyer et al., Acta Mater 2014
Fracture mechanisms at crack tip
Recent progress in the characterization of slant fracture
Localization leading to slant fracture after damage (preexisting or grown) - major effect of second
population and specimen orientation Ueda et al., Acta Mater 2014
… but the transition or not to slant fracture dependi ngon strain hardening capacity, texture, rate dependenc y
etc is still an open issue ….
G or J (kJ/m )
c c2
Specimen thickness (mm)
20
200
100
2 10
Sometimes, even more complex : flip-flap fracture
Fracture mechanisms at crack tip
see recent work by K.L. Nielsen, J.W. Hutchinson
In the absence of slantfracture, major thickness effect
Ductile tearing in bulk materials with undefinedfracture process zone vs plastic zone
Fracture mechanisms at crack tip
Du et al., Acta Mater 2000
« extreme case » : perfectly ductile flow with no fracture
Can we still talk about fracture ? Does it matter ?
Fracture mechanisms at crack tip
Summary of the mechanisms
Fracture mechanisms at crack tip
Benzerga, Pardoen, Pineau, Acta Mater 2016
Outline
1. Experimental characterization of ductile tearinga. The traditional JR curve approachb. The essential work of fracturec. A few words on the mechanisms
2. Modelling strategies
3. Example 1 : Thick components - prediction of JR curves with Gurson type model
4. Example 2 : Thin sheet fracture
5. Example 3 : Application of « stress at a distance » model in adhesive joints
6. Example 4 : Fracture in Al with PF
Starting point : to go beyondfracture mechanics approach
From X.K. Zhu, S.K. Jang, EFM 68 (2001) 285-301
JQ/JT/JA 2 theories … doubts …
Modelling strategies
Modelling strategies
A few references
Benzerga, Pardoen, Pineau, Acta Mater 2016
Hutchinson and Evans, Acta Mater 2000
The big pictureModelling strategies
Benzerga, Pardoen, Pineau, Acta Mater 2016
Outline
1. Experimental characterization of ductile tearinga. The traditional JR curve approachb. The essential work of fracturec. A few words on the mechanisms
2. Modelling strategies
3. Example 1 : Thick components - prediction of JR curves with Gurson type model
4. Example 2 : Thin sheet fracture
5. Example 3 : Application of « stress at a distance » model in adhesive joints
6. Example 4 : Fracture in Al with PF
( )( )
31
0
0
43
=
−f
Tnucl
x
πεε
ε
HRRHRR
HRR
23
exp0.43exp22
exp
Analytical « first order » qualitative model
0
1
2
3
4
5
6
0 0.05 0.1 0.15 0.2 0.25 0.3
σm/σ
e HRR
n
Plane stress
Plane strain
Assume ε0nucl ≈ 0
( )0
HRRN
ln
2
30.exp
3
20.43-
f3
156.0
~~1
0000
+≈
+
TXI
Jxx
NN
εεσαε
αε
( )N
N
xx T
XIJ
1
0
000~~
f31
56.0
+
+≈
HRR
0
NIc
2
3exp
3
20.43-
ln
εεαεσαε
≈ thickness,,, 00
00 fE
NFXJσσIc
Example 1
See models R&T + Brown&E in lecture 3
Simple uncoupled model basedon elastoplastic solution
X0/2
δ0(a)
See earlier works by McMeeking
Example 1
FE model with discrete void in elastoplastic matrix
X0
(b)
Tvergaard & Hutchinson, IJSS, 2002
Two “slightly different” ductile tearing mechanisms
X0
Z0
2R0
2Rz0
δ0
I. Multiple void interaction
Finite strain zone
II. void by void growth
Plastic localization = coalescence = finite strain zone
Plastic localization = coalescence
Example 1
Tvergaard & Hutchinson, IJSS, 2002
Small scale yielding analysis (infinite medium) of ductile fracture with advanced “Gurson” model
K field
Model implemented in "ABAQUS Standard" through a User defined MATerial (UMAT), finite strain setting, fully implicit – or home code (Ph . D. Florence Scheyvaerts)
Pardoen, T., Hutchinson, J.W., 2003 Acta Mater . 51, 133-148.
Example 1
X0
(c)
0
0.5
1
1.5
2
0 0.01 0.02 0.03 0.04
Tvergaard and Hutchinson, 2001
Present work - 1 row of voids
Present work - voids everywhere
1/3
1/2
0
R=
σ0/E=.003, n=0.1, W
0=1, λ
0=1
JIc
/σ0X
0
f0
+ extended Gurson
Dotted lines
ValidationExample 1
Pardoen & Hutchinson, Acta Mater 2003
Tvergaard & Hutchinson, IJSS, 2002
Plane strain fracture toughness of ductile metallic alloys
0
2
4
6
8
10-6 10-5 10-4 10-3 10-2 10-1
σ0/E = 0.003
σ0/E = 0.01
σ0/E = 0.001
f0
JIc
/σ0X
0
n = 0.1, W0 = 1, λ
0 = 1
Negligible effect of σσσσ0/E on JIc/σσσσ0X0
= ... ,,,,, 0000
00 λσσ WfE
nFXJIc
Example 1
Pardoen & Hutchinson, Acta Mater 2003
0
2
4
6
8
10
10-5 10-4 10-3 10-2
f0
0.2
n=
σ0/E = .003, W
0 = 1, λ
0 = 1
JIc
/σ0X
00.1
0.01
Very strong effect of the strain hardening exponent n on JIc/σσσσ0X
Example 1
Pardoen & Hutchinson, Acta Mater 2003
= ... ,,,,, 0000
00 λσσ WfE
nFXJIc
Plane strain fracture toughness of ductile metallic alloys
Large effect of strain hardening exponent partly ex plains why fracture toughness usually decreases with increasin g strength –
not a direct effect of the strength
Plane strain fracture toughness of ductile metallic alloys
σ
ε
↑ σu often means ↓ n
+ also an effect of second population of voids !
Example 1
Pardoen & Hutchinson, Acta Mater 2003
= ... ,,,,, 0000
00 λσσ WfE
nFXJIc
Plane strain fracture toughness of ductile metallic alloys
σ0/E = .003, n = 0.1, λ
0 = 1
0
2
4
6
8
10-5 10-4 10-3 10-2
1/3 1
W0=
1/10
3
10
JIc
/σ0X
0
f0
Very strong effect of the initial void shape on JIc/σσσσ0X0
Example 1
Pardoen & Hutchinson, Acta Mater 2003
= ... ,,,,, 0000
00 λσσ WfE
nFXJIc
Plane strain fracture toughness of ductile metallic alloys
Plane strain ductile tearing resistanceof ductile metallic alloys
0
10
20
30
40
50
60
70
0 20 40 60 80 100
∆a (coalescence)
MODE I, f0=10-3, W0=1, λ0=0, θ0=0σ0/E=3.10-3, n=0.1, non homogeneous
θc=0, θ
c=qcq
NLG=1
NLG=10
NLG=18
The number of « Gurson » rows has an impact on the predicted tearing resistance
Example 1
Scheyvaerts, PhD UCL 2008
Plane strain ductile tearing resistanceof ductile metallic alloys
Relatively small impact of void nucleationstress on tearing modulus
0
10
20
30
40
50
60
70
0 20 40 60 80 100
JR
/(σ0 X
0)
∆a/X0
k=1.5
k=3.8
k=3.5
k=2.5
k=4
k=0
k=4.25
f0=10-3, W
0=1, λλλλ
0=1, PZ=108x18
k=0, PZ=108x30
σnucleation = k σ0
Example 1
Scheyvaerts, PhD UCL 2008
Tearing modulus increases with decreasing initial porosi ty
0
5
10
15
20
25
30
35
40
0 20 40 60 80
JR/(σ
0 X
0)
∆a / X0
n=0.1, W0 = 1, λλλλ
0 = 1.0, θθθθ
0 = 0.0
f0 = 10-2
f0 = 10-3
f0 = 10-4
Example 1
Plane strain ductile tearing resistanceof ductile metallic alloys
Scheyvaerts, PhD UCL 2008
Plastic zone size increases with decreasing initial poro sityand with crack propagation
0
1000
2000
3000
4000
5000
0 20 40 60 80
f0 = 10-2
f0 = 10-3
f0 = 10-4
rp / X0
∆a / X0
n=0.1, W0 = 1.0, λλλλ0 = 1.0, θθθθ0 = 0.0
predictedanalytical
smooth fit of the original curve
ry
Example 1
Plane strain ductile tearing resistanceof ductile metallic alloys
Scheyvaerts, PhD UCL 2008
Effect of initial void shape on tearing modulus issignificant mainly for oblate voids
0
10
20
30
40
50
0 20 40 60 80
JR/(σ
0 X
0)
∆a / X0
n=0.1, f0 = 10-3, λλλλ
0 = 1.0, θθθθ
0 = 0.0
W0 = 1/6
W0 = 1
W0 = 6
Example 1
Plane strain ductile tearing resistanceof ductile metallic alloys
Scheyvaerts, PhD UCL 2008
Significant effect of void distribution
0
5
10
15
20
25
30
35
40
0 20 40 60 80 100
λ0 = 1/2
λ0 = 1
λ0 = 2
JR
/(σ0 X
0)
∆a / X0
n=0.1, f0 = 10-3, W
0 = 1, θθθθ
0 = 0.0
Example 1
Plane strain ductile tearing resistanceof ductile metallic alloys
Scheyvaerts, PhD UCL 2008
CTOA
5 X0
Crack tip opening angle predictions
0°
5°
10°
15°
20°
25°
30°
35°
10 20 30 40 50 60 70 80 90
CTOA / 2
∆a / X0
n=0.1, W0 = 1, λλλλ
0 = 1.0, θθθθ
0 = 0.0
f0 = 10-2
f0 = 10-3
f0 = 10-4
Example 1
Plane strain ductile tearing resistanceof ductile metallic alloys
Scheyvaerts, PhD UCL 2008
The main success of the micromechanics of ductile fracture : predicting constraint effects
in thick components
Example in 3D
Gao, Faleskog, Shih, Dodds, EFM 68 (2001) 285-301
Example 1
See also many studies in the group of A. Pineau & J. B esson from the Ecole des Mines de Paris
Outline
1. Experimental characterization of ductile tearinga. The traditional JR curve approachb. The essential work of fracturec. A few words on the mechanisms
2. Modelling strategies
3. Example 1 : Thick components - prediction of JR curves with Gurson type model
4. Example 2 : Thin sheet fracture
5. Example 3 : Application of « stress at a distance » model in adhesive joints
6. Example 4 : Fracture in Al with PF
Gc
t
?
Mixed mode I-III crackingMode I cracking or
?
?G or J (kJ/m )
c c2
Specimen thickness (mm)
20
200
100
2 10
Ductile cracking in thin components –many open questions
Al alloy result
Example 2
( )( )
31
0
0
4
3
=
−f
Tnucl
x
πεε
ε
HRRHRR
HRR
2
3exp0.43exp22
exp
Analytical « first order » model
0
1
2
3
4
5
6
0 0.05 0.1 0.15 0.2 0.25 0.3
σm/σ
e HRR
n
Plane stress
Plane strain
Assume ε0nucl ≈ 0
( )0
HRRN
ln
2
30.exp
3
20.43-
f3
156.0
~~1
0000
+≈
+
TXI
Jxx
NN
εεσαε
αε
( )N
N
xx T
XIJ
1
0
000~~
f3
156.0
+
+≈
HRR
0
NIc
2
3exp
3
20.43-
ln
εεαεσαε
≈ thickness,,, 00
00 fE
NFXJσσIc
X0 X0
cracking initiation
Example 2
0
0.5
1
1.5
2
0 1 2 3 4 5
Γ0/σ
0X
0
Stress triaxiality T
f0 = 0.01; n = 0.1; σ
0/E = 500
Typical stress triaxiality for "thick" cracked specimens with possible constraint effects
Analytical « first order » model explainswhy fracture toughness is larger in plane
stress compared to plane strain
But, it does not explain why fracture toughness would start first increasing
with increasing thickness
G or J (kJ/m )
c c2
Specimen thickness (mm)
20
200
100
2 10
Example 2
Based on FE void cellsimulations or extended
Gurson model + coalescence
L 0
t0
X0
Diffuse plastic zone
Localized necking zone
Final micro-zone of damage-induced localization
Additional effect in thinplates: crack tip necking
Example 2
L 0
t0
Diffuse plastic zone
L0
Γ = Wtot
L0t0
t0
Γ0
wnwp
)()()( 0 aaa p ∆Γ+∆Γ=∆Γ
In thin plates
In thick plates
Wt = Γ0L0t0 + wnαt02L0 + wpβt0L0
2
In DENT panels
Γ = Γ0 + wnαt0 + wpβL0
Example 2
What are the energy contributions to the total work of facture ?
)()( 0 aa ∆Γ=∆Γ )( an ∆Γ+ )( ap ∆Γ+
0
5 104
1 105
1.5 105
2 105
2.5 105
0 1 2 3 4 5 6 7
Jc (J/m2)
we (J/m2)
t0 (mm)
24300 J/m2
28800 J/m2
33MJ/m3
29MJ/m3
0
100
200
300
400
500
600
0 2000 4000 6000 8000
JR (kJ/m2)
∆atot
(µm)
0.6 mm
6 mm4 mm
2 mm
1 mm
blunting line
Ductile tearing of thin 6082-O aluminium plates
Pardoen, T., Marchal, Y., Delannay, F., 1999. J. Me ch. Phys. Solids 47, 2093-2123.
Example 2
Materials Thickness(mm)
Homog. E(GPa)
σ0(MPa)
n(Swift)
k(Swift)
Stainless steel A316L 0.65 to 3 yes 210 310 0.48 25
Al 6082-O 0.6 to 6 yes 70 50 0.26 265
Brass annealed 0.9 to 2 yes 110 100 0.6 33
Al NS4 // RD 0.57 to 1.5 yes 70 140 0.17 159
Zinc // RD 0.6 to 1.3 yes 61 100 0.15 118
Lead 0.8 to 1.8 yes 16 7 0.25 290
Bronze annealed 0.54 to 1.2 yes 100 120 0.51 38
Bronze ⊥ RD 0.54 to 1.2 yes 100 400 0.01 ? ?
Bronze // RD 0.54 to 1.2 yes 100 410 0.015 ? ?
Brass ⊥ RD 0.9 to 2 no 110 (240) (0.25)
Brass // RD 0.9 to 2 no 110 (210) (0.32)
Al NS4 annealed 0.57 to 1.5 no 70 (80) (0.2)
Al NS 4 ⊥ RD 0.57 to 1.5 no 70 (150) (0.14)
Mild steel ⊥ RD 0.87 to 1.5 no 210 (240) (0.17)
Mild steel // RD 0.79 to 1.5 no 210 (220) (0.17)
Zinc ⊥ RD 1.3 yes 86 140 0.08
Pardoen, T., et al. 2004 J. Mech. Phys. Solids 52, 423.
Same observations for a wide range of metals !!!
Example 2
A316L Brass Annealed Al NS4
Triax = T
TTsides= 0.6 centreε
f
Cup & cup fracture
Example 2
Model for the work of necking
=Γf
0
0
n ,,,, ενσσ E
knF
( )( )∫ ∫=
2
000
maxn
2un
u
ddn
ε ε
εεσ
h hhtlW
( )∫ ∫=Γ
2
0
maxn
2u0
u
ddn
h hh
ε
εεσ
ε u =2nk− 3
3k
Assumption of plane strain tension
active necking region
hnhu
tu
tnnecking region in the initial configuration
h
Example 2
Pardoen et al. J. Mech. Phys. Solids 2004
Model for the work of necking
0
0.1
0.2
0.3
0 0.5 1 1.5 2 2.5
wn/σ
0kn
εf-ε
u
n=0.1
n=0.25
n=0.4
n=0.5
Example 2
Pardoen et al. J. Mech. Phys. Solids 2004
L 0
t0
X0
Diffuse plastic zone
Localized necking zone
Final micro-zone of damage-induced localization
Model for the work of fracture
),,,,( 0000
00
0 λσσ
WfnE
FX
=Γ
Φgrowth ≡C
σ 2s + ησ hgX
2+ 2q g + 1( ) g + f( )cosh κ
σhg
σ
− g+ 1( )2 − q2 g+ f( )2 = 0
Φcoalescence≡σe
σ +
3
2
σ h
σ − F W,χ( )= 0
Example 2
Pardoen et al. J. Mech. Phys. Solids 2004
Extended Gurson model (Gologanu + Thomason)
f0
0.1
1
10
100
1000
10-5 10-4 10-3 10-2 10-1 100
Γ0/σ
0X
0
Γ0/J
Ic
n = 0.1
n = 0.1
n = 0.3
n = 0.5
Example 2
Model for the work of fracture
Extended Gurson model (Gologanu + Thomason)
Pardoen et al. J. Mech. Phys. Solids 2004
Combining the work of neckingand work of fracture
1t/ry
SSY
00Xc
σΓ
stressplaneX00
0
σΓ
Ic
strainplane
JX
=00
0
σΓ
A
B
C
C − Γ0
A Slant fracture (tentative)
B
C
Flat fracture with large σ0 small n
Flat fracture with small σ0 largen
α2
α1
nnn
k
wEk
0030
1
σσα =
C − Γn
Example 2
Pardoen et al. J. Mech. Phys. Solids 2004
… but not the transition or not to slant fracture dep endingon strain hardening capacity, texture, rate dependenc y, etc
As a matter fact, nobody (I take a risk !) has everproduced a convincing quantitative prediction of this kind of curve for the full range of thicknesses !
G or J (kJ/m )
c c2
Specimen thickness (mm)
20
200
100
2 10
Sometimes, even more complex : flip-flap fracture (see recent work by K.L. Nielsen, J.W. Hutchinson)
Example 2
Outline
1. Experimental characterization of ductile tearinga. The traditional JR curve approachb. The essential work of fracturec. A few words on the mechanisms
2. Modelling strategies
3. Example 1 : Thick components - prediction of JR curves with Gurson type model
4. Example 2 : Thin sheet fracture
5. Example 3 : Application of « stress at a distance » model in adhesive joints
6. Example 4 : Fracture in Al with PF
Back to bond line thickness effect in adhesive bonds
0
1000
2000
3000
4000
5000
6000
7000
0 0,5 1 1,5 2 2,5 3
Kinloch and Shaw 1981
Adh
esiv
e jo
int t
ough
ness
(J/
m2 )
Bond line thickness (mm)
XD1493
XD4600
Betamate 73455
ESP110
Example 3
See lecture 1
Cohesive zone based fracture model
• 2D plane strain FE model• Steady-state formulation (see Material flow below)• Far-field plasticity accounted for with von Mises plasticity• Damage mechanisms condensed in a cohesive zone
with zero thickness and with constant parameters
Continuum elements with elastic-plastic behaviour
Continuum elements with elastic-plastic behavior
Cohesive zone elementswith zero thickness
σ
δn
δn
σ
σ
adherend adhesive
Γ0
σ
Example 3
See lecture 1
“3 active plastic zones” whose intensity depends on adhesive thickness
Example 3
See lecture 1
See Martiny et al., IJAA2010, IJF 2012
Yellow Grey Blue
σpeak 87 MPa 99 MPa ~ 100 MPa
σpeak/σ0 2.9 2.7 ~ 3
Γ0 112 J/m2 845 J/m2 ~3500J/m2
Γp ~ 100 J/m2 ~ 40 J/m2 ~1000J/m2
(for interm. h) (OK to capture h effect) (KO to capture h effect) (partly OK h effect)
n 0.13 0.47 0.1
See Martiny et al., IJAA2010, IJF 2012
Example 3
See lecture 1
Betamate 73455 ESP110 Betamate 1493
Important values
Thickness effect in ESP110 not captured
Example 3
See lecture 1
• Elastic-plastic, 2D plane-strain, finite element model• Crack imposed at a predefined location• Failure criterion: the maximum principal stress must reach
a value of σc at a distance rc ahead of the crack tipwith σσσσc and r c kept constant for a given adhesive!
Example 3
Solution : “stress at a distance” model
Martiny et al., EFM 2013
Motivation behind the failure criterion
• Crack propagation is dominated by the nucleationof voids from the second-phase particles
• A maximum principal stress equal to σc is required to nucleate voids by cleavage, debonding, cavitation,…
• The second-phase particles are distant from each other by an average distance equal to rc
• Criterion initially suggested for brittle fracture in metals by Ritchie et al. (1973) and later applied to adhesives a.o. by Clarke and McGregor (1993)
Example 3
“Stress at a distance” model
Identification of the material parameter valuesElastic-plastic properties of the adhesive/adherend:least-square fit of uniaxial tensile stress vs. strain curves with a proper equation for the hardening law
Aluminum alloy
Adhesive ESP 110
Example 3
“Stress at a distance” model
Martiny et al., EFM 2013
Identification of the material parameter values• Elastic-plastic properties of the adhesive/adherend• Parameters, rc and σc, of the failure criterion:
inverse analysis, i.e. chosen to give the best possible agreement between numerical predictions and experimental results
Betamate 73455 ESP 110 Betamate 1493
rc = 18 µm, σc = 98 MPa rc = 49 µm, σc = 210 MPa rc = 6.9 µm, σc = 141 MPa
Example 3
Martiny et al., EFM 2013
“Stress at a distance” model
Attempt to link to the physics of damage
X = 50 to 150 µm
SEM fracture surface ESP110
Example 3
Example 3
Predictions Betamate 73455
Martiny et al., EFM 2013
Example 3
Predictions ESP110
Martiny et al., EFM 2013
Example 3
Martiny et al., EFM 2013
Predictions Betamate 1493
Adhesive fracture energy values• Predicted by the model of the TDCB test
• Normalized by a reference value corresponding to an elastic material (with the same elastic properties) under small-scale yielding conditions
As a function of:• The thickness of the adhesive layer• The elastic-plastic properties of the adhesive assuming the
following stress vs. strain behavior in uniaxial tension:
= if ≤
otherwise
• The parameters, rc and σc, of the failure criterion
Example 3
Parametric study
• Ga varies (increases)non-linearly with hadh
• Ga increases with σc
• Ga increases with smaller values, i.e. lower hardening capabilities
Example 3
Parametric study – effect of thickness, σc and
Martiny et al., EFM 2013
Back to bond line thickness effect in adhesive bonds
0
1000
2000
3000
4000
5000
6000
7000
0 0,5 1 1,5 2 2,5 3
Kinloch and Shaw 1981
Adh
esiv
e jo
int t
ough
ness
(J/
m2 )
Bond line thickness (mm)
XD1493
XD4600
Betamate 73455
ESP110
Example 3
See lecture 1
Martiny et al., EFM 2013
• Ga increases rapidlywith σc at the onsetof plasticity
• Ga increases less rapidly with smaller values, i.e. lower hardening capabilities
Example 3
Parametric study – effect of n
Martiny et al., EFM 2013
See lecture 1
Tvergaard & Hutchinson, JMPS 1992
Outline
1. Experimental characterization of ductile tearinga. The traditional JR curve approachb. The essential work of fracturec. A few words on the mechanisms
2. Modelling strategies
3. Example 1 : Thick components - prediction of JR curves with Gurson type model
4. Example 2 : Thin sheet fracture
5. Example 3 : Application of « stress at a distance » model in adhesive joints
6. Example 4 : Fracture in Al with PFZ
Grain
PFZ
Ee
Se
Grain
PFZ
Se
Ee
Intergranular versus transgranularductile fracture in 7xxx Al alloys
Depending on the mismatch between soft and
hard zones, fracture is either intergranular with very low toughness or
transgranularinter trans
Hard grain core
Soft precipitate free grain
boundary layer
Example 4
Objective : micromechanics based multiscalemodelling to relate inter- versus transgranular
cracking resistance to microstructure
requires proper account of shear !
θ δΣδ
L g1 Dg1
Lp1
Dp1
Dp2
L g2
Dg2d
h
Example 4
Scheyvaerts et al., JMPS 2011
Damage model
Rz
Rx
Extension by Gologanu, Leblond et al. (1993-1997) of Gurson model to spheroidal voids
Φgrowthnew≡C
σ y2 ΣΣΣΣ' +ηΣhX
2 + 2q g +1( ) g + f( )coshκ Σh
σ y
− g +1( )2 − q2 g + f( )2 = 0
( ) pkk
pkkp hhS ε
ε&
&&&
21 :3
12
3 +
−+= Pδδδδεεεε
( ) = py f εεεεσσσσ && −1 ypεσ
( ) piigrowth ff ε&& −= 1nuclgrowth fff &&& +=( )xz RRWS lnln ==
with
Beremin void nucleation criterion Thomason void coalescence criterion
bulkc
erfintc
particleprinc or σσσ max =
( )0maxmax σσσσ −+= eprinc
particleprinc k
( ) ppnucl gf εε && =
( ) ( )
+
−−=χχ
χαχσσ 1
2.11
12
2
Wn
y
z
Pardoen & Hutchinson, JMPS 2000
Example 4
See lecture 3
Void rotation must be taken into accountespecially for inclined PFZ
Example 4
Scheyvaerts et al., JMPS 2011
Fully periodic 3D FE cell calculations under constant stress triaxiality + shear
Kailasam and Ponte Castaneda, JMPS, 1998
22 nωn =& = Pω Ω -Ω
( )1221 nnnnΩPP ⊗+⊗−= ω
( ) ( ) P12212
P1 DAnnnneD:Ce ::
1
1
2
1-
2
2
⊗+⊗−+−=
W
WPω
2L10
2L20
e1
e2
2U12
2U12
U11U11
U22
U22
n2
n1
−θ
0
0.5
1
1.5
2
0 0.1 0.2 0.3 0.4 0.5 0.6
Unit Cell
Model with unit cell data
Extended Gurson model
W0=1/3
W0=3
θ (rad)
γ
Simple shear, n1//e
2 (θ=π/2)
Example 4
Scheyvaerts et al., JMPS 2011
Void rotation
Generalized void coalescence model
( ) ( )( ) ( )( ) ( ) ( )
+
−−=
δχδδχδχ
δχσ
δσeffeffeff
effeffloc
y
n
W
124.1
11.01
2
2
0
0.5
1
1.5
2
2.5
3
3.5
4
-100° -50° 0° 50° 100°
U12
/U22
=0
U12
/U22
=1
U12
/U22
=2
εc
eq
δ
W0=1, f
0=10-3
T=0.57
T=1A
B
Example 4
Scheyvaerts et al., JMPS 2011
Most of the time, coalescence in the ligament oriented perp. to the main loading direction
Coalescence process
Weck et al.∆εn ∆εn
with
and
( ) statestrain ,, ccn WF χε =∆
Example 4
Scheyvaerts et al., IJDM 2010
See lecture 3
Meshing Voronoï-based grain discretization
1 2 3
4 5 6
Random grain dispersion Voronoi tessellation Elementary sub-cell division
Sub-cell meshing Grain boundary merging Material properties allocation
Example 4
Scheyvaerts PhD thesis UCL 2008
Uniaxial tension sample
Small scale yielding conditions
Single grain
Multi-grain box
Crack line
Crack tip
Example 4
Scheyvaerts PhD thesis UCL 2008
Meshing
( )( )( ) ( )( )1expln
1exp
0
RVE deformedon
−∆=
−∆=∆
nnc
nncn
X
Xu
εε
ε
( ) statestrain ,, ccn WF χε =∆
ncσ
Extra strain from coalescence to final fracture of RVE is known
Corresponding extra displacement is known
nσ
nu∆
δ
Example 4
Scheyvaerts et al., JMPS 2011
Introduction of the length scale
GRAINσog/Eg νg ng f0g W0g λog
10-3 0.35 0.05 5 10-3 1 1
PFZσog/ σop νp np f0p W0p λop
2 to 8 0.35 0.3 5.17 10-2 1/3 1
Material parameters
Example 4
Scheyvaerts PhD thesis UCL 2008
Single grain
Grain
PFZ
Σe
Εe
Σe
Grain
PFZ
Εe
Σe
0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1
σ22
/σ0g
ε22
Σ11
/Σ22
=0, Wgrain
=1, (L/D)0p
=1.85,
np=0.3, W
0p=1/3, λ
0p=1, R
0=0.0525
σ0g
/σ0p=2
σ0g
/σ0p=4
σ0g
/σ0p=6
σ0g
/σ0p=8 Transgranular fracture
Intergranular fracture
σ0g
/σ0p=3
σ0g
/σ0p=5
σ0g
/σ0p=7
Example 4
Scheyvaerts PhD thesis UCL 2008
Example 4
Scheyvaerts PhD thesis UCL 2008
Uniaxial tension multigrain specimen
σ0g/ σ0p = 2 leads to transgranular fracture
Example 4
Scheyvaerts PhD thesis UCL 2008
Uniaxial tension multigrain specimen
σ0g/ σ0p = 4 leads to intergranular fracture
0
0.1
0.2
0.3
0.4
0.5
0.6
0% 20% 40% 60% 80% 100%
Wg=1, (L/D)
p=1.844
Wg=1, (L/D)
p=1.5
Wg=1, (L/D)
p=3
Wg=3, θ
g=π/2, (L/D)
p=1.844
Wg=3, θ
g=0, (L/D)
p=1.844
Wg=3, θ
g=π/4, (L/D)
p=1.844
Hexagonal, (L/D)p=1.5
εf
22=ln(A
0/A)
% Intergranular Fracture
Example 4
Scheyvaerts PhD thesis UCL 2008
Uniaxial tension multigrain specimen
COEXISTENCE OF BOTH FRACTURE
MODES
0
0.1
0.2
0.3
0.4
0.5
0.6
0 2 4 6 8 10 12
σ0g
/σ0p
Wg=3, (L/D)
0p=1.85
Wg=3, θ
g=π/4
Wg=1
Wg=3, θ
g=0
Wg=3, θ
g=π/2
εf
22=ln(A
0/A)
Example 4
Scheyvaerts PhD thesis UCL 2008
Strong anisotropy resulting from an elongated grain shape and orientation with respect to the loading direction.
Uniaxial tension multigrain specimen –effect of grain shape
Diapositive 104
FS1 scheyvaerts; 12/12/2008
R100RrY ≤
Crack opens in Mode I
Process zone = multi-grains mesh
Ifield KK =
ry
Example 4
Scheyvaerts PhD thesis UCL 2008
Cracked configuration – SSY conditions
Grain aspect ratio=1
Example 4
SSY cracked multigrain configuration
Scheyvaerts PhD thesis UCL 2008
σ0g/ σ0p = 2 leads to transgranular fracture
Grain aspect ratio=1
Example 4
Scheyvaerts PhD thesis UCL 2008
σ0g/ σ0p = 4 leads to intergranular fracture
SSY cracked multigrain configuration
σ0g/σ0p=2 • important crack tip blunting • path tends to branch and deviate from the horizontal plane
σ0g / σ0p
σ0g/σ0p=3
σ0g/σ0p=4
σ0g/σ0p=6
TRANS
INTER
• small CTOA • path close to the horizontal plane.
Example 4
Scheyvaerts PhD thesis UCL 2008
SSY cracked multigrain configuration
Crack advance = # of cracked material pointsCrack advance = crack projection on the crack plane
0
2
4
6
8
10
12
0 10 20 30 40 50 60 70 80
JR/(σ
0 X
0)
∆a/X0
σ0g
/σ0p
=2
Example 4
Scheyvaerts PhD thesis UCL 2008
SSY cracked multigrain configuration JR curves
Loading in the transverse
direction TD decreases
significantly the tearing resistance,
whatever the fracture mode.
Chaire Francqui 2016
Lectures
04/02/2016: Fracture of interfaces, adhesive joints and welds
18/02/2016: Fracture of coatings and electronic devices
03/03/2016: Fracture of metals and polymers - I. Damage
10/03/2016: Fracture of metals and polymers - II. Crack propagation
17/03/2016: Fracture of composites
24/03/2016: Fracture of nanomaterials
Adresse des lectures
Institut de Mathématiques, Quartier Polytech 1, Allée de la Découverte 12, Bâtiment B37, 4000 Liège - Auditoire 02
Inscription via le site web: http://www.facsa.ulg.ac.be/chairefrancqui/2016
Program of the Chair