Mixed Mode and Interface Fracture Rui Huang The University of Texas at Austin Spring 2008.
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Transcript of Mixed Mode and Interface Fracture Rui Huang The University of Texas at Austin Spring 2008.
![Page 1: Mixed Mode and Interface Fracture Rui Huang The University of Texas at Austin Spring 2008.](https://reader030.fdocuments.net/reader030/viewer/2022032600/56649db35503460f94aa3a10/html5/thumbnails/1.jpg)
Mixed Mode and Interface Fracture
Rui HuangThe University of Texas at Austin
Spring 2008
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Mixed mode fracture• The stress field near a crack tip may be a mixture of modes I,
II, and III crack-tip field.
• In brittle, isotropic, homogeneous materials, cracks advance in the direction that maintains mode I (opening) at the crack tip.
• In anisotropic materials or interfaces between different materials, cracks may grow under mixed mode conditions.
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2D Crack-tip field
jiII
ijIII
ijI
ij Tfr
Kf
r
Kr 11)(
2)(
2),(
• Plane stress or plane strain, homogeneous, isotropic elastic solid.
• The T-stress is important in determining the crack path and its stability
x1
x2r
Energy release rate for straight-ahead growth: '
22
E
KKG III
Phase angle of mode mix:
I
II
K
Karctan
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Crack kinkingCriterion I: maximum hoop stress ();
2tan811
tan2arctan2
0-90°
90°
-70°
70°
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Crack kinking
Hutchinson and Suo, Advances in Applied Mechanics 29, 63-191 (1992).
IIItII
IIItI
KcKcK
KcKcK
)()()(
)()()(
2221
1211
'
22
E
KKG
tII
tIt
• Criterion II: pure mode I direction (KtII = 0);
• Criterion III: maximum energy release rate (Gt).
The three criteria predict similar directions for cracks in homogeneous, isotropic elastic solids.
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Depth of substrate spalling
dd/h
d/h = 2.86
0
In general, the spalling depth depends on the elastic mismatch between the film and the substrate.
Hutchinson and Suo, Advances in Applied Mechanics 29, 63-191 (1992).
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Double cantilever beam
By symmetry, the crack on the mid-plane is in pure mode I and would grow straight ahead.
However, the crack path is unstable. Any slight perturbation to the crack path will cause the crack to deflect further away from the mid-plane.
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Adhesive joint
Different crack trajectories observed in adhesively bonded double cantilever beam specimens (Chen and Dillard, Int. J. Adhesion Adhesives 21, 357-368, 2001 )
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Other crack patterns
Spiral crack in a drying thin layer of precipitate. Neda et al., PRL 88, 095502 (2002).
Oscillating cracks in quenched glass plates. Yuse and Sano, Nature 362, 329-331, 1993.
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Fracture of anisotropic materials
• Examples: crystals, fiber-reinforced composites
• A crack may grow in a mixed-mode path
• Anisotropic fracture toughness, depending on both the crack growth direction and the mode mix.
• Compare the energy release rate, Gt(Ω), with the fracture toughness, Γ(Ω), to determine crack initiation and kink direction.
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Interface fracture - debonding
• A crack may be trapped and grow along an interface between two different materials under mixed mode.
• Crack-tip field depends on the elastic mismatch and may have different singularity.
• Interface fracture resistance (toughness) depends on the interface energy (adhesion) as well as the mode mix at the crack tip.
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Elastic mismatchFor an interface between two elastic materials, the crack behavior depends on the elastic mismatch.
Dundurs parameters:
21
21
1221
1221
11
11
EE
EE
11
11
1221
1221
Plane strain:21
E
E 43
No mismatch: = = 0;
Stiff film on compliant substrate: > 0;
Compliant film on stiff substrate: < 0;
If f = s = 0.5, = 0;
If f = s = 1/3, = /4;
Both and change signs when the materials are switched.
1
-1
-0.25
0.25
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Interface crack tip field
x1
x2r
),(2
]Im[),(
2
]Re[),(
II
ij
iI
ij
i
ij fr
Krf
r
Krr
1
1ln
2
1
Complex stress intensity factor: 21 iKKK
The stress field reduces to that in a homogeneous solid when = 0.
Energy release rate for straight-ahead growth: 2
22
1
2
*
1KK
EG
21 '
1
'
1
2
1
*
1
EEE
E1
E2
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Oscillatory singularityTractions ahead of an interface crack tip:
r
riKKi
i
2212122
x1
x2r
When 0, the opening and shearing tractions are coupled; modes I and II are inseparable.
The ratio between the opening and shear tractions varies with r.
Need a length scale to define the mode mix:
]Re[
]Im[arctan
)(
)(arctan
22
21
i
i
Kl
Kl
lr
lr
1
2arctanK
KWhen = 0:
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Crack face displacements
2)cosh()21(
8*
2112
r
E
iKK
i
ri
i
x1
x2r
• Interpenetration of the crack faces is predicted for 0.
• Contact of the crack surfaces should be considered (not traction free any more!)
• The predicted contact zone is typically small, thus ignored in many applications.
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Example: two semi-infinite blocks22
21 Under the remote loading, at the right crack tip:
2a
ia
aiiiKKK
)2(21212221
Independent of .
Reduce to Griffith’s solution when = 0.
Take l = 2a, then:
2122
2221
2
2tan
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Example: double cantilever beam
M
M
h
h
32
),(
21)1(
32
hh
MeiKKK
i
i
Take l = h, then: ),( ),(
0
4/
Hutchinson and Suo, Advances in Applied Mechanics 29, 63-191 (1992).
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Interface fracture criterion
Following the energy approach by Griffith and Irwin.
Work of adhesion: 12210
Other contributions to the interface fracture toughness include plastic dissipation, interface friction:
fp 0
Interface fracture condition: G
Interface crack often grows under mixed mode, and the interface toughness strongly depends on the mode mix.
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Interface fracture toughnessLiechti and Chai (JAM 59, 295-304, 1992).
h
h
UV
hh
eiUcVEiKK
i
i
),(
21
)(*
h
lln),(
cV
Utan
For epoxy/glass interface:
935.0 188.0
060.0 14
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Choice of the length scale
(I) Specimen size (thickness, crack length, etc.)
(II) Material (intrinsic) length, e.g., size of plastic zone
Rule of transformation: 1
21122 ln
l
lll
1
1
22 ,ln),( l
l
ll
Using a specimen length renders the toughness dependent on the specimen size, while using an intrinsic material length would avoid such artificial size effect.
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Interface toughness measurementVolinsky et al., Acta Mat. 50, 441-466, 2002.
Superlayer method
Nanoindentation test
Scratch test
Sandwich bending methods (Double cantilever, Four-point bending, etc.)
Peel test
Bulge and blister test
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Superlayer test
• Use a superlayer (Cr, epoxy, etc.) to increase the total film thickness and the residual stress without changing the interface.
• Use a thin release layer to introduce an initial debonding.
• Measure the critical thickness to determine the interface toughness.
• Steady-state energy release and the bilayer curvature after debonding
• Phase angle of mode mix?
f
fSS E
hG
2
20
Bagchi et al., J. Mater. Res. 9, 1734-1741, (1994).
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Sandwich specimen
• Easy to load, with variable mode mix.
• Can measure interfacial energy between two thin films when both are sandwiched.
• The effect of residual stress is minimal.
• Control of crack path along the interface of interest?
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Double cantilever test
h
h
P
P
a
b
2
4
32
32
2
8
312 a
EhP
hEb
aG
h
lln),(
• The thin film has little effect on the global energy release rate;
• The local mode mix, however, depends on the film.
• Variable mode mix may be achieved by asymmetric DCB
• Plastic deformation in the film depends on the film thickness
• Various crack paths are possible (in-layer, oscillatory, or alternating)
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Four-point bend test
h
h
PP
PPL L
2
32
2
4
21P
hEb
LGSS
h
lln),(41
• No need to monitor crack length
• Mode mix can be varied by asymmetric bending
P
Steady state
Charalambides et al, 1989; Cao and Evans; 1989; Ma, 1997; Dauskardt et al., 1998.
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Effect of plasticity
Dauskardt et al., Engineering Fracture Mechanics 61, 141-162 (1998).
Lane et al., J. Mater. Res. 15, 2758-2769 (2000).
Intrinsic toughness
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Other effects on interface fracture
• Interface roughness: increased surface area, asperity contact and friction
• Interface chemistry: segregation, bond density• Environment: moisture, stress corrosion or subcritical
debonding
Lane, Annual Rev. Mat. Res. 33, 29-54 (2003).
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Summary
• Under mixed-mode fracture, a crack in a homogeneous, isotropic elastic solid kinks into mode-I path; only mode-I fracture toughness is needed.
• Along an interface, a debonding crack often grows under mixed mode, with oscillatory singularity; interface toughness depends on the mode mix.
• Various methods are available for interface toughness measurement; the effects of plasticity, interface roughness, chemistry, and environment must be carefully considered.
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Additional readings
Freund and Suresh: Chapter 4;
Suo, Reliability of Interconnect Structures. In Comprehensive Structural Integrity (Milne, Ritchie, Karihaloo, Editors-in-Chief), Volume 8: Interfacial and Nanoscale Failure (Gerberich and Yang, Editors), Elsevier, 2003.
Hutchinson and Suo, Advances in Applied Mechanics 29, 63-191 (1992);
Rice, J. Appl. Mech. 55, 98-103 (1988).