Mixed Mode and Interface Fracture

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• Mixed Mode and Interface Fracture
• Rui Huang
• The 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

• Plane stress or plane strain, homogeneous,
  isotropic elastic solid.
• The T-stress is important in determining the
  crack path and its stability

Energy release rate for straight-ahead growth
Phase angle of mode mix
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Crack kinking
Criterion I maximum hoop stress (???)
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Crack kinking

• 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.
Hutchinson and Suo, Advances in Applied Mechanics
29, 63-191 (1992).
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Depth of substrate spalling
d/h
0
d/h 2.86
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(O), with the
  fracture toughness, G(O), 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 mismatch
For an interface between two elastic materials,
the crack behavior depends on the elastic
mismatch.
Dundurs parameters
Plane strain
No mismatch ? ? 0 Stiff film on compliant
substrate ? gt 0 Compliant film on stiff
substrate ? lt 0 If ?f ?s 0.5, ? 0 If ?f
?s 1/3, ? ?/4 Both ? and ? change signs
when the materials are switched.
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Interface crack tip field
E1
E2
Complex stress intensity factor
The stress field reduces to that in a homogeneous
solid when ? 0.
Energy release rate for straight-ahead growth
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Oscillatory singularity
Tractions ahead of an interface crack tip
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
When ? 0
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Crack face displacements

• 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 blocks
Under the remote loading, at the right crack tip
2a
Independent of ?. Reduce to Griffiths solution
when ? 0.
Take l 2a, then
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Example double cantilever beam
Take l h, then
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
Other contributions to the interface fracture
toughness include plastic dissipation, interface
friction
Interface fracture condition
Interface crack often grows under mixed mode, and
the interface toughness strongly depends on the
mode mix.
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Interface fracture toughness
Liechti and Chai (JAM 59, 295-304, 1992).
For epoxy/glass interface
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Choice of the length scale

1. Specimen size (thickness, crack length, etc.)
2. Material (intrinsic) length, e.g., size of
   plastic zone

Rule of transformation
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 measurement
Volinsky 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?

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

• 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
P
Steady state
?

• No need to monitor crack length
• Mode mix can be varied by asymmetric bending

Charalambides et al, 1989 Cao and Evans 1989
Ma, 1997 Dauskardt et al., 1998.
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Effect of plasticity
Intrinsic toughness
Dauskardt et al., Engineering Fracture Mechanics
61, 141-162 (1998). Lane et al., J. Mater. Res.
15, 2758-2769 (2000).
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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).