1. Mixed Mode and Interface Fracture
Rui Huang
The University of Texas at Austin
Spring 2008

2. 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.

3. 2D Crack-tip field x1 x2 r 
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:

4. Crack kinking Criterion I: maximum hoop stress (); 70° -90° 90°
-90°
90°
-70°
70°

5. Crack kinking Criterion II: pure mode I direction (KtII = 0);
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, (1992).

6. Depth of substrate spalling
d/h
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, (1992).

7. 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.

8. Adhesive joint
Different crack trajectories observed in adhesively bonded double cantilever beam specimens (Chen and Dillard, Int. J. Adhesion Adhesives 21, , 2001 )

9. Other crack patterns
Spiral crack in a drying thin layer of precipitate. Neda et al., PRL 88, (2002).
Oscillating cracks in quenched glass plates. Yuse and Sano, Nature 362, , 1993.

10. 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.

11. 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.

12. Elastic mismatch
For an interface between two elastic materials, the crack behavior depends on the elastic mismatch.
Dundurs parameters:
Plane strain: 1 -1
-0.25
0.25
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.

13. Interface crack tip fieldx1 x2 r  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:

14. Oscillatory singularityx1 x2 r 
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:

15. Crack face displacementsx1 x2 r 
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.

16. Example: two semi-infinite blocks
Under the remote loading, at the right crack tip: 2a
Independent of .
Reduce to Griffith’s solution when  = 0.
Take l = 2a, then:

17. Example: double cantilever beamh
Take l = h, then:
Hutchinson and Suo, Advances in Applied Mechanics 29, (1992).

18. 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.

19. Interface fracture toughnessV
Liechti and Chai (JAM 59, , 1992).
For epoxy/glass interface:

20. Choice of the length scale
Specimen size (thickness, crack length, etc.)
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.

21. Interface toughness measurement
Volinsky et al., Acta Mat. 50, , 2002.
Superlayer method
Nanoindentation test
Scratch test
Sandwich bending methods (Double cantilever, Four-point bending, etc.)
Peel test
Bulge and blister test

22. 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, , (1994).

23. 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?

24. Double cantilever testh P a b
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)

25. Four-point bend test P P h  L No need to monitor crack length
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.

26. Effect of plasticity Intrinsic toughness
Dauskardt et al., Engineering Fracture Mechanics 61, (1998).
Lane et al., J. Mater. Res. 15, (2000).

27. 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, (2003).

28. 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.

29. 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, (1992);
Rice, J. Appl. Mech. 55, (1988).