1. Brittle deformation III Terminology The formation of faults and shear cracks Insights from acoustic emission experiments Theoretical versus actual material strength

2. Brittle deformation: Terminology 2 fracture types: Opening fractures: the relative motion is perpendicular to the fracture surface. Shear fractures: the relative motion is parallel to the fracture surface. Faults are shear fractures and joints are opening fractures.

3. Brittle deformation: Terminology Crack propagation modes: Mode I: displacement is perpendicular to the crack plane. Mode II: displacement is in the plane of the crack and normal to the crack edge. Mode III: displacement is in the plane of the crack and parallel to the crack edge. Note that while mode I is an opening fracture, modes II and III are shear fractures.

4. Brittle deformation: Terminology A crack that has component of displacement both parallel and perpendicular to the crack surface is referred to as a mixed-mode crack (for example, a dike intruding a fault).

5. Brittle deformation: The formation of faults and shear cracks It turns out that stress-strain relation is non-trivial! Question:

6. Brittle deformation: The formation of faults and shear cracks Rocks are macroscopically observed to fail in compression by the formation of shear fractures at an acute angle to  1 direction.

7. Brittle deformation: The formation of faults and shear cracks It is not possible for a shear crack to grow in its own plane. Instead, the propagation of a shear crack inclined to  1 occurs by the generation of mode-I cracks parallel to  1. Horri and Nemat-Nasser (1985)

8. Brittle deformation: The formation of faults and shear cracks How to reconcile the two results? Shear cracks form at an acute angle to  1 compressive. A shear crack oriented at an acute angle to  1 cannot propagate in its own plane.

9. Brittle deformation: Insights from acoustic emission experiments Acoustic Emission is the class of phenomena whereby an elastic wave, in the range of ultrasound usually between 20 KHz and 1 MHz, is generated by the rapid release of energy from the source within a material.

10. Brittle deformation: Insights from acoustic emission experiments AE experiments by Lockner et al. (1991): 3 stages: diffuse acoustic emission (a) nucleation (b) propagation (c-f)

11. Brittle deformation: Insights from acoustic emission experiments The evolution of volumetric extension and axial extension: Note that: Acoustic emission starts at stage III During stage III, the sample dilate relative to what would be expected from linear elasticity. In stage IV, the sample dilate (under compression!).

12. Brittle deformation: Insights from acoustic emission experiments In conclusion: 1. Fault formation is preceded by the growth of mode-I cracks throughout the sample. 2. Later, the growth of mode-I is intensified and becomes localized. 3. Finally, a macroscopic fault is formed - probably through the coalescence of micro cracks.

13. Brittle deformation: Fracture coalescence in the lab The following experiment helps to illustrate how the process of crack coalescence works: In the next stage - the sample breaks.

14. Brittle deformation: Fracture coalescence in the field Tensional fractures just before becoming a fault? Figure from from: www.earth.monash.edu.au/Teaching/mscourse/lectures

15. Brittle deformation: Fracture coalescence in the field Tensional fractures just before becoming a fault? 33 11

16. Brittle deformation: Fracture coalescence in the field A close view on the tip of a fault: Figure from Cowie and Scholz, 1992

17. Brittle deformation: Theoretical vs. actual strength Down to the molecular level: During cracking, the bonds between the atoms break: So in principle, it is possible to estimate the theoretical strength of materials.

18. Brittle deformation: Theoretical vs. actual strength Molecular theory: energy versus atomic distance Question: why does heating lead to a volume increase?

19. Brittle deformation: Theoretical vs. actual strength energy bond length Heating increases the potential energy, and owing to the asymmetric shape of the energy curve, it increases the average bond length. Question: how does the bond strength reflected by the energy curve?

20. Brittle deformation: Theoretical vs. actual strength The minimum potential energy of strongly bonded solids is lower than that of weakly bonded solids:

21. Brittle deformation: Theoretical vs. actual strength Potential energy due to attraction between opposite charge ions: Potential energy due to repulsion from electron cloud overlap: Total potential energy is: Differentiation with respect to r gives the force, F:

22. Brittle deformation: Theoretical vs. actual strength Theoretical strength Fracture energy Elastic constant

23. Brittle deformation: Theoretical vs. actual strength From the previous diagram one can estimate the theoretical strength,  th, under tension. It turned out that  th is between 10-50 GPa. Actual strength is 2 orders of magnitude lower. Question: why do materials break under lower applied stresses?

24. Brittle deformation: Theoretical vs. actual strength The modifying effect of a cavity on the distribution of stress in solid (Iglis, 1913): Elliptical cavity:

25. Brittle deformation: Theoretical vs. actual strength Stress concentration for c=3b: Thus, the local stress about a sharp notch or corner could rise to a level several times that of the applied stress, and even submicroscopic flaws my act as stress concentrators that weaken the solid.

26. Brittle deformation: Theoretical vs. actual strength Leonard Cohen too has pointed out that natural materials contain cracks (http://www.azlyrics.com/lyrics/leonardcohen/anthem.html).http://www.azlyrics.com/lyrics/leonardcohen/anthem.html "Anthem" The birds they sang at the break of day Start again I heard them say Don't dwell on what has passed away or what is yet to be. Ah the wars they will be fought again The holy dove She will be caught again bought and sold and bought again the dove is never free. Ring the bells that still can ring Forget your perfect offering There is a crack in everything That's how the light gets in. We asked for signs the signs were sent: the birth betrayed the marriage spent Yeah the widowhood of every government -- signs for all to see. I can't run no more with that lawless crowd while the killers in high places say their prayers out loud. But they've summoned, they've summoned up a thundercloud and they're going to hear from me. Ring the bells that still can ring... You can add up the parts but you won't have the sum You can strike up the march, there is no drum Every heart, every heart to love will come but like a refugee. Ring the bells that still can ring Forget your perfect offering There is a crack, a crack in everything That's how the light gets in. Ring the bells that still can ring Forget your perfect offering There is a crack, a crack in everything That's how the light gets in.