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

Brittle deformation III

• Terminology• The formation of faults and shear cracks• Insights from acoustic emission experiments• Theoretical versus actual material strength

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.


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.


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

Brittle deformation: The formation of faults and shear cracks

It turns out that stress-strain relation is non-trivial!

Question:


Rocks are macroscopically observed to fail in compression by theformation of shear fractures at an acute angle to 1 direction.


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)


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.

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.


AE experiments by Lockner et al. (1991):

3 stages:• diffuse acoustic emission (a)• nucleation (b)• propagation (c-f)


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 (undercompression!).


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.

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.

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


Tensional fractures just before becoming a fault? 3

1


A close view on the tip of a fault:

Figure from Cowie and Scholz, 1992

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.


Molecular theory: energy versus atomic distance

Question: why does heating lead to a volume increase?


ener

gy

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?


The minimum potential energy of strongly bonded solids is lower than that of weakly bonded solids:


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:

€

Uattraction ∝−1

r.

€

Urepulsion ∝1

r12.

€

U total =Uattraction +Urepulsion = −A

r+B

r12.

€

F =dU

dr=A

r2−12B

r13.


• Theoretical strength

• Fracture energy

• Elastic constant


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?


The modifying effect of a cavity on the distribution of stress in solid (Iglis, 1913):

Elliptical cavity:

€

c =σ a 1+ 2c b( ).


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.


Leonard Cohen too has pointed out that natural materials contain cracks (http://www.azlyrics.com/lyrics/leonardcohen/anthem.html).

"Anthem"

The birds they sangat the break of dayStart againI heard them sayDon't dwell on whathas passed awayor what is yet to be.Ah the wars they willbe fought againThe holy doveShe will be caught againbought and soldand bought againthe dove is never free.

Ring the bells that still can ringForget your perfect offeringThere is a crack in everythingThat's how the light gets in.

We asked for signsthe signs were sent:the birth betrayedthe marriage spentYeah the widowhoodof every government --signs for all to see.

I can't run no morewith that lawless crowdwhile the killers in high placessay their prayers out loud.But they've summoned, they've summoned upa thundercloudand they're going to hear from me.

Ring the bells that still can ring ...

You can add up the partsbut you won't have the sumYou can strike up the march,there is no drumEvery heart, every heartto love will comebut like a refugee.

Ring the bells that still can ringForget your perfect offeringThere is a crack, a crack in everythingThat's how the light gets in.

Ring the bells that still can ringForget your perfect offeringThere is a crack, a crack in everythingThat's how the light gets in.That's how the light gets in.That's how the light gets in.

http://www.azlyrics.com/lyrics/leonardcohen/anthem.html

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