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Rheology continued

Lecture 2 Last lecture

• Introduction into topic of course

• Deformation of rocks = material science

• Microstructures are memory of rock

• Rheology describes flow of rocks: flow laws

• Exercise

• Determine flow law from strain rate - stress data

• Determine strength profile of the crust

This lecture

• Discuss exercise

• Determine flow law from strain rate - stress data

• Determine strength profile of the crust

• Have a look at the agents of ductile deformation

• How can a crystal change its shape?

• Introduce dislocations

• Start developing an equation that describes flow rate

• Introduction to the concept of rate controlling process• Exercise of flow rate of traffic

The agents of deformation

• Lattice defects (imperfections)

• 0-dimensional (points): VACANCIES

• 1-dimensional (lines): DISLOCATIONS

• 2-dimensional (planes): TWINS

• Grain boundaries

• GRAIN BOUNDARY SLIDING

• FLUID ON GRAIN BOUNDARIES

• This lecture: dislocations

Flow by glide of dislocations

• An edge dislocation is the edge of an extra half lattice plane

• A dislocation is the edge of a zone along which the crystal has been

translated

• Glide of the edge dislocation through the whole crystal leads to:

• A unit of strain

• Annihilation of the dislocation

Flow by glide of dislocations

• An edge dislocation is the edge of an extra half lattice plane

• A dislocation is the edge of a zone along which the crystal has been

translated

• Glide of the edge dislocation through the whole crystal leads to:

• A unit of strain

• Annihilation of the dislocation

Flow by climb of dislocations

• An edge dislocation can also climb by

adding or removing vacancies

• Adding vacancies gradually removes

the extra half plane, resulting in

• A unit of strain

• Annihilation of the dislocation

A screw dislocation

• A screw dislocation is the edge of a zone along whichthe crystal has been translated parallel to thedislocation line

• Screw dislocations can also glide through the lattice

The Burger's vector

• The Burger's vector (b) is a vector defining

• The distance of slip caused by the dislocation

• The direction of slip

b

b

Edge dislocation Screw dislocation

Type of dislocation and Burger'svector

• Burger's vector normal to slip direction: EDGE

• Burger's vector parallel to slip direction: SCREW

Dislocations in reality

• Dislocations can be revealed

• By etching for a normal microscope

• With a transmission electron microscope

Creating dislocations

• Dislocations are created by deformation

• No deformation: no dislocations

• Main source of dislocations: Frank-Reed source

Dislocation glide and interaction Dislocation glide in reality

Flow by dislocation movement

• Dislocations bound an area where the crystal has beentranslated

• A small "quantum" of strain

• "quantum" size defined by Burger's vector

• How fast does a crystal deform under a certain stress?

• What is the flow law?

• Basically, we need to know

• How many dislocations?

• How strong are they?

• How fast do they glide?

Orowan's equation

• Orowan's equation is a very basic equationfor the rheology of materials that deform bythe movement of dislocations

• Orowan's equation relates the strain rate to

• The Burger's vector

• How much does one dislocation contribute

to strain?

• The dislocation density

• How many dislocations contribute to strain?

• The dislocation velocity

• How fast does one dislocation contribute to

strain?

˙ ! = b"v

b

Orowan's equation

• Take a piece of crystal with volume:

• If one dislocation glides through the whole crystal the added

shear strain is:

• If one dislocation glides through part of the crystal the added

shear strain is:

• If N dislocations glide through part of the crystal the added

shear strain is:

!

V = Lhl

!

" = b /h

!

" =#L

L

b

h

!

" = N#L

L

b

h=Nl

l

#L

L

b

h=Nl#Lb

V

˙ ! = b"v Orowan's equation

• If N dislocation glides through part of the crystal the added

shear strain is:

• Dislocation density [m/m3] is defined by:

• And strain rate by:!

" =Nl#Lb

V

!

" =Nl

V#$ = "%Lb

!

˙ " =#"

#t=# $%Lb( )

#t= $b

# %L( )#t

& ˙ " = b$v

˙ ! = b"v

Orowan's equation

• To solve Orowan's equation, "all" we need to know is:

• Type of dislocations: b

• Density of dislocations as function of stress: !

• Velocity of dislocations as function of stress: v

˙ ! = b"v Dislocation density: !

• From theory and experiment we know that the densityof dislocations is mainly a function of:

• Stress

• NOT temperature

• Equation for dislocation density:

(" = material constant)

Orowan:

!

" =#$

b

%

& '

(

) *

2

!

˙ " = b#v = b$%

b

&

' (

)

* +

2

v, ˙ " =$% 2

v

b

Velocity of dislocations

• We have derived:

• We can determine material properties " and b

• But what is the velocity v?

• The velocity is not a material property

• It may depend on:

• Stress

• Presence of water?

• Impurities?

!

˙ " =#$ 2

v

b

Excursion

• To determine the velocity (v) we must determine whatcontrols that velocity

• We need to know the rate-controlling process

• The rate-controlling process or step is the sloweststep in the whole process

• To illustrate the principle of rate-controlling step, wewill make a little excursion to traffic control

What controls the rate of a car?

• A car can travel at potential speed v0

• However, in a city there are traffic lights

• Every crossing the car may have to wait some time

• Traffic lights are separated an average distance H

• The cycle of a traffic light has a duration of D seconds,with D/2 s red and D/2 s green lights

v0

H

What controls the velocity of a car?

• Question: What is the equation that describes the average

velocity of a car, as a function of

• Its potential speed v0

• Traffic light spacing H

• Traffic light cycle time D

• What is rate controlling? Driving speed or traffic lights?

• Let us ignore acceleration time

• car travels at speed v0 between lights

v0

H

!

v = f v0,H,D( )