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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( )