Strain production and preferred orientation
Groves and Kelly,
Crystallography And Crystal Defects,
1970. Chapter 6.
Wenk, H.-R. Chapter 10,
in Karato and Wenk, Plastic deformation of minerals and rocks, Rev. Mineral. Geochem. Vol. 51, 2002
Tuesday Nov. 4, 2003
12.524 Mechanical properties of rocks
1
Strain during glide ε 12 = b
2h
δl=|b|
z
h
for n dislocations
slipping:
ε ij
z
h
dx
l
Monday, Nov. 24, 2003
total
b
=n
2h
For an increment:
dγ =
b dx t
h l t
bda
dε =
2V
t
12.524 Mechanical properties of rocks
2
Inclined Slip Plane
2
z
1
Strain
z
z
a
b
z
Burgers Vector
Normal to glide plane
Number of dislocations
h
Monday, Nov. 24, 2003
12.524 Mechanical properties of rocks
3
Strain elements from glide
^
n
3
2
1
h
P’
P
r
eij
PP' = α hβ = α (r i
n) β
r’
where is a unit Burgers vector
s
and α =
h
e11 =
Monday, Nov. 24, 2003
dui
≡ ε ij + ωij
dx j
∂
∂
α (r in ) β ) ≡ α
( xk nk ) β1 = α n1β1 = e11
(
∂x1
∂x1
12.524 Mechanical properties of rocks
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Strain and Rotation
 2n1β1 n1β 2 + n2 β1 n1β3 + n3 β1 
α

ε ij =  i
2n2 β 2
n2 β3 + n3 β 2 
2
2n3 β3 
i
 i
z
z
01
n2 β1 − n1β 2

α
ωij =  n1β 2 − n2 β1
0
2
 n1β 3 − n3 β1 n2 β 3 − n3 β 2
Monday, Nov. 24, 2003
n3 β1 − n1β 3 

n3 β 2 − n2 β 3 

03

No component of β
or n in k
direction→εik=0
No climb or diffusion
−n1β1 = n2 β 2 + n3 β 3
z
12.524 Mechanical properties of rocks
Rotation (and strain)
depend on
activity (α)
5
Independent Slip Systems
z
z
z
z
Distinct systems can give rise to same strain. (e.g. interchange n and β)
If strain element unique, then independent.
No more than two β’s on the same plane can
be independent.
Crystallographic symmetry can increase
number of strain elements for a particular slip
systems.
Monday, Nov. 24, 2003
12.524 Mechanical properties of rocks
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Strain from climb
z
For climb only
γ =s
eij =
∂
∂
γ (riβ ) β ) ≡ γ
( xk β k ) β j = γ ( β i ) β j
(
∂x j
∂x j
ε ij =
z
z
z
z
u = γ (riβ ) β
l
γ
ββ
(
2
i
j
+ β j β i ) = γβ i β j
P’ P
l
h
r’
r
Strain is irrotational
Depends only on β not n.
Open system, so 6 ind. s.s.
Three β’s climbing and
gliding give 6 systems.
Monday, Nov. 24, 2003
12.524 Mechanical properties of rocks
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Taylor-von Mises Criterion
z
z
z
z
Low T, glide easier than climb. Dilatancy may result.
For homogeneous, non-dilatant, creep,
5 independent slip systems must be present.
von Mises Criterion
If dilatant, 6 independent slip systems necessary.
If condition not fulfilled
z
z
z
z
twinning
climb or diffusion
void production
inhomogeneous flow
Monday, Nov. 24, 2003
12.524 Mechanical properties of rocks
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Independence of Slip Systems
z
z
Convert vectors to Cartesian system. Choose 5 easiest systems.
If no climb allowed, express strain as 5-
dimensional vector. [ε − ε , ε − ε , ε , ε , ε
]
Form 5x5 matrix, take determinant
11
z
22
33
22
12
23
13
(ε11 − ε 33 ) (ε11 − ε 33 ) (ε11 − ε 33 ) (ε11 − ε 33 ) (ε11 − ε 33 )
I
II
III
IV
V
ε
−
ε
ε
−
ε
ε
−
ε
ε
−
ε
ε
−
ε
( 22 33 ) ( 22 33 ) ( 22 33 ) ( 22 33 ) ( 22 33 )
I
ε12I
ε 23I
ε13I
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II
ε12II
ε 23II
ε13II
III
ε 12III
ε 23III
ε 13III
IV
ε 12IV
ε 23IV
ε 13IV
12.524 Mechanical properties of rocks
V
ε 12V
ε 23V
ε13V
=0
9
Deformation of Polycrystals
z
z
z
z
If 5 independent ss’s available,
homogenous, non-dilatant flow possible.
If inhomogeneous flow possible, then 4 ss’s sufficient.
If dilatancy required, flow is pressure dependent.
With only two ss’s, impossible to get pressure
independent flow.
z
Basal slip, e.g. mica.
Monday, Nov. 24, 2003
12.524 Mechanical properties of rocks
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Texture, Fabric, and Preferred
Orientation
z
z
z
Texture: Geometrical aspects of component
particles of a rock, including size,
shape, and arrangement.
Fabric: Orientation in space of elements of
which rock is composed.
That factor of the texture which depends
on the relative sizes and shapes, and the
arrangement of the component crystals.
Preferred orientation: A rock in which the grains are
more or less systematically oriented by
shape or [by crystallographic orientation].
ƒ
Monday, Nov. 24, 2003
Dictionary of geological terms, Am. Geolog. Inst., Dolphin
Books, 1962.
12.524 Mechanical properties of rocks
11
Methods of measuring
z
z
z
z
z
z
Optical
X-ray pole figure goniometer
Synchrotron X-rays
Neutron diffraction
TEM
EBSD (EBSP)
ƒ
Monday, Nov. 24, 2003
Wenk, H-R. in Plastic Deformation of minerals and rocks,
Rev. Min. and Geochem. Vol.51, 2002.
12.524 Mechanical properties of rocks
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Data representation
z Pole
figures: Density distribution of a single
pole plotted in a stereographic plot relative to
the sample coordinates.
z ODF: An orientation probability distribution
function of three Euler angles
Monday, Nov. 24, 2003
12.524 Mechanical properties of rocks
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Simulations
z
Taylor- equi-strain
z
z
Upper bound ( strength)
Lower bound ( strength)
z
z
Equi-stress-Sachs
z
z
z
z
z
Monday, Nov. 24, 2003
≅Voight elastic bound
fcc bcc metals (hi symm.)
upper bound in strength
≅Voight elastic bound
lower bound
heterogeneous strain
Self-consistent (VPSC)
Finite element
12.524 Mechanical properties of rocks
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Processes
z
z
z
z
Constitutive law
Grain growth/Recrystallization
Metamorphic reactions
Dilatancy
Monday, Nov. 24, 2003
12.524 Mechanical properties of rocks
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