Hello everyone!
Welcome back again, we are in our lecture
number 6 and we will continue what we have
learned in the last lecture, so it is a topic,
concept of strain and deformation and we are
in part 2.
In this lecture will cover strain in 3 dimensions
and we will mostly look at strain ellipsoids,
their shapes and orientations, then we will
move to Flinn diagram and there we will see
constriction, plane strain and flattening
types of deformation.
After that we will see the different ways
to look at progressive deformation from where
we will switch to vorticity and conclude the
lecture.
So strain in three dimensions is very similar
to what we have learned when we understood
the strain in two dimensions, there we had
X and Z and this time because we will be dealing
with three dimensions we will add Y, so homogeneous
deformation without volume change in three
dimensions generally like we described previously
with strain ellipse, this time we will be
described as a strain ellipsoid which is a
shape change from an imaginary or a material
sphere.
Now the sphere becomes ellipse, so we have
to describe its shape and orientation to describe
the strain and the ellipsoid can be described
by this equation.
So the three principal axes of strain are
here and then it is a Cartesian coordinate
system XYZ, so this is the equation of the
strain ellipsoid which is essentially the
function of your three principal axes of strain,
now the three axes of strain we know, the
principal axes of strain are the maximum,
intermediate and minimum principal strain
axes, so they are also mutually perpendicular
to each other and like we have described the
X and Z axes in the 2-D, we have to add here
the intermediate axis which is Y equal to
1 plus epsilon 2 or root over of lambda 2.
The other 2 remains the same that is X equal
to 1 plus epsilon 1 and Z equal to 1 plus
epsilon 3.
Now let us have a look of what do we mean
by strain ellipsoid which is a transformation
from a sphere.
So if we deform a sphere homogeneously then
ideally it should take a shape of an ellipse,
what we see here, this is a sphere and if
we cut it perpendicularly from the middle
then we can get the three ready of the sphere,
in this case all these are units, so this
is one, this is one and this is one.
Therefore this is undeformed and XYZ are all
equal to 1, if it transformed to an ellipse
due to homogeneous deformation, then this
X, Y and Z would change and therefore their
values would change as well and the sphere
would take form of an ellipse.
In that case Z would be 1 plus epsilon 3,
X would be 1 plus epsilon 1 and Y would be
1 plus epsilon 2 and therefore at least from
this visualisation as X has elongated most
that is our considerations, so X is the stretching
axis, Y is the intermediate one compared to
Z, so Y is the intermediate axis and Z which
got compressed most, so therefore Z is the
shortening axis, so these XYZ these three
play a very important role, their deformations,
their shortening, their extension relative
to other two.
Now based on these relative elongations or
relative shortenings of XYZ, we can actually
describe a number of possibilities or end
members of strain ellipsoid, but this time
instead of ellipses we looked with an unit
cube.
So what we see in this diagram, this is the
unit cube highlighted by a circle where X
is vertical, Y and Z are oriented horizontally,
this is un-deformed cube therefore XYZ equal
to 1, now one can deform this cube in many
different ways, but here what is shown by
these five illustrations at these five sides,
these are the some sort of end members of
the deformation in three dimensions.
The first one will take over is uniaxial compression
or compaction, so, before I go to this, go
to the description of this deformation as
you see this blue cube is the un-deformed
cube and what we see in the green colours
these are deformed shaped or from the cube,
it got transformed to the shape of this and
these are all in three dimension.
So what we see here, uniaxial compaction as
Z is the shortening axis, so the compaction
happened along Z, X and Y remain same, as
we can see this is the X direction, so X is
constant, this is the Y direction, Y is constant,
but Z instead of this length, it is now got
shortened and it got shortened in one direction
along one axis, Z axis, so therefore this
is uniaxial compaction.
Now in this side, you can do also uniaxial
extension, in that case, X is the maximum
stretching direction, so extension should
happen along the X direction, so this was
your initial length along X, but now it got
changed to this magnitude, whereas Z and Y
remained same.
Therefore this is uniaxial extension, now
there are three other possibilities, one is
axially symmetric extension, that means it
gets extended in one direction and shortened
in other two directions.
So therefore it got extended along X direction,
but got shortened along Y and along Z direction,
the other possibility is plane strain, this
is very important term that you need to learn.
Plane strain is when in one direction there
is no strain but in other directions yes,
in this case, this illustration what we see
here, this unit cube got extended along X
direction, it got shortened along Z direction,
but it is Y direction remained constant length
as it was, so therefore it is plane strain
that is along one direction, there is no strain.
And the final one is axially symmetric flattening,
what happens in this case that only one axis
shortened very much and other two axes they
extend highly, so what we see here as you
can see the Z direction, it got highly shortened
and this is the X and Y both, this is Y and
this is X, they both got extended significantly
compared to Z and this is known as axially
symmetric flattening.
Now before we switch to the next slide there
are many books or many texts or manual materials,
you may find these X, Y and Z these are oriented
different ways, so you may find that X and
Y these are placed horizontally and Z is vertical.
We may use in some cases these illustrations
as well, but whenever you look at such diagrams,
it is very important that you first make sure
you know where is your X, where is your Y
and where is your Z compared to the deformation
pattern.
Now based on this idea of deformation of an
un-deformed cube to this different possibilities,
you can understand that once these materials
they do deform different ways, you would produce
different types of strains, that means the
rocks would deform differently and at the
same time, because the rocks would be deforming
differently, you will see different types
of features in your rocks and we will see
this now.
So the best way to explain these different
types of deformation is known as Flinn diagram,
the description of three-dimensional strain
ellipsoid is best represented in Flinn diagram.
So in two dimension the shape of the strain
ellipse that we have described by only one
number, if you remember that was R which was
the ratio of long axis versus short axis but
in three-dimension the shape of the strain
ellipsoid is described by two numbers, one
is R1 2, which is the ratio of long axis versus
intermediate axis and another one is R2 3
which is the ratio of intermediate axis versus
short axis.
Now the ratio of these two parameters that
is ratio of R1 2 and R2 3 describes and distinguishes
the different shapes of the strain ellipsoid
and this number is known as K, where K is
defined as R1 2 minus 1 divided by R2 3 minus
1, so from this equation and our understanding
on XYZ we can figure out that minimum values
of R1 2 and R2 3 possible is 1 and K can be
from or could be equal or greater than 0 and
equal or less than infinity.
Now given this condition that K could be 0
to infinity, so you can report the all possibilities
of strain ellipsoids in the Flinn diagram
and the Flinn diagram we will see in the next
slide, it is actually a sort of two-dimensional
Cartesian plot where X axis is represented
by R2 3 and Y axis is represented by R1 2
and you can plot a number of K values in between,
so the diagonal of this plot as we will see
soon that is where R1 2 equal to R2 3 is actually
defining the plane strain condition, that
is Y axis of the finite strain ellipsoid is
constant during the deformation, that means
we have learned it before that there is no
stretching or no shortening along the Y axis.
Whatever deformation happens that happens
along the X and Z axis and therefore on the
XZ plain.
So this is the Flinn diagram we are talking
about, as you can see that along the X axis
we have plotted these and in this case, this
is again long axis which is X and here, this
is Y and again all these things is under log,
now this log is used, sorry, this is LN, so
this log is used to accommodate different
shapes or sizes of the measurements.
So different magnitudes of the measurement
if you have a high value and you have a small
value of the measurements, then you can plot
everything in the same diagram.
Let us try to understand this diagram in detail,
so as we said that along this line which is
running 45 degrees with respect to this X
and Y axis, this is where K equal to 1 and
as we have said K equal to 1 you have plane
strain at constant volume, as you can see
here at this magnitude of strain the Y axis
remained constant, even I increase the strain
magnitude Y axis remained constant, but it
extends along X and shortens along Z direction.
So whatever left on both sides of this K1
value, if I go to the top side that says that
it is under the constriction domain, that
means that in this domain everything would
deform, I mean I am sorry, along X axis it
would extend and other two directions it would
shrink.
Therefore with progressive deformation you
would produce something which is like a pencil
or like a long object, on the other hand if
we go to the flattening side, which is a downside
of this K equal to 1 direction, there the
deformation would happen in axially symmetric
manner, or axially symmetric flattening manner
where X is greater than or equal to Y and
these XY values are much, much greater than
Z, therefore you would produce something like
a pancake or like a flattened disk.
So if I go towards this side K equal to infinity
then I produce rocks characteristically with
some sort of linear features and if I come
to this side K equal to 0, the deformed rock
would show a lot of flattened objects or disk
shaped objects.
Now considering this idea, you can figure
out, if I am in the constriction domain, then
the rocks would tend to produce more linear
features through deformation and at K equal
to 0, the rocks would tend to produce more
flattened or planer fabrics.
Therefore these rocks here would be prone
with schistosity and therefore K equal to
0 or in this domain whatever we produce towards
this side, we call it S-tectonites and here
because we will be producing lineations dominantly
these are known as L-tectonites and whatever
stays in the middle it would have both lineations
and schistosity, therefore these are known
as LS-tectonites.
We will learn more about L-tectonites, LS-tectonites
and S-tectonites when will study foliation
and lineation in one of the next lectures,
so we will come back to this Flinn diagram
again and again for more interpretation of
deformed rocks.
Now let us talk about the volume change, so
we have discussed so far it is homogeneous
deformation without volume change, but volume
change like it happens in two dimensions,
it can also happen in three dimensions.
So pure volume change or volumetric strain
of an object as we, it is a very similar equation
that we have seen, so it is a ratio of the
change of the volume with respect to the initial
volume.
So you can say that volume change or if you
say this is delta then it is they define as
V minus V0 by V0, where V0 and V are volumes
of the object before and after the deformation
respectively.
So the volume factor delta is thus negative
for volume decrease and positive for volume
increase, now volume decrease and increase
whatever happens, it can happen either isotopically
or anisotropically, if it happens isotopically
means that all principal axes of strain are
either extending or shortening equally and
if that does not happen, then it is anisotropic
volume change.
So isotropic volume change therefore could
happen when XYZ, they are all greater than,
there all equal and greater than 1, therefore
you would have volume increase and if these
are all equal, but the values are less than
1, then you have volume decrease.
So here I have tried to show these with some
illustrations, so again we have these undeformed
cubes, undeformed cube here, you have X, you
have Y, you have Z direction here.
As you can see in the first illustration where
volume has decreased isotopically, all these
directions the shortening was equal and along
all these directions the shortenings were
equal.
In the second diagram where the volume has
increased, again along all directions the
extensions or elongations are equal.
Now anisotropic volume decreases essentially
where your XYZ would not remain same of values,
so they could be either greater than 1 or
less than 1, they could be equal to 1 or not
equal to 1, does not matter, but they cannot
be equal to each other.
So here there are three examples, so in the
first case what we see that volume has decrease
anisotopically where X and Y did not change,
so therefore your X remain constant, Y remain
constant but along Z it decreased its volume.
In this case, the second example, which is
also anisotropic volume increase but in this
case, X has extended but Y and Z, this is
your Y and this is your Z, I am sorry, this
is Y, this is Z and this is your X, so you
can see X has extended but Y and Z they remain
constant.
So therefore volume has increased but anisotopically
and in the third and final example what we
see here, volume has decreased again anisotopically
where nothing has changed along the X direction,
but it got shortened along Y direction and
also it got shortened along Z direction.
Now, so we have learned so far that homogeneous
deformation in three-dimension and also volume
deformation that means you can change your
volume in three-dimension as well.
The next topic we discussed in this lecture
is progressive deformation, now if you remember
the first lecture of strain we talked about
initial position, final position and then
one set of examples we talked about displacement
vectors, where we corrected the initial position
with the final position and added these two
points by an arrow and from initial to final
position we added that arrow heads and that
was a vector.
But then there was also another column that
was particle path where we actually could
track that how this point was moving from
its initial position to the next position
or to the final position and depending on
whether you are doing rigid body rotation
or translation or deforming the material by
simple shear, pure shear or any other ways
we can see or we can understand that, this
displacement vectors are not necessarily equal
or they are not necessarily similar to those
of the particle paths.
Now the study of these particle paths actually
fall in the domain of progressive deformation.
So there could be many different ways to achieve
the same deformed state that is the point,
what we see here?
We have initial state in these two rows, both
are circles and the final stage both got in
2 dimension we are looking at, and we have
got a strain ellipse.
Now what we see here, from here to here, if
we look directly then they look similar.
So the displacement vectors if we try to draw
from this position to this position in both
cases, it would be similar but in between
there could be many different possibilities.
So the first example is very straightforward
that I had a circle then little more strain,
even more strain and further more strain I
achieved to this state.
But in the second row what we see that it
took a different shape, it deformed differently
from here to here, from here to here, from
here to here and finally we could reach here.
So displacement vectors here in these two
cases will be very much similar but not the
particle paths, again the study is progressive
deformation.
So in the next three slides, we will have
a look that what could be the possibilities
or what could be the characteristic particle
paths or flow patterns of deforming rock in
terms of pure shear, simple shear and the
combination of these two and before that what
is flow pattern?
A flow pattern of a deforming rock is actually,
it refers the sum of particle paths at different
times steps during deformation, so it is you
add each and every particle path and cumulate
them, what you get is your characteristic
flow pattern of a particular type of deformation.
So what do you see in this slide on the left
side, I particularly would request you to
focus in this side, that here I had in 2 dimensions
I had a square and then progressively I deform
this square to achieve a finite elongation
or finite deformation along X direction of
two, of course you do not see the individual
squares and later rectangles just because
I had to fit them in a single slide, but they
are cascaded in this way.
So the strain progressed this way, so first
one it is undeformed and then slowly 0.1,
0.2, 0.3, 0.5, 0.7, 1, 1.5 and 2, so this
is how it progressively deformed in a pure
shear manner.
Now if I club them together all these images,
then they would appear something like that
and I try to track this little red points
in the form of particle paths, then this point
would move to here and then so on.
So characteristically all these points, at
least from the two sides, the top side and
the bottom side and how they have arrived
to their N shape N deformed shape, if we track
each and every instances of deformation, then
the characteristic flow path would be like
this, what you see here in this diagram by
red arrows.
So tiny red arrows heads and finally you achieve
a pattern, we will analyse this later.
Now let us go to what happens if you have
simple shear deformation, again the plots
are in a very similar way.
So I have a square and then I deform this
square with progressive simple shear, so here
at the shear strain magnitudes given one after
another and again I can club them together
as you can see here and I see the particle
paths or the flow patterns if I try to connect
each and every point from one stage to the
next stage then it would be given as you can
see here with this red arrows in this side
and this side.
And what is interesting that if you remember
the previous slide then it is characteristically
different from pure shear.
Now if we deform the same square, this square
in both pure shear or simple shear manner,
that means I combine pure and simple shear,
so here again in a similar way we have the
cascade of these rectangles and then what
we see here again, if we club them and try
to see the flow patterns then it is extremely
different.
We see that these things are going this way,
these things are going this way and so on,
so again, this is very different to what we
have seen with pure shear and simple shear
separately.
So what do these flow patterns tell us?
Let us have a look.
So if we see the pure shear, this is the simple
form, these yellow arrow heads are showing
that this is your compression direction and
this is your extension direction in pure shear.
What we see here that any material point here
is moving and slowly it is going towards the
extension direction and while doing so initially
this material line or points are in the compression
domain, in this direction they are in compression
and similarly this is symmetric along the
axis.
These are your compression domain and on the
other side, this is your extension domain.
In simple shear it is not so easy to apprehend
the compression or extension looking at the
flow pattern, but you can see the flow patterns
are very much straight and they are moving
opposite to other side of the flow lines.
Now in this case because if I had a circle
here, the circle would deform to an ellipse
of this and therefore I can assume that or
I can conclude that this would be my compression
domain, that means compressions are coming
from this direction, and 
this would be my extensional domain.
Now if you bring in between this combined
pure and simple shear then it would look like
this, here I have both symbols for pure shear
and simple shear.
What we see here that, unlike pure shear here
the material lines initially have some sort
of compression but then again they flow towards
the extension and try to achieve something
that we see in the simple shear pattern.
Now here again, it probably would have a compression
direction in this side as we can see here
or compressive domain like this and these
are extensive domain.
Now because I have compression from one side
and extension on the other side and also all
materials as we see that in these three very
basic examples, they tend to flow towards
the extension direction.
From here, if I consider a material is sitting
in between or then it may have a rotation
or it may stay stationary.
So based on a flow pattern and nature of deformation
if we try to analyse whether the materials
would stay stationary or materials would rotate,
based on this study we can go to another topic
which is known as vorticity.
So vorticity is actually the study of, it
describes how fast a particle do rotate in
a soft medium during deformation and there
is a term which is known as kinematic vorticity
number.
In mathematical expressions, it is expressed
as Wk and Wk is, as I said, it is kinematic
vorticity number is assigned for pure shear
as 0 and 1 for simple shear.
So whatever stays in between is the combination
of pure shear and simple shear, so again we
can have a look of the same diagram that here
Wk is 0, so this is pure shear, here Wk is
1, this is simple shear and whatever stays
in between from 0 to 1, this is your sub simple
shear flow pattern.
Okay and if I increase the Wk value, then
actually I achieve something which is flow
pattern is very similar to rigid body rotation
and 
we see the effects of rigid body rotations
in many different geological structures, including
the shears on features like delta structures
we have seen it in one lectures and will see
it more in when you study the ductile shears.
Now I conclude this lecture, but before concluding
this is important to remind you that 
all this strain and deformation that we have
learned so far, these are just not the descriptions,
this include mathematical analyses which is
not the scope of this lecture series.
But I recommend you to read these books I
recommended and also look and search in the
web to have at least some basic ideas of the
mathematical descriptions of strain.
So in this lecture, I believe and together
with the previous lecture we have some sort
of theoretical basis and more or 
less know the terminologies that we deal with
deformation and strain, particularly in the
context of structural geology.
Now the challenge is how to apply this knowledge
to the natural field, that means if I see
a deformed structure then how to analyse or
how to get strain 
or how to measure strain out of 
it 
and this is the topic of 
the next lecture.
Thank you very much, see 
you 
in 
the 
next lecture.
Bye.
