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Spring 2001
GEO 309 Lecture #6
STRAIN
So, what IS strain?
Things change shape. Strain describes how.
They can be translated or rotated (which is not strain)....
... and dilated or distorted (which is strain)
dilation = volume change
distortion = shape change
<original object translated, rotated, dilated & distorted>
Linear strain = (change of length)/initial length
Start with an arrow-shaped fossil (call him vectorilobite!)
The distance and direction from tail to head is described by a vector.
He and his companion vector-fossils are translated, rotated, and deformed.
Take out the translation - that isn't recorded in the rock.
The rotation does not affects the shape of the fossil.
That leaves deformation
- which is different from point to point, so there’s shape change)
Something that is originally circular will change shape into an ellipse........ and the aspect ratio (ratio of
dimensions) depends on the amount of deformation.
For each starting vector there is an ending vector.
You can list billions of separate start-end vector pairs (pretty dumb), OR.........
............you can write a rule for getting from the start to the finish.
This rule must connect a 3-vector (tail-head at start) to another 3-vector (at end)
...... therefore, the rule requires 9 numbers and it is a tensor.
You can write it as a 3x3 matrix
(but the matrix is just the numbers, you can write it in different ways, with respect to different coordinate
frames, etc. - the tensor is the real thing, the relationship between the start vector and the end vector.
This is the displacement vector.
You can separate the displacement into a whole body rotation + a strain.
whole body rotation = rotation without any internal shape/size change
strain = internal shape/size change
the extension e is the fractional increase in length:
e = (lf-l0)/l0
the stretch S is the ratio of final to starting length:
S = lf/l0
The strain tensor is usually denoted by the Greek letter .
It is simply a way if expressing the fact that the rock changes shape....
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......and that the RELATIVE changes in location of points in the rock as it deforms require
distortions that depend upon orientation.
Thus, to know the distortion vector, you must use the initial orientation vector....
...... each has 3 numbers, so you need 9 numbers to relate them....
Imagine 4 lines (Up-Down, R-L, UR-DL, UL-DR)
Imagine a right-left (x-direction) shortening strain of 10% ,
with 20% stretching in the up-down (y) direction, so
x = -.1x + 0y
y = 0x + .2y
In the x-y frame,
this 2-d strain tensor would be
[ -.10 .20]
so,
(10 , 0) moves (-1,0) to become (9 , 0)
(-10, 0) moves (1, 0)
= (-9, 0)
, 10) moves (-1,2)
= (9,12)
moves (1,-2)
= (-9,-12)
(0 , 10) moves (0, 2)
= (0, 12)
(0, -10) moves (0, -2)
= (0, -12)
(10,-10) moves (-1,-2)
= (9,-12)
(-10, 10) moves (1,2)
= (-9,12)
(10
(-10,-10)
y
(9,12)
(0,11)
(10,10)
(0,10)
(9,0)
(10,0)
X
Note that all lines from the origin rotate (before vs. after) under this strain, EXCEPT for those that run updown and right-left...
These are principal axes of this strain tensor (3 mutually perpendicular directions where the starting vector
and the ending vector are parallel).
In 2-D there are 2 principal axes. In 3-D there are three.
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In one principal axis, the stretching is biggest
In another it is smallest (or, perhaps, the contraction is biggest)
In the third, it is intermediate in value.
In all three of these three special directions, there is no change in direction.
In every other orientation there is a change in direction
For strain, (in 3 dimensions) we call these...
1 (the greatest principal extensional)
2 (the intermediate strain) and
3 the least extensional (or most contractional) strain .
The above is an example of pure shear
....... it involves no rotation
<simple and pure shear>
Pure shear is a symmetric tensor (xy=yx)
With pure shear.....
there are special directions (principal axes) for which there is no rotation.
(for which the angular shear psi =0) .
Simple shear does involve rotation
(so the principal axes themselves rotate with time!)
.....therefore infinitesimal shear and finite shear are somewhat different!
Simple shear is a symmetric tensor
...but the total deformation tensor associated with it is anasymmetric tensor
(exy ≠ eyxbut, as always,xy = yx)
Let's look at an example of simple shear......
x = 0x + 0y
y = .2x + 0y In the x-y frame,
this 2-d deformation tensor would be:
e=
[ .20 00]
=
****** 0 .1 / .1 0
+
0 -.1/.1 0
deformation = strain + rotation
so,
(10 , 0) moves (0,2) to become (10,2)
(10 , 10) moves (0,2)
= (-10,12)
(0 , 10) moves (0, 0)
= (0, 10)
(10,-10) moves (0,2)
= (10,-8)
(-10, 0) moves (0,-2)
= (-10,-2)
(-10,-10) moves (0,-2) = (-10,-12)
(0, -10) moves (0, 0)
= (0, -10)
(-10, 10) moves (0,-2) = (-10,8)
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y
(10,12)
(10,10)
(0,10)
(10,2)
X
(10,0)
------------------------------------------------------What if our strain tensor was


then

[ .1.1 .1.1]
?
x = .1x + .1y , y = .1x + .1y)
(10 ,0) becomes (11,1)
(10,-10) remains fixed, as does (-10, -10)
(0, 10) becomes (1,11)
(10,10) becomes (12,12) etc...
Here, we have a 20% stretch along the (1,1) axis (upper-right/lower-left),
and zero strain in the (1,-1) direction (lower-right/upper’left).....
.....and no rotation at all in either of theses directions
so the principal axes are 45° off from the previous example. (not along axes!)
That is why there are off-diagonal (xy) terms!
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y
(12,12)
(1,11)
(10,10)
(0,10)
(11,1)
(10,0)
X
However, if we wanted to, we could have rewritten this in a new coordinate system, inclined 45°. In
that system, there would be no off-diagonal terms!
Call this the a-b frame (tilted 45° from x-y)
so ...... ‘a’ points up and to the right and ‘b’ points up and to the left....
By inspection.....
the a axis doesn't rotate (so its a principal axis) and it
gets stretched 20%,
the b axis stays put (so it's a principal axis) too, but it
doesn't change length
Thus, in these axes our rotation is:
[ .20
0
0
]
!!!!!
Note: the sum of the diagonals (called the trace) is .2
(that means there is a .2 = 20% dilatation = volume increase)
in the x-y frame the trace was .1+.1 =.2
(the same)
!!!!!!
So, from the trace of a symmetric tensor you can tell something about it
... for strain........ does the volume increase or decrease? By how much?
..... for stress (as we'll see later) is it overall compressive or tensional?
....an increase in volume is a dilation (dilatational strain)
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<negative dilation>
<veins, stylolites, and strain>
All of these rules really apply to infinitesimal (small) strains.
For finite strain, things can get messier because the principal axes rotate.
Stress is measured in bars or MPa,
but strain is just fractional length changes.....
length/length has no units, so strain has no units
("shortening strain of 20%", "or a minimum principal strain of .05")
- not ".05 somethings...)
Strain can happen fast or slow.
If a shortening of 30% happened in 10 years, we'd say that the strain rate was 3%/year = 3*10 -2/yr. (Units are
"per time")
Typically, we describe strain rates as linear rates ( dot), and in units of "per seconds".
Typical geological strain rates are 10-12 to 10-15 s-1
infinitesimal shear = small shear strains for which the principal
axes don't
have time to rotate.
finite shear = large shear strain for which the principal axes may
have rotated
significantly
<sheep, sheared sheep>
To summarize:
• A tensor is a rule relating one vector to another.
• Rotation is an antisymmetric tensor ( Rxy = - Ryx ) relating the starting and
orientations to each other.
ending
• Strain is a symmetric tensor ( xy = yx ) relating start orientation and the
change in relative
position causing displacements within the body.
• A symmetric tensor in 3-d has 3 special directions (principal axes) in which
the starting and ending
vectors are the same. In all other directions they
are different. For strain, that means that there are only
those three
orientations for which something (say, a fossil) is not rotated.
• The three principal axes are mutually perpendicular and two of them are
the maximum and
minimum value directions (for strain, the most and least
extensional)
• The trace of a symmetric tensor tells us the overall sense of the tensor (for
strain, does volume
increase or decrease?)
So, what does strain look like in a rock?
It can be accomodated by rock in many different ways.........
Sometimes markers of known initial shape indicate the amount of strain
a few examples......
....mud cracks
{MUD CRACKS (Baker Quarry, Va.)}
.... here, ≈ 20% shortened in one direction
(elongation = length increase/initial length = -.2 = -20%)
.... note that (unlike geologists)engineers make shortening positive
....distorted inclusions
{STRAIN INDICATORS IN GRANIDIORITE (S. of San Fran.)}
.... (granidiorite = slightly silica-deficient granite)....
..... here, distorted to ≈ 4:1
....sometimes rocks at higher T can be very highly streched and sheared
{EXTREMELY STRETCHED QTZ (T2.20)}
… that strain can differ from mineral to mineral (more on why later!)
{QTZ STRETCHED, FELDSPAR NOT IN GRANITE MYLONITE}
{EXTENSION, SHEAR, MOSTLY DUCTILE, DIFFERENT VISCOSITIES (T3.17)}
....mylonites (strongly foliated metamorphic rocks that have
undergone some recrystallization to small grain sizes:
generally form • at elevated temperature
• in presence of some water
• at moderately low strain rates
(otherwise: more brittle deformation)
{GRANITIC MYLONITE}
....cleavage
....crenulation cleavage shows clear bulk shortening
(accomplished in a somewhat discontinuous manner....)
{ZONAL (KINK) CRENULATION (Everett Schist, rte. 55)}
....solution cleavage
(shortening occurs in highly discrete planes.. ..maybe 10-30%?)
{CLEAVAGE PLANES OVER SHORTENING THRUST (Rte. 23A)}
....folding
bulk-scale shortening strain is often accommodated by folding or faulting
PROJECT FIGURE ONTO WHITE BOARD TO MEASURE LENGTH,
AND CALCULATE THE PERCENTAGE SHORTENING......
{ANGULAR FOLDING IN LORALAI (PA 7.23)}
******* END Lecture #6 *******
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