Transcript Rock Deformation

Strain II
Mohr Circle for Strain
• Recall that for stress, we plotted the normal
stress n against the shear stress s, and we
used equations which represented a circle
• Because geologists deal with deformed
rocks, when using the Mohr circle for strain,
we would like to deal with measures that
represent the deformed state (not the
undeformed state!)
Equations for Mohr circle for strain
• Let’s introduce two new parameters:
´ = 1/ (to represent the abscissa)
´ = / = ´ (to represent the ordinate)
• The two equations for the Mohr circle are in
terms of ´ and ´
´ = (´1+´3)/2–(´3-´1)/2 cos2´
´ = (´3-´1)/2 sin2´
Mohr Circle for Strain
• The coordinates of any point on the circle
satisfy the above two equations
• The Mohr circle always plots to the right
of the origin because we plot the
reciprocal quadratic elongation
’=1/ =(1+e)2, i.e.,
•  =(1+e)2 and ´=1/ are both ‘+’
Mohr Circle for Strain …
• In parametric form, the equations of the
Mohr circle are:
´ = c – r cos2´
´ = r sin2´
Where:
Center, c = (´1+´3)/2 (mean strain)
Radius, r = (´3-´1)/2
Sign Conventions
• The 2 angle is from the c´1 line to the point on the circle
– Points on the circle represent lines in the real world!
• Since we use the reciprocal quadratic elongations ´1 & ´3
– clockwise (cw)  from c´1 is ‘+’, and
– Counterclockwise ccw is ‘-’ (compare it with stress!)
• cw  in real world is cw 2 in the Mohr circle, and vice versa!
• However, ccw , from the O´ line to any point on the circle, is
‘+’, and cw  is ‘-’
Deformed Brachiopod
Lines of no finite elongation - lnfe
• Draw the vertical line of lnfe =1  the ´ axis
• The intersection of this magic line with the Mohr
circle defines the lnfe (there are two of them!).
• Elongation along the lnfe is zero
– They don’t change length during deformation, i.e.,
elnfe=0, and lnfe=1, and therefore ´lnfe=1
• Numerical Solution: tan2 ´ = (1-´1)/(´2-1)
Finding the angular shear
• Because ´=/ and ´=1/ therefore:
´= ´ which yields  = ´/´
• Since shear strain,  =tan , and  = ´/´:
tan  = ´/´
• Note: Mohr circle does not directly provide the
shear strain  or the angular shear , it only
provides ´
– However, notice that ´=  if ´=1!
Angular Shear
 = ´/´
• The above equation means that we can get the
angular shear () for any line (i.e., any point on the
circle) from the ´/´ of the coordinates of that
point
• Thus,  is the angle between the ´ axis and a line
connecting the origin to any point on the circle
• ccw  in the Mohr circle translates into ccw in the
physical world (i.e., same sense)!
Finding the Shear Strain 
• The ordinate of the Mohr diagram is ´, not the shear strain 
• Because  = ´/´, then  = ´ only if ´=1
• This means that for a given deformed line (e.g., a point ‘A’
on the Mohr circle), the ´ coordinate of the intersection of
the ‘magical’ ´=1 line with the OA line (connecting the
origin to ‘A’) is actually  because along the ´=1 line,  and
´ are equal!
• Procedure:
– For any line (which is a point, e.g., ‘A’, on the circle), first
connect the point ‘A’ to the origin (O), and extend the
line OA (if needed), to intersect the ´=1 line
– Read ´along the ´=1 line; this is  for the line!
Finding lines of maximum shear (lms) strain (max)
• Draw two tangents (±) to the Mohr circle
from the origin, and measure the 2´ (±)
where the two lines intersect the circle
• Numerical Solution:
– Orientation:
tan ´lms= (2/1) (Note: these are , not ´)
– Amount:
max =(1-2 )/2 12
Lines of Maximum Shear (lms)
.
Another Example
Example
A unit sphere is shortened by 50% and extended by 100%
e1=1, and e3=-0.5
s1=X=l´/lo =1+e1=2 & s3=Z=l´/lo= 1+e3=0.5
1 = (1+e1)2=s12 =4 &  3 =(1+e3)2 = s32=0.25
 1´ = 1/1 =0.25 & 3´ = 1/3 =4
• Note the area remains constant:
XZ =  1  3 =  4 0.25 =1
c = (´1+ ´3)/2 = (0.25+4)/2=2.125
r = (´3 - ´1)/2 = (4-0.25)/2=1.90
• Having ´c´ and ´r´, we can plot the circle!
Graphic representation of strain ellipse
•
•
Point A (1,1) represents an undeformed circle (1 = 2 = 1)
Because by definition, 1>2 , all strain ellipses fall below
or on a line of unit slope drawn through the origin
•
All dilations fall on the 1 = 2 line through the origin
•
1.
2.
3.
All other strain ellipses fall into one of three fields:
Above the 2=1 line where both principal extensions are +
To the left of the 1=1 where both principal extensions are –
Between two fields where one is (+) and the other (-)
Shapes of the Strain Ellipse
Graphic representation of strain ellipse
• Along AB, the original circle does not change shape
but only change radius
• From A to origin the radius gets smaller (l & 2 <1)
• Along AC: elongation along 1, and no change along
2
• Along AD: shortening along 2 & no change along 1
• Only along the hyperbola through field 3, where 1=
1/2, is the area of the ellipse equal to the area of
the undeformed circle (i.e., constant area)
• Zone 3 is the only field in which there are two lnfes
Flinn Diagram
Volume change on Flinn Diagram
• Recall: S=1+e = l'/lo and ev = v/vo =(v’-vo)/vo
• An original cube of sides 1 (i.e., lo=1), gives vo=1
• Since stretch S=l'/lo, and lo=1, then S=l'
• The deformed volume is therefore: v'=l'. l'. l'
• Orienting the cube along the principal axes
V' =S1.S2.S3 = (1+e1)(1+e2)(1+e3)
Since v =(v’-vo), for vo=1 we get:
v =(1+e1)(1+e2)(1+e3)-1
• Given vo=1, since ev = v/vo, then
ev = v =(1+e1)(1+e2)(1+e3) -1
 1+ev =(1+e1)(1+e2)(1+e3)
If volumetric strain, v = ev = 0, then:
(1+e1)(1+e2)(1+e3) = 1 i.e., XYZ=1
• Express 1+ev =(1+e1)(1+e2)(1+e3) in e & take log:
ln(1+ev) = e1+e2+e3
• Rearrange: (e1-e2)=(e2-e3)-3e2+ln(1+ev)
• Plane strain (e2=0) leads to:
(e1-e2)=(e2-e3)+ln(1+ev)
[straight line: y=mx+b; with slope, m=1]
Ramsay Diagram
Ramsay Diagram
• Small strains are near the origin
• Equal increments of progressive strain (i.e., strain path) plot
along straight lines
• Unequal increments plot as curved plots
• If v=ev is the volumetric strain, then:
• 1+v =(1+e1)(1+e2)(1+e3)  = lnS=ln(1+e)
• It is easier to examine v on this plot
Take log from both sides and substitute  for ln(1+e)
– ln(v +1)= 1+ 2+ 3
• If v>0, the lines intersect the ordinate
• If v<0, the lines intersect the abscissa
Measurement of Strain
• Originally circular objects
• When markers are available that are
assumed to have been perfectly circular and
to have deformed homogeneously, the
measurement of a single marker defines the
strain ellipse
Direct Measurement of Stretches
• Sometimes objects give us the opportunity to
directly measure extension
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•
•
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Examples:
Boudinaged burrow
Boudinaged tourmaline
Boudinaged belemnites
Under these circumstances, we can fit an ellipse
graphically through lines, or we can analytically
find the strain tensor from three stretches
Direct Measurement of Shear Strain
• Bilaterally symmetrical fossils are an
example of a marker that readily gives shear
strain
• Since shear strain is zero along strain axes,
inspection of enough distorted fossils (e.g.
brachiopods, trilobites) can allow us to find
the direction
Wellman's Method
• Relies on a theorem in geometry that says that if
two chords together cover 180° of a circle, the angle
between them is 90°
• In Wellmans method, we draw an arbitrary
diameter of the strain ellipse
• Then we take pairs of lines that were originally at
90° and draw them through the two ends of the
diameter
• The pairs of lines intersect on the edge of the strain
ellipse
Fry’s Method
• Depends on objects that originally were
clustered with a relatively uniform interobject distance.
– After deformation the distribution is non-uniform
• Extension increases the distance between
objects; shortening reduces the distance
– Maximum and minimum distances will be along
S1 and S2, respectively
From:
http://seismo.berkeley.edu/~burgmann/EPS116/labs/lab8_strain/lab8_2009.pdf
Fry’s Method; how to
• Put a tracing paper on top of the objects, and mark
their centers with a dot (this is the centers sheet)
• On a second tracing paper, choose an arbitrary
reference point (this is the reference sheet)
• Place the reference point on top of the one dot
(grain), and mark all other dots from the centers
sheet onto the reference sheet
• Place the reference point on a second dot, and copy
all other dots
• Repeat this for all dots
…
• This lead to many dots, and the strain ellipse is
defined either by:
• an empty elliptical space around the reference point, or
• an elliptical area full of points
• Trace the approximation of the strain ellipse
Pros and cons
• Fry’s Method is fast and easy, and can be used on
rocks that have pressure solution along grain
boundaries, with some original material lost
– Rocks can be sandstone, oolitic limestone, and
conglomerate
• The method requires marking many points (>25)
• The estimation of the strain ellipse’s eccentricity is
subjective and inaccurate
• If grains had an original preferred orientation, the
method cannot be used
Rf/ Method
• In many cases originally, roughly circular
markers have variations in shape that are
random
• In this case the final shape Rf of any one marker
is a function of the original shape Ro and the
strain ratio Rs
Rf,max = Rs.Ro
Rf,min = Ro/Rs
Rf/ Method
http://a1-structural-geology-software.com/The_rf_phi__prog_page.html
http://a1-structural-geology-software.com/The_rf_phi__prog_page.html