Strain II

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Strain II
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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!)

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Equations for Mohr circle for strain

• Lets 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?

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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/? (1e)2, i.e.,
• ? (1e)2 and ?1/? are both

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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

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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 -

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Deformed Brachiopod
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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 dont change length during deformation,
  i.e.,
• elnfe0, and ?lnfe1, and therefore ?lnfe1
• Numerical Solution tan2 ? (1-?1)/(?2-1)

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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!

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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)!

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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!

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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

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Lines of Maximum Shear (lms)

• .

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Another Example
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ExampleA unit sphere is shortened by 50 and
extended by 100

• e11, and e3-0.5
• s1Xl/lo 1e12 s3Zl/lo 1e30.5
• ?1 (1e1)2s12 4 ? 3 (1e3)2 s320.25
• ? 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 (0.254)/22.125
• r (?3 - ?1)/2 (4-0.25)/21.90
• Having c and r, we can plot the circle!

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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
• All other strain ellipses fall into one of three
  fields
• Above the ?21 line where both principal
  extensions are
• To the left of the ?11 where both principal
  extensions are
• Between two fields where one is () and the other
  (-)

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Shapes of the Strain Ellipse
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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
  
• 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

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Flinn Diagram
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Volume change on Flinn Diagram

• Recall S1e l'/lo and ev ?v/vo ?v
  (v-vo)/vo
• An original cube of sides 1 (i.e., lo1), gives
  vo1
• Since Sl'/lo, and lo1, then Sl
• The deformed volume is therefore v'l'.l'.l
• Orienting the cube along the principal axes
• VS1.S2.S3 (1e1)(1e2)(1e3)Since ?v
  (v-vo), for vo1 we get
• ?v (1e1)(1e2)(1e3)-1
• Given vo1, since ev ?v/vo, ev ?v
  (1e1)(1e2)(1e3)-1
• 1ev (1e1)(1e2)(1e3)
• If volumetric strain, ?v ev 0, then
• (1e1)(1e2)(1e3) 1 i.e., XYZ1
• Express 1ev (1e1)(1e2)(1e3) in e take log
• ln(1ev) e1e2e3
• Rearrange (e1-e2)(e2-e3)-3e2ln(1ev)
• Plane strain (e20) leads to
• (e1-e2)(e2-e3)ln(1ev)straight line ymxb
  m1

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Ramsay Diagram
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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 ?vev is the volumetric strain, then
• 1?v (1e1)(1e2)(1e3) ? lnSln(1e)
• It is easier to examine ?v on this plot Take log
  from both sides and substitute ? for ln(1e)
• ln(?v 1) ?1 ?2 ?3
• If ?v0, the lines intersect the ordinate
• If ?v

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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

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Direct Measurement of Stretches

• Sometimes objects give us the opportunity to
  directly measure extension
• 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

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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

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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

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Frys Method

• Depends on objects that originally were clustered
  with a relatively uniform inter-object distance
• We repeatedly trace the pattern of objects,
  shifting the pattern so a different object is
  over the origin
• Shape of the strain ellipse shows up in an
  anticluster at the center

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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
• The Fry method is a graphical technique for
  determining the strain ratio Rs(Rf,max/Rf,min)1
  /2
• Assume an initial uniform point distribution
  (isotopic)
• After deformation, the point distribution is
  non-uniform
• Where extension has occurred, distances between
  points increase where contraction has occurred,
  distances decrease
• The maximum distance between points occurs to
  principal stretch direction, S1, while minimum
  distance occurs to S2

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Rf/? Method - Procedure

• Mark the center of each object on a tracing paper
  overlay (Centers Sheet)
• Copy the dots onto a second overlay and choose a
  central reference point ("Reference Sheet"). You
  should now have two identical pieces of tracing
  paper with a bunch of dots
• Place the Reference Sheet on top of the Centers
  sheet
• Line up the reference point with another point on
  the Centers Sheet
• Trace all the dots from the Centers Sheet onto
  the Reference Sheet. They should show up in
  different locations because youve moved the
  Reference Sheet
• Repeat the process with the Reference point
  lining up with every other point. Your final
  product will have a lot of points ( n2-n points)
  
• If all goes as planned you should clearly see the
  strain ellipse around the reference point, which
  shows up as either an elliptical area devoid of
  points or an elliptical area of concentrated
  points
• Draw in your interpretation of the strain ellipse
  size and orientation

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How good is it?

• Strengths
• Quick and easy
• Can be used on rocks that have pressure solution
  along grain boundaries, i.e. rocks in which some
  original material may have been lost
• Applies to sand grains in sandstone, ooids in
  limestone, and pebbles in a conglomerate
• Weaknesses
• Requires lots of points (at least 25 for more
  precise answers)
• Estimates of ellipticity can be extremely
  subjective and, hence, inaccurate
• Cannot be used if particles being analyzed had
  some preferred axial direction prior to
  deformation