Transcript From Sediment to Rock: Rocks that form near the Earth’s

‫אנליזה קינמטית של דפורמציה‬
‫שמוליק מרקו‬
‫אנליזה קינמטית‬
Reconstruction of movements that occurred during
formation and deformation of rocks.
• Rigid vs. non-Rigid body deformation
• The relative arrangement of points in a body…
• Maintained-> rigid body deformation
– Translation
– Rotation
• Not maintained -> non-rigid body deformation:
change in shape and/or size of original object
– Dilation
– Distortion
‫דפורמציה של גוף קשיח (=צפיד) לעומת גוף לא קשיח‬
Rigid Body Movements:
Translation and Rotation
• All points in a body move along parallel
paths, e.g., sliding book on desk
• Sliding occurs on a discontinuity, e.g.,
fault, bedding plane, desk top
Translation
• Describe translation by a displacement
vector with components of:
– distance of transport
– direction of transport (plunge and azimuth)
– sense of transport
Rotation
Rigid body rotation about an axis
Describe rotation by:
– orientation of axis of rotation (plunge and
azimuth)
– sense of rotation (clockwise vs. counterclockwise, viewed down axis of rotation)
– magnitude of rotation (measured in degrees)
• Examples
‫רוטציה‪ :‬שכבות נטויות (אגף של קמר)‬
‫‪Ramon‬‬
‫‪Wyoming‬‬
Rotation
‫רוטציה של‬
‫בלוקים קשיחים‬
‫וזרימה של‬
‫חרסיות פלסטיות‬
‫חרסית זורמת‬
‫בלחץ‬
‫רוטציה של‬
‫בלוקים בגליל‬
‫‪Ron 1984‬‬
Non-Rigid Body deformation
Dilation
• distance between internal points of reference
increases or decreases but shape remains
uniform
Distortion
• non-uniform changes in distance between points
within a body results in a change in shape
Dilation and/or Distortion =
Strain
Homogeneous Vs. Heterogeneous deformation:
Strain = ‫מעוות‬
Strain results from non-rigid body
deformation, which is
• Change in size - positive or negative dilation.
and/or
• Change in shape - distortion.
Dilation and distortion will result in changes
in line length and angles between points.
Dilation due to shearing
Non-Rigid Body deformation
• Homogeneous deformation: strain is constant
throughout a body:
Straight lines before, are straight after
deformation.
Parallel lines before, are parallel after
deformation.
• Heterogeneous deformation: strain is variable
within a body
Typically we simplify our lives by working on structures that
exhibit homogeneous deformation.
Strain Analysis
Describe changes in shape and size of the original
body of rock using geometrical parameters
(restricted to homogeneous deformation or
parts of heterogenously deformed body that
may be treated as homogeneous deformation)
Rules for uniform strain analysis:
• Lines that were straight prior to deformation
remain straight after deformation
• Lines that were parallel before deformation
remain parallel after deformation
If these rules apply – the strain is uniform
Distortion
Pure distortion is a change in shape
without any change in area (2D) or
volume (3D).
Distortion is usually accompanied by
a change in line length and angles.
In systematic non-rigid deformation
spheres become elipsoids that
embody the full extent of the
deformation.
Circles become ellipses
In 3-D spheres
become ellipsoides
Flexural Slip Folding
At the small scale, individual layers behave rigidly,
but at the large scale the whole fold is enjoying
non-rigid deformation
Rigid- and non-rigid body deformation
commonly occur together
Movement on faults is normally considered to
be a rigid-body motion.
If the faults however are very closely spaced
(smaller than the scale of observation) then
the deformation is considered penetrative, and
therefore it is a non-rigid body deformation.
SCALE OF OBSERVATION IS KEY!
Translation
• All points in the rockmass move in parallel
paths - no motion within the body.
• At the largest scale, tectonic plates are
considered to be rigid bodies.
• At the smallest scale, individual fractured
grains slip on small discontinuities.
Again…
SCALE OF OBSERVATION IS KEY!
Translation of a
rigid plate
‫קו המשווה‬
Shear planes in meso scale (cm-m)
Shear planes in meso scale (cm-m)
Shear planes in micro scale (<1 mm)
0.1 mm
‫מעוות ניתן להגדרה באמצעות מדידת‬
‫שינויים באורך קוים ובאוריינטציה שלהם ב‪:‬‬
‫‪ .1‬אלמנטים קויים שעברו דפורמציה‪,‬‬
‫‪ .2‬צירים גיאומטריים שמוגדרים בתוך‬
‫אלמנטים אליפטיים שהיו קודם לכן‬
‫מעגליים‪.‬‬
‫שינויים באורך קוים‬
‫שינוי באורך קו יחסית לאורכו המקורי ‪• Extension (e):‬‬
‫‪• e = (lf - lo)/lo‬‬
‫‪ = e x 100‬ההתארכות באחוזים •‬
‫‪• +e values = lengthening, lf > lo‬‬
‫)מהי ההתארכות המקסימלית האפשרית?(‬
‫‪• -e values = shortening, lf < lo‬‬
‫)מהי ההתקצרות המירבית האפשרית?(‬
‫מתיחה – האורך הסופי של קו באורך יחידה ‪• Stretch (S):‬‬
‫‪• S = lf/lo = 1 + e‬‬
‫• הערך המירבי ‪ -‬אינסוף‬
‫• הערך המינימלי ‪ -‬אפס‬
Orientation Changes
• Describe changes in the relative orientations of
lines, especially lines that were originally
perpendicular
Angular shear (Y)
• Degree to which two originally perpendicular
lines are deflected from 90o
• +ve = clockwise deflection
• -ve = counter-clockwise deflection
• Range = -90o to +90o
Shear strain (g)
• Shear strain = tan (Y)
• Relates change in orientation to distance moved
by a point along a reoriented line
‫רוטציה של‬
‫בלוקים קשיחים‬
‫סביב ציר אנכי‬
‫רוטציה של‬
‫בלוקים קשיחים‬
‫סביב ציר אנכי‬
‫)‪Y=rotation (angular shear‬‬
‫‪Y‬‬
‫גזירה פשוטה‬
‫שבירה‬
‫נורמלית‬
‫שבירה‬
‫נורמלית‬
‫שבירה‬
‫נורמלית‬
‫‪DL‬‬
Finite Strain Ellipse
Graphic representation of strain in rocks
• Greatest elongation parallel to the long
axis of the ellipse (S1)
• Greatest shortening parallel to the short
axis of the ellipse (S3)
• Angular shear and shear strain zero
parallel to S1 and S3
The Strain Ellipse
Describing changes in the length of lines
Definition:
Stretch = S
Extension=e
lf
S
l0
e=(lf-l0 ) /l0=S-1
S=8/5=1.6
S`=4.8/3=1.6
e=8-5/5=0.6
e=4.8-3/3=0.6
Stretch vs. Extension
e=S-1
Belemnites (Jurassic)
Before
After
Stretched Belemnite
Can this be applied to cross-sections of faulted terranes?
Line changes when circles become ellipses
• Initially circular object will become ellipses
when homogeneously deformed. Fossil burrows
and oolites can be used as a strain gauge.
• By determining the stretch (S) and extension
(e) of the long and short axes of the ellipse we
can describe the amount of lengthening and
shortening the rock containing the burrows or
oolites experienced.
How to get at e and S ?
• Assuming no volume change:
Aellipse = Acircle
pab = pr2
Deformed burrows
What are
a = 2.6/2 = 1.3 e & S?
pab = pr2
b = 2.2/2 = 1.1
Angular Shear
Change of angles between lines
Angular shear: Y (psi) - we need to find a line (L)
that was originally perpendicular to the line in
question.
Angular shear strain describes the departure
of this line from it’s initially perpendicular
relationship with L.
The sign convention is CW = (+) ; CCW = (-),
magnitude is in degrees (°) this is also the
classic right-hand rule.
Angular Shear
Change of angles between lines
Deformed trilobites
Original state
Final state
All lines have changed length,
6 have changed orientation.
‫‪Shear Strain‬‬
‫‪Simple‬‬
‫‪Shear‬‬
‫‪Pure‬‬
‫‪Shear‬‬
‫בגזירה פשוטה כיוון אחד נשאר קבוע וכל השאר‬
‫מסתובבים יחסית אליו‪.‬‬
‫בגזירה טהורה הכיוונים של מקסימום ההתארכות‬
‫ושל מקסימום ההתקצרות קבועים; כיווני הצירים‬
‫הראשיים של אליפסת המעוות לא משתנים וכל שאר‬
‫הקוים מסתובבים יחסית אליהם‪.‬‬
‫שני סוגים של מעוות גזירה‬
‫גזירה פשוטה‬
‫גזירה טהורה‬
Y  about 15
Shear Strain = g (gamma)
Defn: shear strain g = the tangent of the angle Psi (Y ).
‫גזירה פשוטה‬
g = tan Y
Note: The area of the initial box and
final parallelogram is the same.
g = tan (45°) = 1
Every whole number
(1,2,3…) represents
shear of one shear zone
width - i.e. g = 2 means
that the shear zone has
slipped 2 shear zone
width units.
‫הקוים ‪ AB‬ו‪ CD -‬תחילה‬
‫מתקצרים ואחר כך מתארכים‪.‬‬
‫הקוים ‪ CB‬ו‪ AD -‬רק מתארכים‪.‬‬
‫הקו ‪ BD‬לא משתנה כלל‪.‬‬
Shearing
Flattening
‫‪Finite Strain Ellipse‬‬
‫מכל הקוים ששורטטו במעגל המקורי‪ ,‬קו אחד יהיה מקביל‬
‫לציר הארוך ואחד יהיה מקביל לציר הקצר של אליפסת‬
‫המעוות הסופי‪ .‬קוים שהיו מקבילים לקוים אלה במקור יחוו‬
‫את מירב ההתארכות או מירב ההתקצרות‪.‬‬
‫מהו המעוות הזויתי של הקוים הללו?‬
Principal Axes of the Finite Strain Ellipse
Lines that are parallel and perpendicular to S1
and S2 are the only ones that will have
experienced no angular shear and therefore
no shear strain.
These lines are special and are called the
Principal Axes of the finite strain ellipse or
finite stretching axes.
Finite Strain Ellipse
Lines A & B become A’’& B’’
No other lines in the
initial circle will extend
more than A, or shorten
more than B.
Lines A & B are also
directions of zero angular
shear (Y = 0). They must
have been perp. before
defm., but they were not
always that way!
Calibrating the Finite Strain Ellipse
We consider the initial circle to be of unit radius (1).
Therefore S1 = lf1 and S3 = lf3
The bottom line is that strains are relative to the size of
the initial object.
Strain of lines in a body
L = 1.0
All lines in a deformed body
can be evaluated, not just
the principal axes.
Using line L and it’s
perpendicular M, we can
determine e, S and g (tanY).
L’ = 1.11
Fundamental Strain Equations
• Quadratic Elongation: l = S2
• Reciprocal Quadratic Elongation: l’  1/l = 1/S2
Fundamental Strain Equations
Another important parameter is the ratio: g/l
• It describes the mix of angle change versus length
change.
• When g/l approaches zero then we get length
changes with little angular change.
• The ratio g/l is at a maximum for lines that make an
angle of 45° to the S1 direction. Shear strain is
maximum in this orientation.
Fundamental Strain Equations
subscripts refer to principal directions
Fundamental Strain
Equations
If we know q and l1 and l2
we can solve the fundamental
strain equations with only a
calculator.
Variation in Strain
1) Principal axes are directions
of max and min stretch (S1&S3).
2) In any body there are always 2
lines of no finite extension
(lnfe’s) were no change in length
occurs (S = 1).
3) The are 2 directions of
maximum shear (lines that were
initially at 45° to principal axes).
4) S and g increase and decrease
systematically according to
direction; specific values depend
on S1, S2 and qd.
Mohr Circle for Strain
Otto Mohr recognized that the equations for strain could be
represented graphically as a circle.
Another example
Now with
something real!
qd = 60°
q = 72°
S3 = 0.6
S2 = 1.1
Ramsey (1967) gives us an eqn. for rotation of a
line that is rotated by deformation:
tan qd = tan q (S3/S2)
Where:
q = angle between the line of interest (L) and S1 in
the undeformed state, and;
qd = angle between the line of interest (L) and S1
in the deformed state (known).