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. counter-clockwise, 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!

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)

מעוות ניתן להגדרה באמצעות מדידת שינויים באורך קוים ובאוריינטציה שלהם ב: 1. אלמנטים קויים שעברו דפורמציה, 2. צירים גיאומטריים שמוגדרים בתוך אלמנטים אליפטיים שהיו קודם לכן מעגליים.

שינויים באורך קוים Extension (e): שינוי באורך קו יחסית לאורכו המקורי e = (lf - lo)/lo ההתארכות באחוזים = e x 100 +e values = lengthening, lf > lo (מהי ההתארכות המקסימלית האפשרית?) -e values = shortening, lf < lo (מהי ההתקצרות המירבית האפשרית?) Stretch (S): מתיחה – האורך הסופי של קו באורך יחידה 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

Stretch = S Extension=e The Strain Ellipse Describing changes in the length of lines Definition: Stretch = S Extension=e 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

Can this be applied to cross-sections of faulted terranes? 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 e & S? pab = pr2 a = 2.6/2 = 1.3

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 g = tan (45°) = 1 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 AB ו- CD תחילה מתקצרים ואחר כך מתארכים. הקוים CB ו- AD רק מתארכים. הקו BD לא 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 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

Another important parameter is the ratio: g/l 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 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).