Presentation on theme: "Fossen – Chapter 3 How strain is measured and quantified in ductile regime."— Presentation transcript:
Fossen – Chapter 3 How strain is measured and quantified in ductile regime
Strain analysis Allows exploring the state of strain in rocks Find the variation of strain at different scales (microscopic, outcrop, and region) Is used to map shear zones, and estimate the displacement Helps us to find the folding mechanisms in folded rocks The shape of the strain ellipse (e.g., oblate -> flattening; prolate -> constriction) provides information on how deformation occurred. – In slate with reduction spots, for example, it tells us about shortening The orientation of the strain ellipse can tell us if strain was simple shear or pure shear; used for kinematics Strain markers in sedimentary rocks can let us reconstruct original sedimentary thickness.
Strain markers Strain markers reveal the state of strain in a deformed rock. Markers are used to find 1D and 2D strain (later extended into 3D by combining three 2D sections) In 1D, strain analysis measures change in length – This leads to finding the amount and direction of shortening or stretching. For example, from: – Boudinaged dike, elongate minerals, linear fossils
Deformed pebbles in a conglomerate (constriction field), seen in plane in an outcrop. The length of these pebbles my parallel the maximum principal stretch axis (X)
Deformed Bygdin Conglomerate, with quartzite pebbles and quartzite matrix, Norway. Similar pebble and matrix compositions minimize strain partitioning and enhance strain estimates
Part of a stretched belemnite boudins with quartz and calcite infill. The space between the broken pieces of the belemnite are filled with precipitated material (fibers grown parallel to 1 ). The more translucent material in the middle of the gaps is quartz, the material closer to the pieces is calcite. Photo from the root zone of the Morcles nappe in the Rhone valley, Switzerland by Martin Casey 1 quartz calcite
Elongated belemnites in Jurassic limestone in the Swiss Alps. The upper one has enjoyed sinistral shear compared to the lower one which has stretched l’ l o4 l o3
Stretched belemnite. Stretching in the upper right, lower left direction has broken and extended the fossil. The gaps between the pieces are filled with a precipitate. Photo from the root zone of the Morcles nappe, Rhone valley, Switzerland by Martin Casey gallery/belpart.html
Strain in 2D We use strain markers of known initial shape or linear markers oriented variably in a rock 2D strain is done, for example, in sections through conglomerate, oolites, vesicles, pillow lava. – These sections can be oriented thin sectionss, or surfaces of a joint plane in the field.
Block diagrams showing 2D sections through the strain ellipsoid. Flinn diagram represent the shape of the strain ellipse Direction of instantaneous stretching axes (ISA) and fields of instantaneous contraction (black) and extension (white) for dextral (right-lateral) simple shear
Map of the conglomerate layer and variation of strain over a large area.
Change in angle Can be done if the original angular relation between two lines is known. – The change in the angles can be used to find shear strain ( ) from, for example: dikes, foliation, and bedding, which may be seen in neighboring undeformed and deformed zones (e.g., around and in a shear zone). two originally orthogonal lines of symmetry in fossils, such as trilobite, brachiopods, worm burrows (relative to the bedding). – The deviation from orthogonal relation ( ) gives the shear strain.
Deformed Precambrian pillow lava, Superior Province, Ontario, Canada. Can use Rf/ , center-to-center, or Fry method techniques.
Deformed pillow lava.Swamp R., Ontario, Canada. lbot/cdgeol/Structure/Strain/Strai n_in_igrox.html
Measurement of Strain The simplest case: – Originally circular objects Ooids, reduction spots 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
Elliptical reduction spots in a slate from North Wales. The spots were originally round in section and are deformed to ellipses. (photo: Rob Knipe) gallery/belpart.html
Reduction spots in slate. The green spots are reduced from Ferric iron (Fe3+) (Fe 2+ ) by fluids (turned from red to green) and then Fe2+ was leached out. Spots are mostly spherical before deformation. After deformation they are elliptical, and give the strain ellipse.
Direct Measurement of Stretches Sometimes objects give us the opportunity to directly measure extension Examples: Boudinaged burrow, tourmaline, 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 the principal strain axes, inspection of enough distorted fossils (e.g. brachiopods, trilobites) can allow us to find the principal directions! The Wellman and Breddin methods are commonly used!
Undeformed brachiopods originally oriented randomly
Wellman Method Uses deformed, variably oriented pair of lines which were originally perpendicular (e.g., hinge and median lines of brachiopods, trilobites) Procedure: Trace the deformed lines on the image (photo) with a pencil (image is real world) Draw a box around the objects Draw a reference line between two arbitrary points (A and B), preferably parallel to the long edge of the box Put A at the intersection of the two originally perpendicular lines on a fossil, and draw the two lines (e.g., hinge and median lines) While line AB is un-rotated (kept parallel to the box), bring B where A was, and repeat the drawing of the two lines Place a dot ( ) where the pairs of deformed lines, going through A and B, cross Do this for all fossils, while AB is in the same constant orientation For each fossil, the pairs of lines intersect on the edge of the strain ellipse Draw a smooth ellipse through the dots. This is the strain ellipse; measure its long and short semi-axis. Find the strain ratio, R s = (long semi-axis)/(short semi-axis), and the orientation of S 1 relative to line AB
Wellman method used for deformed trilobites and brachiopods with two originally perpendicular lines
Breddin Method Requires presence of many fossils Draw a reference line on the image (photo) of the fossils Measure the angle ( ’) between the hingeline of the fossil w.r.t the reference line (e.g., trace of foliation) Measure the angular shear ( ’) for all fossils (e.g., the angle between the deformed hinge and median lines) Repeat these for all fossils (see next slide) Plot ’ against ’ Compare the plot to an overlay of a transparent standard Breddin Graph centered at ’=0 that shows the R s contours The fossils with the ’=0 give the orientation of the S 1 axis See next slide
Data from two slides before (traced below), plotted on the Breddin graph. Data plot on the curve for R s =2.5, where R s is the strain ellipse
Straight lines are drawn between neighboring grain centers. The line lengths (d’) are plotted vs. the angle ( ) that the lines make with the reference line. Note: reference line may be the trace of foliation! The ratio of the max (X) and min (Y), give the R s = X/Y The center-to-center method
Fry Method Depends on objects that originally were clustered with a relatively uniform inter-object distance. – After deformation the distribution is non-uniform Extension increases the distance between objects Shortening reduces the distance – The maximum and minimum distances will be along S 1 and S 2, respectively
Undeformed medium grained oolitic grainstone dominated by ooliths (70 vol% of allochems), bioclasts (30%) and minor peloids and intraclasts. https://wwwf.imperial.ac.uk/earthscienceandengineering/rocklibrary/viewrecord.php?SampleNo=sp5
Undeformed and deformed oolitic limestone
Fry Method Is a variant of the center-to-center method – Could be used for ooids that may dissolve, and phenocrysts in igneous and metamorphic rocks. Measures the closeness of grains Measurement: On a transparent overlay put a dot ( ) at the center of each grain; number the grains (1, 2, 3,.,., through n, whatever number is) Draw an arbitrary reference line and/or a box around the image Have a transparent overlay, and mark a plus sign (+) at its center Put the overly on the image and trace the reference line on the overlay Put the + sign on 1 (center of grain 1), keep reference lines parallel, and mark all the other points on the overly with dots Put the + sign on 2 (center of grain 2), keep reference lines parallel, and mark all the other points on the overly with dots Repeat for all grains The final product is an empty ellipse, or an elliptical area full of points, which approximates the strain ellipse. Measure the major semi-axes: S 1 and S 3 Determine the strain ratio R s = S 1 /S 3 and the orientations of S 1 and S 3
Grain centers are transferred to an overlay. A central point ( ) on the overlay is defined and moved on the center of grain 1, while copying the other points and overlay’s orientation is kept constant (sides of the boxes remain parallel) An empty ellipse develops which gives the strain ellipse. Fry Method
Center to Center Method Ramsay, J. G., and Huber, M. I., 1983 Modern Structural Geology. Volume 1: Strain Analysis Undeformed Deformed
Pros: 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 Cons: 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, this method cannot be used Fry Method
Moderately deformed Neoproterozoic quartz conglomerate. Strain exposed in sections parallel to the principal planes Newport, Rhode Island, Purgatory conglomerate 08/20/purgatory-conglomerate/
R f / ’ Method In many cases originally, roughly circular markers have variations in shape that are random, – e.g., grains in sandstone or conglomerate In this case the final ratio R f of any one grain is a function of the initial grain ratio R i and the strain ratio R s The final ratio depends on the relative orientation of the long axis of the strain ellipse and that of the grain’s long axis R fmax = R s.R i R fmin = R i /R s
R f / ’ Method Could be used for grains with initial spherical or non- spherical shapes (i.e., initial grain ratio of R i =1 or R i >1) Procedure: Measure the long and short axes of each grain on the deformed rock, or on its image Find the final ratio (R f ) for each grain Find the angle ( ’) between the long axis of each grain and a reference line (e.g., trace of foliation or bedding) Plot the log of R f against ’ Note the pattern (e.g., drop- or onion-shaped) Fit a theoretical curve on a transparent overlay to the distribution. Read the R S and R i.
For pure shear: S 3 = 1/S 1 or 3 = 1/ 1 Apply a pure shear with R s = 1 / 3 = 1.5 (i.e., 1 = 1.5 and 3 = 1/1.5) | S 1 0 | | 1.5 0| | 0 S 3 | or |0 1/ 1.5| Note: Grain #7 has the max R f ; grain #1 has the min R f Or apply a pure shear with R s = 1 / 3 = 3 (i.e., 1 = 3 and 3 = 1/3) Notice the coaxial strain (see strain ellipses ’ is around 0). Rf/ ’ Method Case: Grains had constant R i = La/Sa The plot on the right shows R i =2. R s = 1 / 3 = S 1 /S 3 Undeformed R i > R s R s > R i
Digital restoration of single deformed trilobite Application of digital method on a single fossil. A: Image of deformed trilobite Angelina showing non-orthogonal relationship between hinge line h and median line m. Image has been rotated to make stretching lineation L (long arrow) vertical. White circle is reference circle. Solid square dots 1–8 are dragging handles. B: Shape restoration by pulling handle 4 to right until h and m become orthogonal. R is strain ratio. C: Size restoration
Digital restoration of single deformed brachiopod A: Initial stage. Undeformed brachiopod fossil and reference circle enclosed in square of side l 0. X and Z are directions of maximum and minimum stretching, h is hinge line, m is median line. B: Homogeneous deformation of A results in change in perpendicularity between h and m and transformation of reference circle into strain ellipse (gray) of axial ratio l 2 /l 1. Another reference circle (white) is drawn at this stage. C: Retrodeformation: rectangle in B transforms into square of side l 2. Reference circle and strain ellipse in B transform into reciprocal strain ellipse (white) and circle (gray), respectively.
Mohr Circle – Two deformed brachiopods This method is good when there are only few fossils available Step 1. Measure the angle between the hinge lines of the two brachiopods ( ’). Note: this angle is doubled (2 ’) in the Mohr circle! – (Please read the powerpoint slide on strain Mohr circle!) – Measure the angular shear ( A and B ) for each fossil (see next slide!) Step 2. Plot a circle on a tracing paper of any size with center c. – Draw two radii (A and B) from the center of the circle, with an angle of 2 ’ – Draw (on a graph paper) the Cartesian coordinates of the Mohr Circle ( ’ vs. ’) with an arbitrary scale Step 3. Draw (on the same graph paper) two lines from the origin inclined at the angles to the horizontal axis. Watch for cw and ccw senses! Step 4. Overlay the tracing paper on the graph paper, and put the center (c) of the circle on the x-axis. Rotate the tracing circle, keeping the center on the x- axis, until each of the lines on the graph paper intersects its corresponding radius that emanates from the origin.
Note: A is CCW (+) and B is CW (-) (see next slide) The senses of are the same in the real world and the Mohr circle space! Tracing paper Graph paper Tracing paper overlaid on graph paper photograph BB AA ccw cw c c B A 22 22
Real world Mohr world r CW CW + + O c ++
Example: Three deformed brachiopods see next slide for pictures! Measure the angle between fossils A and B ( ’), and B and C ( ’) Measure the angular shear for each fossil ( A, B, C ) Set up the coordinate system ( ’ vs. ’) with arbitrary scale Draw three lines of any length at A, B, C from the origin Draw a circle of any size on a tracing paper Draw angles 2 ’ (between A & B) and 2 ’ (between B & C) from the center of the circle. Mark points A, B, & C on the circle Move the center of the circle (tracing paper) along the x-axis, and rotate it until lines A, B, C intersect their corresponding points A, B, and C on the circle. Fix the tracing paper with tape. Read the values for and ’ 1 and ’ 3, and S 1 and S 3 (scale does not matter since we want to get R s = S 1 /S 3 Read the amount and sense of the angles 2 ’ A, 2 ’ B,or 2 ’ C Draw 1 from say fossil A on the rock, in the same sense (e.g., cw or ccw) as it is for the 2 in the Mohr circle
AA BB CC A B C 2’2’ 2’2’ ’’ ’ 3 ’ 1 ’ ’’ ’’ AA BB CC cw The hinge and median lines of three brachiopods are traced on a photo The angle between the hinges of neighboring fossils are indicated by ’ and ’, and the angular shears are given by A, B, and A (all are cw) The angles ’ and ’are doubled (2 ’ between A and B, and 2 ’ between B and C), and plotted as two radii (rosette), while c is kept on the x-axis. The three angles are all cw, and plotted on the graph paper. c The dashed circle and the rosette are on the tracing paper
Three 2D section provide data for the 3D strain
Strain obtained from deformed conglomerate plotted on Flinn diagram (Norway)
R f / ’ cont’d If R s < R i (strain ellipticity is < the initial grain ellipticity) R s = (R f max /R f min ) R i max = (R f max R f min ) If R s > R i (strain ellipticity is > the initial grain ellipticity) R s = (R f max R f min ) R i max = (R f max /R f min ) The direction of the maximum is the orientation of S 1