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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!)**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 2angle 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 ‘-’**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 12**ExampleA 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= 40.25 =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 • 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 (-)**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**Volume change on Flinn Diagram**• Recall: S=1+e = l'/lo andev = 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, thenev = 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**• 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=evis thevolumetric 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 forln(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 • 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 inter-object 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 Rfof any one marker is a function of the original shape Ro and the strain ratio Rs Rf,max = Rs.Ro Rf,min = Ro/Rs**http://a1-structural-geology-software.com/The_rf_phi__prog_page.html**http://a1-structural-geology-software.com/The_rf_phi__prog_page.html**http://a1-structural-geology-software.com/The_rf_phi__prog_page.html**http://a1-structural-geology-software.com/The_rf_phi__prog_page.html