GG303 Lab 9 5/8/2023 1

STRAIN

I Main TopicsA General deformationB Homogeneous 2-D strain and the strain ellipseC Homogeneous 3-D strain and the strain ellipsoid (Flinn Diagrams)D Comments on measuring strain

II General deformation (relates changes of position of points in a body)A Rigid body translationB Rigid body rotationC Change in shape (distortional strain)

1 Change in linear dimension

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a Extension

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ε =ΔLLo

=L1 − LoLo

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dimensionless!

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b Stretch

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S =L1Lo

=L1−LoLo

+LoLo

=ε +1Stephen Martel Lab9-6 University of Hawaii

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2 Change in right angles: = tan (engineering shear strain)

Shear strain is positive if the strain decreases the original right angle between two positive axes

D Change in volume (dilational strain)

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a Dilation

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Δ =ΔVVo

=V1 − VoVo

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dimensionless!

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III Homogeneous 2-D strain (plane strain) and the strain ellipseA Equations of homogeneous 2-D strain

1 Lagrangian (final positions in terms of initial positions)a x’ = ax + by (This can be solved for x and for y)b y’ = cx + dy (This can be solved for x and for y)

2 Eulerian (initial positions in terms of final positions)a x = Ax’ + By’

i y = (x’ – ax)/b and y = (y’ – cx)/dii (x’ – ax)/b = (y’ – cx)/diii d(x’ – ax) = b(y’ – cx)iv cbx – adx = by’-dx’v x = (by’-dx’)/(cb-ad) This yields x = x(x’,y’)

b y = Cx’ + Dy’i x = (y’ – dy)/c and x = (x’ – by)/aii (y’ – dy)/c = (x’ – by)/aiii a(y’ – dy) = c(x’ – by)iv cby – ady = cx’-ay’v y = (cx’-ay’)/(cb-ad) This yields y = y(x’,y’)

3 Homogeneous deformation of a unit circlea x2 + y2 = 1 or x2 + y2 – 1 = 0

b

€

by'-dx'cb - ad ⎡ ⎣

⎤ ⎦

2 +

cx' -ay'cb - ad ⎡ ⎣

⎤ ⎦

2- 1 = 0

c

€

x'2d2 + c 2

cb - ad ⎡

⎣ ⎢ ⎤

⎦ ⎥

2

– x'y'2bd + 2ac

cb - ad ⎡ ⎣

⎤ ⎦

2 + y'2

a2 + b2

cb - ad ⎡

⎣ ⎢ ⎤

⎦ ⎥

2

– 1 = 0

d Equation in x,y for an ellipse oblique to the x,y axesC1 x2 + C2 xy + C3 y2 + C4 x + C5 y + C6 = 0 (C1 C3 > 0)

e A unit circle that is homogeneously deformed transforms to an ellipse. This ellipse is called the strain ellipse.

B The strain ellipse1 Strains may be inhomogeneous over a large region but

approximately homogeneous locally

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2 In a general sense, the strain ellipse characterizes 2-D strain at a position in space and a point in time; it can vary with x,y,x,t.

3 Characterization of the strain ellipsea An ellipse has a major semi-axis (a) and a minor semi-axis (b).

An ellipse can be characterized by the (relative) length and orientation of these axes

b Ellipses with perpendicular axes of equal length (i.e., circles) deform homogeneously to become ellipses with perpendicular axes of unequal length

c The set of axes of a circle that have the same orientation before and after the deformation are known as the principal axes for strain. They allow the strain to be described in the most simple form. These are the axes of the strain ellipse.

Direction ofmaximumextension

Direction ofmaximumcontraction

Undeformed Deformedd Coaxial deformation (e.g., pure shear strain)

Principal axes of strain maintain their orientation during an interval of deformation

e Noncoaxial deformation (e.g., simple shear strain)Principal axes of strain do not maintain their orientation during an interval of deformation

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f If the initial radius r of a circular marker is known, then

€

ε1 =a− rr

, ε2 =b− rr

, (extensions or longitudinal strains)

where a and b are the semi-major and semi-minor axes of the ellipse, respectively.

g If the initial radius r of a circular marker is unknown, a stretch ratio independent of r can be found

€

S1S2

=a /rb / r

=ab S1/S2 = stretch ratio

C Wellman’s MethodUses deformed pairs of originally perpendicular lines to determine the strain ellipse. The pairs have different orientations.

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IV Three-dimensional strain and the strain ellipsoidA The 3-D counterpart of the strain ellipsoid (i.e., a unit sphere deforms

into an ellipsoid).B If the initial radius r of a spherical marker is known, then

1

€

ε1 =a− rr

, ε2 =b− rr

, ε3 =c − rr (longitudinal strains)

2

€

S1 =ar

=ε1 +1, S2 =br

=ε2 +1, S3 =cr

= ε3 +1 (stretches)

where a, b, and c are the semi-major, intermediate, and semi-minor axes of the ellipsoid, respectively.

C If the initial radius r of a spherical marker is not known, then stretch ratios independent of r can again be found

€

S1S2

=a /rb / r

=ab

, S2S3

=b / rc / r

=bc

D In a Flinn diagram, S1/S2 is plotted vs. S2/S3

V Comments on measuring strainA Initial geometries and final geometries must both be known.B Commonly the initial geometry is assumed or guessed.C Deformations are commonly not homogeneous

Lab 9 Strain

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Exercise1 Vector fields and lines strained in simple shear (21 pts)A Print a copy of the Matlab function strain1 and read it. (1 point)B Use Matlab function strain1 to produce plots of squares deformed under

simple shear for the following deformation gradients. Type “help strain1” to see how to run this function.F1 = [1 1;0 1] (1 point)F2 = [1 2;0 1] (1 point)

C Assuming that the y-axis is north find the following quantities (show your calculations below or on a separate page) (1 point/box)

Deformation 1 Deformation 2Extension of ABExtension of BCExtension of ACExtension of BDShear strain of ABCShear strain of BADTrend of the axis of greatest extensionTrend of the axis of least extensionDilation of the square

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Exercise 2 Strain field near a hole in a uniaxial tensile stress field 86 pts

PurposeTo produce and examine a two dimensional strain field, where strain varies from point to point, and to set up some questions for the field trip.

To speed things up, I have done steps 1-5 in the experiment already (see attached figures). You will need to complete steps 6-10 to quantify the experimental results and perform the analysis.

Experimental Procedure 1 Lay out a square grid with a 20.0 mm grid spacing on a piece of fabric.

At the grid nodes, draw circles with a diamater of 11.0 mm (I tried my best here, but one of the circles , at x = 60 mm, y = 40 mm has a diameter of about 11.8 mm).

2 Cut a circular hole at the center of the grid with a radius of 32.5 mm.3 Stretch the fabric such that x- and y-axes stay straight.4 Trace the deformed grid lines and deformed circles (i.e., strain ellipses)

onto a transparency, which is available for inspection.5 Scan the transparency, and electronically trace the strain ellipses and

grid lines. Using the attached copies of the electronic scans:

6 Measure the coordinates of the centers of the deformed circles [xp (x’) and yp (y’)] and record them in the tables on the next page.

7 Measure the lengths of the major axes (a or ) and minor axes (b = ) of the strain ellipses and record them in the tables on the next page.

8 Measure the trend in degrees of the minor axes (t or ) of the strain ellipses and record them in the tables on the next page.

Analysis9 Open up MS Word and Matlab.10 Edit the Matlab script “lycra.m” by correctly completing the lines for

calculating e1 and e2, the principal extensions. 2 pts11 Copy the tabulated values into the Word document “lycra_table”12 Then sweep the cursor over the table for “x”, copy it, and paste it into

the Matlab command window. Hit “Enter”. This should transfer all the values in the table for “x” to Matlab.

13 Repeat the procedure for the values in the other seven tables.14 Run the Matlab script [ux,uy,uxi,uyi,e1,e2,e1i,e2i] =

lycra(x,y,d,xp,yp,a,b,t) to get four plots, which are described by their titles.

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xx = [0.0 20.0 40.0 60.0 80.0;0.0 20.0 40.0 60.0 80.0;0.0 20.0 40.0 60.0 80.0;0.0 20.0 40.0 60.0 80.0]

yy = [0.0 0.0 0.0 0.0 0.0;20.0 20.0 20.0 20.0 20.0;40.0 40.0 40.0 40.0 40.0;60.0 60.0 60.0 60.0 60.0]

dd = [11.0 11.0 11.0 11.0 11.0;11.0 11.0 11.0 11.0 11.0;11.0 11.0 11.0 11.8 11.0;11.0 11.0 11.0 11.0 11.0]

xp (1/2 pt/box)xp = [0.0 20.0 ;0.0 20.0 ;

;]

Yp (1/2 pt/box)yp = [0 20.0 ;20.0 40.0 ;

;]

()(1/2 pt/ox)a = [0.0 0.0 ;0.0 0.0 ;

;]

()(1/2 pt/ox)b = [0.0 0.0 ;0.0 0.0 ;

;]

t ()(1/2 pt/ox)t = [0 0 ;0 0 ;

;]

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Discussion (4 pts/question)Provide clear and complete typewritten answers for the following questions:

1 Do straight parallel lines before the deformation remain straight and parallel after the deformation? Explain.

2 Is this homogeneous or inhomogeneous deformation? Why?

3 Suppose the hole were not present. What do you think representative values for , , and would be? Why?

4 Suppose the hole were not present. Do you think straight parallel lines before the deformation would remain straight and parallel after the deformation? Why?

5 Do the grid lines at the grid nodes have the same local orientation as the axes of strain ellipses centered at the nodes? The answer to this question is not a simple yes or no; explain your answer.

6 Does orientation of the grid lines at a particular point reflect the deformation over the whole length of the line, or just the deformation at that point? Explain.

7 Does orientation of the strain ellipse at a particular point reflect the deformation over the whole sheet, or just at that particular point? Explain.

8 Does the displacement at a point determine the orientation of the strain ellipse at that point? If not, what does? Explain.

9 Where does the fabric buckle or pucker?

10 Why does the fabric buckle where it does? Cite at least one of the plots in your answer.

11 If the fabric instead were rock, then fractures might open up where e1 is positive, and their orientation would be given by . Describe clearly and completely where you think the longest and/or widest fractures might open, and what the fracture orientation would be, if fractures formed near a pit crater after the pit crater formed if the pit crater were stretched. Reference your figures as necessary.

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Exercise 4 Flinn diagram problem (Modified from problem 14.8 of Rowland and Duebenforder, 1994). (34 pts total)This exercise shows how geologic data can be used to estimate strain. Consider the attached diagram of an oolitic limestone showing cuts made perpendicular to two principal strain axes. A Measure the dimensions of six ooids in each view (4x6 = 24 pts) B Find a mean semi-major and a mean semi-minor axis for each view (4x1

= 4 pts). C Use the answers from (B) to find ratios for 1+ε2:1+ε3 (1 pt) and 1+ε1:1+ε3

(1 pt). D Use the answers from (C) to find the 1+ε1:1+ε2:1+ε3 ratio for the strain

ellipsoid, assuming the ooids were deformed homogenously (3 pts). E Plot the point on the Flinn diagram that represents this strain ellipsoid (1

pt).

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Exercise 5 Heterogeneous strain around a pressurized hole (30 pts)(Note that the last question, question L, is optional)

Assume that the radial (i.e., outward) displacement (ur) from the center of a pressurized circular cylindrical hole is given by the following equation:

€

ur = u0ar

where (1)

€

u0 is the radial (outward) displacement away from the center of the hole that occurs at the perimeter of the hole.(2) a is the original radius of the hole.(3) r is the original distance of a point from the center of the hole (i.e., the distance before deformation).

€

ur = r'−r where r’ is the radial distance after deformation

We want to see how the square deforms if the hole is inflated such that its radius doubles (i.e., u0 = a), given the displacement relationship above.

A In the center of a page, draw a circle with a radius of 2 cm (1 pt) and a square 2 cm on a side (1 pt) that is tangent to circle, and label the points on the square as shown below (1 pt):

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B Fill in the table below assuming that u0 = a.Point r (cm)

(1 pt/box)ur (cm)(1 pt/box)

r’ (cm)(1 pt/box)

ABCDEFGHC Draw the hole showing it after it has been inflated. (3 pts)D Draw the deformed “square” A’B’C’E’H’G’F’D’A’. (8 pts)E Is the set of parallel lines ABC and FGH parallel to A’B’C’ and F’G’H’?

(1 pt)F Is the set of parallel lines ADF and CEH parallel to A’D’F’ and C’E’H’?

(2 pts)G What is the shear strain for the pair of lines DA and AB? To answer this

you need to know angles DAB and D’A’B’. (2 pts)H What is the shear strain for the pair of lines BC and CE? To answer this

you need to know angles BCE and B’C’E’.(2 pts)

I Draw a circle with a 1cm radius inside the square ACHF and draw its deformed counterpart as best you can in figure A’B’C’E’H’G’F’D. (4 pts)

J Is the deformed counterpart an ellipse? (1 pt)K Is the deformation homogenous or inhomogenous? (1 pt)L (Bonus* - not required) Determine F and Ju for this deformation. Show

all your work. (20 pts)

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