GG303 Lab 9
2/12/2016
1
STRAIN
I
Main Topics
A General deformation
B Homogeneous 2-D strain and the strain ellipse
C 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 translation
B Rigid body rotation
C Change in shape (distortional strain)
1 Change in linear dimension
L L1  Lo

Lo
Lo
a Extension

b Stretch
L
L  Lo Lo
S 1  1

  1
Lo
Lo
Lo
dimensionless!
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)
a Dilation
Stephen Martel

V V1  V o

Vo
Vo
dimensionless!
Lab9-1
University of Hawaii
GG303 Lab 9
2/12/2016
2
III Homogeneous 2-D strain (plane strain) and the strain ellipse
A 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
ii
(x’ – ax)/b = (y’ – cx)/d
iii
d(x’ – ax) = b(y’ – cx)
iv
cbx – adx = by’-dx’
v
x = (by’-dx’)/(cb-ad)
y = (y’ – cx)/d
and
This yields x = x(x’,y’)
b y = Cx’ + Dy’
i
x = (y’ – dy)/c
ii
(y’ – dy)/c = (x’ – by)/a
iii
a(y’ – dy) = c(x’ – by)
iv
cby – ady = cx’-ay’
v
y = (cx’-ay’)/(cb-ad)
x = (x’ – by)/a
and
This yields y = y(x’,y’)
3 Homogeneous deformation of a unit circle
x2 + y2 – 1 = 0
a
x2 + y2 = 1
b
by' -dx'  cx' -ay' 
+
- 1 = 0
cb - ad  cb - ad 
or
2
c
2 d
2
2
2
2
2
2
 c 2 
2bd + 2ac 
2 a  b 
x'
– x' y'
+ y'
–1= 0
cb - ad 
cb - ad 
 cb - ad 




2
d Equation in x,y for an ellipse oblique to the x,y axes
C1 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 ellipse
1 Strains may be inhomogeneous over a large region but approximately
homogeneous locally
Stephen Martel
Lab9-2
University of Hawaii
GG303 Lab 9
2/12/2016
3
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 ellipse
a 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
of
maximum
extension
Undeformed
Direction of
maximum
contraction
Deformed
d 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
Stephen Martel
Lab9-3
University of Hawaii
GG303 Lab 9
2/12/2016
f
4
If the initial radius r of a circular marker is known, then
1 
ar
br
, 2 
,
r
r
(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
S1 a /r a


S2 b/ r b
S1/S2 = stretch ratio
C Wellman’s Method
Uses deformed pairs of originally perpendicular lines to determine the strain
ellipse. The pairs have different orientations.
Stephen Martel
Lab9-4
University of Hawaii
GG303 Lab 9
Stephen Martel
2/12/2016
Lab9-5
5
University of Hawaii
GG303 Lab 9
2/12/2016
6
IV Three-dimensional strain and the strain ellipsoid
A 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
1 
ar
br
cr
, 2 
, 3 
r
r
r
2
S1 
a
b
c
 1 1, S2    2 1, S3   3 1
r
r
r
(longitudinal strains)
(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
S1 a /r a S2 b/ r b

 ,


S2 b/ r b S3 c / r c
D In a Flinn diagram, S1/S2 is plotted vs. S2/S3
V Comments on measuring strain
A 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
Stephen Martel
Strain
Lab9-6
University of Hawaii
GG303 Lab 9
2/12/2016
7
Exercise1 Vector fields and lines strained in simple shear
A Print a copy of the Matlab function strain1 and read it.
(21 pts)
(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 2
Extension of AB
Extension of BC
Extension of AC
Extension of BD
Shear strain  of ABC
Shear strain  of BAD
Trend of the axis of greatest
extension
Trend of the axis of least
extension
Dilation of the square
Stephen Martel
Lab9-7
University of Hawaii
GG303 Lab 9
Exercise 2
2/12/2016
8
Strain field near a hole in a uniaxial tensile stress field 86 pts
Purpose
To 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
2
3
4
5
6
7
8
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).
Cut a circular hole at the center of the grid with a radius of 32.5 mm.
Stretch the fabric such that x- and y-axes stay straight.
Trace the deformed grid lines and deformed circles (i.e., strain ellipses) onto a
transparency, which is available for inspection.
Scan the transparency, and electronically trace the strain ellipses and grid lines.
Using the attached copies of the electronic scans:
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.
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.
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.
Analysis
9 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 pts
11 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.
Stephen Martel
Lab9-8
University of Hawaii
GG303 Lab 9
Stephen Martel
2/12/2016
Lab9-9
9
University of Hawaii
GG303 Lab 9
Stephen Martel
2/12/2016
Lab9-10
10
University of Hawaii
GG303 Lab 9
2/12/2016
11
x
x = [0.0
0.0
0.0
0.0
20.0
20.0
20.0
20.0
40.0
40.0
40.0
40.0
60.0
60.0
60.0
60.0
80.0;
80.0;
80.0;
80.0]
y
y = [0.0
20.0
40.0
60.0
0.0
20.0
40.0
60.0
0.0
20.0
40.0
60.0
0.0
20.0
40.0
60.0
0.0;
20.0;
40.0;
60.0]
d
d = [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
11.0
11.8
11.0
11.0;
11.0;
11.0;
11.0]
xp (1/2 pt/box)
xp = [0.0
0.0
20.0
20.0
;
;
;
]
Yp (1/2 pt/box)
yp = [0
20.0
20.0
40.0
;
;
;
]
a ()(1/2 pt/box)
a = [0.0
0.0
0.0
0.0
;
;
;
]
b ()(1/2 pt/box)
b = [0.0
0.0
0.0
0.0
;
;
;
]
)(1/2 pt/box)
t = [0
0
0
0
;
;
;
]
Stephen Martel
Lab9-11
University of Hawaii
GG303 Lab 9
2/12/2016
12
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.
Stephen Martel
Lab9-12
University of Hawaii
GG303 Lab 9
Stephen Martel
2/12/2016
Lab9-13
13
University of Hawaii
GG303 Lab 9
Exercise 4
2/12/2016
14
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+2:1+3 (1 pt) and 1+1:1+3 (1 pt).
D Use the answers from (C) to find the 1+1:1+2:1+3 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).
Stephen Martel
Lab9-14
University of Hawaii
GG303 Lab 9
Stephen Martel
2/12/2016
Lab9-15
15
University of Hawaii
GG303 Lab 9
Exercise 5
2/12/2016
16
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:
a
ur  u
0r
where
(1) u is the radial (outward) displacement away from the center of the
0
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):
Stephen Martel
Lab9-16
University of Hawaii
GG303 Lab 9
2/12/2016
17
B Fill in the table below assuming that u0 = a.
Point
r (cm)
ur (cm)
r’ (cm)
(1 pt/box)
(1 pt/box)
(1 pt/box)
A
B
C
D
E
F
G
H
C 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?
K Is the deformation homogenous or inhomogenous?
(1 pt)
(1 pt)
L (Bonus* - not required) Determine F and Ju for this deformation. Show all your
work.
Stephen Martel
(20 pts)
Lab9-17
University of Hawaii