OREGON STATE UNIVERSITY
ME 453 – STRUCTURE AND MECHANICS LABORATORY
Lab 3 Experiment – Winter 2012
Full-Field Displacement and Strain Exercise – Dot Tracking
This is a simple experiment to introduce the concepts of full-field discrete displacement
measurement and strain calculation. The set-up represents a generic deformable object with
discrete surface markers; it is actually a piece of stretchable fabric with printed dots.
Data Collection:
For each experiment described here, make sure you have calibrated the measurement plane.
1. In-plane Translation: Take an initial image of the sample, then move it laterally within the
measurement plane in three displacement steps of know magnitude. Image at each position.
2. In-plane Rotation: Take an initial image of the sample, then rotate within the measurement
plane in three small rotation steps of known magnitude. For ease of data processing later, do
not rotate enough to get rows of dots to overlap. Image at each position.
3. In-plane Strain: Take an initial image of the sample, then separate the tops of the vertical
supports in three steps of known magnitude. Image at each position.
4. Optical Axis Translation: Take an initial image of the sample, then move it toward the
camera, along the optical axis, in three displacement steps of know magnitude. Keep the
steps small enough to maintain focus of the dots. Image at each position.
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Data Processing:
The initial data processing step for each exercise is to measure the location of each dot in each
image, then determine the displacements for each dot. Note that you will have to know the identity of
each dot. A sorting routine for the dots, first by vertical position then by horizontal, is helpful.
Now prepare the data for import into and analysis with Matlab. There is no single way to do this, but
I suggest the following. Each experiment contains the initial (x,y) position of a point, and a set of
three (dx,dy) displacement values. Create a simple text file organized as follows:
x
y
dx1
dy1
dx2
dy2
dx3
dx4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
etc.
Analysis:
Read this file into Matlab using the “csvread” function or the “Import data” control in the workspace.
If the variable created is simply called “data”, then data(:,1) refers to the x-positions, data(:,2) the ypositions, data(:,3) the x-displacements from target image 1, etc. The following sequence will create
an interesting plot:
quiver(data(:,1),data(:,2),data(:,3),data(:,4));
hold on;
quiver(data(:,1),data(:,2),data(:,5),data(:,6));
quiver(data(:,1),data(:,2),data(:,7),data(:,8));
hold off;
Evaluating strain from the discrete displacements is a bit more involved. There is no single way to do
this, but the following procedure is relatively simple. The primary difficulty is that your data is not
from a regular (perfectly aligned) grid of points, which complicates the calculation of derivatives. We
will overcome this difficulty by creating a regular grid of points that sits just inside of your data field,
then re-sampling the data onto the regular grid through an interpolation procedure. Matlab makes
this relatively simple.
inc = ###;
// first select and define an increment size for the grid. Don’t get carried away.
[mx, my] = meshgrid ( xmin : inc : xmax , ymin : inc : ymax );
// create the regular grid
u1 = griddata (data(:,1), data(:,2), data(:,3), mx, my, 'cubic');
// resample onto the regular grid
v1 = griddata (data(:,1), data(:,2), data(:,4), mx, my, 'cubic');
[e1xx, u1y] = gradient (u1, inc, inc);
// use Matlab gradient function
[v1x, e1yy] = gradient (v1, inc, inc);
e1xy = 0.5*( u1y + v1x);
You now have the complete strain tensor for target image 1.
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Another plot type that is useful:
surf (mx, my, e1xx);
// plot x-component normal strain for target image 1
Also calculate principal strains and maximum shear strain magnitude, and do some simple statistical
evaluations. Since your data is now in matrix form, do the following to find mean and standard
deviation of your data sets. The example here finds mean and standard deviation of variable u1:
mean2 ( u1 );
// this returns the mean of a 2d grid of values
std2 ( u1 );
// this returns the standard deviation of a 2d grid of values
Turn In:
For each of the four experiments, please prepare:
1. A single quiver plot that contains displacement vectors for the three target images.
2. Descriptive statistics for the displacement magnitudes:
a. Mean and standard deviation.
b. Maximum and minimum.
c. A brief discussion about measured vs. expected.
3. Calculate maximum principal, minimum principal, and maximum shear strain for each target
image in each experiment. As above, summarize with descriptive statistics:
a. Mean and standard deviation.
b. Maximum and minimum.
c. A brief discussion about measured vs. expected.
4. Prepare contour plots that show the variation of maximum principal, minimum principal, and
maximum shear strain for the last target image of each experiment (12 plots). Put all three
plots for a given experiment on a single page (4 pages.)
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