Understanding tensors (of engineering sciences)—part 2: yet another DIY experiment

I continue from my last post.

There is another simple DIY experiment that you can perform at home. The idea of this experiment had occurred quite some time back, but I had completely forgotten it. (I had forgotten it even while delivering lectures for the FEM courses which I taught in 2009 and 2012). Last night, I happened to recall the idea once again, and thought of immediately sharing it with you via this blog post.

A Fun DIY Experiment # 2:

Get that piece of men’s innerware which is known in India as the “banian” or “banyan,” and in American English as the “vest” [^].  (If not sure, check out the “aaraam kaa maamla” ads.) Basically, a banyan (at least these days) is a cotton garment like a T-shirt, but it’s bit smaller in size, and as an inner-ware, it is also meant to be more closely fitting to the body. That also makes it more easily stretchable, and therefore, better suited to our purposes. It’s also very inexpensive.

Start with a new (i.e. unused and never washed (i.e. never stretched/wrinkled)) banyan. The cloth should be easily stretchable. The fabric should be plain and simple, and without any special knitting pattern; e.g. no “self-stripes” etc. Cut it open and lay the cloth flat on a table. Mark a set of regular Cartesian grid-points on it with the help of an ink pen. You can easily make a bigger grid (say of the size 15 cm X 15 cm) at a regular spacing, say of 1 cm.

Lay the cloth flat and unstretched on the glass surface of a computer scanner (or even a Xerox machine), and obtain an image, say PH1. Next, with the help of a friend, stretch the cloth non-uniformly, by pulling unevenly along many directions. Make sure that the stretch is non-uniform but completely planar, and, of course, that there are no wrinkles. Scan it in this stretched state, and thus obtain the second image, PH2.

Advantages of this second experiment are easy to see: (i) As compared to the balloon rubber, is easier to lay the banyan cloth flat and without wrinkles. (ii) It is easier to stretch it in many directions. (iii) It is easier to mark out a regular grid—the regularity of the fabric of the cloth actually helps in ensuring regularity.

Also, even if you manage to get a good piece of a large rubber balloon, it should anyway be easier to obtain the image of a grid on it using an image scanner/Xerox machine rather than using a digital camera—the issues of having to maintain the same zoom and distance don’t arise.

Process the images as mentioned in the previous post, and keep them ready.

In the meanwhile, also consult the references mentioned in the last post, and make sure to go through the following concepts in particular: (i) position vector for a point-particle; (ii) displacement vector for a point-particle; (iii) the position vectors for an infinity of points in a continuum—i.e. the position vector field; (iv) the line segment, i.e., the relative position vector (i.e., the difference between two position vectors); (v) the translation and rotation of a line segment; (vi) the relative displacement vector of a line segment (i.e., the relative displacement vector of a relative position vector!); (vii) the rigid-body translation and rotation vs. the change of size and shape of a continuum body; (viii) the displacement gradient tensor at a point in a continuum; (ix) the rotation tensor at a point in a continuum body vs. its rigid-body rotation as a whole; (viii) the strain tensor at a point in a continuum body; etc. …

We will of course look into all these concepts—in fact, we will calculate the particular values that all these quantities assume in our simple experiment, using the basic data of the two images that our simple experiment generates. That will be our topic in the next post.

But before coming to it, let’s take a pause for a moment to recall what the purpose of this whole exercise is. It is: to know the physical meanings/correspondents of the mathematical concepts; to try and develop a proper hierarchical order the concepts; to develop a physical “feel” for the more abstract concepts involved. And, as far as the last is concerned (developing a physical feel for abstract concepts), there’s no substitute to realizing what the more concrete context of a given more abstract concept is. In understanding the proper context of mathematical concepts, there is no substitute to physical observation. That’s why, no matter how ridiculously simple these experiments might look like, do not skip the step of actually performing one of these two experiments.

And, BTW, in this series, more DIY experiments and fun ideas are going to follow.

More, later.

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A Song I Like:

(Hindi) “ek ghar banaoonga, tere ghar ke saamane…”
Singers: Mohammad Rafi and Lata Mangeshkar
Music: S. D. Burman
Lyrics: Hasrat Jaipuri