       tensors, matrices, and rotation

• To: mathgroup at smc.vnet.net
• Subject: [mg66888] tensors, matrices, and rotation
• From: "Chris Chiasson" <chris at chiasson.name>
• Date: Fri, 2 Jun 2006 04:09:44 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```To the MathGroup,
I know this post is a little off topic, but a lot of you are very good
at math, physics and programming, so you might know how to answer the
questions herein.

Context:
1. change of basis via rotation of a stress tensor or matrix
2. eigenvalues and normalized eigenvectors, the matrix of which is a
rotation matrix, to obtain principal stresses
3. other rotations that can yield important properties of the stress
tensor, such as the maximum shear stress
4. determination of Euler angles from rotation matrices

Questions:
Do you know of a book that completely covers the creation of a stress
tensor (and matrix) from a given force vector and basis (coordinate
system)? I really need something that explains everything from first
principles and does not leave any implicit relations left for me to
figure out. It needs to cover everything so I can write an algorithm
that takes a coordinate system and a given force vector and can
construct the stress tensor at any point. I really don't want a mess
of tensor equations appended with QED (lol). Furthermore, I would like
to be able to _fully_ interpret the normalized column eigenvectors of
the stress tensor and their relation to the coordinate system
rotations (change of basis) that will give me principal stresses
(eigenvalues) on the diagonal of the stress tensor.

I find one point particularly confusing, probably because I do not
know how to mathematically (with matrix operations) obtain the stress
tensor as mentioned above. In order to change the basis of the stress
tensor it must be pre and post multiplied by two different versions of
the rotation matrix (one regular and the other the transpose/inverse).
I do not know which goes on which side of the stress matrix, and I am
guessing that the order could more easily be figured out if I knew the
mathematical definition of the stress tensor.

A confusing issue is tensor notation. I do not know how to interpret
tensor notation, nor how to go from a set of tensor equations to a
series of matrix operations. If you could recommend an excellent
reference on this I would be grateful.

One other objective is that I would like to be able to determine the
maximum shear stress rotation (if such a thing exists).

For all of these rotation matrices, I would like to also be able to
obtain the series of angular rotations about a given set of axes.

I would like to derive all properties that could be gleaned from the
(three) Mohr's circle(s) from matrix operations instead.

Here are some links to the better references I have found so far, but
which I am just too thick to assimilate into the required knowledge
above:
David Park's rotations notebooks:
These two pdfs on the stress tensor:
http://www.math.psu.edu/yzheng/m597k/m597kL9.pdf
http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-001Mechanics---Materials-ISpring2003/FCF19507-2728-4202-9C54-3F72CAC1D101/0/stresstransformations.pdf