Re: tensors, matrices, and rotation

*To*: mathgroup at smc.vnet.net*Subject*: [mg66913] Re: tensors, matrices, and rotation*From*: bghiggins at ucdavis.edu*Date*: Sat, 3 Jun 2006 03:26:21 -0400 (EDT)*References*: <e5otge$i9p$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

A book I can suggest is Introduction to the mechanics of continuous media by L E. Malvern Likely out of print but a good library should have it. Brian Chris Chiasson wrote: > 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: > http://home.earthlink.net/~djmp/Mathematica.html > 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 > > Thank you for your expertise, > -- > http://chris.chiasson.name/

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**tensors, matrices, and rotation**

**Re: tensors, matrices, and rotation**