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MathGroup Archive 2006

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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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