Question in linear algebra, derivation of a certain relation

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Discussion Overview

The discussion revolves around the derivation of a specific relation in linear algebra related to strain tensors, particularly focusing on equations 9.30 and 9.32 from a referenced book. Participants are attempting to understand the meaning and implications of these equations and how to derive one from the other.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant identifies a potential mistake in equation 9.31, suggesting that the terms cos(psi) and cos(phi)sin(psi) are incorrectly placed.
  • The same participant outlines a method for deriving equation 9.30 by grouping strain tensor elements and projecting them onto a unit vector in the phi-psi direction, but expresses confusion about the outcome not matching the expected result.
  • Another participant provides a link to a PDF thesis that may contain a clearer representation of the equations, although they do not engage with the mathematical content directly.

Areas of Agreement / Disagreement

There is no consensus reached among participants regarding the derivation or the correctness of the equations discussed. The initial participant expresses uncertainty about their approach, while others have not contributed clarifying responses.

Contextual Notes

The discussion lacks clarity on the assumptions underlying the equations and the definitions of the terms involved. There are also unresolved mathematical steps in the derivation process that may contribute to the confusion.

antonni
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Hello good people,
please refer to this:
Pic.png


(notice the mistake in 9.31: cos(psi) switches places with cos(phi)sin(psi) to the best of my understanding)

Now, I am trying to derive 9.30 and for this, according to the book, we solve 9.32. The problem is I can not understand 9.32, the meaning of it.

I see it like this:

1) Group the strain tensor elements in each direction:
ST1=(s11+s21+s31)*e1 ; ST2=(s22+s12+s32)*e2 ; ST3=(s33+s13+s23)*e3
then the "s " stands for epsilon (strain) and the numbers are the strain element of the tensor.
e1,2,3 are unit vectors in S1,2,3 directions respectively.
2) Now, simply project the ST1,2,3 on the unit vector in the phi-psi direction, the h of 9.31. That way you get the magnitude of the strain in the phi-psi direction, which is what we want and what should be 9.30 according to the book. (of course s21=s12, s13=s31 & s23=s32)

I do it the like this:
component of ST1 on h = (h dot ST1) / |h| ; similarly for ST2 and ST3.
component of ST1 on h = (h dot ST1) / |h| = h1*(s11+s21+s31) ; similarly for ST2 and ST3.
As you can see I do not get the 9.30.

Where is my mistake?

Thank you very much

Anton
 

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Hi there,

no one?
 
hohohoo
 

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