Derivation of equations of stress tensor transformation

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SUMMARY

The discussion focuses on the derivation of equations for the 2D transformation of the stress tensor, specifically relating to a plane ABCD under rotation. The user has derived equations connecting the stress tensor components for two cases, where only the plane rotates while the coordinate system remains fixed. The primary goal is to establish a clear expression that links the Sigma1 tensor to the Sigma2 tensor. The user seeks validation of their equations and simpler expressions for the transformation process.

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  • Basic principles of rotational transformations in physics
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Mike_bb
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Hello !

I derived equations of stress tensor 2D transformation.
Some details: I have plane ABCD in two cases (see top on the pic) and I know tensor components for case 1 only. Only plane ABCD rotate in two cases (top of the picture) but not coordinate system. Coordinate system rotates only on the bottom of picture.
I want to obtain expression that connects tensor for case 1 and tensor for case 2.

My attempt:

1.webp


Are these equations correct? Is there more easier expression for stress tensor 2D transformation for my case?



Thanks.
 
Last edited:
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2.webp


What does expression exist for this case? If we know only Sigma1 tensor then how to find Sigma2 tensor?
 
Solved:

2.webp
 

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