Undergrad Derivation of equations of stress tensor transformation

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The discussion focuses on the derivation of equations for the 2D transformation of the stress tensor, specifically relating to a plane ABCD that rotates while the coordinate system remains fixed. The user seeks to connect the stress tensor components from case 1 to case 2, asking if their derived equations are correct and if there is a simpler expression for this transformation. They inquire about the method to determine the Sigma2 tensor given only the Sigma1 tensor. The conversation emphasizes the need for clarity in the transformation equations and potential simplifications. Ultimately, the user is looking for guidance on this mathematical relationship.
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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