Derivation of equations of stress tensor transformation

[Mike_bb]

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:

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



Thanks.

 

[Mike_bb]

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

 

[Mike_bb]

Solved: