So the transformation rule is $A′=Q^tAQ$ where A is any tensor. However I am having difficulty finding Q.
The question we have is "Consider a unit cube embedded with a set of orthonormal material‐fibre base‐vectors, (f,s,n), oriented such that the f‐axis is parallel to the Y‐axis, and the n‐axis is parallel to the Z‐axis."
I have two potential Q's to rotate tensors between the reference system and the material system. The first is $Q=\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix}$
the second is $Q=\begin{bmatrix}0&-1&0\\1&0&0\\0&0&1\end{bmatrix}$
Here is a picture of the deformation and the reference coordinate system
the deformation is given by x = X+0.5Z ,y=Y ,z=Z
The questions ask us to transform between the Right Cauchy-Green deformation tensor in the reference coordinates and the material coordinates. Both Q's give the same numbers just one has negative deformations.
My question is does it matter which Q I use? or can I choose depending on if I define the material coordinates to positive or negative in the same direction as the reference system?
Thanks for any help/input