Hello to all of you,
I have one basic doubt, As I understood from my basic continuum mechanics course that any orthogonal or say proper orthogonal tensor Q or R ( in some books/notes ) is used to change basis vector by rotation or transformation between vector components.Now when I was reading polar decomposition theorem, that is F=RU=VR. Further the objectivity of deformation gradient that is F_star=F.Q and this is due to F is two point tensor same as for 1st.P.K stress tensor. Now when we try to check objectivity of U or V that is also ok and understandable. But when we say due to uniqueness of polar decomposition R_star=QR then I am confused because this is only possible when R is two point tensor ( as per definition of F_star or P_star). When I checked some continuum mechanics books and online lecture notes, I found that some prof. or say most of them says. Q and R is proper orthogonal or orthogonal tensor. In my, understanding (which i am confused) That if Q is uniform tensor means same basis from one co-ordinate system, like Cauchy stress tensor. Then why it is like, if Q and R tensors are having same name and then R_star = QR (i.e objectivity of rotation tensor from polar decomposition ) which simply means that R_star is two point tensor and what does it mean physically ?
In essence, my questions are,
1. what is difference between Q ( that is orthogonal or say proper orthogonal tensor) and R rotation tensor ?
2. Are they same ? As some books define/mention them having same name(notation point of view) ? or is there any slight technical difference between them ,even having same name?
3. If R_Star=QR, which is true due to uniqueness of polar decomposition then how R_star become two point tensor? And what does it mean?
I hope my question is not so stupid and I have put in correct frame of understanding. Thanks for your time and waiting for reply.
Regards,
Nilesh
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