Double inner product of derivative of a 2nd order tensor with another 2ndorder tensor

josh_machine

Some one please help me how to prove the following:

[tex]\dot{A}:B + A:\dot{B}=A^{\nabla J}:B+A:B^{\nabla J}[/tex]

A and B are II order tensors and : represents the inner product.

josh_machine

How do I prove the following:

[tex]\dot{A}:B + A:\dot{B}=A^{\nabla J}:B+A:B^{\nabla J}[/tex]

arkajad

Where did you get this notation from? What is your dot? What is your J? What do you mean by "inner product", for what kind of tensors? Any reference to some place where your original notation is defined?

josh_machine

The dot represents material time derivative. A and B are second order tensors, eg Stress.
I myself am not clear what [tex]\nabla J[/tex] means here. However, I guess it represents divergence.

This was as a homework question for a Continuum Mechanics course. I have not got any luck trying to understand or prove this expression. Any insight will be greatly appreciated.