Going to the Basis...

Since I enter in the "fabulous" word of  Poromechanics and Homogenization theories I started to see that their formulation is written in tensor notation. (More information of tensor algebra can be found at http://www.foamcfd.org/Nabla/guides/ProgrammersGuidese3.html)

Every engineer remember that one in some lecture the professor told to him/her:

"The stiffness tensor is a  4th order tensor that has 81 constants. Due to symmetry when can reduce the number of this constants and we can write it as matrix 6x6". 

After that the word "tensor" is complete disappeared from our world and we keep write the stress strain relationship as:

or better:

Of course everything is fine till we go from strain to stress and the other way around. But what happen if I need to multiply the stiffness or the compliance tensors for other tensors? Can i still use the matrix notation?

My thought was, maybe yes. But actually is not totally true... I mean we need to pay attention to how we write the matrix form. And we have to remember that matrix operations and tensor operations are not equivalent!

A clear explanation of this issue can be found in the presentation of Klaus Helbig, titled "The Structure of the Elastic Tensor. A study of the possibilities opened up by Kelvin 150 years ago." (if you are interested in, just google the title!)

Conclusion: some time we forgot where some assumptions are coming from, but still they can make a lot of difference!