Manipulating 2nd Order Tensor Expression: Isolating Q

[afallingbomb]

Given the expression

<br /> \textbf{A} = \textbf{Q} : \textbf{B}<br />

where A and B are second order tensors of rank 2 and Q is a second order tensor of rank 4.

How can I manipulate this expression to calculate for Q given A and B? In other words, isolate Q on one side of the equation. Thank you very much!

 

[arkajad]

The notation you are using

A=Q:B

is not one of the many standard textbooks. Perhaps you can explain it or provide a reference to some easily available text.

 

[afallingbomb]

Sure, Arkajad.

The double dot product or double contraction is the inner product of the two tensors. If they were both second order tensors, their double dot product would yield a scalar. As is described http://www.google.com/url?sa=t&sour...SV4NbzjQ&sig2=r-g3vmH4tcR9bwM82WJ7Jw&cad=rja".

Wikipedia has a small section on it: http://en.wikipedia.org/wiki/Dyadics#Double-dot_product

Thanks!

 

[afallingbomb]

The double dot product between a fourth order tensor and a second order tensor is a second order tensor (now I'm referring to order as the number of subscripts... rank/order are used interchangeably in the literature). I can work it out using dyadics, but I'm not sure how to move around terms in the equation to isolate Q. There are many products to choose from and I'm not very comfortable with the rules, especially using an inverted second order tensor and perhaps (single or double?) contracting it on the left.

 

[arkajad]

Alright, make it simpler. Let A,B be vectors and Q a second order tensor. So what you have then is an equation of the form

a = Qb

where a and b are vectors. Can you calculate nxn coefficients form n equations? You can't. But you can try to find a general solution for Q, with arbitrary coefficients. Is that what you mean? If so, try to do it for matrices and vectors - you will see what kind of algebra is needed.

The idea is: you have an inhomogeneous linear equation (for Q). Therefore a general solution is a sum of a particular solution of this equation and a general solution of the homegeneous one.

 

[afallingbomb]

arkajad, thank you very much. You made me see that my problem was not well posed.

The physical motivation of this problem stemmed from my curiosity on determining a unique material property (stiffness or compliance) from a specified stress and strain state at a material point. In the most general, anisotropic case, there is no unique answer to this problem.

 

[arkajad]

You are welcome!