Finding the inverse of 4th rank elasticity tensor

[Pilou115]

Homework Statement

I need to find how to obtain the expression of the inverse of a rank four elasticity tensor.

Relevant Equations

C = k 1x1 + 2µ[I-1/3*1x1] where C in the foutrth order tensor
C^-1 = k^(-1)/9 1x1 + 2µ^(-1)[I-1/3*1x1]

I'm desperately trying to understand how to get from 2.7.11 to 2.7.16 and cannot find any reference online on how to find the inverse of an elastic tangent modulus (fourth_order tensor). Can someone help me or give me a reference I can check where they do a similar thing? I would really appreciate it !

 

[docnet]

If you know how to do tensor algebra, you should be able to show that ##C/otimes C^{-1}=\text{ the identity tensor.}## (I don't know how to do it.)