Chapter 10.7, Problem 6P

If P and S are 2^nd-rank tensors, show that $9^2=81$ coefficients are needed to write each component of $P$ as a linear combination of the components of $S$.

Show that $81=3^4$ is the number of components in a $4^{\text{th }}$-rank tensor. If the components of the $4^{\text{th }}$-rank tensor are $C_{ijkm},$ then equation (7.5) gives the components of $P$ in terms of the components of S. If $P$ and $S$ are both symmetric, show that we need only 36 different non-zero components in $C_{ijkm}.$ Hint: Consider the number of different components in P and S when they are symmetric. Comment: The stress and strain tensors can both be shown to be symmetric. Further symmetry reduces the 36 components of $C$ in (7.5) to 21 or less.