Show that any second rank Cartesian tensor $P_{i j}$ in $\mathbb{R}^{3}$ can be written as a sum of a symmetric tensor and an antisymmetric tensor. Further, show that $P_{i j}$ can be decomposed into the following terms

$$P_{i j}=P \delta_{i j}+S_{i j}+\epsilon_{i j k} A_{k},$$

where $S_{i j}$ is symmetric and traceless. Give expressions for $P, S_{i j}$ and $A_{k}$ explicitly in terms of $P_{i j}$.

For an isotropic material, the stress $P_{i j}$ can be related to the strain $T_{i j}$ through the stress-strain relation, $P_{i j}=c_{i j k l} T_{k l}$, where the elasticity tensor is given by

$$c_{i j k l}=\alpha \delta_{i j} \delta_{k l}+\beta \delta_{i k} \delta_{j l}+\gamma \delta_{i l} \delta_{j k}$$

and $\alpha, \beta$ and $\gamma$ are scalars. As in $(\dagger)$, the strain $T_{i j}$ can be decomposed into its trace $T$, a symmetric traceless tensor $W_{i j}$ and a vector $V_{k}$. Use the stress-strain relation to express each of $T, W_{i j}$ and $V_{k}$ in terms of $P, S_{i j}$ and $A_{k}$.

Hence, or otherwise, show that if $T_{i j}$ is symmetric then so is $P_{i j}$. Show also that the stress-strain relation can be written in the form

$$P_{i j}=\lambda \delta_{i j} T_{k k}+\mu T_{i j}$$

where $\mu$ and $\lambda$ are scalars.