State the transformation law for an $n$ th-rank tensor $T_{i j \cdots k}$.

Show that the fourth-rank tensor

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

is isotropic for arbitrary scalars $\alpha, \beta$ and $\gamma$.

The stress $\sigma_{i j}$ and strain $e_{i j}$ in a linear elastic medium are related by

$$\sigma_{i j}=c_{i j k l} e_{k l} .$$

Given that $e_{i j}$ is symmetric and that the medium is isotropic, show that the stress-strain relationship can be written in the form

$$\sigma_{i j}=\lambda e_{k k} \delta_{i j}+2 \mu e_{i j}$$

Show that $e_{i j}$ can be written in the form $e_{i j}=p \delta_{i j}+d_{i j}$, where $d_{i j}$ is a traceless tensor and $p$ is a scalar to be determined. Show also that necessary and sufficient conditions for the stored elastic energy density $E=\frac{1}{2} \sigma_{i j} e_{i j}$ to be non-negative for any deformation of the solid are that

$$\mu \geq 0 \quad \text { and } \quad \lambda \geq-\frac{2}{3} \mu .$$