Dictionary:Isotropic media

A medium is isotropic if all directions of wave propagation are equivalent, and that case of isotropy has the most simple symmetry. In this case, the material has only two independent elastic moduli, called Lame's and shear modulus, $\lambda$ and $G$, which are related to the stiffness tensor $C_{ijkl}$ by: \begin{equation} C_{ijkl}=[\lambda\delta_{ij}\delta_{kl}+G(\delta_{ij}\delta_{kl}+\delta_{ij}\delta_{kl})] \end{equation} where $\delta$ is the Kronecker delta function, $\delta _{ij}\equiv 1$ for $i=j$ , and $\delta _{ij}\equiv 0$ for $i\neq j$, with i,j,k,l=1,2,3. The stiffness tensor $C_{ijkl}$ of isotropic media is explicitly written as the stiffness matrix (in Voigt notation ) $C_{IJ}$: \begin{equation} C_{IJ}^{iso}= \begin{bmatrix} C_{33} & C_{12} & C_{12} & 0 & 0 & 0\\ C_{12} & C_{33} & C_{12} & 0 & 0 &0\\ C_{12} & C_{12} & C_{33} & 0 & 0 & 0\\ 0 &0 &0 &C_{44} &0 &0\\ 0 &0 &0 &0 &C_{44} &0\\ 0 &0 &0 &0 &0 &C_{44}\\ \end{bmatrix} \equiv \begin{bmatrix} \lambda+2G & \lambda & \lambda & 0 & 0 & 0\\ \lambda & \lambda+2G & \lambda & 0 & 0 &0\\ \lambda & \lambda & \lambda+2G & 0 & 0 & 0\\ 0 &0 &0 &G &0 &0\\ 0 &0 &0 &0 &G &0\\ 0 &0 &0 &0 &0 &G\\ \end{bmatrix} \end{equation} where λ+2G = M = K +4G/3 = ρV_P² is the longitudinal modulus, and G = ρV_S², with ρ the density, and K the bulk modulus.