Inhomogeneity

While simple solutions are possible for a homogeneous isotropic model, a general inhomogeneous elastic medium is described by a fourth-order elasticity tensor $c_{ijkl}(\mathbf{x})$ that depends on position $\mathbf{x}$. In the isotropic case, this generally twenty-one independent component tensor can be reduced to one depending on just $\lambda(\mathbf{x})$ and $\mu(\mathbf{x})$, as in the test done at the end of subsection 2.2 below.