On the Backus average of a layered medium with elasticity tensors in random orientations

As shown by Backus (J Geophys Res 67(11):4427–4440, 1962), the average of a stack of isotropic layers results in a transversely isotropic medium. Herein, we consider a stack of layers consisting of randomly oriented anisotropic elasticity tensors, which—one might reasonably expect—would result in an isotropic medium. However, we show—by means of a fundamental symmetry of the Backus average—that the corresponding Backus average is only transversely isotropic and not, in general, isotropic. In the process, we formulate, and use, an analogy between the Backus and Gazis et al. (Acta Crystallogr 16(9):917–922, 1963) averages.