Bos, Len; Slawinski, Michael A.; Stanoev, Theodore On the Backus average of a layered medium with elasticity tensors in random orientations. (English) Zbl 1415.86031 Z. Angew. Math. Phys. 70, No. 3, Paper No. 84, 15 p. (2019). Summary: As shown by G. E. Backus [J. Geophys. Res. 67, 4427–4440 (1962; Zbl 1369.86005)], 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 D. C. Gazis et al. [“The elastic tensor of given symmetry nearest to an anisotropic elastic tensor”, Acta Crystallogr. 16, No. 9, 917–922 (1963; doi:10.1107/S0365110X63002449)] averages. MSC: 86A15 Seismology (including tsunami modeling), earthquakes 74Q20 Bounds on effective properties in solid mechanics 74E15 Crystalline structure Keywords:Backus average; elasticity theory; quaternion rotation; inhomogeneity; anisotropy Citations:Zbl 1369.86005 PDFBibTeX XMLCite \textit{L. Bos} et al., Z. Angew. Math. Phys. 70, No. 3, Paper No. 84, 15 p. (2019; Zbl 1415.86031) Full Text: DOI References: [1] Backus, G.E.: Long-wave elastic anisotropy produced by horizontal layering. J. Geophys. Res. 67(11), 4427-4440 (1962) · Zbl 1369.86005 [2] Gazis, D.C., Tadjbakhsh, I., Toupin, R.A.: The elastic tensor of given symmetry nearest to an anisotropic elastic tensor. Acta Crystallogr. 16(9), 917-922 (1963) [3] Slawinski, M.A.: Waves and Rays in Elastic Continua, 3rd edn. World Scientific, Singapore (2015) · Zbl 1316.74002 [4] Bóna, A., Bucataru, I., Slawinski, M.A.: Space of SO(3)-orbits of elasticity tensors. Arch. Mech. 60(2), 123-138 (2008) · Zbl 1162.74322 [5] Slawinski, M.A.: Waves and Rays in Seismology: Answers to Unasked Questions, 2nd edn. World Scientific, Singapore (2018) · Zbl 1390.86001 [6] Bóna, A., Bucataru, I., Slawinski, M.A.: Coordinate-free characterization of the symmetry classes of elasticity tensors. J. Elast. 87(2-3), 109-132 (2007) · Zbl 1151.74309 [7] Haar Measure. https://en.wikipedia.org/wiki/Haar_measure. Accessed 19 Apr 2019 [8] Kumar, D.: Applying Backus averaging for deriving seismic anisotropy of a long-wavelength equivalent medium from well-log data. J. Geophys. Eng. 10(5), 055001 (2013) [9] Dalton, D.R., Slawinski, M.A.: Numerical examination of commutativity between Backus and Gazis et al. averages. arXiv (2016) [10] Bos, L., Dalton, D.R., Slawinski, M.A., Stanoev, T.: On Backus average for generally anisotropic layers. J. Elast. 127(2), 179-196 (2017) · Zbl 1368.86005 [11] Bos, L., Danek, T., Slawinski, M.A., Stanoev, T.: Statistical and numerical considerations of Backus-average product approximation. J. Elast. 132(1), 141-159 (2018) · Zbl 1395.74008 [12] Dewangan, P., Grechka, V.: Inversion of multicomponent, multiazimuth, walkaway VSP data for the stiffness tensor. Geophysics 68(3), 1022-1031 (2003) [13] Thomsen, L.: Weak elastic aniostropy. Geophysics 51(10), 1954-1966 (1986) [14] Adamus, F.P., Slawinski, M.A., Stanoev, T.: On effects of inhomogeneity on anisotropy in Backus average. arXiv:1802.04075 [physics.geo-ph] (2018) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.