×

Longitudinal normals and the existence of acoustic axes in crystals. (English) Zbl 1167.74368

Summary: We obtain three conditions on the phase speeds of the longitudinal and transverse waves propagating along the longitudinal normals in a crystal so that each of these conditions guarantees existence of acoustic axes in this crystal. The result is based on the properties of the rational-valued topological degree and of the index of a singular point for some classes of discontinuous mappings. In addition, we give an upper estimate of the number of acoustic axes in a crystal and show some interrelation between their indices.

MSC:

74E15 Crystalline structure
74N05 Crystals in solids
74B05 Classical linear elasticity
74E10 Anisotropy in solid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alshits, V. I.; Sarychev, A. V.; Shuvalov, A. L., Classification of degeneracies and analysis of their stability in theory of elastic waves in crystals, J. Exp. Theor. Phys., 89, 3(9), 922-938 (1985), (in Russian)
[2] Borisovich, Yu. G.; Darinskii, B. M.; Kunakovskaya, O. V., Application of topological methods to estimate the number of longitudinal elastic waves in crystals, Theoret. Math. Phys., 94, 1, 104-108 (1993) · Zbl 0812.58021
[3] Darinskii, B. M., Acoustic axes in crystals, Cristallogr. Rep., 39, 697-703 (1994)
[4] Darinskii, B. M., Polarization of plane elastic waves in anisotropic solids, Cristallogr. Rep., 40, 531-537 (1995)
[5] Fedorov, F. I., Theory of Elastic Waves in Crystals (1968), Plenum Press: Plenum Press New York
[6] Garcia, C. B.; Li, T. Y., On the number of solutions to polynomial systems of equations, SIAM J. Numer. Anal., 17, 4, 540-546 (1980) · Zbl 0445.65037
[7] Horn, R. A.; Johnson, C. R., Matrix Analysis (1985), Cambridge University Press · Zbl 0576.15001
[8] Khatkevich, A. G., The acoustic axis in crystals, Sov. Phys. Crystallogr., 7, 601-604 (1962) · Zbl 0127.23306
[9] Krasnosel’skii, M. A.; Perov, A. I.; Povolockii, A. I.; Zabreiko, P. P., Plane Vector Fields (1966), Academic Press: Academic Press New York
[10] Krasnosel’skii, M. A.; Zabreiko, P. P., (Geometrical Methods of Nonlinear Analysis. Geometrical Methods of Nonlinear Analysis, A Series of Comprehensive Studies in Mathematics, 263 (1984), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York, Tokio) · Zbl 0546.47030
[11] Lloyd, N. G., Degree Theory (1978), Cambridge University Press · Zbl 0367.47001
[12] Norris, A. N., Acoustic axes in elasticity, Wave Motion, 40, 4, 315-328 (2004) · Zbl 1163.74414
[13] Prasolov, V. V., Intuitive Topology (1995), AMS, Providence: AMS, Providence RI · Zbl 0817.57001
[14] Vorotnikov, D. A.; Darinskii, B. M.; Zvyagin, V. G., Topological approach to investigation of acoustic axes in crystals, Cristallogr. Rep., 51, 1, 104-109 (2006)
[15] van der Waerden, B. L., Die Alternative bei nichtlinearen Gleichungen, Nachr. Gesells. Wiss. Göttingen, Math. Phys. Klasse, 77-87 (1928) · JFM 54.0140.05
[16] van der Waerden, B. L., Modern Algebra, vol. II (1931), F. Ungar Publishing Co.: F. Ungar Publishing Co. New York · JFM 57.0153.03
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.