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Elastodynamics of radially inhomogeneous spherically anisotropic elastic materials in the Stroh formalism. (English) Zbl 1364.74018
Summary: A method for solving elastodynamic problems in radially inhomogeneous elastic materials with spherical anisotropy is presented, i.e. materials having \(c_{ijkl}=c_{ijkl}(r)\) in a spherical coordinate system \(\{r,\theta,\phi\}\). The time-harmonic displacement field \(\mathbf u(r, \theta,\phi)\) is expanded in a separation of variables form with dependence on \(\theta,\phi\) described by vector spherical harmonics with \(r\)-dependent amplitudes. It is proved that such separation of variables solution is generally possible only if the spherical anisotropy is restricted to transverse isotropy (TI) with the principal axis in the radial direction, in which case the amplitudes are determined by a first-order ordinary differential system. Restricted forms of the displacement field, such as \(\mathbf u(r,\theta)\), admit this type of separation of variables solution for certain lower material symmetries. These results extend the Stroh formalism of elastodynamics in rectangular and cylindrical systems to spherical coordinates.

MSC:
74B05 Classical linear elasticity
74H10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of dynamical problems in solid mechanics
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References:
[1] Journal of Elasticity 18 pp 131– (1987) · Zbl 0631.73016
[2] J ELASTICITY 62 pp 85– (2001) · Zbl 1005.74019
[3] INTERNATIONAL JOURNAL OF MECHANICAL SCIENCES 43 pp 667– (2001) · Zbl 1010.74025
[4] SIAM J APPL MATH 55 pp 1345– (1995) · Zbl 0844.35128
[5] ARCH APPL MECH INGENIEUR ARCHIV 79 pp 97– (2009) · Zbl 1190.74012
[6] Scientia Sinica, Series B: Chemistry, Life Sciences, & Earth Sciences 3 pp 247– (1954)
[7] MESS MATH 49 pp 125– (1919)
[8] Journal of Applied Physiology 47 pp 428– (1976)
[9] The Quarterly Journal of Mechanics and Applied Mathematics 63 (4) pp 401– (2010) · Zbl 1242.74038
[10] Scandrett, Journal of the Acoustical Society of America 111 (2) pp 893– (2002)
[11] Shuvalov, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 456 (2001) pp 2197– (2000) · Zbl 0996.74048
[12] Shuvalov, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 459 (2035) pp 1611– (2003) · Zbl 1058.74044
[13] Shuvalov, The Quarterly Journal of Mechanics and Applied Mathematics 56 (3) pp 327– (2003) · Zbl 1127.74336
[14] Wave Motion 40 pp 413– (2004) · Zbl 1163.74442
[15] INT APPL MECH 24 pp 439– (1988)
[16] Torrent, Physical Review Letters 103 (6) pp 064301– (2009)
[17] J ELASTICITY 53 pp 47– (1998) · Zbl 0946.74014
[18] J ELASTICITY 51 pp 213– (1998) · Zbl 0927.74010
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