Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Shuvalov A. L. 2000On the theory of wave propagation in anisotropic platesProc. R. Soc. Lond. A.4562197–2222http://doi.org/10.1098/rspa.2000.0609SectionRestricted accessResearch articleOn the theory of wave propagation in anisotropic plates A. L. Shuvalov A. L. Shuvalov Department of Civil and Structural Engineering, Nottingham Trent University, Burton Street, Nottingham NG1 4BU, UK Google Scholar Find this author on PubMed Search for more papers by this author A. L. Shuvalov A. L. Shuvalov Department of Civil and Structural Engineering, Nottingham Trent University, Burton Street, Nottingham NG1 4BU, UK Google Scholar Find this author on PubMed Search for more papers by this author Published:08 September 2000https://doi.org/10.1098/rspa.2000.0609AbstractA theoretical framework is developed which describes wave propagation in infinite homogeneous elastic plates of unrestricted anisotropy. The approach exploits the propagator matrix which is the exponential of the fundamental elasticity matrix underlying Stroh's formalism of anisotropic elastodynamics. The matrices of plate impedance and admittance are introduced, and their analytical properties are established, which appear fruitful for computing and analysing the plate wave spectra. On this basis, the dispersion equation can be cast into the form of a real equation involving the monotonic function, whose zeros and poles are the wave velocities in a given plate subjected to different boundary conditions (traction–free or clamped faces). It is proved that three fundamental wave branches exist for any orientation of wave propagation in an anisotropic plate with traction–free faces, and that those branches are missing if one or both faces are clamped. The intrinsic symmetry of the wave motion in an arbitrary plate is revealed. The general formalism is applied to elaborate the long–wavelength low–frequency approximation for a (thin) free plate of unrestricted anisotropy. The frequency–dispersive wave velocities, displacements and tractions at the onset of the fundamental wave branches are derived explicitly. The conditions for the extreme velocity values and some other useful universal connections are determined for an arbitrary thin plate, and exemplified for specific anisotropy cases. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Puurtinen T and Maasilta I (2023) Effective medium theory for the low-temperature heat capacity of a metasolid plate, Communications Materials, 10.1038/s43246-022-00329-0, 4:1 Hu W, Xiao C and Deng Z (2023) Structure-Preserving Analysis of Impact Dynamic Systems Geometric Mechanics and Its Applications, 10.1007/978-981-19-7435-9_5, (277-330), . Darinskii A (2022) Nonreciprocal propagation of surface electromagnetic waves in structures comprising magneto-optical materials, Physical Review A, 10.1103/PhysRevA.106.033513, 106:3 Hu W, Xu M, Jiang R, Zhang C and Deng Z (2021) Wave propagation in non-homogeneous asymmetric circular plate, International Journal of Mechanics and Materials in Design, 10.1007/s10999-021-09556-8, 17:4, (885-898), Online publication date: 1-Dec-2021. Hu W, Xu M, Jiang R, Zhang F, Zhang C and Deng Z (2021) Wave Propagation in Non-homogeneous Centrosymmetric Damping Plate Subjected to Impact Series, Journal of Vibration Engineering & Technologies, 10.1007/s42417-021-00355-1, 9:8, (2183-2196), Online publication date: 1-Nov-2021. Kaur G, Singh D and Tomar S (2021) Lamb waves in nonlocal elastic plate with voids, Journal of Mechanics of Materials and Structures, 10.2140/jomms.2021.16.389, 16:4, (389-405) Darinskii A (2021) Surface plasmon polaritons in metal films on anisotropic and bianisotropic substrates, Physical Review A, 10.1103/PhysRevA.104.023507, 104:2 Anh V, Vinh P, Linh N and Thang L (2021) Explicit transfer matrix for an incompressible orthotropic elastic layer and applications, Zeitschrift für angewandte Mathematik und Physik, 10.1007/s00033-021-01579-7, 72:4, Online publication date: 1-Aug-2021. Linh N, Vinh P, Thang L and Giang P (2021) Lamb waves in sandwich orthotropic elastic plates, Waves in Random and Complex Media, 10.1080/17455030.2021.1940352, (1-19) Xia G, Su Y and Chen W (2021) Instability of compressible soft electroactive plates, International Journal of Engineering Science, 10.1016/j.ijengsci.2021.103474, 162, (103474), Online publication date: 1-May-2021. Dudchenko A (2021) Identification and Assessment of Zones with Interfacial Partially Healed Cracks Using Lamb Waves Dispersion, IOP Conference Series: Materials Science and Engineering, 10.1088/1757-899X/1079/4/042059, 1079:4, (042059), Online publication date: 1-Mar-2021. Conroy Broderick H, Dorfmann L and Destrade M (2020) Electro-elastic Lamb waves in dielectric plates, Extreme Mechanics Letters, 10.1016/j.eml.2020.100782, 39, (100782), Online publication date: 1-Sep-2020. Conroy Broderick H, Righi M, Destrade M and Ogden R (2020) Stability analysis of charge-controlled soft dielectric plates, International Journal of Engineering Science, 10.1016/j.ijengsci.2020.103280, 151, (103280), Online publication date: 1-Jun-2020. Vinh P, Tuan T, Hue L, Anh V, Dung T, Linh N and Malischewsky P (2019) Exact formula for the horizontal-to-vertical displacement ratio of Rayleigh waves in layered orthotropic half-spaces, The Journal of the Acoustical Society of America, 10.1121/1.5122184, 146:2, (1279-1289), Online publication date: 1-Aug-2019. Su Y, Conroy Broderick H, Chen W and Destrade M (2018) Wrinkles in soft dielectric plates, Journal of the Mechanics and Physics of Solids, 10.1016/j.jmps.2018.07.001, 119, (298-318), Online publication date: 1-Oct-2018. Carfagna M, Destrade M, Gower A and Grillo A (2017) Oblique wrinkles, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 375:2093, Online publication date: 13-May-2017. Ilyashenko A and Kuznetsov S (2017) Theoretical Aspects of Applying Lamb Waves in Nondestructive Testing of Anisotropic Media, Russian Journal of Nondestructive Testing, 10.1134/S1061830917040039, 53:4, (243-259), Online publication date: 1-Apr-2017. Hernando Quintanilla F, Lowe M and Craster R (2017) The symmetry and coupling properties of solutions in general anisotropic multilayer waveguides, The Journal of the Acoustical Society of America, 10.1121/1.4973543, 141:1, (406-418), Online publication date: 1-Jan-2017. Kuznetsov S (2015) Lamb waves in a clamped and a partially clamped elastic layer, Mechanics of Solids, 10.3103/S0025654415010082, 50:1, (81-95), Online publication date: 1-Jan-2015. Pant S, Laliberte J, Martinez M and Rocha B (2014) Derivation and experimental validation of Lamb wave equations for an n-layered anisotropic composite laminate, Composite Structures, 10.1016/j.compstruct.2014.01.034, 111, (566-579), Online publication date: 1-May-2014. Kuznetsov S (2014) Lamb waves in anisotropic plates (review), Acoustical Physics, 10.1134/S1063771014010084, 60:1, (95-103), Online publication date: 1-Jan-2014. Kuznetsov S (2013) Cauchy Six-Dimensional Formalism for Lamb Waves in Multilayered Plates, ISRN Mechanical Engineering, 10.1155/2013/698706, 2013, (1-11), Online publication date: 16-Sep-2013. Norris A and Shuvalov A (2011) Elastodynamics of radially inhomogeneous spherically anisotropic elastic materials in the Stroh formalism, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468:2138, (467-484), Online publication date: 8-Feb-2012. Mukhomodyarov R and Rogerson G (2011) Long-wave dispersion phenomena in a layer subject to elastically restrained boundary conditions, Zeitschrift für angewandte Mathematik und Physik, 10.1007/s00033-011-0161-0, 63:1, (171-188), Online publication date: 1-Feb-2012. Lancioni G (2011) Dynamics of incoherent elastic layers, European Journal of Mechanics - A/Solids, 10.1016/j.euromechsol.2011.06.005, 30:6, (962-975), Online publication date: 1-Nov-2011. DESTRADE M and MERODIO J (2012) COMPRESSION INSTABILITIES OF TISSUES WITH LOCALIZED STRAIN SOFTENING, International Journal of Applied Mechanics, 10.1142/S1758825111000877, 03:01, (69-83), Online publication date: 1-Mar-2011. Shuvalov A and Le Clezio E (2010) Low-frequency dispersion of fundamental waves in anisotropic piezoelectric plates, International Journal of Solids and Structures, 10.1016/j.ijsolstr.2010.07.018, 47:25-26, (3377-3388), Online publication date: 1-Dec-2010. Kuznetsov S (2010) Love waves in nondestructive diagnostics of layered composites. Survey, Acoustical Physics, 10.1134/S1063771010060126, 56:6, (877-892), Online publication date: 1-Nov-2010. Moukhomodiarov R, Pichugin A and Rogerson G (2009) The Transition between Neumann and Dirichlet Boundary Conditions in Isotropic Elastic Plates, Mathematics and Mechanics of Solids, 10.1177/1081286509103781, 15:4, (462-490), Online publication date: 1-Jun-2010. Ting T (2009) Existence of one-component Rayleigh waves, Stoneley waves, Love waves, slip waves and one-component waves in a plate or layered plate, Journal of Mechanics of Materials and Structures, 10.2140/jomms.2009.4.631, 4:4, (631-647), Online publication date: 1-Jun-2009. Ting T (2009) Steady Waves in an Anisotropic Elastic Layer Attached to a Half-Space or Between Two Half-Spaces—A Generalization of Love Waves and Stoneley Waves, Mathematics and Mechanics of Solids, 10.1177/1081286508092602, 14:1-2, (52-71), Online publication date: 1-Jan-2009. Shuvalov A, Le Clezio E and Feuillard G (2008) The state-vector formalism and the Peano-series solution for modelling guided waves in functionally graded anisotropic piezoelectric plates, International Journal of Engineering Science, 10.1016/j.ijengsci.2008.03.007, 46:9, (929-947), Online publication date: 1-Sep-2008. Alshits V and Maugin G (2008) Dynamics of anisotropic multilayers, Wave Motion, 10.1016/j.wavemoti.2007.07.006, 45:5, (629-640), Online publication date: 1-Apr-2008. Ting T (2007) Dispersion equations for steady waves in an anisotropic elastic plate or a layered plate, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 464:2091, (613-629), Online publication date: 8-Mar-2008. Feuillard G, Clezio E, Levassort F, Hue L, Delaunay T and Lethiecq M (2007) New tools for electromechanical characterisation of piezoceramics, Journal of Electroceramics, 10.1007/s10832-007-9177-4, 19:4, (419-426), Online publication date: 1-Dec-2007. Singh D and Tomar S (2007) Rayleigh–Lamb waves in a microstretch elastic plate cladded with liquid layers, Journal of Sound and Vibration, 10.1016/j.jsv.2006.12.002, 302:1-2, (313-331), Online publication date: 1-Apr-2007. Fu Y Linear and Nonlinear Wave Propagation in Coated or Uncoated Elastic Half-spaces Waves in Nonlinear Pre-Stressed Materials, 10.1007/978-3-211-73572-5_4, (103-127) Poncelet O, Shuvalov A and Kaplunov J (2006) Approximation of the flexural velocity branch in plates, International Journal of Solids and Structures, 10.1016/j.ijsolstr.2005.07.031, 43:21, (6329-6346), Online publication date: 1-Oct-2006. Shuvalov A, Poncelet O and Deschamps M (2006) Analysis of the dispersion spectrum of fluid-loaded anisotropic plates: leaky-wave branches, Journal of Sound and Vibration, 10.1016/j.jsv.2005.12.059, 296:3, (494-517), Online publication date: 1-Sep-2006. Shuvalov A (2006) On the impact of anisotropy on dispersion spectra of acoustic waves in plates, Comptes Rendus Mécanique, 10.1016/j.crme.2006.03.001, 334:4, (243-251), Online publication date: 1-Apr-2006. Shuvalov A, Poncelet O and Deschamps M (2006) Analysis of the dispersion spectrum of fluid-loaded anisotropic plates: flexural-type branches and real-valued loops, Journal of Sound and Vibration, 10.1016/j.jsv.2005.05.015, 290:3-5, (1175-1201), Online publication date: 1-Mar-2006. Alshits V, Deschamps M and Lyubimov V (2005) Dispersion anomalies of shear horizontal guided waves in two- and three-layered plates, The Journal of the Acoustical Society of America, 10.1121/1.2046807, 118:5, (2850-2859), Online publication date: 1-Nov-2005. Shuvalov A, Poncelet O, Deschamps M and Baron C (2005) Long-wavelength dispersion of acoustic waves in transversely inhomogeneous anisotropic plates, Wave Motion, 10.1016/j.wavemoti.2005.04.005, 42:4, (367-382), Online publication date: 1-Oct-2005. Tomar S (2016) Wave Propagation in a Micropolar Elastic Plate with Voids, Journal of Vibration and Control, 10.1177/1077546305054788, 11:6, (849-863), Online publication date: 1-Jun-2005. Wu C, Chang J and Wu K (2011) Analysis of Wave Propagation in Infinite Piezoelectric Plates, Journal of Mechanics, 10.1017/S1727719100004573, 21:2, (103-108), Online publication date: 1-Jun-2005. Alshits V and Maugin G (2005) Dynamics of multilayers: elastic waves in an anisotropic graded or stratified plate, Wave Motion, 10.1016/j.wavemoti.2004.09.002, 41:4, (357-394), Online publication date: 1-Apr-2005. Alshits V On the Role of Anisotropy in Crystalloacoustics Surface Waves in Anisotropic and Laminated Bodies and Defects Detection, 10.1007/1-4020-2387-1_1, (3-68) Shuvalov A, Poncelet O and Deschamps M (2004) General formalism for plane guided waves in transversely inhomogeneous anisotropic plates, Wave Motion, 10.1016/j.wavemoti.2004.02.008, 40:4, (413-426), Online publication date: 1-Oct-2004. Shuvalov A (2004) General relationships for guided acoustic waves in anisotropic plates, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 460:2049, (2671-2679), Online publication date: 8-Sep-2004. Le Clezio E and Shuvalov A Transmission of acoustic waves through piezoelectric plates: modeling and experiment IEEE Ultrasonics Symposium, 2004, 10.1109/ULTSYM.2004.1417785, 0-7803-8412-1, (553-556) Shuvalov A (2003) A sextic formalism for three–dimensional elastodynamics of cylindrically anisotropic radially inhomogeneous materials, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 459:2035, (1611-1639), Online publication date: 8-Jul-2003. Alshits V, Deschamps M and Maugin G (2003) Elastic waves in anisotropic plates: short-wavelength asymptotics of the dispersion branches vn(k), Wave Motion, 10.1016/S0165-2125(02)00078-1, 37:3, (273-292), Online publication date: 1-Mar-2003. Shuvalov A and Every A (2002) Some properties of surface acoustic waves in anisotropic-coated solids, studied by the impedance method, Wave Motion, 10.1016/S0165-2125(02)00013-6, 36:3, (257-273), Online publication date: 1-Sep-2002. Shuvalov A (2002) Theory of plane subsonic elastic waves in fluid-loaded anisotropic plates, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 458:2022, (1323-1352), Online publication date: 8-Jun-2002. Shuvalov A and Every A (2002) Characteristic features of the velocity dispersion of surface acoustic waves in anisotropic coated solids, Ultrasonics, 10.1016/S0041-624X(02)00235-4, 40:1-8, (939-942), Online publication date: 1-May-2002. SHUVALOV A and KRYLOV V (2000) GEOMETRICAL-ACOUSTICS CONSIDERATION OF THE FLEXURAL MODES IN IMMERSED ANISOTROPIC WEDGES, Journal of Sound and Vibration, 10.1006/jsvi.2000.3068, 237:3, (427-434), Online publication date: 1-Oct-2000. This Issue08 September 2000Volume 456Issue 2001 Article InformationDOI:https://doi.org/10.1098/rspa.2000.0609Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/09/2000Published in print08/09/2000 License: Citations and impact Keywordsfundamental branchesthin platesanisotropic plateselastic wavesStroh formalismimpedance matrix