×

Explicit transfer matrix for an incompressible orthotropic elastic layer and applications. (English) Zbl 1476.74081

In this paper an explicit transfer matrix for an incompressible orthotropic elastic layer is derived. As an application the authors obtain explicit formulas for the reflection coefficients of SV-waves from an incompressible orthotropic elastic layer. The dispersion equation of Lamb waves propagating in a two-layered incompressible orthotropic plate is studied as well. Necessary and sufficient conditions for the existence of one or two reflected waves are proved. Numerical examples and simulations are presented at the end of the paper.

MSC:

74J20 Wave scattering in solid mechanics
74E30 Composite and mixture properties
74E10 Anisotropy in solid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Rokhlin, SI; Wang, L., Stable recursive algorithm for elastic wave propagation in layered anisotropic media: stiffness matrix method, J. Acoust. Soc. Am., 112, 822-834 (2002) · doi:10.1121/1.1497365
[2] Hosten, B.; Castaings, M., Surface impedance matrices to model the propagation in multilayered media, Ultrasonics, 41, 501-507 (2003) · doi:10.1016/S0041-624X(03)00167-7
[3] Chen, X., A systematic and efficient method of computing normal modes for multilayered half-space, Geophys. J. Int., 115, 391-409 (1993) · doi:10.1111/j.1365-246X.1993.tb01194.x
[4] Knopoff, L., A matrix method for elastic waves problems, Bull. Seismol. Soc. Am., 54, 431-438 (1964)
[5] Lowe, MJS, Matrix techniques for modeling ultrasonic waves in multilayered media, IEEE Trans., 42, 525-542 (1995) · doi:10.1109/16.372080
[6] Thomson, WT, Transmission of elastic waves through a stratified solid medium, J. Appl. Phys., 21, 89-93 (1950) · Zbl 0036.13304 · doi:10.1063/1.1699629
[7] Haskell, NA, The dispersion of surface waves on multilayered media, Bull. Seismol. Soc. Am., 43, 17-34 (1953) · doi:10.1785/BSSA0430010017
[8] Green, WA; Green, ER, Stress vibration due to an impact line load on a four-ply fibre composite plate, Int. J. Solids Struct., 28, 567-595 (1991) · doi:10.1016/0020-7683(91)90172-C
[9] Solyanik, FI, Transmission of plane waves through a layered medium of anisotropic materials, Sov. Phys. Acoust., 23, 533-536 (1977)
[10] Rokhlin, SI; Wang, YJ, Equivalent boundary conditions for thin orthotropic layer between, J. Acoust. Soc. Am., 91, 1875-1887 (1992) · doi:10.1121/1.403717
[11] Vinh, PC; Anh, VTN; Linh, NTK, On a technique for deriving the explicit secular equation of Rayleigh waves in an orthotropic half-space coated by an orthotropic layer, Waves Random Complex Med., 26, 176-188 (2016) · Zbl 1378.74033 · doi:10.1080/17455030.2015.1132859
[12] Rogerson, GA; Sandiford, KJ, Some comments on the dispersion relation for periodically layered pre-stressed elastic media, Int. J. Eng. Sci., 40, 23-49 (2002) · doi:10.1016/S0020-7225(01)00051-9
[13] Wijeyewickrema, AC; Leungvichcharoen, S., Wave propagation in pre-stressed imperfectly bonded compressible elastic layered composites, Mech. Mater., 41, 1192-1203 (2009) · Zbl 1045.74551 · doi:10.1016/j.mechmat.2009.04.004
[14] Vinh, PC; Anh, VTN; Merodio, J.; Hue, LT, Explicit transfer matrices of pre-stressed elastic layers, Int. J. Non-linear Mech., 106, 288-296 (2018) · doi:10.1016/j.ijnonlinmec.2018.05.011
[15] Chadwick, P., Wave propagation in incompressible transversely isotropic elastic media II, Proc. R. Ir. Acad., 94A, 85-104 (1994) · Zbl 0807.73016
[16] Kaplunov, J.; Prikazchikov, DA; Prikazchikova, LA, Dispersion of elastic waves in a strongly inhomogeneous three-layered plate, Int. J. Solids Struct., 113-114, 169-179 (2017) · doi:10.1016/j.ijsolstr.2017.01.042
[17] Keith, CM; Crampin, S., Seismic body waves in anisotropic media: reflection and refraction at a plane interface, Geophys. J. Int., 49, 181-208 (1977) · doi:10.1111/j.1365-246X.1977.tb03708.x
[18] Rokhlin, SI; Belland, TK; Adler, L., Reflection and refraction of elastic waves on a plane interface between two generally anisotropic media, J. Acoust. Soc. Am., 79, 906-918 (1986) · doi:10.1121/1.393764
[19] Nair, S.; Sotiropoulos, DA, Elastic waves in orthotropic incompressible materials and reflection from an interface, J. Acoust. Soc. Am., 102, 102-109 (1997) · doi:10.1121/1.419767
[20] Chattopadhyay, A., Reflection of three-dimensional plane waves in triclinic crystalline medium, Appl. Math. Mech., 28, 1309-1316 (2007) · Zbl 1231.74069 · doi:10.1007/s10483-007-1004-y
[21] Chatterjee, M.; Dhua, S.; Sahu, SA; Chattopadhyay, A., Reflection in a highly anisotropic medium for three-dimensional plane waves under initial stresses, Int. J. Eng. Sci., 85, 136-149 (2014) · doi:10.1016/j.ijengsci.2014.08.010
[22] Chattopadhyay, A.; Kumari, P.; Sharma, VK, Reflection and refraction at the interface between distinct generally anisotropic half spaces for three-dimensional plane quasi-P waves, J. Vib. Control, 21, 493-508 (2015) · doi:10.1177/1077546313488615
[23] Paswan, B.; Sahu, SA; Chattopadhyay, A., Reflection and transmission of plane wave through fluid layer of finite width sandwiched between two monoclinic elastic half-spaces, Acta Mech., 227, 3687-3701 (2016) · Zbl 1380.86012 · doi:10.1007/s00707-016-1684-4
[24] Modi, C.; Kumari, P.; Sharma, VK, Reflection/refraction of qP/qSV wave in layered self-reinforced media, Appl. Math. Model., 40, 8737-8749 (2016) · Zbl 1471.86006 · doi:10.1016/j.apm.2016.05.015
[25] Kumari, P.; Sharma, VK; Modi, C., Reflection/refraction pattern of quasi- (P/SV) waves in dissimilar monoclinic media separated with finite isotropic layer, J. Vib. Control, 22, 2745-2758 (2016) · doi:10.1177/1077546314548911
[26] Sahu, SA; Paswan, B.; Chattopadhyay, A., Reflection and transmission of plane waves through isotropic medium sandwiched between two highly anisotropic half-spaces, Waves Random Complex Med., 26, 42-67 (2016) · Zbl 1367.74016 · doi:10.1080/17455030.2015.1102361
[27] Kumari, P.; Sharma, VK, Dynamics of seismic waves in highly anisotropic triclinic media with intermediate monoclinic layer, Appl. Math. Model., 71, 375-393 (2019) · Zbl 1481.86018 · doi:10.1016/j.apm.2019.02.029
[28] Chadwick, P., Wave propagation in incompressible transversely isotropic elastic media I, Proc. R. Ir. Acad., 93A, 231-253 (1993) · Zbl 0788.73024
[29] Lamb, H., On waves in an elastic plate, Proc. R. Soc. Lond. A, 93, 114-128 (1917) · JFM 46.1232.01 · doi:10.1098/rspa.1917.0008
[30] Mindlin, RD, An Introduction to the Mathematical Theory of Vibrations of Elastic Plates, Edited by J. Yang (2006), Singapore: World Scientific Publishing Co. Pte. Ltd, Singapore · Zbl 1113.74001 · doi:10.1142/6309
[31] Gazis, DC, Exact analysis of the plane-strain vibrations of thick-walled hollow cylinders, J. Acoust. Soc. Am., 30, 786-794 (1958) · doi:10.1121/1.1909761
[32] Viktorov, IA, Rayleigh and Lamb Waves (1967), New York: Plenum Press, New York · doi:10.1007/978-1-4899-5681-1
[33] Kaplunov, JD; Kossovich, LY; Nolde, EV, Dynamics of Thin Walled Elastic Bodies (1998), Cambridge: Academic Press, Cambridge · Zbl 0927.74001
[34] Abubakar, I., Free vibrations of a transversely isotropic plate, Q. J. Mech. Appl. Math., 15, 129-136 (1962) · Zbl 0107.19405 · doi:10.1093/qjmam/15.1.129
[35] Baylis, ER; Green, WA, Flexural waves in fiber reinforced laminated plates, J. Sound Vib., 110, 1-26 (1986) · doi:10.1016/S0022-460X(86)80070-0
[36] Kaplunov, JD; Kossovich, LY; Rogerson, GA, Direct asymptotic integration of the equations of transversely isotropic elasticity for a plate near cut-off frequencies, Q. J. Mech. Appl. Math., 53, 323-341 (2000) · Zbl 0984.74030 · doi:10.1093/qjmam/53.2.323
[37] Kossovich, LY; Moukhomodiarov, RR; Rogerson, GA, Analysis of the dispersion relation for an incompressible transversely isotropic elastic plate, Acta Mech., 153, 89-111 (2002) · Zbl 1066.74034 · doi:10.1007/BF01177053
[38] Solie, LP; Auld, BA, Elastic waves in free anisotropic plates, J. Acoust. Soc. Am., 54, 50-65 (1973) · doi:10.1121/1.1913575
[39] Baid, HK, Detection of Damage in a Composite Structure Using Guided Waves (2012), Los Angeles: University of California, Los Angeles
[40] Baid, H.; Schaal, C.; Samajder, H.; Mail, A., Dispersion of Lamb waves in a honeycomb composite sandwich panel, Ultrasonics, 56, 409-416 (2015) · doi:10.1016/j.ultras.2014.09.007
[41] Nayfeh, AH; Chimenti, DE, Free wave propagation in plates of general anisotropic media, ASME J. Appl. Mech., 56, 881-886 (1989) · Zbl 0724.73053 · doi:10.1115/1.3176186
[42] Shuvalov, AL, On the theory of wave propagation in anisotropic plates, Proc. R. Soc. Lond. A, 456, 2197-2222 (2000) · Zbl 0996.74048 · doi:10.1098/rspa.2000.0609
[43] Kuznetsov, SV, Lamb waves in anisotropic plates (review), Acoust. Phys., 60, 95-103 (2014) · doi:10.1134/S1063771014010084
[44] Ogen, RW; Roxburgh, DG, The effect of pre-stress on the vibration and stability of elastic plates, Int. J. Eng. Sci., 31, 1611-1639 (1993) · Zbl 0780.73038 · doi:10.1016/0020-7225(93)90079-A
[45] Ogen, RW; Roxburgh, DG, Stability and vibration of pre-stressed compressible elastic plates, Int. J. Eng. Sci., 32, 427-454 (1994) · Zbl 0801.73046 · doi:10.1016/0020-7225(94)90133-3
[46] Rogerson, GA, Some asymptotic expansions of the dispersion relation for an incompressible elastic plate, Int. J. Solids Struct., 34, 2785-2802 (1997) · Zbl 0939.74546 · doi:10.1016/S0020-7683(96)00218-1
[47] Kaplunov, JD; Nolde, EV; Rogerson, GA, A low-frequency model for dynamic motion in pre-stressed incompressible elastic structures, Proc. R. Soc. Lond. A, 456, 2589-2610 (2000) · Zbl 1049.74583 · doi:10.1098/rspa.2000.0627
[48] Kaplunov, JD; Nolde, EV; Rogerson, GA, An Asymptotically consistent model for long-wave height-frequency motion in a pre-stressed elastic plate, Math. Mech. Solids, 7, 581-606 (2002) · Zbl 1062.74018 · doi:10.1177/108128602029660
[49] Kaplunov, JD; Nolde, EV; Rogerson, GA, Short wave motion in a pre-stressed incompressible elastic plate, IMA J. Appl. Mech., 67, 383-399 (2002) · Zbl 1136.74329
[50] Kaplunov, JD; Nolde, EV, Long-wave vibrations of a nearly incompressible isotropic plate with fixed faces, Q. J. Mech. Appl. Math., 55, 345-356 (2002) · Zbl 1036.74024 · doi:10.1093/qjmam/55.3.345
[51] Jones, JP, Wave propagation in a two-layered medium, ASME J. Appl. Mech., 31, 213-222 (1964) · Zbl 0126.41707 · doi:10.1115/1.3629589
[52] Yao-Jun, W.; Wei, N.; Xian-Hua, O., Lamb wave modes in a two-layered solid medium with weak interface, Acta Phys. Sin., 3, 561-566 (1994)
[53] Demcenko, A.; Mazeika, L., Calculation of Lamb waves dispersion curves in multi-layered planar structures, Ultragarsas, 44, 3, 15-17 (2002)
[54] Liu, GR; Tani, J.; Watanabe, K.; Ohyoshi, T., Lamb wave propagation in anisotropic laminates, ASME J. Appl. Mech., 57, 923-929 (1990) · doi:10.1115/1.2897662
[55] Nayfeh, AH, The general problem of elastic wave propagation in multilayered anisotropic media, J. Acout. Soc. Am., 60, 1521-1531 (1991) · doi:10.1121/1.400988
[56] Verma, KL, On the wave propagation in layered plates of general anisotropic media, World Acad. Sci. Eng. Tech., 13, 487-493 (2008)
[57] Nafchi, A.M., Zangeneh, V., Moradi, A., Mohamadirad, H.: Modeling of ultrasonic wave propagation in two-layer plate and drawing the dispersion curves. In: International Conference on Mechanical Engineering and Advanced Technology-ICMEAT2012 1012 October, 2012, Abbasi International Hotel, Isfahan, Iran, ICMEAT2012-1106
[58] Bratton, R.; Datta, S.; Thompson, DO; Chimenti, DE, Analysis of guided waves in a bilayered plate, Review of Progress in Quantitative Nondestructive Evaluation, 193-200 (1992), New York: Plenum Press, New York · doi:10.1007/978-1-4615-3344-3_24
[59] Wang, Z.; Jen, C-K; Cheeke, J., D, : Analytical solutions for sagittal plane waves in three-layer composites, IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 4, 293-301 (1993) · doi:10.1109/58.251277
[60] Lutianov, M.; Rogerson, GA, Long wave motion in layered elastic media, Int. J. Eng. Sci., 48, 1856-1871 (2010) · Zbl 1231.74181 · doi:10.1016/j.ijengsci.2010.07.003
[61] Ogden, RW; Hopkins, HG; Sewell, MJ, Elastic deformations of rubberlike solids, Mechanics of Solids. The Rodney Hill 60th Anniversary Volume, 499-537 (1982), Oxford: Pergamon Press, Oxford · Zbl 0491.73045
[62] Amabil, M.; Breslavsky, ID; Reddy, JN, Nonlinear higher-order shell theory for incompressible biological hyperelastic materials, Comput. Methods Appl. Mech. Eng., 346, 841-861 (2019) · Zbl 1440.74063 · doi:10.1016/j.cma.2018.09.023
[63] Stroh, AN, Steady state problems in anisotropic elasticity, J. Math. Phys., 41, 77-103 (1962) · Zbl 0112.16804 · doi:10.1002/sapm196241177
[64] Vinh, PC; Anh, VTN, Effective boundary condition method and approximate secular equations of Rayleigh waves in orthotropic half-spaces coated by a thin layer, J. Mech. Mater. Struct., 11, 259-277 (2016) · doi:10.2140/jomms.2016.11.259
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.