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Axisymmetric time-harmonic response of a surface-stiffened transversely isotropic half-space

Abstract

This study deals with the elastodynamic response of a surface-stiffened transversely isotropic half-space subjected to a buried time-harmonic normal load. The half-space is reinforced by a Kirchhoff thin plate on its surface. By virtue of a displacement potential function and appropriate time-harmonic Green’s functions of transversely isotropic half-spaces, a robust solution corresponding to two plate-medium bonding assumptions, namely (a) frictionless interface, and (b) perfectly bonded interface is obtained for the first time. All elastic fields of the problem are expressed explicitly in the form of semi-infinite line integrals. Results of some limiting cases including isotropic materials, static loading, and surface loading are recovered from the obtained solutions and subsequently have been verified with those available in the literature. Effects of anisotropy, depth of loading, bonding assumption, and frequency of excitation on the results are precisely discussed. Based on the proposed numerical scheme, some plots of practical importance are depicted.

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References

  1. 1.

    Wolf JP, Deeks AJ (2004) Foundation vibration analysis: a strength of materials approach. Butterworth-Heinemann, Oxford

    Google Scholar 

  2. 2.

    Tu KN, Rosenberg R (2013) Analytical techniques for thin films: treatise on materials science and technology, vol 27. Elsevier, Amsterdam

    Google Scholar 

  3. 3.

    Rajapakse RKND (1988) The interaction between a circular elastic plate and a transversely isotropic elastic half-space. Int J Numer Anal Methods Geomech 12:419–436. doi:10.1002/nag.1610120406

    Article  Google Scholar 

  4. 4.

    Wang YH, Tham LG, Cheung YK (2005) Beams and plates on elastic foundations: a review. Prog Struct Eng Mater 7:174–182. doi:10.1002/pse.202

    Article  Google Scholar 

  5. 5.

    Gladwell GML (1980) Contact problems in classical theory of elasticity. Sijthoff and Noordhoff, Dordrecht

    Book  MATH  Google Scholar 

  6. 6.

    Mura T (1987) Micromechanics of defects in solids, vol 3. Springer, Berlin

    MATH  Google Scholar 

  7. 7.

    Shodja HM, Ahmadi SF, Eskandari M (2014) Boussinesq indentation of a transversely isotropic half-space reinforced by a buried inextensible membrane. Appl Math Model 38:2163–2172. doi:10.1016/j.apm.2013.10.048

    MathSciNet  Article  Google Scholar 

  8. 8.

    Fabrikant VI (2011a) Contact problems for several transversely isotropic elastic layers on a smooth elastic half-space. Meccanica 46:1239–1263. doi:10.1007/s11012-010-9378-9

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Fabrikant VI (2011b) Application of generalized images method to contact problems for a transversely isotropic elastic layer on a smooth half-space. Arch Appl Mech 81:957–974. doi:10.1007/s00419-010-0448-1

    Article  MATH  Google Scholar 

  10. 10.

    Fabrikant VI (2006) Elementary solution of contact problems for a transversely isotropic elastic layer bonded to a rigid foundation. ZAMP 57:464–490. doi:10.1007/s00033-005-0041-6

    ADS  MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Fabrikant VI (2009) Solution of contact problems for a transversely isotropic elastic layer bonded to an elastic half-space. Proc Inst Mech Eng Part C J Mech Eng Sci 223:2487–2499. doi:10.1243/09544062JMES1643

    Article  Google Scholar 

  12. 12.

    Ahmadi SF, Eskandari M (2014) Axisymmetric circular indentation of a half-space reinforced by a buried elastic thin film. Math Mech Solids 9:703–712. doi:10.1177/1081286513485085

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Hogg AHA (1938) Equilibrium of a thin plate, symmetrically loaded, resting on an elastic foundation of infinite depth. Philos Mag Ser 7(25):576–582. doi:10.1080/14786443808562039

    Article  MATH  Google Scholar 

  14. 14.

    Holl DL (1938) Thin plates on elastic foundations. In: Proceedings of 5th international congress of applied mechanics. Wiley, New York, pp 71–74

  15. 15.

    Selvadurai APS, Gaul L, Willner K (1999) Indentation of a functionally graded elastic solid: application of an adhesively bonded plate model. WIT Trans Eng Sci 24:3–14. doi:10.2495/CON990011

    Google Scholar 

  16. 16.

    Selvadurai APS (2001) Mindlin’s problem for a half-space with a bonded flexural surface constraint. Mech Res Commun 28:157–164. doi:10.1016/S0093-6413(01)00157-4

    Article  MATH  Google Scholar 

  17. 17.

    Selvadurai APS (2014) Mechanics of contact between bi-material elastic solids perturbed by a exible interface. IMA J Appl Math 79:739–752. doi:10.1093/imamat/hxu001

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Rahman M, Newaz G (1997) Elastostatic surface displacements of a half-space reinforced by a thin film due to an axial ring load. Int J Eng Sci 35:603–611. doi:10.1016/S0020-7225(96)00096-1

    Article  MATH  Google Scholar 

  19. 19.

    Rahman M, Newaz G (2000) Boussinesq type solution for a transversely isotropic half-space coated with a thin film. Int J Eng Sci 38:807–822. doi:10.1016/S0020-7225(99)00052-X

    Article  MATH  Google Scholar 

  20. 20.

    Argatov II, Sabina FJ (2012) Spherical indentation of a transversely isotropic elastic half-space reinforced with a thin layer. Int J Eng Sci 50:132–143. doi:10.1016/j.ijengsci.2011.08.009

    Article  Google Scholar 

  21. 21.

    Eskandari M, Ahmadi SF (2012) Green’s functions of a surface-stiffened transversely isotropic half-space. Int J Solids Struct 49:3282–3290. doi:10.1016/j.ijsolstr.2012.07.001

    Article  Google Scholar 

  22. 22.

    Senjuntichai T, Sapsathiarn Y (2003) Forced vertical vibration of circular plate in multilayered poroelastic medium. J Eng Mech 129:1330–1341. doi:10.1061/(ASCE)0733-9399(2003)129:11(1330)

    Article  Google Scholar 

  23. 23.

    Liou GS (2009) Vibrations induced by harmonic loadings applied at circular rigid plate on half-space medium. J Sound Vib 323:257–269. doi:10.1016/j.jsv.2008.12.025

    ADS  Article  Google Scholar 

  24. 24.

    Ahmadi SF, Eskandari M (2014) Vibration analysis of a rigid circular disk embedded in a transversely isotropic solid. J Eng Mech 140:04014048. doi:10.1061/(ASCE)EM.1943-7889.0000757

    Article  Google Scholar 

  25. 25.

    Hamidzadeh HR, Dai L, Jazar RN (2014) Wave propagation in solid and porous half-space media. Springer, Berlin

    Book  MATH  Google Scholar 

  26. 26.

    Khojasteh A, Rahimian M, Eskandari M, Pak RYS (2008) Asymmetric wave propagation in a transversely isotropic half-space in displacement potentials. Int J Eng Sci 46:690–710. doi:10.1016/j.ijengsci.2008.01.007

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Sneddon IN (1972) The use of integral transforms. McGraw-Hill, New York

    MATH  Google Scholar 

  28. 28.

    Achenbach JD (1978) Wave propagation in elastic solids. North-Holland, Amsterdam

    MATH  Google Scholar 

  29. 29.

    Lekhnitskii SG (1981) Theory of elasticity of an anisotropic body. MIR Publishers, Moscow

    MATH  Google Scholar 

  30. 30.

    Ooura T, Mori M (1991) The double exponential formula for oscillatory functions over the half infinite interval. J Comput Appl Math 38(1):353–360. doi:10.1016/0377-0427(91)90181-I

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Morteza Eskandari.

Appendices

Appendix 1: Defined coefficients

$$\begin{aligned} a&=\frac{1}{2}\left( s_1^2+s_2^2\right) ,\\ b&=-\frac{1}{2}\rho \,\omega ^2\Big (\frac{1}{c_{33}} + \frac{1}{c_{44}}\Big ),\\ c&=\left( s_1^2-s_2^2\right) ^2,\\ d&=-2\rho \, \omega ^2\Big [\Big (\frac{1}{c_{33}}+\frac{1}{c_{44}} \Big )\left( s_1^2+s_2^2\right) -2\,\frac{c_{11}}{c_{33}} \left( \frac{1}{c_{11}}+\frac{1}{c_{44}}\right) \Big ],\\ e&=\rho ^2\omega ^4\Big (\frac{1}{c_{33}}-\frac{1}{c_{44}}\Big )^2, \\ \eta _i&=c_{13}\lambda _i^2+c_{11}(\xi ^2-k_P^2),\\ \vartheta _i&=(c_{13}+c_{44})\lambda _i^2-\eta _i, \\ \nu _i&=\Big [\eta _i-(c_{13}+c_{44})\big (\frac{c_{13}}{c_{33}}\, \xi ^2+\lambda _i^2\big )\Big ]\lambda _i,\\ I^{-}&=\eta _2\,\nu _1-\eta _1\,\nu _2,\quad i=1,2. \end{aligned}$$

Appendix 2: Internal forces and couples of plate

$$\begin{aligned} M_r^f(r) &=\int _0^{\infty } \frac{F_z D c_{11}(c_{13}+c_{44})\left( \xi ^2-k_P^2\right) }{2\pi c_{33}c_{44}I^-}\left( \frac{\nu _1}{\lambda _1}e^{-\lambda _1 s}-\frac{\nu _2}{\lambda _2}e^{-\lambda _2s}\right) \\&\quad\times\,\frac{\xi ^2}{1+{\varPsi }_f (\xi )}\Big [\xi J_0(r\xi ) + \Big (\frac{\nu _p-1}{r}\Big )J_1(r\xi )\Big ]d\xi , \\ M_\theta ^f(r)&=\int _0^{\infty } \frac{F_z D c_{11}(c_{13}+c_{44})\left( \xi ^2-k_P^2\right) }{2\pi c_{33}c_{44}I^-}\left( \frac{\nu _1}{\lambda _1}e^{-\lambda _1 s}-\frac{\nu _2}{\lambda _2}e^{-\lambda _2s}\right) \\&\quad\times\,\frac{\xi ^2}{1+{\varPsi }_f (\xi )}\Big [\nu _p\,\xi J_0(r\xi )-\Big (\frac{\nu _p-1}{r}\Big )J_1(r\xi )\Big ]d\xi , \\ Q_r^f(r)&=\int _0^{\infty } \frac{F_z D c_{11}(c_{13}+c_{44})\left( \xi ^2-k_P^2\right) }{2\pi c_{33}c_{44}I^-}\left( \frac{\nu _1}{\lambda _1}e^{-\lambda _1 s}-\frac{\nu _2}{\lambda _2}e^{-\lambda _2s}\right) \\&\quad\times\,\frac{\xi ^4 J_1(r \xi )}{1+{\varPsi }_f (\xi )}d\xi , \\ M_r^b(r)&=\int _0^{\infty } \frac{F_z D}{2\pi c_{33}c_{44}(\lambda _1^2-\lambda _2^2)}\left( \frac{\vartheta _1}{\lambda _1} e^{-\lambda _1 s}-\frac{\vartheta _2}{\lambda _2}e^{-\lambda _2s}\right) \\&\quad\times\,\frac{\xi ^2}{1+{\varPsi }_b (\xi )}\Big [\xi J_0(r\xi )+\Big (\frac{\nu _p-1}{r}\Big )J_1(r\xi )\Big ]d\xi , \\ M_\theta ^b(r)&=\int _0^{\infty } \frac{F_z D}{2\pi c_{33}c_{44}(\lambda _1^2-\lambda _2^2)}\left( \frac{\vartheta _1}{\lambda _1} e^{-\lambda _1 s}-\frac{\vartheta _2}{\lambda _2}e^{-\lambda _2s}\right) \\&\quad\times\,\frac{\xi ^2}{1+{\varPsi }_b (\xi )}\Big [\nu _p \,\xi J_0(r\xi )-\Big (\frac{\nu _p-1}{r}\Big )J_1(r\xi )\Big ]d\xi , \\ Q_r^b(r)&=\int _0^{\infty } \frac{F_z D}{2\pi c_{33} c_{44}(\lambda _1^2-\lambda _2^2)}\left( \frac{\vartheta _1}{\lambda _1} e^{-\lambda _1 s}-\frac{\vartheta _2}{\lambda _2} e^{-\lambda _2s}\right) \\&\quad\times\,\frac{\xi ^4 J_1(r \xi )}{1+{\varPsi }_b(\xi )}d\xi . \end{aligned}$$

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Eskandari, M., Samea, P. & Ahmadi, S.F. Axisymmetric time-harmonic response of a surface-stiffened transversely isotropic half-space. Meccanica 52, 183–196 (2017). https://doi.org/10.1007/s11012-016-0387-1

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Keywords

  • Transverse isotropy
  • Reinforced half-space
  • Time-harmonic response
  • Coating
  • Kirchhoff plate