Study of Love Waves in a Clamped Viscoelastic Medium with Irregular Boundaries

Document Type : Research Paper


1 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore -632014, TN, India

2 Department of Mathematics, Madanapalle Institute of Technology & Science Madanapalle-517325, AP, India



A mathematical model is presented to investigate the effects of sandiness, irregular boundary interfaces, heterogeneity and viscoelasticity on the phase velocity of Love waves. Geometry of the problem is consisting of an initially stressed viscoelastic layer with corrugated irregular boundaries, which is sandwiched between heterogeneous orthotropic semi-infinite half-space with initial stress and pre-stressed dry sandy half-space. Heterogeneity arises in the upper half-space is due to trigonometric variation in elastic parameters of the orthotropic medium. Inclusion of the concept of corrugated irregular viscoelastic layer clamped between two dissimilar half-spaces under different physical circumstances such as initial stress and heterogeneity brings a novelty to the existing literature related to the study of Love wave. Dispersion equation for Love wave is obtained in closed form. The obtained dispersion relation is found to be in well agreement with classical Love wave equation. Numerical example and graphical illustrations are made to demonstrate notable effect of initial stress, internal friction, wave number and amplitude of corrugations on the phase velocity of Love waves.


[1] Barak M. S., Kaliraman V., 2018, Reflection and transmission of elastic waves from an imperfect boundary between micropolar elastic solid half space and fluid saturated porous solid half space, Mechanics of Advanced Materials and Structures 26(14): 1226-1233.
[2] Miglani A., Kumari N., 2013, Plane strain deformation of a poroelastic half-space lying over another poroelastic half-space, Journal of Solid Mechanics 5: 418-435.
[3] Kundu S., Alam P., Gupta S., 2017, Shear waves in Magnetoelastic Transversely Isotropic (MTI) layer bonded between two heterogeneous elastic media, Mechanics of Advanced Materials and Structures 26(5): 407-415.
[4] Tomar S. K., Saini S. L., 1997, Reflection and refraction of SH-waves at a corrugated interface between two-dimensional transversely isotropic half spaces, Journal of Physics of the Earth 45: 347-362.
[5] Selim M. M., 2007, Static deformation of an irregular initially stressed medium, Applied Mathematics and Computation 188: 1274-1284.
[6] Singh S. S., Tomar S. K., 2008, qP-wave at a corrugated interface between two dissimilar pre-stressed elastic half-spaces, Journal of Sound and Vibration 317: 3-5.
[7] Alam P., Kundu S., Gupta S., 2018, Dispersion study of SH-wave propagation in an irregular magneto-elastic anisotropic crustal layer over an irregular heterogeneous half-space, Journal of King Saud University – Science 30: 301-310.
[8] Liu H., Wang Z. K., Wang T. J., 2001, Effect of initial stress on the propagation behavior of Love waves in a layered piezoelectric structure, International Journal of Solids and Structures 38: 37-51.
[9] Qian J., Jin F., Wang Z., Kishimoto K., 2004, Love waves propagation in a piezoelectric layered structure with initial stresses, Acta Mechanics 171: 41-47.
[10] Biot M. A., 1940, The influence of initial stress on elastic waves, Journal of Applied Physics 11: 522-530.
[11] Dey S., Chakraborty M., 1983, Influence of gravity and initial stress on the Love waves in a transversely isotropic medium, Geophysical Research Bulletin 21: 311-323.
[12] Akbarov S., Ilhan N., 2008, Dynamics of a system comprising a pre-stressed orthotropic layer and pre-stressed orthotropic half-plane under the action of a moving load, International Journal of Solids and Structures 45: 4222-4235.
[13] Kundu S., Alam P., Gupta S., Pandit D.K., 2017, Impacts on the propagation of SH-waves in a heterogeneous viscoelastic layer sandwiched between an anisotropic porous layer and an initially stressed isotropic half space, Journal of Mechanics 33: 13-22.
[14] Alam P., Kundu S., Gupta S., 2017, Dispersion and attenuation of torsional wave in a viscoelastic layer bonded between a layer and a half-space of dry sandy media, Applied Mathematics and Mechanics 38: 1313-1328.
[15] Gupta S., Bhengra N., 2017, Dispersion study of propagation of torsional surface wave in a layered structure, Journal of Mechanics 33: 303-315.
[16] Abd-Alla A. M., Ahmed S.M., 1999, Propagation of Love wave in a nonhomogeneous orthotropic elastic layer under initial stress overlying semi-infinite media, Applied Mathematics and Computation 106: 265-275.
[17] Gupta S., Ahmed M., 2017, Influence of prestress and periodic corrugated boundary surfaces on Rayleigh waves in an orthotropic medium over a transversely isotropic dissipative semi-infinite substrate, The European Physical Journal Plus 132: 8.
[18] Singhal A., Sahu S. A., 2017, Transference of rayleigh waves in corrugated orthotropic layer over a pre-stressed orthotropic half-space with self weight, Procedia Engineering 173: 972-979.
[19] Barak M. S., Kaliraman V., 2018, Propagation of elastic waves at micropolar viscoelastic solid/fluid saturated incompressible porous solid interface, International Journal of Computational Methods 15: 1850076.
[20] Kielczynski P., Szalewski M., Balcerzak A., 2012, Effect of a viscous liquid loading on Love wave propagation, International Journal of Solids and Structures 49: 2314-2319.
[21] Kumari P., Sharma V. K., 2014, Propagation of torsional waves in a viscoelastic layer over an inhomogeneous half space, Acta Mechanica 225: 1673-1684.
[22] Kumari N., 2014, Reflection and transmission of longitudinal wave at micropolar viscoelastic solid/fluid saturated incompressible porous solid interface, Journal of Solid Mechanics 6: 240-254.
[23] Vishwakarma S. K., Xu R., 2016, Rayleigh wave dispersion in an irregular sandy earth’s crust over orthotropic mantle, Applied Mathematical Modeling 40: 8647-8659.
[24] Dey S., Chandra A., 1983, Surface waves in a dry sandy medium under gravity, Acta Geophysics 31: 395-404.
[25] Alam P., Kundu S., Gupta S., Saha A., 2018, Study of torsional wave in a poroelastic medium sandwiched between a layer and a half-space of heterogeneous dry sandy media, Waves in Random and Complex Media 28: 182-201.
[26] Vishwakarma S. K., Gupta S., Kundu S., 2013, Torsional wave propagation in a substratum over a dry sandy Gibson half-space, International Journal of Geomechanics 14: 06014002.
[27] Love A. E. H., 1920, Mathematical Theory of Elasticity, Cambridge University Press, UK.
[28] Prosser William H., Robert E., Green Jr., 1990, Characterization of the nonlinear elastic properties of graphite/epoxy composites using ultrasound, Journal of Reinforced Plastics and Composites 9: 162-173.
[29] Gubbins D., 1990, Seismology and Plate Tectonics, Cambridge University Press, Cambridge.