Rayleigh Wave Propagation in a Rotating Functionally Graded Fiber-Reinforced Medium Subject to Magnetic and Gravitational Fields

Rayleigh Wave Propagation in a Rotating Functionally Graded Fiber-Reinforced Medium Subject to Magnetic and Gravitational Fields

Madhumita Kundu

Department of Mathematics, Makhla Debiswari Vidyaniketan, Hooghly, West Bengal 712245, India

Biswajit Saha

Department of Physics, Gobardanga Hindu College, 24-Parganas (N), 743273, India

Sakti Pada Barik

Department of Mathematics, Gobardanga Hindu College, 24-Parganas (N), 743273, India

Abstract – An analytical investigation is conducted on Rayleigh wave propagation in a rotating ber- reinforced functionally graded (FG) half-space subjected to magneto-gravitational eects. The orientation of the magnetic eld is congured to facilitate a two-dimensional treatment of the governing equations. A transcendental dispersion equation is established, which is shown to reduce to well-known classical solutions under limiting conditions. Finally, parametric studies are performed to quantify the inuence of rotation, gravity, and material grading on the phase velocity, with the ndings presented graphically.

Keywords: Rayleigh waves; Fiber-reinforced medium; Functionally graded material; Mag- netic permeability; Electrical conductivity; Rotational eect; Gravity eld; Wave velocity.

Introduction

The analysis of stress and deformation in ber-reinforced composites has remained a focal point of solid mechanics research for over thirty years. Although reinforcing techniques have historical precursors, the evolution of modern materials science has necessitated more so- phisticated applications for advanced engineering structures. The core design objective is the optimization of tensile properties and load-carrying capacity without incurring signicant

mass penalties. An FRC typically comprises a dispersed ber phase and a continuous matrix phase, separated by an interphase region; the specic spatial orientation of the bersbe it unidirectional, random, or wovendictates the materials macroscopic anisotropy. The- oretical foundations in the continuum modeling of such media were established by Beleld et al.[1], following foundational work by Spencer [2], Pipkin [3], and Rogers [4, 5]. In many instances, these composites are modeled as transversely isotropic elastic media, a framework supported by the variational approaches of Hashin and Rosen [15] for deriving eective elas- tic moduli.

The mechanical performance of ber-reinforced composites can be signicantly enhanced through the incorporation of functional gradation. Over recent decades, functionally graded materials (FGMs) have undergone rapid development and are increasingly utilized across diverse engineering sectors. Unlike conventional layered laminates, FGMs are spatially het- erogeneous composites characterized by a continuous and gradual variation in the volume fraction of their constituent phases. First conceptualized in 1984 by researchers in Sendai, Japan (Yamanouchi et al. [7]; Koviani [6]), these materials have since garnered extensive academic attention. By virtue of their continuously varying macroscopic properties, FGMs oer superior mechanical advantages over traditional laminates, particularly in the mitiga- tion of interfacial thermal stress concentrations. Consequently, they are uniquely suited for extreme operating environments, with applications spanning aerospace thermal protection systems, thermoelectric generators, automotive braking components, and biocompatible im- plants. Notable contributions include the work of Abd-Alla et al. [37], who examined radial vibrations in rotating, functionally graded orthotropic half-spaces under gravitational inu- ence, and Gunghas et al. [38], who explored the synergistic eects of rotation and magnetic elds on thermoelastic solids. Furthermore, Barik et al. [35, 36] addressed various contact mechanics problems within this framework. The convergence of mechanical anisotropy in ber-reinforced media with functional gradation remains a critical frontier in modern en- gineering research, as underscored by the studies of Sahu [28],, Deresiewicz [30], Markham [31], and Zorammuana [32].

The propagation of mechanical disturbances in solid media represents a foundational pillar of physics and engineering. The evolution of wave dynamics is supported by a dis- tinguished historical framework, with seminal contributions from Poisson, Cauchy, Green, Lam´e, and Stokes, as documented in Loves classic treatise on the mathematical theory of elasticity. While classical elasticity provides a robust starting point, it often falls short in characterizing the elastic response of materials with complex internal microstructures. In geophysics and seismology, the study of seismic wavesenergy disturbances generated by tectonic shifts or anthropogenic explosionsis essential for modeling Earths interior and op- timizing resource recovery. These waves are fundamentally categorized into body waves and

surface waves, the latter being primarily responsible for structural damage during seismic events. Among these, Rayleigh waves (1885) are particularly signicant due to their conne- ment to the free surface and their substantial energy density. Consequently, understanding the inuence of initial stress on Rayleigh wave propagation is of paramount importance for seismic risk mitigation and structural integrity assessment. Following Rayleighs pioneering work, extensive research has addressed wave behavior in half-spaces and stratied systems involving inhomogeneous or non-homogeneous media.

The literature regarding surface wave propagation is comprehensive [8, 9, 10, 11]. Unlike body waves, surface waves are characterized by higher energy concentrations at the interface and slower propagation speeds. Research by Acharya and Sengupta [12] and others [13, 14] has examined these characteristics within ber-reinforced anisotropic media. Because large- scale geophysical systems are inherently non-inertial, the study of wave motion in rotating mediapioneered by Schoenberg and Censor [34] is of paramount importance. Addi- tionally, magneto-elastic wave propagation represents a critical area of study in earthquake science, with Acharya and Roy [33] investigating these phenomena in electrically conducting reinforced media. Theoretical developments by Jassim [16] regarding the inhomogeneous wave equation, and by Pradhan et al.[19] on anisotropic dynamics, have provided deeper insights into material reinforcement. The inuence of gravitational elds and initial stresses, rst identied by Bromwich [20], Love [21] and Biot [22], remains a focal point in contem- porary models. Recent studies, such as those by Sethi et al. [40] and Abd-Alla et al. [41], have integrated rotation, magnetism, and gravity to evaluate Rayleigh wave behavior in or- thotropic and functionally graded media [39].

This study investigates the propagation of Rayleigh waves within a rotating, ber-reinforced, functionally graded elastic medium, accounting for the inuences of a magnetic eld and gravity. The magnetic eld orientation is assumed to permit a two-dimensional formulation of the problem. A characteristic wave velocity equation for Rayleigh waves is derived, and numerical simulations are performed using MATLAB to illustrate the relationship between wave velocity and wave number. The inuence of various governing parameters is analyzed and represented graphically. Finally, the generalized results are compared with established literature, with graphical comparisons highlighting the specic impacts of the additional parameters introduced in this model.

Formulation of the Problem:

Consider a semi-innite elastic medium comosed of a functionally graded ber-reinforced material (FGFRM) bounded by a planar surface. A Cartesian coordinate system xyz is established such that the origin O lies on the boundary surface, with the medium occupying the region z 0. The physical model is governed by the following assumptions:

the medium is subjected to a uniform external magnetic eld H0 = (0, H0, 0) and a constant

Figure 1: Geometry of the problem

gravitational force acting in the positive z-direction. Furthermore, the system undergoes uniform rotation with an angular velocity = (0, , 0). The reinforcing bers are oriented parallel to the x-axis, dened by the unit vector a = (1, 0, 0). Finally, the analysis focuses on elastic wave propagation along the x-axis, where the disturbance is localized near the free surface z = 0 and vanishes asymptotically as z .

Based on the aforementioned assumptions, the displacement eld is uniform along any line parallel to the y-axis. Consequently, all eld variables are independent of the y coordinate, rendering the problem two-dimensional in the x z plane. Since the medium is assumed to be rotating with angular velocity in an applied magnetic eld of intensity H0, an induced magnetic eld h = (0, h, 0), an induced electric eld E and a current density J will be

developed. If e is the magnetic permeability of the medium then the total magnetic eld in the medium is B = eH, where H = H0 + h is the magnetic eld arising from applied magnetic eld H0 and induced eld h. Denoting the displacement vector by u = u(x, t), the simplied system of the equations of electrodynamics for a slowly moving homogeneous electrically conducting medium, may be written as

where is the Hamiltons operator, 0 is the electrical permeability, and u is the dynamic displacement vector. Here we ignore the small eect of temperature radiant on the current density vector J . The deformation is supposed to be small and dynamic displacement vector is actually measured from a steady state reformed position.

Following Beleld et al. [1], the stressstrain relations for linearly berreinforced elastic medium may be expressed in tensor form as

where ij are the Cartesian components of the stress tensor, Eij are the strain components

related to the displacement vector ui. , T are elastic constants, , , (L T ) are rein- forcement parameters, and a = (a1, a2, a3) such that a2 + a2 + a2 = 1.

In the absence of body forces, the elastodynamic equations for the rotating medium take the following form:

where denotes the material density. The body force components arising from the applied magnetic eld and gravitational eects are given by

By combining equations (1)(3), and upon neglecting the cross-products of h and u (along with their derivatives), the governing elastodynamic equations for the rotating, ber-reinforced medium subjected to magneto-gravitational eects are given by:

To account for the functionally graded nature of the medium, the constitutive elastic param- eters, the reinforcement coecients, and the mass density are assumed to vary exponentially with the vertical coordinate z. This spatial distribution is modeled as:

where k is real constant.

Assuming the bers are aligned with the x-axis a = (1, 0, 0), the constitutive equations

provide the relevant components of the stress tensor as follows:

For brevity, the hats on the dimensionless parameters are suppressed hereafter. Upon ap- plying equations (9)(12) to (5)(7), and taking

the governing dynamical equations are obtained as follows:

For Rayleigh waves we concentrate only on eqn (14) and eqn (16) which reduces to

CA = alfven velocity, C4 = P-wave velocity, C0 = velocity of light in vacuum.

By introducing the dimensionless parameters x¯, z¯, u¯, w¯, t¯ such that

where denotes the wavenumber, the Equations (17) and (18) for the half-space reduce

Boundary Conditions

Assuming that the disturbance is propagating near the surface of the half space in the form of Rayleigh waves, the boundary conditions at the free surface of the half-space can be written as:

Solution of the Problem:

To solve the governing equations, we seek a harmonic wave solution of the form:

(u¯, 0, w¯) = {u(z), 0, w(z)} exp{i(x ct)} (22) where u(z), w(z) are are depth-dependent amplitudes. In this expression, is the wave

number associated with a wave length of 2

and c is the wave speed. By substituting

equation (22) into the equations (19) and (20), we obtain

From equations (23) and (24) we get the following equations to determining u(z) or w(z):

Since u, w represent surface waves, we assume that they vanish as z . Accordingly, we seek a solution of the form

Using Eqs. (26) and (27) in Eqs. (23) and (24), and equating the coecients of ei1z and

ei2z to zero we obtain

where

and kt = k .

Imposing the boundary conditions from Eq. (21) at z = 0 yields the following system of equations:

For a non-trivial solution for the constants A1 and B1, the determinant of the coecients

must vanish, leading to the characteristic equation:

Equation (33) represents the dispersion relation (wave velocity equation) for Rayleigh waves propagating in a rotating, ber-reinforced, functionally graded medium subjected to mag- netic and gravitational elds. In the absence of gravitational eects, this frequency equation is consistent with the results obtained by Acharya [12]. Furthermore, by neglecting the inu- ences of gravity, rotation, and the magnetic eld, Eq. (33) reduces to the classical Rayleigh wave velocity equation for an isotropic medium:

Numerical Results and Discussions:

The present research examines the collective impact of functionally graded parameters, mag- netic elds, rotational forces, and gravitational eects on the propagation characteristics of Rayleigh waves in ber-reinforced media. To evaluate these eects numerically, three dis- tinct parameter setsdesignated as Fiber-1, Fiber-2, and Fiber-3were adopted from the established literature [25, 31, 32] as detailed below.

Although the theoretical framework accommodates arbitrary propagation directions, the numerical computations are restricted to specic orientations to facilitate calculation and highlight key trends in wave velocity under bre-reinforced functionally graded character- istics of the media. Utilizing the aforementioned parameter sets, this numerical analysis examines the propagation behavior of Rayleigh waves under varying media conditions. Fig- ures 26 present the Raleigh wave velocity plotted against the wavenumber for dierent congurations of reinforcement and functional grading.

Figure 2: Variation of Rayleigh wave velocity for dierent ber reinforced media with xed value of graded parameter k

Figure 2 displays the variation Rayleigh wave velocity with respect to wavenumber for dif-ferent bre reinforce media with a xed functionally gradation. It indicates that the Rayleigh wave velocity decreases when the value of wave number increases. We also observe that for a ber-reinforced medium, the Rayleigh

wave velocity is aected signicantly by the rein-forcing parameter. The dispersion curves indicate that the Rayleigh wave velocity vanishes in the short-wavelength limit as the wavenumber increases ( ): the

wave velocity approaches to zero. Figure 3 illustrates the inuence of the magnetic eld on Rayleigh wave velocity. The velocity decreases with increasing wave number. Furthermore, at a constant wave number, the Rayleigh wave velocity is found to decrease as the applied magnetic eld intensity increases.

Figure 3: Eect of magnetic eld on Rayleigh wave velocity.

Figure 4: Eect of rotation on Rayleigh wave velocity.

The eects of rotation and material density are captured in Figures 4 and 5 respectively, where an increase in either parameter yields a reduction in wave speed. Figure 6 is plotted to observe the inuence of dierent gravity parameter on the Rayleigh wave velocity with respect wave number. It is observed that for a particular value of wave number, Rayleigh wave velocity decreases with the increase of gravitational eects.

Figure 5: Eect of density on Rayleigh wave velocity.

Figure 6: Eect of gravity on Rayleigh wave velocity.

Figure 7: Eect of functionally graded parameter k on Rayleigh wave velocity

Finally, Figure 7 demonstrates that the functional gradation parameter (k) signicantly alters the velocity magnitude; specically, the intensication of material gradation (increased k) results in a consistent attenuation of the Rayleigh wave velocity.

Conclusion:

The study focuses on how a plane surface wave propagating in a rotating ber-reinforced functionally graded (FG) half-space is aected by an applied magnetic eld, rotation, density,

gravity, ber-reinforcing and functionally gradation. A signicant observation across these gures is that as the wavenumber increases (i.e., in the high-frequency limit, ),the

wave velocity asymptotically approaches zero. The results indicate that the direct eects of dierent parameters on Rayleigh wave velocity are very pronounced.

Conict of Interests:

The authors declare that there is no conict of interests regarding the publication of this paper.

Acknowledgment

The authors wish to express their sincere gratitude to Professor P. K. Chaudhuri for his unwavering guidance, valuable insights, and sustained encouragement throughout the course of this work. His support and mentorship were pivotal in shaping the direction and enhancing the depth of this research.

References

1. Beleld, A. J., Rogers, T. G. and Spencer, A. J. M., Stress in elastic plates reinforced by bers lying in concentric circles, Journal of the Mechanics and Physics of Solids, vol. 31, pp. 2554, 1983.
2. Spencer, A. J. M., Deformation of Fiber-Reinforced Materials, Clarendon Press, Oxford, 1941.
3. Pipkin, A. C., Finite deformations of ideal ber-reinforced composites, in Composite Materials, Sendeckyj, G. P. (ed.), pp. 251308, Academic Press, New York, 1973.
4. Rogers, T. G., Finite deformations of strongly anisotropic materials, in Theoretical Rheology, Hutton, J. F., Pearson, R.

   A. and Walters, K. (eds.), pp. 141168, Applied Science Publishers, London, 1975.

5. Rogers, T. G., Anisotropic elastic nd plastic materials, in Continuum Mechanics Aspects of Geodynamics and Rock Fracture, Christensen, R. M. (ed.), pp. 177200, Reidel Publishing, Dordrecht, 1975.
6. Koviani, M., Concept of FGM, Ceramic Transactions, vol. 34, no. 3, pp. 110, 1993.
7. Yamanouchi, M., Koizumi, M. and Shiota, I. (eds.), Proceedings of the 1st Interna-tional Symposium on Functionally Gradient Materials, Functionally Gradient Materials Forum, Kyoto, 1990.
8. Bullen, K. E., An Introduction to the Theory of Seismology, Cambridge University Press, Cambridge, 1965.
9. Ewing, W. M. and Jardetzky, W., Elastic Waves in Layered Media, McGrawHill, New York, 1957.
10. Rayleigh, Lord, On waves propagated along the plane surface of an elastic solid, Proceedings of the London Mathematical Society, vol. 17, pp. 411, 1885.
11. Stoneley, R., Elastic waves at the surface of separation of two solids, Proceedings of the Royal Society A, vol. 106, pp. 416 428, 1924.
12. Acharya, D. P. and Sengupta, P. R., Magneto-thermo-elastic surface waves in initially stressed conducting media, Acta Geophysica, vol. 26, pp. 299311, 1978.
13. Pal, P. C. and Sengupta, P. R., Surface waves in layered anisotropic media, Journal of the Acoustical Society of America, vol. 81, pp. 111118, 1987.
14. Sengupta, P. R. and Nath, S., “Surface waves in fiber-reinforced anisotropic elastic media,” Sadhana, vol. 26, pp. 363-370, 2001.
15. Hashin, Z. and Rosen, B. W., “The elastic moduli of fiber-reinforced materials,” Journal of Applied Mechanics, vol. 31, pp. 223-232, 1964.
16. Jassim, H. K., “Analytical approximate solution for inhomogeneous wave equation on Cantor sets by local fractional variational iteration method,” International Journal of Advanced Applied Mathematics and Mechanics, vol. 3, no. 1, pp. 57-61, 2015.
17. Abo-Dahab, S. M. and Abd-Alla, A. M., “Surface waves propagation in fiber-reinforced anisotropic elastic media subjected to gravity field,” International Journal of Physical Sciences, vol. 8, pp. 574-584, 2013.
18. Das, A., Singh, A., Patel, P., Mistri, K. and Chattopadhyay, A., “Reflection and re-fraction of plane waves at the loosely bonded common interface of piezoelectric fiber-reinforced and fiber-reinforced composite media,” Ultrasonics, vol. 94, pp. 131-144, 2019.
19. Pradhan, A., Samal, S. K. and Mahanti, N. C., “Influence of anisotropy on the Love waves in a self-reinforced medium,” Tamkang Journal of Science and Engineering, vol. 17, no. 3, pp. 173-178, 2013.
20. Bromwich, T. J. I., “On the influence of gravity on elastic waves, with special reference to vibrations of an elastic globe,” Proceedings of the London Mathematical Society, vol. 30, pp. 98-165, 1898.
21. Love,A.E.H., Some Problems of Geodynamics, Dover Publications, New York, 1911.
22. Biot, M. A., Mechanics of Incremental Deformations, John Wiley & Sons, New York, 1965.
23. Ranjan, C. and Samal, S. K., “Love waves in fiber-reinforced layer over a gravitating porous half-space,” Acta Geophysica, vol. 61, no. 5,

    pp. 1170-1183, 2013.

24. Khan, A., Abo-Dahab, S. M. and Abd-Alla, A. M., “Gravitational effect on surface waves in a homogeneous fiber-reinforced anisotropic thermo-viscoelastic media with voids,” International Journal of Physical Sciences, vol. 10, no. 24, pp. 604-613, 2015.
25. Maity, N., Barik, S. P. and Chaudhuri, P. K., “Propagation of plane waves in a rotating magnetothermoelastic fiber-reinforced medium with voids under G-N theory,” IOSR Journal of Engineering, vol. 8, pp. 25-35, 2018.
26. Maity, N., Barik, S. P. and Chaudhuri, P. K., “Wave propagation in a rotating fiber-reinforced poroelastic solid under the action of uniform magnetic field,” International Journal of Computational Science, vol. 4, pp. 901-917, 2016.
27. Manna, S., Kundu, S. and Gupta, S., “Love wave propagation in a piezoelectric layer overlying an inhomogeneous elastic half- space,” Journal of Vibration and Control, vol. 21, no. 13, pp. 2553-2568, 2015.
28. Sahu, S. A., Saroj, P. K. and Paswan, B., “Shear waves in a heterogeneous fiber-reinforced layer over a half-space under gravity,”

    International Journal of Geomechanics, vol. 15, no. 2, Article ID 04014048, 2014.

29. Ke, L. L., Wang, Y. S. and Zhang, Z. M., “Love waves in an inhomogeneous fluid-saturated porous layered half-space with linearly varying properties,” Soil Dynamics and Earthquake Engineering, vol. 26, pp. 574-581, 2006.
30. Deresiewicz, H., “A note on Love waves in a homogeneous crust overlying an inhomo-geneous substratum,” Bulletin of the Seismological Society of America, vol. 52, no. 3, pp. 639-645, 1962.
31. Markham, M. F., “Measurement of the elastic constants of fiber composites by ultra-sonics,” Composites, vol. 1, pp. 145-149, 1969.
32. Zorammuana, C. and Singh, S. S., “SH-wave at a plane interface between homogeneous and inhomogeneous fiber-reinforced elastic half- spaces,” Indian Journal of Materials Science, pp. 1-8, 2015.
33. Acharya, D. P. and Roy, I., “Magnetoelastic surface waves in electrically conducting fiber-reinforced media,” Bulletin of the Institute of Mathematics, Academia Sinica, vol. 4, pp. 333-352, 2009.
34. Schoenberg, M. and Censor, D., “Elastic waves in rotating media,” Journal of the Acoustical Society of America, vol. 54, no. 2, pp. 396-403, 1973.
35. Barik, S. P., Kanoria, M. and Chaudhuri, P. K., “Steady state thermoelastic contact problem in a functionally graded material,”

    International Journal of Engineering Sci-ence, vol. 46, no. 8, pp. 775-789, 2008.

36. Barik, S. P. and Chaudhuri, P. K., “Thermoelastic contact between a functionally graded elastic cylindrical punch and a half-space involving frictional heating,” Journal of En-gineering Mathematics, vol. 36, no. 1, pp. 123-138, 2012.
37. Abd-Alla, A. M., Abo-Dahab, S. M., Al-Tamali, T. A. and Mahmoud, S. R., “Influence of the rotation and gravity field on Stoneley waves in a non-homogeneous orthotropic elastic medium,” Journal of Computational and Theoretical Nanoscience, vol. 10, no. 2, pp. 297-305, 2013.
38. Gunghas, A., Kumar, R., Deswal, S. and Kalkal, K. K., “Influence of rotation and magnetic fields on a functionally graded thermoelastic solid subjected to a mechanical load,” Journal of Mathematics, vol. 2019, Article ID 1016981, 16 pages, 2019.
39. Bo Zhang, Honghang Tu, Liangjuan Li, Jiangong Yu and Jun Dai, “Rayleigh Waves Propagating in the Functionally Graded One- Dimensional Hexagonal Quasicrystal Half-Space ,” Crystals, vol. 13, no. 8, pp. 1-16, 2023.
40. M Sethi, K C Gupta and D Manisha, “Surface waves in fibre-reinforced anisotropic solid elastic media under the influence of gravity,” Int.

    J. Appl. Mech. and Engg., vol. 18, pp. 177-188, 2013.

41. A M Abd-Alla, S M Abo-Dahab, S R Mamoud, M I Helmy, ” Influence of rotation, magnetic field, initial stress and gravity on Rayleigh waves in a homogeneous orthotropic elastic half-space,” Appl. Math. Sci, vol. 4, pp. 91-108, 2010.