# Rayleigh Waves in a Rotating Orthotropic Micropolar Elastic Solid Half-Space

Authors: Baljeet Singh; Ritu Sindhu; Jagdish Singh
Journal: International Journal of Geophysics
 (2013)

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## Abstract

A problem on Rayleigh wave in a rotating half-space of an orthotropic micropolar material is considered. The governing equations are solved for surface wave solutions in the half space of the material. These solutions satisfy the boundary conditions at free surface of the half-space to obtain the frequency equation of the Rayleigh wave. For numerical purpose, the frequency equation is approximated. The nondimensional speed of Rayleigh wave is computed and shown graphically versus nondimensional frequency and rotation-frequency ratio for both orthotropic micropolar elastic and isotropic micropolar elastic cases. The numerical results show the effects of rotation, orthotropy, and nondimensional frequency on the nondimensional speed of the Rayleigh wave.


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## Body

## 1. Introduction

Material response to external stimuli depends heavily on the motions of its inner structures. Classical elasticity does not include this effect, where only translation degrees of freedom of material point of body are considered. Eringen [1] developed the linear micropolar theory of elasticity, which included the intrinsic rotations of the microstructure. It provides a model which can support body and surface couples and display high frequency optical branch of the wave spectrum. For engineering applications, it can model composites with rigid chopped fibres, elastic solid with rigid granular inclusions, and other industrial materials such as liquid crystals.The assumptions of isotropy in a solid medium may not capture some of significant features of the continuum responses of soils, geological materials, and composites. Iesan [2–4] studied some static problems in orthotropic micropolar elasticity. Kumar and Choudhary [5, 6] studied the mechanical sources and dynamic behaviour of orthotropic micropolar elastic medium. Kumar and Chaudhary [7] studied the plane strain problem in a homogeneous orthotropic micropolar elastic solid. Kumar and Ailawalia [8] studied the response of a micropolar cubic crystal due to various sources. Kumar and Gupta [9] studied the propagation of waves in transversely isotropic micropolar generalized thermoelastic half-space. Singh [10] investigated the two-dimensional plane wave propagation in an orthotropic micropolar elastic solid.Surface waves in elastic solids were first studied by Rayleigh [11] for an isotropic elastic solid. The extension of surface wave analysis and other wave propagation problems to anisotropic elastic materials has been the subject of many studies; see, for example, [12–21]. The aim of the present paper is to study the propagation of Rayleigh wave in a rotating orthotropic micropolar elastic solid half space. The frequency equation of the Rayleigh wave is obtained. The speed of Rayleigh wave is computed with the help of approximated frequency equation. The effects of orthotropy, non-dimensional frequency, and rotation are shown graphically on the non-dimensional speed of the Rayleigh wave.


## 2. Formulation of the Problem and Solution

We consider a homogeneous and orthotropic medium of an infinite extent with Cartesian coordinate system(x,y,z). We restrict our study to the plane strain parallel to xy-plane, with the displacement vector u=(u1,u2,0) and microrotation vector ϕ=(0,0,ϕ3). Following Eringen [22] and Schoenberg and Censor [23], the field equations in xy-plane for homogeneous and rotating orthotropic micropolar solid in absence of body forces and couples are written as
(1)A11u1,11+(A12+A78)u2,12+A88u1,22-K1ϕ3,2=ρ[∂2u1∂t2-Ω2u1-2Ω∂u2∂t],(2)(A12+A78)u1,12+A77u2,11+A22u2,22-K2ϕ3,1=ρ[∂2u2∂t2-Ω2u2+2Ω∂u1∂t],(3)B66ϕ3,11+B44ϕ3,22-χϕ3+K1u1,2+K2u2,1=ρjϕ¨3,
where
(4)K1=A78-A88,K2=A77-A78,χ=K2-K1.We consider the following surface wave solutions(5){u1,u2,ϕ3}={u-1(y),u-2(y),ϕ-3(y)}eik(x-ct),
where k is the wave number, c is phase velocity of the wave, and ω=kc is the angular frequency. Making use of (5) in (1) to (3), we obtain three homogeneous equations in u-1(y),u-2(y), and ϕ-3(y), which have nontrivial solutions if
(6)αD6-βD4+γD2-δ=0,
where D=d/dy and α,β,γ,andδ are given in Appendix.Letm1,m2, and  m3 be the roots of auxiliary equation (6). Then, the general solutions of (6) are written as
(7)u1=(A1e-m1y+A2e-m2y+A3e-m3y+A4em1y+A5em2y+A6em3y)eik(x-ct),u2=(ζ1A1e-m1y+ζ2A2e-m2y+ζ3A3e-m3y+ζ1A4em1y+ζ2A5em2y+ζ3A6em3y)eik(x-ct),ϕ3=(η1A1e-m1y+η2A2e-m2y+η3A3e-m3y+η1A4em1y+η2A5em2y+η3A6em3y)eik(x-ct),
where
(8)m12+m22+m32=βα,(9)m12m22+m22m32+m32m12=γα,(10)m12m22m32=δα,
and the expressions for ζ1, ζ2, and ζ3 and η1, η2, and η3 are given in the Appendix.With the use of the radiation conditionsu1→0,u2→0,φ3→0 as y→∞, we obtain the particular solutions for medium (y>0) as
(11)u1=(A1e-m1y+A2e-m2y+A3e-m3y)eik(x-ct),(12)u2=(ζ1A1e-m1y+ζ2A2e-m2y+ζ3A3e-m3y)eik(x-ct),(13)ϕ3=(η1A1e-m1y+η2A2e-m2y+η3A3e-m3y)eik(x-ct).


## 3. Boundary Conditions

The mechanical boundary conditions aty=0 are the vanishing of normal force stress tangential force stress; and tangential couple stress that is,
(14)t22=0,t21=0,m23=0,
where
(15)t22=A12∂u1∂x+A22∂u2∂y,t21=A78∂u2∂x+A88∂u1∂y+(A88-A78)ϕ3,m23=B44∂ϕ3∂y.The solutions given by (11) to (13) satisfy the boundary conditions (14) at y=0, and we obtain the following frequency equation:
(16)A12A22A78A88∑m1kη1k(ζ2-ζ3)+iA12A22∑m1km2k(η1k-η2k)+iA12A22K1A88∑m1kη1k(η2k-η3k)+iA78A88∑m1km2kζ3(η2kζ1-η1kζ2)+K1A88∑m1km2kη3k(η1kζ2-η2kζ1)+∑m1km2km3k(η1kζ2-η2kζ1)=0.


## 4. Particular Case

The frequency equation (16) reduces to the frequency equation for an isotropic rotating micropolar elastic case, if we take
(17)A11=A22=λ+2μ+κ,A77=A88=μ+κ,A12=λ,A78=μ,B44=B66=γ,-K1=K2=χ2=κ.


## 5. Numerical Results and Discussion

From relations (8) to (10), we obtain the following approximated roots:
(18)m12k2≅(A11-ρc2Ω*)A88,(19)m22k2≅(A77-ρc2Ω*)A22,(20)m32k2≅(B66+ρjc2+(χ/k2))B44.
With the help of (18) to (20), the frequency equation (16) reduces to the approximated frequency equation for an orthotropic rotating micropolar elastic case. The approximated frequency equation is used to compute the non-dimensional speed of the Rayleigh wave in orthotropic micropolar solid half-space for the following arbitrary physical constants:
(21)A11=11.65×1010Nm-2,A22=11.71×1010Nm-2,A12=7.69×1010Nm-2,A77=1.99×1010Nm-2,A78=1.98×1010Nm-2,A88=2.01×1010Nm-2,B44=0.036×1010N,B66=0.037×1010N,ρ=2.19×103Kgm-3,j=0.000196m2.The non-dimensional speed of the Rayleigh wave is also computed for isotropic micropolar elastic case with following relevant parameters [24]:
(22)λ=7.59×1010Nm-2,μ=1.89×1010Nm-2,κ=0.0149×1010Nm-2,γ=0.0268×109N,ρ=2.19×103Kgm-3,j=0.000196m2.The non-dimensional speedc*(=ρc2/A22) of Rayleigh wave is computed for orthotropic micropolar elastic case and isotropic micropolar elastic case for different values of non-dimensional frequency ω*(=ω2/[χ/ρj]) and rotation-frequency ratio Ω/ω.The non-dimensional speedc* is plotted against the rotation-frequency ratio Ω/ω, when non-dimensional frequency ω*=5,10, and 20. The speed c* decreases with the increase in value of rotation-frequency ratio Ω/ω. For each value of rotation-frequency ratio Ω/ω, the speed c* increases with the increase in value of non-dimensional frequency ω*. The effect of non-dimensional frequency ω* on non-dimensional speed c* decreases with the increase in values of Ω/ω. The variations showing the effect of orthotropy on non-dimensional speed of Rayleigh wave are shown in Figure 1, where solid lines and dotted lines correspond to the orthotropic micropolar elastic case and isotropic micropolar elastic case, respectively.Figure 1
Variations of the nondimensional speed of Rayleigh wave against rotation-frequency ratio (solid lines:orthotropic micropolar elastic case, dotted lines:isotropic micropolar elastic case).The speedc* is also plotted against the non-dimensional frequency ω*, when rotation-frequency ratio Ω/ω=0.2,0.4, and 0.6. The speed c* of Rayleigh wave increases sharply with the increase in value of non-dimensional frequency ω* for both orthotropic micropolar elastic case and isotropic micropolar elastic case. Beyond ω*=4, it increases slowly in the both cases. Here, the speed c* decreases with the increase in values of Ω/ω in each case. The effect of rotation on non-dimensional speed c* increases with the increase in value of non-dimensional frequency ω*. The variations showing the effect of orthotropy on non-dimensional speed of Rayleigh wave are shown in Figure 2, where solid lines and dotted lines correspond to the orthotropic micropolar elastic case and isotropic micropolar elastic case, respectively.Figure 2
Variations of the non-dimensional speed of Rayleigh wave against non-dimensional frequencyω* (solid lines:orthotropic micropolar elastic case, dotted lines:isotropic micropolar elastic case).


## 6. Conclusion

The propagation of Rayleigh wave is studied in an orthotropic micropolar elastic solid half-space, where we obtained the required approximated frequency equation of Rayleigh wave. The non-dimensional speedc*(=ρc2/A22) is computed for certain ranges of non-dimensional frequency ω* and rotation-frequency ratio Ω/ω. The comparison of solid and dotted line curves in the figures reveals the effect of orthotropy, rotation, and non-dimensional frequency ω* on the non-dimensional speed c* of the Rayleigh wave in an orthotropic micropolar elastic solid half-space.



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