# Anti-Plane Wave Propagation in the Functionally Graded Hybrid Structure under an External Impulsive Force: A Green’s Function Approach

Authors: Uma Bharti; Pramod Kumar Vaishnav
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

To propagate the anti-plane wave, a hybrid layered-half-space schematic of fiber-reinforced and orthotropic materials is taken into account. Due to its importance in engineering applications, anti-plane wave propagation through functionally graded orthotropic materials has attracted a lot of attention. The functionally graded orthotropic materials in the half-space and the reinforced materials under high initial stress are utilized in the superficial layer of finite depth. To examine the effect of heterogeneity on wave phase velocity, the elastic characteristics of orthotropic materials are defined in terms of hyperbolic functions. Moreover, a rectangular plate is installed at the interface of the materials to expose the effect of irregularity in the materials. An external force (impulsive force) is applied to the wave at the point source of the disturbance to understand the influence of such a force on the phase velocity of the wave. The nonhomogeneous equations of motion are deduced by Fourier transformation and are solved analytically by Green’s function technique along with possible applications of Dirac delta function. The dispersion relation of the anti-plane wave propagation is obtained analytically. This study presents a graphic explanation of the proposed schematic and discusses the stability of the model.


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## Body

## 1. Introduction

Anti-plane wave propagation through layered schematic gives immense knowledge about the earth’s interior for possible natural minerals. The dispersion relation of an anti-plane wave depends on the chemical properties (fluid-solid interaction, strength, hardness, etc.) of the composite materials. Consequently, obtaining the dispersion curve of the Love wave to understand the characteristics of the materials present in the earth’s interior is intriguing. The propagation of Love waves in reinforced-orthotropic heterogeneous composite materials under the influence of impulsive forces and uneven interfaces is a new study field in mineral exploration, civil, mining and structural engineering, hydrology, geophysics, etc.Immense studies on Love wave propagation in composite materials have been carried out by many researchers like Lamb wave, Love wave, and other elastic waves in isotropic and anisotropic composite materials that have been studied in [1–4]. The effect of the reinforcement, different heterogeneities (viz., quadratic, linear, exponential, and hyperbolic), initial stress, several irregular boundaries (viz., parabolic, periodic, and rectangular), rigid and soft mountain surfaces, and viscosity and porosity on the phase velocity of the Love wave in several composite materials (including fiber-reinforced, self-reinforced, void porous, and orthotropic) have been investigated in [5–15]. Mal [16] demonstrated the influence of rectangular irregularity on the phase velocity of the Love wave at the interface of two distinct materials. Furthermore, the perturbation approach was used to solve the non-vanishing governing equation of motion. Because of the existence of impulse forces (body forces associated with composite materials) in the earth’s interior, the governing equation of motion becomes nonhomogeneous, making it harder to solve. Subsequently, the Green’s function technique was used to solve the nonhomogeneous wave equation. In the extant literature, there are very few publications on using Green’s function to address elastic wave propagation issues. Chattopadhyay et al. [17], Kundu et al. [18], and Manna et al. [19, 20] employed the Green’s function approach to solve the elastic wave propagation in the elastic medium with regular boundaries. Singh et al. [21, 22] later employed the aforementioned approach to solve the Love wave propagation in piezoelectric and piezomagnetic structures. Recently, Deliktas and Teymur [23] examined the elastic wave propagation in an irregular layer with an effect of a nonlinear surface. Rayleigh wave propagation in irregular subspace of the porous medium has been explained by Xiao et al. [24]. Kumari et al. [25] studied the influence of the imperfect boundaries and abrupt thickening on the phase velocity of the surface waves. SH waves characteristics in Cosserat isotropic medium with scattering have been studied by Chaki and Singh [26]. Mei et al. [27] examined the band gaps of SH waves of metamaterials.Apart from existing research, we considered a superficial layer of reinforced materials of finite thicknessH resting over an initially stressed heterogeneous orthotropic substrate as a medium of anti-plane wave propagation. At the interface of these materials, a rectangular plate of length h and depth h1 is installed to investigate the effect of irregularity on the phase velocity of the Love wave, we focused on the behavior of the phase velocity of the Love wave only. Impulse forces due to point source are associated along the depth of the substrate to examine the behavior of the phase velocity of the Love wave. Moreover, the sine hyperbolic function is considered as a variation in the rigidities, densities, and initial stress which represent the diversity of the materials. The dispersion relation of the Love wave is derived, the governing equations of motion are deduced by Fourier transformation and solved analytically by Green’s function technique and its well-known properties. However, a number of methods have been developed for solving the problem of wave propagation analytically in different mediums with irregular interfaces such as the integral method and the power series method. In spite of that, the Green’s function is a useful mathematical tool to solve the equations of motion analytically in the presence of point source phenomena. The obtained dispersion relation is fairly matched with the conventional form of the Love wave dispersion which validates the present study.To the best of the author’s knowledge, no study has been made so far to analyze the anti-plane wave propagation in the proposed schematic. The effect of varying parameters, initial stress, irregularity, and impulsive forces on the phase velocity of the Love wave are studied numerically and manifested graphically. The mechanism of the wave propagation and the numerical data for the elastic constants of the materials are taken from [28–32]. Our finding observed that the phase velocity of the Love wave was affected significantly in the presence of irregularity, heterogeneity, body forces (impulse forces due to point source), initial stress, and magnify parameters. Moreover, it is also noticed that the change in the length of the installed rectangular plate at the interface of the schematic affects the phase velocity of the Love wave significantly.This paper is organized as follows: in Section2, we present the schematic of the problem, discuss the mechanism of irregularity at the interface of materials and describe the hyperbolic variation in the orthotropic half-space. Section 3, highlights the dynamics of reinforced materials and orthotropic materials under impulsive force. In this section, we deduce the governing equations of motion by using Fourier transformation for the materials. Section 4 presents the boundary conditions and the solution of the problem. Stability of the earth model is discussed in Section 5. Numerical results and discussion are provided in Section 6. Finally, the main conclusions are reported in Section 7.


## 2. Schematic of the Problem

Hybridization in the propagation medium is considered with the hyperbolic heterogeneity in an anisotropic elastic substrate with the irregular interface of two mediums (layer and half-space). Anti-plane wave propagates in a hybrid structure with the velocityc and the wave number k along x−direction, the mechanical displacement of the particle is observed in the y−direction only. A two-dimensional earth model has been described in the Cartesian coordinate system as shown in Figure 1.Figure 1
Schematic of hybrid structure with an impulsive point source.A plate of the rectangular shape has been installed at the contact interface of the guiding layer and half-space to measure the effect on the phase velocity and wave numbers. Length and depth of the plate are taken ash1 and h, respectively. Source of energy disturbance (impulsive point source) has been considered along the z−direction in the orthotropic half-space.Interface of the two mediums has been formulated mathematically as(1)z=ηfx−H,fx=h,x<h2,0,x⩾h2.η=h1/h is a perturbation parameter which is a very small positive quantity η<<1.Apart from the established results, we will take the hyperbolic heterogeneity in the orthotropic medium with irregular interface and an impulsive point source effect into account when dealing the anti-plane wave propagation. The shear moduli of the orthotropic medium has been assumed as(2)L=L1+ϵsinhbz−H,N=N1+ϵsinhbz−H,T=T1+ϵcoshbz−H,ρ=ρ1+ϵcoshbz−H.,where N, L, ρ and L1, N1, ρ1 are the shear moduli of the homogeneous and heterogeneous mediums, respectively. T and T1 are the associated initial stresses of the homogeneous and heterogeneous orthotropic medium, respectively, and ϵ and b are the heterogeneity parameters. The aim of this paper is to obtain the dispersion relation of the anti-plane wave in the hybrid structure to demonstrate the effect of initial stress, heterogeneity, reinforcement, irregularity, and anisotropy on the phase velocity and wave numbers for the fundamental mode of propagation.


## 3. Dynamics of the Materials

### 3.1. Dynamics for the Upper Layer

The governing equation of motion with the source of disturbance in the medium is(3)τi,j+Fi=ρ∂2uj∂xj2,where τi,j represents the stress components in jth direction j=1,2,3, ρ represents the density of the material found in the earth’s interior layer, and Fi are the body forces.As wave is propagating horizontally along thex-axis: u=w=0, v=vx,z,t and ∂/∂y=0.Considering the impact of the point source in the reinforced medium, the equation of motion is(4)P∂2v1∂z2+Q∂2v1∂x2+R∂2v1∂x∂z−ρ1∂2v1∂t2=4πσ1r,t,where(5)P=μT+μL−μTa32,Q=μT+μL−μTa12,R=a1a3μL−μT.μT is transverse shear modulus and μL is longitudinal shear modulus considered in a preferred direction. σ1r,t is the force density disturbance with the effect of the point source. Force is acting at a distance of r from origin at a time t.a1,a2,a3 are the components of a⟶ in the direction of reinforcement such that a12+a22+a32=1.Considering(6)v1x,z,t=v1x,zeiωt,σ1r,t=σ1reiωt,in (4), we obtain(7)P∂2v1∂z2+Q∂2v1∂x2+R∂2v1∂x∂z+ρ1v1x,zω2=4πσ1r,where ω=kc is the angular frequency.Here the impulsive forceσ1r cause some disturbances which can be represented as σ1r=δxδz−H.So the above equation of motion is(8)P∂2v1∂z2+Q∂2v1∂x2+R∂2v1∂x∂z+ρ1v1x,zω2=4πδxδz−H.Defining the Fourier transformv¯rξ,z of vrx,z as(9)v¯rξ,z=12π∫−∞∞vrx,zeιξxdx,and inverse Fourier transform as(10)vrx,z=∫−∞∞v¯rξ,ze−ιξxdξ.Imposing Fourier transform on (9), we get(11)d2v¯1dz2+2Mdv¯1dz+β12v¯1=2Pδz−H,where(12)β12=ρ1Pω2−Qξ2P,M=RιξP.In order to make the equation free of a first derivative term, setv¯1=w¯1e−Mz in (12), it becomes(13)d2w¯1dz2−α2w¯1=2PeMzδz−H,where(14)α2=M2−β12.


### 3.2. Dynamics for the Orthotropic Layer

The equation of motion for the orthotropic medium of infinite depth is(15)∂∂xN∂v2∂x+∂∂zL∂v2∂z−T∂∂x12∂v2∂x−ρ∂2v2∂t2=0,where N and L are directional rigidity, T represents initial stress, and ρ represents density of the half-space. Using (2), we get(16)v2x,z,t=v2x,zeιωt,in (16), and we get(17)N1−T12∂2v2∂x2+L1∂2v2∂z2+ρ1ω2v2,=−ϵsinhbz−H−12coshbz−H∂2v2∂x2+bcoshbz−H∂v2∂z,+sinhbz−H∂2v2∂z2+ω2coshbz−Hv2.With Fourier transform onv¯2x,z, (18) takes the form of ODE as(18)d2v¯2dz2−m2v¯2=4πσ2z,where(19)m2=ξ2L1N1−T12−ρ1L1ω2,4πσ2z=−ϵL1−ξ2sinhbz−H+ξ22coshbz−H+ω2coshbz−Hv¯2+bcoshbz−Hdv¯2dz+sinhbz−Hd2v¯2dz2.


## 3.1. Dynamics for the Upper Layer

The governing equation of motion with the source of disturbance in the medium is(3)τi,j+Fi=ρ∂2uj∂xj2,where τi,j represents the stress components in jth direction j=1,2,3, ρ represents the density of the material found in the earth’s interior layer, and Fi are the body forces.As wave is propagating horizontally along thex-axis: u=w=0, v=vx,z,t and ∂/∂y=0.Considering the impact of the point source in the reinforced medium, the equation of motion is(4)P∂2v1∂z2+Q∂2v1∂x2+R∂2v1∂x∂z−ρ1∂2v1∂t2=4πσ1r,t,where(5)P=μT+μL−μTa32,Q=μT+μL−μTa12,R=a1a3μL−μT.μT is transverse shear modulus and μL is longitudinal shear modulus considered in a preferred direction. σ1r,t is the force density disturbance with the effect of the point source. Force is acting at a distance of r from origin at a time t.a1,a2,a3 are the components of a⟶ in the direction of reinforcement such that a12+a22+a32=1.Considering(6)v1x,z,t=v1x,zeiωt,σ1r,t=σ1reiωt,in (4), we obtain(7)P∂2v1∂z2+Q∂2v1∂x2+R∂2v1∂x∂z+ρ1v1x,zω2=4πσ1r,where ω=kc is the angular frequency.Here the impulsive forceσ1r cause some disturbances which can be represented as σ1r=δxδz−H.So the above equation of motion is(8)P∂2v1∂z2+Q∂2v1∂x2+R∂2v1∂x∂z+ρ1v1x,zω2=4πδxδz−H.Defining the Fourier transformv¯rξ,z of vrx,z as(9)v¯rξ,z=12π∫−∞∞vrx,zeιξxdx,and inverse Fourier transform as(10)vrx,z=∫−∞∞v¯rξ,ze−ιξxdξ.Imposing Fourier transform on (9), we get(11)d2v¯1dz2+2Mdv¯1dz+β12v¯1=2Pδz−H,where(12)β12=ρ1Pω2−Qξ2P,M=RιξP.In order to make the equation free of a first derivative term, setv¯1=w¯1e−Mz in (12), it becomes(13)d2w¯1dz2−α2w¯1=2PeMzδz−H,where(14)α2=M2−β12.


## 3.2. Dynamics for the Orthotropic Layer

The equation of motion for the orthotropic medium of infinite depth is(15)∂∂xN∂v2∂x+∂∂zL∂v2∂z−T∂∂x12∂v2∂x−ρ∂2v2∂t2=0,where N and L are directional rigidity, T represents initial stress, and ρ represents density of the half-space. Using (2), we get(16)v2x,z,t=v2x,zeιωt,in (16), and we get(17)N1−T12∂2v2∂x2+L1∂2v2∂z2+ρ1ω2v2,=−ϵsinhbz−H−12coshbz−H∂2v2∂x2+bcoshbz−H∂v2∂z,+sinhbz−H∂2v2∂z2+ω2coshbz−Hv2.With Fourier transform onv¯2x,z, (18) takes the form of ODE as(18)d2v¯2dz2−m2v¯2=4πσ2z,where(19)m2=ξ2L1N1−T12−ρ1L1ω2,4πσ2z=−ϵL1−ξ2sinhbz−H+ξ22coshbz−H+ω2coshbz−Hv¯2+bcoshbz−Hdv¯2dz+sinhbz−Hd2v¯2dz2.


## 4. Boundary Conditions and Solution of the Problem

Our aim is to get the displacement relation for both the mediums from (14) and (19) by the Green’s function method. These equations also satisfies the boundary conditions at the prescribed boundaries at z = 0 and at z=ηfx−H.The boundary conditions are as follows:(i)
The surfacez=0 is stress free.(20)R∂w¯1∂x+P∂w¯1∂z=0.(ii)
The stress is continuous at the common interfacez=ηfx−H.(21)R∂w¯1∂x+P∂w¯1∂z=L∂v¯2∂z.(iii)
The displacement is continuous atz=ηfx−H.(22)w¯1z=v¯2z.LetG1z/z0 be Green’s function for the reinforced medium satisfying the conditions dG1/dz=0 at z=0 and z=ηfx−H. Equation (14) satisfied by G1z/z0 is(23)d2G1dz2−α2G1=δz−z0.Multiplying (14) by G1z/z0 and (25) by w¯1z, subtracting and integrating from 0 to ηfx−H, then replacing z0 by z and using the symmetric property Gz/ηfx−H=Gηfx−H/z, we get(24)w¯1z=2PeMηfx−HG1zηfx−H−G1zηfx−Hdw¯1dzz=ηfx−H.For half-space of the orthotropic medium,G2z/z0 be Green’s function that satisfies (19), then G2z/z0 is the solution given by(25)d2G2dz2−m2G2=δz−z0.G2z/z0 satisfies dG2/dz = 0 at z = 0 and z=ηfx−H, and also dG2/dz⟶0 as z⟶∞.(26)v¯2z=G2zηfx−Hdv¯2dzz=ηfx−H+∫ηfx−H∞4πσ2z0G2zz0dz0.Applying boundary condition (24), (26), and (28) collectively implies(27)2PeMηfx−HG1ηfx−Hηfx−H−G1ηfx−Hηfx−Hdw¯1dzz=ηfx−H,=G2ηfx−Hηfx−Hdv¯2dzz=ηfx−H+∫ηfx−H∞4πσ2z0G2zz0dz0.Using (23), (29) gives us(28)dv¯2dzz=ηfx−H=2/PeMηfx−HG1ηfx−H/ηfx−H−∫ηfx−H∞4πσ2z0G2ηfx−H/z0dz0G2ηfx−H/ηfx−H+A1/PG1ηfx−H/ηfx−H,where A1 is defined in the appendix.According to (23), (29) yields(29)dw¯1dzz=ηfx−H=2/PeMηfx−HG1ηfx−H/ηfx−H−∫ηfx−H∞4πσ2z0G2ηfx−H/z0dz0G1ηfx−H/ηfx−H+P/A1G2ηfx−H/ηfx−H.With the aid of (31), the value of dw¯1/dz and (21), and the value of 4πσ2z0, (26) results in(30)w¯1z=2/A1eMηfx−HG2ηfx−H/ηfx−HG1z/ηfx−HG1ηfx−H/ηfx−H+P/A1G2ηfx−H/ηfx−H−ϵ/L1G1z/ηfx−HG1ηfx−H/ηfx−H+P/A1G2ηfx−H/ηfx−H×∫ηfx−H∞−ξ2sinhbz−H+ξ22coshbz−H+ω2coshbz−Hv¯2+bcoshbz−Hdv¯2dz+sinhbz−Hd2v¯2dz2G2ηfx−Hz0dz0.Applying (30) in (28), we have(31)v¯2z=2/PeMηfx−HG1ηfx−H/ηfx−HG2z/ηfx−HG2ηfx−H/ηfx−H+A1/PG1ηfx−H/ηfx−H,+∫ηfx−H∞−ξ2sinhbz0−H+ξ22coshbz0−H+ω2coshbz−Hv¯2+bcoshbz−Hdv¯2dz0+sinhbz−Hd2v¯2dz02G2zηfx−Hdz0−G2ηfx−H/ηfx−HG2ηfx−H/ηfx−H+A1/PG1ηfx−H/ηfx−H×∫ηfx−H∞−ξ2sinhbz0−H+ξ22coshbz0−H+ω2coshbz0−Hv¯2+bcoshbz0−Hdv¯2dz0+sinhbz0−Hd2v¯2dz02G2ηfx−Hz0dz0.The value ofv¯2z can be obtained from (33) by the method of successive approximations. Taking the first approximation of v¯2z and neglecting the higher powers of ϵ, (33) reduces to(32)v¯2z=2/PeMηfx−HG1ηfx−H/ηfx−HG2z/ηfx−HG2ηfx−H/ηfx−H+A1/PG1ηfx−H/ηfx−H.This is the displacement at any point in the half-space.Using (34) in (32), we obtain(33)w¯2z=2/A1eMηfx−HG2ηfx−H/ηfx−HG1z/ηfx−HG1ηfx−H/ηfx−H+P/A1G2ηfx−H/ηfx−H−2ϵ/PL1G1z/ηfx−HG1ηfx−H/ηfx−HG1ηfx−H/ηfx−H+A1/PG2ηfx−H/ηfx−HG2ηfx−H/ηfx−H+A1/PG1ηfx−H/ηfx−H,+∫ηfx−H∞−ξ2sinhbz0−H+ξ22coshbz0−H+ω2coshbz−HG2z0ηfx−H+bcoshbz−Hd2G2z0/ηfx−Hdz0+sinhbz0−Hd2G2z0/ηfx−Hdz0G2ηfx−Hz0dz0.The value ofw¯1z can be evaluated by first determining the values of G1z0/ηfx−H and G2z0/ηfx−H. For this, we will find the solution satisfying (25). Let us consider two independent solutions for the following equation:(34)d2G1z/z0dz2−α2G1zz0=0,which vanish at z=−∞ and z=∞ as(35)E1z=eαz,E2z=e−αz.Therefore, the solution of (36) for an infinite medium is(36)E1zE2z0Wforz<z0,E1z0E2zWforz>z0,where(37)W=E1zE2′z−E2zE1′z=−2α.So, we can express the solution of (36) as(38)G1zz0=−e−αz−z02α.The solution of (25) can be written as(39)G1zz0=−e−αz−z02α+Aeαz+Be−αz.SinceG1z/z0 satisfy the condition dG1/dz=0 at z=0 and at z=ηfx−H. Using this, evaluating A and B, we obtain(40)G1zz0=−e−αz−z02α+eαz+e−αz2αeαηfx−H−eαηfx−H×e−αηfx−H−z0H+z0−ηfxηfx−H−z0−eαηfx−H−z0.In view of (42), the values of Green’s function for the reinforced medium are as follows:(41)G1zηfx−H=−eαz+e−αzαeαηfx−H−e−αηfx−H,(42)G1ηfx−Hηfx−H=−eαηfx−H+e−αηfx−Hαeαηfx−H−e−αηfx−H.Adopting the same procedure, we find the solution of (27) as(43)G2zz0=−e−mz−z02m+e−mz0+z−2ηfx−H2m,G2ηfx−Hz0=−e−mz0+z−2ηfx−Hm,G2ηfx−Hηfx−H=−1m.Substituting the values from (43), (44), and (45) in (35), it may be deduced to(44)w¯1z=−2/mA1e−Mz−ηfx+Heαz+e−αz/αeαηfx−H−e−αηfx−HM0+N0×1+ϵ/αL1eαηfx−H+e−αηfx−H/eαηfx−H−e−αηfx−HFmM0+N0,where M0=αeαηfx−H+e−αηfx−H/eαz−e−αz and N0=P/mA1, and F is given in appendix.Neglecting higher powers ofϵ, we obtain(45)w¯1z=−2/mA1e−Mz−ηfx+Heαz+e−αz/αeαηfx−H−e−αηfx−HM0+N0−ϵ/αL1eαηfx−H+e−αηfx−H/eαηfx−H−e−αηfx−HFme−ιξxdξ.Applying inverse Fourier transform on (47) defined in (11), we have(46)w1z=−2∫−∞∞e−Mz−ηfx+H1/mA1eαz+e−αz/αeαηfx−H−e−αηfx−HM0+N0−ϵ/αL1eαηfx−H+e−αηfx−H/eαηfx−H−e−αηfx−HFme−ιξxdξ.Settingw1z=v1zeMz in (48), we get(47)v1z=−2∫−∞∞e−Mz−ηfx+H1/mA1eαz+e−αz/αeαηfx−H−e−αηfx−HM0+N0−ϵ/αL1eαηfx−H+e−αηfx−H/eαηfx−H−e−αηfx−HFme−ιξxdξ.The value of the integral in (49) depends on the pole of the integrand. So, here it needs to equate the denominator to zero.(48)M0+N0−ϵαL1eαηfx−H+e−αηfx−Heαηfx−H−e−αηfx−HFm=0.Puttingα=ιkR2/P2+c2/P1c12−Q/P, c1=μT/ρ1, and m=kN1/L1−T1/2L1−c2/c22, c2=L1/ρ1 and replacing ξ by k, equation (50) may be simplified to(49)tankH−ηkh2R2P2+c2P1c12−QP,=L1A2PA3−ϵA2PA3,×−1A2+12A5A2+bA4k,+12A2+c2A2−bkA22A2A4+bA5k+ϵsinhbA5PA3,where A2, A3, A4, A5, and P1 are shown in the appendix.This is the dispersion relation of propagation of the Love wave in the reinforced medium lying over orthotropic medium under the impact of point source and irregularity.


## 5. Stability of the Model

These are special cases considered to validate the dispersion relation of Love wave propagation under the influence of the point source and irregularity obtained in (51) with the dispersion relation for propagation of classical Love-wave.Case 1.
When there is no reinforcementN1⟶L1⟶μ2 and μL⟶μT⟶μ1, we get(50)tankH−ηfxc2c12−1=μ2μ11−c2/c22c2/c12−1−ϵμ11−c2/c22c2/c22−1×−11−c2/c12+121−c2c22A5+bA4k+121−c2/c22+c21−c2/c22−bk1−c2/c222A41−c2c22+bA5k+ϵμ1sinhA5c2/c12−1.Case 2.
When the surfaces are smooth, there is no irregularity present at the interface of the mediums, i.e.,η=0, A4=1, and A5=−2b/kkH, (52) becomes(51)tankHc2c12−1=μ2μ11−c2/c22c2/c22−1−ϵμ11−c2/c22c2/c22−1×−11−c2/c22+1−4bH1−c2c22+bk+121−c2/c22+c21−c2/c22−bk1−c2/c2221−c2c22+bk−2bkkH−ϵμ1sinh2bHc2/c22−1.Case 3.
If heterogeneity becomes homogeneity in the medium, then (52) is obtained as(52)tankHc2c12−1=μ2μ11−c2/c22c2/c22−1.
So, (52) verifies the dispersion relation of this problem with the classical Love-wave dispersion relation.


## 6. Numerical Computations and Discussions

Numerical computations are performed for (51) using the data provided in Table 1.Table 1
Rigidity and density of the anisotropic materials gubbins [32].
ReinforcedOrthotropicRigidityμL=5.66×1010N/m2N1=5.82×1010N/m2μT=2.46×1010N/m2L1=3.99×1010N/m2Densityρ1=7.8×103kg/m2ρ1=4.5×103kg/m2Dispersion curve (51) reveals the chemical properties of the reinforced and orthotropic materials with frequency and phase velocity of the wave. Figures 2–9 demonstrate the relation of frequency and wave number for different chemical properties of the materials under the consideration of irregularity in the materials, while Figures 10–13 demonstrated in the absence of irregularity at the interface of the materials. Figure 2 shows the influence of irregularity (between the materials) on the phase velocity c2/c22 with wave number kH. According to this graph, the phase velocity of the wave increases as the length of the rectangular plate (irregularity) grows kh/2=0.2,0.4,0.6,0.8. The phase velocity has been found to be influenced by the size of the interfacial irregularity, and hence the chemical characteristics of the materials are also influenced. One of the most important factors in understanding the chemical processes of orthotropic materials during anti-plane wave propagation is inhomogeneity. Figure 3 demonstrates the effect of functionally graded elastic properties of orthotropic materials on propagation speed. The phase velocity of the wave falls with respect to wave numbers as the magnitude of the parameter b/k grows. It has been visually examined in Figures 4 and 10 that the influence of the heterogeneity parameter ϵ/μT on the phase velocity remains invariant in the presence and absence of the irregularity. In each figures, the phase velocity increases for low frequency and reduces for high frequency. Figures 5 and 11 demonstrate the effects of initial stress parameter T1/2L1 on the phase velocity of the wave. The moderate effect of the parameter T1/2L1 is seen in these figures, phase velocity increases slightly as the magnitude of the initial stress increases. In the absence and presence of rectangular irregularity at the materials’ interface, the influence of the parameter T1/2L1 remains invariant, according to our findings.Figure 2
Effect of the depthkh/2 of irregularity on the phase velocity c2/c22 against frequency kH.Figure 3
Variation of dimensionless phase velocityc2/c22 against nondimensional frequency kH under the impact of the heterogeneity parameter (b/k=0.1,0.3,0.5).Figure 4
Variation of dimensionless phase velocityc2/c22 against nondimensional frequency kH under the impact of the heterogeneity parameter with ϵ/μT=0.2,0.4,0.6,0.8.Figure 5
Dimensionless phase velocityc2/c22 against dimensionless frequency kH under the effect of initial stress present with T1/2L1=0.2,0.4,0.6,0.8.Figure 6
Dimensionless phase velocityc2/c22 against nondimensional kH with varying values of reinforced parameters.Figure 7
Dimensionless phase velocityc2/c22 against nondimensional kH with varying values of reinforced parameters.Figure 8
Dimensionless phase velocityc2/c22 against nondimensional frequency kH with varying values of the perturbation parameter (η=0.2,0.4,0.6,0.8).Figure 9
Variation of dimensionless phase velocityc2/c22 against nondimensional frequency kH under the effect of the heterogeneity parameter b/k=0.1,0.3,0.5 without irregularity.Figure 10
Dimensionless phase velocityc2/c22 against nondimensional frequency kH in the absence of irregularity under the effect of the heterogeneity parameter ϵ/μT=0.2,0.4,0.6,0.8.Figure 11
Variation of dimensionless phase velocity against nondimensional frequencykH in the absence of irregularity and η=0 under the effect of the initial stress parameter T1/2L1=0.2,0.4,0.6,0.8.Figure 12
Dimensionless phase velocityc2/c22 against nondimensional kH with the varying values of reinforced parameters in the absence of irregularity.Figure 13
Dimensionless phase velocityc2/c22 against nondimensional kH with different values of reinforced parameters without irregularity.Figures6 and 12 show the major effect of chemical characteristics of reinforce materials under the effect of irregularity and without irregularity. The wave’s phase velocity increases rapidly as the reinforce parameter a32 is increased. Geologists can use this relationship between reinforcing parameters and phase velocity to determine the composition of natural minerals. The fluctuation of phase velocity with respect to frequency is depicted in Figures 7 and 13. With rising values of a12 and lowering values of a32, the reinforced parameters have a significant effect on phase velocity, i.e., (0.75, 0.25), 0.65,0.35, 0.55,0.45, and 0.45,0.55 are the values of a12,a32 for curve 1, curve 2, curve 3, and curve 4, respectively. For these settings, the phase velocity declines significantly for kH≤3.5 and then decreases monotonically with rising frequency values. The phase velocity for the lowest value of a12 and the greatest value of a32 declines quicker than for the other values.Figure8 is an attempt to study the impact of irregularity on phase velocity of the Love wave. In this case, phase velocity increases slightly as the amount of the irregularity parameter η increases. Figure 9 renders the effect of heterogeneity parameter b/k in the absence of the irregularity η=0. Effect of the parameter b/k on the phase velocity is significantly influenced by the irregularity parameter. In the presence of irregularity, the phase velocity decreases slightly as the magnitude of the parameter b/k grows, while the phase velocity increases as the value of b/k increases in the absence of the irregularity parameter. It has been noticed that, the irregularity between the materials must be considered during the exploration of the natural minerals.


## 7. Conclusions

Anti-plane wave propagation in a hybrid earth structure consisting of a reinforced layer and functionally graded orthotropic half-space with irregularity and external impulsive forces is theoretically investigated in this paper. External impulsive force is taken in terms of Dirac delta function and the nonhomogeneous equation of motion has been solved by using the Green’s function approach. The dispersion relation of anti-plane wave propagation has been derived and deduced into the standard form. A discussion of the dispersion relation in the absence and presence of the irregularity was carried out in detail. Graphical results indicate that the chemical properties of the materials really influenced by irregularity of the interface. Reinforced parametersa12 and a32, initial stress T1/2L1, coefficient of heterogeneity ϵ/μT, and heterogeneity parameter b/k have been observed in the presence and absence of the irregularity. It has been noticed that b/k has a profound effect on the phase velocity with and without irregularity. The performance of the parameter b/k on the phase velocity is recorded different in case of irregular and regular interface of the materials, whereas the performance of other parameters on the phase velocity is moderate in both cases.It is also pointed out that the irregularity between materials should not be ignored when analyzing the chemical properties of materials through dispersion relation of the anti-plane wave propagation in the hybrid structure. Due to computational complexity, the group velocity of the wave is not discussed in this study. The obtained results of the current study are fundamental and can provide the sufficient path for exploring the possible natural minerals in such hybrid structures.



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