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

To propagate the anti-plane wave, a hybrid layered-half-space schematic of ber-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. e functionally graded orthotropic materials in the half-space and the reinforced materials under high initial stress are utilized in the supercial layer of nite depth. To examine the eect of heterogeneity on wave phase velocity, the elastic characteristics of orthotropic materials are dened in terms of hyperbolic functions. Moreover, a rectangular plate is installed at the interface of the materials to expose the eect 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 inuence of such a force on the phase velocity of the wave. e 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. e dispersion relation of the anti-plane wave propagation is obtained analytically. is study presents a graphic explanation of the proposed schematic and discusses the stability of the model.


Introduction
Anti-plane wave propagation through layered schematic gives immense knowledge about the earth's interior for possible natural minerals. e dispersion relation of an antiplane wave depends on the chemical properties ( uid-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. e propagation of Love waves in reinforced-orthotropic heterogeneous composite materials under the in uence of impulsive forces and uneven interfaces is a new study eld 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][2][3][4]. e e ect of the reinforcement, di erent 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 ber-reinforced, self-reinforced, void porous, and orthotropic) have been investigated in [5][6][7][8][9][10][11][12][13][14][15]. Mal [16] demonstrated the in uence 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 thickness H 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 h 1 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. e 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. e 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. e effect of varying parameters, initial stress, irregularity, and impulsive forces on the phase velocity of the Love wave are studied numerically and manifested graphically. e mechanism of the wave propagation and the numerical data for the elastic constants of the materials are taken from [28][29][30][31][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. is paper is organized as follows: in Section 2, 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.

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 velocity c 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.
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 as h 1 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 η � h 1 /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. e shear moduli of the orthotropic medium has been assumed as where N, L, ρ and L 1 , N 1 , ρ 1 are the shear moduli of the homogeneous and heterogeneous mediums, respectively. T and T 1 are the associated initial stresses of the homogeneous and heterogeneous orthotropic medium, respectively, and ϵ and b are the heterogeneity parameters. e aim of this paper is to obtain the dispersion relation of the anti-plane wave in the hybrid structure to demonstrate the e ect of initial stress, heterogeneity, reinforcement, irregularity, and anisotropy on the phase velocity and wave numbers for the fundamental mode of propagation.

Dynamics of the Materials
where τ (i,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 F (i) are the body forces. As wave is propagating horizontally along the x-axis: u w 0, v v(x, z, t) and z/zy 0.
Considering the impact of the point source in the reinforced medium, the equation of motion is where μ T is transverse shear modulus and μ L is longitudinal shear modulus considered in a preferred direction. σ (1) (r, t) is the force density disturbance with the e ect of the point source. Force is acting at a distance of r from origin at a time t.
(a 1 , a 2 , a 3 ) are the components of a → in the direction of reinforcement such that a 2 1 + a 2 2 + a 2 in (4), we obtain where ω kc is the angular frequency.
Here the impulsive force σ (1) (r) cause some disturbances which can be represented as σ (1) So the above equation of motion is and inverse Fourier transform as Imposing Fourier transform on (9), we get where In order to make the equation free of a rst derivative term, set v (1) (12), it becomes where 3.2. Dynamics for the Orthotropic Layer. e equation of motion for the orthotropic medium of in nite depth is where N and L are directional rigidity, T represents initial stress, and ρ represents density of the half-space. Using (2), we get v (2) in (16), and we get With Fourier transform on v (2) (x, z), (18) takes the form of ODE as where

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. ese equations also satisfies the boundary conditions at the prescribed boundaries at z � 0 and at z � ηf(x) − H. e boundary conditions are as follows: (ii) e stress is continuous at the common interface Let G 1 (z/z 0 ) be Green's function for the reinforced medium satisfying the conditions dG 1 Multiplying (14) by G 1 (z/z 0 ) and (25) by w (1) (z), subtracting and integrating from 0 to ηf(x) − H, then replacing z 0 by z and using the symmetric property For half-space of the orthotropic medium, G 2 (z/z 0 ) be Green's function that satisfies (19), then G 2 (z/z 0 ) is the solution given by Applying boundary condition (24), (26), and (28) collectively implies Using (23), (29) gives us where A 1 is defined in the appendix. According to (23), (29) yields With the aid of (31), the value of [dw (1) /dz] and (21), and the value of 4πσ (2) (z 0 ), (26) results in Applying (30) in (28), we have Mathematical Problems in Engineering v (2) e value of v (2) (z) can be obtained from (33) by the method of successive approximations. Taking the first approximation of v (2) (z) and neglecting the higher powers of ϵ, (33) reduces to v (2) is is the displacement at any point in the half-space. Using (34) in (32), we obtain , e value of w (1) (z) can be evaluated by first determining the values of G 1 (z 0 /ηf(x) − H) and G 2 (z 0 /ηf(x) − H). For this, we will find the solution satisfying (25). Let us consider two independent solutions for the following equation: which vanish at z � −∞ and z � ∞ as erefore, the solution of (36) for an infinite medium is where 6 Mathematical Problems in Engineering So, we can express the solution of (36) as e solution of (25) can be written as Since G 1 (z/z 0 ) satisfy the condition dG 1 /dz � 0 at z � 0 and at z � ηf(x) − H. Using this, evaluating A and B, we obtain In view of (42), the values of Green's function for the reinforced medium are as follows: Adopting the same procedure, we find the solution of (27) as 2m , Substituting the values from (43), (44), and (45) in (35), it may be deduced to where M 0 � (α(e α(ηf(x)− H) + e − α(ηf(x)− H) )/e αz − e −αz ) and N 0 � P/mA 1 , and F is given in appendix.
Neglecting higher powers of ϵ, we obtain Applying inverse Fourier transform on (47) defined in (11), we have Setting w (1) Putting α � ιk ���� � L 1 /ρ 1 and replacing ξ by k, equation (50) may be simplified to where A 2 , A 3 , A 4 , A 5 , and P 1 are shown in the appendix. is 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.

Stability of the Model
ese 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 reinforcement
Case 3. If heterogeneity becomes homogeneity in the medium, then (52) is obtained as So, (52) verifies the dispersion relation of this problem with the classical Love-wave dispersion relation.

Numerical Computations and Discussions
Numerical computations are performed for (51) using the data provided in Table 1.
Dispersion curve (51) reveals the chemical properties of the reinforced and orthotropic materials with frequency and phase velocity of the wave.  Figure 2 shows the influence of irregularity (between the materials) on the phase velocity c 2 /c 2 2 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).
e 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. e 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 in uence of the heterogeneity parameter ϵ/μ T on the phase velocity remains invariant in the presence and absence of the irregularity. In each gures, the phase velocity increases for low frequency and reduces for high frequency. Figures 5 and 11 demonstrate the e ects of initial stress parameter T 1 /2L 1 on the phase velocity of the wave. e moderate e ect of the parameter T 1 /2L 1 is seen in these gures, phase velocity increases Table 1: Rigidity and density of the anisotropic materials gubbins [32].  slightly as the magnitude of the initial stress increases. In the absence and presence of rectangular irregularity at the materials' interface, the in uence of the parameter T 1 /2L 1 remains invariant, according to our ndings. Figures 6 and 12 show the major e ect of chemical characteristics of reinforce materials under the e ect of irregularity and without irregularity. e wave's phase velocity increases rapidly as the reinforce parameter a 2 3 is increased. Geologists can use this relationship between reinforcing parameters and phase velocity to determine the composition of natural minerals. e uctuation of phase velocity with respect to frequency is depicted in ) for curve 1, curve 2, curve 3, and curve 4, respectively. For these settings, the phase velocity declines signi cantly for kH ≤ 3.5 and then decreases monotonically with rising frequency values. e phase velocity for the lowest value of a 2 1 and the greatest value of a 2 3 declines quicker than for the other values. Figure 8 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 e ect of heterogeneity parameter b/k in the absence of the irregularity (η 0). E ect of the parameter b/k on the phase velocity is signi cantly in uenced 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.

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. e 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 parameters (a 2 1 and a 2 3 ), initial stress (T 1 /2L 1 ), coefficient of heterogeneity (ϵ/μ T ), 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. e 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. e obtained results of the current study are fundamental and can provide the sufficient path for exploring the possible natural minerals in such hybrid structures.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.