Effects of a magnetic field and initial stress on reflection of SV-waves at a free surface with voids under gravity

1 Introduction

Seismic wave propagation has gained much attention in the last few years for study, because of their relevance to many difference applications in several branches of science and technology, such as earthquake science, geophysics and optics. The general equation of reflection and refraction of elastic waves at a plane half-space was first derived by Knott [1] and subsequently developed by Jefferys [2], Gutenberg [3] and many others but they did not take initial stress in the solid half-space, into account. Generally, several media are initially stressed due to many physical causes. The earth is also an initially stressed medium. Biot [4] has proved that the initial stress has significant effect on the propagation of elastic waves. Latterly, other authors like Dey and Addy [5] included the effect of initial stress on the reflection of waves at the solid half-space.

In the classical theory of elasticity, voids are an important generalization. Nunziato and Cowin [6] and Cowin and Nunziato [7] discussed the theory in elastic media with voids. Puri and Cowin [8] studied the effects of voids on plane waves in linear elastic media and it is evident that pure shear waves remain unaffected by the presence of pores. Chandrasekharaiah [9] and [10] discussed the effects of voids on propagation of plane and surface waves. Abo-Dahab [11] and Atwa et al. [12] investigated the propagation of P-waves from stress-free surface elastic half-space with voids.

Wave propagation in the presence of initial stress and voids is of great significance in numerous fields such as earthquake engineering, soil dynamics, composite materials in aeronautics, astronautics and nuclear reactors, etc. The investigation of wave propagation in an isotropic generalized solid in the presence of initial stress, voids and a magnetic field under the influence of gravity yields information about the existence of new or modified waves. This information may be efficacious for experimental seismologists in correcting earthquake estimation.

This paper investigates the problem of the reflection of a plane SV-wave incident on an initially stressed, homogenous, isotropic solid half-space with voids under the influence of gravity and magnetic field. In this study, firstly, the stress-strain relations with incremental isotropy introduced by Biot [4] are used, and then a modified void equation was used. Furthermore, the governing equations are solved analytically for two-dimensional motions in the x₁x₂-plane to obtain the expressions for displacement potentials and velocities. Applying the free surface boundary conditions, expressions for the reflection coefficients are derived. Numerical computation for a chosen material is also discussed.

2 Formulation and solution of the problem

Governing equations with initial stress and a magnetic field under the influence of gravity in a homogenous elastic medium are as follows:

1. The equations of motion:

   τij,j+F⇀i+G⇀i=ρu¨i,

   where F⇀=μ0(J_×H_),G⇀i=(ρgε3jkuj,k)δij,In component form, equation of motion becomes,

   (1)τ11,1+τ12,2+F1−ρgu2,1=ρu¨1,

   (2)τ21,1+τ22,2+F2+ρgu1,1=ρu¨1.

2. The equation for voids:

   (3)αφ,ii−ω0φ−υφ˙−βui,i=ρκφ¨,

3. Constitutive relations:

   (4)τij=−P(δij+ϖij)+λεkkδij+2μεij+βδijφ, where εij=12(ui,j+uj,i),ϖij=12(ui.j−uj,i).

We take the linearized Maxwell equations governing the electromagnetic field for a perfectly conducting medium as follows:

where H = H₀ + h, h is the induced magnetic force and ε_o is the electric permeability. Initial applied magnetic field H₀ = (0, 0, H₀), i.e. taken along the x₃ axis and the material lies in x₁x₂ plane.

Thus, H = H₀ + h = (h₁, h₂, h₃ + H₀).

Then the magnetic force is as follows

where e = u_1.1 + u_2,2.

In these equations, F_i represents the magnetic force, J is the current density, H is the magnetic field vector and μ_o is the magnetic permeability. φ is the so-called volume fraction field. α, β, ω₀, ν and κ are new material constants characterizing the presence of voids. Where ε_ijk is the Levi-Civita tensor, τ_ij are the components of stress, ρ is the mass density and u_i is the displacement vector. λ and μ are elastic constants and u_i is the displacement component. A comma followed by the index shows a partial derivative with respect to a coordinate. Also, Einstein’s summation convention over repeated indexes is used.

Here we consider a half space which is a homogenous and isotropic elastic solid. The x₁x₂ half-space is chosen to coincide with the free surface with initial compressive stress P in the x₁ direction. A plane wave incident at “0” on the boundary surface in the x₁x₂ plane at x_{2 =0}, making an angle θ₀, with the normal to the boundary as shown in Figure 1.

Figure 1:

Schematic of the problem.

Substituting equations (3) into (1), we have

The modified voids equation is as follows:

Using Helmholtz’s theorem, the displacement vector u→ can be written in the displacement potentials ϕ and ψ form

which reduces to

By using (8 and 9) into equation (5), we have

By using (8 and 9) in equation (6), we have

where

Substituting (8 and 9) into (7), we have

The solutions of equations (10–13) can be taken as:

Using (14–16) in (10), (11) and (13), we have

Eliminating ϕ₀, ψ₀, and φ₀ from equations (17–19), we have

where

It is obvious from (20) that it has three roots which corresponds three velocities for reflected waves.

3 Reflection coefficients

There are three reflected waves, P-wave, SV-wave and a wave due to voids. Thus, if a rotational wave falls on boundary x₂ = 0 from the solid half space we have one reflected rotational wave and two reflected compressional waves traveling with two different velocities. Accordingly, if the normal wave of the incident rotational wave makes an angle θ₀ with the positive x₂-axis and those of reflected SV-, P- and void-waves make angles θ₁, θ₂ and θ₃ with the same direction. The displacement potential and the volume fraction field take the following forms:

where

where, A₀ is the amplitude of the incident P-wave and A₁, A₂ and A₃ are the amplitudes of reflected SV-, P- and wave due to voids, respectively.

4 Boundary conditions

As the boundary at x₂ = 0 is adjacent to a vacuum, it is free from surface tractions, therefore

where, Maxwell’s stresses are as follows:

It is also assumed that there is no change in volume traction, φ, along the x₂-direction, thus

Using equations (21–23) in (24–26), we get

where

and

5 Numerical results and discussion

With the view of computational work, we take the following physical constants.

λ=5.65×1010 Nm−2,μ=2.46×1010 Nm−2,ρ=2.66×103 kgm−3,α=−1.28×1010 Nm−2,β=220.90×1010 Nm−2.

Using these values, the modulus of the reflection coefficients for the SV-wave and P-wave have been calculated for different angles of incidence.

We plotted a graph to show the variation of reflection coefficient R_C1 of the P-wave with the variation of magnetic field H₀ with respect to the angle of incidence θ₀ keeping κ, ω, α, Ω, P, ε₀, ω₀ constant (see Figure 2A). From this graph, it can be seen, that R_C1 decreases with the increase in θ₀. It is also observed that reflection coefficient decreases as H₀ increases. Figure 2B shows that reflection coefficient R_C2 of the SV-wave is unaffected by the magnetic field, whereas Figure 2C shows that reflection coefficient decreases as H₀ increases. There will be no reflection for the angle of incidence θ0=π2 and reflection coefficients have a maximum value as the angle of incidence θ₀→0 and reflection does not exist at θ₀ = 0.

Figure 2:

Variations of a magnetic field H = 0.1___, 0.3–.–., 0.5– – – on the magnitude of amplitude ratios with respect to the angle of incidence.

Furthermore, in Figure 3A we show that the variation of reflection coefficient R_C1 of the P-wave with the variation of initial stresses P with respect to the angle of incidence θ₀ keeping κ, ω, α, H₀, Ω, ε₀, ω₀ as constants. From this figure, it can be observed that R_C1 decreases with the increase in θ₀. It is also shown that reflection coefficient increases as P increases. Figure 3B shows that reflection coefficient R_C2 of the SV-wave is unaffected by initial stresses, whereas Figure 3C shows that reflection coefficient increases as P increases. There will be no reflection for angle of incidence θ0=π2 and reflection coefficients are maximum for angle of incidence θ₀ = 0.

Figure 3:

Variations of initial stress P = 10¹⁰___, 2(10¹⁰)–.–., 3(10¹⁰)– – – on the magnitude of amplitude ratios with respect to the angle of incidence.

In Figure 4A, we show the variation of reflection coefficient R_C1 of the P-wave with the variation of electric field ε₀ with respect to the angle of incidence θ₀ keeping κ, ω, α, H₀ Ω, P, ω₀ constant. It is observed that R_C1 decreases with the increase in ε₀. It is also noted that reflection coefficient increases as ε₀ increases. In Figure 4A and B, we plotted these graphs to show that reflection coefficient R_C2 of the SV-wave is unaffected by an electric field, whereas, Figure 4C shows that reflection coefficient increases as ε₀ increases. There will be no reflection for angle of incidence θ0=π2 and reflection coefficients have maximum value near the angle of incidence θ₀ = 0 and reflection does not exist at θ₀ = 0.

Figure 4:

Variations of electric field ε = 0.01____, 0.02–.–., 0.04– – – on the magnitude of amplitude ratios with respect to the angle of incidence.

6 Conclusion

The reflection of SV-waves on a free surface under a magnetic field, initial stress and electric field with voids was studied. In addition, expressions for reflection coefficients for P-waves, SV-waves and waves due to voids are derived. Numerical results for a chosen material, aluminum, for different parameters are given and illustrated graphically. It is observed that a magnetic field, initial stress and electric field significantly affect the reflection coefficients. In the absence of voids, the results reduce to well-known isotropic medium.