SH-Wave at a Plane Interface between Homogeneous and Inhomogeneous Fibre-Reinforced Elastic Half-Spaces

The problem of reflection and refraction of SH-waves at a plane interface between the homogeneous and inhomogeneous fibrereinforced elastic half-spaces has been investigated. Amplitude and energy ratios corresponding to the reflected and refracted SHwaves are derived using appropriate boundary conditions. These ratios are computed numerically for a particular model and the results are depicted graphically.


Introduction
Fibre-reinforced composite material with the reinforcement distributed continuously in concentric circles is a locally transversely isotropic material, with the circumferential direction as the preferred direction coinciding with fibres directions.The fibres may be continuous, in which each fibre extends through a body from one boundary to another, or discontinuous, but in the discontinuous case the length of the fibre must be large compared to its diameter.The elastic fibrereinforced composite materials, for example, typical carbon fibre-epoxy resin composites, are not just anisotropic but also strongly anisotropic in the sense that the modulus for extension in the fibre direction greatly exceeds the moduli for extension in the transverse direction and for shear in the fibre or transverse direction.Fibres have an excellent potential to improve the mechanical properties of rapid-setting materials and could be used effectively to improve the performance of repairs.The behavior of fibre-reinforced rapid-setting materials is similar to that of Portland cement fibre-reinforced concrete.Spencer [1] gave the concept of deformation in fibre-reinforced elastic materials.Belfield et al. [2] discussed the problem of stress in elastic plates reinforced by fibers lying in concentric circles and explained the anisotropic characters of fibre-reinforced materials.Sengupta and Nath [3] studied the problem of surface waves in fibre-reinforced anisotropic elastic media and derived the frequency equation for surface wave.Chattopadhyay et al. [4] investigated the problem of reflection of quasi- and quasi- waves at the plane-free and rigid boundaries of a fibre-reinforced elastic medium and obtained the phase velocity of quasi- and quasi- waves.B. Singh and S. J. Singh [5] discussed the problem of reflection of plane waves at the free surface of a fibrereinforced elastic half-space and obtained the closed form expression of the amplitude ratios for reflected  and  waves at the free surface of a fibre-reinforced, anisotropic, homogeneous, elastic half-space.Singh [6] studied the problem of propagation of plane waves in thermally conducting linear fibre-reinforced composite materials and derived the frequency equation.Abbas [7] discussed a two-dimensional problem for a fibre-reinforced anisotropic thermoelastic halfspace with energy dissipation and the results with energy dissipation and without energy dissipation are compared.Singh and Zorammuana [8] solved the problem of incident longitudinal wave at a fibre-reinforced thermoelastic halfspace and obtained the amplitude and energy ratios of the reflected waves.
A number of researchers have attempted the problem of SH-waves in elastic media.Gutenberg [9,10] explored the existence of low velocity layer in the earth mantle.Chattopadhyay and Keshri [11] discussed the propagation of shear waves along the plane surface between two different elastic media with initial stress.Singh and Tomar [12] investigated the problem of shear waves at a corrugated interface between two dissimilar fiber-reinforced elastic half-spaces using Rayleigh's method of approximation.Chattopadhyay and Singh [13] studied the problem of -type seismic waves in fibre-reinforced media and obtained the dispersion equation for the propagation of -type seismic wave in fibre-reinforced layer lying over an inhomogeneous fibre-reinforced elastic half-space.Literatures showed many problems of propagation of SH-waves and notable among them are Bath and Arroyo [14], Gupta [15], Tomar et al. [16], Kumar et al. [17], Chaudhary et al. [18,19], Emets et al. [20], Chattopadhyay and Michel [21], Tomar and Singh [22], Tomar and Kaur [23,24], Abd-Alla and Alsheikh [25], Chattopadhyay et al. [26], Singh [27], Wang and Zhao [28], and Sahu et al. [29].
Fibre-reinforced composite materials are very attractive in many engineering applications due to their high strength and low weight.In this paper, the problem of reflection and refraction of SH-waves in the homogeneous/inhomogeneous fibre-reinforced elastic half-spaces has been studied.We have obtained the amplitude and energy ratios corresponding to the reflected and refracted SH-waves using appropriate boundary conditions.These ratios are computed numerically for a particular model.

Basic Equations
The constitutive relation for a linearly fibre-reinforced elastic medium is given by Belfield et al. [2] as where   is stress tensor,   is strain tensor and is defined by ,   , , , and   are elastic constants in which   is identified as the shear modulus in transverse shear across the preferred direction and   as the shear modulus in longitudinal shear in the preferred direction,  and  are specific stress components depending upon the concrete part of the composite materials,   is Kronecker delta, and   is the component of a unit vector a = ( 1 ,  2 ,  3 ) which gives the preferred direction of fibre-reinforcement and  2 1 + 2 2 + 2 3 = 1.The equation of motion for the fibre-reinforced elastic material without body forces may be written as where   = ( 1 ,  2 ,  3 ) and  1 = ,  2 = ,  3 = .
Let us take two-dimensional problem of SH-wave propagation in -plane and the preferred direction of fibre-reinforcement is taken as ( 1 , 0,  3 ).We may take  1 =  3 = 0,  2 =  2 (, , ).With the help of ( 1) and (3), we have where It may be noted that the second term in the expression of the stress tensors ( 12 and  23 ) contributes due to the effect of direction of reinforcement, that is, fibre orientation in the self-reinforced material.
Using the stress tensors into (4), we obtain the equation of motion for SH-wave propagation in the fibre-reinforced elastic medium as where

Problem Formulation
Let and -axes of the Cartesian coordinate system be on the horizontal plane and let -axis be pointing vertically downward.Consider a homogeneous fibre-reinforced elastic half-space, {; 0 ≤  < ∞} with  as density and   ,   as elastic constants and a nonhomogeneous fibre-reinforced elastic half-space, {  ; −∞ <  ≤ 0} with   as density and    ,    as elastic constants.The elastic constants and density in the nonhomogeneous fibre-reinforced elastic half-space   are defined as (see [13]) where  is small positive constant and  is real depth parameter.
The stress tensors in the nonhomogeneous fibre-reinforced elastic half-space (  ) are Indian Journal of Materials Science 3 The equation of motion for SH-wave in the half-space,   , may be written as where The displacement components in the half-spaces,  and   , may be represented by where ⟨, ⟩ are amplitude constants, (= ) is angular frequency, and p = ( 1 , 0,  3 ) is the unit propagation vector.Using ( 12) into ( 6) and (10), the phase velocities of SHwaves in the half-spaces,  and   , are, respectively, given by The expression of   2 depending on  may be written as (i) zeroth order, ( 0 ): (ii) first order, ( 1 ): (iii) second order, ( 2 ): and so on.We will use expression (15) of   2 for the numerical computation.

Wave Propagation
Suppose a plane SH-wave with amplitude,  0 , making an angle  0 with the normal be incident at the plane interface ( = 0) between homogeneous and inhomogeneous fibrereinforced elastic half-spaces.Such an incident SH-wave, due to the interface, give rises a reflected SH-wave in the halfspace, , and a refracted SH-wave in the half-space,   .The geometry of the problem is given in Figure 1.
The total displacement due to incident and reflected SHwaves in the half-space, , is given by where  is the amplitude constant of the reflected SH-wave at an angle  1 ,   = /  ( = 0, 2) is the wavenumber,  is angular frequency, and   is the phase speed.
The displacement due to refracted SH-wave in the halfspace,   , is given by where   2 = /  2 is the wavenumber with   2 as the phase speed and  is the amplitude constant of the refracted SH-wave at an angle  2 .Snell's law of this problem is given by We assume that the angle of incidence is equal to the angle of reflection.

Boundary Conditions
The boundary conditions are the continuities of the displacements and stress tensors at the interface.Mathematically, these conditions at  = 0 are Equation ( 22) may be written in terms of displacement as Using ( 18)-( 20) into boundary conditions ( 21) and ( 23), we have where These equations will give the amplitude ratios corresponding to the reflected and refracted SH-waves.

Amplitude and Energy Ratio
The amplitude ratios of the reflected and refracted SH-waves are defined as the ratio of the amplitudes corresponding to the reflected and refracted waves to that of the incident wave.Solving (24), we have where / 0 is the amplitude ratios corresponding to the reflected SH-wave and / 0 is that of the refracted SH-wave.
We have observed that these ratios depend on the elastic constants, fibre orientation, inhomogeneity parameter, and the angle of incidence.Let us consider the energy partition of the reflected and refracted SH-waves due to the plane interface,  = 0.The energy transmission per unit area may be given as (see [30]) The expression of the energy due to incident wave is given by where The energy ratio of the reflected and refracted SH-waves may be defined as the ratios of the energy corresponding to the reflected and refracted SH-waves to that of the incident wave.The modulus of energy ratios of the reflected and refracted SH-waves is given as Indian Journal of Materials Science 5 where  1 is the energy ratio of the reflected SH-wave and  2 is that of refracted SH-wave and the expressions of  1 and  2 are given by We come to know that the energy ratios are functions of the amplitude ratios, elastic constants, fibre orientation, inhomogeneity parameter, and the angle of incidence of the incident SH-wave.

Particular Cases
7.1.Case I.When the inhomogeneous fibre-reinforced elastic half-space,   , reduces to homogeneous fibre-reinforced elastic medium, the problem reduces to the reflection and refraction of SH-waves at the plane interface between the two dissimilar homogeneous fibre-reinforced elastic half-spaces.Under this condition,  = 0 and ( 14) reduces to The amplitude and energy ratios corresponding to the reflected and refracted SH-waves are given by ( 26) and ( 29) with the following modified values: These results are similar to those of Singh and Tomar [12].

Case II.
When the two half-spaces  and   reduce to homogeneous isotropic elastic media, the problem reduces to the reflection and refraction of SH-waves between two dissimilar isotropic elastic half-spaces.Under this condition, The amplitude and energy ratios corresponding to the reflected and refracted SH-waves are given by ( 26) and ( 29) with the following modified values: These results exactly match with that of Achenbach [30].

Numerical Results and Discussion
For the numerical computations, we take the following relevant parameters for fibre-reinforced elastic half-space, , and the nonhomogeneous fibre-reinforced elastic half-space,   [13].(36) The variation of the modulus of amplitude ratios corresponding to the reflected and refracted SH-waves with the angle of incidence,  0 , is depicted in Figures 2 and 3    values and decreases with the increase of  0 and attains the minimum value at the grazing angle of incidence.We have observed that the sum of energy ratios is close to unity.

Conclusion
The problem of reflection and refraction of SH-waves at the plane interface between the homogeneous fibre-reinforced half-space and the nonhomogeneous fibre-reinforced halfspace has been investigated.The amplitude and energy ratios corresponding to the reflected and refracted SH-waves have  been obtained and computed numerically.We may conclude the following points: (i) The amplitude and energy ratios corresponding to the reflected and refracted SH-waves are functions of elastic constants, fibre orientation, inhomogeneity parameter, and angle of incidence.
(ii) The amplitude ratio, / 0 , and energy ratio,  1 , attain the maximum value at grazing angle of incidence.
(iii) The values of / 0 and  1 decrease with the increase of .
(iv) The values of / 0 and  1 with fibre-reinforcement are greater than those values without fibre-reinforcement.
(v) The values of / 0 and  2 increase with the increase of .
(vi) The sum of energy ratios is close to unity.

Figure 1 :
Figure 1: Geometry of the problem.
with and without fibre-reinforcement at different values of inhomogeneity parameter, .In Figures 2(a) and 2(b), the amplitude Indian Journal of Materials Science  = 0.0 II:  = 0.1 III:  = 0.2 (b) In the absence of reinforcement

Figure 4 :
Figure 4: Variation of the modulus of energy ratio,  1 with  0 .

Figure 5 :
Figure 5: Variation of the modulus of energy ratio,  2 with  0 .