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

Authors: C. Zorammuana; S. S. Singh
Journal: Indian Journal of Materials Science
 (2015)

---

## Abstract

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


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## Body

## 1. 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 fibre-reinforced 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-P and quasi-SV waves at the plane-free and rigid boundaries of a fibre-reinforced elastic medium and obtained the phase velocity of quasi-P and quasi-SV waves. B. Singh and S. J. Singh [5] discussed the problem of reflection of plane waves at the free surface of a fibre-reinforced elastic half-space and obtained the closed form expression of the amplitude ratios for reflected qSV and qP 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 half-space 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 half-space and obtained the amplitude and energy ratios of the reflected waves.A number of researchers have attempted the problem ofSH-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 G-type seismic waves in fibre-reinforced media and obtained the dispersion equation for the propagation of G-type seismic wave in fibre-reinforced layer lying over an inhomogeneous fibre-reinforced elastic half-space. Literatures showed many problems of propagation ofSH-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 ofSH-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 refractedSH-waves using appropriate boundary conditions. These ratios are computed numerically for a particular model.


## 2. Basic Equations

The constitutive relation for a linearly fibre-reinforced elastic medium is given by Belfield et al. [2] as(1)τij=λekkδij+2μTeij+αakamekmδij+ekkaiaj+2μL-μTaiakekj+ajakeki+βakamekmaiaj,i,j,k,m=1,2,3,where τij is stress tensor, eij is strain tensor and is defined by (2)eij=12ui,j+uj,i,λ, μT, α, β, and μL are elastic constants in which μT is identified as the shear modulus in transverse shear across the preferred direction and μL as the shear modulus in longitudinal shear in the preferred direction, α and β are specific stress components depending upon the concrete part of the composite materials, δij is Kronecker delta, and ai is the component of a unit vector a=(a1,a2,a3) which gives the preferred direction of fibre-reinforcement and a12+a22+a32=1.The equation of motion for the fibre-reinforced elastic material without body forces may be written as(3)τij,j=ρ∂2ui∂t2,i,j=1,2,3,where ui=(u1,u2,u3) and x1=x, x2=y, x3=z.Let us take two-dimensional problem ofSH-wave propagation in xz-plane and the preferred direction of fibre-reinforcement is taken as (a1,0,a3). We may take u1=u3=0, u2=u2(x,z,t). With the help of (1) and (3), we have(4)∂τ12∂x+∂τ23∂z=ρ∂2u2∂t2,where(5)τ12=μT∂u2∂x+μL-μTa1a1∂u2∂x+a3∂u2∂z,τ23=μT∂u2∂z+μL-μTa3a1∂u2∂x+a3∂u2∂z.It may be noted that the second term in the expression of the stress tensors (τ12 and τ23) 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 forSH-wave propagation in the fibre-reinforced elastic medium as(6)P∂2u2∂z2+Q∂2u2∂x2+R∂2u2∂x∂z=ρ∂2u2∂t2,where(7)P=μT+a32μL-μT,Q=μT+a12μL-μT,R=2a1a3μL-μT.


## 3. Problem Formulation

Letx- and y-axes of the Cartesian coordinate system be on the horizontal plane and let z-axis be pointing vertically downward. Consider a homogeneous fibre-reinforced elastic half-space, {H;0≤z<∞} with ρ as density and μL, μT as elastic constants and a nonhomogeneous fibre-reinforced elastic half-space, {H′;-∞<z≤0} with ρ′ as density and μL′, μT′ as elastic constants. The elastic constants and density in the nonhomogeneous fibre-reinforced elastic half-space H′ are defined as (see [13])(8)μL′=μL01-ɛcos⁡sz,μT′=μT01-ɛcos⁡sz,ρ′=ρ01-ɛcos⁡sz,where ɛ is small positive constant and s is real depth parameter.The stress tensors in the nonhomogeneous fibre-reinforced elastic half-space (H′) are(9)τ12′=μT′∂u2′∂x+μL′-μT′a1a1∂u2′∂x+a3∂u2′∂z,τ23′=μT′∂u2′∂z+μL′-μT′a3a1∂u2′∂x+a3∂u2′∂z.The equation of motion forSH-wave in the half-space, H′, may be written as(10)P01-ɛcos⁡sz∂2u2′∂z2+ɛssin⁡sz∂u2′∂z+Q01-ɛcos⁡sz∂2u2′∂x2+R021-ɛcos⁡sz∂2u2′∂x∂z+ɛssin⁡sz∂u2′∂x=ρ01-ɛcos⁡sz∂2u2′∂t2,where(11)P0=μT0+a32μL0-μT0,Q0=μT0+a12μL0-μT0,R0=a1a3μL0-μT0.The displacement components in the half-spaces, H and H′, may be represented by(12)u2,u2′x,z,t=A,Bexp⁡ıωt-kxp1+zp3,where A,B are amplitude constants, ω(=kc) is angular frequency, and p=(p1,0,p3) is the unit propagation vector.Using (12) into (6) and (10), the phase velocities ofSH-waves in the half-spaces, H and H′, are, respectively, given by(13)ρc22=Pp32+Qp12+Rp1p3,(14)ρ0c2′2=P0p32+ıɛsp3sin⁡sz1-ɛcos⁡sz-1k-1+Q0p12+R02p1p3+ıɛsp1sin⁡sz1-ɛcos⁡sz-1k-1.The expression of c2′ depending on ɛ may be written as(i)
zeroth order,O(ε0):(15)ρ0c2′2=P0p32+Q0p12+R02p1p3,(ii)
first order,O(ε1):(16)
ρ
0
c
2
′
2
=
P
0
p
3
2
+
ı
ɛ
s
p
3
s
z
-
s
z
3
3
!
+
s
z
5
5
!
-
s
z
7
7
!
+
⋯
k
-
1
+
Q
0
p
1
2
+
R
0
2
p
1
p
3
+
ı
ɛ
s
p
1
s
z
-
s
z
3
3
!
+
s
z
5
5
!
-
s
z
7
7
!
+
⋯
k
-
1
,(iii)
second order,O(ε2):(17)
ρ
0
c
2
′
2
=
P
0
p
3
2
+
ı
ɛ
s
p
3
s
z
-
s
z
3
3
!
+
s
z
5
5
!
-
s
z
7
7
!
+
⋯
k
-
1
+
ı
ɛ
2
s
p
3
s
z
-
s
z
3
2
!
+
s
z
5
4
!
-
⋯
+
-
s
z
3
3
!
+
s
z
5
3
!
2
!
-
s
z
7
3
!
4
!
+
⋯
+
s
z
5
5
!
-
s
z
7
5
!
2
!
+
s
z
9
5
!
4
!
⋯
k
-
1
+
Q
0
p
1
2
+
R
0
2
p
1
p
3
+
ı
ɛ
s
p
1
s
z
-
s
z
3
3
!
+
s
z
5
5
!
-
s
z
7
7
!
+
⋯
k
-
1
+
ı
ɛ
2
s
p
1
s
z
-
s
z
3
2
!
+
s
z
5
4
!
-
⋯
+
-
s
z
3
3
!
+
s
z
5
3
!
2
!
-
s
z
7
3
!
4
!
+
⋯
+
s
z
5
5
!
-
s
z
7
5
!
2
!
+
s
z
9
5
!
4
!
⋯
k
-
1
,and so on. We will use expression (15) of c2′ for the numerical computation.


## 4. Wave Propagation

Suppose a planeSH-wave with amplitude, A0, making an angle θ0 with the normal be incident at the plane interface (z=0) between homogeneous and inhomogeneous fibre-reinforced elastic half-spaces. Such an incidentSH-wave, due to the interface, give rises a reflectedSH-wave in the half-space, H, and a refractedSH-wave in the half-space, H′. The geometry of the problem is given in Figure 1.Figure 1
Geometry of the problem.The total displacement due to incident and reflectedSH-waves in the half-space, H, is given by(18)u2=A0exp⁡ıωt-k0xsin⁡θ0+zcos⁡θ0+Aexp⁡ıωt-k2xsin⁡θ1-zcos⁡θ1,where A is the amplitude constant of the reflectedSH-wave at an angle θ1, ki=ω/ci (i=0,2) is the wavenumber, ω is angular frequency, and ci is the phase speed.The displacement due to refractedSH-wave in the half-space, H′, is given by(19)u2′=Bexp⁡ıωt-k2′xsin⁡θ2+zcos⁡θ2,where k2′=ω/c2′ is the wavenumber with c2′ as the phase speed and B is the amplitude constant of the refractedSH-wave at an angle θ2. Snell’s law of this problem is given by(20)k0sin⁡θ0=k2sin⁡θ1=k2′sin⁡θ2.We assume that the angle of incidence is equal to the angle of reflection.


## 5. Boundary Conditions

The boundary conditions are the continuities of the displacements and stress tensors at the interface. Mathematically, these conditions atz=0 are(21)u2=u2′,(22)τ23=τ23′.Equation (22) may be written in terms of displacement as(23)μT∂u2∂z+μL-μTa3a1∂u2∂x+a3∂u2∂z=μT′∂u2′∂z+μL′-μT′a3a1∂u2′∂x+a3u2′∂z.Using (18)–(20) into boundary conditions (21) and (23), we have(24)AA0-BA0=-1,r1AA0+r2BA0=r0,where(25)
r
0
=
μ
T
k
0
cos
⁡
θ
0
+
μ
L
-
μ
T
a
3
a
3
k
0
cos
⁡
θ
0
+
a
1
k
0
sin
⁡
θ
0
,
r
1
=
μ
T
k
2
cos
⁡
θ
1
+
μ
L
-
μ
T
a
3
a
3
k
2
cos
⁡
θ
1
-
a
1
k
0
sin
⁡
θ
0
,
r
2
=
μ
T
0
k
2
′
cos
⁡
θ
2
+
μ
L
0
-
μ
T
0
a
3
a
3
k
2
′
cos
⁡
θ
2
+
a
1
k
0
sin
⁡
θ
0
1
-
ɛ
.These equations will give the amplitude ratios corresponding to the reflected and refractedSH-waves.


## 6. Amplitude and Energy Ratio

The amplitude ratios of the reflected and refractedSH-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(26)AA0=r0-r2r1+r2,BA0=r0+r1r1+r2,where A/A0 is the amplitude ratios corresponding to the reflectedSH-wave and B/A0 is that of the refractedSH-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 refractedSH-waves due to the plane interface, z=0. The energy transmission per unit area may be given as (see [30])(27)℘∗=τ23·u˙2+τ23′·u2′˙.The expression of the energy due to incident wave is given by(28)Einc=J0ωA02exp⁡2ıωt-k0xsin⁡θ0+zcos⁡θ0,where J0=k0{μTcosθ0+(μL-μT)a3(a3cosθ0+a1sinθ0)}.The energy ratio of the reflected and refractedSH-waves may be defined as the ratios of the energy corresponding to the reflected and refractedSH-waves to that of the incident wave. The modulus of energy ratios of the reflected and refractedSH-waves is given as(29)E1=J1J0AA02,E2=J2J0BA02,where E1 is the energy ratio of the reflectedSH-wave and E2 is that of refractedSH-wave and the expressions of J1 and J2 are given by (30)
J
1
=
-
k
2
μ
T
cos
⁡
θ
1
+
μ
L
-
μ
T
a
3
a
3
cos
⁡
θ
1
-
a
1
sin
⁡
θ
1
,
J
2
=
k
2
′
μ
T
0
cos
⁡
θ
2
+
μ
L
0
-
μ
T
0
a
3
a
3
cos
⁡
θ
2
+
a
1
sin
⁡
θ
2
1
-
ɛ
.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 incidentSH-wave.


## 7. Particular Cases

### 7.1. Case I

When the inhomogeneous fibre-reinforced elastic half-space,H′, reduces to homogeneous fibre-reinforced elastic medium, the problem reduces to the reflection and refraction ofSH-waves at the plane interface between the two dissimilar homogeneous fibre-reinforced elastic half-spaces. Under this condition, ɛ=0 and (14) reduces to (31)ρ0c2′2=P0p32+Q0p12+2R0p1p3.The amplitude and energy ratios corresponding to the reflected and refractedSH-waves are given by (26) and (29) with the following modified values: (32)
r
0
=
μ
T
k
0
cos
⁡
θ
0
+
μ
L
-
μ
T
a
3
a
3
k
0
cos
⁡
θ
0
+
a
1
k
0
sin
⁡
θ
0
,
r
1
=
μ
T
k
2
cos
⁡
θ
1
+
μ
L
-
μ
T
a
3
a
3
k
2
cos
⁡
θ
1
-
a
1
k
2
sin
⁡
θ
1
,
r
2
=
μ
T
0
k
2
′
cos
⁡
θ
2
+
μ
L
0
-
μ
T
0
a
3
a
3
k
2
′
cos
⁡
θ
2
+
a
1
k
2
′
sin
⁡
θ
2
,
J
1
=
-
k
2
μ
T
cos
⁡
θ
1
+
μ
L
-
μ
T
a
3
a
3
cos
⁡
θ
1
-
a
1
sin
⁡
θ
1
,
J
2
=
k
2
′
μ
T
0
cos
⁡
θ
2
+
μ
L
0
-
μ
T
0
a
3
a
3
cos
⁡
θ
2
+
a
1
sin
⁡
θ
2
.These results are similar to those of Singh and Tomar [12].


### 7.2. Case II

When the two half-spacesH and H′ reduce to homogeneous isotropic elastic media, the problem reduces to the reflection and refraction ofSH-waves between two dissimilar isotropic elastic half-spaces. Under this condition, (33)P0=Q0=μT0=μL0=μ0,P=Q=μT=μL=μ,R0=R=0,c22=μρ,c2′2=μ0ρ0.The amplitude and energy ratios corresponding to the reflected and refractedSH-waves are given by (26) and (29) with the following modified values: (34)r0=μk0cos⁡θ0,r1=μk2cos⁡θ1,r2=μ0k2′cos⁡θ2,J1=-k2μcos⁡θ1,J2=k2′μ0cos⁡θ2.These results exactly match with that of Achenbach [30].


## 7.1. Case I

When the inhomogeneous fibre-reinforced elastic half-space,H′, reduces to homogeneous fibre-reinforced elastic medium, the problem reduces to the reflection and refraction ofSH-waves at the plane interface between the two dissimilar homogeneous fibre-reinforced elastic half-spaces. Under this condition, ɛ=0 and (14) reduces to (31)ρ0c2′2=P0p32+Q0p12+2R0p1p3.The amplitude and energy ratios corresponding to the reflected and refractedSH-waves are given by (26) and (29) with the following modified values: (32)
r
0
=
μ
T
k
0
cos
⁡
θ
0
+
μ
L
-
μ
T
a
3
a
3
k
0
cos
⁡
θ
0
+
a
1
k
0
sin
⁡
θ
0
,
r
1
=
μ
T
k
2
cos
⁡
θ
1
+
μ
L
-
μ
T
a
3
a
3
k
2
cos
⁡
θ
1
-
a
1
k
2
sin
⁡
θ
1
,
r
2
=
μ
T
0
k
2
′
cos
⁡
θ
2
+
μ
L
0
-
μ
T
0
a
3
a
3
k
2
′
cos
⁡
θ
2
+
a
1
k
2
′
sin
⁡
θ
2
,
J
1
=
-
k
2
μ
T
cos
⁡
θ
1
+
μ
L
-
μ
T
a
3
a
3
cos
⁡
θ
1
-
a
1
sin
⁡
θ
1
,
J
2
=
k
2
′
μ
T
0
cos
⁡
θ
2
+
μ
L
0
-
μ
T
0
a
3
a
3
cos
⁡
θ
2
+
a
1
sin
⁡
θ
2
.These results are similar to those of Singh and Tomar [12].


## 7.2. Case II

When the two half-spacesH and H′ reduce to homogeneous isotropic elastic media, the problem reduces to the reflection and refraction ofSH-waves between two dissimilar isotropic elastic half-spaces. Under this condition, (33)P0=Q0=μT0=μL0=μ0,P=Q=μT=μL=μ,R0=R=0,c22=μρ,c2′2=μ0ρ0.The amplitude and energy ratios corresponding to the reflected and refractedSH-waves are given by (26) and (29) with the following modified values: (34)r0=μk0cos⁡θ0,r1=μk2cos⁡θ1,r2=μ0k2′cos⁡θ2,J1=-k2μcos⁡θ1,J2=k2′μ0cos⁡θ2.These results exactly match with that of Achenbach [30].


## 8. Numerical Results and Discussion

For the numerical computations, we take the following relevant parameters for fibre-reinforced elastic half-space,H, and the nonhomogeneous fibre-reinforced elastic half-space, H′ [13].For the homogeneous half-spaceH,(35)μL=7.07×109N/m2,μT=3.5×109N/m2,ρ=1600Kg/m3.For the nonhomogeneous half-space H′,(36)μL0=5.66×109N/m2,μT0=2.46×109N/m2,ρ0=7800Kg/m3,a1=0.00316227,ɛ=0.0,0.1,0.2.The variation of the modulus of amplitude ratios corresponding to the reflected and refractedSH-waves with the angle of incidence, θ0, is depicted in Figures 2 and 3 with and without fibre-reinforcement at different values of inhomogeneity parameter, ɛ. In Figures 2(a) and 2(b), the amplitude ratio, A/A0, corresponding to the reflectedSH-wave starts from certain values and increases with the increase of θ0 which attains the maximum value at the grazing angle of incidence. The values of A/A0 decrease with the increase of inhomogeneity parameter.Figure 2
Variation of the modulus of amplitude ratio,A/A0 with θ0.
(a)
In the presence of reinforcement
(b)
In the absence of reinforcementFigure 3
Variation of the modulus of amplitude ratio,B/A0 with θ0.
(a)
In the presence of reinforcement
(b)
In the absence of reinforcementAll Curves I, II, and III in Figures3(a) and 3(b) show that B/A0 corresponding to the refractedSH-wave decreases with the increase of θ0 and attains the minimum value at the grazing angle of incidence. The values of B/A0 increase with the increase of inhomogeneity parameter. Figures 4 and 5 show the variation of the modulus of energy ratios corresponding to the reflected and refractedSH-waves with the angle of incidence with and without fibre-reinforcement at different values of inhomogeneity parameter, ɛ. In Figures 4(a) and 4(b), E1 corresponding to the reflectedSH-wave starts from certain values and increases up to θ0=88° and decreases to θ0=89° which increases thereafter with the increase of θ0.Figure 4
Variation of the modulus of energy ratio,E1 with θ0.
(a)
In the presence of reinforcement
(b)
In the absence of reinforcementFigure 5
Variation of the modulus of energy ratio,E2 with θ0.
(a)
In the presence of reinforcement
(b)
In the absence of reinforcementThe modulus of energy ratio,E2, corresponding to the refractedSH-wave in Figures 5(a) and 5(b) starts from certain values and decreases with the increase of θ0 and attains the minimum value at the grazing angle of incidence. We have observed that the sum of energy ratios is close to unity.


## 9. Conclusion

The problem of reflection and refraction ofSH-waves at the plane interface between the homogeneous fibre-reinforced half-space and the nonhomogeneous fibre-reinforced half-space has been investigated. The amplitude and energy ratios corresponding to the reflected and refractedSH-waves have been obtained and computed numerically. We may conclude the following points:(i)
The amplitude and energy ratios corresponding to the reflected and refractedSH-waves are functions of elastic constants, fibre orientation, inhomogeneity parameter, and angle of incidence.(ii)
The amplitude ratio,A/A0, and energy ratio, E1, attain the maximum value at grazing angle of incidence.(iii)
The values ofA/A0 and E1 decrease with the increase of ɛ.(iv)
The values ofA/A0 and E1 with fibre-reinforcement are greater than those values without fibre-reinforcement.(v)
The values ofB/A0 and E2 increase with the increase of ɛ.(vi)
The sum of energy ratios is close to unity.



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*Source: 532939-2015-07-09.xml*