# Influence of Initial Stress and Gravity Field on Propagation of Rayleigh and Stoneley Waves in a Thermoelastic Orthotropic Granular Medium

Authors: S. M. Ahmed; S. M. Abo-Dahab
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

The propagation of Rayleigh and Stoneley waves in a thermoelastic orthotropic granular half-space supporting a different layer under the influence of initial stress and gravity field is studied. The frequency equation of Rayleigh waves in the form of twelfth-order determinantal expression and the frequency equation of Stoneley waves in the form of eighth-order determinantal expression are obtained. The standard equation of dispersion is discussed to obtain Rayleigh and Stoneley waves that have complex roots; the real part gives the velocity of Rayleigh or Stoneley waves but the imaginary part gives the attenuation coefficient. Finally, the numerical results have been given and illustrated graphically, and their physical meaning has been explained.


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## Body

## 1. Introduction

The propagation of thermoelastic waves in a granular medium under initial stress and gravity field has applications in soil mechanics, earthquake science, geophysics, mining engineering, and so forth. The theoretical outline of the development of the subject from the mid-thirties was given by Paria [1]. The present paper, however, is based on the dynamics of granular media as propounded by Oshima [2, 3]. The medium under consideration is discontinuous such as one composed numerous large or small grains. Unlike a continuous body, each element or grain cannot only translate but also rotate about its centre of gravity. This motion is the characteristics of the medium and has an important effect upon the equation of motion to produce internal friction. It is assumed that the medium contains so many grains that they will never be separated from each other during the deformation, and that the grain has perfect elasticity. The propagation of Rayleigh waves in granular medium was given by many authors such as Bhattacharyya [4], El-Naggar [5], Ahmed [6, 7], and others. Ahmed [8] discussed Stoneley waves in a nonhomogeneous granular medium under the influence of gravity.The problem of Stoneley waves plays an important role in the earthquake science, optics, geophysics, and plasma physics. Many authors such as Abd-Alla and Ahmed [9, 10], El-Naggar et al. [11], Das et al. [12], and others investigated the effect of gravity of the propagation of surface waves (Stoneley waves and Rayleigh waves) in an elastic solid medium. Goda [13] illustrated the effect of inhomogeneity and anisotropy on Stoneley waves; Abd-Alla et al. [14] discussed Rayleigh waves in a magnetoelastic half-space of orthotropic material under influence of initial stress and gravity field.  Sharma et al. [15] studied the propagation characteristics of Rayleigh waves in transversely isotropic piezothermoelastic half-space with rotation and thermal relaxation. Sharma et al. [16] investigated the problem of propagation characteristics of Rayleigh waves in transversely isotropic piezothermoelastic materials. Sharma and Walia [17] investigated Rayleigh waves in piezothermoelastic materials. Ting [18] illustrated the surface waves in a rotating anisotropic elastic half-space. Abd-Alla and Abo-Dahab [19] investigated Rayleigh waves in magneto-thermo-viscoelastic solid with thermal relaxation times. Recently, Ahmed and Abo-Dahab [20] pointed out Propagation of Love waves in an orthotropic granular layer under initial stress overlying a semi-infinite granular medium. Vinh and Seriani [21] illustrated explicit secular equations of Rayleigh waves in a nonhomogeneous orthotropic elastic medium under the influence of gravity. Vinh and Seriani [22] discussed the explicit secular equations of Stoneley waves in a nonhomogeneous orthotropic elastic medium under the influence of gravity. Vinh [23] discussed the explicit secular equations of Rayleigh waves in elastic media under the influence of gravity and initial stress. Lo [24] investigated the propagation and attenuation of Rayleigh waves in a semi-infinite unsaturated poroelastic medium. El-Naggar [25] investigated the dynamical problem of a generalized thermoelastic granular infinite cylinder under initial stress. Recently, Abd-Alla et al. [26] investigated Rayleigh waves in generalized magneto-thermo-viscoelastic granular medium under the influence of rotation, gravity field.This paper is devoted to study the effect of gravity field and the initial stress on the propagation of Rayleigh and Stoneley waves in thermoelastic orthotropic granular half-space supporting a different layer under initial stress and gravity field. The frequency equations are obtained: the frequency equation of Rayleigh waves in the form of twelfth-order determinantal expression and the frequency equation of Stoneley waves in the form of eighth-order determinantal expression. The standard equation of dispersion is discussed to obtain the Rayleigh and Stoneley waves that have complex roots; the real part gives the velocity of Rayleigh or Stoneley waves but the imaginary part gives the attenuation coefficient. The results obtained are displayed graphically and their physical meaning has been explained.


## 2. Formulation of the Problem

Let us consider an initially stressed orthotropic granular layer of finite thicknessH overlaying a semi-infinite orthotropic granular medium. The upper surface of the upper layer is assumed to be free and horizontal. We take a set of orthogonal Cartesian axes Ox1x2x3 such that the interface and the free surface of the granular layer resting on the granular half-space of different material are the planes x3=H and  x3=0, respectively, with the origin O being any point on the interface surface; x3-axis is positive in the direction towards the exterior of the half-space, and x1-axis is positive along the direction of Rayleigh waves and Stoneley waves propagation. Let the both media be under initial compression stress P along x1-axis, with the influence of gravity and at initial temperature T0. It is assumed that both media exchange heat freely with their surroundings; an initial stress is produced by a slow process of creep, where the shear stresses tend to become small or vanish after a long interval of time.In view of the two-dimensional nature of the problem, we assume that the state of initial stress is(2.1)τ11=τ33=τ,τ13=0,
the equilibrium conditions of the initial stress field are given by [12](2.2)∂τ∂x1=0,∂τ∂x3-ρg=0.The state of deformation in the granular medium is described by the displacement vectorU⃗=(u1,0,u3) of the centre of gravity of a grain and the rotation vector ξ⃗=(ξ,η,ζ) of the grain about its centre of gravity. There exist a stress tensor and a couple stress which are nonsymmetric, that is,(2.3)τij≠τji,Mij≠Mji(i,j=1,2,3).The stress tensorτij can be expressed as the sum of symmetric and antisymmetric tensors(2.4)τij=σij+σij′,
where (2.5)σij=12(τij+τji),σij′=12(τij-τji).The symmetric tensorσij=σji is related to the symmetric strain tensor (2.6)eij=eji=12(∂ui∂xj+∂uj∂xi)
by the Hook's law.The antisymmetric stressσij′ is given by (2.7)σ23′=-F∂ξ∂t,σ31′=-F∂η∂t,σ12′=-F∂ζ∂t,σ11′=σ22′=σ33′=0,
the couple stress Mij is given by (2.8)Mij=Mνij,ν11=∂ξ∂x1,ν22=0,ν33=∂ζ∂x3,ν23=0,ν31=∂ξ∂x3,ν12=∂∂x1(η+ω2),ν32=∂∂x3(η+ω2),ν13=∂ζ∂x1,ν21=0,
where,   ω2=∂u1/∂x3-∂u3/∂x1.The six equations of motion are [9, 11, 12](2.9)∂τ11∂x1+∂τ31∂x3-P2∂ω2∂x3-ρg∂u3∂x1=ρ∂2u1∂t2,∂τ12∂x1+∂τ32∂x3=0,∂τ13∂x1+∂τ33∂x3-P2∂ω2∂x1+ρg∂u1∂x1=ρ∂2u3∂t2,τ23-τ32+∂M11∂x1+∂M31∂x3=0,τ31-τ13+∂M12∂x1+∂M32∂x3=0,τ12-τ21+∂M13∂x1+∂M33∂x3=0.The components of stress for orthotropic body, under the effect of an initial compression stressP, are given by [11] (2.10)τ11=(c11+P)∂u1∂x1+(c13+P)∂u3∂x3-υ1T,τ33=c13∂u1∂x1+c33∂u3∂x3-υ3T,τ31=c55(∂u1∂x3+∂u3∂x1)-F∂η∂t,τ13=c55(∂u1∂x3+∂u3∂x1)+F∂η∂t,
where (2.11)υ1=(c11+c12)α1+c13α2,υ3=2c13α1+c33α2.Substituting (2.9) into (2.10), we obtain(2.12)(c11+P)∂2u1∂x12+(c55+P2)∂2u1∂x32+(c13+c55+P2)∂2u3∂x1∂x3-∂∂x1(υ1T)-ρg∂u3∂x1-F∂∂t(∂η∂x3)=ρ∂2u1∂t2,F∂∂t(∂ξ∂x3-∂ζ∂x1)=0,(c55+c13+P2)∂2u1∂x1∂x3+(c55-P2)∂2u3∂x12+c33∂2u3∂x32-∂∂x3(υ3T)+ρg∂u1∂x1+F∂∂t(∂η∂x1)=ρ∂2u3∂t2,-2F∂ξ∂t+∇2(Mξ)=0,-2F∂η∂t+∇2[M(η+∂u1∂x3-∂u3∂x1)]=0,-2F∂ζ∂t+∇2(Mζ)=0.
The heat conduction equation is given by [11](2.13)∇2T-1χ∂T∂t-ε∇⋅(∂U⃗∂t)=0,
where χ=(δ1+δ2)/2ρs,ε=T0(υ1+υ3)/(δ1+δ2).


## 3. Solution of the Problem

By Helmholtz's theorem [27], the displacement vector u⃗ can be written in the form of the potentials ϕ(x1,x3,t) and ψ(x1,x3,t) which are related to the displacement components u1 and u3 by the relations(3.1)u1=∂φ∂x1-∂ψ∂x3,u3=∂φ∂x3+∂ψ∂x1.Substituting (3.1) into (2.12) and (2.13), we get the following wave equations satisfied by φ,ψ,ξ,η, and ζ:(3.2)(c11+P)∂2φ∂x12+(c13+2c55+P)∂2φ∂x32-ρg∂ψ∂x1-υ1T=ρ∂2φ∂t2,(3.3)∂∂t(∂ζ∂x1-∂ξ∂x3)=0,(3.4)(c55-P2)∂2ψ∂x12+(c33-c31-c55-P2)∂2ψ∂x32+ρg∂ϕ∂x1+F∂η∂t=ρ∂2ψ∂t2,(3.5)∇2ξ-S∂ξ∂t=0,(3.6)∇2η-S∂η∂t-∇4ψ=0,(3.7)∇2ζ-S∂ζ∂t=0,(3.8)∇2T-1χ∂T∂t-ε∇2(∂ϕ∂t)=0,(3.9)S=2FM.
Eliminating η from (3.4) and (3.6), we get(3.10)(∇2-S∂∂t)[(c55-P2)∂2ψ∂x12+(c33-c31-c55-P2)∂2ψ∂x32-ρ∂2ψ∂t2+ρg∂ϕ∂x1]+F∇4(∂ψ∂t)=0.
Also, T can be eliminated by using (3.8) and (3.9) as follows:(3.11)(∇2-1χ∂∂t)[(c11+P)∂2φ∂x12+(c13+2c55+P)∂2φ∂x32-ρg∂ψ∂x1-ρ∂2φ∂t2]-υ1ε∇2(∂ϕ∂t)=0.Assuming that(3.12)(φ,ψ)={φ1(x3),ψ1(x3)}exp⁡{i(Lx1-bt)},(3.13)(ξ,η,ζ)={ξ1(x3),η1(x3),ζ(x3)}exp⁡{i(Lx1-bt)},
where b is real positive and L is in general complex.Substituting (3.13) into (3.3), (3.5), and (3.7), we get(3.14)Dξ1-iLζ1=0,(3.15)D2ξ1+h2ξ1=0,(3.16)D2ζ1+h2ζ1=0,
where h2=ibS-L2,D≡d/dx.Solutions of (3.15) and (3.16) are(3.17)ξ1=A1eihx3+A2e-ihx3,ζ1=B1eihx3+B2e-ihx3,
respectively.From (3.14) and (3.17), we obtain(3.18)h(A1eihx3-A2e-ihx3)-L(B1eihx3-B2e-ihx3)=0.
Equating the coefficients of eihx3 and e-ihx3 to zero in (3.18), we get(3.19)A1=LhB1,A2=-LhB2.Substituting (3.12) and (3.13) into (3.10) and (3.11), we get(3.20)[(c33-c13-c55-P2-ibF)D4+{(ibS-L2)(c33-c13-c55-P2)-L2(c55-P2)+ρb2+2ibFL2}D2+(ibS-L2)(ρb2-L2(c55-P2)-ibFL4)]ψ1+iρgL[D2-L2+ibS]ϕ1=0,(3.21){ibχ(c13+2c55+P)D4+[ρb2-L2(c11+c13+2c55+2P)+ibυ1ε+ibχ(c13+2c55+P)]D2+[(ibχ-L2)(ρb2-L2(c11+P))-ibυ1εL2]}ϕ1-iρgL[D2-L2+ibχ]ψ1=0.The solution of (3.20) and (3.21) has the form(3.22)ϕ1=Aje-iλjx3+Bjeiλjx3,ψ1=Eje-iλjx3+Fjeiλjx3(j=3,4,5,6),
where the constants Aj,Bj are related with the constants Ej,Fj, respectively, by means of (3.20) or (3.21),  and λj(j=3,4,5,6) are taken to be imaginary.Equating the coefficients ofe-iλjx3,eiλjx3(j=3,4,5,6) to zero, we have using(3.21)(3.23)Ej=mjAj,Fj=mjBj(j=3,4,5,6),
where (3.24)mj=1iρgL[-λj2-L2+ib/χ]×{ibχibχ(c13+2c55+P)λj4-[ρb2-L2(c11+c13+2c55+2P)+ibυ1ε+ibχ(c13+2c55+P)]λj2+(ibχ-L2)(ρb2-L2(c11+P))-ibυ1εL2},
where λ3,λ4,λ5,λ6 are the imaginary roots of the equation(3.25)k8λ8+k6λ6+k4λ4+k2λ2+k0=0,(3.26)k8=(c13+2c55+P)(c33-c13-c55-P2-ibF),k6=-(ρb2-L2(c11+c13+2c55+2P)+ibυ1ε+ibχ(c13+2c55+P))×(c33-c13-c55-P2-ibF)-(c13+2c55+P)×[(ibS-L2)(c33-c13-c55-P2)-L2(c55-P2)+ρb2+2ibFL2],k4=(ibχ-L2)(ρb2-L2(c11+P)-ibυ1εL2)(c33-c13-c55-P2-ibF)+(c13+2c55+P)(ibS-L2)[ρb2-L2(c55-P2)-ibFL4]+[(ibS-L2)(c33-c13-c55-P2)+ρb2-L2(c55-P2)+2ibFL2]×[ρb2-L2(c11+c13+2c55+2P)+ibυ1ε+ibχ(c13+2c55+P)]-ρ2g2L2,k2=[(ibS-L2)(c33-c13-c55-P2)-L2(c55-P2)+ρb2+2ibFL2]×[(ibχ-L2)(ρb2-L2(c11+P))-ibυ1εL2]+[ρb2-L2(c11+c13+2c55+2P)+ibυ1ε+ibχ(c13+2c55+P)]×(ibS-L2)[ρb2-L2(c55-P2)-ibFL4]-ρ2g2L2(2L2-ibs-ibχ).k0=(ibS-L2)[ρb2-L2(c55-P2)-ibFL4]×[(ibχ-L2)(ρb2-L2(c11+P))-ibυ1εL2]-ρ2g2L2(ibχ-L2)(ibS-L2).Using (3.2), (3.3), (3.12), (3.13), (3.22), and (3.23), we get    (3.27)ϕ=(Aje-iλjx3+Bjeiλjx3)exp⁡(i(Lx1-bt)),ψ=mj(Aje-iλjx3+Bjeiλjx3)exp⁡(i(Lx1-bt))(j=3,4,5,6)
From (3.2) and (3.27), the temperature T has the form(3.28)T=nj(Aje-iλjx3+Bjeiλjx3)exp⁡(i(Lx1-bt))(j=3,4,5,6),
where(3.29)nj=-1υ1[(c13+2c55+P)λj2+L2(c11+P)+iLρgmj-ρb2]
also, from (3.6) and (3.27),  η has the form(3.30)η=Ωjmj(Aje-iλjx3+Bjeiλjx3)exp⁡(i(Lx1-bt))(j=3,4,5,6),
where  (3.31)Ωj=-(λj2+L2)2λj2+L2-ibS(j=3,4,5,6).We use the symbols with a bar for the quantities in the lower medium (exceptx3,L,b,P,g), and by assuming that the solution of (3.20) and (3.21) satisfies the condition that the corresponding stresses vanish as x3→-∞, we obtain(3.32)ξ1¯=-Lh¯B2¯e-ih¯x3,ζ1¯=B2¯e-ih¯x3,η1¯=Ω¯mj¯Aj¯e-iλj¯x3,ϕ1¯=Aj¯e-iλj¯x3,ψ1¯=mj¯Aj¯e-iλj¯x3,T¯=nj¯Aj¯exp⁡(i(Lx1-bt-λj¯x3))(j=3,4,5,6).


## 4. Boundary Conditions and Frequency Equation

The boundary conditions on the interfacex3=0  are(4.1)(i)u1=u1¯,(ii)u3=u3¯,(iii)ξ=ξ¯,(iv)η=η¯,(v)ζ=ζ¯,(vi)M33=M33¯,(vii)M31=M31¯,(viii)M32=M32¯,(ix)τ33=τ33¯,(x)τ31=τ31¯,(xi)τ32=τ32¯,(xii)T=T¯,(xiii)∂T∂x3+θT=∂T¯∂x3+θ¯T¯.
The boundary conditions on the free surface x3=H are(4.2)(xiv)M33=0,(xv)M31=0,(xvi)M32=0,(xvii)τ33=0,(xviii)τ31=0,(xix)τ32=0,(xx)∂T∂x3+θT=0,
where(4.3)M33=M∂ζ∂x3,M32=M∂∂x3(η-∇2ψ),M31=M∂ξ∂x3,τ33=c13∂2φ∂x12+c33∂2φ∂x32+(c33-c13)∂2ψ∂x1∂x3-υ3T,τ32=-F∂ξ∂t,τ31=c55(∂2ψ∂x12-∂2ψ∂x32+2∂2φ∂x1∂x3)-F∂η∂t,θ is the ratio of the coefficients of heat transfer to the thermal conductivity.From the boundary conditions(iii),(v),(vi), and (vii), we get(4.4)B1eihH-B2e-ihH=-B2¯e-ih¯H,B1eihH+B2e-ihH=-B2¯eih¯H,M(B1eihH-B2e-ihH)=-M¯B2¯e-ih¯H,M(B1eihH+B2e-ihH)=-M¯B2¯eih¯H,
hence(4.5)B1=B2=B2¯=0,ξ=ζ=ξ¯=ζ¯=0.The other significant boundary conditions are responsible for the following relations:(4.6)(xvi)(λj2+Ωj+L2)(e-iλjHAj-eiλjHBj)=0,(xvii)(c13L(L+mjλj)+c33λj(λj-Lmj)+υ3nj)e-iλjHAj+(c13L(L-mjλj)+c33λj(λj+Lmj)+υ3nj)eiλjHBj=0,(xviii)[c55(λj2-L2)mj+2Lλj+ibFΩjmj]e-iλjHAj+[c55(λj2-L2)mj-2Lλj+ibFΩjmj]eiλjHBj=0,(i)(L+mjλj)Aj+(L-mjλj)Bj=(L+mj¯λj¯)Aj¯,(ii)(Lmj-λj)Aj+(Lmj+λj)Bj=(Lmj¯-λj¯)Aj¯,(iv)Ωjmj(Aj+Bj)=Ωj¯mj¯Aj¯,(viii)Mmjλj(λj2+Ωj+L2)(Aj-Bj)=M¯mj¯λj¯(λj2¯+Ωj¯+L2),(ix)[c13L(L+mjλj)+c33λj(λj-Lmj)+υ3nj]Aj+[c13L(L-mjλj)+c33λj(λj+Lmj)+υ3nj]Bj=[c13¯L(L+mj¯λj¯)+c33¯λj¯(λj¯-Lmj¯)+υ3¯nj¯]Aj¯,(x){c55[mj(L2+λj2)-2Lλj]-ibFmjΩj}Aj+{c55[mj(L2+λj2)+2Lλj]-ibFmjΩj}Bj={c55¯[mj¯(L2+λj2¯)-2Lλj¯]-ibF¯mj¯Ωj¯}Aj¯,(xii)njAj+njBj=nj¯Aj¯,(xiii)nj(θ-iλj)Aj+nj(θ+iλj)Bj=nj¯(θ¯-iλj¯)Aj¯,(xx)nj(θ-iλj)e-iλjHAj+nj(θ+iλj)eiλjHBj=0,(j=3,4,5,6).Elimination ofAj,Bj,Aj¯(j=3,4,5,6) gives the wave velocity equation in the form(4.7)det⁡⋅(drc)=0(r,c=1,2,…,12),
where the nonvanishing entries of the twelfth-order determinant of drc are given by(4.8)d11=(λ32+Ω3+L2)e-iλ3H,d12=(λ42+Ω4+L2)e-iλ4H,d13=(λ52+Ω5+L2)e-iλ5H,d14=(λ62+Ω6+L2)e-iλ6H,d15=-(λ32+Ω3+L2)eiλ3H,d16=-(λ42+Ω4+L2)eiλ4H,d17=-(λ52+Ω5+L2)eiλ5H,d18=-(λ62+Ω6+L2)eiλ6H,d21=(c13L(L+m3λ3)+c33λ3(λ3-Lm3)+υ3n3)e-iλ3H,d22=(c13L(L+m4λ4)+c33λ4(λ4-Lm4)+υ3n4)e-iλ4H,d23=(c13L(L+m5λ5)+c33λ5(λ5-Lm5)+υ3n5)e-iλ5H,d24=(c13L(L+m6λ6)+c33λ6(λ6-Lm6)+υ3n6)e-iλ6H,d25=(c13L(L-m3λ3)+c33λ3(λ3+Lm3)+υ3n3)eiλ3H,d26=(c13L(L-m4λ4)+c33λ4(λ4+Lm4)+υ3n4)eiλ4H,d27=(c13L(L-m5λ5)+c33λ5(λ5+Lm5)+υ3n5)eiλ5H,d28=(c13L(L-m6λ6)+c33λ6(λ6+Lm6)+υ3n6)eiλ6H,d31=[c55(λ32-L2)m3+2Lλ3+ibFΩ3m3]e-iλ3H,d32=[c55(λ42-L2)m4+2Lλ4+ibFΩ4m4]e-iλ4H,d33=[c55(λ52-L2)m5+2Lλ5+ibFΩ5m5]e-iλ5H,d34=[c55(λ62-L2)m6+2Lλ6+ibFΩ6m6]e-iλ6H,d35=[c55(λ32-L2)m3-2Lλ3+ibFΩ3m3]eiλ3H,d36=[c55(λ42-L2)m4-2Lλ4+ibFΩ4m4]eiλ4H,d37=[c55(λ52-L2)m5-2Lλ5+ibFΩ5m5]eiλ5H,d38=[c55(λ62-L2)m6-2Lλ6+ibFΩ6m6]eiλ6H,d41=L+m3λ3,d42=L+m4λ4,d43=L+m5λ5,d44=L+m6λ6,d45=L-m3λ3,d46=L-m4λ4,d47=L-m5λ5,d48=L-m6λ6,d49=-(L+m3¯λ3¯),d410=-(L+m4¯λ4¯),d411=-(L+m5¯λ5¯),d412=-(L+m6¯λ6¯),d51=Lm3-λ3,d52=Lm4-λ4,d53=Lm5-λ5,d54=Lm6-λ6,d55=Lm3+λ3,d56=Lm4+λ4,d57=Lm5+λ5,d58=Lm6+λ6,d59=-Lm3¯+λ3¯,d510=-Lm4¯+λ4¯,d511=-Lm5¯+λ5¯,d512=-Lm6¯+λ6¯,d61=Ω3m3,d62=Ω4m4,d63=Ω5m5,d64=Ω6m6,d65=Ω3m3,d66=Ω4m4,d67=Ω5m5,d68=Ω6m6,d69=-Ω3¯m3¯,d610=-Ω4¯m4¯,d611=-Ω5¯m5¯,d612=-Ω6¯m6¯,d71=Mm3λ3(λ32+Ω3+L2),d72=Mm4λ4(λ42+Ω4+L2),d73=Mm5λ5(λ52+Ω5+L2),d74=Mm6λ6(λ62+Ω6+L2),d75=-Mm3λ3(λ32+Ω3+L2),d76=-Mm4λ4(λ42+Ω4+L2),d77=-Mm5λ5(λ52+Ω5+L2),d78=-Mm6λ6(λ62+Ω6+L2),d79=-M¯m3¯λ3¯(λ32¯+Ω3¯+L2),d710=-M¯m4¯λ4¯(λ42¯+Ω4¯+L2),d711=-M¯m5¯λ5¯(λ52¯+Ω5¯+L2),d712=-M¯m6¯λ6¯(λ62¯+Ω6¯+L2),d81=c13L(L+m3λ3)+c33λ3(λ3-Lm3)+υ3n3,d82=c13L(L+m4λ4)+c33λ4(λ4-Lm4)+υ3n4,d83=c13L(L+m5λ5)+c33λ5(λ5-Lm5)+υ3n5,d84=c13L(L+m6λ6)+c33λ6(λ6-Lm6)+υ3n6,d85=c13L(L-m3λ3)+c33λ3(λ3+Lm3)+υ3n3,d86=c13L(L-m4λ4)+c33λ4(λ4+Lm4)+υ3n4,d87=c13L(L-m5λ5)+c33λ5(λ5+Lm5)+υ3n5,d88=c13L(L-m6λ6)+c33λ6(λ6+Lm6)+υ3n6,d89=-[c13¯L(L+m3¯λ3¯)+c33¯λ3¯(λ3¯-Lm3¯)+υ3¯n3¯],d810=-[c13¯L(L+m4¯λ4¯)+c33¯λ4¯(λ4¯-Lm4¯)+υ3¯n4¯],d811=-[c13¯L(L+m5¯λ5¯)+c33¯λ5¯(λ5¯-Lm5¯)+υ3¯n5¯],d812=-[c13¯L(L+m6¯λ6¯)+c33¯λ6¯(λ6¯-Lm6¯)+υ3¯n6¯],d91=c55[m3(L2+λ32)-2Lλ3]-ibFm3Ω3,d92=c55[m4(L2+λ42)-2Lλ4]-ibFm4Ω4,d93=c55[m5(L2+λ52)-2Lλ5]-ibFm5Ω5,d94=c55[m6(L2+λ62)-2Lλ6]-ibFm6Ω6,d95=c55[m3(L2+λ32)+2Lλ3]-ibFm3Ω3,d96=c55[m4(L2+λ42)+2Lλ4]-ibFm4Ω4,d97=c55[m5(L2+λ52)+2Lλ5]-ibFm5Ω5,d98=c55[m6(L2+λ62)+2Lλ6]-ibFm6Ω6,d99=-{c55¯[m3¯(L2+λ32¯)-2Lλ3¯]-ibF¯m3¯Ω3¯},d910=-{c55¯[m4¯(L2+λ42¯)-2Lλ4¯]-ibF¯m4¯Ω4¯},d911=-{c55¯[m5¯(L2+λ52¯)-2Lλ5¯]-ibF¯m5¯Ω5¯},d912=-{c55¯[m6¯(L2+λ62¯)-2Lλ6¯]-ibF¯m6¯Ω6¯},d101=n3,d102=n4,d103=n5,d104=n6,d105=n3,d106=n4,d107=n5,d108=n6,d109=n3¯,d1010=n4¯,d1011=n5¯,d1012=n6¯,d111=n3(θ-iλ3),d112=n4(θ-iλ4),d113=n5(θ-iλ5),d114=n6(θ-iλ6),d115=n3(θ+iλ3),d116=n4(θ+iλ4),d117=n5(θ+iλ5),d118=n6(θ+iλ6),d119=-n3¯(θ¯-iλ3¯),d1110=-n4¯(θ¯-iλ4¯),d1111=-n5¯(θ¯-iλ5¯),d1112=-n6¯(θ¯-iλ6¯),d121=n3(θ-iλ3)e-iλ3H,d122=n4(θ-iλ4)e-iλ4H,d123=n5(θ-iλ5)e-iλ5H,d124=n6(θ-iλ6)e-iλ6H,d125=n3(θ+iλ3)eiλ3H,d126=n4(θ+iλ4)eiλ4H,d127=n5(θ+iλ5)eiλ5H,d128=n6(θ+iλ6)eiλ6H.Equation (4.7) determines the wave velocity equation for the Rayleigh waves in an orthotropic thermoelastic granular medium under the influence of initial stress and gravity field.


## 5. Stoneley Waves

To investigate the possibility Stoneley waves in thermoelastic granular medium under the influence of initial stress and gravity field, we replace the layer by a half-space(H→∞), in the preceding problem, in this case eiλjH→0, and the coefficients of e-iλjH(j=3,4,5,6) must vanish. Hence, the wave velocity equation (4.7) reduces to(5.1)det⁡⋅(drc′)=0(r,c=1,2,…,8),
where (5.2)d11′=L+m3λ3,d12′=L+m4λ4,d13′=L+m5λ5,d14′=L+m6λ6,d15′=L-m3λ3,d16′=L-m4λ4,d17′=L-m5λ5,d18′=L-m6λ6,d21′=Lm3-λ3,d22′=Lm4-λ4,d23′=Lm5-λ5,d24′=Lm6-λ6,d25′=Lm3+λ3,d26′=Lm4+λ4,d27′=Lm5+λ5,d28′=Lm6+λ6,d31′=Ω3m3,d32′=Ω4m4,d33′=Ω5m5,d34′=Ω6m6,d35′=-Ω3m3,d36′=-Ω4m4,d37′=-Ω5m5,d38′=-Ω6m6,d41′=Mm3λ3(λ32+Ω3+L2),d42′=Mm4λ4(λ42+Ω4+L2),d43′=Mm5λ5(λ52+Ω5+L2),d44′=Mm6λ6(λ62+Ω6+L2),d45′=-Mm3λ3(λ32+Ω3+L2),d46′=-Mm4λ4(λ42+Ω4+L2),d47′=-Mm5λ5(λ52+Ω5+L2),d48′=-Mm6λ6(λ62+Ω6+L2),d51′=c13L(L+m3λ3)+c33λ3(λ3-Lm3)+υ3n3,d52′=c13L(L+m4λ4)+c33λ4(λ4-Lm4)+υ3n4,d53′=c13L(L+m5λ5)+c33λ5(λ5-Lm5)+υ3n5,d54′=c13L(L+m6λ6)+c33λ6(λ6-Lm6)+υ3n6,d55′=c13L(L-m3λ3)+c33λ3(λ3+Lm3)+υ3n3,d56′=c13L(L-m4λ4)+c33λ4(λ4+Lm4)+υ3n4,d57′=c13L(L-m5λ5)+c33λ5(λ5+Lm5)+υ3n5,d58′=c13L(L-m6λ6)+c33λ6(λ6+Lm6)+υ3n6,d61′=c55[m3(L2+λ32)-2Lλ3]-ibFm3Ω3,d62′=c55[m4(L2+λ42)-2Lλ4]-ibFm4Ω4,d63′=c55[m5(L2+λ52)-2Lλ5]-ibFm5Ω5,d64′=c55[m6(L2+λ62)-2Lλ6]-ibFm6Ω6,d65′=c55[m3(L2+λ32)+2Lλ3]-ibFm3Ω3,d66′=c55[m4(L2+λ42)+2Lλ4]-ibFm4Ω4,d67′=c55[m5(L2+λ52)+2Lλ5]-ibFm5Ω5,d68′=c55[m6(L2+λ62)+2Lλ6]-ibFm6Ω6,d71′=n3,d72′=n4,d73′=n5,d74′=n6,d75′=n3,d76′=n4,d77′=n5,d78′=n6,d81′=n3(θ-iλ3),d82′=n4(θ-iλ4),d83′=n5(θ-iλ5),d84′=n6(θ-iλ6),d85′=n3(θ+iλ3),d86′=n4(θ+iλ4),d87′=n5(θ+iλ5),d88′=n6(θ+iλ6).Equation (5.1) determines the wave velocity equation for the Stoneley waves in an orthotropic thermoelastic granular medium under the influence of initial stress and gravity field.


## 6. Special Cases

The transcendental equations (4.7) and (5.1), in the determinant form, have complex roots. The real part gives the velocity of Rayleigh waves and Stoneley waves, respectively, while the imaginary part gives the attenuation due to the granular nature of the medium. It is clear from the frequency equations (4.7) and (5.1) that the phase velocity depends on the initial stress P, the friction F, the gravity field, and the coupling factor ε.When the gravity field is neglected, there is no coupling between the constantsAj,Bj, and Ej,Fj(j=3,4,5,6), the equations (3.10), (3.11) become, respectively,  (6.1)(∇2-S∂∂t)[(c55-P2)∂2ψ∂x12+(c33-c31-c55-P2)∂2ψ∂x32-ρ∂2ψ∂t2]+F∇4(∂ψ∂t)=0,(∇2-1χ∂∂t)[(c11+P)∂2φ∂x12+(c13+2c55+P)∂2φ∂x32-ρ∂2φ∂t2]-υ1ε∇2(∂ϕ∂t)=0.Equation (6.1) are in agreement with the corresponding equations obtained by Ahmed [7].In the absence of initial stress and when there is no coupling between the temperature and strain fields, (4.7) takes the form as obtained by Oshima [2].


## 7. Numerical Results and Discussions

For a numerical computational work, Potash material is considered as an upper granular media with thicknessH and Fe as lower granular media [28](7.1)ρ=1.098,M=0.4,F=0.5,Q1=0.4,Q2′=0.6,ρ′=4.499,M′=0.6,F′=0.6,Q1′=0.6,Q2′=0.8.
A Mathcad program is used to invert the transform in order to obtain the results in the physical domain. From Figure 1, It is obvious that the determinant of Stoneley waves velocity increases and decreases with an increasing of the various values of the initial stress P, also with an increasing of the density ρ; the real and imaginary parts of the determinant of Stoneley waves frequency equation increase. It is seen that magnitude of determinant of the frequency equation for Stoneley waves takes a large decreasing with an increasing of the initial stress and density.Effect of the densityρ on (real, imaginary, and magnitude) of Stoneley waves determinant with respect to initial stress.
(a)(b)(c)Figure2 displays the influence of parameter b on the frequency equation of Stoneley waves. It is concluded that the determinant of frequency equation of Stoneley waves increases with an increasing of the initial stress P. Also, it is clear that Re(Δ) increases with an increasing of parameter b but Im(Δ) decreases.Effect of the parameterb on (real, imaginary, and magnitude) of Stoneley waves determinant with respect to initial stress.
(a)(b)(c)Finally, it appears that the magnitude ofΔ increases with the increased values of the parameter b.


## 8. Conclusion

From the results obtained, we concluded the following.(1)
The determinant of frequency of Rayleigh and Stoneley waves is affected by the influences of initial stress, density, gravity, and orthotropic of material and very pronounced on the waves propagation phenomena that indicates their utilitarian aspects in diverse fields as Geophysics, Geology, Acoustics, Plasma, and so forth.(2)
From Figures1 and 2, it is obvious that the magnitude of Stoneley waves increases clearly with the influences of the density ρ and parameter b.(3)
If the media considered are isotropic, the relevant results obtained deduce to the results obtained by El-Naggar [25].



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*Source: 245965-2012-02-22.xml*