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Acta Mechanica ISSN 0001-5970Volume 227Number 4 Acta Mech (2016) 227:1199-1212DOI 10.1007/s00707-015-1527-8

Rayleigh surface waves problem in linearthermoviscoelasticity with voids

Andreea Bucur

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Acta Mech 227, 1199–1212 (2016)DOI 10.1007/s00707-015-1527-8

ORIGINAL PAPER

Andreea Bucur

Rayleigh surface waves problem in linearthermoviscoelasticity with voids

Received: 24 July 2015 / Revised: 9 November 2015 / Published online: 29 December 2015© Springer-Verlag Wien 2015

Abstract In this paper, we study the propagation of the Rayleigh surface waves in a half-space filled byan exponentially functionally graded thermoviscoelastic material with voids. We take into consideration thedissipative character of the porous thermoviscoelastic models upon the propagation waves and study thedamped in time wave solutions. The propagation condition is established in the form of an algebraic equationof tenth degree whose coefficients are complex numbers. The eigensolutions of the dynamical system areexplicitly obtained in terms of the characteristic solutions. The concerned solution of the Rayleigh surfacewave problem is expressed as a linear combination of the five analytical solutions, while the secular equationis established in an implicit form. The explicit secular equation is obtained for an isotropic and homogeneousthermoviscoelastic porous half-space, and some numerical simulations are given for a specific material.

1 Introduction

The theory of a thermoelastic material with voids is a recent generalization of the classical theory of ther-moelasticity, by taking into account the porous and the memory effects. The intended applications to thistheory may be found in many areas of applied sciences and engineering: soil mechanics, petroleum engi-neering, construction engineering, material sciences as well as in biomechanics. Iesan [1] developed a lineartheory of thermoviscoelastic materials with voids in which the time derivative of the strain tensor and the timederivative of the gradient of the volume fraction are included in the set of independent constitutive variables.This theory represents an extension to the theory of elastic material with voids, formulated by Cowin andNunziato [2], and to the theory of thermoelastic materials with voids, formulated by Iesan [3], by consideringthe viscous effects upon the materials.

There are several papers concerning various problems based on the theory of thermoviscoelastic materialswith voids (see, e.g., Chirita [4]; Chirita and Danescu [5]; Quintanilla et al. [6]; Bucur [7,8]; Svanadze [9]).

Recently, Sharma and Kumar [10] and Tomar et al. [11] studied the time harmonic steady state vibrationsof assigned frequency in an infinite thermoviscoelastic material with voids. They found four basic wavestraveling with distinct speeds, out of which one is a shear wave, and the remaining three are dilatational waves.All the dilatational waves are found to be coupled due to the presence of voids and thermal properties of thematerial, while the shear wave is uncoupled and travels independently with the speed that exists in a linearviscoelastic medium. Chirita [4] proved that the positivenesses of the viscoelastic and thermal dissipationenergies are sufficient for characterizing the spatial decay and growth properties of the harmonic vibrations ina right cylinder, without any limiting restriction upon their frequencies.

ARayleigh surfacewaves propagation problemfinds its principal application in the interpretation of groundmotions due to earthquakes or explosions. Also, many other applications have been found in the industrial

A. Bucur (B)Faculty of Mathematics, Al. I. Cuza University of Iasi, Blvd. Carol I. No. 11, 700506 Iasi, RomaniaE-mail: bucur.andreea.v@gmail.com

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world (electronic components, soil dynamics, filters and sensors, resonators, detectors, etc.). The propagationof surfacewaveswas investigated in a series of papers, as, for example, thosewritten byRayleigh [12], Destrade[13], Bucur et al. [14], Chirita et al. [15,16,18], Ciarletta et al. [19] and Ting [20,21].More recently, Chirita andDanescu [5] studied the propagation of the surface waves for an isotropic and homogeneous thermoviscoelasticmaterial with voids. They proved that the thermal and viscous dissipation energies influence the attenuation intime and in deep of the half-space for the surface waves solutions.

The present paper discusses the thermal and the memory effects upon the propagation of Rayleigh surfacewaves in a half-space filled with an exponentially functionally graded thermoviscoelastic material with voids.We study the propagation of Rayleigh surface waves in an anisotropic and inhomogeneous porous thermovis-coelastic half-space. Therefore, using an idea of Destrade [13], we consider that the characteristic coefficientsand the mass density are exponentially varying with depth, proportional to a common factor exp(−τ x2), whereτ is the inverse of an inhomogeneous characteristic length. The presence of the dissipation energy implies thatthe wave solutions have to decay to zero when time tends to infinity.

We consider a surface wave that is propagating in the x1-direction in the half-space x2 ≥ 0 made by athermoviscoelastic material with voids. In order to solve the Rayleigh surface waves propagation problem, weseek wave solutions in the class of damped in time displacement, volume fraction, and temperature fields ofthe form

{u1, u2, u3, ϕ, θ} = {U1,U2,U3, Φ, Θ}eiχ(x1−vt+px2)

whereUi , Φ, andΘ are complex parameters,χ is thewave number, v is a complex parameter so that Re(v) > 0is giving the wave speed, and exp[χ Im(v)] is giving the damping in time of the wave. The complex parameterp satisfies Im(p) > 0 and assures the asymptotic decay when x2 tends to infinity. The propagation condition isestablished in a form of an algebraic ten degree equation with complex coefficients. The results are illustratedfor the case of an isotropic and inhomogeneous thermoviscoelastic half-space with voids. For the case of anisotropic and homogeneous material, the analytical solutions are explicitly obtained in terms of characteristicsolutions to the propagation equation. The concerned solution of the Rayleigh surface waves problem isexpressed as a linear combination of the five eigensolutions. The implicit and the explicit expression for thesecular equation is also obtained, and some numerical simulations are given for a specific material.

2 Basic equations

Throughout this paper, we refer the motion of the continuum to a fixed system of Cartesian axes Oxi , (i =1, 2, 3). We shall employ the usual summation and differentiation conventions: Latin subscripts have therange 1, 2, 3, greek subscripts have the range 1, 2, summation over repeated subscripts is implied, subscriptspreceded by a comma denote partial differentiation with respect to the corresponding Cartesian coordinate,and a superposed dot denotes time differentiation.

Throughout this section, we suppose that the regular region B is filled by an anisotropic and inhomogeneousporous thermoviscoelasticmaterial. Therefore, according to Iesan [1], the fundamental systemoffield equationsconsists of

– the equations of motion

t j i, j + ρ fi = ρui , (1)

Hi,i + g + ρ = ρκϕ, (2)

– the energy equation

ρT0η = Qi,i + ρS, (3)

in B × (0,∞),

– the constitutive equations

ti j = Ci jrsers + Bi jϕ + Di jkϕ,k − βi jθ + S∗i j ,

Hi = Ai jϕ, j + Drsi ers + diϕ − aiθ + H∗i ,

g = −Bi j ei j − ξϕ − diϕ,i + mθ + g∗, (4)

ρη = βi j ei j + aθ + mϕ + aiϕ,i ,

Qi = ki jθ, j + firs ers + bi ϕ + ai j ϕ, j ,

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with S∗i j , H

∗i , and g∗ given by

S∗i j = C∗

i jrs ers + B∗i j ϕ + D∗

i jk ϕ,k + M∗i jkθ,k,

H∗i = A∗

i j ϕ, j + G∗rsi ers + d∗

i ϕ + P∗i jθ, j , (5)

g∗ = −F∗i j ei j − ξ∗ϕ − γ ∗

k ϕ,k − R∗j θ, j ,

in B × [0,∞), and

– the geometrical relations

ei j = 1

2

(ui, j + u j,i

), (6)

in B × [0,∞).Here, ui are the components of the displacement vector, ϕ is the void volume fraction, θ is the variation

of temperature from the uniform reference absolute temperature T0 > 0, ti j are the components of the stresstensor, Hi are the components of the equilibrated stress vector, g is the intrinsic equilibrated force, Qi are thecomponents of the heat flux vector, κ is the equilibrated inertia, η is the entropy density per unit mass, ρ isthe mass density, fi are the components of the body force vector, is the extrinsic equilibrated body force perunit mass, and S is the heat supply per unit mass.

The constitutive coefficients Ci jrs, Bi j , . . . , firs, and ai j are prescribed functions depending on the spatialvariable x, continuous differentiable on B, and satisfying the following symmetries:

Ci jrs = C jirs = Crsi j , Bi j = Bji , Di jk = Djik, βi j = β j i , Ai j = A ji , (7)

C∗i jrs = C∗

j irs = C∗rsi j , B∗

i j = B∗j i , D∗

i jk = D∗j ik, A∗

i j = A∗j i , ki j = k ji ,

M∗i jk = M∗

j ik, G∗rsi = G∗

sri , F∗i j = F∗

j i , P∗i j = P∗

j i , firs = fisr , ai j = a ji . (8)

If the material is homogeneous, then the coefficients are constant functions.It is worth noting that the internal energy for an anisotropic thermoviscoelastic porous material is given by

W = 1

2Ci jrsei j ers + 1

2Ai jϕ,iϕ, j + 1

2ξϕ2 + Bi j ei jϕ + Di jkei jϕ,k + diϕϕ,i . (9)

The dissipation energy density in this case is given by

Λ = C∗i jrs ei j ers + ξ∗ϕ2 + A∗

i j ϕ,i ϕ, j + 1

T0ki jθ,iθ, j + (B∗

i j + F∗i j )ei j ϕ

+(D∗i jk + G∗

i jk)ei j ϕ,k +(M∗

i jk + 1

T0fki j

)ei jθ,k + (d∗

i + γ ∗i )ϕϕ,i (10)

+(R∗j + 1

T0b j

)ϕθ, j +

(P∗i j + 1

T0a ji

)ϕ,iθ, j .

If we consider that the region B is filled by an isotropic porous thermoviscoelasticmaterial, then the constitutiveequations become:

ti j =λemmδi j + 2μei j + bϕδi j − βθδi j + λ∗emmδi j + 2μ∗ei j + b∗ϕδi j ,

Hi =αϕ,i + α∗ϕ,i + τ ∗θ,i ,

g = − bemm − ξϕ + mθ − γ ∗emm − ξ∗ϕ, (11)

ρη =βemm + aθ + mϕ,

Qi =kθ,i + ζ ϕ,i

where λ,μ, α, ξ, β,m, k, ζ, a, λ∗, μ∗, τ ∗, b∗, α∗, and γ ∗ are the constitutive coefficients and δi j are the com-ponents of the Kronecker delta. In this case, the dissipation energy density becomes

Λ = λ∗emmenn + 2μ∗ei j ei j + α∗ϕ,r ϕ,r + ξ∗ϕ2 + 1

T0kθ,rθ,r + (b∗ + γ ∗)emm ϕ

+(τ ∗ + 1

T0ζ)ϕ,rθ,r , (12)

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1202 A. Bucur

while the mechanical energy can be written as follows:

W = 1

2λerr ess + μersers + 1

2αϕ,rϕ,r + 1

2ξϕ2 + berrϕ. (13)

In what follows, we assume that the energy density and the dissipation energy density are positive definitequadratic forms. Therefore, we suppose that

μ > 0, ξ > 0, b2 < ξ(λ + 2

3μ

), α > 0, (14)

and

μ∗ > 0, ξ∗ > 0,1

4

(b∗ + γ ∗)2 > ξ∗(λ∗ + 2

3μ∗),

k > 0, T0(τ ∗ + 1

T0ζ)2

< 4α∗k. (15)

If we consider the case of an isotropic and homogeneous material, then by substituting the constitutive Eq.(11) into the field Eqs. (2) and (3), we obtain the following partial differential equation system in terms of thedisplacement ui , the volume fraction ϕ, and temperature variation θ :

μ1ui, j j + (λ1 + μ1)ur,ri + b1ϕ,i − βθ,i = ρui ,

α1ϕ,rr − γ1ur,r − ξ1ϕ + τ ∗θ,rr + mθ = ρκϕ, (16)

kθ,rr − βT0ur,r + ζ ϕ,rr − mT0ϕ = cθ ,

where

λ1 = λ + λ∗ ∂

∂t, μ1 = μ + μ∗ ∂

∂t, b1 = b + b∗ ∂

∂t,

α1 = α + α∗ ∂

∂t, γ1 = b + γ ∗ ∂

∂t, ξ1 = ξ + ξ∗ ∂

∂t, (17)

and c = aT0.

3 Surface waves in an anisotropic and inhomogeneous thermoviscoelastic porous half-space

Throughout this section, we assume B to be the half-space x2 ≥ 0. We consider that the half-space is filledby an anisotropic thermoviscoelastic porous material that is inhomogeneous in the sense that the constitutivecoefficients and the mass density are depending only on x2, in the following form:

Ci jkl = C0i jkl e

−τ x2 , Bi j = B0i j e

−τ x2 , Di jk = D0i jke

−τ x2 , βi j = β0i j e

−τ x2 ,

Ai j = A0i j e

−τ x2 , di = d0i e−τ x2 , ai = a0i e

−τ x2 , ξ = ξ0e−τ x2 , m = m0e−τ x2 , (18)

a = a0e−τ x2 , ki j = k0i j e−τ x2 , fi jk = f 0i jke

−τ x2 , bi = b0i e−τ x2 , ai j = a0i j e

−τ x2 ,

C∗i jkl = C∗0

i jkl e−τ x2 , B∗

i j = B∗0i j e

−τ x2 , Di jk = D∗0i jke

−τ x2 , Mi jk = M∗0i jke

−τ x2 ,

A∗i j = A∗0

i j e−τ x2 , G∗

i jk = G∗0i jke

−τ x2 , d∗i = d∗0

i e−τ x2 , P∗i j = P∗0

i j e−τ x2 , (19)

F∗i j = F∗0

i j e−τ x2 , ξ∗ = ξ∗0e−τ x2 , R∗

i = R∗0i e−τ x2 , γ ∗

i = γ ∗0i e−τ x2 , ρ = �e−τ x2 .

Here τ , C0i jkl , B0

i j , . . . , γ∗0i , and � are real constants. If τ = 0, then we have a homogeneous material.

We also assume that the half-space x2 ≥ 0 is free of body forces, equilibrated body force, and heat supply,and on its surface x2 = 0 has zero traction, zero equilibrated traction, and zero heat flux.Moreover, we considerthat the body is free to exchange heat with the region x2 < 0.

In what follows, we consider a surfacewave that is propagating in the direction of x1−axis, in the half-spacex2 ≥ 0. The surface traction, the equilibrated traction, and heat flux at x2 = 0 must vanish; therefore, we havethe following boundary conditions:

t2i (x1, 0, x3, t) = 0, H2(x1, 0, x3, t) = 0, Q2(x1, 0, x3, t) = 0, (20)

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for all x1, x3 ∈ R, t ≥ 0. To these boundary conditions, we adjoin the asymptotic conditions

limx2→∞ ui (x1, x2, x3, t) = 0, lim

x2→∞ ϕ(x1, x2, x3, t) = 0, limx2→∞ θ(x1, x2, x3, t) = 0,

limx2→∞{t11, t22, t12, t13, t23, h1, h2, g, q1, q2}(x1, x2, x3, t) = 0, (21)

for all x1, x3 ∈ R, and t ≥ 0.TheRayleigh surfacewave propagation problem for the anisotropic and inhomogeneous thermoviscoelastic

material with voids half-space x2 ≥ 0 consists of finding solutions to the boundary value problem defined bythe basic Eqs. (2)–(6), the boundary conditions (20), and the asymptotic conditions (21). To solve this problem,we first seek solutions U = {u1, u2, u3, ϕ, θ} of the field Eqs. (2)–(6) in the form

U(x1, x2, x3, t) = Veiχ(x1−vt+px2) (22)

where V = {U1,U2,U3, Φ, �} is a constant vector, χ is the wave number, p is a complex scalar such that

Im(p) > 0, (23)

and v is a constant complex parameter so that

Re(v) > 0, Im(v) ≤ 0. (24)

The quantity Re(v) is giving the wave speed, while exp[χ Im(v)t] is giving the damping in time of the surfacewave.

According to the relations (18), (19), and (22), we can write the constitutive Eqs. (4) and (5) as follows:

ti j = iχTi j (V, p, v)eiχ

[x1−vt+

(p+ iτ

χ

)x2

]

,

Hi = iχhi (V, p, v)eiχ

[x1−vt+

(p+ iτ

χ

)x2

]

,

g = iχG(V, p, v)eiχ

[x1−vt+

(p+ iτ

χ

)x2

]

, (25)

ρη = iχ�N (V, p, v)eiχ

[x1−vt+

(p+ iτ

χ

)x2

]

,

Qi = iχqi (V, p, v)eiχ

[x1−vt+

(p+ iτ

χ

)x2

]

where

Ti j =(Ci jr1 + pCi jr2

)Ur +

(Di j1 + pDi j2 − i

χBi j

)Φ +

(M∗0

i j1 + pM∗0i j2 + i

χβ0i j

)Θ,

hi =(Gr1i + pGr2i

)Ur +

(Ai1 + p Ai2 − i

χdi

)Φ +

(P∗01i + pP∗0

i2 + i

χa0i

)Θ,

G = −[(Fr1 + pFr2)Ur +

(γ1 + pγ2 − i

χξ)Φ +

(R∗01 + pR∗0

2 + i

χm0

)Θ

], (26)

�N = (β0r1 + pβ0

r2)Ur +(a01 + pa02 − i

χm0

)Φ − i

χa0Θ,

qi = −iχv( f 0ir1 + f 0ir2)Ur − iχv(a0i1 + pa0i2 − i

χb0i

)Φ + (k0i1 + pk0i2)Θ,

and

Cmnpq = C0mnpq − iχvC∗0

mnpq , Bmn = B0mn − iχvB∗0

mn, Amn = A0mn − iχvA∗0

mn,

Dmnp = D0mnp − iχvD∗0

mnp, Gmnp = D0mnp − iχvG∗0

mnp, dm = d0m − iχvd∗0m , (27)

Fmn = B0mn − iχvF∗0

mn, ξ = ξ0 − iχvξ∗0, γm = d0m − iχvγ ∗0m .

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In view of the relations (22), (25), and (26), the evolution Eqs. (2)–(6) become

T1i +(p + iτ

χ

)T2i = �v2Ui ,

h1 +(p + iτ

χ

)h2 − i

χG = �κv2Φ, (28)

q1 +(p + iτ

χ

)q2 + vT0�N = 0.

Therefore, according to the relations (26), we can write (28) in the following form:

HirUr + GiΦ + TiΘ = 0,

IrUr + KΦ + LΘ = 0, (29)

JrUr + MΦ + NΘ = 0

where

Hi j = Qi j + pRi j +(p + iτ

χ

)(R ji + pSi j ) − �v2δi j ,

Gi = D1i1 + pD1i2 − i

χB1i +

(p + iτ

χ

)(D2i1 + pD2i2 − i

χB2i

),

Ti = M∗01i1 + pM∗0

1i2 + i

χβ01i +

(p + iτ

χ

)(M∗0

2i1 + pM∗02i2 + i

χβ021

), (30)

I j = G1 j1 + pG2 j1 +(p + iτ

χ

)(G1 j2 + pG2 j2

)+ i

χ

(F1 j + pF2 j

),

J j = −iχv[f 01 j1 + p f 01 j2 +

(p + iτ

χ

)(f 02 j1 + p f 02 j2

)+ iT0

χ

(β0j1 + pβ0

j2

)],

K = A11 + p A12 − i

χd1 +

(p + iτ

χ

)(A21 + p A22 − i

χd2

)+ i

χ

(γ1 + pγ2 − i

χξ)

− ��v2,

L = P∗011 + pP∗0

12 + i

χa01 +

(p + iτ

χ

)(P∗021 + pP∗0

22 + i

χa02

)+ i

χ

(R∗01 + pR∗0

2 + i

χm0

),

M = −iχv[a011 + pa012 − i

χb01 +

(p + iτ

χ

)(a021 + pa022 − i

χb02

)+ iT0

χ

(a01 + pa02 − i

χm0

)], (31)

N = k011 + pk012 +(p + iτ

χ

)(k021 + pk022

)− ivT0

χa0,

and

Qi j = C1i j1, Ri j = C1i j2, Si j = C2i j2. (32)

Since we seek a non-trivial solution V = {U1,U2,U3, Φ, Θ} of the homogeneous system of algebraic Eq.(29), it follows that its determinant must vanish, and hence, the characteristic equation takes the form

det(H ) = 0 (33)

where

H =

⎛

⎜⎜⎜⎝

H11 H12 H13 G1 T1H21 H22 H23 G2 T2H31 H32 H33 G3 T3I1 I2 I3 K LJ1 J2 J3 M N

⎞

⎟⎟⎟⎠

. (34)

Further, we assume that all the solutions of the propagation conditions are real or genuine complex quan-tities. We will denote by pn, n = 1, 2, 3, 4, 5, the eigenvalues with the following property:

Im(pn) > 0, (35)

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so that the asymptotic conditions (21) are satisfied. This assumption can always be fulfilled by means ofappropriate constraints upon the constitutive coefficients and by taking some appropriate restrictions upon thewave speed Re(v) > 0, and on the damping rate � Im(v) ≤ 0.

In the general case of an anisotropic thermoviscoelastic material with voids, such assumptions cannotbe written in an explicit form, but for particular symmetries (like isotropic materials), they can be explicitlyexpressed.

Let us considerV(n) = {U (n)1 ,U (n)

2 ,U (n)3 , Φ(n), Θ(n)

}as the eigensolution corresponding to the eigenvalue

pn, n = 1, 2, 3, 4, 5. In view of the relation (22), we will seek a solution U(x1, x2, t) of the Rayleigh surfacewave propagation problem, as a linear combination of five eigensolutions U (n)(x1, x2, t)=V(n)eiχ(x1−vt+pnx2),n=1, 2, 3, 4, 5, in the following form:

U(x1, x2, t) =5∑

n=1

ϑnV(n)eiχ(x1−vt+pnx2) (36)

whereϑ = (ϑ1, ϑ2, ϑ3, ϑ4, ϑ5) is a nonzero constant vector to be determined such that the boundary conditions(20) hold true.

If we substitute the relation (36) in (25) and (26), we will obtain

ti j = iχ5∑

n=1

ϑnT(n)i j e

iχ[x1−vt+

(pn+ iτ

χ

)x2

]

,

Hi = iχ5∑

n=1

ϑnh(n)i e

iχ[x1−vt+

(pn+ iτ

χ

)x2

]

,

g = iχ5∑

n=1

ϑnG(n)e

iχ[x1−vt+

(pn+ iτ

χ

)x2

]

, (37)

Qi = iχ5∑

n=1

ϑnq(n)i e

iχ[x1−vt+

(pn+ iτ

χ

)x2

]

,

ρη = iχ5∑

n=1

ϑn�N(n)e

iχ[x1−vt+

(pn+ iτ

χ

)x2

]

.

Thus, in view of the relation (37), the boundary conditions (20) imply that

ϑ1T(1)21 + ϑ2T

(2)21 + ϑ3T

(3)21 + ϑ4T

(4)21 + ϑ5T

(5)21 = 0,

ϑ1T(1)22 + ϑ2T

(2)22 + ϑ3T

(3)22 + ϑ4T

(4)22 + ϑ5T

(5)22 = 0,

ϑ1T(1)23 + ϑ2T

(2)23 + ϑ3T

(3)23 + ϑ4T

(4)23 + ϑ5T

(5)23 = 0, (38)

ϑ1h(1)2 + ϑ2h

(2)2 + ϑ3h

(3)2 + ϑ4h

(4)2 + ϑ5h

(5)2 = 0,

ϑ1q(1)2 + ϑ2q

(2)2 + ϑ3q

(3)2 + ϑ4q

(4)2 + ϑ5q

(5)2 = 0.

We observe that the system (38) is a homogeneous algebraic linear system. Therefore, we will obtain a non-trivial solution if and only if its characteristic determinant is zero, that is, if

∣∣∣∣∣∣∣∣∣∣∣

T (1)21 T (2)

21 T (3)21 T (4)

21 T (5)21

T (1)22 T (2)

22 T (3)22 T (4)

22 T (5)22

T (1)23 T (2)

23 T (3)23 T (4)

23 T (5)23

h(1)2 h(2)

2 h(3)2 h(4)

2 h(5)2

q(1)2 q(2)

2 q(3)2 q(4)

2 q(5)2

∣∣∣∣∣∣∣∣∣∣∣

= 0 (39)

where T (n)i j = Ti j (V(n), pn, v), h(n)

i = hi (V(n), pn, v), G(n) = G(V(n), pn, v), q(n)i = qi (V(n), pn, v),

�N (n) = �N (V(n), pn, v).

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Relation (39) represents the secular equation for the complex parameter v whose real part gives the wavespeed and whose imaginary part gives the rate of damping in time. Therefore, we have to select the solutionsof the secular equation that satisfy the conditions (24) and (35). It is not easy to see that the secular equationhas such a solution for characterizing Rayleigh waves. Actually, a general result concerning the existence anduniqueness of solutions of the secular Eq. (39), under the restrictions (24) and (35), is not possible to proveat this stage. So, this remains an open problem to be studied. However, for particular materials, the secularequation can be solved by means of numerical methods.

4 Application for an isotropic and inhomogeneous thermoviscoelastic porous half-space

In this section, we illustrate the above results for the class of isotropic and inhomogeneous porous thermovis-coelastic media. Therefore, we consider that the half-space x2 ≥ 0 is filled by an isotropic thermoviscoelasticmaterial with voids that is inhomogeneous in the sense that the constitutive coefficients and the mass densityare depending only on x2, in the following form:

λ = λ0e−τ x2 , μ = μ0e

−τ x2 , b = b0e−τ x2 , α = α0e

−τ x2 , ξ = ξ0e−τ x2 , λ∗ = λ∗

0e−τ x2 ,

μ∗ = μ∗0e

−τ x2 , b∗ = b∗0e

−τ x2 , γ ∗ = γ ∗0 e

−τ x2 , ξ∗ = ξ∗0 e

−τ x2 , ζ = ζ0e−τ x2 , β = β0e

−τ x2 , (40)

α∗ = α∗0e

−τ x2 , m = m0e−τ x2 , a = a0e

−τ x2 , k = k0e−τ x2 , ρ = �e−τ x2 .

Therefore, using the constitutive equations we can write the relations (26) in the following form:

T11 =(λ + 2μ

)U1 + pλU2 − i b

χΦ + iβ0

χΘ,

T22 = λU1 + p(λ + 2μ)U2 − i b

χΦ + iβ0

χΘ,

T33 = λU1 + pλU2 − i b

χΦ + iβ0

χΘ,

T12 = T21 = pμU1 + μU2, T13 = T31 = μU3, T23 = T32 = μpU3,

h1 = αΦ + τ ∗0 Θ, h2 = p(αΦ + τ ∗

0 Θ), h3 = 0,

G = −γU1 − pγU2 + i ξ

χΦ − im0

χΘ,

�N = β0U1 + pβ0U2 − im0

χΦ − ia0

χΘ,

q1 = −iχvζ0Φ + k0Θ, q2 = −iχvζ0 pΦ + k0 pΘ (41)

where we have used the following notations:

λ = λ0 − iχvλ∗0, μ = μ0 − iχvμ∗

0, α = α0 − iχvα∗0 ,

ξ = ξ0 − iχvξ∗0 , b = b0 − iχvb∗

0, γ = b0 − iχvγ ∗0 . (42)

If we insert relations (41) into (28), we obtain the following algebraic system:

HirUr + GiΦ + TiΘ = 0,

IrUr + KΦ + LΘ = 0, (43)

JrUr + MΦ + NΘ = 0

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Surface waves problem in linear thermoviscoelasticity with voids 1207

where

H11 = λ + 2μ − �v2 + μp(p + iτ

χ

), H12 = λp + μ

(p + iτ

χ

),

H21 = λ(p + iτ

χ

)+ μp, H22 = μ − �v2 + (λ + 2μ)p

(p + iτ

χ

),

H33 = μp(p + iτ

χ

)+ μ − �v2, H13 = H31 = H23 = H32 = 0,

G1 = − i b

χ, G2 = − i b

χ

(p + iτ

χ

), G3 = T3 = I3 = J3 = 0, T1 = iβ0

χ, (44)

T2 = iβ0

χ

(p + iτ

χ

), I1 = i γ

χ, I2 = i γ

χp, J1 = vT0β0, J2 = vT0β0 p,

K = α p(p + iτ

χ

)+ α + ξ

χ2 − �κv2, L = τ ∗0 p

(p + iτ

χ

)+ τ ∗

0 − m0

χ2 ,

M = −iχv[ζ0 p

(p + iτ

χ

)+ ζ0 + m0T0

χ2

], N = k0 p

(p + iτ

χ

)+ k0 − ia0T0v

χ.

Since we seek non-trivial solutions V = {U1,U2,U3, Φ, Θ} for the homogeneous algebraic system (43),we must impose that

det(H ) = 0 (45)

where

H =

⎛

⎜⎜⎜⎝

H11 H12 H13 G1 T1H21 H22 H23 G2 T2H31 H32 H33 G3 T3I1 I2 I3 K LJ1 J2 J3 M N

⎞

⎟⎟⎟⎠

.

Therefore, the characteristic Eq. (45) becomes

[μp

(p + iτ

χ

)+ μ − �v2

]Γ = 0 (46)

where Γ is a complicated function depending on p, v, τ that will not be reported here.

5 Application for an isotropic and homogeneous thermoviscoelastic material with voids

In this section, we consider that the half-space x2 ≥ 0 is filled with homogeneous and isotropic thermovis-coelastic materials with voids (we consider τ = 0). The propagation of Rayleigh surface waves in an isotropicthermoviscoelastic material with voids was studied by Chirita and Danescu [5].

In this case, we can write the characteristic Eq. (46) in the following form:

[μ(p2 + 1) − ρv2]2Γ (p, v) = 0 (47)

where

Γ (p, v) =[(

λ + 2μ)(p2 + 1) − ρv2

]{(αk + iχvτ ∗ζ

)(p2 + 1)2 +

[k( 1

χ2 ξ − ρκv2)

− iv

χ

(αc + mζ − mT0τ

∗)](p2 + 1) − iv

χ

[c( 1

χ2 ξ − ρκv2)

+ m2T0χ2

]}

+ 1

χ2

{−kbγ + iχvβ

[γ ζ − T0(bτ

∗ + βα)]}

(p2 + 1)2 + iv

χ3

[βmT0(b + γ ) (48)

+ bγ c − β2T0(ξ − ρκχ2v2)](p2 + 1).

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In what follows, we will denote by p1, p2, p3, p4, and p5 the solutions of the characteristic Eq. (47)that satisfy the condition (23). Like in [5], for p = p2 and p = p3 we obtain the following eigensolutions:V(2) = {U (2)

1 ,U (2)2 ,U (2)

3 , Φ(2), �(2)} and V(3) = {U (3)1 ,U (3)

2 ,U (3)3 , Φ(3),�(3)} of the form

V(2) ={− p2

χ,1

χ, 0, 0, 0

}, V(3) =

{0, 0,

1

χ, 0, 0

}, (49)

Moreover, for p = p1 we obtain the following eigensolution: V(1) = {U (1)1 ,U (1)

2 ,U (1)3 , Φ(1), Θ(1)} of the

algebraic system (43) given by

U (1)1 = 1

χ(αk − χτ ∗ζw)

{(αk − χτ ∗ζw)ν21 +

[k( ξ

χ2 + ρκw2)

+ w

χ(αc − mT0τ

∗ + mζ )]ν1

+ cw

χ

( ξ

χ2 + ρκw2)

+ m2T0ω

χ3

},

U (1)2 = p1U

(1)1 , U (1)

3 = 0, (50)

Φ(1) = −iν1χ2(αk − χτ ∗ζw)

[(γ k − χβT0τ

∗w)ν1 + w

χ(γ c + mT0β)

],

Θ(1) = iwν1

χ(αk − χτ ∗ζw)

[(γ ζ − βT0α)ν1 + mT0γ

χ2 − βT0( ξ

χ2 + ρκw2)]

,

for p = p4 we have the following eigensolution: V(4) = {U (4)1 ,U (4)

2 ,U (4)1 , Φ(4), Θ(4)} with

U (4)1 = i

kχ(λ + 2μ)

[(bk + χβζw)ν4 + w

χ(bc + βmT0)

],

U (4)2 = p4U

(4)1 , U (4)

3 = 0, (51)

Φ(4) = 1

k (λ + 2μ)

{k (λ + 2μ)ν24 +

[ρkw2 + cw

χ(λ + 2μ) + β2T0w

χ

]ν4 + ρcw3

χ

},

Θ(4) = −χw

k (λ + 2μ)

{ζ (λ + 2μ)ν24 +

[mT0χ2 (λ + 2μ) + ρζw2 − βbT0

χ2

]ν4 + mT0

χ2 ρw2},

and, for p = p5 we have V(5) = {U (5)1 ,U (5)

2 ,U (5)3 , Φ(5), Θ(5)} given by

U (5)1 = −iT0

αχ (λ + 2μ)

[(bτ ∗ + αβ)ν5 + β

( ξ

χ2 + ρκw2)

− mb

χ2

],

U (5)2 = p5U

(5)1 , U (5)

3 = 0,

Φ(5) = −T0α(λ + 2μ)

{τ ∗(λ + 2μ)ν25 +

[ρw2τ ∗ + βγ

χ2 − m

χ2 (λ + 2μ)]ν5 − m

χ2 ρw2}, (52)

Θ(5) = T0α(λ + 2μ)

[α(λ + 2μ)ν25 +

[αρw2 + (λ + 2μ)

( ξ

χ2 + ρκw2)

− bγ

χ2

]ν5

+ ρw2( ξ

χ2 + ρκw2)}

,

with

w = −iv, (53)

and

λ = λ + χwλ∗, μ = μ + χwμ∗, b = b + χwb∗,α = α + χwα∗, γ = b + χwγ ∗, ξ = ξ + χwξ∗. (54)

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Surface waves problem in linear thermoviscoelasticity with voids 1209

The corresponding state of stress, equilibrated stress, and heat flux is obtained by substituting relations(49), (50), (51), and (52) into (41). Thus, we obtain the following relations:

T (1)21 = 2μp1

χ(αk − χτ ∗ζw)

{(αk − χτ ∗ζw)ν21 +

[k( ξ

χ2 + ρκw2)

+ w

χ(αc − mT0τ

∗ + mζ )]ν1

+ cw

χ

( ξ

χ2 + ρκw2)

+ m2T0w

χ3

},

T (1)22 = 1

χ(αk − χτ ∗ζw)

{[(αk − χτ ∗ζw)(λ + (λ + 2μ)p21) − b

χ2 (γ k − βT0τ∗χw) (55)

− βw

χ(γ ζ − βT0α)

]ν21 + [(λ + 2μ)p21 + λ]

[k( ξ

χ2 + ρκw2)

+ w

χ(αc − mT0τ

∗ + mζ )]ν1

− w

χ3

[βT0mγ − β2T0(ξ + ρκχ2w2

)+ b(γ c + mT0β)

]ν1 + w

χ3

[c(ξ + ρχ2κw2) + m2T0

]

· [(λ + 2μ)p21 + λ]},

h(1)2 = i p1ν1

χ(αk − χτ ∗ζw)

{γ(τ ∗ζw − αk

χ

)ν1 + τ ∗T0w

χ2

[mγ − β(ξ + ρκχ2w2)

]− αw

χ2 (γ c + mT0β)},

(56)

q(1)2 = − i p1ν1w

χ(αk − χτ ∗ζw)

[βT0(αk − χτ ∗ζw)ν1 − mT0kγ

χ2 + βT0k( ξ

χ2 + ρκw2)

+ ζw

χ(γ c + mT0β)

],

T (2)21 = μ

χ(1 − p22), T (2)

22 = 2μ

χp2, T (2)

23 = 0, h(2)2 = 0, q(2)

2 = 0,

T (3)21 = 0, T (3)

22 = 0, T (3)23 = μ

χp3, h(3)

2 = 0, q(3)2 = 0, (57)

T (4)21 = 2iμp4

kχ(λ + 2μ)

[(bk + βχζw)ν4 + w

χ(bc + βmT0)

],

T (4)22 = −i

kχ(λ + 2μ)

{(λ + 2μ)(kb + χβζw)ν24 + (kb + χβζw)[ρw2 − (λ + 2μ)p24 − λ]ν4 (58)

+ w

χ(λ + 2μ)(bc + mT0β)ν4 + w

χ(bc + mT0β)[ρw2 − (λ + 2μ)p24 − λ]

},

h(4)2 = p4

k (λ + 2μ)

{(λ + 2μ)(αk − χζτ ∗w)ν24 +

[ρw2(αk − χζτ ∗w) + w

χ(λ + 2μ)

· (αc − mT0τ∗) + βT0w

χ(βα + τ ∗b)

]ν4 + ρw3

χ(αc − mT0τ

∗)}, (59)

q(4)2 = p4w

k (λ + 2μ)

{[(λ + 2μ)

(cζw − mT0k

χ

)+ βT0

(βζw + kb

χ

)]ν4 + ρw2(cζw − mT0k

χ

)},

and

T (5)21 = −2iμT0 p5

αχ (λ + 2μ)

[(τ ∗b + βα)ν5 + β

( ξ

χ2 + ρκw2)

− mb

χ2

],

T (5)22 = iT0

αχ (λ + 2μ)

{(λ + 2μ)(bτ ∗ + βα)ν25 + [ρw2 − (λ + 2μ)p25 − λ](τ ∗b + βα)ν5 + (λ + 2μ) (60)

·[β( ξ

χ2 + ρκw2)

− mb

χ2

]ν5 +

[β( ξ

χ2 + ρκw2)

− mb

χ2

][ρw2 − (λ + 2μ)p25 − λ]

},

h(5)2 = T0 p5

α(λ + 2μ)

{(λ + 2μ)

[τ ∗( ξ

χ2 + ρκw2)

+ mα

χ2

]ν5 − γ

χ2 (bτ ∗ + βα)ν5

+ ρw2[τ ∗( ξ

χ2 + ρκw2)

+ mα

χ2

]},

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1210 A. Bucur

q(5)2 = T0 p5

α(λ + 2μ)

{(λ + 2μ)(αk − χζτ ∗w)ν25 +

[ρw2(αk − χζτ ∗w) − γ

χ2 (kb + χζβw)]ν5 (61)

+ (λ + 2μ)[k( ξ

χ2 + ρκw2)

+ mζw

χ

]ν5 + ρw2

[k( ξ

χ2 + ρκw2)

+ mζw

χ

],

T (1)23 = T (4)

23 = T (5)23 = 0.

According to the boundary conditions (20) and to relation (38), we must impose the condition (39), whichcan be written in the form

T (3)23 (T (2)

22 Δ1 − T (2)21 Δ2) = 0 (62)

where

Δ1 =

∣∣∣∣∣∣∣

T (1)21 T (4)

21 T (5)21

h(1)2 h(4)

2 h(5)2

q(1)2 q(4)

2 q(5)2

∣∣∣∣∣∣∣, Δ2 =

∣∣∣∣∣∣∣

T (1)22 T (4)

22 T (5)22

h(1)2 h(4)

2 h(5)2

q(1)2 q(4)

2 q(5)2

∣∣∣∣∣∣∣, (63)

and T (s)2i , h(s)

2 and q(s)2 , s = 1, 2, . . . , 5 are given by relations (55)–(61).

Consequently, we can rewrite the secular equation (62) in the following form:

F(w) ≡ T (3)23 [(T (2)

22 T (1)21 − T (2)

21 T (1)22 )Γ45 + (T (2)

21 T (4)22 − T (2)

22 T (4)21 )Γ15 + (T (2)

22 T (5)21 − T (2)

21 T (5)22 )Γ14] = 0

where

Γ45 = h(4)2 q(5)

2 − h(5)2 q(4)

2 , Γ15 = h(1)2 q(5)

2 − h(5)2 q(1)

2 , Γ14 = h(1)2 q(4)

2 − h(4)2 q(1)

2 .

It is important to mention that in [5] the secular equation is obtained for the case of a Kelvin-Voigtthermoviscoelastic material and also for a viscoelastic material with voids. Here we have obtained the secularequation for the general case of a thermoviscoelastic material with voids.

6 Numerical simulations

In this section, we will present some numerical simulations that are important in order to understand thetheoretical results presented in Sect. 5. For this purpose, we consider the half-space x2 ≥ 0, made of anisotropic and homogeneous thermoviscoelastic material with voids. Thus, we take the values of the relevantparameters for a copper material as in [5], that is,

λ = 7.76 × 1011 dyn/cm2, μ = 3.86 × 1011 dyn/cm2, c = 3.4303 × 104 dyn/cm2◦C,

β = 0.4 × 10−1 dyn/cm2◦C, ξ = 1.475 dyn/cm2, m = 0.2 × 107 dyn/cm2◦C,

b = 2 × 103 dyn/cm2, α = 1.688 dyn, k = 0.386 × 108 dyn/s ◦C,

T0 = 19.8 ◦C, ρ = 8.954 gm/cm3, κ = 1.75 × 10−11cm2,

and

λ∗ = 0.1 dyns/cm2, μ∗ = 0.2 dyns/cm2, b∗ = 0.1 × 10−3 dyns/cm2,

ξ∗ = 0.3 dyns/cm2, α∗ = 0.1 dyns, γ ∗ = 0.5 × 10−7 dyns/cm2,

τ ∗ = 0.3 × 10−7 dyn/◦C, ζ = 1.5 × 1011 dyn, χ = 1 cm−1.

If we substitute the assigned values of the relevant parameters in the above equations, then we can solvethe secular equation (62) using graphical methods with the software package Wolfram Mathematica.

For computing convenience and in order to outline the expected solution, we introduce the followingfunctions:

F (Re(w), Im(w)) = |F(w1, w2) × 10−70| =√

|F(w1, w2) × 10−70|2 + |F(w1, w2) × 10−70|2.

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Surface waves problem in linear thermoviscoelasticity with voids 1211

Fig. 1 The graphic of F (Re(w), Im(w)) in the complex plane for a thermoviscoelastic material with voids for Re(w) ∈(−500.0, −2000.0) and Im(w) ∈ (0.0, −1500.0)

Fig. 2 The graphic of F (Re(w), Im(w)) in the complex plane for a thermoviscoelastic material with voids for Re(w) ∈(−500.0, −2000.0) and Im(w) ∈ (0.0, −1500.0)

−1800 −1600 −1400 −1200 −1000 −800 −600

2.0 1010

4.0 1010

6.0 1010

8.0 1010

1.0 1011

1.2 1011

Fig. 3 The graphic ofF (Re(w), Im(w)) for Re(w) ∈ (−500.0, −2000.0) and Im(w) = −0.1

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1212 A. Bucur

Fig. 4 The graphic ofF (Re(w), Im(w))2 for Re(w) ∈ (−500.0, −1500.0) and Im(w) ∈ (0.0, −500.0)

As can be seen in Figs. 1, 2, and 3, there is a pointw = w1+iw2 whereF (Re(w), Im(w)) = 0, andw1 is nearto −1100 and w2 is near to zero. To highlight this, we have also computed the graphic ofF (Re(w), Im(w))2

in Fig. 4. We can conclude that the Rayleigh wave travels with an amplitude and at a speed lower than in theisotermal case. This can be explained by the fact that the thermoelastic dissipation is not sufficiently stronger.

Acknowledgments The author would like to thank the reviewers for their valuable comments and suggestions to improvethe quality of the paper. This work was supported by a grant of the Romanian National Authority for Scientific Research andInnovation, CNCS-UEFISCDI, Project Number PN-II-RU-TE-2014-4-0320.

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