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Wave Motion 54 (2015) 100–114

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Wave Motion

journal homepage: www.elsevier.com/locate/wavemoti

Surface waves problem in a thermoviscoelastic poroushalf-spaceStan Chiriţă a,b,∗, Alexandre Danescu c

a Faculty of Mathematics, Al. I. Cuza University of Iaşi, 700506 – Iaşi, Romaniab Octav Mayer Mathematics Institute, Romanian Academy, 700505 – Iaşi, Romaniac Lyon Institute of Nanotechnology, Ecole Centrale de Lyon, 69131 Ecully, France

h i g h l i g h t s

• The surface waves problem is studied for a thermoviscoelastic model for porous materials with dissipation energy.• The secular equation is explicitly obtained for an isotropic thermoviscoelastic porous half space.• The thermal and viscous dissipation energies influence the attenuation in time and in deep of the half space for the surface wave

solutions.• Numerical simulation reveals that there can exist more than one solution of the surface wave propagation problem.• The present analysis leads to surface wave solutions with finite internal energy, instead to that with infinite energy existent in the

literature.

a r t i c l e i n f o

Article history:Received 13 October 2014Received in revised form 28 November2014Accepted 29 November 2014Available online 8 December 2014

Keywords:Surface wavesThermoviscoelastic porous half spaceSecular equationDamped in time surface wave solutions

a b s t r a c t

In this paper we analyze the surface Rayleigh waves in a half space filled by a linear ther-moviscoelastic material with voids. We take into account the effect of the thermal and vis-cous dissipation energies upon the corresponding waves and, consequently, we study thedamped in time wave solutions. The associated characteristic equation (the propagationcondition) is a ten degree equation with complex coefficients and, therefore, its solutionsare complex numbers. Consequently, the secular equation results to be with complex co-efficients, and therefore, the surface wave is damped in time and dispersed. We obtain theexplicit form of the solution to the surface wave propagation problem and we write thedispersion equation in terms of the wave speed and the thermoviscoelastic homogeneousprofile. The secular equation is established in an implicit form and afterwards an explicitform is written for an isotropic and homogeneous thermoviscoelastic porous half-space.Furthermore, we use numerical methods and computations to solve the secular equationfor some special classes of thermoviscoelastic materials considered in the literature.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

The concept of porous media is used in many areas of applied science and engineering: filtration, mechanics (acous-tics, geomechanics, soil mechanics, rock mechanics), engineering (petroleum engineering, bio-remediation, constructionengineering), geosciences (hydrogeology, petroleum geology, geophysics), biology and biophysics, material science, etc.

∗ Corresponding author at: Faculty of Mathematics, Al. I. Cuza University of Iaşi, 700506 – Iaşi, Romania.E-mail addresses: [email protected] (S. Chiriţă), [email protected] (A. Danescu).

http://dx.doi.org/10.1016/j.wavemoti.2014.11.0140165-2125/© 2014 Elsevier B.V. All rights reserved.

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S. Chiriţă, A. Danescu / Wave Motion 54 (2015) 100–114 101

Thermoviscoelastic porous materials play an important role in many branches of civil engineering, seismology, nano-technology and biomaterials. Fundamentals-based scientific knowledge can be a great help in exploring many avenues ofthermoviscoelasticmaterials research and development. It is imperative thatwe should know the fundamentals ofmaterialsbefore we can utilize them properly and efficiently. Study of bone viscoelasticity is best placed in the context of strain levelsand frequency components associated with normal activities andwith applications of diagnostic tools. The investigations ofthe solutions of viscoelastic wave equations and the attenuation of seismic wave in the viscoelastic media are very impor-tant for geophysical prospecting technology. Moreover, the behavior of viscoelastic porousmaterials can be understood andpredicted in great detail using nano-mechanics. There are numerous applications of these materials as outlined in Polarzand Smarsly [1] and in the books by Voyiadjis and Song [2], Park and Lakes [3] and Lakes [4] and references therein.

Ieşan [5] developed a linear theory of thermoviscoelastic materials with voids, in which the set of independentconstitutive variables includes the timederivative of the strain tensor, the timederivative of the volume fraction field and thetime derivative of the gradient of the volume fraction field. The theory represents an extension of the Cowin and Nunziato’stheory of elastic material with voids [6] and that of Ieşan’s theory of thermoelastic material with voids [7] to incorporatethe memory effects. Recently, a continuum theory for a thermoviscoelastic composite was developed in [8] as a mixture ofa microstretch viscoelastic material of Kelvin–Voigt type and a microstretch elastic solid.

Several papers concerning various problems based on the theories of elasticity and thermoelasticitywith voids have beenappeared in the literature. Some notable results are established by Ieşan [9], Ciarletta and Scalia [10], Chiriţă and Scalia [11],Chiriţă et al. [12] for dynamical problems, Ieşan and Nappa [13], Chiriţă and D’Apice [14,15] for plane strain of porous elasticbodies and Ciarletta et al. [16] for solids with double porosity. For a review of the literature on elastic materials with voidsthe reader is referred to the book by Ieşan [17].

The propagation of plane waves in an infinite thermoelastic medium with voids was studied in a series of papers, as, forexample, those written by Puri and Cowin [18], Chandrasekharaiah [19,20] and, more recently, by Singh and Tomar [21],Ciarletta and his coworkers [22,23] and Bucur et al. [24].

As regards the linear theory of thermoviscoelastic materials, in [5] is established a uniqueness result and the continuousdependence of solution upon the initial data and supply terms. A solution of field equations is also presented. Recently,Sharma and Kumar [25], Svanadze [26] and Tomar et al. [27] study the steady state time harmonic waves of assignedfrequency in an infinite thermoviscoelastic material with voids. Chiriţă [28] proves that the positiveness of the viscoelasticand thermal dissipation energies is sufficient for characterizing the spatial decay and growth properties of the harmonicvibrations in a cylinder, without any limiting restriction upon their frequencies.

The present paper takes into account the dissipative character of the thermoviscoelastic model developed in [5] for dis-cussing the surface wave propagation problem in an anisotropic and homogeneous thermoviscoelastic porous half space.The presence of the dissipation energy implies that the wave solutions have to decay to zero when the time tends to infin-ity. On this basis and considering the half space x2 > 0, we are lead to seek wave solutions in the class of damped in timedisplacement–volume fraction–temperature variation fields {ur , ϕ, θ} defined like

ur (x, t) = ReArei~(nsxs−σ x2−vt) ,

ϕ (x, t) = ReBei~(nsxs−σ x2−vt) ,

θ (x, t) = ReCei~(nsxs−σ x2−vt) ,

where Ar , B and C are complex parameters, ~ is the real wave number and ns are the components of a real unit vectorgiving the propagation direction, σ is a complex parameter with Im(σ ) < 0, assuring the asymptotic decay when x2 → ∞,and v is a complex parameter so that Re (v) > 0 is giving the wave speed and exp [~ Im (v) t] is giving the damping intime of the wave and hence we have to assume that Im (v) ≤ 0. Moreover, the above representation allows us to havewave solutions with finite internal energy. We obtain the explicit form of the solution to the surface wave propagationproblem andwewrite the dispersion equation in terms of the wave speed and the thermoviscoelastic homogeneous profile.The secular equation is established in an implicit form and afterwards an explicit form is written for an isotropic andhomogeneous thermoviscoelastic porous half-space. Numerical computations have been performed for some special classesof thermoviscoelastic materials considered in the literature and various interesting results have been depicted graphicallyand discussed.

2. Basic equations

Throughout this paper, we refer the motion of a continuum to a fixed system of rectangular Cartesian axes Oxk, (k =

1, 2, 3). We shall employ the usual summation and differentiation conventions: Latin subscripts are understood to rangeover the integers (1, 2, 3), whereas Greek subscripts are confined to the range (1, 2), summation over repeated subscriptsis implied, subscripts preceded by a comma denote partial differentiation with respect to the corresponding Cartesiancoordinate, and a superposed dot denotes time differentiation. Throughout this section we suppose that a regular region Bis filled by a homogeneous and anisotropic thermoviscoelastic material. Considering the linear theory of thermoviscoelasticmaterials with voids and assuming that the initial body is free from stresses and has zero intrinsic equilibrated body forceand entropy, the system of field equations consists of (cf. Ieşan [5])

102 S. Chiriţă, A. Danescu / Wave Motion 54 (2015) 100–114

– the equations of motion

trs,r + ϱfs = ϱus,

Hr,r + g + ϱℓ = ϱκϕ, in B × (0, ∞), (1)– the equation of energy

ϱT0η = Qr,r + ϱs, in B × (0, ∞), (2)– the constitutive equations

trs = Crspqepq + Brsϕ + Drspϕ,p − βrsθ + S∗

rs,

Hr = Arsϕ,s + Dpqrepq + drϕ − arθ + H∗

r ,

g = −Brsers − ξϕ − drϕ,r + mθ + g∗,

ϱη = βrsers + aθ + mϕ + arϕ,r ,

Qr = krsθ,s + frpqepq + br ϕ + arsϕ,s, in B × [0, ∞)

(3)

with

S∗

rs = C∗

rspqepq + B∗

rsϕ + D∗

rspϕ,p + M∗

rspθ,p,

H∗

r = A∗

rsϕ,s + G∗

pqr epq + d∗

r ϕ + P∗

rsθ,s,

g∗= −F∗

rsers − ξ ∗ϕ − γ ∗

r ϕ,r − R∗

r θ,r ,

(4)

and– the geometrical relations

ers =12

ur,s + us,r

, in B × [0, ∞). (5)

Here we have used the following notations: trs are the components of the stress tensor, Hr are the components of the equi-librated stress vector, g is the intrinsic equilibrated body force, η is the entropy per unit mass, Qr are the components of theheat flux vector, ers are the components of the strain tensor, ϱ is themass density of themedium, κ is the equilibrated inertia,ur are the components of the displacement vector, ϕ is the void volume fraction, θ is the change in temperature from theconstant ambient temperature T0 > 0, fr are the components of the body force per unit mass, ℓ is the extrinsic equilibratedbody force per unit mass, s is the heat supply per unit mass. The constitutive coefficients have the following symmetries

Crspq = Cpqrs = Csrpq, βrs = βsr , Dpqr = Dqpr ,

Ars = Asr , Brs = Bsr ,(6)

C∗

rspq = C∗

pqrs = C∗

srpq, B∗

rs = B∗

sr , D∗

pqr = D∗

qpr ,

M∗

pqr = M∗

qpr , A∗

rs = A∗

sr , G∗

rsp = G∗

srp, P∗

rs = P∗

sr ,

F∗

rs = F∗

sr , krs = ksr , fpqr = fprq, ars = asr .(7)

Furthermore, in view of the second law of thermodynamics, the Clausius–Duhem inequality must be satisfied, which pro-vides the positive semi-definiteness of the total dissipation energy Λ, that is

Λers, ϕ, ϕ,p, θ,q

≥ 0, (8)

where


= C∗

pqrsepqers + A∗

rsϕ,r ϕ,s + ξ ∗ϕ2+

1T0

krsθ,rθ,s

+B∗

rs + F∗

rs

ersϕ +

D∗

pqr + G∗

pqr

epqϕ,r

+

M∗

pqr +1T0

frpq

epqθ,r +

d∗

r + γ ∗

r

ϕϕ,r

+

R∗

r +1T0

br

ϕθ,r +

P∗

rs +1T0

asr

ϕ,rθ,s. (9)

We note that for an isotropic and homogeneous body, the constitutive equations (3) and (4) reduce totrs = λemmδrs + 2µers + bϕδrs − βθδrs + λ∗emmδrs + 2µ∗ers + b∗ϕδrs,

Hr = αϕ,r + α∗ϕ,r + τ ∗θ,r ,

g = −bemm − ξϕ + mθ − γ ∗emm − ξ ∗ϕ,

ϱη = βemm + aθ + mϕ,

Qr = kθ,r + ζ ϕ,r , in B × [0, ∞), (10)


where δrs are the components of the Kronecker delta, λ and µ are well known Lame’s constant parameters, b, α, ξ and ξ ∗

are the constant parameters corresponding to voids present in the medium, β, τ ∗,m, k, ζ and a are the constant thermalparameters and λ∗, µ∗, b∗, α∗ and γ ∗ are the constant viscoelastic parameters. Furthermore, the dissipation energy Λ

becomes


= λ∗emmenn + 2µ∗ersers + α∗ϕ,r ϕ,r + ξ ∗ϕ2

+1T0

kθ,rθ,r +b∗

+ γ ∗emmϕ +

τ ∗

+1T0

ζ

ϕ,rθ,r (11)

and it is a positive semi-definite form if the following inequalities among the various thermoviscoelastic moduli hold true

µ∗≥ 0, ξ ∗

≥ 0,14

b∗

+ γ ∗2

≤ ξ ∗

λ∗

+23µ∗

,

k ≥ 0, T0

τ ∗

+1T0

ζ

2

≤ 4α∗k. (12)

By substituting the relations (4) and (5) into (3) and, further, the result into (1) and (2) and assuming that the body forcesand heat source density are zero, we obtain the following system of differential equations

µ0us,rr + (λ0 + µ0) um,ms + b0ϕ,s − βθ,s = ϱus,

α0ϕ,rr − γ0ur,r − ξ0ϕ + τ ∗θ,rr + mθ = ϱκϕ,

kθ,rr − βT0ur,r + ζ ϕ,rr − mT0ϕ = cθ , (13)

where c = aT0 and

λ0 = λ + λ∗∂

∂t, µ0 = µ + µ∗

∂

∂t,

b0 = b + b∗∂

∂t, α0 = α + α∗

∂

∂t,

γ0 = b + γ ∗∂

∂t, ξ0 = ξ + ξ ∗

∂

∂t. (14)

3. Constitutive hypotheses

Throughout this paper we will assume the following constitutive hypotheses:(H1) ϱ, ~ and a are strictly positive;(H2) the constitutive coefficients satisfy the symmetry relations (6) and (7) and the dissipation energyΛ is a positive definite

quadratic form in terms of ers, ϕ, ϕ,p, that is there exist the strictly positive constants µ∗m, a∗

m, ξ ∗m, km and µ∗

M , a∗

M ,ξ ∗

M , kM so that

µ∗

mζrsζrs + a∗

mω2+ ξ ∗

mνpνp + kmτqτq ≤ Λζrs, ω, νp, τq

≤ µ∗

Mζrsζrs + a∗

Mω2+ ξ ∗

Mνpνp + kMτqτq (15)

for all real ζrs = ζsr , ω, νp and τq;(H3) The mechanical energy U , defined as

Uers, ϕ, ϕ,p

=

12Cpqrsepqers +

12

ξϕ2+

12Arsϕ,rϕ,s + Brsersϕ + Dpqrepqϕ,r + drϕϕ,r , (16)

is a positive definite quadratic form in terms of ers, ϕ andϕ,p. Thismeanswe assume that there exist the strictly positiveconstants µm, am, ξm and µM , aM , ξM so that

µmζrsζrs + amω2+ ξmτpτp ≤ 2U

ζrs, ω, τp

≤ µMζrsζrs + aMω2

+ ξMτpτp (17)

for all real ζrs = ζsr , ω and τp.

The above hypotheses are used in [5] in order to establish the uniqueness theorem for the boundary–initial-value prob-lems associated with the thermoviscoelastic model of porous materials. We note that the assumption (17) implies that Cpqrsis a positive definite tensor and hence we will have

µmξrsξrs ≤ Cpqrsξpqξrs ≤ µMξrsξrs, for all real ξrs = ξsr (18)

whenµm andµM are theminimumandmaximumelasticmoduli for the elasticity tensor Cpqrs (see, for example, Gurtin [29]).When the thermoviscoelastic porous material is isotropic then the mechanical energy is

U =12λemmenn + µersers +

12αϕ,rϕ,r +

12ξϕ2

+ bemmϕ, (19)


and it is a positive definite form if we assume that

µ > 0, ξ > 0, b2 < ξ

λ +

23µ

, α > 0. (20)

Further, the dissipation energy is a positive definite form if

µ∗ > 0, ξ ∗ > 0,14

b∗

+ γ ∗2

< ξ ∗

λ∗

+23µ∗

,

k > 0, T0

τ ∗

+1T0

ζ

2

< 4α∗k. (21)

4. Surface waves in an anisotropic thermoviscoelastic porous half space

Throughout this section we consider the following surface wave problem (P ): the half space x2 > 0 is free of body force,equilibrated body force and body heat supply and free of traction and equilibrated traction on its surface x2 = 0; moreover,the poroviscoelastic medium is free to exchange heat with the contents of the region x2 < 0, and prior to the appearance ofa disturbance both media are everywhere at constant temperature T0 > 0. For a surface wave propagating in the directionof the x1-axis in the half space x2 > 0, the surface traction, the equilibrated surface traction and the heat flux at x2 = 0mustvanish, that is

t2r = 0, H2 = 0, Q2 = 0 at x2 = 0. (22)We search for a solution of the surface wave problem (P ) in the class of displacement, volume fraction and temperature

variation {ur , ϕ, θ} characterized by the following requirements(i) the internal and dissipation energies associated with {ur , ϕ, θ} to be finite;(ii) {ur , ϕ, θ}, together with the corresponding state of stress, equilibrated stress and heat flux, decay asymptotically with

the deep in the inside of the considered half space, that is they decay to zero when x2 tends to infinity;(iii) {ur , ϕ, θ} is damped in time, that is it decreases with respect to time variable.

More precisely, we first search for a solution U = {ur , ϕ, θ} of the basic equations (1)–(5) in the following formur (x, t) = Re

Arei~(x1−σ x2−vt) ,

ϕ (x, t) = ReBei~(x1−σ x2−vt) ,

θ (x, t) = ReCei~(x1−σ x2−vt) , (23)

where i =√

−1 is the imaginary unit, Re{·} is the real part, ~ > 0 is the real wave number and π = {A1, A2, A3, B, C} is anon-zero complex vector. Moreover, according with the requirement (ii), we assume σ to be a complex parameter assuringthe asymptotic decay to zero of the solution when x2 → ∞, that is

Im(σ ) < 0. (24)While the constant parameter v is allowed to be complex

v = Re (v) + i Im (v) (25)so that, according with the requirement (iii),

Re (v) > 0 (26)represents the speed of propagation and exp [~ Im (v) t] is giving the damping in time of the wave and hence we assumethat

Im (v) ≤ 0. (27)If it happens that the imaginary part of v is vanishing, that is Im (v) = 0, thenwe have an undamped harmonic in timewave.Otherwise, that is when Im (v) < 0, we have a damped in time wave. Finally, we note that the requirement (i) is fulfilled bythe assumption that the wave number ~ to be a positive real number.

In what follows we are searching solutions of the form (23) for the basic equations (1)–(5) and in this respect we willtry to determine the parameters A1, A2, A3, B and C and σ so that the basic equations (1)–(5) to be satisfied. To this end wesubstitute (23) into (3)–(5) to obtain

trs = ReTrs (π, σ , v) ei~(x1−σ x2−vt) ,

Hr = Rehr (π, σ , v) ei~(x1−σ x2−vt) ,

g = ReG (π, σ , v) ei~(x1−σ x2−vt) ,

ϱη = ReϱN (π, σ , v) ei~(x1−σ x2−vt) ,

Qr = Reqr (π, σ , v) ei~(x1−σ x2−vt) ,

(28)


where

Trs (π, σ , v) = i~C0rsp1 − σC0

rsp2

Ap +

B0rs + i~

D0rs1 − σD0

rs2

B −

βrs − i~

M∗

rs1 − σM∗

rs2

C, (29)

hr (π, σ , v) = i~G0p1r − σG0

p2r

Ap +

d0r + i~

A0r1 − σA0

r2

B −

ar − i~

P∗

r1 − σP∗

r2

C, (30)

G (π, σ , v) = −i~F 0r1 − σ F 0

r2

Ar −

ξ 0

+ i~γ 01 − σγ 0

2

B +

m − i~

R∗

1 − σR∗

2

C, (31)

ϱN (π, σ , v) = i~ (βr1 − σβr2) Ar + [m + i~ (a1 − σa2)] B + aC, (32)

qr (π, σ , v) = ~2vfrp1 − σ frp2

Ap − i~v [br + i~ (ar1 − σar2)] B + i~ (kr1 − σkr2) C, (33)

and

C0rspq = Crspq − i~vC∗

rspq, B0rs = Brs − i~vB∗

rs,

D0rsp = Drsp − i~vD∗

rsp, A0rs = Ars − i~vA∗

rs,

G0pqr = Dpqr − i~vG∗

pqr , d0r = dr − i~vd∗

r ,

F 0rs = Brs − i~vF∗

rs, ξ 0= ξ − i~vξ ∗, γ 0

r = dr − i~vγ ∗

r .

(34)

Furthermore, we substitute the relations (23) and (28) into basic equations (1) and (2) to obtain

i~ (T1s − σT2s) = −ϱ~2v2As,

i~ (h1 − σh2) + G = −ϱκ~2v2B,i~ (q1 − σq2) = −i~vT0ϱN.

(35)

Finally,we replace the relations (29)–(33) into Eq. (35) in order to obtain for the unknownA1, A2, A3, B andC the followingalgebraic linear system

HrsAs + Hr4B + Hr5C = 0,H4sAs + H44B + H45C = 0,H5sAs + H54B + H55C = 0,

(36)

where

Hrs = −~2 C01rs1 − σ

C01rs2 + C0

2rs1

+ σ 2C0

2rs2 − ϱv2δrs,

Hr4 = i~B01r − σB0

2r

− ~2

D01r1 − σ

D01r2 + D0

2r1

+ σ 2D0

2r2

,

Hr5 = −i~ (β1r − σβ2r) − ~2 M∗

1r1 − σM∗

1r2 + M∗

2r1

+ σ 2M∗

2r2

,

(37)

H4r = −i~F 0r1 − σ F 0

r2

− ~2

G0r11 − σ

G0r21 + G0

r12

+ σ 2G0

r22

,

H44 = −ξ 0+ i~

d01 − γ 0

1 − σd02 − γ 0

2

− ~2

A011 − σ

A012 + A0

21

+ σ 2A0

22 − ϱκv2 ,

H45 = m + i~−

a1 + R∗

1

+ σ

a2 + R∗

2

− ~2

P∗

11 − σP∗

12 + P∗

21

+ σ 2P∗

22

,

(38)

H5r = −~2T0v (βr1 − σβr2) + i~3vf1r1 − σ (f1r2 + f2r1) + σ 2f2r2

,

H54 = i~vT0m − ~2 [T0v (a1 − σa2) − v (b1 − σb2)] + i~3va11 − σ (a12 + a21) + σ 2a22

,

H55 = i~vT0a − ~2 k11 − σ (k12 + k21) + σ 2k22

.

(39)

From the condition to have non-trivial solutions of the form (23), it follows from (36) the following characteristic equation

det (Hmn)(5×5) = 0, (40)

and such a form of the propagation condition is written as a vanishing 5× 5 determinant. In what follows, we shall assumethat all the solutions of the propagation condition are genuine complex quantities. Further, we denote by σn, n = 1, 2, 3,4, 5, the eigenvalues with a negative imaginary part, that is

Im(σn) < 0, (41)

so that the asymptotic conditions (i) and (ii) are satisfied. This assumption can always be fulfilled by means of appropriateconstraints upon the thermoviscoelastic coefficients and by appropriate restrictions upon the wave speed Re(v) > 0 andon the damping rate ~ Im(v) ≤ 0. While in the general case of anisotropic thermoviscoelastic porous materials, such as-sumptions cannot be expressed in an explicit form, for particular symmetries (like orthotropic or isotropic materials), theycan be explicitly expressed.

Letπ (n)= {A(n)

1 , A(n)2 , A(n)

3 , B(n), C (n)}be the eigensolution of the algebraic system (36) corresponding to the eigenvalueσn,

for n = 1, 2, 3, 4, 5, so that the eigensolution U (n)= {u(n)

1 , u(n)2 , u(n)

3 , ϕ(n), θ (n)} of the differential system (1)–(5) is given by

U (n)= Re{π (n) ei~(x1−σnx2−vt)

}. (42)


Then the general solution U = {u1, u2, u3, ϕ, θ} of Eqs. (1)–(5) can be expressed by superimposing the five eigensolutionsU (n)(x1, x2, t), n = 1, 2, 3, 4, 5. Thus, we seek for solution of the surface wave problem (P ) the following form

U(x1, x2, t) = Re

5

n=1

ωnπ(n) ei~(x1−σnx2−vt)

, (43)

where ω1, ω2, . . . , ω5 are constant parameters, at least one different from zero, to be determined in order for the boundaryconditions (22) to be satisfied. Then we have

trs (U) = Re

5

n=1

ωnTrsπ (n), σn, v

ei~(x1−σnx2−vt)

,

Hr (U) = Re

5

n=1

ωnhrπ (n), σn, v


,

g (U) = Re

5

n=1

ωnGπ (n), σn, v


,

ϱη (U) = Re

5

n=1

ωnϱNπ (n), σn, v


,

Qr (U) = Re

5

n=1

ωnqrπ (n), σn, v


,

(44)

and hence, a non-trivial solution for (P ) exists if and only if

∆ (v) ≡

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, (45)

which represents the secular equation for the complex parameter v whose real part gives the wave speed and whose imag-inary part gives the rate of damping in time. In the above relation we have used the notation

T (n)rs = Trs

π (n), σn, v

, h(n)

r = hrπ (n), σn, v

, q(n)

r = qrπ (n), σn, v

, n = 1, 2, 3, 4, 5. (46)

Thus, we have to select the solutions of the secular equation (45) that satisfy the conditions (26), (27) and (41). It isnot obvious that the secular equation (45) has such a solution for characterizing the Rayleigh waves. A general analyticalresult concerning the existence and uniqueness of solutions of the secular equation (45), under restrictions generated by theinequalities (26), (27) and (41), seems to be not possible at this stage, and so, it remains an open problem to be studied infuture research. However, for particular materials with numerical values for the thermoviscoelastic coefficients, the secularequation can be solved by using numerical methods.

5. Application for an isotropic thermoviscoelastic porous half space

In the case of an isotropic and homogeneous thermoviscoelastic porous half space the relations (29)–(33) become

T11 (π, σ , v) = i~

λ0+ 2µ0 A1 − λ0σA2

+ b0B − βC,

T22 (π, σ , v) = i~λ0A1 −

λ0

+ 2µ0 σA2+ b0B − βC,

T33 (π, σ , v) = i~λ0 (A1 − σA2) + b0B − βC,

T12 (π, σ , v) = i~µ0 (A2 − σA1) , T23 (π, σ , v) = −i~σµ0A3, T31 (π, σ , v) = i~µ0A3,

(47)

h1 (π, σ , v) = i~α0B + τ ∗C

, h2 (π, σ , v) = −i~σ

α0B + τ ∗C

, h3 (π, σ , v) = 0,

G (π, σ , v) = −γ 0i~ (A1 − σA2) − ξ 0B + mC,(48)

ϱN (π, σ , v) = i~β (A1 − σA2) + mB + aC,

q1 (π, σ , v) = i~ (kC − i~ζvB) , q2 (π, σ , v) = i~σ (i~ζvB − kC) , q3 (π, σ , v) = 0,(49)

where

λ0= λ − i~vλ∗, µ0

= µ − i~vµ∗, b0 = b − i~vb∗,

α0= α − i~vα∗, ξ 0

= ξ − i~vξ ∗, γ 0= b − i~vγ ∗.

(50)


Moreover, we have

H11 = −~2 λ0

+ 2µ0− ϱv2

+ µ0σ 2 , H22 = −~2 µ0

− ϱv2+

λ0

+ 2µ0 σ 2 ,

H33 = −~2 µ0

1 + σ 2− ϱv2 , H44 = −

ξ 0

− ϱκ~2v2+ ~2

1 + σ 2 α0 ,

H55 = i~vT0a − ~2 1 + σ 2 k, H12 = H21 = ~2

λ0+ µ0 σ ,

H24 = −σH14 = −i~b0σ , H25 = −σH15 = i~βσ, H42 = −σH41 = i~γ 0σ ,

H52 = −σH51 = ~2vT0βσ, H45 = m − ~2 1 + σ 2 τ ∗,

H54 = i~vT0m + i~3v1 + σ 2 ζ , H13 = H31 = H23 = H32 = 0,

H34 = H43 = H35 = H53 = 0,

(51)

and the characteristic equation (40) becomes

− ~6 µ0

1 + σ 2− ϱv22 ∆ = 0, (52)

where

∆ =

λ0+ 2µ0

1 + σ 2− ϱv2

~4 α0k + i~vζτ ∗

1 + σ 22

+ ~2 kξ 0

− ϱκ~2v2− i~v

α0T0a + mζ − mT0τ ∗

1 + σ 2

− i~vT0aξ 0

− ϱκ~2v2+ m2

+ ~2 −b0γ 0k + i~vβ

γ 0ζ − T0

b0τ ∗

+ βα0 1 + σ 22

+ i~vT0ab0γ 0

+ βmγ 0

+ b0− β2

ξ 0− ϱκ~2v2

1 + σ 2 . (53)It can be seen that Eq. (52) implies

σ 22 = σ 2

3 = −1 +ϱv2

µ0, (54)

and moreover, we have the following eigensolutions of the algebraic system (36)

π (2)=

σ2

~,1~

, 0, 0, 0

, π (3)=

0, 0,

1~

, 0, 0

, (55)

with the corresponding eigensolutions of the differential system (13) given by

U (2)= Re

σ2

~,1~

, 0, 0, 0

ei~(x1−σ2x2−vt)

,

U (3)= Re

0, 0,

1~

, 0, 0, 0

ei~(x1−σ2x2−vt)

.

(56)

Further, the non-zero components of Trs and hr are

T11π (2), σ2, v

= −T22

π (2), σ2, v

= 2iµ0σ2, T12

π (2), σ2, v

= iµ0

1 − σ 22

, (57)

and

T23π (3), σ3, v

= −iµ0σ3, T31

π (3), σ3, v

= iµ0. (58)

So in what follows we will consider the solutions corresponding to the Eq. (53). To this end we introduce the notations

v = iw, 1 + σ 2= ν, (59)

and note that Eqs. (52) and (53) imply

P0ν3+ P1ν2

+ P2ν + P3 = 0, (60)where

P0 = ~4 λ0

+ 2µ0 α0k − ~wζτ ∗

, (61)

P1 = ~4ϱw2 α0k − ~wζτ ∗

+ ~2

λ0+ 2µ0

kξ 0

+ ϱκ~2w2+ ~α0T0aw

+ ~wmζ − T0τ ∗m

− ~2

b0γ 0k + ~wβγ 0ζ − ~T0wβb0τ ∗

+ βα0 , (62)

P2 = ~2ϱw2 kξ 0

+ ϱκ~2w2+ ~w

T0α0a + mζ − T0τ ∗m

+ ~wT0

λ0

+ 2µ0 aξ 0

+ ϱκ~2w2+ m2

−ab0 + βm

γ 0

− βmb0 − β

ξ 0

+ ϱκ~2w2 , (63)

P3 = ~ϱw3T0aξ 0

+ ϱκ~2w2+ m2 , (64)


and now we have

λ0= λ + ~wλ∗, µ0

= µ + ~wµ∗, b0 = b + ~wb∗,

α0= α + ~wα∗, ξ 0

= ξ + ~wξ ∗, γ 0= b + ~wγ ∗.

(65)

Let us denote by ν1, ν4 and ν5 the roots of the above equation. We shall further mark by σ1, σ4 and σ5 the solutions ofequations

1 + σ 2= ν1, 1 + σ 2

= ν4, 1 + σ 2= ν5, (66)

respectively, whose imaginary parts are positive. Obviously, these solutions are dependent on the coupling thermoviscoelas-tic porous coefficients T = {b, β, b∗, τ ∗,m, γ ∗, ζ }. We choose the order of these solutions so that when T = {b, β, b∗, τ ∗,m, γ ∗, ζ } = 0 we have

σ 21 (T = 0) = −1 −

ϱw2

λ0 + 2µ0,

σ 24 (T = 0) = −1 −

ξ 0+ ϱκ~2w2

~2α0,

σ 25 (T = 0) = −1 −

T0aw~k

.

(67)

Further, for the eigensolution σ1 we have the following π (1)= {A(1)

1 , A(1)2 , A(1)

3 , B(1), C (1)} as given by

A(1)1 =

1~

kα0 − ~wζτ ∗

kα0

− ~wζτ ∗ν21 + ~−2

kξ 0

+ ϱκ~2w2+ ~w

T0α0a − T0τ ∗m + ζm

ν1 + ~−3wT0

aξ 0

+ ϱκ~2w2+ m2 ,

A(1)2 = −σ1A

(1)1 , A(1)

3 = 0,

B(1)=

−iν1

~2kα0 − ~wζτ ∗

kγ 0

− ~wT0βτ ∗ν1 + ~−1wT0

mβ + aγ 0 ,

C (1)=

−iwν1

~kα0 − ~wζτ ∗

T0βα0

− ζγ 0 ν1 + ~−2T0β

ξ 0

+ ϱκ~2w2− γ 0m

,

(68)

while, for the eigensolution σ4 we have the following π (4)= {A(4)

1 , A(4)2 , A(4)

3 , B(4), C (4)} as given by

A(4)1 =

i~k

λ0 + 2µ0

b0k + ~wβζ

ν4 + ~−1T0w

b0a + βm

,

A(4)2 = −σ4A

(4)1 , A(4)

3 = 0,

B(4)=

1kλ0 + 2µ0

kλ0

+ 2µ0 ν24 +

kϱw2

+ ~−1wT0aλ0

+ 2µ0+ ~−1wT0β2 ν4 + ϱ~−1w3T0a

,

C (4)= −

~w

kλ0 + 2µ0

ζ

λ0

+ 2µ0 ν24 + ~−2

T0mλ0

+ 2µ0+ ϱ~2w2ζ − b0T0β

ν4 + ϱ~−2w2T0m

,

(69)

and for σ5 we have the following π (5)= {A(5)

1 , A(5)2 , A(5)

3 , B(5), C (5)} as given by

A(5)1 = −

iT0~α0

λ0 + 2µ0

b0τ ∗

+ βα0 ν5 + ~−2 β

ξ 0

+ ϱκ~2w2− mb0

,

A(5)2 = −σ5A

(5)1 , A(5)

3 = 0,

B(5)= −

T0α0

λ0 + 2µ0

τ ∗

λ0

+ 2µ0 ν25 − ~−2

mλ0

+ 2µ0− ϱ~2w2τ ∗

− βγ 0 ν5 − ϱ~−2w2m,

C (5)=

T0α0

λ0 + 2µ0

α0

λ0+ 2µ0 ν2

5 +ϱw2α0

− b0γ 0+ ~−2

λ0+ 2µ0

ξ 0+ ϱκ~2w2 ν5

+ ϱ~−2w2 ξ 0

+ ϱκ~2w2 .

(70)


Furthermore, we have

T12π (n), σn, v

= −2i~µ0σnA

(n)1 ,

T22π (n), σn, v

= i~

λ0

+λ0

+ 2µ0 σ 2n

A(n)1 + b0B(n)

− βC (n),

T23π (n), σn, v

= 0,

h2π (n), σn, v

= −i~σn

α0B(n)

+ τ ∗C (n) ,

q2π (n), σn, v

= −i~σn

~wζB(n)

+ kC (n) , for n ∈ {1, 4, 5},

(71)

and hence the secular equation (45) becomes1 − σ 2

2

∆1 + 2σ2∆2 = 0, (72)

where

∆1 =

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

, ∆2 =

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

. (73)

At this stage one can check that if we neglect the presence of some thermal, voids or viscoelastic characteristics from thecontinuousmedium, thenwehave to be leftwithwell known results in the classical elasticity, thermoelasticity, Kelvin–Voigtviscoelasticity or porous elasticity.

I. Kelvin–Voigt viscoelastic materials. When we neglect the presence of thermal and porous properties from the medium,then the relation (72) reduces to the well known secular equation of the Kelvin–Voigt viscoelastic half space

1 − σ 22

2+ 4σ1σ2 = 0, (74)

where now we have

σ 21 = −1 −

ρw2

λ0 + 2µ0, σ 2

2 = −1 −ρw2

µ0. (75)

II. Kelvin–Voigt thermoviscoelastic materials. If we neglect the presence of porous properties from the medium, then thepropagation condition implies

σ 22 = −1 −

ρw2

µ0, (76)

and

ν1 =1

2~2kλ0 + 2µ0

−~wT0

β2

+ aλ0

+ 2µ0− ~2ρkw2

+√D

,

ν5 =1

2~2kλ0 + 2µ0

−~wT0

β2

+ aλ0

+ 2µ0− ~2ρkw2

−√D

,

(77)

where

D =~wT0

β2

+ aλ0

+ 2µ0+ ~2ρkw22

− 4~3w3ρT0akλ0

+ 2µ0 . (78)

Further, we now have

A(1)1 =

1~

ν1 +

aT0~k

w

, C (1)

= −iT0βw

kν1, (79)

and

A(5)1 = −

iT0β~

λ0 + 2µ0

, C (5)= T0

ν5 +

ρw2

λ0 + 2µ0

. (80)

Consequently, the secular equation for the Kelvin–Voigt thermoviscoelasticity is written in the form

F (w) ≡ σ5C (5)

1 − σ 22

T (1)22 + 2σ2T

(1)21

− σ1C (1)

1 − σ 2

2

T (5)22 + 2σ2T

(5)21

= 0. (81)

III. Viscoelastic materials with voids. We neglect the presence of thermal properties from the medium and then the prop-agation condition implies

~2α0(λ0+ 2µ0)ν2

+~2α0ρw2

+ (λ0+ 2µ0)(ξ 0

+ ρκ~2w2) − b0γ 0 ν + ρw2(ξ 0+ ρκ~2w2) = 0, (82)


with solutions

ν1 =−

~2α0ρw2

+ (λ0+ 2µ0)(ξ 0

+ ρκ~2w2) − b0γ 0+

√E

2~2α0(λ0 + 2µ0),

ν4 =−

~2α0ρw2

+ (λ0+ 2µ0)(ξ 0

+ ρκ~2w2) − b0γ 0−

√E

2~2α0(λ0 + 2µ0),

(83)

where

E =~2α0ρw2

+ (λ0+ 2µ0)(ξ 0

+ ρκ~2w2) − b0γ 02− 4~2α0ρw2(λ0

+ 2µ0)(ξ 0+ ρκ~2w2). (84)

Moreover, we now have

A(1)1 =

1α0~3

ξ 0

+ ~2(α0ν1 + ρκw2), B(1)

= −iγ 0ν1

α0~2,

A(4)1 =

ib0

~(λ0 + 2µ0), B(4)

= ν4 +ρw2

λ0 + 2µ0,

(85)

and

T (1)22 = i~

λ0

+ (λ0+ 2µ0)(ν1 − 1)

A(1)1 + b0B(1),

T (1)12 = −2i~µ0σ1A

(1)1 , h(1)

2 = −i~σ1α0B(1),

(86)

T (4)22 = i~

λ0

+ (λ0+ 2µ0)(ν4 − 1)

A(4)1 + b0B(4),

T (4)12 = −2i~µ0σ4A

(4)1 , h(4)

2 = −i~σ4α0B(4).

(87)

Then the secular equation is

E(w) ≡ σ4B(4)(1 − σ 2

2 )T (1)22 + 2σ2T

(1)21

− σ1B(1)

(1 − σ 2

2 )T (4)22 + 2σ2T

(4)21

= 0. (88)

6. Numerical applications

With the aim of illustrating theoretical results obtained in the preceding section we examine in detail, by means ofnumerical computations, the thermoviscoelastic material with voids considered by Sharma and Kumar in [25]. Thus, inwhat follows we take for the values of the thermoviscoelastic coefficients for copper material (thermoviscoelastic solid) asin [25], that is1

λ = 7.76 × 1011 dyn/cm2, µ = 3.86 × 1011 dyn/cm2, ϱ = 8.954 gm/cm3,

c = 3.4303 × 104 dyn/cm2 °C, b = 2 × 103 dyn/cm2, α = 1.688 dyn,

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

k = 0.386 × 108 dyn/s °C, T0 = 293 K, κ = 1.75 × 10−11 cm2,

(89)

and we setλ∗

= 0.1 dyn s/cm2, µ∗= 0.2 dyn s/cm2, b∗

= 0.1 × 10−3 dyn s/cm2,

ξ ∗= 0.3 dyn s/cm2, α∗

= 0.1 dyn s, γ ∗= 0.5 × 10−7 dyn s/cm2,

τ ∗= 0.3 × 10−7 dyn/°C, ζ = 1.5 × 10−11 dyn.

(90)

All calculations will be made by using the software package WolframMathematica and by considering the wave number tobe equal to 1 cm−1.

I. Kelvin–Voigt viscoelastic materials. Here we neglect the thermal and voids effects and hence we take


λ∗= 0.1 dyn s/cm2, µ∗

= 0.2 dyn s/cm2.(91)

If we introduce

c1 =

λ + 2µ

ρ, c2 =

µ

ρ, c∗

1 =

λ∗ + 2µ∗

ρ, c∗

2 =

µ∗

ρ,

C1 =~c∗2

1

c1, C2 =

~c∗22

c2,

(92)

1 In order to capture same order of magnitude effects from thermoviscoelasticity and porosity, we significantly modify the numerical values of b and ξ

of [25] and use b = 2 × 103 dyn/cm2 and ξ = 1.475 dyn/cm2 instead of b = 1.139 × 1011 dyn/cm2 and ξ = 1.475 × 1011 dyn/cm2 .


Fig. 1. The domains of viscoelastic materials characterized by C1 and C2 with the number of Rayleigh wave solutions.

Fig. 2. The graphic of |F (w)| in (98) in the complex plane for a thermoviscoelastic material with λ⋆= 0.1 dyn s/cm2, µ∗

= 0.2 dyn s/cm2 .

Fig. 3. Graphic of |F (w)| near the first root for w = −iv for a thermoviscoelastic material with λ∗= 1.49206 × 106 dyn s/cm2 and µ∗

= 2.23092 ×

106 dyn s/cm2 .


Fig. 4. Graphic of |F (w)| near the second root for w = −iv for a thermoviscoelastic material with λ∗= 1.49206 × 106 dyn s/cm2 and µ∗

= 2.23092 ×

106 dyn s/cm2 .

then the numerical calculation shows that

c1 = 415793 cm/s, c∗

1 = 0.236307 cm2/s

c2 = 207628 cm/s, c∗

2 = 0.149454 cm2/s,

C1 = 1.343 × 10−7 cm3/s, C2 = 1.07579 × 10−7 cm3/s,

(93)

so that the secular equation (74) has the unique value

v = (193636 − 0.0092358 i) cm/s. (94)

For the general case of a Kelvin–Voigt viscoelastic material, the secular equation (74) becomes2 +

w2

1 + C2w

2

− 4

1 +w2

c21c22

+ C1w

1 +

w2

1 + C2w= 0, (95)

which,when C1 and C2 vary in [0, 2], can have one or two solutions. In fact, as it can be seen in Fig. 1, in the class of viscoelasticmaterials with the viscosity coefficients so that (C1; C2) ∈ D1 there is a unique surfacewave solution, while for (C1; C2) ∈ D2there are two surface wave solutions. This agrees with the results reported by Currie et al. [30]. Numerical values for whichthere are two surfacewave solutions occur, for example (see Fig. 1), when C1 = 1 cm3/s and C2 = 1.2 cm3/s, that is for λ∗

=

1.49206 × 106 cm2/s and µ∗= 2.23092 × 106 cm2/s. The corresponding two values for v are

v1 = (491622 − 132535 i) cm/s, v2 = (167104 − 110819 i) cm/s. (96)

II. Kelvin–Voigt thermoviscoelastic materials. We neglect now the porous effects and hence, for the Kelvin–Voigt thermo-viscoelastic materials, we take the following constitutive coefficients


λ∗= 0.1 dyn s/cm2, µ∗

= 0.2 dyn s/cm2,

c = 3.4303 × 104 dyn/cm2 °C, β = 0.4 × 10−1 dyn/cm2 °C,k = 0.386 × 108 dyn/s °C, T0 = 293 K.

(97)

When we substitute the assigned values of the relevant parameters from the above equations, the secular equation (81), forw, is solved by means of the graphical method.2 For computing convenience we introduce the following function

F(Re(w), Im(w)) = |F (w)| =

|Re[F (w)]|2 + |Im[F (w)]|2, (98)

and we generate it graphically for Re(w) ∈ (−104, 0.0), Im(w) ∈ (−2 · 105, 0.0).

2 We used the software package Wolfram Mathematica.


Fig. 5. The contour plots of the |F (w)| along a path (λ∗(s), µ∗(s)) joining the numerical values of (λ∗, µ∗) = (0.1, 0.2) dyn s/cm2 (illustrated in Fig. 2)and (1.49, 2.23)×106 dyn s/cm2 (illustrated in Figs. 3 and 4).We notice (i) the evolution of the locus of the first non-zero root situated along the right-handside of each sub-figure and (ii) the occurrence of a second non-zero root near the upper-side of the sub-figures. In each contour plot Rew ∈ (−1.5c2, 0) (inthe horizontal direction) and Imw ∈ (−1.5c2, 0) (in the vertical one) so that the point w = 0 is located in the upper-right corner. Pictures are obtained byusing a linear path on the logarithmic scale joining the above numerical values used in Figs. 2–4.

As it can be seen in Fig. 2, there is a point v0 = v1 + v2i, where F(Re(w), Im(w)) = 0 and v1 is near to 193636 cm/s andv2 is near to zero. So, the attenuation in time factor is very small and the surface wave therefore travels at a speed and withan amplitude slightly lower than that in the isothermal case. However, for a thermoviscoelastic material having constitutivecoefficients given by (97), but λ∗

= 1.49206×106 dyn s/cm2 andµ∗= 2.23092×106 dyn s/cm2 the surfacewave problem

has two non-zero solutions as shown in Figs. 3 and 4. Otherwise stated, accounting for thermal effects lead to a new surfacewave independently of the number of surface wave solutions in the viscoelastic case. Fig. 3 shows the graphical illustrationof |F(Re(w), Im(w))| for w2 = Im(w) ∈ (0, 104) and w1 = Re(w) ∈ (0, 2 × 105). We notice a solution of the secularequation with w1 ≃ 2 × 105 (i.e., rapid attenuation in time) and small w2 ≃ 0.0 (i.e., low velocity). In Fig. 4 we illustrate|F(Re(w), Im(w))| for Re(w) ∈ (104, 2 × 104) and Im(w) ∈ (0, 2 × 105). Another solution is visible with large time factor(large w1) and high velocity (large w2).

The transition between a low viscosity regime (λ∗ and µ∗ small, one surface wave solution in the isothermal case) anda large one (large λ∗ nd µ∗, two surface waves solutions in the isothermal case) is illustrated in the sequence of contourplots of |F (w)|, (Fig. 5), as a function of (Re(w), Im(w)) for (λ∗, µ∗) along a path (linear on a logarithmic scale) joining thenumerical values used in Figs. 2–4. We notice (i) the shift of the surface wave solution from small attenuation factor in thecase of low viscosity to high attenuation factor for large values of (λ∗, µ∗) and (ii) the appearance of a second non-zerosolution near the axis w2 = 0 (or equivalent v1 = 0) in agreement with the result illustrated in Fig. 1 (for the isothermalcase). We also notice that, due to different time scales involved in viscoelastic and thermal phenomena, in order to captureboth effects we need to (slightly) modify the numerical values of material parameters b and ξ .


7. Concluding remarks

We investigated the surface wave propagation problem in a half space made of a thermoviscoelastic mediumwith voidswithin the context of the theory developed by Ieşan [5]. The results concluded from the above analysis can be summarizedas follows:

1. The secular equation is explicitly obtained for an isotropic thermoviscoelastic porous half space;2. The possible solutions of the surface wave problem are damped in time;3. {ur , ϕ, θ}, together with the corresponding state of stress, equilibrated stress and heat flux, decay asymptotically with

the deep in the inside of the considered half space, that is they decay to zero when x2 tends to infinity;4. The internal and dissipation energies associated with {ur , ϕ, θ} are finite;5. The thermal and viscous dissipation energies influence the attenuation in time and in deep of the half space of the

solutions in a thermoviscoelastic porous half space;6. Numerical simulation reveals that there can exist more than one solution of the surface wave propagation problem;7. The secular equation for Kelvin–Voigt is rediscovered;8. The secular equation is established for Kelvin–Voigt thermoviscoelastic materials, as well as for viscoelastic materials

with voids.

Acknowledgment

Support of the LEA Franco-Roumain Math-Mode is gratefully acknowledged.

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