Proc. R. Soc. A (2010) 466, 439–458doi:10.1098/rspa.2009.0360

Published online 21 October 2009

Strong ellipticity and progressive wavesin elastic materials with voids

BY STAN CHIRITA1 AND IONEL-DUMITREL GHIBA2,*1Faculty of Mathematics, Al. I. Cuza University of Iasi, Blvd. Carol I, No. 11,

700506 Iasi, Romania2‘Octav Mayer’ Mathematics Institute, Romanian Academy of Science,

Iasi Branch, Blvd. Carol I, No. 8, 700506 Iasi, Romania

In the present paper, we investigate a model for propagating progressive waves associatedwith the voids within the framework of a linear theory of porous media. Owing to theuse of lighter materials in modern buildings and noise concerns in the environment, suchmodels for progressive waves are of much interest to the building industry. The analysisof such waves is also of interest in acoustic microscopy where the identification of materialdefects is of paramount importance to the industry and medicine. Our analysis is based onthe strong ellipticity of the poroelastic materials. We illustrate the model of progressivewave propagation for isotropic and transversely isotropic porous materials. We also studythe propagation of harmonic plane waves in porous materials including the thermal effect.

Keywords: strong ellipticity; progressive waves; elastic materials with voids;thermoelastic materials with voids

1. Introduction

Wave motion in an elastic material containing voids is an area with immensepotential for practical applications. Many theories have been developed whichdescribe the behaviour of materials with voids and which have proved to be usefulin practice. In seismic zones, buildings are constructed with much lighter porousmaterials and typically have thinner walls. As a consequence, there is a great needto study the acoustic properties of porous materials including the nature of thesolid elastic matrix and the gas filling the pores, and the influence of temperatureon these quantities. But seismology represents only one of the many fields wherethe theory of materials with voids is applied. Medicine, various branches of biologyand the oil exploration industry are other important fields of application.

In this paper, we consider the theory of materials with voids introduced byCowin & Nunziato (1983). The theory of elastic materials with voids is thesimplest extension of the classical theory of elasticity. The basic idea of this theoryis to suppose that there is a distribution of voids throughout the elastic body.Consequently, the bulk density is written as the product of two fields: the matrixmaterial density field and the volume fraction field. This representation introduces

*Author for correspondence (ghiba_dumitrel@yahoo.com).

Received 13 July 2009Accepted 21 September 2009 This journal is © 2009 The Royal Society439

440 S. Chirita and I.-D. Ghiba

an additional degree of kinematic freedom and it was employed previously byGoodman & Cowin (1972) to develop a continuum theory of granular materials.The first investigations in the theory of thermoelastic materials with voids aredue to Nunziato & Cowin (1979) and Iesan (1986). The Nunziato–Cowin theoryhas enjoyed much success in predicting various effects especially within the remitof linear theory, e.g. Ciarletta & Iesan (1993), Iesan (2004) and the recent work ofQuintanilla (2001), Casas & Quintanilla (2005), Magaña & Quintanilla (2006a,b,2007), Ghiba (2008a) and the references therein. The intended applications ofthe theory concern the manufacture of building materials, e.g. plasterboard, brickand concrete.

The first aim of the present study is to establish the necessary and sufficientconditions characterizing the strong ellipticity of materials with voids. Suchconditions are important in discussing the uniqueness, wave propagation (e.g.Gurtin 1972, p. 86), loss of ellipticity in the context of the nonlinear elasticity offibre-reinforced materials (Merodio & Ogden 2003a) and also in the study of thespatial behaviour of the constrained anisotropic cylinders (Chirita & Ciarletta2006) or of constrained rectangular plates (Ghiba 2008b).

Simpson & Spector (1983) treated the strong ellipticity condition for isotropicnonlinearly elastic materials. They established necessary and sufficient conditionsfor the strong ellipticity of the equations governing an isotropic (compressible)nonlinearly elastic material at equilibrium. The main tool in their analysis consistsof a representation theorem for copositive matrices. A review on the strongellipticity for isotropic nonlinearly elastic materials is given by Dacorogna (2001).

Notable contribution concerning the strong ellipticity for the class of transverseisotropic linearly elastic materials is the work by Payton (1983) and the recentresults by Padovani (2002), Merodio & Ogden (2003b) and Chirita (2006). Forother symmetries of elastic materials, the strong ellipticity is studied by Chiritaet al. (2007), Chirita & Danescu (2008) and Han et al. (2009).

In this paper, the general necessary and sufficient conditions are establishedfor anisotropic centro-symmetric materials with voids but two types of materialsare discussed in detail: isotropic and transversely isotropic elastic materials withvoids. The theory of isotropic materials is a good approach for many problems inengineering and biological sciences. Ding et al. (2006) have presented methods forstudying the different types of problems which arise in the theory of transverselyisotropic elastic materials. Besides the well-known applications of this type ofmaterial in the mechanics of rocks (Wyllie 1999; Jaeger et al. 2007), transverselyisotropic materials are very useful in many branches of biology (Humphrey 2002;Dong & Guo, 2004; Goldmann et al. 2005). The recent studies of fibre-reinforcedcomposites (Spencer 1992; Bunsell & Renard 2005) and the modern technologiesalso encourage the study of transversely isotropic materials.

In the second part of this paper, we study the propagation of progressivewaves in elastic materials with voids. We also consider the propagation of wavesin thermoelastic isotropic materials with voids. We outline that the problemof wave propagation in materials with voids was studied in various works(Nunziato & Walsh 1977, 1978; Nunziato et al. 1978; Cowin & Nunziato 1983;Ciarletta & Straughan 2006, 2007a,b; Ciarletta et al. 2007) and a comprehensivereview of the results can be found in the book by Straughan (2008). The useof acceleration waves and related analysis has proved extremely useful in therecent investigations of wave motion in various dispersive and random media,

Proc. R. Soc. A (2010)

Strong ellipticity and progressive waves 441

in a variety of thermodynamic states, e.g. Ostoja-Starzewski & Trebicki (1999),Puri & Jordan (2004), Quintanilla & Straughan (2004), Christov & Jordan (2005),Jordan & Christov (2005), Jordan & Puri (2005) and Christov et al. (2006).

The present mathematical results can be used in laboratories for studying thespecific properties of materials with voids.

2. Basic equations of elastic model with voids

Throughout this section, B is a bounded regular region of three-dimensionalEuclidean space. We let B denote the closure of B, call ∂B the boundary ofB, and designate by n the outward unit normal of ∂B. We assume that the bodyoccupying B is a linearly elastic solid with voids. The body is referred to a fixedsystem of rectangular Cartesian axes Oxi (i = 1, 2, 3). Throughout this paper,Latin indices have the range 1–3, Greek indices have the range 1, 2 and the usualsummation convention is employed. Moreover, subscripts preceded by a commadenote partial differentiation with respect to the corresponding coordinate, whilea superposed dot means the time differentiation.

Let u be the displacement vector field over B, and let ϕ be the volume fractionfield over B. Then the components ers of the linear strain tensor are given by

ers = 12(ur ,s + us,r ) (2.1)

and the constitutive equations for a homogeneous and anisotropic elastic solidwith voids are (cf. Cowin & Nunziato 1983)

trs = Crsmnemn + Brsϕ + Drskϕ,k ,

hr = Arsϕ,s + Dmnremn + drϕ

and g = −Brsers − ξϕ − drϕ,r .

⎫⎪⎬⎪⎭ (2.2)

Here, we have used the notations: trs are the components of the stress tensor, hrare the components of the equilibrated stress vector, g is the intrinsic equilibratedbody force and Ars, Brs, Crsmn, Drsk, dr and ξ are constant constitutive coefficientssatisfying the following symmetries:

Ars = Asr, Brs = Bsr, Drsk = Dsrk, Cmnrs = Crsmn = Cnmrs. (2.3)

The specific internal energy W is given by

W (e, ϕ, grad ϕ) = 12Cmnrsemners + 1

2ξϕ2 + 1

2Arsϕ,rϕ,s

+ Brsersϕ + Drskersϕ,k + drϕϕ,r . (2.4)

The specific internal energy W0 for an isotropic and homogeneous elastic materialwith voids is defined by

W0(e, ϕ, grad ϕ) = μersers + 12λerress + 1

2ξϕ2 + 1

2αϕ,rϕ,r + βϕerr, (2.5)

where λ, μ, α, β and ξ are constant constitutive coefficients.

Proc. R. Soc. A (2010)

442 S. Chirita and I.-D. Ghiba

The transverse isotropy is characterized by the symmetry group consisting ofunit tensor 1 and the rotations R

θe3

(0 < θ < 2π), that is, the orthogonal tensorcorresponding to a right-handed rotation through the angle θ , 0 < θ < 2π , aboutthe axis in the direction of the unit vector e3. For a transversely isotropic elasticmaterial with voids, the specific internal energy W1 is defined by

W1(e, ϕ, grad ϕ) = 12c11(e2

11 + e222) + 1

2c33e2

33 + 2c55(e213 + e2

23) + 2c66e212

+ 12ξϕ2 + 1

2a11(ϕ

2,1 + ϕ2

,2) + 12a33ϕ

2,3 + c12e11e22

+ c13(e11 + e22)e33 + b11(e11 + e22)ϕ + b33e33ϕ, (2.6)

where

c11 = C1111, c33 = C3333, c55 = C1313, c13 = C1133, c12 = C1122,

c66 = 12(c11 − c12), a11 = A11, a33 = A33, b11 = B11, b33 = B33. (2.7)

The other constitutive coefficients are zero.In the absence of the body force and the extrinsic equilibrated body force, the

equations of motion of the elastic material with voids are given by (cf. Cowin &Nunziato 1983)

trs,r = us

and hr ,r + g = �ϕ.

}(2.8)

If we substitute equations (2.1) and (2.2) into equation (2.8), we obtain thebasic equations in terms of the displacement and volume fraction fields in the form

(Cmnrsur ,s + Bmnϕ + Dmnkϕ,k),m = un

and (Amnϕ,n + Drsmur ,s + dmϕ),m − Brsur ,s − ξϕ − drϕ,r = �ϕ,

}(2.9)

which for isotropic materials with voids becomes

μ�ur + (λ + μ)us,sr + βϕ,r = ur

and α�ϕ − ξϕ − βur ,r = �ϕ,

}(2.10)

where � is the Laplace operator.We have to note that the specific internal energy W0 is a positive definite

quadratic form in terms of the variables (e11, e22, e33, e12, e23, e31, ϕ, ϕ,1, ϕ,2, ϕ,3) ifand only if

μ > 0, α > 0, ξ > 0,(

λ + 23μ

)ξ > β2, (2.11)

Proc. R. Soc. A (2010)

Strong ellipticity and progressive waves 443

while W1 is a positive definite quadratic form in terms of the variables(e11, e22, e33, e12, e23, e31, ϕ, ϕ,1, ϕ,2, ϕ,3) if and only if

c11 > 0, c33 > 0, c55 > 0, ξ > 0, a11 > 0, a33 > 0,

|c12| < c11, |c13| <√

12(c11 + c12)c33

and(

c13 − b11b33

ξ

)2

<

(c33 − b2

33

ξ

)[12(c11 + c12) − b2

11

ξ

].

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(2.12)

3. Strongly elliptic elastic materials with voids

We will say that the elastic material with voids is strongly elliptic if

W (A, η, ζ) > 0 for all (A, η, ζ) �= 0, (3.1)

with A = m ⊗ n, m, n and ζ arbitrary vectors and η an arbitrary scalar.The strongly elliptic condition (3.1) can be written in an explicit way in thefollowing form:

12Cmnrsmmmrnnns + 1

2ξη2 + 1

2Arsζrζs + Brsmrnsη

+ Drskmrnsζk + drηζr > 0 for all (m ⊗ n, η, ζ) �= 0, (3.2)

where m = (m1, m2, m3), n = (n1, n2, n3) and ζ = (ζ1, ζ2, ζ3).We can observe that the strong ellipticity condition (3.1) assures the strong

ellipticity of the system of partial differential equations (2.9) (Fichera 1965). Wehave to note that the strong ellipticity condition (3.2) implies that

Cmnrsmmmrnnns > 0 for all unit vectors n, m, (3.3)

ξ > 0 (3.4)

and

Arsζrζs > 0 for all unit vectors ζ. (3.5)

The inequality (3.2) is the general strong ellipticity condition for an arbitraryanisotropic poroelastic body. Our objective here consists of finding explicitnecessary and sufficient conditions characterizing the strong ellipticity of elasticmaterials with voids. However, we believe that the results one obtains when theporous body has a centre of symmetry are revealing. When the poroelastic bodypossesses a centre of symmetry, the terms Drsk and dm are zero. Thus, for thecentro-symmetric elastic materials with voids, the strong ellipticity condition(3.2) reduces to

12Cmnrsmmmrnnns + 1

2ξη2 + 1

2Arsζrζs + Brsmrnsη > 0 for all (m ⊗ n, η, ζ) �= 0.

(3.6)

Proc. R. Soc. A (2010)

444 S. Chirita and I.-D. Ghiba

This condition is equivalent to the following inequalities:

Arsζrζs > 0 for all unit vectors ζ (3.7)

and

ξη2 + 2Brsmrnsη + Cmnrsmmmrnnns > 0

for all real scalar η and unit vectors m, n.(3.8)

Further, regarding equation (3.8) as a quadratic in η ∈ R, we deduce that it isequivalent to the following inequalities:

ξ > 0 (3.9)

and(Brsmrns)

2 < ξCmnrsmmmrnnns for all unit vectors m, n. (3.10)

This last condition can be written as

(ξCmnrs − BmnBrs)mmmrnnns > 0 for all unit vectors m, n, (3.11)

that is, the tensorCmnrs = ξCmnrs − BmnBrs (3.12)

is strongly elliptic.Concluding, we deduce that a centro-symmetric elastic material with voids is

strongly elliptic if and only if Ars is a positive definite tensor, ξ > 0 and Cmnrs isstrongly elliptic tensor.

In what follows, we will establish the explicit necessary and sufficient conditionscharacterizing the strong ellipticity of the isotropic and transversely isotropicelastic materials with voids.

(a) Isotropic elastic materials with voids

In this subsection, we characterize the strong ellipticity for the class of isotropicelastic materials with voids. To this end, we note that the strong ellipticitycondition (3.6) becomes

μmrmrnsns + 12λmrnrmsns + βηmrnr + 1

2ξη2 + 1

2αζrζr > 0, (3.13)

and it is equivalent toα > 0, ξ > 0 (3.14)

and

β2mrnrmsns < ξ [2μmrmrnsns + λmrnrmsns], for all unit vectors m, n. (3.15)

Further, we recall that we can write equation (3.15) in the form

2ξμ[(m1n2 − m2n1)

2 + (m2n3 − m3n2)2 + (m3n1 − m1n3)

2]+ [ξ(λ + 2μ) − β2](m1n1 + m2n2 + m3n3)

2 > 0, (3.16)

and note that it is equivalent to the following inequalities:

ξμ > 0

and ξ(λ + 2μ) − β2 > 0.

}(3.17)

Proc. R. Soc. A (2010)

Strong ellipticity and progressive waves 445

Concluding, we can say that an isotropic elastic material with voids is stronglyelliptic if and only if the elastic coefficients satisfy the following inequalities:

μ > 0,

ξ > 0,

α > 0

and ξ(λ + 2μ) > β2.

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

(3.18)

It appears clear from the relations (2.11) and (3.18) that the positivedefiniteness assumption of the specific internal energy W0 implies the strongellipticity of the elastic material with voids.

(b) Transversely isotropic elastic materials with voids

Many natural and man-made materials are classified as transversely isotropic.Such materials are characterized by the fact that one can find a line, which allowsa rotation of the material about it without changing its elastic properties. Theplane, which is perpendicular to this line (the axis of rotational symmetry), iscalled a plane of elastic symmetry or a plane of isotropy. Modern examples forsuch materials are laminates made of randomly oriented chopped fibres that arein general placed in a certain plane. The effective elastic material propertiesfor a bundled structure have no preferred direction in that plane, which thenbecomes a plane of elastic symmetry. Hence, each plane that contains the axisof rotation is a plane of symmetry, and therefore transversely isotropic elasticmaterial admits an infinite number of elastic symmetries. Thus, the transverseisotropy is appropriate to real materials having a laminate or a bundled structure.

The strong ellipticity condition (3.6) reduces to

c66(n1m2 − n2m1)2 + c11(n1m1 + n2m2)

2 + c33n23m

23 + c55(n2

3m21 + n2

1m23

+ n23m

22 + n2

2m23) + ξη2 + a11(ζ

21 + ζ 2

2 ) + a33ζ23 + 2(c13 + c55)(n1m1

+ n2m2)n3m3 + 2b11(n1m1 + n2m2)η + 2b33n3m3η > 0, (3.19)

and it is equivalent to the inequalities

a11 > 0, a33 > 0, ξ > 0, (3.20)

and the strong ellipticity of the elasticity tensor given by⎛⎜⎜⎜⎜⎜⎝

ξc11 − b211 ξc12 − b2

11 ξc13 − b11b33 0 0 0ξc12 − b2

11 ξc11 − b211 ξc13 − b11b33 0 0 0

ξc13 − b11b33 ξc13 − b11b33 ξc33 − b233 0 0 0

0 0 0 ξc55 0 00 0 0 0 ξc55 00 0 0 0 0 ξc66

⎞⎟⎟⎟⎟⎟⎠. (3.21)

In view of the results established by Merodio & Ogden (2003b), Chirita (2006) andChirita et al. (2007), the elasticity tensor defined in equation (3.21) is strongly

Proc. R. Soc. A (2010)

446 S. Chirita and I.-D. Ghiba

elliptic if and only if

ξc11 − b211 > 0, ξc33 − b2

33 > 0, (3.22)

ξc55 > 0, ξ(c11 − c12) > 0 (3.23)

and ∣∣∣∣c13 + c55 − b11b33

ξ

∣∣∣∣< c55 +√(

c11 − b211

ξ

)(c33 − b2

33

ξ

). (3.24)

Concluding, we deduce that a transversely isotropic elastic material with voidsis strongly elliptic if and only if the constitutive coefficients satisfy the relations(3.20), (3.22), (3.23) and (3.24).

4. Progressive waves in centro-symmetric elastic materials with voids

Throughout this section, we will consider plane progressive waves in elasticmaterials with voids having a centre of symmetry, that is, materials for whichwe have

Dijk = 0, dr = 0. (4.1)

By a plane progressive wave in a centro-symmetric elastic material with voids,we mean a couple of functions (ur , ϕ) on R

3 × (−∞, ∞) of the following form:

ur (x, t) = UrΛ(nsxs − ct)

and ϕ(x, t) = ΦΓ (nsxs − ct),

}(4.2)

where

(i) Λ and Γ are the real-valued functions of class C 2 on (−∞, ∞) with

d2Γ

ds2= −Γ �= 0,

dΛ

ds= Γ ; (4.3)

(ii) U = (U1, U2, U3) is the displacement amplitude vector, n = (n1, n2, n3)is a unit vector which represents the direction of propagation and(U1, U2, U3, Φ) �= 0;

(iii) c is the speed of propagation.

We say that the progressive wave is longitudinal if U and n are linearlydependent, that is, U × n = 0. It is transverse when U and n are perpendicular,that is, U · n = 0. For any given constant γ and for any given time t, thedisplacement and volume fraction fields (4.2) are constants on the plane nsxs −ct = γ . This last plane is perpendicular to n and it is moving with the speed c inthe direction n.

These plane waves are important in that they furnish valuable informationconcerning the propagation characteristics of elastic materials with voids.

If we substitute the plane progressive wave described by equation (4.2) intothe basic equations (2.9) and then we take into consideration the assumptions

Proc. R. Soc. A (2010)

Strong ellipticity and progressive waves 447

described into equations (4.1) and (4.3), we see that such waves are possible inan elastic material with voids if and only if

(Cmnrsnsnm − c2δrn)Ur + BmnnmΦ = 0

and BrsnsUr + (ξ + Arsnrns − �c2)Φ = 0.

}(4.4)

Such a linear algebraic system admits a non-zero solution (U1, U2, U3, Φ) if andonly if c2 satisfies the following algebraic equation:

det

⎛⎜⎝

Q11(n) Q12(n) Q13(n) Q14(n)Q21(n) Q22(n) Q23(n) Q24(n)Q31(n) Q32(n) Q33(n) Q34(n)Q41(n) Q42(n) Q43(n) Q44(n)

⎞⎟⎠= 0, (4.5)

whereQrs(n) = Cmsrnnnnm − c2δrs,

Qr4(n) = Q4r (n) = Bmrnm for r , s = 1, 2, 3

and Q44(n) = ξ + Arsnrns − �c2.

⎫⎪⎬⎪⎭ (4.6)

It is easy to see that all the solutions of equation (4.5) are real numbers. In fact,from equation (4.4), we have

c2(U 21 + U 2

2 + U 23 + �Φ2) = CmnrsUrUnnsnm

+ (ξ + Arsnrns)Φ2 + 2BrsUrnsΦ. (4.7)

In view of the strong ellipticity condition (3.6), we can see that the right-handside of equation (4.7) is a positive definite term and the corresponding roots c2 arepositive. Thus, all solutions c1, c2, c3 and c4 of equation (4.5) are real numbers.

Therefore, for every direction of propagation n = (n1, n2, n3) in a centro-symmetric elastic material with voids, there exist three orthogonal directionsof motion and four progressive waves propagating with the associated speeds ofpropagation c1, c2, c3 and c4.

(a) Isotropic poroelastic materials

For an isotropic poroelastic material, the progressive wave (4.2) can propagatewith the speeds c1 = c2, c3 and c4, where

c21 = c2

2 = μ

, (4.8)

c23 = 1

2�

{ξ + α + �(λ + 2μ) +

√[ξ + α − �(λ + 2μ)]2 + 4�β2

}(4.9)

and

c24 = 1

2�

{ξ + α + �(λ + 2μ) −

√[ξ + α − �(λ + 2μ)]2 + 4�β2

}. (4.10)

Proc. R. Soc. A (2010)

448 S. Chirita and I.-D. Ghiba

Moreover, for c = c1 = c2 =√μ/, we have the corresponding linear

independent solutions (U1, U2, U3, Φ)(1) and (U1, U2, U3, Φ)(2) of the algebraicsystem (4.4)

(U1, U2, U3, Φ)(1) = (−n3, 0, n1, 0)

and (U1, U2, U3, Φ)(2) = (−n2, n1, 0, 0),

}(4.11)

for the direction of propagation n = (n1, n2, n3). Thus, the characteristic space forc2 = c2

1 = c22 is generated by the linear independent vectors U(1) and U(2). It is

easy to see that we have

U(1) · n = 0, U(2) · n = 0, (4.12)

where U(1) = (−n3, 0, n1) and U(2) = (−n2, n1, 0). Therefore, the progressive wavespropagating with speeds c = c1 = c2 = √

μ/ are transverse waves.For c = c3, we have, up to a scalar factor, the corresponding solution

(U1, U2, U3, Φ)(3) of the algebraic system (4.4) given by

(U1, U2, U3, Φ)(3) = (n1c, n2c, n3c, β), (4.13)

where

c = �c23 − ξ − α, (4.14)

while for c = c4, we have the corresponding solution (U1, U2, U3, Φ)(4) of thealgebraic system (4.4) given by

(U1, U2, U3, Φ)(4) = (n1a, n2a, n3a, β), (4.15)

with

a = �c24 − ξ − α. (4.16)

Thus, it is easy to see that we have

U(3) × n = 0, U(4) × n = 0, (4.17)

where U(3) = cn and U(4) = an and hence the progressive waves propagating withspeeds c = c3 and c = c4 are longitudinal waves. The lines spanned by U(3) andU(4), respectively, are the characteristic spaces for c2

3 and c24.

Concluding, we can formulate the following result:

Proposition 4.1. For a strongly elliptic isotropic elastic material with voids, theprogressive wave defined in equation (4.2) is either one transverse propagating withthe speed c = c1 = c2 or it is one longitudinal propagating with the speeds c = c3or c = c4.

Thus, for strongly elliptic isotropic elastic materials with voids, there are onlytwo types of progressive waves: longitudinal and transverse.

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Strong ellipticity and progressive waves 449

(b) Transverse isotropic poroelastic materials

In this section, we consider the case of strongly elliptic transverse isotropicporoelastic materials. For these types of materials, the matrix Q defined byequation (4.5) is given by

Q11 = c11n21 + c66n2

2 + c55n23 − c2,

Q22 = c66n21 + c11n2

2 + c55n23 − c2,

Q33 = c55(n21 + n2

2) + c33n23 − c2,

Q44 = ξ + a11(n21 + n2

2) + a33n23 − �c2,

Q12 = Q21 = (c12 + c66)n1n2,

Q13 = Q31 = (c13 + c55)n1n3,

Q14 = Q41 = b11n1,

Q23 = Q32 = (c13 + c55)n2n3,

Q24 = Q42 = b11n2

and Q34 = Q43 = b33n3.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(4.18)

Then equation (4.5) reduces to

[c66(n21 + n2

2) + c55n23 − c2]D = 0, (4.19)

where

D = det(qij), (4.20)

with

q11 = c11(n21 + n2

2) + c55n23 − c2,

q22 = c55(n21 + n2

2) + c33n23 − c2,

q33 = ξ + a11(n21 + n2

2) + a33n23 − �c2,

q12 = (c13 + c55)n3, q21 = (c13 + c55)(n21 + n2

2)n3,

q13 = b11, q31 = b11(n21 + n2

2)

and q23 = q32 = b33n3.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(4.21)

Thus, we can observe that the progressive wave can propagate in an arbitrarydirection n with the first speed

c21 = 1

[c66 + (c55 − c66)n2

3 ]. (4.22)

The other speeds c are the solutions of the equation

c6 + Λ1c4 + Λ2c2 + Λ3 = 0, (4.23)

Proc. R. Soc. A (2010)

450 S. Chirita and I.-D. Ghiba

where

Λ1 = −1

[ξ

�+(a11

�+ c11 + c55

)+(

a33 − a11

�+ c33 − c11

)n2

3

],

Λ2 = 12

{[(c11 + c55) + (c33 − c11)n2

3 ](

ξ

�+ a11

�+ a33 − a11

�n2

3

)

+ [c11 + (c55 − c11)n23 ][c55 + (c33 − c55)n2

3 ]

− (c13 + c55)2(1 − n2

3)n23 − b2

11

�− b2

33 − b211

�n2

3

},

Λ3 = − 13

(ξ

�+ a11

�+ a33 − a11

�n2

3

){[c11 + (c55 − c11)n2

3 ][c55 + (c33 − c55)n23 ]

− (c13 + c55)2(1 − n2

3)n23} + 1

3�{b2

11[c55 + (c33 − c55)n23 ] × (1 − n2

3)

+ b233[c11 + (c55 − c11)n2

3 ]n23 − 2b11b33(c13 + c55)(1 − n2

3)n23}.

In what follows, we will discuss the following two directions of propagation: (i)the progressive wave propagates in the plane of isotropy and (ii) the progressivewave propagates along the axis of rotational symmetry Ox3.

Case (i): n3 = 0In this case, the speeds of propagation are given by

c21 = 1

c66,

c22 = 1

c55,

c23 = 1

2�

[ξ + a11 + �c11 −

√(ξ + a11 − �c11)2 + 4�b2

11

]

and c24 = 1

2�

[ξ + a11 + �c11 +

√(ξ + a11 − �c11)2 + 4�b2

11

].

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(4.24)

The solutions of the system (4.4) corresponding to the speeds c1, c2, c3 and c4are, respectively, given by

(U1, U2, U3, Φ)(1) = (−n2, n1, 0, 0), (4.25)

(U1, U2, U3, Φ)(2) = (0, 0, 1, 0), (4.26)

(U1, U2, U3, Φ)(3) = ((�c23 − ξ − a11)n1, (�c2

3 − ξ − a11)n2, 0, b11) (4.27)

and (U1, U2, U3, Φ)(4) = ((�c24 − ξ − a11)n1, (�c2

4 − ξ − a11)n2, 0, b11). (4.28)

Proc. R. Soc. A (2010)

Strong ellipticity and progressive waves 451

In conclusion, we have the following result:

Proposition 4.2. For a strongly elliptic transversely isotropic elastic materialwith voids, the progressive waves which propagate in the plane of isotropy are twotransverse propagating with the speeds c1 or c2 and two longitudinal propagatingwith the speeds c3 or c4.

Case (ii): n1 = 0, n2 = 0If the direction of propagation is along of the axis of rotational symmetry, then

the speeds of propagation are

c21 = c2

2 = 1

c55,

c23 = 1

2�

[ξ + a33 + �c33 −

√(ξ + a33 − �c33)2 + 4�b2

33

]

and c24 = 1

2�

[ξ + a33 + �c33 +

√(ξ + a33 − �c33)2 + 4�b2

33

].

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

(4.29)

For the double solution of equation (4.5), the corresponding solutions of thealgebraic system (4.4) are given by

(U 1, U 2, U 3, Φ)(1) = (1, 0, 0, 0)

and (U 1, U 2, U 3, Φ)(2) = (0, 1, 0, 0),

}(4.30)

while the corresponding solutions for the speeds c3 and c4 are

(U1, U2, U3, Φ)(3) = (0, 0, � c23 − ξ − a33, b33)

and (U1, U2, U3, Φ)(4) = (0, 0, � c24 − ξ − a33, b33).

}(4.31)

Thus, we can give the following result:

Proposition 4.3. For a strongly elliptic transversely isotropic elastic materialwith voids, the progressive waves which propagate along the axis of isotropy aretwo transverse propagating with the speed c1 = c2 and two longitudinal propagatingwith the speeds c3 or c4.

5. Thermal effects

In this section, we study the plane progressive waves in thermoelastic isotropicmaterials with voids. In the absence of the body force, the extrinsic equilibratedbody force and the heat source, the equations which describe the behaviour ofthe thermoelastic material with voids are given by (cf. Iesan 2004)

μ�ur + (λ + μ)uj ,jr + βϕ,r − bθ,r = ur ,

α�ϕ − βuj ,j − ξϕ + mθ = �ϕ

and k�θ − T0buj ,j − T0mϕ = T0aθ ,

⎫⎪⎬⎪⎭ (5.1)

Proc. R. Soc. A (2010)

452 S. Chirita and I.-D. Ghiba

where λ, μ, α, β, ξ , a, b, m and k are constitutive coefficients which describe thethermal properties of material and T0 is the ambient absolute temperature. Wesuppose the material to be strongly elliptic. As a consequence of the entropyinequality, we have that k is a positive constant.

By a plane harmonic wave in a thermoelastic material with voids, we mean aset of functions (ur , ϕ, θ) on R

3 × (−∞, ∞) of the following form:

ur (x, t) = Re{Ur exp[i(nsxs − ct)]},ϕ(x, t) = Re{Φ exp[i(nsxs − ct)]}

and θ(x, t) = Re{Θ exp[i(nsxs − ct)]},

⎫⎪⎬⎪⎭ (5.2)

where U = (U1, U2, U3) denotes the direction of motion, Φ, Θ are complexconstants, n = (n1, n2, n3) is the direction of propagation and (U1, U2, U3, Φ, Θ) �=0 and Re[c] is the speed of propagation. If Im[c] less than 0, then the waves aredamped, whereas if c is a real number the waves are undamped.

By a direct substitution of the expressions of the plane progressivethermoelastic wave described by equation (5.2) into the basic equation (5.1),we obtain that such waves are possible in a thermoelastic material with voids ifand only if

[(μ − c2)δrj + (λ + μ)nrnj ]Uj − iβnrΦ + ibnrΘ = 0,

iβnjUj + (α + ξ − �c2)Φ − mΘ = 0

and T0bcnjUj − iT0mcΦ + (k − iT0ac)Θ = 0.

⎫⎪⎬⎪⎭ (5.3)

This linear algebraic system admits a non-zero solution (U1, U2, U3, Φ, Θ) ifand only if c is a solution of the following equation:

(c2 − μ)2

∣∣∣∣∣∣λ + 2μ − c2 −iβ ib

iβ ξ + α − �c2 −mT0bc −iT0cm k − iT0ac

∣∣∣∣∣∣= 0, (5.4)

that is,

(c2 − μ)2(iΥ5c5 + Υ4c4 + iΥ3c3 + Υ2c2 + iΥ1c + Υ0) = 0, (5.5)

where

Υ5 = −aT0�2, Υ4 = k�2,

Υ3 = T0{m2 + aξ + b2� + a[α + �(λ + 2μ)]},Υ2 = −k[ξ + α + �(λ + 2μ)],Υ1 = −T0{−2bmβ + m2(λ + 2μ) + b2ξ + a[−β2 + (λ + 2μ)ξ ]

+ α[b2 + a(λ + 2μ)]}and Υ0 = k[−β2 + (λ + 2μ)(ξ + α)].

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(5.6)

Obviously, the above equation has the double root

c = c1 = c2 =√

μ

. (5.7)

Proc. R. Soc. A (2010)

Strong ellipticity and progressive waves 453

For these values of c, we have the following solution of system (5.3):

(U1, U2, U3, Φ, Θ) = (−(n2 + n3), n1, n1, 0, 0) if n1 �= 0,

(U1, U2, U3, Φ, Θ) = (n2, −(n1 + n3), n2, 0, 0) if n2 �= 0

and (U1, U2, U3, Φ, Θ) = (n3, n3, −(n1 + n2), 0, 0) if n3 �= 0.

⎫⎪⎬⎪⎭ (5.8)

It is easy to see that these progressive waves are transverse waves. If wetake into consideration the case of transverse waves (U · n = 0), then fromequations (5.3) we find that the speeds defined by equation (5.7) are the onlyadmissible speeds for this type of waves.

In the case of longitudinal waves, we have U · n = U , where U = |U|. For thecase of longitudinal waves the system (5.3) becomes

(λ + 2μ − c2)U − iβΦ + ibΘ = 0,

iβU + (α + ξ − �c2)Φ − mΘ = 0

and T0bcU − iT0mcΦ + (k − iT0ac)Θ = 0.

⎫⎪⎬⎪⎭ (5.9)

This system has a non-trivial solution if and only if c is a solution of the equation

iΥ5c5 + Υ4c4 + iΥ3c3 + Υ2c2 + iΥ1c + Υ0 = 0. (5.10)

If b �= 0 or m �= 0, then the solutions of this equation are not real numbers. Infact, if we suppose that there is a real solution of the above equation, then it hasto satisfy both equations below

c(Υ5c4 + Υ3c2 + Υ1) = 0

and

Υ4c4 + Υ2c2 + Υ0 = 0.

⎫⎪⎬⎪⎭ (5.11)

But the poroelastic material is strongly elliptic and thus we have that c = 0 cannotbe the solution of equation (5.11). Actually, for b �= 0 or m �= 0, the equation (5.11)do not have any common solution. In consequence, in this case, equation (5.10)has only the solutions in C\R.

If b → 0 and m → 0, then equations (5.11) have two common solutions. Thiscase will be discussed as a particular case.

In view of the above discussion, we can conclude as follows:

Proposition 5.1. For a thermoelastic isotropic material with voids for whichb �= 0 or m �= 0, the progressive wave defined in equation (5.2) is either oneundamped transverse wave propagating with the speeds c = c1 = c2 or there are threedamped longitudinal waves propagating with the speeds c which are the solutionsof the equation (5.10).

In the linear theory of thermoleastic materials with voids, there are threecoupling coefficients: β, b and m. The pair (b, m) couples equation (5.1) describingthe poroelastic behaviour of the material with the heat equation (5.1), while thepair (β, b) couples equation (5.1) which describes purely the elastic behaviour ofmaterial with the porosity and heat equation (5.1).

Proc. R. Soc. A (2010)

454 S. Chirita and I.-D. Ghiba

In what follows, we use the convergence results established by Chirita &Ciarletta (2008), in order to discuss how the thermoelastic progressive wavebehaves as the coupling coefficients β, b and m tend to zero. This discussionhelps us to compare the results obtained in thermoporoelastic theory withthose established in the poroelastic theory and also with the results from theclassical elasticity.

Case (i): b → 0, m → 0 and β �= 0In this case, the possible propagating speeds are

c21 = c2

2 = μ

,

c23 = 1

2�

{ξ + α + �(λ + 2μ) +

√[ξ + α − �(λ + 2μ)]2 + 4�β2

},

c24 = 1

2�

{ξ + α + �(λ + 2μ) −

√[ξ + α − �(λ + 2μ)]2 + 4�β2

}

and c5 = − ikaT0

.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(5.12)

For the first four speeds, we retrieve the poroelastic case discussed in the §4a.The thermal wave is missing for these speeds and the poroelastic waves are notaffected by the thermal field.

For the fifth speed, we can observe that we cannot have a poroelastic wave butwe have a damped thermal wave which is not affected by the poroelastic field.

This behaviour is possible because the equations are uncoupled. In the coupledcase, the thermal field affects the poroelastic waves and the poroelastic field affectsthe thermal waves.

Case (ii): β → 0, b → 0 and m �= 0In this uncoupled case, the possible propagating speeds are

c21 = c2

2 = μ

,

c23 = λ + 2μ

,

c4 = i3aT0

[−k + 1

�

(3√

2pq

− q3√

2

)],

c5 = i3aT0

⎡⎣−k − 1

2�

⎛⎝ 3

√2(1 + i

√3)

p

q−(1 − i

√3)

q3√

2

⎞⎠⎤⎦

and c6 = i3aT0

⎡⎣−k − 1

2�

⎛⎝ 3

√2(1 − i

√3)

p

q−(1 + i

√3)

q3√

2

⎞⎠⎤⎦

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(5.13)

Proc. R. Soc. A (2010)

Strong ellipticity and progressive waves 455

where

p = �{−k2� + 3aT 2[m2 + a(ξ + α)]},� = 4[−k2� + 3aT 2

0 (m2 + a(ξ + α))]3+ k2�[9aT 2

0 (−m2 + 2aξ) + 2(9a2T 20 α + k2�)]2

and q ={{[−9am2T 2

0 + 2k2� + 18a2T 20 (ξ + α)] + �

√�}1/3 if � ≥ 0

{[−9am2T 20 + 2k2� + 18a2T 2

0 (ξ + α)] + i�√−�}1/3 if � < 0.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

(5.14)The waves with the propagating speed c2

1 are discussed in the general case.Let us now consider the progressive wave corresponding to the speed c2

3. For thisspeed, up to a scalar factor, we find

(U1, U2, U3, Φ, Θ) = (n1, n2, n3, 0, 0). (5.15)

We can observe that we retrieve the longitudinal progressive wave from theclassical elasticity. This fact is a consequence of the decoupling of equations whichdescribe the mechanical behaviour by the porosity equation and heat equation.

Regarding the progressive waves which propagate with the speeds c4,5,6,we obtain

(U1, U2, U3, Φ(4,5,6), Θ) = (0, 0, 0, k − iT0ac4,5,6, T0mc4,5,6). (5.16)

These are porothermal waves which are not affected by the classical mechanicalfield but are affected by the presence of voids.

Case (iii): β → 0, b → 0 and m → 0This is a fully uncoupled case. In this case, the waves are purely mechanical

waves with the speedsc21 = c2

2 = μ

and c23 = λ + 2μ

⎫⎪⎪⎬⎪⎪⎭ (5.17)

or purely porous waves with the speed

c4 = α + ξ

�(5.18)

or purely thermal waves with the speed

c5 = − ikaT0

. (5.19)

Moreover, each field (mechanical, porosity, thermal field) do not affect thewaves from the other fields.

6. Concluding remarks

The main purpose of this paper was to study the strong ellipticity condition inthe theory of materials with voids. To this end, the strong ellipticity conditionwas formulated for general anisotropic elastic materials with voids and relevant

Proc. R. Soc. A (2010)

456 S. Chirita and I.-D. Ghiba

results were reported concerning the explicit necessary and sufficient conditionscharacterizing the strong ellipticity of the isotropic and transversely isotropicelastic materials with voids. We have to outline that similar results can beobtained for other elastic materials with voids with special symmetries as, forexample, those with rhombic symmetry.

Supposing the material to be strongly elliptic, we studied the propagation ofprogressive waves in isotropic and transversely isotropic poroelastic bodies. It isshown that there are only transverse and longitudinal undamped waves and theexplicit expressions are presented for the speeds of propagation. In the contextof strongly elliptic poroelastic materials, we can also give a good description ofthe surface waves. The analysis of the behaviour of surface waves in poroelasticmaterials will be treated in a forthcoming paper.

In the last section, we studied the propagation of harmonic plane waves inisotropic thermoelastic materials with voids. Then, there are two undampedtransverse harmonic waves and three damped longitudinal harmonic waves. Whenthe thermal effect is taken into consideration, there are three coefficients couplingthe equations that describe purely the elastic behaviour of material with theporosity and heat equations. In the fully coupled case, we give a characterizationresult and then we show how the coupling coefficients affect the propagationof harmonic plane waves. This discussion proves that the present results are inconcordance with the results from the classical elasticity and also with the resultsestablished for poroelastic materials.

This research of the authors was supported by the Romanian Ministry of Education and Research,grant ID-401, 15/28.09.2007. The authors would like to express their gratitude to the referees fortheir careful assessment as well as for their fruitful suggestions regarding the initial version ofthis paper.

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