On the Theory of Viscoelasticity for Materials with Double Porosity

DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2014.19.2335 DYNAMICAL SYSTEMS SERIES B Volume 19, Number 7, September 2014 pp. 2335–2352

ON THE THEORY OF VISCOELASTICITY FOR MATERIALS WITH DOUBLE POROSITY

Merab Svanadze Institute for Fundamental and Interdisciplinary Mathematics Research Ilia State University K. Cholokashvili Ave., 3/5, 0162, Tbilisi, Georgia

Abstract. In this paper the linear theory of viscoelasticity for Kelvin-Voigt materials with double porosity is presented and the basic partial differential equations are derived. The system of these equations is based on the equations of motion, conservation of fluid mass, the effective stress concept and Darcy’s law for materials with double porosity. This theory is a straightforward generalization of the earlier proposed dynamical theory of elasticity for materials with double porosity. The fundamental solution of the system of equations of steady vibrations is constructed by elementary functions and its basic properties are established. Finally, the properties of plane harmonic waves are studied. The results obtained from this study can be summarized as follows: through a Kelvin-Voigt material with double porosity three longitudinal and two transverse plane harmonic attenuated waves propagate.

1. Introduction. The concept of porous media is used in many areas of applied science (e.g., biology, biophysics, biomechanics) and engineering. There are a number of theories which describe mechanical properties of porous materials. The general 3D theory of consolidation for materials with single porosity was formulated in [7]. One important generalization of this theory that has been studied extensively started with the work [3], where a fissured porous medium is modeled as two com- pletely overlapping flow regions: one representing the porous matrix, and the other the fissure network. The theory of consolidation for elastic materials with double porosity was presented in [6, 34, 55]. This theory unifies the earlier proposed models of Barenblatt for porous media with double porosity [3] and Biot for porous media with single porosity [7]. With regard to the Aifantis’ quasi-static theory, the cross-coupled terms were included in the equations of fluid mass conservation in [35, 36]. The phenomenolog- ical equations of the quasi-static theory for double porosity media were established and the method to determine the relevant coefficients was presented in [4, 5, 39]. The governing system of equations for an anisotropic material with double porosity was obtained in [56]. In the governing equations of the above mentioned theories of poroelasticity the inertial term was neglected and quasi-static problems were investigated (see [4 - 7, 34 - 36, 39, 56]). On the other hand, the inertial effect plays a pivotal role in the investigation of various problems of vibrations and wave propagation through double porosity media. Therefore, it is important to study a full dynamic model

2010 Mathematics Subject Classiﬁcation. Primary: 74D05, 74F10; Secondary: 74A60, 74J05. Key words and phrases. Viscoelasticity, double porosity, Kelvin-Voigt material.

2335 2336 MERAB SVANADZE for materials with double porosity. The full dynamical system able to describe the deformation in single porosity media was developed in [8 - 10]. Recently, the linear theory of elastodynamics for the anisotropic nonhomogeneous materials with double porosity was considered in [44], and the uniqueness and stability of solution of the initial-boundary value problem were proved. The boundary value problems in the quasi-static and full dynamical cases of the theory of elasticity for double porosity materials were investigated by using the potential method (boundary integral method) and the theory of singular integral equations in [49, 50, 54]. The fundamental solutions of the system of equations of this theory were constructed by elementary functions in [48, 51, 52]. The basis properties of the plane harmonic waves were established and uniqueness theorems in the coupled linear theory of elasticity for solids with double porosity were proved in [13]. The double porosity model would consider the bone fluid pressure in the vascular porosity and the bone fluid pressure in the lacunar-canalicular porosity. The extensive review of the results in the theory of bone poroelasticity can be found in the survey papers [16, 17, 41]. In reality, bone is a viscoelastic material with triple porosity [16, 17, 33, 41]. There are three levels of bone porosity within cortical bone and within the trabec- ulae of cancellous bone, all containing a fluid. The pore sizes in cortical bone are of approximately three discrete magnitudes: the largest pore size (approx. 50 µm diameter) is associated with the vascular porosity, the second largest pore size (approx. 0.3 µm diameter) is associated with the lacunar-canalicular porosity, and the smallest pore size (approx. 10 nm diameter) is in the collagen-apatite porosity [16, 17, 41]. In the double porosity models the bone fluid pressure in the vascular porosity and the bone fluid pressure in the lacunar-canalicular porosity are considered, and the movement of the bone fluid in the collagen-apatite porosity is neglected. The classical theories of viscoelasticity was initiated by Maxwell, Meyer, Boltz- mann, and studied by Voigt, Kelvin, Zaremba, Volterra and others. An account of the historical developments of the theory of viscoelasticity as well as references to various contributions may be found in the books [1, 12, 26, 33], papers [2, 20-25, 27] and references therein. In the last decade, several mechanical theories of viscoelasticity and thermoviscoelasticity for Kelvin-Voigt materials were formulated. A nonlinear theory for a viscoelastic composite as a mixture of a porous elastic solid and a Kelvin-Voigt material was developed in [29]. A linear variant of this theory was presented in [40]. Some exponential decay estimates of solutions of equations of steady vibrations in the theory of viscoelastricity for Kelvin-Voigt materials were obtained in [11]. A theory of thermoviscoelastic composites modelled as interacting Cosserat continua was developed in [30]. In [31], the basic equations of the nonlinear theory of thermoviscoelasticity for Kelvin-Voigt materials with voids were established. Recently, the theory of thermoviscoelasticity for Kelvin-Voigt microstretch composite materials was presented in [38]. The main results in the theories of viscoelasticity and thermoviscoelasticity of differential and integral types were obtained in the series of papers [14, 15, 18, 19, 32, 42, 43, 45 - 47, 53]. In this paper the linear theory of viscoelasticity for Kelvin-Voigt materials with double porosity is presented and the basic partial differential equations are derived. The system of these equations is based on the equations of motion, conservation of fluid mass, the effective stress concept and Darcy’s law for material with double porosity. This theory is a straightforward generalization of the dynamical theory THEORY OF VISCOELASTICITY 2337 of elasticity for materials with double porosity [13, 44, 49, 52]. The fundamental solution of the system of equations of steady vibrations is constructed by elementary functions and its basic properties are established. Finally, the properties of plane harmonic waves are studied. The results obtained from this study can be summarized as follows: a Kelvin-Voigt material with double porosity admits three longitudinal and two transverse plane harmonic attenuated waves.

2. Basic equations. Let x = (x1, x2, x3) be a point of the Euclidean three- dimensional space R3, let t denote the time variable, t ≥ 0, uˆ(x, t) is the displacement vector in the solid, uˆ = (û1, uˆ2, uˆ3);p ˆ1(x, t) andp ˆ2(x, t) are the pore and fissure fluid pressures, respectively. We assume that the subscripts preceded by a comma denote partial differentiation with respect to the corresponding Cartesian coordinate, repeated indices are summed over the range (1,2,3), and the dot denotes differentiation with respect to t. The governing system of field equations in the linear theory of viscoelasticity for Kelvin-Voigt material with double porosity consists of the following equations: a) The equations of motion [8 - 10] ¨ 0 tlj,j = ρ(uˆl − Fl ), l = 1, 2, 3, (1) where tlj are the components of the total stress tensor, ρ is the reference mass 0 0 0 0 density, ρ > 0, F = (F1,F2,F3) is the body force per unit mass. b) The equations of fluid mass conservation [35, 36] (1) ˙ div v + ζ1 + β1e˙rr + γ(ˆp 1 − pˆ2) = 0, (2) and (2) ˙ div v + ζ2 + β2e˙rr − γ(ˆp 1 − pˆ2) = 0, (3) where v(1) and v(2) are the fluid flux vectors for the pores and fissures, respectively; elj are the components of the strain tensor, 1 e = (û +u ˆ ) , l, j = 1, 2, 3, (4) lj 2 l,j j,l β1 and β2 are the effective stress parameters, γ is the internal transport coefficient (leakage parameter) and corresponds to a fluid transfer rate respecting the intensity of flow between the pores and fissures, γ > 0; ζ1 and ζ2 are the increments of fluid (volumetric strain) in the pores and fissures, respectively, and defined by

ζ1 = α1 pˆ1 + α12 pˆ2, ζ1 = α21 pˆ1 + α2 pˆ2, (5)

α1 and α2 measure the compressibilities of the pore and fissure systems, respectively; α12 and α21 are the cross-coupling compressibilities for fluid flow at the interface between the two pore systems at a microscopic level [35, 36]. However, the coupling effect (α12 and α21) is often neglected in the literature [6, 34, 55]. In the following 2 2 we assume that β1 + β2 > 0 (the case β1 = β2 = 0 is too simple to be considered). c) The equations of effective stress concept (extending Terzaghi’s effective stress concept to double porosity) [4, 5, 35, 36, 55] 0 tlj = tlj − (β1pˆ1 + β2pˆ2) δlj, l, j = 1, 2, 3, (6) where 0 ∗ ∗ tlj = 2µelj + λerrδlj + 2µ e˙lj + λ e˙rrδlj 2338 MERAB SVANADZE are the components of effective stress tensor, λ, µ, λ∗ and µ∗ are the constitutive coefficients, δlj is the Kronecker’s delta. d) The Darcy’s law for materials with double porosity [4, 5, 39] 1 v(1) = − (κ gradp ˆ + κ gradp ˆ ) − ρ s(1), µ0 1 1 12 2 1 (7) 1 v(2) = − (κ gradp ˆ + κ gradp ˆ ) − ρ s(2), µ0 21 1 2 2 2 0 where µ is the fluid viscosity, κ1 and κ2 are the macroscopic intrinsic permeabilities associated with matrix and fissure porosity, respectively; κ12 and κ21 are the cross- coupling permeabilities for fluid flow at the interface between the matrix and fissure (1) (2) phases; ρ1, s and ρ2, s are the densities of fluid and the external forces (such as gravity) for the pore and fissure phases, respectively. The cross-coupling terms of (7) with coefficients κ12 and κ21 are considered by several authors [4, 5, 39]. However, the latter coupling effect (κ12 and κ21) is neglected in the literature [6, 34, 55]. Substituting equations (4) - (7) into (1) - (3), we obtain the following system of equations of motion in the full coupled linear theory of viscoelasticity for Kelvin- Voigt materials with double porosity expressed in terms of the displacement vector uˆ and pressuresp ˆ1 andp ˆ2: µ ∆uˆ + (λ + µ) grad div uˆ + µ∗∆uˆ˙ + (λ∗ + µ∗) grad div uˆ˙ ¨ 0 −β1 gradp ˆ1 − β2 gradp ˆ2 = ρ (uˆ − F ), (8) ˙ ˙ ˙ (1) k1 ∆p ˆ1 + k12 ∆p ˆ2 − α1 pˆ1 − α12 pˆ1 − γ(ˆp 1 − pˆ2) − β1 div uˆ = −ρ1 div s , ˙ ˙ ˙ (2) k21 ∆p ˆ1 + k2 ∆p ˆ2 − α21 pˆ1 − α2 pˆ2 + γ(ˆp 1 − pˆ2) − β2 div uˆ = −ρ2 div s , κ κ κ where ∆ is the Laplacian operator, k = j (j = 1, 2), k = 12 , k = 21 . j µ0 12 µ0 21 µ0 If the body force F0 and the external forces s(1) and s(2) are assumed to be absent, and the displacement vector uˆ and the pressuresp ˆ1 andp ˆ2 are postulated to have a harmonic time variation, that is, −iωt {uˆ, pˆ1, pˆ2} (x, t) = Re {u, p 1, p 2} (x) e , then from the system (8) we obtain the following system of frequency-domain equations describing plane waves under the full coupled linear theory of viscoelasticity for Kelvin-Voigt materials with double porosity 2 µ1 ∆u + (λ1 + µ1) grad div u − β1 grad p 1 − β2 grad p 2 + ρ ω u = 0,

(k1 ∆ + a1)p 1 + (k12 ∆ + a12)p 2 + iω β1 div u = 0, (9)

(k21 ∆ + a21)p 1 + (k2 ∆ + a2)p 2 + iω β2 div u = 0, ∗ ∗ where λ1 = λ − iωλ , µ1 = λ − iωµ , aj = iω αj − γ, alj = iω αlj + γ (l, j = 1, 2); ω is the oscillation frequency, ω > 0. Obviously, neglecting inertial effect in the first equation of (8), from (9) we obtain the system of homogeneous equations of steady vibrations in the full coupled linear quasi-static theory of viscoelasticity for solids with double porosity. We introduce the second order matrix differential operators with constant coefficients: THEORY OF VISCOELASTICITY 2339

2 1) 2 ∂ A(Dx) = (Alj(Dx))5×5 ,Alj(Dx) = (µ1∆ + ρω )δlj + (λ1 + µ1) , ∂xl∂xj ∂ ∂ Al;m+3(Dx) = −βm ,Am+3;l(Dx) = iωβm , ∂xl ∂xl

A44(Dx) = k1∆ + a1,A45(Dx) = k12∆ + a12,A54(Dx) = k21∆ + a21,

A55(Dx) = k2∆ + a2, m = 1, 2, l, j = 1, 2, 3. 2) 2 (o) (o) (o) ∂ A (Dx) = Alj (Dx) ,Alj (Dx) = µ1∆δlj + (λ1 + µ1) , 5×5 ∂xl∂xj (o) (o) (o) A44 (Dx) = k1 ∆,A45 (Dx) = k12 ∆,A54 (Dx) = k21 ∆, (o) (o) (o) A55 (Dx) = k2 ∆,Al;m+3(Dx) = Am+3;l(Dx) = 0, m = 1, 2, l, j = 1, 2, 3.

It is easily seen that the system (9) can be written as A(Dx)U(x) = 0, where 3 U = (u, p 1, p 2) is a ﬁve-components vector function and x ∈ R . The matrix (o) diﬀerential operator A (Dx) is called the principal part of the operator A(Dx).

Definition 2.1. The operator A(Dx) is said to be elliptic if [28] det A(o)(ξ) 6= 0, where ξ = (ξ1, ξ2, ξ3), |ξ|= 6 0. Obviously, we have (o) 2 10 det A (ξ) = µ1µ0 k |ξ| , where µ0 = λ1 + 2µ1, k = k1k2 − k12k21. Hence, A(Dx) is an elliptic differential operator if and only if µ1 µ0 k 6= 0. (10) Definition 2.2. The fundamental solution of system (9) (the fundamental matrix of operator A(Dx)) is the matrix Γ(x) = (Γlj(x))5×5 satisfying condition (in the class of generalized functions) [28]

A(Dx)Γ(x) = δ(x)J, (11) 3 where δ(x) is the Dirac delta, J = (δlj)5×5 is the unit matrix, x ∈ R . In the next section the matrix Γ is constructed in terms of elementary functions and some of its basic properties are established.

3. Fundamental solution of the system of steady vibrations equations. We consider the system of nonhomogeneous equations 2 µ1 ∆u + (λ1 + µ1) grad div u + iωβ1 grad p 1 + iωβ2 grad p 2 + ρ ω u = f,

(k1 ∆ + a1)p 1 + (k21 ∆ + a21)p 2 − β1 div u = f1, (12)

(k12 ∆ + a12)p 1 + (k2 ∆ + a2)p 2 − β2 div u = f2, where f is a three-components vector function, f1 and f2 are scalar functions on R3. As one may easily verify, the system (12) may be written in the form T A (Dx)U(x) = F(x), (13) T where A is the transpose of matrix A, F = (f, f1, f2) is a ﬁve-components vector function and x ∈ R3. 2340 MERAB SVANADZE

Applying the operator div to (12)1 from system (12) we obtain 2 (µ0∆ + ρ ω ) div u + iωβ1 ∆ p 1 + iωβ2 ∆ p 2 = div f,

(k1∆ + a1)p 1 + (k21∆ + a21)p 2 − β1 div u = f1, (14)

(k12∆ + a12)p 1 + (k2∆ + a2)p 2 − β2 div u = f2. From (14) we have B(∆)V (x) = ϕ (x) , (15) where V = (div u, p 1, p 2), ϕ = (ϕ1, ϕ2, ϕ3) = (div f, f1, f2) and  2  µ0∆ + ρω iωβ1 ∆ iωβ2 ∆   B(∆) = (Blj(∆)) =  −β1 k1∆ + a1 k21∆ + a21  . 3×3   −β k ∆ + a k ∆ + a 2 12 12 2 2 3×3 We introduce the notation 1 Λ1(∆) = det B (∆) . kµ0

It is easily seen that Λ1(−τ) = 0 is a cubic algebraic equation and there exist three 2 2 2 roots τ1 , τ2 and τ3 (with respect to τ). Then we have 2 2 2 Λ1(∆) = (∆ + τ1 )(∆ + τ2 )(∆ + τ3 ). The system (15) implies

Λ1(∆) V = Φ, (16) where 3 1 X ∗ Φ = (Φ1, Φ2, Φ3), Φj = Blj ϕl, j = 1, 2, 3 (17) kµ0 l=1 ∗ and Blj is the cofactor of element Blj of the matrix B. Now applying the operator Λ1(∆) to (12)1 and taking into account (16), we obtain Λ2(∆)u = F1, (18) 2 2 2 ρω where Λ2(∆) = Λ1(∆)(∆ + τ4 ), τ4 = and µ1 1 F1 = [Λ1(∆)f − (λ1 + µ1)grad Φ1 − iωβ1grad Φ2 − iωβ2grad Φ3] . (19) µ1 On the basis of (16) and (18) we get Λ(∆)U(x) = ψ(x), (20) where ψ = (F1, Φ2, Φ3) is a ﬁve-components vector and

Λ(∆) = (Λlj (∆))5×5 , Λ11(∆) = Λ22(∆) = Λ33(∆) = Λ2(∆),

Λ44(∆) = Λ55(∆) = Λ1(∆), Λlj(∆) = 0, l, j = 1, 2, ··· , 5, l 6= j. We introduce the notations

1 ∗ ∗ ∗ nj1(∆) = − (λ1 + µ1)Bj1(∆) + iωβ1Bj2(∆) + iωβ2Bj3(∆) , kµ1µ0 (21) 1 ∗ njl(∆) = Bjl(∆), j = 1, 2, 3, l = 2, 3. kµ0 THEORY OF VISCOELASTICITY 2341

In view of (17) and (21), from (19) it follows that 1 F1 = Λ1(∆)I + n11(∆) grad div f + n21(∆) gradf1 + n31(∆) gradf2, µ1 (22)

Φm = n1m(∆) divf + n2m(∆)f1 + n3m(∆)f2, m = 2, 3, where I = (δlj)3×3 is the unit matrix. Thus, from (22) we have T ψ (x) = L (Dx) F (x) , (23) where 1 ∂2 L (Dx) = (Llj (Dx))5×5 ,Llj (Dx) = Λ1(∆) δlj + n11(∆) , µ1 ∂xl∂xj ∂ ∂ Ll;m+2 (Dx) = n1m(∆) ,Lm+2;l (Dx) = nm1(∆) , (24) ∂xl ∂xl

Lm+2;4 (Dx) = nm2(∆),Lm+2;5 (Dx) = nm3(∆), l, j = 1, 2, 3, m = 2, 3. By virtue of (13) and (23), from (20) it follows that ΛU = LT AT U. It is obvious that LT AT = Λ and, hence,

A(Dx)L(Dx) = Λ(∆). (25) 2 2 We assume that τl 6= τj , where l, j = 1, 2, 3, 4 and l 6= j. Let 4 X Y(x) = (Ylm(x))5×5 ,Y11(x) = Y22(x) = Y33(x) = η2jγj(x), j=1

3 X (26) Y44(x) = Y55(x) = η1jγj(x),Ylm(x) = 0, j=1 l 6= m, l, m = 1, 2, ··· , 5, where eiτj |x| γ (x) = − (27) j 4π |x| 2 is the fundamental solution of Helmholtz’ equation, i.e., (∆ + τj )γj(x) = δ (x) and 3 4 Y 2 2 −1 Y 2 2 −1 η1m = (τl − τm) , η2j = (τl − τj ) , l=1, l6=m l=1, l6=j m = 1, 2, 3, j = 1, 2, 3, 4. Lemma 3.1. The matrix Y is the fundamental solution of operator Λ(∆), that is, Λ(∆)Y (x) = δ (x) J, (28) where x ∈ R3.

Proof. It suﬃces to show that Y11 and Y44 are the fundamental solutions of operators Λ2(∆) and Λ1(∆), respectively, i.e.,

Λ2(∆)Y11 (x) = δ (x) (29) 2342 MERAB SVANADZE and

Λ1(∆)Y44 (x) = δ (x) . Taking into account the equalities 2 2 2 2 η11 + η12 + η13 = 0, η12(τ1 − τ2 ) + η13(τ1 − τ3 ) = 0, 2 2 2 2 2 2 2 η13(τ1 − τ3 )(τ2 − τ3 ) = 1, (∆ + τl )γj(x) = δ (x) + (τl − τj )γj(x), l, j = 1, 2, 3, x ∈ R3, we have 3 2 2 X 2 2 Λ1(∆)Y44 (x) = (∆ + τ2 )(∆ + τ3 ) η1j δ (x) + (τ1 − τj )γj(x) j=1

3 2 2 X 2 2 = (∆ + τ2 )(∆ + τ3 ) η1j(τ1 − τj )γj(x) j=2

3 2 X 2 2 2 2 = (∆ + τ3 ) η1j(τ1 − τj ) δ (x) + (τ2 − τj )γj(x) j=2 2 = (∆ + τ3 )γ3(x) = δ (x) . Equation (29) is proved quite similarly.

We introduce the matrix

Γ (x) = L (Dx) Y (x) . (30) Using identities (25) and (28) from (30) we get

A (Dx) Γ (x) = A (Dx) L (Dx) Y (x) = Λ (∆) Y (x) = δ (x) J. Hence, Γ (x) is the solution of (11). We have thereby proved the following theorem. Theorem 3.2. If the condition (10) is satisﬁed, then the matrix Γ(x) deﬁned by (30) is the fundamental solution of system (9), where the matrices L (Dx) and Y (x) are given by (24) and (26), respectively.

Obviously, each element Γlj (x) of the matrix Γ (x) is represented in the following form Γlj (x) = Llj (Dx) Y11 (x) , Γlm (x) = Llm (Dx) Y44 (x) , (31) l = 1, 2, ··· , 5, j = 1, 2, 3, m = 4, 5.

∗ ∗ ∗ ∗ Remark 1. In the cases λ = µ = 0 and λ = µ = k12 = k21 = α12 = α21 = ρ = 0, the matrix Γ(x) is constructed in [52] and [48], respectively.

Remark 2. On the basis of operator L (Dx) and (30) we can obtain the Galerkin type representation of solution of system (9).

Remark 3. Obviously, the operator A (Dx) is not self adjoined. It is possible to construct the fundamental solution of adjoined operator in a quite similar manner. Remark 4. The matrix Γ (x) is constructed by 4 metaharmonic functions (solutions of the Helmholtz’ equation) γm (m = 1, 2, 3, 4) (see (27)). THEORY OF VISCOELASTICITY 2343

4. Basic properties of fundamental solution. Theorem 3.2 leads to the following results. Theorem 4.1. Each column of the matrix Γ (x) is a solution of homogeneous equation A(Dx)U(x) = 0 at every point x ∈ R3 except the origin. Theorem 4.2. If condition (10) is satisﬁed, then the fundamental solution of the system (o) A (Dx)U(x) = 0 is the matrix Ψ (x) = (Ψlj (x))5×5 , where 2 1 λ1 + µ1 ∂ Ψlj (x) = ∆ δlj − γ2(x) µ1 µ0 ∂xl∂xj

1 1 0 δlj 0 xlxj = γ2,lj(x) − Rljγ2(x) = λ + µ 3 , µ0 µ1 |x| |x|

k2 k12 Ψ44 (x) = γ1(x), Ψ45 (x) = − γ1(x), k k (32) k k Ψ (x) = − 21 γ (x), Ψ (x) = 1 γ (x), 54 k 1 55 k 1 ∂2 Ψlm = Ψml = 0,Rlj(Dx) = − ∆δlj, ∂xl∂xj λ + 3µ λ + µ λ0 = − 1 1 , µ0 = − 1 1 , l, j = 1, 2, 3, m = 4, 5. 8πµ1µ0 8πµ1µ0

It is easy to verify that R(Dx) = (Rlj(Dx))3×3 = curl curl. Obviously, theorem 4.2 leads to the following result. Corollary 1. The relations −1 −1 Ψlj (x) = O |x| , Ψmn (x) = O |x| (33) hold in the neighborhood of the origin, where l, j = 1, 2, 3 and m, n = 4, 5. We shall use the following lemma. Lemma 4.3. If condition (10) is satisﬁed, then

1 1 2 ∗ ∆n11(∆) = − Λ1(∆) + (∆ + τ4 )B11, µ1 kµ0

iω 2 n21(∆) = (∆ + τ4 )[β1(k2∆ + a2) − β2(k12∆ + a12)] , (34) kµ0

iω 2 n31(∆) = − (∆ + τ4 )[β1(k21∆ + a21) − β2(k1∆ + a1)] . kµ0 Proof. Taking into account the equalities (21) and 2 ∗ ∗ ∗ (µ0∆ + ρω )B11(∆) + iωβ1∆B12(∆) + iωβ2∆B13(∆) = detB we have 1 2 ∗ ∆n11(∆) = − detB − (µ1∆ + ρω )B11(∆) kµ1µ0

1 1 2 ∗ = − Λ1(∆) + (∆ + τ4 )B11. µ1 kµ0 2344 MERAB SVANADZE

The formulae (34)2 and (34)3 are proven in a quite similar manner. We introduce the notations (m) 1 ∗ 2 (4) 1 d11 = − 2 η1mB11(−τm), d11 = 2 , kµ0τm ρ ω (m) 2 (m) 2 (35) dq1 = η2mnq1(−τm), dq1 = η1mn1q(−τm), (m) 2 dqr = η1mnqr(−τm), m = 1, 2, 3, q, r = 2, 3. On the basis of lemma 4.3 we can rewrite the fundamental solution Γ (x) in the simple form for x 6= 0. We have the following result. Theorem 4.4. If x 6= 0, then 3 X (m) (4) Γlj (x) = d11 γm,lj(x) + d11 Rljγ4(x), m=1 3 3 X (m) X (m) Γl;q+2 (x) = d1q γm,l(x), Γq+2;l (x) = dq1 γm,l(x), (36) m=1 m=1 3 X (m) Γq+2;r+2 (x) = dqr γm(x), l, j = 1, 2, 3, q, r = 2, 3. m=1 Proof. Let x 6= 0. It is easy to verify that 2 2 1 ∂ ∆γm(x) = −τmγm(x), δljγm(x) = − 2 − Rlj γm(x), τm ∂xl∂xj (37) l, j = 1, 2, 3, m = 1, 2, 3, 4.

On the second hand from (34)1 it follows that

2 1 2 1 2 2 ∗ 2 n11(−τm) − 2 Λ1(−τm) = − 2 (τ4 − τm)B11(−τm), m = 1, 2, 3, 4. µ1τm kµ0τm (38) By virtue of (24), (26), (37) and (38) from (31) we have 4 1 ∂2 X Γ (x) = Λ (∆) δ + n (∆) η γ (x) lj µ 1 lj 11 ∂x ∂x 2m m 1 l j m=1 4 X 1 ∂2 = η Λ (−τ 2 ) δ + n (−τ 2 ) γ (x) 2m µ 1 m lj 11 m ∂x ∂x m m=1 1 l j 4 X 1 ∂2 ∂2 = η − Λ (−τ 2 ) − R + n (−τ 2 ) γ (x) 2m µ τ 2 1 m ∂x ∂x lj 11 m ∂x ∂x m m=1 1 m l j l j 4 X 1 ∂2 1 = η − (τ 2 − τ 2 )B∗ (−τ 2 ) + Λ (−τ 2 ) R γ (x). 2m kµ τ 2 4 m 11 m ∂x ∂x µ τ 2 1 m lj m m=1 0 m l j 1 m (39) On the basis of (35) and identities 2 2 (τ4 − τj )η2j = η1j, j = 1, 2, 3, 0 for m = 1, 2, 3 η Λ (−τ 2 ) = 2m 1 m 1 for m = 4 THEORY OF VISCOELASTICITY 2345 from (39) we obtain

3 2 4 X (m) ∂ X 1 Γ (x) = d γ (x) + η Λ (−τ 2 )R γ (x) lj 11 ∂x ∂x m µτ 2 2m 1 m lj 4 m=1 l j m=1 m 3 X (m) (4) = d11 γm,lj(x) + d11 Rljγ4(x). m=1 The other formulae of (36) can be proven quite similarly. Theorem 4.4 leads to the following result. Theorem 4.5. The relations −1 −1 Γlj (x) = O |x| , Γmq (x) = O |x| , (40) Γmj (x) = O (1) , Γjm (x) = O (1) hold in the neighborhood of the origin, where l, j = 1, 2, 3, m, q = 4, 5. Lemma 4.6. If condition (10) is satisﬁed, then 3 3 X (m) 1 X (m) 1 d = − , τ 2 d = − . (41) 11 ρω2 m 11 µ m=1 m=1 0 Proof. It is easy to verify that ∗ 2 4 2 B11(−τm) = kτm + (a12k21 + a21k12 − a1k2 − a2k1)τm + a, 2 (42) 2 2 2 ρω a τ1 τ2 τ3 = , kµ0 where a = a1a2 − a12a21. By virtue of (42) we obtain 3 X 1 B∗ (−τ 2) B∗ (−τ 2) η B∗ (−τ 2 ) = 11 1 + 11 2 τ 2 1m 11 m τ 2 (τ 2 − τ 2)(τ 2 − τ 2) τ 2 (τ 2 − τ 2)(τ 2 − τ 2) m=1 m 1 2 1 2 1 2 1 2 3 2

∗ 2 B11(−τ3 ) a kµ0 + 2 2 2 2 2 = 2 2 2 = 2 . τ3 (τ1 − τ3 )(τ2 − τ3 ) τ1 τ2 τ3 ρω Hence, from (35) we have 3 3 X (m) 1 X 1 1 d = − η B∗ (−τ 2 ) = − . 11 kµ τ 2 1m 11 m ρω2 m=1 0 m=1 m Similarly, by virtue of (42) we obtain 3 X B∗ (−τ 2) B∗ (−τ 2) η B∗ (−τ 2 ) = 11 1 + 11 2 1m 11 m (τ 2 − τ 2)(τ 2 − τ 2) (τ 2 − τ 2)(τ 2 − τ 2) m=1 2 1 3 1 1 2 3 2

∗ 2 B11(−τ3 ) + 2 2 2 2 = k. (τ1 − τ3 )(τ2 − τ3 ) Finally, from (35) we get 3 3 X (m) 1 X 1 τ 2 d = − η B∗ (−τ 2 ) = − . m 11 kµ 1m 11 m µ m=1 0 m=1 0 2346 MERAB SVANADZE

Now we can establish the singular part of the matrix Γ (x) in the neighborhood of the origin. Theorem 4.7. The relations

Γlj (x) − Ψlj (x) = const + O (|x|) (43) hold in the neighborhood of the origin, where l, j = 1, 2, ··· , 5. Proof. Let x 6= 0. In view of (32) and (36) we obtain

Γlj (x) − Ψlj (x) 2 " 3 # ∂ X (m) 1 1 1 (44) = d γ (x) − γ (x) + R γ (x) + γ (x) ∂x ∂x 11 m µ 2 lj ρω2 4 µ 2 l j m=1 0 1 for l, j = 1, 2, 3. In the neighborhood of the origin from (27) have

∞ n 1 X (iτm|x|) iτm γ (x) = − = γ (x) − − τ 2 γ (x) +γ ˜ (x), (45) m 4π|x| n! 1 4π m 2 m n=0 ∞ n 1 X (iτm|x|) whereγ ˜ (x) = − , m = 1, 2, 3, 4. Obviously, m 4π|x| n! n=3 2 γ˜m(x) = O |x| , γ˜m,j(x) = O (|x|) , γ˜m,lj(x) = const + O (|x|) , (46) l, j = 1, 2, 3, m = 1, 2, 3, 4. On the basis of (45) from (44) we get 3 3 X (m) 1 X (m) iτm d γ (x) − γ (x) = d γ (x) − +γ ˜ (x) 11 m µ 2 11 1 4π m m=1 0 m=1 (47) 3 ! X (m) 1 − τ 2 d + γ (x). m 11 µ 2 m=1 0 By virtue of equalities (41) from (47) it follows that 3 3 3 X (m) 1 1 i X (m) X (m) d γ (x) − γ (x) = − γ (x) − τ d + d γ˜ (x). 11 m µ 2 ρω2 1 4π m 11 11 m m=1 0 m=1 m=1 (48) Similarly, from (45) we have 1 1 1 iτ 1 γ (x) + γ (x) = γ (x) − 4 − τ 2γ (x) +γ ˜ (x) + γ (x) ρω2 4 µ 2 ρω2 1 4π 4 2 4 µ 2 1 1 (49) 1 iτ 1 = γ (x) − 4 + γ˜ (x). ρω2 1 4πρω2 ρω2 4

Taking into account (46), (48), (49) and ∆γ1(x) = 0 (x 6= 0) from (44) we obtain

2 3 (s) 1 ∂ X (m) 1 Γ (x) − Ψ (x) = − − R γ (x) + d γ˜ (x) + γ˜ (x) lj lj ρω2 ∂x ∂x lj 1 11 m ρω2 4 l j m=1 1 = − ∆γ (x) + const + O (|x|) = const + O (|x|) , l, j = 1, 2, 3. ρω2 1 The other formulae of (43) can be proven in a quite similar manner. THEORY OF VISCOELASTICITY 2347

Thus, on the basis of corollary 1 and theorem 4.7 the matrix Ψ (x) is the singular part of the fundamental solution Γ (x) in the neighborhood of the origin (see (33), (40) and (43)).

5. Plane harmonic waves. In this section we assume that k12 = k21 and α12 = α21 and these values are denoted by k3 and α3, respectively. We suppose that the following inequalities are true ∗ ∗ ∗ 2 µ > 0, λ + 2µ > 0, k1 > 0, k1k2 > k3, (50) 2 α1 > 0, α1α2 > α3, γ > 0. Let us suppose that plane harmonic waves corresponding to a wave number ξ and to an angular frequency ω propagate in the x1-direction through the porous solid with double porosity. Then displacement vector and pressures have the following form i(ξx1−ωt) i(ξx1−ωt) uˆ(x, t) = C e , pˆj(x, t) = Cj+3 e , (51) where C = (C1,C2,C3) is a constant vector, C4 and C5 are constant values, ω > 0, j = 1, 2. On the basis of (51), from the system of homogeneous equations µ ∆uˆ + (λ + µ) grad div uˆ + µ∗∆uˆ˙ + (λ∗ + µ∗) grad div uˆ˙ ¨ −β1 gradp ˆ1 − β2 gradp ˆ2 − ρ uˆ = 0, ˙ ˙ ˙ k1 ∆p ˆ1 + k3 ∆p ˆ2 − α1 pˆ1 − α3 pˆ1 − γ(ˆp 1 − pˆ2) − β1 div uˆ = 0, ˙ ˙ ˙ k3 ∆p ˆ1 + k2 ∆p ˆ2 − α3 pˆ1 − α2 pˆ2 + γ(ˆp 1 − pˆ2) − β2 div uˆ = 0, it follows that 2 2 [µ1 + (λ1 + µ1) δ1l] ξ − ρω Cl + iβ1ξδ1lC4 + iβ2ξδ1lC5 = 0,

2 2 β1ωξC1 + k1ξ − a1 C4 + k3ξ − a3 C5 = 0, (52)

2 2 β2ωξC1 + k3ξ − a3 C4 + k2ξ − a2 C5 = 0, where a3 = iωα3 + γ, l = 1, 2, 3. By virtue of (52) for C1,C4 and C5 we have the system 2 2 µ0ξ − ρω C1 + iβ1ξC4 + iβ2ξC5 = 0,

2 2 β1ωξC1 + k1ξ − a1 C4 + k3ξ − a3 C5 = 0, (53)

2 2 β2ωξC1 + k3ξ − a3 C4 + k2ξ − a2 C5 = 0. From (53) it follows that 2 L ξ Cl = 0, l = 1, 4, 5, (54) where 2 0 6 2 0 4 2 2 2 L ξ = µ0k ξ − µ0r + ρω k + iωr1 ξ + µ0a + ρω r + iωr2 ξ − ρω a (55) and 2 0 2 a = a1a2 − a3, k = k1k2 − k3, r = a1k2 + a2k1 − 2a3k3, 2 2 2 2 (56) r1 = k2β1 + k1β2 − 2k3β1β2, r2 = a2β1 + a1β2 − 2a3β1β2. 2348 MERAB SVANADZE

If ξ is a solution of the equation L ξ2 = 0, (57) then (54) has non-trivial solutions (C1,C4,C5). Similarly, from (52) we have the following equation for C2 and C3 2 T ξ Cj = 0, j = 2, 3, (58) 2 2 2 where T (ξ ) = µ1ξ − ρω . If ξ is a solution of the equation T ξ2 = 0, (59) then (58) has non-trivial solutions C2 and C3. The relations (57) and (59) will be called the dispersion equations of longitudinal and transverse plane harmonic waves, respectively. Obviously, (59) is the dispersion equation of transverse plane harmonic waves in the classical theory of viscoelasticity for Kelvin-Voigt materials without porosity (see [45, 46]) and, consequently, transverse waves in the theory of viscoelasticity for Kelvin-Voigt materials with double porosity have the same characteristics as the corresponding waves in the classical theory of viscoelasticity. Obviously, if ξ > 0, then the corresponding plane wave has constant amplitude, and if ξ is complex with Im ξ > 0, then the plane wave is attenuated as x1 → +∞. 2 2 2 2 Let ξ1 , ξ2 , ξ3 and ξ4 be roots of the equation L (ξ) = 0 (60) 2 2 −1 and T (ξ) = 0, respectively. From (59) it follows that ξ4 = ρω µ1 . Obviously, ξ1, ξ2, ξ3 and ξ4 are the wave numbers of longitudinal and transverse plane harmonic waves, respectively. We denote the longitudinal plane wave with wave number ξj (j = 1, 2, 3) by Pj (P-primary), and the transverse horizontal and vertical plane waves with wave number ξ4 by SH and SV , respectively (S-secondary). In what follows we use the following result. Lemma 5.1. If the condition (50) is satisﬁed, then the equation (60) has not a positive root. Proof. On the basis of (56) we have 2 2 r = −γk0 + iωα4, r2 = −γβ0 + iωβ3, a = −ω α − iωγα0, (61) where 2 α = α1α2 − α3, α0 = α1 + α2 + 2α3, α4 = α1k2 + α2k1 − 2α3k3, (62) 2 2 k0 = k1 + k2 + 2k3, β0 = β1 + β2, β3 = α1β2 + α2β1 − 2α3β1β2. By virtue of (50), from (56) and (62) we get 0 k > 0, k0 > 0, α > 0, α0 > 0, α4 > 0, β3 > 0. (63) On the other hand, keeping in mind (56), (61) and (62) we obtain 2 0 2 0 2 µ0r + ρω k + iωr1 = ρω k + ω µ2α4 − γλ2k0 + iω(λ2α4 + µ2γk0 + r1), 2 2 µ0a + ρω r + iωr2 = −ω (αλ2 + µ2γα0 + ργk0 + β3) (64) 2 2 2 +iω(ω αµ2 + ρω α4 − γα0λ2 − γβ0 ), ∗ ∗ where λ2 = λ + 2µ, µ2 = λ + 2µ . THEORY OF VISCOELASTICITY 2349

It is easy to see that (60) is an equation with complex coeﬃcients. Let ξ0 be the real root of (60). Separating real and imaginary parts in (60), on the basis of (55), (61) and (64) we obtain the following system 0 3 2 0 2 2 k λ2ξ0 − (ρω k + ω µ2α4 − γλ2k0)ξ0 2 4 −ω (αλ2 + µ2γα0 + ργk0 + β3)ξ0 + αρω = 0, (65) 0 3 2 k µ2ξ0 + (λ2α4 + µ2γk0 + r1)ξ0 2 2 2 2 −(ω αµ2 + ρω α4 − γα0λ2 − γβ0 )ξ0 − ρω α0γ = 0. As one may easily verify, the system (65) may be written in the form 2 0 2 2 2 (λ2ξ0 − ρω ) k ξ0 + γk0ξ0 − αω = ω ξ0 [µ2(α4ξ0 + γα0) + β3] , 2 0 2 2 2 (λ2ξ0 − ρω )(α4ξ0 + α0γ) + µ2ξ0 k ξ0 + γk0ξ0 − αω = −ξ0(r1ξ0 + γβ0 ). (66) 2 It is obvious that (λ2ξ0 − ρω )ξ0 6= 0 and, hence, from the system (66) it follows that 0 2 2 0 2 2 2 k ξ0 + γk0ξ0 − αω µ2 k ξ0 + γk0ξ0 − αω + r1ξ0 + γβ0 2 = −ω [µ2(α4ξ0 + γα0) + β3](α4ξ0 + γα0), and we have 4 3 2 c1ξ0 + c2ξ0 + c3ξ0 + c4ξ0 + c5 = 0, (67) where 0 0 c1 = µ2k , c2 = 2µ2k k0γ, 2 2 0 2 2 2 0 c3 = µ2γ k0 + γ r1k0 + k β0 + ω µ2 α4 − 2k α , (68) 2 2 2 2 c4 = ω (β3α4 − αr1) + 2ω γµ2 (α4α0 − k0α) + γ k0β0 , 2 2 2 2 2 2 2 c5 = ω µ2 α ω + α0γ + ω γ α0β3 − αβ0 . Taking into account (50), (63) and 2 0 2 2 2 2 2 2 2 α4 − 2k α = 2(k1k2 + k3)α3 − 4k3(k1α2 + k2α1)α3 + k1α2 + k2α1 + 2k3α1α2

2 2 2 = 2 (k1k2 + k3)α3 − k3(k1α2 + k2α1) k1k2 + k3 0 k 2 0 + 2 (k1α2 − k2α1) + 2k α1α2 > 0, k1k2 + k3 2 2 β3α4 − αr1 = k1(β1α2 − β2α3) + k2(β1α3 − β2α1)

−2k3(β1α2 − β2α3)(β1α3 − β2α1) ≥ 0 2 2 α4α0 − k0α = k1(α2 + α3) + k2(α1 + α3) − 2k3(α1 + α3)(α2 + α3) > 0 2 2 α0β3 − αβ0 = [β1(α2 + α3) − β2(α1 + α3)] ≥ 0 from (68) we obtain

cj > 0, j = 1, 2, ··· , 5, i.e. (67) is the equation with positive coeﬃcients and it has not a positive root.

Hence, the longitudinal and the transverse plane wave numbers ξ1, ξ2, ξ3 and ξ4 are complex values. We assume that Imξj > 0 (j = 1, 2, 3, 4). The amplitude change of a decaying plane wave can be expressed as

0 −Imξj x1 Cl = Cl e , j = 1, 2, 3, l = 1, 4, 5. 2350 MERAB SVANADZE

0 In this expression the amplitude Cl is the reduced amplitude after the wave has traveled a distance x1 from the initial location, Cl is the unattenuated amplitude of the propagating wave at the same location, the quantity Im ξj is the attenuation coefficient of the wave traveling in the x1-direction. Lemma 5.1 leads to the following result. Theorem 5.2. If condition (50) is satisfied, then through a Kelvin-Voigt material with double porosity five plane harmonic waves propagate: i) three longitudinal plane waves P1,P2,P3 with wave numbers ξ1, ξ2 and ξ3, respectively; these are attenuated waves as x1 → +∞, and ii) two transverse plane waves SH and SV with wave number ξ4; these are attenuated waves as x1 → +∞. Remark 5. In the quasi-static theory (the inertial term is neglected in the first equation of (8)), plane harmonic waves have the same properties (these properties are given in theorem 5.2) as in the above considered dynamical theory (the values c1, c2, ··· , c5 are independent on the density ρ). Remark 6. The basic properties of plane harmonic waves in elastic materials with double porosity and viscoelastic materials with voids are established in [13, 49] and [43, 46], respectively.

Acknowledgments. The author is grateful to the reviewers for useful comments.

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