Potential Method in the Linear Theory of Viscoelastic Materials with Voids

In the present paper the linear theory of viscoelasticity for Kelvin–Voigt materials with voids is considered and some basic results of the classical theory of elasticity are generalized. Indeed, the basic properties of plane harmonic waves are established. The explicit expression of fundamental solution of the system of equations of steady vibrations is constructed by means of elementary functions. The Green’s formulas in the considered theory are obtained. The uniqueness theorems of the internal and external basic boundary value problems (BVPs) are proved. The representation of Galerkin type solution is obtained and the completeness of this solution is established. The formulas of integral representations of Somigliana type of regular vector and regular (classical) solution are obtained. The Sommerfeld-Kupradze type radiation conditions are established. The basic properties of elastopotentials and singular integral operators are given. Finally, the existence theorems for classical solutions of the internal and external basic BVPs of steady vibrations are proved by using of the potential method (boundary integral method) and the theory of singular integral equations.

Viscoelastic materials play an important role in many branches of civil engineering, geotechnical engineering, technology and, in recent years, biomechanics. Viscoelastic materials, such as amorphous polymers, semicrystalline polymers, and biopolymers, can be modeled in order to determine their stress or strain interactions as well as their temporal dependencies. Study of bone viscoelasticity is best placed in the context of strain levels and frequency components associated with normal activities and with applications of diagnostic tools (see, Lakes [5]). The investigations of the solutions of viscoelastic wave equations and the attenuation of seismic wave in the viscoelastic media are very important for geophysical prospecting technology. In addition, the behavior of viscoelastic porous materials can be understood and predicted in great detail using nano-mechanics. The applications of these materials are many. One of the applications may be to the NASA space program, such as the prediction of soils behavior in the Moon and Mars (for details, see, Voyiadjis and Song [6], Polarz and Smarsly [7], Chen et al. [8] and references therein).
A great attention has been paid to the theories taking into account the viscoelastic effects (see, Amendola et al. [4], Fabrizio and Morro [9], Di Paola and Zingales [10,11]). The existence and the asymptotic stability of solutions in the linear theory of viscoelasticity for solids is investigated by Fabrizio and Lazzari [12], and Appleby et al. [13]. The main results on the free energy in the linear viscoelasticity are obtained in the series of papers [14][15][16][17][18][19][20][21]. A general way to provide existence of solutions of the initial and boundary value problems for linear viscoelastic bodies is provided without the need of appealing to transient solutions is presented by Fabrizio and Morro [9], Fabrizio and Lazzari [12], and Deseri et al. [14].
Material having small distributed voids may be called porous material or material with voids. The intended application of the theory of elastic material with voids may be found in geological materials like rocks and soils, in biological and manufactured porous materials for which the theory of elasticity is inadequate. But seismology represents only one of the many fields where the theories of elasticity and viscoelasticity of materials with voids is applied. Medicine, various branches of biology, the oil exploration industry and nanotechnology are other important fields of application.
Various theories of viscoelastic materials with voids of integral type have been proposed by Cowin [22], Ciarletta and Scalia [23], De Cicco and Nappa [24], and Martínez and Quintanilla [25]. In the last decade there are been interest in formulation of the mechanical theories of viscoelastic materials with voids of differential type. In this connection, Ieşan [26] has developed a nonlinear theory for a viscoelastic composite as a mixture of a porous elastic solid and a Kelvin-Voigt material. A linear variant of this theory was developed by Quintanilla [27], and existence and exponential decay of solutions are proved. Ieşan and Nappa [28] introduced a nonlinear theory of heat conducting mixtures where the individual components are modelled as Kelvin-Voigt viscoelastic materials. Some exponential decay estimates of solutions of equations of steady vibrations in the theory of viscoelastricity for Kelvin-Voigt materials are obtained by Chiriţȃ et al. [29]. A theory of thermoviscoelastic composites modelled as interacting Cosserat continua is presented by Ieşan [30].
In [31], Ieşan extends theory of elastic materials with voids (see, Nunziato and Cowin [32,33]), the basic equations of the nonlinear theory of thermoviscoelasticity for "virgin", namely in the absence pre-existing stresses (see, Fabrizio and Morro [9], Deseri et al. [13]), Kelvin-Voigt materials with voids are established, the linearized version of this theory is derived, a uniqueness result and the continuous dependence of solution upon the initial data and supply terms are proved. Recently, the theory of thermoviscoelasticity for Kelvin-Voigt microstretch composite materials is presented by Passarella et al. [34].
For a review of the literature on elastic materials with voids the reader is referred to [35][36][37][38][39][40][41][42][43][44][45][46][47][48] and the references therein. A new approach may be found in Amendola et al. [4], Fabrizio and Morro [9], although this is not limited just to voids. An account of the historical developments of the theory of porous media as well as references to various contributions may be found in the books by de Boer [49] and Ieşan [50].
In this paper the linear theory of viscoelasticity for Kelvin-Voigt materials with voids (see, Ieşan [31]) is considered and some basic results of the classical theory of elasticity are generalized. Indeed, the basic BVPs of steady vibrations are investigated using the potential method and the theory of singular integral equations.
The investigation of BVPs of mathematical physics by the classical potential method has a hundred year history. The application of this method to the 3D basic BVPs of the theory of elasticity reduces these problems to 2D singular integral equations (see, Kupradze et al. [51]). Owing to the works of Mikhlin [52], Kupradze [53], and Burchuladze and Gegelia [54], the theory of multidimensional singular integral equations has presently been worked out with sufficient completeness. An extensive review of works on the potential method can be found in Gegelia and Jentsch [55].
This work is articulated as follows. Section 2 is devoted to basic equations of steady vibrations of the linear theory of viscoelasticity for isotropic and homogeneous Kelvin-Voigt materials with voids with experienced no past strain histories prior of the instant of observation of the evolution of the body (see Ieşan [31]). The basis for generalizing the present analysis to pre-existing stresses may be found in [20,21], where the state of the material is characterized for viscoelastic media. In Sect. 3 the basic properties of plane harmonic waves are established. In Sect. 4 the fundamental solution of the system of equations of steady vibrations is constructed by means of elementary functions, and its some basic properties are established. In Sect. 5 the Sommerfeld-Kupradze type radiation conditions are given and basic BVPs are formulated. In Sect. 6 the uniqueness theorems of these BVPs are proved. In Sect. 7 the Green's formulas in the considered theory are obtained, the formulas of integral representations of Somigliana type of regular vector and regular (classical) solution are presented, the representation of Galerkin type solution is obtained and the completeness of this solution is established. In Sect. 8 the basic properties of the elastopotentials and the singular integral operators are given. Finally, in Sect. 9 the existence theorems of the BVPs of steady vibrations are proved.
On the basis of the potential method the uniqueness and existence theorems in the classical theories of viscoelasticity and thermoviscoelasticity for Kelvin-Voigt materials without voids are proved by Svanadze [56].

Basic Equations
We consider an isotropic homogeneous viscoelastic Kelvin-Voigt material with voids that occupies the region Ω of the Euclidean three-dimensional space , and let t denote the time variable. In the absence of the body force and the extrinsic equilibrated body force, the system of homogeneous equations of motion in the linear theory of viscoelasticity for Kelvin-Voigt materials with voids has the following form (see, Ieşan [31]) is the displacement vector, ϕ is the volume fraction field, ρ is the reference mass density (ρ > 0), ρ 0 = ρκ, κ is the equilibrated inertia (κ > 0); λ, μ, b, α, ξ , λ * , μ * , b * , α * , ν * , ξ * are the constitutive coefficients, and a superposed dot denotes differentiation with respect to t :u = ∂u ∂t ,ü = ∂ 2 u ∂t 2 . The system (2.1) we can rewritten as If the displacement vector u and the volume fraction function ϕ are postulated to have a harmonic time variation, that is, then from system of equations of motion (2.2) we obtain the following system of homogeneous equations of steady vibrations where ω is the oscillation frequency (ω > 0), (2.5) Obviously, (2.4) is the system of partial differential equations with complex coefficients in with are 14 real parameters: λ, λ * , μ, μ * , b, b * , α, α * , ξ, ξ * , ν * , ω, ρ and ρ 0 .
We introduce the matrix differential operator where δ lj is the Kronecker delta, and l, j = 1, 2, 3. The system (2.4) can be written as where U = (u, ϕ) and x ∈ Ω.
On the other hand the system of nonhomogeneous equations of steady vibrations in the linear theory of viscoelastic materials with voids can be written as follows where F and s are is the body force and the extrinsic equilibrated body force per unit mass, respectively. The system (2.7) can be written as (2.8) where F = (−ρF , −ρs). Throughout this article, we suggest that ξ 2 = 0 (the case ξ 2 = 0 is to simple to be considered).

Plane Harmonic Waves
We introduce the notation (3.1) In this section, it is assumed that On the basis of (3.2) from (3.1) we get Suppose that plane harmonic waves corresponding to the wave number τ and angular frequency ω are propagated in the x 1 -direction through the viscoelastic Kelvin-Voigt material with voids. Then From (3.5) for B 0 and B 1 we have (3.6) For the system (3.6) to have a non-trivial solution for B 0 and B 1 we must set the determinant of their coefficients equal to zero, thus In the same way from (3.5) for B 2 and B 3 we have and if τ is the solution of equation then (3.8) have non-trivial solution. The relations (3.7) and (3.9) will be called the dispersion equations of longitudinal and transverse plane waves in the linear theory of viscoelasticity for Kelvin-Voigt materials with voids, respectively. It is obvious that if τ > 0, then the corresponding plane wave has the constant amplitude, and if τ is complex with Im τ > 0, then the plane wave is attenuated as x 1 → +∞.
Theorem 3.1 If the conditions (3.2) are satisfied, then through a Kelvin-Voigt material with voids 4 plane harmonic plane waves propagate: two longitudinal plane waves P 1 and P 2 with wave numbers λ 1 , λ 2 and two transverse plane waves SH and SV with wave number λ 3 ; these are attenuated waves as x 1 → +∞.

Remark 3.1
It is obvious that if plane harmonic waves are propagated in an arbitrary direction through a Kelvin-Voigt material with voids, then we obtain the same result as given in Theorem 3.1.

Fundamental Solution
In what follows we assume that τ 2 1 = τ 2 2 = τ 2 3 = τ 2 1 . In the sequel we use the matrix differential operators: We have the following result.

Lemma 4.1
If Lemma 4.1 is proved by direct calculation. We introduce the notations Obviously, Y is the fundamental matrix of operator , that is, where δ(x) is the Dirac delta, J = (δ pq ) 4×4 is the unit matrix, and x ∈ R 3 . We define the matrix = (Γ pq ) 4×4 by In view of (4.4)-(4.6) we get Hence, (x) is the fundamental matrix of differential operator A(D x ). We have thereby proved the following theorem. We are now in a position to establish basic properties of matrix (x). Theorem 4.1 leads to the following results.
Clearly (see, Kupradze et al. [51]), the relations hold in a neighborhood of the origin. On the basis of Theorem 4.1 and Corollary 4.2 we obtain the following result.

Theorem 4.2 If condition (4.3) is satisfied, then the relations
Thus, (x) is the singular part of the fundamental matrix (x) in the neighborhood of the origin.
Equalities in (5.2) are the Sommerfeld-Kupradze type radiation conditions in the linear theory of viscoelasticity for materials with voids (see, Sommerfeld [58], Kupradze [59]). We note that (5.1) and (5.2) imply (for details see, Vekua [60]) It is easy to see that each column of the matrix (x) is a regular vector in the domains Ω + and Ω − . Indeed, each element of the matrix (x) satisfies the radiation conditions (5.2) and (5.3).
In the sequel we use the matrix differential operators where n = (n 1 , n 2 , n 3 ) is the unit vector, ∂ ∂n is the derivative along the vector n and l, j = 1, 2, 3.
The basic internal and external BVPs of steady vibration in the theory of viscoelastic materials with voids are formulated as follows.
Find a regular (classical) solution to system (2.8) for x ∈ Ω + satisfying the boundary condition are satisfied, then the internal BVP (K) + F,f admits at most one regular solution, where K = I, II.

Green's Formulas. Representations of General Solutions
In this section, first, we establish the Green's formulas in the linear theory of viscoelastic materials with voids, then we obtain the integral representation of regular vector (representation of Somigliana-type) and the Galerkin-type solution of the system (2.8), and finally, we establish the representation of the general solution of the system of homogeneous equations (2.4) by solutions of Helmholtz equations (metaharmonic functions).
In the sequel we use the matrix differential operatorsÃ(D x ) andP(D x , n), wherẽ The identities (7.2) and (7.3) are the Green's formulas in the linear theory of viscoelastic materials with voids for domains Ω + and Ω − , respectively.
Keeping in mind (7.1) from (7.2) and (7.3) we obtain the formulas of integral representation of regular vector (representation of Somigliana-type) for the domains Ω + and Ω − .
The next two theorems provide a Galerkin-type solution to system (2.8).

Theorem 7.6 If
Proof Let U be a solution of system (2.8). Obviously, if (x) is the fundamental matrix of the operator L(D x ), then the vector function is a solution of (7.8).
On the other hand, by virtue of (2.8), (4.4) and (7.8) we have Hence, W is a solution of (7.9).
Proof Applying the operator div to the first equation of (2.4), from system (2.4) we have (7.14) Clearly, we obtain from (7.14) Now applying the operator curl to the first equation of (2.4) it follows that + τ 2 3 curl u = 0. (7.16) We introduce the notation Taking into account (7.15)-(7.17), the function ϕ j and vector function ψ are the solutions of (7.10) and (7.11), respectively, and the function ϕ is represented by (7.12). Now we prove the first relation of (7.12). Obviously, on the basis of (7.10) we may rewrite the second equation of (7.14) in the form div u = c 3 ϕ 1 + c 4 ϕ 2 , (7.18) where Keeping in mind (7.17), (7.18) and identity u = grad div u − curl curl u, from (2.4) we obtain (7.19) Finally, from (7.19) we get the first relation of (7.12).
Hence, on the basis of Theorems 7.7 and 7.8 the completeness of solution of the homogeneous equation (2.4) is proved.

Basic Properties of Elastopotentials
On the basis of a Somigliana-type integral representation of a regular vector (see, (7.4)) we introduce the following notations where g and φ are four-component vectors. As in the classical theory of elasticity (see, e.g., Kupradze et al. [51]), the vector functions Z (1) (x, g), Z (2) (x, g) and Z (3) (x, φ, Ω ± ) are called a single-layer, a double-layer and volume potentials in the linear theory of viscoelasticity for Kelvin-Voigt materials with voids, respectively.

respectively.
First we establish the basic properties of elastopotentials.
where Ω is a finite domain in R 3 andΩ − 0 ⊂ Ω − . Theorems 8.1 to 8.4 can be proved similarly to the corresponding theorems in the classical theory of thermoelasticity (for details, see, Kupradze et al. [51], Chap. X).

Equation ind K
Thus, the singular integral operator K (j ) (j = 1, 2, 3, 4) is of the normal type with an index equal to zero. Consequently, the Fredholm's theorems are valid for K (j ) (for details, see, e.g., Mikhlin [52]).

Remark 8.1
The definitions of a normal type singular integral operator, the symbol and the index of operator are given in Kupradze et al. [51] and Mikhlin [52].

Existence Theorems
Obviously, by Theorems 8.3 and 8.4 the volume potential Z (3) (x, F, Ω ± ) is a regular solution of (2.8), where F ∈ C 0,p (Ω ± ), 0 < p ≤ 1; supp F is a finite domain in Ω − . Therefore, further we will consider problem (K) ± 0,f for K = I, II. In addition, we assume that the conditions (6.1) are satisfied. Now we prove the existence theorems of a regular (classical) solution of problems (K) + where g is the required four-component vector function.
By Theorem 8.2 the vector function U is a solution of (2.6) for x ∈ Ω + . Keeping in mind the boundary condition (5.4) and using (8.2), from (9.1) we obtain, for determining the unknown vector g, a singular integral equation (9.2) By Lemma 8.3 the Fredholm's theorems are valid for operator K (1) . We prove that (9.2) is always solvable for an arbitrary vector f. Let us consider the associate homogeneous equation i.e., the vector V(x) is a regular solution of problem (I ) + 0,0 . Using Theorem 6.1, the problem (I ) + 0,0 has only the trivial solution, that is, By virtue of (9.4), (9.5) and identity (8.1) we obtain Thus, the homogeneous equation (9.3) has only the trivial solution and therefore (9.2) is always solvable for an arbitrary vector f. We have thereby proved where h is the required four-component vector function.
Obviously, by Theorem 8.1 the vector function U is a solution of (2.6) for x ∈ Ω − . Keeping in mind the boundary condition (5.7) and using (8.1), from (9.6) we obtain, for determining the unknown vector h, a singular integral equation (9.7) It has been proved above that the corresponding homogeneous equation (9.3) has only the trivial solution. Hence, it follows that (9.7) is always solvable. We have thereby proved Theorem 9.2 If S ∈ C 2,p , f ∈ C 0,p (S), 0 < p < p ≤ 1, then a regular solution of problem (II) − 0,f exists, is unique and is represented by single-layer potential (9.6), where h is a solution of the singular integral equation (9.7) which is always solvable for an arbitrary vector f. Problem (II) + 0,f : We seek a regular solution to problem (II) + 0,f in the form where g is the required four-component vector function.
Obviously, by Theorem 8.1 the vector function U is a solution of (2.6) for x ∈ Ω + . Keeping in mind the boundary condition (5.5) and using (8.1), from (9.8) we obtain, for determining the unknown vector g, a singular integral equation (9.9) By Lemma 8.3 the Fredholm's theorems are valid for operator K (2) . We prove that (9.9) is always solvable for an arbitrary vector f. Let us consider the corresponding homogeneous equation K (2) g 0 (z) = 0 for z ∈ S, (9.10) where g 0 is the required four-component vector function. Now we prove that (9.10) has only the trivial solution. Indeed, let g 0 be a solution of the homogeneous equation (9.10). On the basis of Theorem 8.1 and (9.10) the vector V(x) = Z (1) (x, g 0 ) is a regular solution of problem (II) + 0,0 . Using Theorem 6.1, the problem (II) + 0,0 has only the trivial solution, that is V(x) = 0 for x ∈ Ω + . By virtue of (9.11), (9.12) and identity (8.1) we obtain g 0 (z) = P(D z , n)V(z) − − P(D z , n)V(z) + = 0 for z ∈ S.
Thus, the homogeneous equation (9.10) has only a trivial solution and therefore (9.9) is always solvable for an arbitrary vector f. We have thereby proved where h is the required four-component vector function.
Obviously, by Theorem 8.2 the vector function U is a solution of (2.6) for x ∈ Ω − . Keeping in mind the boundary condition (5.6) and using (8.2), from (9.13) we obtain, for determining the unknown vector h, a singular integral equation (9.14) It has been proved above that the corresponding homogeneous equation (9.10) has only the trivial solution. Hence, it follows that (9.14) is always solvable. We have thereby proved Theorem 9.4 If S ∈ C 2,p , f ∈ C 1,p (S), 0 < p < p ≤ 1, then a regular solution of problem (I ) − 0,f exists, is unique and is represented by double-layer potential (9.13), where h is a solution of the singular integral equation (9.14) which is always solvable for an arbitrary vector f. Kelvin-Voigt materials with voids (see, Ieşan [31]) by using potential method and the theory of singular integral equations. 4. By virtue of Theorems 9.1 to 9.4 it is possible to obtain numerical solutions of the BVPs of the linear theory of viscoelasticity for Kelvin-Voigt materials with voids by using of the boundary element method.