ON THE LINEAR EQUILIBRIUM THEORY OF ELASTICITY FOR MATERIALS WITH TRIPLE VOIDS

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1 ON THE LINEAR EQUILIBRIUM THEORY OF ELASTICITY FOR MATERIALS WITH TRIPLE VOIDS by MERAB SVANADZE (Institute for Fundamental and Interdisciplinary Mathematics Research, Ilia State University, K. Cholokashvili Ave., 3/5, 0162 Tbilisi, Georgia) [Received 22 October Revise 11 March Accepted 21 May 2018] Summary In this article, the linear equilibrium theory of elasticity for materials with a triple (macro, meso and micro) voids structure is considered and the following results are obtained. The fundamental solution of the system of equilibrium equations is constructed explicitly and its basic properties are established. The uniqueness theorems for classical solutions of the basic boundary value problems (BVPs) of the considered theory are proved. The basic properties of the surface and volume potentials and singular integral operators are established. These BVPs are reduced to the equivalent singular integral equations for which Fredholm s theorems are valid. Finally, the existence theorems for classical solutions of the above mentioned BVPs are proved by using the potential method and the theory of singular integral equations. 1. Introduction The prediction of the mechanical properties of materials with single and multiple porosity has been one of hot topics of continuum mechanics for more than 100 years. In this connection, Nunziato and Cowin (1) presented a nonlinear theory for the behavior of porous solids with (single) voids or vacuous pores, where matrix material is elastic and the interstices are void of material. In this article the bulk density is written as the product of two fields, the matrix material density field and the volume fraction field. This theory differs from the theory of finite elasticity in that the volume fraction corresponding to the void volume is taken as an independent kinematical variable, called the volume distribution function. This representation of the bulk density of the material was employed previously by Goodman and Cowin (2) to develop a continuum theory for granular materials. Further, Cowin and Nunziato (3) considered the linear theory of elastic materials with single voids. Ieşan (4) extended this theory and presented a linear theory of thermoelastic materials with single voids. In this article some general theorems (uniqueness, reciprocal and variational theorems) are established and the problems of equilibrium are studied. A theory of hyperbolic thermoelasticity for elastic materials with single voids is formulated by Dhaliwal and Wang (5). The theories of micropolar elasticity and thermoelasticity for materials with voids are developed by Ieşan (6) and Passarella (7), respectively. A theory of thermoelastic diffusion for solids with single voids is presented by Aouadi (8). Nowadays, these mathematical theories for materials with single voids are extensively investigated by several authors and the basic results may be found in the books of Ciarletta and Ieşan (9), Ieşan (10), Straughan (11) and references therein. <svanadze@gmail.com> Q. Jl Mech. Appl. Math, Vol. 71. No. 3 The Author(s), Published by Oxford University Press; all rights reserved. For Permissions, please journals.permissions@oup.com Advance Access publication 16 July doi: /qjmam/hby008

2 330 M. SVANADZE Recently, Ieşan and Quintanilla (12) have developed the theory of thermoelastic deformable materials with double voids structure by using the mechanics of materials with single voids (see (1) and (2)). Moreover, in this new theory the independent variables are the displacement vector, the volume fractions of pores and fissures and the variation of temperature. The basic boundary value problems (BVPs) of equilibrium theory of elasticity for materials with double voids are investigated by using the potential method by Ieşan (13). The uniqueness and existence theorems for classical solutions of the BVPs of steady vibrations in theories of elasticity and thermoelasticity for materials with double voids are proved by Svanadze (14) (16). In the series of articles by Kumar et al. (17) (19), the plane waves and the BVP in thermoelastic material with double porosity due to thermomechanical sources are studied, the fundamental solution in the theory of micropolar thermoelastic materials with double voids structure is constructed and reflection of plane waves in thermoelastic medium with double voids is established. Most recent results in the theories of materials with double voids are given in the new book of Straughan (20). The mathematical models for materials with single or multiple voids have found applications in many branches of civil and geotechnical engineering, technology and biomechanics. The intended applications of the theories for elastic and thermoelastic materials with multiple voids are to geological materials such as rocks and soils, manufactured porous materials such as ceramics and pressed powders, and biomaterials such as bone (for details see Bai et al. (21), Cowin (22) and references therein). In this article, by using the concept of the mechanics of materials with voids the linear equilibrium theory of elasticity for materials with a triple voids structure is considered. This work is articulated as follows. In section 2, the governing field equations of the considered theory are given. In section 3, the fundamental solution of the system of equilibrium equation is constructed explicitly by means of elementary functions and its basic properties are established. In section 4, the basic internal and external BVPs are formulated. In section 5, the uniqueness theorems for these problems are proved. In section 6, the basic properties of the surface (single-layer and double-layer) and volume potentials and singular integral operators are established. Finally, in section 7, these BVPs are reduced to the equivalent singular integral equations for which Fredholm s theorems are valid and the existence theorems for regular (classical) solutions of the internal and external BVPs of equilibrium are proved by using the potential method and the theory of singular integral equations. 2. Basic equations In what follows we consider an isotropic and homogeneous elastic solid with macro-, meso- and microporosity (first, second and third porosity) structure that occupies the region of the Euclidean three-dimensional space R 3. Let u = (u 1, u 2, u 3 ) be the displacement vector in solid, ϕ 1 (x),ϕ 2 (x) and ϕ 3 (x) are the changes of volume fractions from the reference configuration corresponding to macro-, meso- and microporosity, respectively. We assume that subscripts preceded by a comma denote partial differentiation with respect to the corresponding Cartesian coordinate and repeated indices are summed over the range (1,2,3). Within a dual porosity conceptual framework (see Ieşan and Quintanilla (12)), the governing field equations in the linear equilibrium theory of elastic materials with a triple voids structure may be expressed as the following equations: Constitutive equations t lj = λe rr δ lj + 2μe lj + b r ϕ r δ lj, σ (l) j = a lr ϕ r,j, l, j = 1, 2, 3; (2.1)

3 LINEAR EQUILIBRIUM THEORY FOR TRIPLE VOIDS MATERIALS 331 Equations of equilibrium t lj,j = ρf (1) l, σ (l) j,j + ξ (l) = ρf (2) l, (2.2) where t lj is the component of total stress tensor, F (1) = (F (1) 1, F(1) 2, F(1) 3 ) is the body force per unit mass, F (2) l and σ (l) j are the extrinsic equilibrated body forces per unit mass and the components of the equilibrated stresses associated to the l-th porosity network, respectively; ρ is the reference mass density, ρ>0; δ lj is Kronecker s delta, e lj are the components of the strain tensor, e lj = 1 2 ( ul,j + u j,l ), l, j = 1, 2, 3, (2.3) λ, μ, b j, a lj,α lj are the constitutive coefficients, a lj = a jl and α lj = α jl, the functions ξ (l) (l = 1, 2, 3) are the intrinsic equilibrated body forces and defined by ξ (l) = b l e rr α lj ϕ j. (2.4) Substituting (2.1), (2.3) and (2.4) into (2.2) we obtain the following system of equations in the linear equilibrium theory of elasticity for materials with a triple voids structure expressed in terms of the displacement vector u and the volume fraction vector ϕ = (ϕ 1,ϕ 2,ϕ 3 ): μ u + (λ + μ) divu + b j ϕ j = ρf (1), (a α) ϕ b divu = ρf (2), where is the Laplacian operator, F (2) = (F (2) 1, F(2) 2, F(2) 3 ), a = (a lj) 3 3, α = (α lj ) 3 3 and b = (b 1, b 2, b 3 ). We introduce the matrix differential operator A(D x ) = ( A lj (D x ) ) 6 6, where 2 A lj (D x ) = μ δ lj + (λ + μ), x l x j A l+3;j+3 (D x ) = a lj α lj, D x = A l;j+3 (D x ) = A j+3;l (D x ) = b j, x l ( ),,, l, j = 1, 2, 3. x 1 x 2 x 3 It is easily seen that the system (2.5) we can rewritten in the following matrix form (2.5) A(D x )U(x) = F(x), (2.6) where U = (u, ϕ) and F = ( ρf (1), ρf (2) ) are six-component vector functions. The purpose of this article is to investigate the internal and external BVPs for (2.6) by means of the potential method and the theory of singular integral equations. For this we need some basic properties of the fundamental solution of (2.6) and the surface and volume potentials.

4 332 M. SVANADZE 3. Fundamental solution Definition 3.1. The fundamental solution of system (2.6) (the fundamental matrix of the operator A(D x )) is the matrix Ɣ(x) = ( Ɣ lj (x) ) 6 6 satisfying the following equation in the class of generalized functions A(D x )Ɣ(x) = δ(x)j, (3.1) where δ(x) is the Dirac delta, J = ( δ lj )6 6 is the unit matrix and x R3. We introduce the notation: 1) 1 ( ) = 1 det B ( ), a 0 μ 0 where μ 0 = λ + 2μ, a 0 is the determinant of the matrix a and 2) μ 0 b 1 b 2 b 3 b 1 a 11 α 11 a 12 α 12 a 13 α 13 B ( ) = b 2 a 12 α 12 a 22 α 22 a 23 α 23 b 3 a 13 α 13 a 23 α 23 a 33 α 33 n l1 ( ) = 1 a 0 μμ 0 [ (λ + μ)b l1 ( ) b jb l;j+1 ( ) ], n lm ( ) = 1 B a 0 μ lm ( ), l = 1, 2, 3, 4, m = 2, 3, 4, 0 where Blj is the cofactor of the element B lj of matrix B. 3) We can consider μ 0 b 1 b 2 b 3 b 1 a 11 ξ α 11 a 12 ξ α 12 a 13 ξ α 13 det b 2 a 12 ξ α 12 a 22 ξ α 22 a 23 ξ α 23 b 3 a 13 ξ α 13 a 23 ξ α 23 a 33 ξ α 33 as an algebraic equation of the third degree with respect to ξ which admits three roots τ1 2,τ2 2 and τ3 2. Then we have 3 1 ( ) = ( τj 2 ). j= = 0. We assume that τ 1,τ 2 and τ 3 are distinct and positive values.

5 LINEAR EQUILIBRIUM THEORY FOR TRIPLE VOIDS MATERIALS 333 4) L(D x ) = ( L lj (D x ) ) 6 6, L l;j+3 (D x ) = n 1;j+1 ( ) x l, L l+3;j+3 (D x ) = n l+1;j+1 ( ), l, j = 1, 2, 3. L lj(d x ) = 1 μ 2 1( ) δ lj + n 11 ( ), x l x j L j+3;l (D x ) = n j+1;1 ( ) x l, (3.2) 5) where and Y(x) = (Y lm (x)) 6 6, Y 11 (x) = Y 22 (x) = Y 33 (x) = Y 44 (x) = Y 55 (x) = Y 66 (x) = 4 β 1j γ j (x), j=1 Y lm (x) = 0, l = m, l, m = 1, 2,, 6, γ j (x) = e τ j x 4π x, γ 4(x) = 1 4π x, β 1j = τ 2 j 3 (τl 2 τj 2 ) 1, l=1, l =j β 2j = τ 4 j 5 β 2j γ j (x), j=1 (3.3) γ 5(x) = x, j = 1, 2, 3 (3.4) 8π 3 (τl 2 τj 2 ) 1, l=1, l =j β 14 = β 25 = 1 τ1 2τ 2 2τ 3 2, β 24 = τ 1 2τ τ 1 2τ τ 2 2τ 3 2 τ1 4τ 2 4τ 3 4, j = 1, 2, 3. We have the following Theorem 3.1. If then the matrix Ɣ(x) defined by a 0 μμ 0 = 0, (3.5) Ɣ(x) = L(D x )Y(x) (3.6) is the fundamental solution of (2.6), where the matrices L(D x ) and Y(x) are given by (3.2) and (3.3), respectively. Proof. On the basis of identities A(D x )L(D x ) = ( ), ( )Y(x) = δ(x)j,

6 334 M. SVANADZE where ( ) = ( lj ( ) ) 6 6, 11 ( ) = 22 ( ) = 33 ( ) = 1 ( ), 44 ( ) = 55 ( ) = 66 ( ) = 1 ( ), lj ( ) = 0, l = j, l, j = 1, 2,, 6, it follows the relation (3.1). Hence, the matrix Ɣ (x) is constructed explicitly by five elementary functions γ (j) (j = 1, 2,, 5) (see (3.4)). Theorem 3.1 directly leads to the following basic properties of Ɣ (x). Theorem 3.2. Each column of the matrix Ɣ (x) is a solution of the homogeneous equation A(D x ) U(x) = 0 (3.7) at every point x R 3 except the origin. Theorem 3.3. If condition (3.5) is satisfied, then the fundamental solution of the system is the matrix (x) = ( lj (x) ) 6 6, where μ u + (λ + μ) div u = 0, a ϕ = 0 ( ) lj (x) = 1 δ lj λ + μ 2 γ (5) (x) = λ + 3μ δ lj μ μ 0 x l x j 8πμμ 0 x λ + μ x l x j 8πμμ 0 x 3, l+3;j+3 (x) = a lj a 0 γ (4) (x), l;j+3 (x) = l+3;j (x) = 0, l, j = 1, 2, 3 and a lj is the cofactor of the element a lj of matrix a. Obviously, (3.6) and (3.8) imply the following results. Corollary 3.1. The relations (3.8) lj (x) = O ( x 1), l+3;j+3 (x) = O ( x 1) (3.9) hold in the neighborhood of the origin, where l, j = 1, 2, 3. Theorem 3.4. The relations Ɣ lj (x) = O ( x 1), Ɣ l+3;j+3 (x) = O ( x 1), Ɣ l;j+3 (x) = O (1), Ɣ l+3;j (x) = O (1), hold in the neighborhood of the origin, where l, j = 1, 2, 3.

7 LINEAR EQUILIBRIUM THEORY FOR TRIPLE VOIDS MATERIALS 335 On the basis of Theorem 3.4 and Corollary 3.1 we can prove the following Theorem 3.5. The relations Ɣ lj (x) lj (x) = const + O ( x ) (3.10) hold in the neighborhood of the origin, where l, j = 1, 2,, 6. Thus, in view of (3.9) and (3.10), matrix (x) gives the singular part of the fundamental solution Ɣ (x) in the neighborhood of the origin. 4. Boundary value problems Let S be the closed surface surrounding the finite domain + in R 3, S C 1,ν, 0 <ν 1, + = + S, = R 3 \ +, = S; n(z) is the external (with respect to + ) unit normal vector to S at z. Definition 4.1. A vector function U = (U 1, U 2,, U 6 ) is called regular in (or + ) if (i) (ii) U l C 2 ( ) C 1 ( ) (or U l C 2 ( + ) C 1 ( + )), for x 1, where l = 1, 2,, 6 and j = 1, 2, 3. In the sequel, we use the matrix differential operator where U l (x) = O( x 1 ), U l,j (x) = o( x 1 ) (4.1) P(D x, n) = (P lj (D x, n)) 6 6, P lj (D x, n) = μδ lj n + μn j + λn l, P l;j+3 (D x, n) = b j n l, x l x j P l+3;j (D x, n) = 0, P l+3;j+3 (D x, n) = a lj, l, j = 1, 2, 3 n and is the derivative along the vector n. n The basic internal and external BVPs in the equilibrium theory of elasticity for materials with a triple voids structure are formulated as follows. Find a regular (classical) solution to (2.6) for x + satisfying the boundary condition lim U(x) + x z S {U(z)}+ = f(z) (4.3) in the internal Problem (I) + F,f, and lim P(D x, n(z))u(x) {P(D z, n(z))u(z)} + = f(z) (4.4) + x z S in the internal Problem (II) + F,f, where F and f are prescribed six-component vector functions. (4.2)

8 336 M. SVANADZE Find a regular (classical) solution to (2.6) for x satisfying the boundary condition lim U(x) x z S {U(z)} = f(z) (4.5) in the external Problem (I) F,f, and lim x z S P(D x, n(z))u(x) {P(D z, n(z))u(z)} = f(z) (4.6) in the external Problem (II) F,f, where F and f are prescribed six-component vector functions and supp F is a finite domain in. 5. Uniqueness theorems ( In what follows ) we assume that the constitutive coefficients satisfy the conditions: a, α and μ1 α lj b l b j 3 3 are positive definite matrices and μ>0, where μ 1 = 1 3 (3λ + 2μ). Obviously, this assumption implies μ 0 >μ 1 > 0. In this section the uniqueness of regular solutions of the BVPs (K) + F,f and (K) F,f will be studied, where K = I, II. In the sequel we use the matrix differential operators: 1) ( A (0) (D x ) = A (0) ) lj (D x ), A(0) lj (D x ) = μ δ lj + (λ + μ), ( x l x j A (1) (D x ) = A (1) ) lr (D x), 3 6 A(1) lr (D x) = A lr (D x ), ( A (2) (D x ) = A (2) ) lr (D x), 3 6 A(2) lr (D x) = A l+3;r (D x ); 2) ( P (0) (D x, n) = P (0) ) lj (D x, n), 3 3 P(0) lj (D x, n) = P lj (D x, n), ( P (1) (D x, n) = P (1) ) lr (D x, n), 3 6 P(1) lr (D x, n) = P lr (D x, n), ( P (2) (D x, n) = P (2) ) lj (D x, n), 3 3 P(2) lj (D x, n) = P l+3;j+3 (D x, n), where l, j = 1, 2, 3 and r = 1, 2,, 6. We introduce the notation W 0 (u) = μ 1 div u 2 + μ 3 u j 2 + u l 2 x l,j=1; l =j l x + μ 3 u l j 3 u j 2 x l,j=1 l x, j W 1 (U) = W 0 (u) + b j ϕ j div u, W 2 (U) = a lj ϕ l,k ϕ j,k + α lj ϕ l ϕ j + b j ϕ j div u, W 3 (U) = μ 1 div u 2 + 2b j ϕ j div u + α lj ϕ l ϕ j, W 4 (U) = a lj ϕ l,k ϕ j,k + μ 3 u j 2 + u l 2 x l,j=1; l =j l x + μ 3 u l j 3 u j 2 x l,j=1 l x. j (5.1) (5.2) (5.3)

9 LINEAR EQUILIBRIUM THEORY FOR TRIPLE VOIDS MATERIALS 337 We have the following Lemma 5.1. If U = (u, ϕ) is a regular vector in +, then [ ] A (1) (D x ) U(x) u(x) + W 1 (U) dx = P (1) (D z, n)u(z) u(z) d z S, + S [ ] A (2) (D x ) U(x)ϕ(x) + W 2 (U) dx = P (2) (D z, n)ϕ(z) ϕ(z) d z S. + Proof. On the basis of the divergence Theorem the following identities are valid (see for example Kupradze et al. (23)) [ ] A (0) (D x ) u(x) u(x) + W 0 (u) dx = P (0) (D z, n)u(z) u(z) d z S, + S [ ϕl (x) ϕ j (x) + ϕ l (x) ϕ j (x) ] ϕ l (z) dx = n(z) ϕ j(z) d z S, + S [ ϕ l (x) u(x) + ϕ l (x)divu(x)] dx = ϕ l (z)n(z) u(z) d z S. + S Keeping in mind (5.3), from (5.5) we obtain the identities (5.4). Lemma 5.1 directly leads to the following Lemma 5.2. If U = (u, ϕ) is a regular vector in +, then [A(D x ) U(x) U(x) + W(U)] dx = P(D z, n)u(z) U(z) d z S, + S where W(U) = W 3 (U) + W 4 (U). Similarly, on the basis of Lemma 5.2 and the condition at infinity (4.1) we obtain the following Lemma 5.3. If U = (u, ϕ) is a regular vector in, then [A(D x ) U(x) U(x) + W(U)] dx = P(D z, n)u(z) U(z) d z S. (5.7) S We are now in a position to study the uniqueness of regular solutions of the BVPs (K) + F,f and (K) F,f, where K = I, II. We have the following results. Theorem 5.1. The internal BVP (I) + F,f admits at most one regular solution. S (5.4) (5.5) (5.6) Proof. Suppose that there are two regular solutions of problem (I) + F,f. Then their difference U is a regular solution of the internal homogeneous BVP (I) + 0,0. Hence, U is a regular solution of the

10 338 M. SVANADZE homogeneous equation (3.7)in + satisfying the homogeneous boundary condition {U(z)} + = 0 for z S. (5.8) Clearly, by virtue of (3.7) and (5.8), from (5.6) we obtain + W(U)dx = 0, j = 1, 2, 3, 4. (5.9) On the basis of the assumption on the constitutive coefficients from (5.3) wehave W 3 (U) 0, W 4 (U) 0 and consequently, W(U) 0. Hence, from (5.9) it follows that W(U) = 0, that is and μ 1 div u 2 + 2b j ϕ j div u + α lj ϕ l ϕ j = 0 (5.10) a lj ϕ l,k ϕ j,k + μ 3 u j 2 + u l 2 x l,j=1; l =j l x + μ 3 u l j 3 u j 2 x l,j=1 l x = 0. (5.11) j Obviously, the relation (5.10) implies div u(x) = 0 and ϕ(x) 0 (5.12) for x +. On the other hand, by virtue of (5.12) from (5.11) it follows that u is the rigid displacement vector (see for example Kupradze et al. (23)) and has the following form u(x) = c + d x, (5.13) where c and d are arbitrary three-component constant vectors, d x is the vector product of the vectors d and x. In view of homogeneous boundary condition (5.8) from (5.13) wegetu(x) 0 for x +. Thus, U(x) 0 for x + and we have desired result. Quite similarly, the following result is proved. Theorem 5.2. Two regular solutions of the internal BVP (II) + F,f, may differ only for an additive vector U = (u, ϕ), where the vectors u and ϕ satisfies the conditions (5.13) and (5.12), respectively for x +. Theorem 5.3. The external BVP (K) F,f has one regular solution, where K = I, II. Theorem 5.3 can be proved similarly to Theorems 5.1 and 5.2 using the condition at infinity (4.1) and the identity (5.7).

11 LINEAR EQUILIBRIUM THEORY FOR TRIPLE VOIDS MATERIALS Basic properties of potentials and singular integral operators In this section, first we present the basic properties of the surface (single-layer and double-layer) and volume potentials. Then some singular integral operators are studied. These results are used in the next section. We introduce the potential of a single-layer Z (1) (x, g) = Ɣ(x y)g(y)d y S, the potential of a double-layer Z (2) (x, g) = and the potential of volume S S [ P(D y, n(y))ɣ (x y)] g(y)dy S, Z (3) (x, φ, ± ) = ± Ɣ(x y)φ(y)dy, where Ɣ is the fundamental matrix of the operator A(D x ) and defined by (3.6), the operator P is givenby(4.2), g and φ are six-component vectors, and the superscript denotes transposition. We have the following results. Theorem 6.1. If S C m+1,ν, g C m,ν (S), 0 <ν <ν 1, and m is a nonnegative whole number, then: a) Z (1) (, g) C 0,ν (R 3 ) C m+1,ν ( ± ) C ( ± ), b) A(D x ) Z (1) (x, g) = 0, x ±, c) P(D z, n(z)) Z (1) (z, g) is a singular integral for z S, d) { } ± P(D z, n(z)) Z (1) 1 (z, g) = 2 g(z) + P(D z, n(z)) Z (1) (z, g), z S, (6.1) e) for x 1 and l = 1, 2, 3. Z (1) (x, g) = O( x 1 ), x l Z (1) (x, g) = O( x 2 ) (6.2) Theorem 6.2. If S C m+1,ν, g C m,ν (S), 0 <ν <ν 1, then:

12 340 M. SVANADZE a) b) c) Z (2) (z, g) is a singular integral for z S, d) for the nonnegative integer m, e) Z (2) (, g) C m,ν ( ± ) C ( ± ), A(D x ) Z (2) (x, g) = 0, x ±, { } ± Z (2) 1 (z, g) =± 2 g(z) + Z(2) (z, g), z S (6.3) Z (2) (x, g) = O( x 2 ), x l Z (2) (x, g) = O( x 3 ) (6.4) for x 1and l = 1, 2, 3, f) { + P(D z, n(z)) Z (2) (z, g)} ={P(Dz, n(z)) Z (2) (z, g)} (6.5) for the natural number m and z S. Theorem 6.3. If S C 1,ν, φ C 0,ν ( + ), 0 <ν <ν 1, then: a) b) Z (3) (, φ, + ) C 1,ν (R 3 ) C 2 ( + ) C 2,ν ( + 0 ), where + 0 is a domain in R3 and A(D x ) Z (3) (x, φ, + ) = φ(x), x +, Theorem 6.4. If S C 1,ν, suppφ =, φ C 0,ν ( ), 0 <ν <ν 1, then: a) b) Z (3) (, φ, ) C 1,ν (R 3 ) C 2 ( ) C 2,ν ( 0 ), A(D x ) Z (3) (x, φ, ) = φ(x), x, where is a finite domain in R 3 and 0. Theorems can be proved similarly to the corresponding theorems in the theory of elasticity for materials with double voids (for details see Svanadze (16)).

13 LINEAR EQUILIBRIUM THEORY FOR TRIPLE VOIDS MATERIALS 341 Remark 6.1. Obviously, the surface potentials are regular vectors in + and. Moreover, Z (1) (x, g) and Z (2) (x, g) satisfy the conditions (6.2) and (6.4) at infinity, respectively. Clearly, the double-layer potential Z (2) (x, g) satisfies more strong condition (6.4) that (4.1). We introduce the notation K (1) g(z) 1 2 g(z) + Z(2) (z, g), K (2) g(z) 1 2 g(z) + P(D z, n(z))z (1) (z, g), K (3) g(z) 1 2 g(z) + Z(2) (z, g), K ς g(z) 1 2 g(z) + ς Z(2) (z, g), z S, K (4) g(z) 1 2 g(z) + P(D z, n(z))z (1) (z, g), where ς is an arbitrary complex number. Obviously, Theorems 6.1 and 6.2 imply that K (m) (m = 1, 2, 3, 4) and K ς are the singular integral operators. In the sequel we need the following Lemma 6.1. The singular integral operator K (m) is of the normal type with an index equal to zero, where m = 1, 2, 3, 4. Proof. Let σ (m) = (σ (m) lj ) 6 6 be the symbol of the operator K (m) (m = 1, 2, 3, 4). From (6.6) we have ( det σ (m) = det σ (m+2) = 1 ) ) 6 (1 μ2 (λ + μ)(λ + 3μ) 2 μ 2 = 0 64μ 2 > 0. (6.7) 0 Hence, the operator K (m) is of the normal type, where m = 1, 2. Let σ ς and ind K ς be the symbol and the index of the operator K ς, respectively. It may be easily shown that det σ ς = μ2 0 μ2 ς 2 64μ 2 0 and det σ ς vanishes only at two points ς 1 and ς 2 of the complex plane. By virtue of (6.7) and det σ 1 = det σ (1) we get ς j = 1(j = 1, 2) and ind K 1 = ind K (1) = ind K 0 = 0. Quite similarly we obtain ind K (2) = ind K (1) = 0 and ind K (3) = ind K (4) = 0. Thus, the singular integral operator K (m) (m = 1, 2, 3, 4) is of the normal type with an index equal to zero. Consequently, Fredholm s theorems are valid for K (m). Remark 6.2. The basic theory of the multidimensional singular integral equations is given in Mikhlin (24). For the definitions of a normal type singular integral operator, the symbol and the index of the 2D singular integral operators see, for example Kupradze et al. (23). 7. Existence theorems In this section, we establish the existence of regular solutions of the BVPs (K) + F,f and (K) F,f by means of the potential method and the theory of 2D singular integral equations, where K = I, II. (6.6)

14 342 M. SVANADZE By Theorems 6.3 and 6.4 the volume potential Z (3) (x, F, ± ) is a particular solution of the nonhomogeneous equation (2.6), where F C 0,ν ( ± ), 0 <ν 1; supp F is a finite domain in. Therefore, further we will consider problems (K) + 0,f and (K) 0,f, where K = I, II. Now we prove the existence theorems of a regular (classical) solution of these BVPs. Problem (I) + 0,f. We seek a regular solution to this problem in the form of the double-layer potential U(x) = Z (2) (x, g) for x +, (7.1) where g is the required six-component vector function. Obviously, by Theorem 6.2 the vector function U is a solution of (3.7) for x +. Keeping in mind the boundary condition (4.3) and using (6.3) from (7.1) we obtain, for determining the unknown vector g, a singular integral equation K (1) g(z) = f(z) for z S. (7.2) We prove that the (7.2) is always solvable for an arbitrary vector f. The homogeneous adjoint integral equation of (7.2) has the following form K (2) h(z) = 0 for z S, (7.3) where h is the required six-component vector function. Now we prove that (7.3) has only the trivial solution. Indeed, let h 0 be a solution of the homogeneous equation (7.3). On the basis of Theorem 6.1 and (7.3) the vector function V(x) = Z (1) (x, h 0 ) is a regular solution of the external homogeneous BVP (II) 0,0. Using Theorem 5.3, the problem (II) 0,0 has only the trivial solution, that is On the other hand, by Theorem 6.1 and (7.4) weget V(x) 0 for x. (7.4) {V(z)} + ={V(z)} = 0 for z S, that is, the vector V(x) is a regular solution of problem (I) + 0,0. Using Theorem 5.1 the problem (I)+ 0,0 has only the trivial solution, that is By virtue of (7.4), (7.5) and identity (6.1) we obtain V(x) 0 for x +. (7.5) h 0 (z) ={P(D z, n)v(z)} {P(D z, n)v(z)} + = 0 for z S. Thus, the homogeneous equation (7.3) has only the trivial solution and therefore on the basis of Fredholm s theorem the integral equation (7.2) is always solvable for an arbitrary vector f. We have thereby proved

15 LINEAR EQUILIBRIUM THEORY FOR TRIPLE VOIDS MATERIALS 343 Theorem 7.1. If S C 2,ν, f C 1,ν (S), 0 <ν <ν 1, then a regular solution of the internal BVP (I) + 0,f exists, is unique and is represented by double-layer potential (7.1), where g is a solution of the singular integral equation (7.2) which is always solvable for an arbitrary vector f. Problem (II) 0,f. We seek a regular solution to this problem in the form of the single-layer potential U(x) = Z (1) (x, h) for x, (7.6) where h is the required six-component vector function. Obviously, by Theorem 6.1 the vector function U is a solution of (3.7) for x. Keeping in mind the boundary condition (4.6) and using (6.1), from (7.6) we obtain the following singular integral equation for determining the unknown vector h K (2) h(z) = f(z) for z S. (7.7) It has been proved above that the corresponding homogeneous equation (7.3) has only the trivial solution. Hence, it follows that (7.7) is always solvable. We have thereby proved Theorem 7.2. If S C 2,ν, f C 0,ν (S), 0 <ν <ν 1, then a regular solution of the external BVP (II) + 0,f exists, is unique and is represented by single-layer potential (7.6), where h is a solution of the singular integral equation (7.7) which is always solvable for an arbitrary vector f. Problem (I) 0,f. We seek a regular solution to this problem in the sum of the double-layer and single-layer potentials U(x) = Z (2) (x, g) + Z (1) (x, g) for x, (7.8) where g is the required six-component vector function. Obviously, by Theorems 6.1 and 6.2 the vector function U is a solution of (3.7) for x. Keeping in mind the boundary condition (4.5) and using (6.3) from (7.8) we obtain, for determining the unknown vector g, a singular integral equation K (5) g(z) K (4) g(z) + Z (1) (z, g) = f(z) for z S. (7.9) We prove that the (7.9) is always solvable for an arbitrary vector f. Clearly, the singular integral operator K (5) is of the normal type and ind K (5) = ind K (4) = 0. Now we prove that the homogeneous equation K (5) g 0 (z) = 0 for z S (7.10) has only a trivial solution. Indeed, let g 0 be a solution of the homogeneous equation (7.10). Then the vector V(x) Z (2) (x, g 0 ) + Z (1) (x, g 0 ) for x (7.11) is a regular solution of problem (I) 0,0. Using Theorem 5.3 we have (7.4). On the other hand, by Theorems 6.1 and identities (6.3) and (6.5) from (7.11) weget {V(z)} + {V(z)} = g 0 (z), {P(D z, n)v(z)} + {P(D z, n)v(z)} = g 0 (z), for z S. (7.12)

16 344 M. SVANADZE On the basis of (7.4) from (7.12) it follows that {P(D z, n)v(z) + V(z)} + = 0 for z S. (7.13) Obviously, the vector V is a solution of (3.7) in + satisfying the boundary condition (7.13). Now applying identity (5.6) for vector V we obtain {V(z)} + = 0 for z S. (7.14) Finally, by virtue of (7.4) and (7.14) from the first equation of (7.12) wegetg 0 (z) 0 for z S. Thus, the homogeneous equation (7.10) has only the trivial solution and therefore on the basis of Fredholm s theorem the integral equation (7.9) is always solvable for an arbitrary vector f. Wehave thereby proved Theorem 7.3. If S C 2,ν, f C 1,ν (S), 0 <ν <ν 1, then a regular solution of the external BVP (I) 0,f exists, is unique and is represented by sum of double-layer and single-layer potentials (7.8), where g is a solution of the singular integral equation (7.9) which is always solvable for an arbitrary vector f. Remark 7.1. On the basis of Remark 6.1 the double-layer potential cannot reduce the external BVP (I) 0,f to the equivalent always solvable singular integral equation. Problem (II) + 0, f. The regular solution of problem (II)+ 0,f is sought in the form of a potential of single-layer U(x) = Z (1) (x, g) for x +, (7.15) where g is the required six-component vector function. Taking into account the boundary property of potential of single-layer (6.1) and boundary condition (4.4) from (7.15) we obtain, for determining the vector g, the following singular integral equation K (3) g(z) = f(z) for z S. (7.16) To investigate the solvability of (7.16) we consider the homogeneous equation K (3) g(z) = 0 for z S. (7.17) Clearly, the adjoint homogeneous integral equation of (7.17) has the form K (4) h(z) = 0 for z S. (7.18) First it will be proved the following Lemma 7.1. The homogeneous equations (7.17) and (7.18) have six linearly independent solutions each and they constitute complete system of solutions. Proof. On the basis of (5.12) and (5.13) we introduce the following six-component vectors ψ (1) (x) = (1, 0, 0, 0, 0, 0), ψ (2) (x) = (0, 1, 0, 0, 0, 0), ψ (3) (x) = (0, 0, 1, 0, 0, 0), ψ (4) (x) = (0, x 3, x 2, 0, 0, 0), ψ (5) (x) = (x 3, 0, x 1, 0, 0, 0), ψ (6) (x) = ( x 2, x 1, 0, 0, 0, 0). (7.19)

17 LINEAR EQUILIBRIUM THEORY FOR TRIPLE VOIDS MATERIALS 345 Obviously, { ψ (j) (x) } 6 is the system of linearly independent vectors, and by Theorem 5.2 each j=1 vector ψ (j) (x) is a regular solution of problem (II) + 0, 0 and the homogeneous singular integral equation (7.18), that is A(D x ) ψ (j) (x) = 0 for x +, { P(D z, n) ψ (j) (z)} + = 0, K (4) ψ (j) (z) = 0 for z S, j = 1, 2,, 6. Hence, (7.17) and (7.18) have six linearly independent solutions each. { } 6 It will now be show that ψ (j) (x) is a complete system of linearly independent solutions j=1 of (7.18). Let { g (j) (z) } m j=1 is the complete system of linearly independent solutions of homogeneous Eq. (7.17), where m 6. We construct potentials of single-layer Z (1) ( x, g (j)) (j = 1, 2,, m). From identity m ( 0 = d j Z (1) x, g (j)) m = Z (1) x, d j g (j) for x + we get j=1 j=1 m d j g (j) (z) = 0 for z S, (7.20) j=1 where d 1, d 2,, d m are arbitrary constants, and from (7.20) wehaved 1 = d 2 = = d m = 0. Hence, { Z (1) ( x, g (j))} m j=1 is the system of linearly independent vectors. On the other hand, it is easy to see that each vector Z (1) ( x, g (j)) (j = 1, 2,, m) is a regular solution of problem (II) + 0, 0. On the basis of (7.19) and Theorem 5.2, vector ( Z(1) x, g (j)) can be written as follows ( Z (1) x, g (j)) 6 (x) = d jl ψ (j) (x), where d jl (j = 1, 2,, m, l = 1, 2,, 6) are constants. Hence, each vector of system { Z (1) (x, g (j)} m j=1 is represented by six linearly independent vectors ψ(1) (x), ψ (2) (x),, ψ (6) (x). Thus, m = 6. By Fredholm s theorem the necessary and sufficient condition for (7.16) to be solvable has the form f(z) ψ (j) (z)d z S = 0, j = 1, 2,, 6, (7.21) S l=1 where ψ (j) is determined by (7.19).

18 346 M. SVANADZE Obviously, if f = (f 1, f 2,, f 6 ) and f (0) = (f 1, f 2, f 3 ), then by virtue of (7.19) the condition (7.21) can be rewritten as f (0) (z)d z S = 0, z f (0) (z)d z S = 0. (7.22) S S It is well known (see, for example Kupradze et al. (23), Ch. VI) that the conditions (7.22) are necessary and sufficient for solvability of the second internal BVP of the classical theory of elasticity A (0) (D x ) u(x) = 0, { P (0) (D z, n) u(z) } + = f (0) (z) for x +, z S, where the matrix differential operators A (0) (D x ) and P (0) (D z, n) are defined by (5.1) and (5.2), respectively. Clearly, the conditions (7.22) show that the resultant vector and moment of the external force are equal to zero. We have thereby proved the following Theorem 7.4. If S C 2,ν, f C 0,ν (S), 0 <ν <ν 1, then problem (II) + 0, f is solvable only when conditions (7.22) are fulfilled. The solution of this problem is represented by a potential of single-layer (7.15) and is determined to within an additive vector of Ũ = ( ũ, ϕ ), where g is a solution of the singular integral equation (7.16) and ũ(x) = c + d x, ϕ(x) 0 for x +, c and d are arbitrary three-component constant vectors. 8. Concluding remarks (1) In this article, the linear equilibrium theory of elasticity for materials with a triple (macro, meso and micro) voids structure is considered and the following results are obtained: (i) the fundamental solution of the system of equilibrium equations is constructed explicitly by means of elementary functions and its basic properties are established; (ii) the uniqueness theorems for classical solutions of the basic internal and external BVPs of the considered theory are proved; (iii) the basic properties of the surface (single-layer and double-layer) and volume potentials are established; (iv) on the basis of the surface potentials the BVPs are reduced to the equivalent singular integral equations; (v) the existence theorems for classical solutions of the above mentioned BVPs are proved by using the potential method and the theory of singular integral equations. (2) On the basis of results of this article are possible to construct the fundamental solution and to prove the uniqueness and existence theorems in the linear theory of thermoelasticity for materials with a multiple porosity structure by using the potential method and the theory of singular integral equations.

19 LINEAR EQUILIBRIUM THEORY FOR TRIPLE VOIDS MATERIALS 347 (3) The BVPs of the classical theories of elasticity and thermoelasticity are investigated by using the potential method in the books of Kupradze et al. (23), Kupradze (25), Burchuladze and Gegelia (26). An extensive review of works on this method can be found in the survey article of Gegelia and Jentsch (27). (4) In the last three decades, on the basis of the extended Darcy s law several new triple porosity models for solids with a hierarchical structure are presented by Bai and Roegiers (28), Straughan (29), Svanadze (30) and studied in (31) (35). References 1. J. W. Nunziato and S. C. Cowin, A nonlinear theory of elastic materials with voids, Arch. Rat. Mech. Anal. 72 (1979) S. C. Cowin and J. W. Nunziato, Linear elastic materials with voids, J. Elasticity 13 (1983) M. A. Goodman and S. C. Cowin, A Continuum theory for granular materials, Arch. Rat. Mech. Anal. 44 (1972) D. Ieşan, A theory of thermoelastic materials with voids, Acta Mechanica 60 (1986) R. S. Dhaliwal and J. Wang, A heat-flux dependent theory of thermoelasticity with voids, Acta Mechanica 110 (1995) D. Ieşan, Shock waves in micropolar elastic materials with voids, An. St. Univ. Al. I. Cuza Iasi 81 (1985) F. Passarella, Some results in micropolar thermoelasticity, Mech. Res. Comm. 23 (1996) M. Aouadi, A theory of thermoelastic diffusion materials with voids, Z. Angew. Math. Phys. 61 (2010) M. Ciarletta and D. Ieşan, Non-Classical Elastic Solids. (Longman Scientific and Technical, John Wiley & Sons, Inc., New York, NY, Harlow, Essex, UK 1993). 10. D. Ieşan, Thermoelastic Models of Continua. (Kluwer Academic Publishers, London, UK: 2004). 11. B. Straughan, Stability and Wave Motion in Porous Media. (Springer, New York 2008). 12. D. Ieşan and R. Quintanilla, On a theory of thermoelastic materials with a double porosity structure, J. Thermal Stres. 37 (2014) D. Ieşan, Method of potentials in elastostatics of solids with double porosity, Int. J. Engng. Sci. 88 (2015) M. Svanadze, Plane waves, uniqueness theorems and existence of eigenfrequencies in the theory of rigid bodies with a double porosity structure. Continuous Media with Microstructure 2. (eds. B. Albers & M. Kuczma; Springer International Publishing Switzerland 2016) M. Svanadze, Boundary value problems of steady vibrations in the theory of thermoelasticity for materials with double porosity structure, Arch. Mechanics 69 (2017) M. Svanadze, Steady vibrations problems in the theory of elasticity for materials with double voids, Acta Mechanica 229 (2018) R. Kumar, R. Vohra and M. G. Gorla, State space approach to boundary value problem forthermoelastic material with double porosity, Appl. Math. Comp. 271 (2015) R. Kumar, R. Vohra and M. G. Gorla, Some considerations of fundamental solution in micropolar thermoelastic materials with double porosity, Arch. Mechanics 68 (2016) R. Kumar, R. Vohra and M. G. Gorla, Reflection of plane waves in thermoelastic medium with double porosity, Multidis. Model. Mater. Struc. 12 (2016)

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