PAMM · Proc. Appl. Math. Mech. 17, 471 – 472 (2017) / DOI 10.1002/pamm.201710205 External boundary value problems in the quasi static theory of triple porosity thermoelasticity Merab Svanadze 1, ∗ 1 Institute for Fundamental and Interdisciplinary Mathematics Research, Ilia State University, K. Cholokashvili Ave., 3/5, 0162 Tbilisi, Georgia In this paper the quasi static linear theory of thermoelasticity for materials with triple porosity is considered. Basic external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness and existence theorems for regular (classical) solutions of these BVPs are established. c  2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Mathematical formulation of flow through triple porosity media with macro-, meso- and microporosity structure is first intro- duced three decades ago. The several new triple porosity models for solids with this hierarchical structure are presented by Bai and Roegiers [1], Straughan [2], Svanadze [3]. The BVPs of steady vibrations in the theory of triple porosity elasticity are investigated by means of the potential method and the theory of singular integral equations in Svanadze [4]. The BVPs of quasi-static theory of thermoelasticity for solids with triple porosity are studied by Svanadze [5]. The general models for anisotropic thermoelastic solids with triple, quadruple and quintuple porosity structures are presented by Straughan [2]. Re- cently, the mathematical models of double porosity materials are studied by several authors [6-13]. For an extensive review of the works and basic results on the multiple porosity media see Straughan [14]. The intended applications of the theories of elasticity and thermoelasticity for materials with a multi-porosity structure are to geological materials such as oil and gas reservoirs, rocks and soils, manufactured porous materials such as ceramics and pressed powders, and biomaterials such as bone. In this paper the quasi static linear 3D theory of thermoelasticity for materials with triple porosity [2] is considered. Basic external BVPs of steady vibrations are formulated. The uniqueness and existence theorems for regular (classical) solutions of these BVPs are established. 2 Basic equations and boundary value problems Let x =(x 1 ,x 2 ,x 3 ) be a point of the Euclidean three-dimensional space R 3 . We consider an isotropic and homogeneous elastic solid with macro-, meso- and microporosity structure occupies a region of R 3 . The governing system of homogeneous equations of steady vibrations in the quasi static linear theory of thermoelasticity for triple porosity materials with macro-, meso- and micropores consists of the following equations [2]: μ Δu +(λ + μ) ∇divu −∇ (β p) − ε 0 ∇θ = 0, (K Δ+ iωα)p + iωβ divu + iωε θ = 0, (kΔ+ iωaT 0 )θ + iωT 0 (ε 0 div u + ε p)=0, (1) where Δ is the Laplacian operator, u =(u 1 ,u 2 ,u 3 ) is the displacement vector, p 1 , p 2 and p 3 are the macro-, meso- and micropore fluid pressures, respectively; θ is the temperature measured from some constant absolute temperature T 0 (T 0 > 0), p =(p 1 ,p 2 ,p 3 ), β =(β 1 ,β 2 ,β 3 ), βp = ∑ 3 j=1 β j p j ; K =(k lj ) 3×3 , α =(α lj ) 3×3 ,ω is the oscillation frequency, ω> 0, ε =(ε 1 ,ε 2 ,ε 3 ); λ, μ, a, β j ,ε 0 ,ε j ,k,k lj ,α lj (l, j =1, 2, 3) are the constitutive coefficients. In the follows we assume that μ> 0, 3λ +2μ> 0,a> 0,k> 0, K and α are positive definite matrices. Let S be the closed surface surrounding the infinite domain Ω − in R 3 , Ω − =Ω − ∪ S, n(z) be the internal (with respect to Ω − ) unit normal vector to S at z, n =(n 1 ,n 2 ,n 3 ), ∂ ∂n is the derivative along the vector n. Definition 2.1 A vector function U =(u, p,θ)=(U 1 ,U 2 , ··· ,U 7 ) is called regular in Ω − if U l ∈ C 2 (Ω − ) ∩ C 1 ( Ω − ) and U l (x)= O(|x| −1 ), ∂ ∂xj U l (x)= o(|x| −1 ) for |x|≫ 1, where l =1, 2, ··· , 7,j =1, 2, 3. The basic external BVPs of steady vibrations in the quasi static linear theory of thermoelasticity for triple porosity materials are formulated as follows: find in Ω − a regular (classical) solution U =(u, p,θ) to system (1) satisfying the boundary condi- tion lim Ω − ∋x→ z∈S U(x) ≡{U(z)} − = f (z) in the BVP (I ) − f and lim Ω − ∋x→ z∈S R(D x , n(z))U(x) ≡{R(D z , n(z))U(z)} − = ∗ Corresponding author: e-mail svanadze@gmail.com, phone +995 577 553384 c  2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim