On the certain asymptotic approach to the modelling of microheterogeneous media

The object of analysis is mathematical modelling of thermomechanical problems of microheterogeneous media. The aim of this contribution is to propose a certain asymptotic modeling technique. The proposed approach is realized both in consistent and semiconsistent form. Introduction The well known asymptotic homogenization theory leads to the solutions to what are called the periodic cell problems, [1]. Solution to these problems makes it possible to define the effective modulae of the medium under considerations. In most cases obtaining the solution to the cell problem is not an easy task and can not be realized in the analytical form. On the other hand the asymptotic modelling of differential equations introduced in this paper does not involve any periodic (or locally periodic) cell problem. However, the proposed asymptotic procedure introduces the extra unknown functions which are called fluctuation amplitudes. At the same time the consistent asymptotic modelling makes it possible to define effective modulae by eliminating these extra unknowns. Simultaneously, the semiconsistent asymptotic procedure describes the effect of the cell size on the overall behaviour of the medium under consideration in contrast to the asymptotic homogenization technique. 1. Basic concepts and notions To make this paper self-consistent in this section we summarise the main concepts introduced in [2]. Let Ξ × Ω be a bounded domain in n R such that m R ⊂ Ω and m n R − ⊂ Ξ provided that m n > . Points from domain Ω will be denoted by ( ) m x x x x ,..., , 2 1 = or ( ) m z z z z ,..., , 2 1 = and points from Ξ by ( ) m n− ξ ξ ξ = ξ ,..., , 2 1 . If m n = then Ξ and ξ drop out from considerations. We introduce gradient operators of a smooth funcPlease cite this article as: Jolanta Borowska, Jowita Rychlewska, Czeslaw Woźniak, On the certain asymptotic approach to the modelling of microheterogeneous media, Scientific Research of the Institute of Mathematics and Computer Science, 2009, Volume 8, Issue 1, pages 5-12. The website: http://www.amcm.pcz.pl/ J. Borowska, J. Rychlewska, C. Woźniak 6 tion f defined on Ξ × Ω setting ( ) ξ , z f grad f z ≡ ∂ , ( ) ξ ≡ ∇ ξ , z f grad for m n > such that . f f f ∇ + ∂ ≡ ∇ When m n = then ∇ drop out from considerations and the total gradient of f is denoted by f f ∂ = ∇ . Moreover, we define the basic cell setting [ ] [ ] 2 / , 2 / ... 2 / , 2 / 1 1 m m λ λ λ λ − × × − ≡ □ where 0 ,..., 1 > m λ λ . By λ we denote diameter of □ , □ diam ≡ λ , and assume that Ω << λ L , where Ω L is the smallest characteristic length dimension of domain Ω . Hence ( ) □ □ + ≡ x x is a cell with centre at . m R x ∈ We also define ( ) ( ) z x z x □ □ ∈ ∩ Ω ≡ Ω , Ω ∈ x as a cluster of m 2 cells having common sides. Family of cells ( ) ( ) { } Ω ∈ = Ω x x , , □ □ will be referred to as the uniform cell distribution assigned to Ω . In order to present an unified approach to the asymptotic modelling all subsequent considerations will be restricted to Sobolev spaces ( ) Ω k H for . 1 , 0 = k By ( ) □ 0 H we denote the space of □ periodic square-integrable functions defined in m R . Moreover let ( ( ) ⋅ ⋅, ~ k f be a function defined in , m R × Ω . ,..., 1 , 0 α = k For the sake of simplicity the above denotations are assumed to hold both for scalar as well as vector functions. Function ( ) Ω ∈ 1 H f will be called the tolerance periodic function (with respect to cell □ and tolerance parameter δ), ( ) □ , 1 Ω ∈ δ TP f , if for 1 , 0 = k the following conditions hold (i) ( ) ( ( ) ( ) ( ) □ 0 , ~ H x f x k ∈ ⋅ ∃ Ω ∈ ∀ ( ) ( ( ) ( ) , , ~ 0     ≤ ⋅ − ⋅ ∂ Ω Ω δ x x H k k x f f (ii) ( ( ) ( ) ( ) ∫ ⋅ Ω ∈ ⋅ □ 0 , ~ C dz z f k . Function ( ( ) ⋅ , ~ x f k will be referred to as the periodic approximation of f k ∂ in ( ) , , Ω ∈ x x □ 1 , 0 = k . Function ( ) Ω ∈ 1 H h will be called the highly oscillating function (with respect to the cell □ and tolerance parameter δ), ( ) □ , 1 Ω ∈ δ HO h , if (i) ( ) □ , 1 Ω ∈ δ TP h , (ii) ( ) ( ( ) ( ) ( )     = ⋅ ∂ = ⋅ Ω ∈ ∀ 1 , 0 , , ~ , ~ ฀ k x h x h x k x k . On the certain asymptotic approach to the modelling of microheterogeneous media 7 Let ( ) ( ) { } N A HO h h A ,..., 1 , , 1 = Ω ∈ ⋅ = δ □ be a system of N linear independent functions which is assumed to be postulated a priori in every modelling problem under consideration. Functions ( ) ⋅ A h , N A ,..., 1 = are referred to as fluctuation shape functions. We shall assume that for every Ω ∈ x condition ( ) 0 = ρ x h is satisfied for a certain given a priori positive function ( ) □ , 0 Ω ∈ ρ δ TP . In a special case ( ) 0 ≡ x h , N A ,..., 1 = . The text of the subsequent sections is strictly related to that presented in the first part of monograph [3]. 2. Consistent asymptotic averaging of integral functionals Let us assume that ( ) ( ) □ , 0 Ω ∈ ⋅ δ HO f . We recall that if ( ) ( ) □ , 0 Ω ∈ ⋅ δ HO f then for every Ω ∈ x there exists function ( ) z x f , ~ , ( ) x z □ ∈ . Function ( ) ⋅ , ~ x f is called a periodic approximation of highly oscillating function ( ) ⋅ f in ( ) Ω ∩ x □ . Firstly we introduce parameter n 1 = e , ,... 2 , 1 = n . Moreover we define       − × ×       − ≡ Ω 2 , 2 ... 2 , 2 1 1 m m λ e λ e λ e λ e e as a scaled cell and ( ) e e + ≡ □ □ x x as a scaled cell with a centre at Ω ∈ x . Let ( ) ( ) □ 1 , ~ H x f ∈ ⋅ for every Ω ∈ x . We shall denote by ( ) ( ) ( ) □ □ 1 1 , ~ H H x f ⊂ ∈ ⋅ e e family of functions defined by ( )       e ≡ e z x f z x f , ~ , ~ where ( ) x z e ∈□ , . Ω ∈ x Moreover, for an arbitrary function ( ) Ω ∈ g we have ( ) ( ) ( ) ( ) x x x → ∀ ⇒ → e e Ω ∈ □ 0 We start with lagrangian ( ) ( ) □ , , , 0 Ω ∈ ∇ ⋅ = δ HO w w L L , where for the time being it is assumed that ( ) Ξ × Ω ∈ 1 C w . Due to the fact that lagrangian L is highly oscillating in Ω there exists for every Ω ∈ x lagrangian ( ) w w x , , , ~ ∇ ⋅ L which is ( ) x □ periodic in m R and constitutes a periodic approximation of lagrangian L in ( ) x □ . Let ( ) ⋅ A h , N A ,..., 1 = be a system of linear independent highly oscillating functions, ( ) ( ) □ , 1 Ω ∈ ⋅ δ HO h A . It follows that there exist functions ( ) ⋅ , ~ x h A , N A ,..., 1 = J. Borowska, J. Rychlewska, C. Woźniak 8 for every Ω ∈ x . Moreover, let ( ) ⋅ e , ~ x h A , N A ,..., 1 = be a family of functions given by ( )       e ≡ e z x h z x h A A , ~ , ~ , ( ) x z e ∈□ (1) The fundamental assumption of the proposed procedure is that we introduce family of fields ( ) ( ) ( ) ( ) ( ) Ξ ∈ ξ ∈ ξ e + ξ = ξ e e e , , , , ~ , , , x z z z x h z z x w A A □ (2) where summation over N A ,..., 1 = holds. It is assumed that functions , A are continuously bounded in Ω together with their first derivatives. Formula (2) will be referred to as the consistent asymptotic decomposition of field ( ) ξ , , z x w , ( ) x z □ ∈ , Ξ ∈ ξ . Setting ( ) ( ) e = e ∂ e ≡ ∂ z z A A z x h z x h , ~ 1 , ~ (3) from formula (1) and (2) we obtain ( ) ( ) ( ) ( ) ( ) ( ) ( ) x z z z x h z z x h z z x w A A A A e e e e ∈ ξ ∇ e + ξ ∂ + ξ ∇ = ξ ∇ □ , , , ~ , , ~ , , , (4) for an arbitrary but fixed . Ω ∈ x Here and subsequently in the proposed approach functions ( ) ⋅ , ( ) ⋅ A , A=1,...,N are assumed to be independent of e . This is the main difference between the asymptotic approach under consideration and approach which is used in the homogenisation theory [1, 4]. We have to keep in mind that by means of property of mean value, [1], functions ( ) z x h A , e , ( ) x z e ∈□ , are weakly bounded and have under 0 → e weak limit. Taking into account 0 → e , by virtue of ( ) x z e ∈□ , Ω ∈ x , we obtain ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) e + ξ ∇ = ξ ∇ e + ξ = ξ e + ξ ∇ = ξ ∇ e + ξ = ξ O x z O x z O x z O x z A A A A , , , , , , , ,