Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 45, 2, pp. 239-257, Warsaw 2007 ON THE MODELLING OF HEAT CONDUCTION IN A NON-PERIODICALLY LAMINATED LAYER Jarosław Jędrysiak Alina Radzikowska Department of Structural Mechanics, Technical University of Łódź e-mail: jarek@p.lodz.pl, alla radzikowska@o2.pl Heat conduction in a thick layer made of non-periodically distributed micro-laminas of two rigid conductors is considered. Macroscopic pro- perties of the layer are continuously graded across its thickness (FGL layer). The aim of the paper is to propose an averaged model of non- stationary heat conduction in the thick layer with functionally graded macroscopicproperties.Model equations are derived in the frameworkof the modified tolerance averaging technique, cf.Woźniak and Wierzbicki (2000).Moreover, themodel is applied to showthemicrostructural effect on heat conduction in the FGL layer. Key words: heat conduction, functionally gradedmaterial, non-periodic layer 1. Introduction Themainobject of considerations is a thick layermadeof two rigid conductors, non-periodically distributed in the form of micro-laminas along the thickness of the layer. It is assumed that this laminated composite has macroscopic properties continuously varying across its thickness. Materials of this kind are called Functionally Graded Materials (FGM), cf. Suresh and Mortensen (1998). The geometry of microstructure of those composites cannot be described exactly. Thus, thermomechanical behaviour of such composites can be analy- sed onlywithinmicromechanicalmodelswith idealised geometries.Todescribe FGM-type media, methods proposed to investigate macroscopically homoge- neous composites, e.g. periodic composites (laminates), can be applied. De- spite of the fact that FGM-type media are notmacroscopically homogeneous, those methods are modified and adapted to analyse their overall behaviour. Somemethods were discussed by Reiter et al. (1997) and in a monograph by 240 J. Jędrysiak, A. Radzikowska Suresh and Mortensen (1998). Between these methods, we have to mention methods based on the asymptotic homogenization proposed and developed for periodic composites and structures, cf. Bakhvalov and Panasenko (1984), Ji- kov et al. (1994), and applied to heat conduction problems e.g. byGałka et al. (1994). Heat transfer problemswere also analysed in the framework ofmodels with microlocal parameters, cf. Matysiak (1991). However, models based on the asymptotic homogenization as well as microlocal models neglect usually the effect of microstructure size (called the length-scale effect or the effect of period lengths) on the overall behaviour of laminates withmicroperiodic struc- ture. Homogenization procedures were applied to investigate problems of heat conduction and/or thermal stresses in functionally graded materials in many papers, see e.g. Itoh et al. (1996), where also the aforementioned effect was neglected. In order to circumvent the above drawback, one can apply the tolerance averaging technique, proposed and discussed for periodic composites in a book by Woźniak and Wierzbicki (2000). This method was used to analyse special problems of different periodic structures in a series of papers, e.g. for thin plates in Jędrysiak (2001), for wavy plates in Michalak (2001), for laminates in Szymczyk andWoźniak (2006). Heat conduction problems of periodic com- posites were investigated in the framework of this technique in some papers by e.g. Woźniak et al. (1996), Ignaczak and Baczyński (1997), Woźniak and Wierzbicki (2000), Woźniak M. et al. (2001), Łaciński (2005). This approach leads from equations with functional, periodic, highly-oscillating and, in ge- neral, non-continuous coefficients to a system of differential equations with constant coefficients, describing the effect of microstructure size on the overall behaviour of a periodic composite. In the last time, the tolerance averaging techniquewasmodified and adop- ted to investigate mechanical problems of FGM-type structures, e.g. for FGL plates by Jędrysiak et al. (2005), shells byWoźniak et al. (2005), for laminates with microdefects by Rychlewska et al. (2006). The aim of this paper is twofold. Firstly, to propose a certain averaged microstructural model of heat conduction in a non-periodically laminated lay- er, which has macroscopic properties functionally varying along its thickness. Secondly, to apply the newmodel to show somemicrostructural effects in heat conduction of that layer. 2. Preliminaries Subscripts i,j, . . . run over 1,2,3 and are related to the coordinate system Ox1x2x3; subscripts α,β,. . . run over 2,3 and are related to the system On the modelling of heat conduction... 241 Ox2x3; summation convention holds. Denote by x ≡ (x2,x3), x ≡ x1 and introduce t as the time coordinate. It is assumed that the non-periodically la- minated layer under consideration occupies the region (0,L1)×(0,L2)×(0,L3) in the physical space. Denote by H =L1 the thickness of the layer along the x-axis. The layer ismade of twomaterials distributed in m laminas having the same thickness l; hence H =ml. It is assumed that the number m of laminas is large (m−1 ≪ 1). Thus, the condition l≪H is satisfied and the thickness l will be called the microstructure parameter. The nth lamina is defined as the region In×Π, n=1, . . . ,m, where In ≡ ((n−1)l,nl),Π ≡ (0,L2)× (0,L3). Moreover, every lamina consists of two homogeneous sub-laminas with thick- nesses l′n, l ′′ n. Properties of the sub-laminas are described by specific heats c ′, c′′ andheat conduction tensors k′ij, k ′′ ij, i,j =1,2,3.Material volume fractions in the nth lamina are denoted by ν′n ≡ l′n/l, ν′′n ≡ l′′n/l. Assuming the sequence {ν′n}, n=1, . . . ,m, to bemonotone and for every n=1, . . . ,m−1 to satisfy condition |ν′n+1−ν′n|≪ 1, the thick layer can be treated as made of a FGM and will be called the functionally graded laminated layer (the FGL layer). Similar conditions are satisfied by the sequence {ν′′n} because of ν′n+ν′′n =1. Under the above requirements, the sequences {ν′n}, {ν′′n}, n = 1, . . . ,m, can be approximated by continuous functions ν′(·), ν′′(·), which describe the gra- dation of material properties across the layer thickness. The choice of these functions is very important in design and optimization problems forFGL com- posites.The functions ν′(·), ν′′(·)will be called the fraction ratios ofmaterials. Hence, the non-homogeneity ratio ν can be introduced as a function defined by the formula ν(·) ≡ √ ν′(·)ν′′(·). The aforementioned functions of fraction ratios are assumed to be slowly varying (this concept is defined in the book by Woźniak andWierzbicki (2000) andwill be recalled in the subsequent section). A fragment of the macrostructure of the FGL layer (as the layer made of a functionally graded material) is shown in Fig.1; however, a fragment of the FGL layer on the micro level is presented in Fig.2. Assuming small oscillations of an unknown temperature field θ and intro- ducing the thermal load q (the intensity of heat sources), the heat conduction in the FGL layer can be considered in the framework of the Fourier model, described by the governing equation −(kijθ,j),i+ cθ̇= q (2.1) It has to be emphasized that the above equation has coefficients kij = kij(x), c= c(x), being highly-oscillating, non-continuous functions in x. In order to replace differential equation (2.1) by a system of differential equations with continuous, slowly varying functional coefficients, the tolerance averaging me- thod will be adopted. This method was proposed for periodic structures by Woźniak andWierzbicki (2000). 242 J. Jędrysiak, A. Radzikowska Fig. 1. A fragment of the macrostructure of the non-periodically laminated layer Fig. 2. A fragment of the microstructure of the non-periodically laminated layer 3. Tolerance averaging technique 3.1. Foundations Denote by f an arbitrary integrable function defined in [0,H], which can also depend on x and t as parameters.The averaging operatorwill be defined by 〈f〉(x)= l−1 x+ l 2∫ x− l 2 f(x) dx x∈ [ l 2 ,H− l 2 ] (3.1) where l is the constant thickness of every lamina. Let Ψ be a differentiable function, Ψ ∈C1([0,H]), which can depend on x and t as parameters. The function Ψ will be called a slowly varying func- tion (for a certain tolerance ε ≪ 1 and with respect to the microstructure parameter l) and denoted Ψ ∈ SV , if functions l∂Ψ (where ∂(·) is differen- On the modelling of heat conduction... 243 tiation operation with respect to x) and O(εΨ) are of the same order, i.e. l∂Ψ ∈ O(εΨ), 0 < ε ≪ 1. We also introduce the concept of the fluctuation shape function g(x), x∈ [0,H], defined by the following formula g(x) =    −l √ 3 ν(x) ν′′(x) [ 2 x l +ν′(x) ] for x∈ ( − l 2 ,− l 2 + lν′′(x) ) l √ 3 ν(x) ν′(x) [ 2 x l −ν′′(x) ] for x∈ ( l 2 − lν′(x), l 2 ) where x is the centre of the interval (−l/2, l/2). This function is assumed to be continuous, linear across every sub-lamina thickness and of an order O(l). Moreover, it has values l √ 3ν(x) at the interfaces between laminas, and it takes values −l √ 3ν(x) at the interfaces between the adjacent sub-laminas within the lamina. Because the non-homogeneity ratio ν(·) is a slowly varying function, it can be shown that the mean value of the function g(·) in every lamina is equal to zero. An example of the fluctuation shape function is shown in Fig.3. Fig. 3. An example of the fluctuation shape function g(x) (1 – the 1st conductor, 2 – the 2nd conductor) Adetaileddiscussionof theabove concepts is shown inWoźniak andWierz- bicki (2000). 3.2. Modelling assumptions In the framework of the tolerance averaging technique, additional assump- tions are formulated, cf.Woźniak andWierzbicki (2000). The first modelling assumption is the thermal assumption, which states that the temperature field θ = θ(x,x, t), x ∈ [0,H], x ∈ Π, is restricted by the following formula θ(x,x, t)=Θ(x,x, t)+g(x)ϑ(x,x, t) (3.2) 244 J. Jędrysiak, A. Radzikowska where Θ(·,x, t) ∈ SV is called the averaged temperature, ϑ(·,x, t) ∈ SV is called the fluctuation variable of temperature. The functions Θ(·), ϑ(·) are basic unknowns describing temperature in the FGL layer. The second assumption, called the tolerance approximation, states that for an arbitrary slowly varying function Ψ, Ψ ∈ SV , the approximation Ψ +O(εΨ) ∼= Ψ will be employed, which neglects terms of the order O(ε) as small when compared to 1. 3.3. The outline of the modelling procedure Themodelling procedure for FGM-type composites is similar to the procedure of the tolerance averaging, cf. Woźniak and Wierzbicki (2000), and can be divided into three steps: 1) to average equation (2.1), use formula (3.1), 2) to formulate a variational condition (cf. the aforementioned book), mul- tiply equation (2.1) by the test functionwhichhas characteristic features as the fluctuation shape function, and then to average the resulting equ- ation use formula (3.1), 3) substitute formula (3.2) into theaforementioned averaged equations, and after some calculations, differential equations for the averaged tempera- ture Θ and the fluctuation variable ϑ are obtained. 4. Governing equations of the microstructural model Applying the aforementioned modelling procedure and denoting Kij ≡〈kij〉 K̃ij ≡〈kijg,j〉 K̆ij ≡〈kijg,ig,j〉 K̂ij ≡ l−1〈kijg〉 Kij ≡ l−2〈kijgg〉 C ≡〈c〉 Ĉ ≡ l−1〈cg〉 C ≡ l−2〈cgg〉 Q≡〈q〉 Q̂≡ l−1〈qg〉 (4.1) we arrive at the following equations of the microstructural model of the he- at conduction in the thick FGL layer (the thick layer with an internal non- periodically laminated structure) Kαj(x)Θ,jα+[K1j(x)Θ,j],1−C(x)Θ̇+[K̃11(x)ϑ],1+ K̃α1(x)ϑ,α+ +l{[K̂1α(x)ϑ,α],1+ K̂αi(x)ϑ,αi− Ĉ(x)ϑ̇}=−Q (4.2) K̃1j(x)Θ,j + l[Ĉ(x)Θ̇− K̂αj(x)Θ,jα]+ +K̆11(x)ϑ+ l 2[C(x)ϑ̇−Kαβ(x)ϑ,αβ] =−lQ̂ with terms dependent on the microstructure parameter l. On the modelling of heat conduction... 245 It has to be emphasized that model equations (4.2) have three characteri- stic features: 1) coefficients of these equations are slowly varying functions in x, because the functions ν′(·), ν′′(·), ν(·) are slowly varying, 2) since equations (4.2) depend on the microstructure parameter l, the proposedmodel describes certain microstructural phenomena, 3) boundary conditions for thefluctuation variable ϑhave to be formulated only on that part of the boundary, which intersects lamina interfaces, because the equation for ϑ is independent of the derivative of ϑ with respect to x. Summarizing, the microstructural heat conduction model is determined by: • equations (4.2) for the unknowns Θ(·,x, t), ϑ(·,x, t), • conditions of applicability of themodel, i.e. equations (4.2) have physical sense for theunknowns Θ(·,x, t),ϑ(·,x, t) being slowly varying functions in x for every x and twhich can be treated as parameters, • the temperature field of the non-periodically laminated layer can be ap- proximated by means of formula (3.2). At the end of this section, let us observe that the formal passage l→ 0 in equations (4.2) makes it possible to eliminate the fluctuation variable ϑ from (4.2), which is given by the formula ϑ=−K̃1j(x) K̆11(x) Θ,j (4.3) After substituting (4.3) into equation (4.2)1, a simplified equation repre- senting a certain asymptoticmodel, called themacrostructural heat conduction model of the FGL layer, is derived Kαj(x)Θ,jα+[K1j(x)Θ,j],1−C(x)Θ̇− [K̃11(x)K̃1j(x) K̆11(x) Θ,j ] ,1 + (4.4) − K̃α1(x)K̃1j(x) K̆11(x) Θ,jα =−Q where the effect of the microstructure parameter l is not taken into account. It can be observed that for constant values of the parameters ν′, ν′′, ν′ + ν′′ = 1, equations (4.2) or (4.4) represent certain heat conduction mo- dels of a periodically laminated layer. It is necessary to emphasize that the governing equations of the proposed microstructural model can be employed to periodically laminated layers. 246 J. Jędrysiak, A. Radzikowska 5. Applications 5.1. Heat conduction in a thick FGL layer Let us consider a thick FGL layer subjected to a thermal gradient in the directionparallel to laminas.Hence, a planeproblemof heat conduction in this layer boundedby the planes x=0,x=H and x2 =0,x2 =L is investigated. Sub-laminas of the layer are made of two different isotropic materials. Properties of the sub-laminas are described by specific heats c′, c′′ and heat conduction tensors with elements k′11 = k ′ 22 = k ′ 33 = k ′, k′′11 = k ′′ 22 = k ′′ 33 = k ′′, where k′, k′′ are heat conduction coefficients. Assume that heat sources are zero, Q = Q̂ = 0 and denote K1 ≡ K11, K2 ≡ K22, K̆ ≡ K̆11, K ≡ K22, K̃ ≡ K̃11 and z≡x2. All unknowns in equations (4.2) are functions of x, z, t, i.e. Θ=Θ(x,z,t), ϑ=ϑ(x,z,t). Hence, themicrostructural model equations take the formof twodifferential equations [K1(x)Θ,1],1+K2(x)Θ,22−C(x)Θ̇=−[K̃(x)ϑ],1 (5.1) K̆(x)ϑ+ l2[C(x)ϑ̇−K(x)ϑ,22] =−K̃(x)Θ,1 In the framework of the macrostructural model we obtain from (4.4) {( K1(x)− [K̃(x)]2 K̆(x) ) Θ,1 } ,1 +K2(x)Θ,22−C(x)Θ̇=0 (5.2) which describes only the macrotemperature Θ. Let us introduce a new unknown Ψ instead of ϑ given by the formula Ψ ≡ϑ+ K̃(x) K̆(x) Θ,1 (5.3) It can be observed that in the framework of the macrostructural model the function Ψ is equal to zero. It can be called the microstructural variable. Because themacrotemperature Θ(·,z,t) and the fluctuation variable ϑ(·,z,t) are slowly varying functions in x for every z and t, the new unknown – the microstructural variable Ψ(·,z,t) is also slowly varying in x. Substituting (5.3) into (5.1), we obtain modified equations C(x)Θ̇−K2(x)Θ,22− {( K1(x)− [K̃(x)]2 K̆(x) ) Θ,1 } ,1 = [K̃(x)Ψ],1 (5.4) K̆(x)Ψ+ l2[C(x)Ψ̇ −K(x)Ψ,22] = l2[C(x)Θ̇,1−K(x)Θ,122] K̃(x) K̆(x) On the modelling of heat conduction... 247 The right hand side of (5.4)2 can be rewritten as l2[C(x)Θ̇,1−K(x)Θ,122] K̃(x) K̆(x) = = l2[ν(x)]2[C(x)Θ̇,1−K2(x)Θ,122] K̃(x) K̆(x) Using equation (5.4)1 and taking into account that all coefficients are slowly varying functions in x, we can write the above equation in the form l2[C(x)Θ̇,1−K(x)Θ,122] K̃(x) K̆(x) = = l2[ν(x)]2 K̃(x) K̆(x) {[( K1(x)− [K̃(x)]2 K̆(x) ) Θ,1+ K̃(x)Ψ ] ,11 } Hence, the right hand side of (5.4)2 is small comparing to the left hand side of this equation, bearing in mind that the macrotemperature Θ(·,z,t) and the microstructural variable Ψ(·,z,t) are slowly varying functions in x. The right hand side of (5.4)2 can be neglected in the first approximation of this model, and equations (5.4) can be written in the form {( K1(x)− [K̃(x)]2 K̆(x) ) Θ,1 } ,1 +K2(x)Θ,22−C(x)Θ̇=−[K̃(x)Ψ],1 (5.5) K̆(x)Ψ+ l2[C(x)Ψ̇ −K(x)Ψ,22] = 0 Hence, for the microstructural variable, we obtain independent equation (5.5)2. Similar formulas for vibrations were derived: for FGL plates by Jędry- siak et al. (2005), for FGL shells by Woźniak et al. (2005). Equations (5.5) can be treated as the first approximation of equations (5.1). They describe a certain approximated microstructural model. At the same time, formula for the temperature (3.2) takes the following form θ=Θ−g(x)K̃(x)[K̆(x)]−1Θ,1+g(x)Ψ Below, our considerations are restricted to the problem of changeability of the microstructural variable Ψ, described by equation (5.5)2. Hence, the effect of the microstructure size on the microtemperature in the FGL layer is analysed. 248 J. Jędrysiak, A. Radzikowska 5.2. The effect of microstructure size on the microstructural variable Ψ The effect of microstructure size on the microtemperature, described by equation (5.5)2, is investigated. Let us assume that the solution to this equ- ation has to satisfy the following initial-boundary conditions: — the initial condition Ψ(x,z,0)= f(x,z) (5.6) — the boundary conditions (only for the boundaries z=0, z=L) Ψ(x,0, t)= f(x,0)exp(−ωt)= f0(x)exp(−ωt) Ψ(x,L,t)= 0 (5.7) It can be observed that the boundary conditions for the function Ψ can be formulated only for the z-axis. Hence, the solution to equation (5.5)2 has the form Ψ(x,z,t)= f(x,z)exp(−ωt) (5.8) Denoting χ2 ≡ K(x) C(x) η2 ≡ K̆(x) l2C(x) and substituting solution (5.8) into equation (5.5)2, we obtain a differential equation for the function f f,22−f(η2−ω)χ−2 =0 (5.9) It can be observed that the argument x can be treated as a parameter. Following (5.7), the boundary conditions for the function f have the form f(x,0)= f0(x) f(x,L)= 0 (5.10) Introduce non-dimensional denotations ζ and φ such that ζ ≡ z L f(x,z)= f0(x)φ(x,ζ) Hence, boundary conditions (5.10) take the form φ(0)= 1 φ(1)= 0 Denoting ∂(·)≡ ∂(·) ∂ζ λ≡ l L ρ̆2 ≡ K̆ K Ω≡ ω η2 differential equation (5.9) can be written as ∂2φ−φ(1−Ω) ρ̆ 2 λ2 =0 (5.11) On the modelling of heat conduction... 249 After some manipulations, we arrive at the following cases of solutions to equation (5.11): 1. if Ω=0 and setting κ2 ≡ ρ̆2λ−2 then φ(ζ)= exp(−κζ)1− exp[2κ(ζ−1)] 1− exp(−2κ) (5.12) 2. if 0<Ω< 1 and setting κ2Ω ≡ (1−Ω)ρ̆2λ−2 then φ(ζ)= exp(−κΩζ) 1− exp(−2κΩ) + exp(κΩζ) 1− exp(2κΩ) (5.13) 3. if Ω=1 then φ(ζ)= 1− ζ (5.14) 4. if Ω> 1 and γ2 ≡ (Ω−1)ρ̆2λ−2 6=n2π2 (n=1,2, . . .) then φ(ζ)= sin[γ(1− ζ)] sinγ (5.15) 5. if Ω> 1 and γ2 =n2π2 (n=1,2, . . .) then Ωn =n 2π2λ2ρ̆−2+1 (5.16) Hence, using formulas (5.12)-(5.16), the microstructural variables Ψ of temperature, i.e. solutions to equation (5.5)2, can be obtained: 1. if Ω=0 and setting κ2 ≡ ρ̆2λ−2 then Ψ(x,z)= f0(x)exp ( −κz L )1− exp [ 2κ ( z L −1 )] 1− exp(−2κ) 2. if 0<ω<η2 and setting κ2Ω ≡ (1−Ω)ρ̆2λ−2 then Ψ(x,z)= f0(x) [ exp ( −κΩz L ) 1−exp(−2κΩ) + exp ( κΩz L ) 1− exp(2κΩ) ] exp(−ωt) 3. if ω= η2 then Ψ(x,z)= f0(x) ( 1− z L ) exp(−ωt) 4. if ω>η2 and γ2 ≡ (Ω−1)ρ̆2λ−2 6=n2π2 (n=1,2, . . .) then Ψ(x,z)= f0(x) sin [ γ ( 1− z L )] sinγ exp(−ωt) 250 J. Jędrysiak, A. Radzikowska 5. if ω>η2 and γ2 =n2π2 (n=1,2, . . .) then ωn =n 2π2χ2L−2+η2 It can be observed that the microstructural variable Ψ related to the mi- crostructural effect depends on the frequency ω. Themicrostructural variable decays exponentially for cases 1-2 and linearly for case 3. Certain values of the frequency cause a non-decayed formof themicrostructural variable Ψ (ca- se 4). Moreover, the solution to equation (5.5)2 does not exist (the case 5) for certain frequencies. It has tobe emphasized that themicrostructural variable has tobea slowly varying function in x. Hence, some solutions donot satisfy the aforementioned modelling condition, because they can have points of singularity, e.g. case 4. The above microstructural effects cannot be analysed in the framework of the macrostructural heat conduction model, described by equation (4.4) or (5.2). Some calculational results illustrating the above formulas are shown in the subsequent section. 6. Calculational results Let us assume two different cases of the material distribution across the layer thickness: 1. the first case of the fraction ratios of materials – linear functions ν′(x)= x H ν′′(x)= 1−ν′(x) (6.1) 2. the second case of the fraction ratios ofmaterials – exponential functions ν′(x)= 1− exp ( 2x H ) 1− exp(2) ν′′(x)= 1−ν′(x) (6.2) These functions and proper non-homogeneity ratios ν = √ ν′ν′′ are shown in Fig.4. Some calculational results are shown in Figs. 5-8. In Figs. 5a,b, plots of solutions to equation (5.11), i.e. non-dimensional parts φ of microstructural variables Ψ versus the non-dimensional coordinate ζ ∈ [0,1] are presented. These curves are found for the following parameters: l/L=0.1, k′′/k′ =0.5, x/H =0.5. Decaying solutions (Ω¬ 1:Ω=0,Ω=0.9, Ω=1) are shown in Fig.5a, and oscillating solutions (Ω> 1:Ω=1.1) in Fig.5b. On the modelling of heat conduction... 251 Fig. 4. The fraction ratios of materials ν′, ν′′ and the non-homogeneity ratio ν versus the non-dimensionless coordinate ξ=x/H; (1) – for formulas (6.1), (2) – for formulas (6.2) Fig. 5. Diagrams of non-dimensional parts φ of microstructural variables Ψ versus the non-dimensional coordinate ζ; (a) decaying solutions (Ω¬ 1:Ω=0, 0.9, 1), (b) oscillating solutions (Ω> 1:Ω=1.1) (for parameters: l/L=0.1, k′′/k′ =0.5, x/H =0.5) Plots of non-dimensional functions φ versus the ratio x/H ∈ [0,1] (the non-dimensional parameter ξ) are presented in Figs. 6a,b. These diagrams are determined for the following parameters: l/L=0.1, k′′/k′ =0.5, ζ = z/L= 0.05. Plots for the exponentially decaying solutions (Ω < 1: Ω =0, Ω =0.9) are shown in Fig.6a, and for the oscillating solutions (Ω > 1: Ω = 1.1) in Fig.6b. It can be observed that the oscillating solutions (for Ω > 1: Ω =1.1) are not slowly varying functions. Hence, they cannot be treated as solutions in the framework of the microstructural model, because they do not satisfy the modelling conditions of the tolerance averaging.Below, onlydecaying solutions will be considered. In Fig.7, we have diagrams of non-dimensional parts φ versus the ra- tio k′′/k′ ∈ [0,1]. These curves are calculated for the following parameters: l/L = 0.1, x/H = 0.5, ζ = z/L = 0.05. The presented plots are given only for the exponentially decaying solutions (Ω< 1:Ω=0,Ω=0.9). 252 J. Jędrysiak, A. Radzikowska Fig. 6. Diagrams of non-dimensional parts φ of microstructural variables Ψ versus the ratio x/H; (a) for exponentially decaying solutions (Ω< 1:Ω=0,Ω=0.9), (b) for oscillating solutions (Ω> 1:Ω=1.1) (for parameters: l/L=0.1, k′′/k′ =0.5, z/L=0.05) Fig. 7. Diagrams of non-dimensional parts φ of microstructural variables Ψ versus the ratio k′′/k′ for exponentially decaying solutions, i.e. Ω< 1:Ω=0,Ω=0.9 (for parameters: l/L=0.1, x/H =0.5, z/L=0.05) Curves of non-dimensional functions φ, being the exponentially decaying solutions (Ω < 1: Ω = 0, Ω = 0.9), versus the ratio l/L ∈ [0,1] are shown in Fig.8. These plots correspond to the following parameters: x/H = 0.5, ζ = z/L=0.05, k′′/k′ =0.5. Analysing the obtained results, some remarks can be formulated: 1. the shape of fluctuation variables depends on the non-dimensional para- meter Ω, describing fluctuation frequencies, see Fig.5, i.e.: (a) for 0¬Ω< 1 the fluctuations decay exponentially, (b) for Ω=1 they decay linearly, (c) for Ω> 1 they oscillate; On the modelling of heat conduction... 253 Fig. 8. Diagrams of non-dimensional parts φ of microstructural variables Ψ versus the ratio l/L for exponentially decaying solutions, i.e. Ω< 1:Ω=0,Ω=0.9 (for parameters: k′′/k′ =0.5, x=0.5, z/L=0.05) 2. it can be observed for parameters which are not taken account in Fig.5, that: (a) for small values of the parameter Ω from the interval [0,1), the microstructural variables strongly decay (Fig.5a and Fig.6a), (b) for small values of the ratios l/L, k′′/k′ and for 0 ¬ Ω < 1, the microstructural variables strongly decay, (c) for fraction ratios (6.2), the solutions more decay than for fraction ratios (6.1), see Fig.5a, (d) with increasing values of the parameter Ω (for Ω> 1), the micro- structural variables strongly oscillate; 3. from the results shown in Fig.6, it can be observed that: (a) the maximum values of the decaying microstructural variables for the exponential functions of the fraction ratios of materials (6.2) are obtained for greater values of the ratio x/H than for the linear functions of fraction ratios (6.1), (b) for fixed values of the ratios: l/L, k′′/k′, z/L, the microstructural variables do not exist for certain values of the ratio x/H, determi- ned by formula (5.16), (see points of singularity in Fig.6b), (c) for parameters Ω > 1 (e.g. Ω = 1.1), we obtain the oscillating solutions which have points of singularity (Fig.6b); hence, these solutions are not slowly varying functions in x and they do not hold the modelling conditions; 4. analysing the results presented in Fig.7, we can observe that decaying microstructural variables for the exponential functions of the fraction 254 J. Jędrysiak, A. Radzikowska ratios of materials (6.2) increase with the increasing of the ratio k′′/k′, in contrast to the variables for linear fraction ratios (6.1), which are constant; 5. from the results shown inFig.8, it follows that decayingmicrostructural variables for linear (6.1) and exponential (6.2) functions of the fraction ratios of materials increase with the increasing ratio l/L. 7. Final remarks Applying themodified tolerance averagingmethod proposed for periodic com- posites byWoźniak andWierzbicki (2000) to the equation of heat conduction in thick non-periodically laminated layers (FGL layers), governing equations of the non-asymptotic model of such layers have been derived. Summarizing our considerations, the following general remarks regarding some results concerning the heat conduction inFGL layers can be formulated. • Derived non-asymptotic heat conductionmodel of FGL layers takes into account the effect of microstructure size (the lamina thickness) in con- trast to the asymptoticmodelwhichdescribes only the overall behaviour of the layers. Hence, the proposed model is called the microstructural model and the asymptotic one – the macrostructural. • Both models are governed by equations with functional, continuous co- efficients, being slowly varying functions of the argument describing the layer along the direction perpendicular to laminas. • The microstructural model makes it possible to formulate initial and boundary conditions not only for the averaged temperature but also for fluctuations of the temperature. Analysing the application of the proposed model, some remarks can be for- mulated. • Onexemplaryapplications of themicrostructuralmodel, the effect ofmi- crostructure size on the problemof heat conduction in a thick FGL layer hasbeenanalysed. It hasbeen shown that changes of themicrostructural variable (hence, the fluctuation of the temperature) can be investigated independently of the changes of macrotemperature (the averaged tem- perature). Hence, this effect is called the intrinsic microstructural effect. However, changes of the averaged temperature cannot be analysed inde- pendently of the changes of temperature fluctuation, but this problem will be analysed in forthcoming papers. On the modelling of heat conduction... 255 • The microstructural variables are different for different values of fluc- tuation frequencies, i.e. there exists a special value of the fluctuation frequency, such that: – the microstructural variables decay exponentially for frequencies smaller than this value, – the variables decay linearly for the frequency equal to the special value, – the variables oscillate for frequencies greater than this value (5.15) and they have points of singularity, – the variables do not exist for the frequencies determined from for- mula (5.16). • It can be observed that the microstructural variables are not slowly va- rying functions for frequencies greater than this special value of the fluc- tuation frequency, and these variables are not solutions in the framework of the model assumptions. • The changeability of the microstructural variables depends on: – different functions of the fraction ratios of materials, e.g. exponen- tial or linear functions, – differences between the heat conduction coefficients k′, k′′. Other problems of heat conduction in FGL layers in the framework of the microstructural model will be considered in forthcoming papers, e.g. the effect of changes of the fluctuation of temperature on changes of the averaged temperature. References 1. BakhvalovN.C.,PanasenkoG., 1984,Averaging Processes in PeriodicMe- dia, Nauka, Moscow [in Russian] 2. GałkaA., Telega J., WojnarR., 1994, Thermodiffusion in heterogeneous elastic solids and homogenisation,Arch. Mech., 46, 267-314 3. Ignaczak J., Baczyński Z.F., 1997, On a refined heat-conduction theory of micro-periodic layered solids, J. Thermal Stresses, 20, 749-771 4. Itoh Y., Takahashi M., Takano H., 1996, Design of tungsten/copper gra- ded composite for high heat flux components, Fusion Engng. Design, 31, 4, 279-289 256 J. Jędrysiak, A. Radzikowska 5. Jędrysiak J., 2001, Dispersion models of thin periodic plates. Theory and applications, habilitation thesis, Sci. Bull. of Łódź Techn. Univ., 872, series: Transactions, 289, Łódź [in Polish] 6. Jędrysiak J., Rychlewska J., Woźniak C., 2005, Microstructural 2D-models of functionally graded laminated plates, in: Shell Structures: The- ory andApplications,W.Pietraszkiewicz,C.Szymczak (edit.), series:Balkema – Proceedings and Monographs in Engineering, Water and Earth Science, Taylor and Francis, London-Leiden, 119-123 7. Jikov V.V., Kozlov S.M., Oleinik O.A., 1994, Homogenization of Diffe- rential Operators and Integral Functionals, Springer-Verlag 8. Łaciński Ł., 2005, Numerical verification of twomathematical models for the heat transfer in a laminated rigid conductor,J.Theor. Appl.Mech., 43, 367-384 9. Matysiak S.J., 1991, On certain problems of heat conduction in periodic composites, ZAMM, 71, 524-528 10. Michalak B., 2001, Dynamics and stability of the wavy plates, habilitation thesis, Sci. Bull. of Łódź Techn. Univ., 881, series:Transactions, 295, Łódź [in Polish] 11. Reiter T., Dvorak G.J., Tvergaard V., 1997, Micromechanical models for graded composite materials, J. Mech. Phys. Solids, 45, 1281-1302 12. Rychlewska J., Woźniak C., Woźniak M., 2006, Modelling of func- tionally graded laminates revisited, Electronic J. Polish Agric. Univ., 6, 2 (www.ejpau.media.pl) 13. Suresh S., Mortensen A., 1998,Fundamentals of Functionally Graded Ma- terials, Cambridge, The University Press 14. Szymczyk J., Woźniak C., 2006, A contribution to the modelling of periodically laminated solids, Electronic J. Polish Agric. Univ., 6, 1 (www.ejpau.media.pl) 15. Woźniak C., Baczyński Z.F., Woźniak M., 1996, Modelling of nonsta- tionary heat conduction problems in micro-periodic composites, ZAMM, 76, 223-229 16. WoźniakC., Rychlewska J.,Wierzbicki E., 2005,Modelling and analysis of functionally graded laminated shells, in: Shell Structures: Theory and Appli- cations,W.Pietraszkiewicz, C. Szymczak (edit.), series:Balkema – Proceedings andMonographs in Engineering,Water and Earth Science, Taylor andFrancis, London-Leiden, 187-190 17. Woźniak C., Wierzbicki E., 2000,Averaging Techniques in Thermomecha- nics of Composite Solids, Częstochowa,WydawnictwoPolitechnikiCzęstochow- skiej 18. Woźniak M., Wierzbicki E., Woźniak C., 2001, A macroscopic model of the diffusion and heat transfer processes in a periodicallymicro-stratified solid layer,Acta Mech., 157, 175-185 On the modelling of heat conduction... 257 O modelowaniu przewodnictwa ciepła w nieperiodycznie laminowanej warstwie Streszczenie W pracy rozpatrywane jest przewodnictwo ciepła w grubej warstwie zbudowanej z nieperiodycznie rozmieszczonychmikrolamin, wykonanych z dwóch przewodników. Własności makroskopowe warstwy zmieniają się w sposób ciągły wzdłuż jej grubo- ści (tzw. warstwa typu FGL – ang. functionally graded laminate), por. rys.1. Ce- lem pracy jest przedstawienie uśrednionego modelu niestacjonarnego przewodnictwa ciepła w grubej warstwie o funkcyjnie zmieniających się własnościach makroskopo- wych. Równania modelu otrzymano w ramach zmodyfikowanej techniki tolerancyj- nego uśredniania, por.Woźniak iWierzbicki (2000). Ponadto, zaproponowanymodel będzie wykorzystany w przykładzie dla warstwy typu FGL, który pozwoli pokazać wpływmikrostruktury. Manuscript received May 31, 2006; accepted for print October 18, 2006