Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 44, 4, pp. 783-795, Warsaw 2006 ON THE MODELLING AND OPTIMIZATION OF FUNCTIONALLY GRADED LAMINATES Jowita Rychlewska Institute of Mathematics and Computer Sciences, Częstochowa University of Technology e-mail: rjowita@imi.pcz.pl The object of investigations are Functionally Graded Materials (FGM) which on the microstructural level are made of two kinds of very thin laminae. These FGM will be referred to as the Functionally Graded Laminates (FGL). The aim of this contribution is to formulate discrete- continuum and continuum 3D-models of elastodynamics of FGL. The proposedmodelling procedure constitutes a certain generalization of the approach to the modelling of periodic structures leading to a system of finite difference equations and then to their continuum approximation, Rychlewska andWoźniak (2003).The obtained results are applied to the analysis of a certain layered structure with a FGL transition zone. The optimization problem related to the position of the transition zone is discussed. Key words: functionally graded laminates, dynamics, modelling 1. Introduction Functionally Graded Materials (FGM) are usually regarded as heterogeneous composites having effective (macroscopic) properties varying smoothly in spa- ce. A review of researches on FGM can be found in Suresh and Mortensen (1998). In this paper, the object of considerations are micro-layered linear elastic solids made of two materials and having macroscopic properties conti- nuously varying in the direction normal to the layering. These solids will be referred to as the Functionally Graded Laminates (FGL). A fragment of FGL on themacro andmicro-level is shown in Fig.1. Themodelling approach pre- sented in this contribution takes into account some concepts and assumptions of the tolerance averaging technique formulated and applied in Woźniak and Wierzbicki (2000) for periodic structures. This technique was also used in the modelling of elastodynamics of functionally graded laminated shells,Woźniak et al. (2005), functionally graded laminated plates, Jędrysiak et al. (2005) and 784 J. Rychlewska functionally graded laminates with interlaminar microdefects, Rychlewska et al. (2006). The purpose of this contribution is to formulate discrete-continuum models ofFGLwhich state abasis for continuummodels.To this end, a certain generalization of the knownapproach to themodelling of periodic structures is applied. For the aforementioned periodic structures, the periodic simplicial di- vision techniquewas used inRychlewska andWoźniak (2003). In this case, the system of finite difference equations is obtained. These equations constitute foundations of different continuummodels represented by equations with con- stant coefficients. For functionally graded laminates, the proposed continuum model equations have slowly-varying, smooth coefficients. Notations. The index n run over 1, . . . ,N unless otherwise stated and is assigned to the n-th layer of FGL. Subscripts α,β,γ,δ run over the sequ- ence 1,2 and subscripts i,j,k, l over 1,2,3. For an arbitrary sequence {fm}, m=0, . . . ,N, we define the difference operators ∆fm = fm+1−fm l m=0, . . . ,N −1 ∆fm = fm−fm−1 l m=1, . . . ,N where the superscript m is related to the interface z=ml between the m-th and (m+1)-th layer (provided that m= 1, . . . ,N −1) and m= 0, m=N are related to the boundary planes z=0, z=Nl, respectively. In the physical space, we introduce the cartesian orthogonal coordinate system 0x1x2x3 with the x3 axis normal to the lamina interfaces. Let ∂αf and ∂kf stand for partial differentiation of the function f(x1,x2,x3) with respect to xα and xk, respectively. We also use gradient operators ∇ = (∂1,∂2,∂3), ∇=(∂1,∂2,0), gradient-difference operators D=(∂1,∂2,∆), D=(∂1,∂2,∆) and introduce notations z=x3,x=(x1,x2). The time coordinate is denoted by t and time differentiation by the overdot. Small bold-face letters represent vectors andpoints in 3D space, capital bold-face letters stand for second order tensors, and block letters are used for higher order tensors. In the paper, the absolute notations one used. 2. Preliminaries Let Ω× (0,L), Ω⊂R2, stand for a region occupied in the physical space by the laminated medium under consideration in its natural configuration. The subject of analysis is a FGL medium composed of two linear-elastic materials distributed in N layers Λ1, . . . , ,ΛN of the same thickness l. It is assumed that N−1 ≪ 1. Every layer Λn ismade of two homogeneous laminae Λ′n,Λ′′n having On the modelling and optimization... 785 different thicknesses l′n, l ′′ n, respectively, n = 1, . . . ,N. By ρ ′, C′, ρ′′, C′′ we denote mass densities and tensors of elastic moduli in every pair of adjacent laminae, cf. Fig.1. The material volume fractions in the laminae Λ′n, Λ ′′ n are denotedby ν′n = l ′ n/l andby ν ′′ n = l ′′ n/l, respectively, ν ′ n+ν ′′ n =1.Moreover,we introduce the phase distribution sequence {νn} setting νn = √ ν′nν ′′ n. By ν ′(·), ν′′(·)wedenote smooth functionsdefinedon [0,L] representingdistributionsof themean volume fractions of laminamaterials, ν′(z)+ν′′(z)= 1, z∈ [0,L]. It means that ν′(zn)= ν ′ n, ν ′′(zn)= ν ′′ n for some zn ∈ [(n−1)l,nl],n=1, . . . ,N. We also define ν(z)= √ ν′(z)ν′′(z), z∈ [0,L]. Fig. 1. Fragments of FGL on the macro- andmicro-structural level together with a scheme of the n-th layer, n=1, . . . ,N The sequence {fm} will be referred to as slowly-varying (with a certain tolerance 0 < ε ≪ 1) if the condition l|∆fm| ¬ εmax{|f0|, . . . , |fN|} holds for every m = 0,1, . . . ,N − 1. In this case, we shall write {fm} ∈ SVε. For a detailed discussion of the concept of slowly-varying function cf. Woźniak and Wierzbicki (2000). Crucial assumptions related to the material volume sequences are: 1◦ sequence {ν′n}, n=1, . . . ,N, is strongly monotone; 2◦ sequence {ν′n}, n=1, . . . ,N, is slowly varying. Similar requirements are satisfied by the sequence {ν′′n}, n=1, . . . ,N. Under the above conditions, the laminatedmedium represents a certain Functionally Graded Laminate (FGL). Moreover, it is assumed that the lamina materials have elastic symmetry planes parallel to the lamina interfaces and that the laminae are perfectly bonded. 786 J. Rychlewska 3. Modelling assumptions Let w = w(x,z,t), x = (x1,x2) ∈ Ω, z ∈ [0,L] stand for the displacement field at time t. The restriction of this field to the n-th layer Λn will bedenoted by wn i. e., wn = w(x,z,t), x ∈Ω, z ∈ [(n−1)l,nl], n= 1, . . . ,N. Let us also denote w′n =w(x,z ′ n, t),x∈Ω, z′n ∈ [(n−1)l, l′n+(n−1)l],n=1, . . . ,N and w′′n =w(x,z ′′ n, t),x∈Ω, z′′n ∈ [l′n+(n−1)l,nl], n=1, . . . ,N. Moreover, let us denote by w̃n(x,z,t), z = c̃n, n = 0,1, . . . ,N, the restriction of the displacement field to interfaces between the layers Λn and by wn(x,z,t), z= cn,n=1, . . . ,N, the restriction of this field to interfaces between laminae in Λn, n = 1, . . . ,N, cf. Fig.2, where c̃n = nl, cn = nl+ l ′ n. We introduce functions un(x, t), vn(x, t) satisfying conditions w̃n =un n=1, . . . ,N wn =un+2 √ 3lνnvn+ l ′ n∆un n=0,1, . . . ,N−1 Fig. 2. A scheme of the n-th layer and displacements on interfaces, c̃ n =nl, c n =nl+ l′ n , n=1, . . . ,N,N−1 ≪ 1 The first modelling assumption states that displacements in every laminae belonging to the n-th layer, n=1, . . . ,N, are linear functions of z. This sta- tement is satisfied if for every pair of laminae Λ′n,Λ ′′ n, pertinent displacements w ′ n,w ′′ n are assumed respectively in the form w ′ n = ( 2 √ 3 νn ν′n vn+∆un ) z′n+un (3.1) w ′′ n = ( ∆un−2 √ 3 νn ν′′n vn ) z′′n+un+2l √ 3νnvn+ l ′ n∆un where un = un(x, t), vn = vn(x, t), ∆un = ∆un(x, t), z ′ n ∈ [(n− 1)l, l′n + (n−1)l], z′′n ∈ [l′n+(n−1)l,nl], n=1, . . . ,N. Moreover, conditions v0 = v1 and vN−1 = vN are implied by the postulated homogeneity of the layers Λ1 and ΛN, respectively. The secondmodelling assumption states that the continuous mass distribu- tion in the laminate can be approximated by a proportional mass distribution On the modelling and optimization... 787 only on the interfaces between adjacent layers. This assumption can be applied if macroscopic wavelengths are large when compared to the lamina thicknes- ses. Let ρn, n=1, . . . ,N, stand for the mass density on the interface z= c̃n between layers. Then, the above assumption implies that the kinetic energy density will take the form κn = 1 2 ρn(u̇n) 2 (3.2) where ρn = 1 2 ρ′[(ν′n−1) 2+ν′n(2−ν′n)]+ 1 2 ρ′′[ν′′n−1(2−ν′′n−1)+(ν′′n)2] (3.3) Themean strain energy density in the n-th layer is given by σn = ν ′ nσ ′ n+ν ′′ nσ ′′ n (3.4) where σ′n = 1 2 E ′ : C′ :E′ σ′′n = 1 2 E ′′ : C′′ :E′′ and where E ′ = 1 2 ( ∇w′n+(∇w′n)⊤ ) E ′′ = 1 2 ( ∇w′′n+(∇w′′n)⊤ ) stand for pertinent linearized strain tensors. 4. Model equations 4.1. Discrete-continuum model The governing equations for the basic unknowns un(x, t), n=0,1, . . . ,N and vn(x, t), n=0,1, . . . ,N,x∈Ω, t∈R, will be derived from the principle of stationary action for the action functional A= N∑ n=1 t1∫ t0 ∫ Ω Ln(vn,∇un,∆un,u̇n) dxdt (4.1) Ln =κn−σn where κn and σn are determined by formulae (3.2), (3.4). Let us introduce the following denotations 〈Cn〉= ν′nC′+ν′′nC′′ [Cn] = 2 √ 3νn(C ′− C′′) · i3 [Cn]⊤ =2 √ 3νni3 · (C′− C′′) {Cn}=12i3 · (ν′′nC′+ν′nC′′) · i3 (4.2) 788 J. Rychlewska where i3 =(0,0,1) is a unit normal to the layering. It can be shown that the Euler-Lagrange equations related to (4.1) take the form ρnü n−D · (〈Cn〉 :Dun+[Cn] ·vn)=0 (4.3) {Cn} ·vn+[Cn]⊤ :Dun =0 Equations (4.3) represent the discrete- continuum model of the FGL under consideration. The abovemodel equations have to be considered together with relevant boundary and initial conditions. After obtaining a solution to the specific boundary/initial value problem, the distribution of displacements in the laminae Λ′n,Λ ′′ n, n=1, . . . ,N, is described by formulae (3.1). Let us observe that the unknowns vn, n=0,1, . . . ,N, can be eliminated from the governing equations. We obtain v n =−{Cn}−1 · [Cn]⊤ :Dun (4.4) Introducing the following tensors of effective elastic moduli C n 0 = 〈Cn〉− [Cn] · {Cn}−1 · [Cn]⊤ we obtain equations ρnü n−D · (Cn0 :Dun)=0 (4.5) Equations (4.5) and (4.4) represent an alternative form of the general model equations (4.3). It has tobe emphasized that the solutions un,n=0,1, . . . ,N, have a physical sense only if the sequences {ν′n}, {ν′′n} are slowly-varying. Then, the mass density ρn reduces to the form ρn = ρ ′ν′n+ρ ′′ν′′n Equations (4.4), (4.5) constitute the foundationsof subsequentanalysis leading to a continuummodel of the FGL under consideration. 4.2. Continuum model We shall assume that for the finite sequence {fn}, n= 1, . . . ,N, in equ- ations (4.5), (4.4) there exists a continuous function f(z), z ∈ [0,L] such that fn are approximated by f(nl) for n=1, . . . ,N.Moreover, formacrosco- pic deformation wavelengths large when compared to the lamina thicknesses we assume that the function f(·) is differentiable, and we shall approximate ∆fn by ∂3f(nl). Under the above conditions, equations (4.4), (4.5) can be interpreted as a certain finite difference approximation of the equations ρü−∇· (Ch :∇u)=0 (4.6) On the modelling and optimization... 789 and v=−{C}−1 · [C]⊤ :∇u (4.7) where Ch is the tensor of effective elastic moduli C h = 〈C〉− [C] · {C}−1 · [C]⊤ and ρ= ρ′ν′+ρ′′ν′′ Equations (4.6) and (4.7) represent the continuummodel equations of theFGL under consideration. In the subsequent section, the proposed models will be comparedwithmodels obtained byusing a similar discretization approach and presented in Rychlewska (2006). 5. Comparison of models Themodelling procedure proposed in Rychlewska (2006) is based on the con- cepts of the tolerance averaging technique formulated and applied inWoźniak and Wierzbicki (2000) for periodic composites. Moreover, this approach is a certain generalization of the modelling technique leading to a system of finite difference/differential equations.Tomake this paper self-consistent, we outline below the basic concepts and results presented in Rychlewska (2006). Instead of (3.1), the displacements w′n, w ′′ n are assumed respectively in the form w ′ n = [(un− l √ 3νnvn)z ′ n+(un− l′n∆un+ l √ 3νnvn)(l ′ n−z′n)] 1 l′n (5.1) w ′′ n = [(un− l √ 3νnvn)(l ′′ n−z′′n)+ +(un+ l ′′ n∆un+ l √ 3νnvn+ l 2 √ 3νn∆vn− l∆(l′n∆un))z′′n] 1 l′′n where un = un(x, t), vn = vn(x, t), ∆un = ∆un(x, t), ∆vn = ∆vn(x, t), z′n ∈ [(n−1)l, l′n+(n−1)l], z′′n ∈ [l′n+(n−1)l,nl], n=1, . . . ,N. On the assumption that the sequences {ν′n}, {ν′′n} of component volume fractions in the FGL are slowly-varying, it was stated that {un}, {νnvn}, {l′n∆un} are slowly-varying (in a certain tolerance ε). Hence, the displace- ments on interfaces between the adjacent layers are w̃n ∼=un+l √ 3νnvn wn ∼=un−l √ 3νnvn w̃n+1 ∼=un+l √ 3νnvn (5.2) 790 J. Rychlewska and strains in the laminae Λ′n,Λ ′′ n of the n-th layer are obtained in the form ε′n =∆un−2l √ 3νn(ν ′ n) −1 vn ε ′ n ∼=∆un+2l √ 3νn(ν ′′ n) −1 vn (5.3) The strain energy density is taken in the form analogous to that given by (3.4), while the kinetic energy density is represented by κn = 1 2 l2ρn(νn) 2(v̇n) 2+ 1 2 ρn(u̇n) 2 (5.4) where ρn = ρ ′ν′n + ρ ′′ν′′n. Under denotations (4.2), the discrete-continuum model is represented by equations ρnü n−D ·Sn =0 l2ρnν2nv̈n+hn =0 n=2, . . . ,N −1 (5.5) where S n = 〈Cn〉 :Dun+[Cn] ·vn hn = {Cn} ·vn+[Cn]⊤ :Dun (5.6) n=1, . . . ,N In the framework of continuummodels, one can bemention here the tolerance averaged model equations ρü−∇·S=0 l2ρν2v̈+h=0 (5.7) where ρ= ρ′ν′+ρ′′ν′′ S = 〈C〉 :∇u+[C] ·v h= {C}·v+[C]⊤ :∇u (5.8) and the asymptotic approximation model equations are ρü−∇· (Ch :∇u)=0 v=−{C}−1 · [C]⊤ :∇u (5.9) where Ch is the tensor of effective elastic moduli. It can be easily observed that continuum model equations (4.6), (4.7) ob- tained in this paper have the same form like equations (5.9). It has to be emphasized that, contrary to discrete-continuum model equations (4.3), mo- del equations (5.5) and (5.6) describe the microstructure length-scale effect on the overall behaviour of the FGL. It follows that also continuum tolerance averaged models take into account the effect of the layer thickness l on the dynamic behaviour of the FGL. The proposed continuummodel neglects this effect. Equations (4.6), (4.7) represent the continuummodel corresponding to that of the linear elasticity theory and described by equations obtained by the known homogenization approach. However, the form of equations (4.6), (4.7) is relatively simple and it can be applied to the analysis of special problems in which the length-scale effect can be neglected. An example of such a case will be shown in the subsequent section. On the modelling and optimization... 791 6. Example of applications As an example of applications we shall investigate the problem of harmonic vibration along the x3-axis of a laminated solid consisting of two isotropic homogeneous layers interconnected by a functionally graded layer, see Fig.3. Let us denote by E′, E′′ the elastic moduli of component materials in the uniaxial extension and/or compression.By ρ′, ρ′′ mass densities of component materials will be denoted. The problem will be treated as independent of x, and hence (4.6) implies that 〈ρ〉ü− (Eeffuz)z =0 (6.1) where u=u3(z,t), z∈ [0,L], t∈R and 〈ρ〉= [1− ν̃(z)]ρ′+ ν̃(z)ρ′′ Eeff = E′E′′ ν̃(z)E′+[1− ν̃(z)]E′′ (6.2) Fig. 3. The laminated solid consisting of two homogeneous layers and a graded interlayer with inertial loading of the mass m The function ν̃(·) is defined on [0,L] and determines the gradation of material properties for the component material with the superscript ”bis”. It is assumed that the graded layer has the thickness H and L = L1 +L2, where L1,L2 are thicknesses from the midplane of the transition zone to the boundary planes, see Fig.3. The distribution of the volume fraction is shown in Fig.4. It is postulated in the following form ν̃(z)=    0 if z∈ [ −L1,− H 2 ] 1 2 + z H if z∈ [ − H 2 , H 2 ] 1 if z∈ [H 2 ,L2 ] (6.3) 792 J. Rychlewska Fig. 4. The distribution of the volume fraction ν̃(z), z∈ [−L1,L2] Let us denote c2 = E′ ρ′ = E′′ ρ′′ k= E′ E′′ −1= ρ ′ ρ′′ −1­ 0 Theparameter kwill be called the coefficient of inhomogeneity (k­ 0).Using this parameter, we obtain 〈ρ〉= ρ′ ( 1− k k+1 ν̃(z) ) Eeff = E′ 1+kν̃(z) (6.4) For the sake of simplicity, let us restrict the considerations to the laminated solid with the inertial loading as shown in Fig.3. Let us also assume that L1 =L2 =L/2. In this case, the governing equations have the form (E′uz)z =0 if z∈ [ −L 2 ,−H 2 ] ( 1 1+kν̃(z) uz ) z =0 if z∈ [ −H 2 , H 2 ] (E′′uz)z =0 if z∈ [H 2 , L 2 ] (6.5) with boundary conditions u (L 2 , t ) =0 mü ( − L 2 , t ) = p(t)−E′uz ( − L 2 , t ) (6.6) and jump (continuity) conditions u (H 2 +0, t ) =u (H 2 −0, t ) u ( − H 2 +0, t ) =u ( − H 2 −0, t ) uz (H 2 +0, t ) =uz (H 2 −0, t ) uz ( − H 2 +0, t ) =uz ( − H 2 −0, t ) (6.7) On the modelling and optimization... 793 We shall investigate the eigenvalue problem setting p(t)= p0cos ω̆t u(z,t)=w(z)cos ω̆t In the subsequent analysis, it is assumed that p0 = 0. Then equations (6.5)-(6.7) are transformed to the form wzz =0 if z∈ [ −L 2 ,−H 2 ] ( 1 1+kν̃(z) wz ) z =0 if z∈ [ −H 2 , H 2 ] wzz =0 if z∈ [ H 2 , L 2 ] (6.8) with boundary conditions w (L 2 ) =0 − ω̆2mw ( −L 2 ) +E′wz ( −L 2 ) =0 (6.9) and jump conditions w (H 2 +0 ) =w (H 2 −0 ) w ( − H 2 +0 ) =w ( − H 2 −0 ) wz (H 2 +0 ) =wz (H 2 −0 ) wz ( − H 2 +0 ) =wz ( − H 2 −0 ) (6.10) Let us transformequations (6.8)-(6.10) to a dimensionless formby introducing the argument ζ = z L where ζ ∈ [−1/2,1/2]. Let us also denote δ= H L Ω2 = ω̆2Lm E′ Hence, we obtain equations (6.8)-(6.10) in the dimensionless form wζζ =0 if ζ ∈ [ − 1 2 ,− δ 2 ] ( 1 1+kν̃(ζ) wζ ) ζ =0 if ζ ∈ [ −δ 2 , δ 2 ] wζζ =0 if ζ ∈ [δ 2 , 1 2 ] (6.11) with boundary conditions w (1 2 ) =0 −Ω2w ( − 1 2 ) +wζ ( − 1 2 ) =0 (6.12) 794 J. Rychlewska and jump conditions w (δ 2 +0 ) =w (δ 2 −0 ) w ( − δ 2 +0 ) =w ( − δ 2 −0 ) wζ (δ 2 +0 ) =wζ (δ 2 −0) wζ ( −δ 2 +0 ) =wζ ( −δ 2 −0 ) (6.13) We shall solve the optimization problem of finding the position of the graded layer.Tothis end, for theknown k,k> 0,we shall look for λ=minΩ2, λ=λk(δ), and finally we shall find δ0 =maxλk(δ), δ∈ [0,1]. Hence Ω2 = 1 1 2 (1− δ)(k+2)+(k+1)δ The analysis of the above optimization problemwas carried out for k=1, 10, 20. The results are shown in Table 1. The optimization result was obtained for δ=1 (H =L) and k=1 (E′ =2E′′). Table 1. Results of the analysis of the optimization problem k δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 20 31 5 8 20 33 10 17 4 7 5 9 20 37 10 19 20 39 1 2 10 2 13 1 7 2 15 1 8 2 17 1 9 2 19 1 10 2 21 1 11 20 1 12 1 13 1 14 1 15 1 16 1 17 1 18 1 19 1 20 1 21 7. Conclusions Themain results of this contribution are: • An averaged mathematical model for analysis of dynamic behaviour of FGL is formulated. The obtained model equations are represented by a system of finite-difference/differential equations. • It is shown that it is possible to eliminate the unknowns vn, n = 1, . . . ,N, from the governing equations. Then, we arrive at model equations depending on certain effective smoothly varying coefficients. • Thepossible applications of the proposedmodel are illustrated by analy- sis of an optimization problem for a FGL subjected to inertial loadings. On the modelling and optimization... 795 • It can be observed that for periodic laminated structures coefficients in continuum model equations (4.6), (4.7) are constant. In this case, the obtained results coincide with those derived by the asymptotic approxi- mation, Woźniak andWierzbicki (2000). References 1. Jędrysiak J., Rychlewska J., Woźniak C., 2005, Microstructural 2D- models in functionally graded laminated plates, Shell Structures: Theory and Applications, Taylor & Francis Group, 119-125 2. Rychlewska J., 2006, Discrete and continuum modelling in elastodynamics of functionally graded laminates, accepted for publication 3. Rychlewska J., Woźniak C., 2003, On continuum modelling of dynamic problems in composite solids with periodic microstructure, J. Theor. Appl. Mech., 41, 4, 735-753 4. Rychlewska J.,WoźniakC.,WoźniakM., 2006,Modelling of functionally graded laminates revisited,EJPAU, 9, 2 5. Suresh S., Mortensen A., 1998,Fundamentals of Functionally Graded Ma- terials, The University Press, Cambridge 6. WoźniakC., Rychlewska J.,Wierzbicki E., 2005,Modelling and analysis of functionally graded laminated shells, Shell Structures: Theory and Applica- tions, Taylor & Francis Group, 187-191 7. Woźniak C., Wierzbicki E., 2000,Averaging Techniques in Thermomecha- nics of Composite Solids, Wydawnictwo Politechniki Częstochowskiej, Często- chowa, Poland Modelowanie i optymalizacja laminatów o strukturze gradientowej Streszczenie Przedmiot rozważań stanowi szczególna klasa materiałów gradientowych, tzw. laminatów o strukturze gradientowej, które na poziomie mikrostrukturalnym złożo- ne są z dużej liczby bardzo cienkich warstewek. Celem pracy jest zaproponowanie modelu dyskretno-ciągłego i ciągłego zagadnień elastodynamiki takich laminatów. Sformułowanymodel ciągły został zastosowany do analizy drgań ośrodka obciążone- go inercyjnie, złożonego z dwóch jednorodnych warstw, pomiędzy którymi znajduje się strefa przejściowa. Przedyskutowano zagadnienie optymalizacji położenia strefy przejściowej. Manuscript received April 20, 2006; accepted for print October 8, 2006