JOURNAL OF THEORETICAL AND APPLIED MECHANICS 44, 1, pp. 3-18, Warsaw 2006 ON MODELLING OF PERIODIC PLATES WITH INHOMOGENEITY PERIOD OF AN ORDER OF THE PLATE THICKNESS Eugeniusz Baron Department of Building Structures Theory, Silesian University of Technology, Gliwice e-mail: eugeniusz.baron@polsl.pl In this contribution, a new averaged non-asymptotic model of Reissner- type plateswith aperiodic non-homogeneous structure is proposed.This model is obtainedby the toleranceaveragingtechnique (TAT)andmakes it possible to investigate the effect of the period length parameter on the overall plate behaviuor (the length-scale effect). A new element is applyingTATdirectly to the equation of 3D-theory of elasticity of solids with periodic structures. Then, taking into account the Hencky-Bolle kinematicassumption, anon-asymptotic2D-model ofplateswithperiods of an order of the plate thickness is derived. The proposed model is applied to the analysis of some vibration problems. Key words: modelling, composite plates, dynamics 1. Introduction The subject of analysis are medium thickness (Reissner-type) rectangular elastic plates with a periodic non-homogeneous structure in directions parallel to theplatemid-plane.Thegeometry of the aboveplates, apart fromtheglobal mid-plane length dimensions L1, L2 and constant thickness d, is characteri- sed by the length l which determines the period of structure inhomogeneity, l =min(l1, l2). A fragment of the aforementioned plate is shown in Fig.1. Direct applications of Reissner-type plate equations to the analysis of spe- cial problems of periodic plates are rather difficult due to the highly oscil- lating and possibly non-continuous form of the coefficients, cf. Bensousson et al. (1978). Thus, a problem arises here how to formulate an approximate 2D-model of a periodic plate described by an equation with certain averaged 4 E.Baron Fig. 1. An example of a medium thickness periodic plate constant coefficients. This problem can be solved by using the homogenisation theory of PDEs with periodic coefficients, see Caillerie (1984), Kohn and Vo- gelius (1984). Homogenized models of the Reissner-type plates were studied by Lewiński and Telega (2000) and Lewiński (1991, 1992). However, homoge- nized equations cannot describe the effect of the period l on the overall plate behaviour, the so called length-scale effect. The main aim of this paper is to formulate a new non-asymptotic mo- del of medium thickness periodic plates which is free of the aforementioned drawback. To this end, we shall take into account the modelling approach proposed by Woźniak et al. (2004), which is a certain generalization of the tolerance averaging technique presented by Woźniak and Wierzbicki (2000). So far, the tolerance averaging technique has been applied to themodelling of medium thickness plates, cf. Woźniak and Baron (1995), Baron (2002, 2005). In contrast to the results derived in the aforementioned papers, where the period of plate inhomogeneity was assumed to be large when compared to the plate thickness, the obtained model describe the behaviour of Reissner- type prestressed plates with periods of an order of the plate thickness. This model is obtained by the tolerance averaging technique, applied directly to 3D-equations of linear elastodynamics. Using the Hencky-Bolle kinematic as- sumption, we shall derive a non-asymptotic 2D-model of medium thickness periodic plates. In contrast to the homogenized 2D-model, it takes into acco- unt the effect of plate rotational inertia on the dynamic response and enables one to determine higher-order vibration frequencies caused by the plate perio- dic inhomogeneity. Thepresentedgeneral results are illustratedby the analysis of some vibration problems. Throughout the paper, subscripts α,β,. . . (i,j, . . .), run over 1,2 (1,2,3), where superscripts A,B,. . . take the values 1,2, . . . ,N. The summation co- nvention holds related to all aforementioned indices. On modelling of periodic plates... 5 2. Preliminaries Let 0x1x2x3 be an orthogonal Cartesian coordinate system in a physical space E3, and Ω is a region occupied by the solid under consideration in its reference state. Let ∆(x) = ∆+x be a periodic cell of the central of point x∈ E3. By li we denote the period of the solid inhomogeneity in the direction of the xi-axis. It will be assumed that li are sufficiently small when compared to theminimumcharacteristic lengthdimension of Ω. It is possible to consider three special cases of the non-homogeneity, cf. Woźniak et al. (2004). In this paper, considerations will be restricted to the bending of plates with a bi- directional periodic structure. Therefore, for a solid periodic in the x1 and x2-axis directions, we shall introduce the averaging operator 〈f〉(x)= 1 l1l2 ∫ ∆(x) f(y1,y2,x3) dy1dy2 (2.1) x= {xi}∈ Ω0 Ω0 = {x∈E 3, ∆(x)⊂ Ω} for an arbitrary integrable function f defined on Ω. Thebasic concept is that of a slowly varying function of the argument x. It is a function satisfying the following tolerance averaging approximation (TAA) 〈Ef〉(x)'〈f〉(x)F(x) (2.2) whichhas to hold for every integrable function f;where ' is a certain toleran- ce relation, seeWoźniak andWierzbicki (2000). If condition (2.2) holds for all continuous derivatives of F (which exist) thenwe shall write F(·)∈ SV∆(T). By T we denote the set of all tolerance relations in the problem under consi- deration. 3. Modelling technique Let ui(xj, t), xj ∈ Ω, be a displacement field at the time t from the reference configuration of the periodic solid. The solid material is assumed to be elastic and the components Aijkl of the elastic moduli tensor as well the mass densityρ depend on xj and are periodic functions with respect to the x1 and x2 coordinates. Let σ0 = σ0ji be a tensor of the initial stress, bi stands for body forces. From the principle of stationary action for the functional depending on the 6 E.Baron displacement field components, we obtain the following linearized equations of motion for a prestressed solid (Aijkluk,l),j +σ 0 klui,kl−ρüi+ρbi =0 (3.1) Equations (3.1) have highly oscillating (frequently non-continuous) coefficients Aijkl and ρ. In most cases, the prestressing field tensor σ 0 ji is also periodic and non-continuous. Treating Eqs. (3.1) as a starting point, we formulate an approximate mo- del of the solid under consideration, which will be represented by equations with constant coefficients. The proposedmodelling technique is based on two assumptions. To formulate these assumptions, we introduce the following de- composition of displacements ui(x, t)= u 0 i(x, t)+ri(x, t) x∈ Ω0 (3.2) where u0 is an averaged part of the displacement defined by u0i(x, t)= 〈ui〉(x, t)= [〈ρ〉(x3)] −1〈ρui〉(x, t) (3.3) and ri(·, t) is a part of the residual displacement field. The first assumption states that in the macroscopic description of these class of considered problems, the averaged displacement field is slowly-varying for every t u0i(·,x3, t)∈ SV∆(T) On the ground of (2.2), we obtain 〈ρri〉' 0. It follows that ri can be interpre- tedasafluctuationdisplacementfieldcausedbytheperiodicnon-homogeneous structure of the solid. The second assumption states that the fluctuation of the displacement field, representedby ri andcausedbythenon-homogeneousperiodic structure, conforms to this structure. It means that in every cell ∆(x), x ∈ Ω0, these fluctuations can be approximated by periodic functions in the form of finite sums ri(x, t)= h A i (xα)V A(x, t) A =1,2, . . . ,N summation convention holds where V A(·,x3, t) for every t are slowly-varying functions V A(·,x3, t) ∈ ∈ SV∆(T) and h A i (xα) are periodic linear independent functions such that 〈hAi 〉=0. The functions V A(·,x3, t) constitute new kinematical variables called fluc- tuation amplitudes, and hAi (·) are assumed to be known a priori and are On modelling of periodic plates... 7 referred to as mode-shape functions. In general, hAi (·) represent free periodic vibrations of the 3D-periodic cell and can be treated as eigenvectors related to a certain eigenvalue problem. An alternative specification of the mode-shape functions based onmass discretization of the periodic cell is also possible. In order to derive the governing equations for fields u0i , V A, we shall intro- duce a displacement field ui in the formgiven below into the action functional ui(x, t)= u 0 i(x, t)+h A i (xα)V A(x, t) (3.4) Taking into account that u0i(·,x3, t)∈ SV∆(T), V A(·,x3, t)∈ SV∆(T) we shall use in calculations the tolerance averaging approximation given by (2.2). We will also restrict considerations to the problem in which Aijkl(·) and ρ(·) are even, and hAi (·) are odd functions. In such a case, applying the principle of stationary action, for bi = const, we obtain the following system of equations for u0i and V A 〈ρ〉ü0i − (〈Aijkl〉u 0 k,l+ 〈Aijkαh A k,α〉V A),j −〈σ 0 kl〉u 0 i,kl−〈ρ〉bi =0 (3.5) 〈ρhAi h B i 〉V̈ B − (〈Ai3k3h A i h B k 〉V B ,3 ),3+ 〈Aiαkβh A i,αh B k,β〉V A+ +〈Aijkαh A k,α〉u 0 i,j + 〈σ 0 αβh A k,αh B k,β〉V B =0 Equations (3.5) have constant coefficients and hence represent a certain ma- croscopic model of a prestressed periodic solid. The solutions u0i(·, t), V A(·, t) have physical sense only if u0i(·, t) ∈ SV∆(T)and V A(·, t) ∈ SV∆(T) for eve- ry t. These equations cannot be used in analysis of boundary-value problems. The boundary conditions for V A may not be derived as approximations of boundary conditions for the displacement ui = u 0 i +h A i V A. 4. Applications to medium thickness plates Let Π = (0,L1)× (0,L2) be a rectangle with the dimensions L1 and L2 on the plane 0x1x2. Assume that equations (3.5) hold in a region Ω = Π × (−d/2,d/2) occupied by a Reissner-type un-deformed plate with a constant thickness d. Let us also assume that the plate has a plane periodic structure, and hence ∆ = (−l1/2, l−1/2)× (−l2/2, l2/2) is a 2D-periodicity cell on the 0x1x2-plane. Moreover, let the plate be homogeneous in the di- rection of the x3-axis and bemade of periodically distributedmaterials along themid-plane. The dimensions lα are of an order of the plate thickness d and sufficiently small with respect to Lα, d � Lα. 8 E.Baron Setting x=(x1,x2), z =x3, we shall use denotation ∆(x)= ∆+x Π0 = {x∈ Π : ∆(x)⊂ Π} Ω0 = {(x,z)∈ Ω : ∆(x,z)⊂ Ω} Instead of operator (2.1), we introduce the following two kinds of averaging of an arbitrary integrable function f(·) 〈f〉(x,z)= 1 l1l2 ∫ ∆(x) f(y1,y2,z) dy1dy2 x ∈ Π0 − d 2 ¬ z ¬ d 2 (4.1) 〈f〉(x)= 1 d d/2 ∫ −d/2 〈f〉(x,z) dz For ∆-periodic function f, 〈f〉 is constant. Assuming that the planes z = const are elastic symmetry planes, we define Cαβγ3 = Aαβγ3−Aαβ33A33γδ(A3333) −1 Bαβ = KAα3β3 where K is the shear coefficient of the medium-thickness plate theory. We introduce the Hencky-Bolle kinematics assumption in the known form uα(x,z,t) = zϑα(x, t) u3(x,z,t)= w(x, t) (4.2) where w(·, t) are displacements of points of themid-plane Π, whereas ϑα(·, t) are independent rotations. Taking into account the modelling assumptions, outlined in the previous Section, there exist decompositions of ϑα and w into slowly varying avera- ged parts ϑ0α, w 0 and residual displacements approximated by finite sums hAi (x)V A(xj, t). Assuming that h A 3 (·) = 0 and V A = zψA(x, t), we obtain uα(x,z,t) = zϑ 0 α(x, t)+zh A α(x)ψ A(x, t) (4.3) u3(x,z,t) = w 0 α(x, t) where ϑ0α, w 0, ψA are basic unknowns which are slowly varying for every time t. On modelling of periodic plates... 9 Substituting the right-hand sides of (4.3) into the action functional, we obtain from(3.5) the following systemof equations for themid-planedeflection w0(x, t), rotations vt0α(x, t) and 2D-fluctuation amplitudes ψ A(x, t) j〈ρ〉ϑ̈0α− j(〈Cαβγδ〉ϑ 0 γ,δ),β + 〈Bαβ〉(ϑ 0 α+w 0 ,α)− j(〈Cαβγδh A γ,δ〉ψ A),β + −j(〈σ0γβ〉ϑ 0 α,γ),β + 〈zσ 0 γ3〉ϑ 0 α,γ =0 〈ρ〉ẅ0− [〈Bαβ〉(ϑ 0 α+w 0 ,α)],β − (〈σ 0 γβ〉w 0 ,γ),β =0 (4.4) j〈ρhAαh B α〉ψ̈ B +(j〈Cαβγδh B γ,δ〉+ 〈Bαβh A αh B β 〉)ψ B + j〈Cαβγδh A γ,δ〉ϑ 0 α,β + +j〈σ0αβh A γ,αh B γ,β〉ψ B =0 where j = d2/12 and, for the sake of simplicity, we have neglected the body forces. The characteristic feature of the derived system of equations (4.4) is that the fluctuation amplitudes ψA are governed by the system of ordinary diffe- rential equations involving only time derivatives of ψA. Hence, these variables do not enter into the boundary conditions and play the role of certain internal variables. Let us observe that the underlined coefficients 〈ρhAαh B α〉, 〈Bαβh A αh B β 〉 are values of an order of the period length. Thus, equations (4.4) describe the effect of the period length on the overall behaviour of the plate. This inho- mogeneity period is of an order of the plate thickness. Neglecting the terms involving the period length, we can eliminate fluctuation variables ψA from (4.4) and hence obtain a system of equations for ϑ0α and w 0 as the basic unk- nowns. It can be shown that this system represents a certain approximation of the homogenized 2D-model of the periodic plate under consideration. For a homogeneous plate ρ, Cαβγδ, Bαβ are constant and hence 〈Cαβγδh A γ,δ〉=0. In this case, equation (4.4)3 yields ψ A = 0 provided that σ0ij = 0, the ini- tial conditions for ψA are homogeneous and (4.4) takes the form of known Hencky-Bolle plate equations. Equations (4.4) represent the non-asymptotic averaged 2D-model of the Reissner-type prestressed plates with a plane periodic structure. In contrast to the 2D-models of plates obtained from the equations of the plate theory by Baron and Woźniak (1995), Baron (2000, 2002, 2005), the above model was derived from the macroscopic 3D-model of a periodic solid, and hence can be applied to problems inwhich period lengths are of the same order as the plate thickness. 10 E.Baron 5. Dynamic behaviour of medium-thickness plates The aforementioned results will be now applied to the analysis of free bending vibrations of a plate with a periodic non-homogeneous structure on- ly in the direction of the x2-axis. The plate under consideration is simply supported on the edges x2 = 0, x2 = L and subjected to the initial stress N = ∫d/2 −d/2 σ022 dz, see Fig.2. The plate is made of two linear elastic, isotropic andhomogeneousmaterials. Itwill be assumed that all unknown functions de- pend on time and variable x2, exclusively. It is a plate with a one-directional periodic structure which can be treated as a special case of plates with bi- directional periodic structures. Fig. 2. A plate with a one-directional periodic structure Considering the isotropy (in this special case also orthotropy) of the plate, it is denoted C = C2222, D = B22 = K2A2323. In the first approximation, we can introduce only one vector of shape func- tions h1 = (0,h(x2)), where h(·) is a saw-like l-periodic function shown in Fig.3. Thus, in this example, we shall deal with only one fluctuation amplitude ψ1(x2, t)=ψ(x2, t), and formulae (4.2) in the form u1(x2,z,t)= 0 u2(x2,z,t)= zϑ 0 α(x2, t)+zh(x2)ψ(x2, t) (5.1) u3(x2,z,t)= w 0(x2, t) Under these conditions, equations (4.4) reduce to the system of three equ- ations for the averaged plate deflection w0(x2, t), rotation ϑ = ϑ 0 2(x2, t) and On modelling of periodic plates... 11 Fig. 3. Diagrams of the function h(·) fluctuation amplitude ψ(x2, t) j〈ρ〉ϑ̈0−〈C〉ϑ0,22− j〈Ch,2〉ψ,2+ 〈D〉(ϑ 0+w,2)− jNϑ 0 ,22 =0 〈ρ〉ẅ0−〈D〉(ϑ0+w,2),2−Nw 0 ,22 =0 (5.2) j〈ρh2〉ψ̈+(j〈Ch2,2〉+ 〈Dh 2〉)ψ + j〈Ch,2〉ϑ 0 ,2+ jN〈h 2 ,2〉ψ =0 where N = const, j = d2/12. It can be seen that the coefficients 〈D h2〉 = l2〈D〉/12, 〈ρh2〉 = l2〈ρ〉/12 depend explicitly on the period length l and describe the length-scale effect. Assuming the unknown functions in the form w0(x2, t)= e iωtw(x2) ϑ 0(x2, t)= e iωtϑ(x2) ψ(x2, t)= e iωtψ(x2) where ω is a vibration frequency, we obtain from (5.2)3 ψ =− 〈Ch,2〉 Rω ϑ,2 (5.3) Rω = 〈Ch 2 ,2〉+ j −1〈Dh2〉+ 〈h2,2〉N −〈ρh 2〉ω2 Substituting (5.3) into (5.2)1, taking into account the previous assumption, we can look for a solution to (5.2) in the well known form w(x2)= wn sinknx2 ϑ(x2)= ϑncosknx2 12 E.Baron where kn = nπ/L, n =1,2, . . ., and wn, ϑn are arbitrary constants, summa- tion convention holds. In this case, we arrive at a system of linear algebraic equations for wn, ϑn [ k2nj ( 〈C〉− 〈Ch,2〉 2 Rω +N ) + 〈D〉−〈ρ〉ω2 knj〈D〉 knj〈D〉 k 2 n(〈D〉+N)−〈ρ〉ω 2 ][ ϑn wn ] = [ 0 0 ] (5.4) Equations (5.4) have a nontrivial solution provided that their determinant is equal to zero. In thisway, bearing inmind that h, lh,2 and d are of an order of the period l and l � L, we obtain the following approximate formulae for the first three free vibration frequencies ω21 = k2nN 〈ρ〉 + k4nH 〈ρ〉 +O(ε6) ω22 = 〈D〉 j〈ρ〉 + 1 〈ρ〉 [k2nH0+k 2 nj(N + 〈D〉)]+O(ε 4) (5.5) ω23 = 〈Ch2,2〉+N〈h 2 ,2〉+ j −1〈Dh2〉 〈ρh2〉 + k2n 〈ρ〉 〈Ch,2〉 2 〈Ch2,2〉+N〈h 2 ,2〉 +O(ε4) where H = j ( 〈C〉− 〈Ch,2〉 2 〈Ch2,2〉+N〈h 2 ,2〉+ j −1〈Dh2〉 ) H0 = j ( 〈C〉− 〈Ch,2〉 2 〈Ch2,2〉+N〈h 2 ,2〉 ) Formedium thickness plates, relations (5.5) have a physicalmeaning provided that ε2 = k2nj � 1. Now let us discuss an asymptotic approximation of Eqs. (5.2). By formal transition l → 0, Eqs. (5.2)3 lead to an algebraic equation for ψ 〈Ch2,2〉ψ =−〈Ch,2〉ϑ 0 ,2−N〈h 2 ,2〉 and to a system of equations for themid-plane deflection w0 and rotation ϑ0 j〈ρ〉ϑ̈0−H0ϑ 0 ,22+ 〈D〉(ϑ 0+w0,2)− jNϑ 0 ,22 =0 (5.6) 〈ρ〉ẅ−〈D〉(ϑ0+w0,2),2−Nw,22 =0 On modelling of periodic plates... 13 Substituting w0 = eiωtwn sin(knx2), ϑ 0 = eiωtϑncos(knx2) into (5.6), we arrive at the single frequency equation j〈ρ〉ω4−〈ρ〉[(k2nH0)+ 〈D〉+k 2 nj(2N + 〈D〉)]ω 2+ (5.7) +k2nN(k 2 nH0+ 〈D〉)+k 2 n(k 2 nH0+k 2 njN) = 0 Taking into account that k2nj � 1 and using the approximation √ 1+k2nj ≈ 1+ 1 2 k2nj we obtain from (5.7) the following formulae for the free vibration frequencies ω21 = k2nN 〈ρ〉 + k4n 〈ρ〉 H0〈D〉 〈D〉(1+k2nj)+k 2 nH0 (5.8) ω22 = 〈D〉 j〈ρ〉 + 1 j〈ρ〉 [k2nH0+k 2 nj(N + 〈D〉)]+ k4n 〈ρ〉 H0〈D〉 〈D〉(1+k2nj)+k 2 nH0 One should remember that d is of an order of l, then it is possible to neglect the terms involving j in formulae (5.6) and (5.7). This assumption is equivalent to the neglecting of the rotational inertia in themodel described by (5.6). In that case, considering that k2nH0/〈D〉� 1, we obtain an asymptotic model equation ω20 = k2n(N +k 2 nH0) 〈ρ〉 (5.9) In the course of numerical calculations, the analysis of interrelations be- tween the non-dimensional lower free vibration frequency and geometrical pa- rameter κ = l/d is carried out. The obtained results are comparedwith those corresponding to the asymptotic model. Let the orthotropic constituents of the plate havemass densities ρ′, ρ′′ and elastic moduli C′, C′′ and D′, D′′, Fig.3. In this case, by denoting x = l′/l, x ∈ (0,1), the averaging operator reduces to the form 〈f〉= xf ′+(1−x)f ′′ and 〈fh2〉= l2 12 〈f〉 〈fh,2〉= f ′−f ′′ 〈fh2,2〉= f ′ x + f ′′ 1−x For simplicity, we assume that N = 0, ρ′ = ρ′′ = ρ. Next we introduce parameters η = C′′/C′, ζ = D′′/D′ and ν = D′/C′.Multiplyingbothrelations 14 E.Baron (5.5)1 and (5.9) by ρ(C ′k4nj) −1 and taking into account the above denotations and assumptions, we arrive at the following formulae for the non-dimensional free vibration frequencies Ω0 = η (1−x)+xη Ω1 = R[x− (1−x)η]− (1−η)2 R (5.10) R = 1 x + 1 1−x +κ2ν[x+(1−x)ζ] Calculations were performed for three values of the parameters η = 2; 10; 20 and κ = 0.5; 1.0; 2.0. We found that ν = 0.3 and ζ = η. Diagrams representing the interrelation between non-dimensional frequencies Ω and the size of the periodicity cell (given by x and κ) as well the parameter η are shown in Fig.4. Commenting the obtained results it should be stated that, with assump- tions made regardless of the material η, and geometrical κ parameters, the asymptoticmodel gives the lowest values of the vibration frequency.The influ- ence of κ on the frequency values rises with the growth of η. For the given η, the highest frequency is obtained when the period l is of an order the pla- te thickness. The differences in the vibration frequency depending on κ are the highest when the material of greater material parameters fills up the cell periodicity by about 2/3 of its volume. The calculation assumptions are fulfilled by glued timber plates which are composed of elements cut along and across the fibres, see Fig.5. Mechanical properties of timber can be treated in different ways. According to PN-B- 03150-2000, timber is a quasi-isotropic material with elastic moduli: C′ = E90 =430MPa C ′′ =E =13000MPa D′ =D′′ = G =810MPa (for timber GL-35) According to Neuhaus (2004), timber can also be treated as an anisotro- pic material for which, after certain calculations, we obtain C ′ = 428MPa, C′′ =12290MPa, D′ =558MPa, D′′ =37MPa. In Fig.6, diagrams of relation between the vibration frequency and the parameter κ for the both mentioned cases are presented. It is clearly seen that no significant differences in the values of vibration frequencies can be observed both for quasi-isotropic and anisotropic timber if the period l is of an order of the plate thickness or lower. The conclusions obtained beforehand have also been confirmed as far as the comparison to the asymptotic model is concerned. On modelling of periodic plates... 15 Fig. 4. Diagrams of free vibration frequencies versus differentmaterial characteristics Fig. 5. An example of glued timber plate 16 E.Baron Fig. 6. Interrelation between the frequency andmechanical properties for timber 6. Summary of new results The following new results and remarks on composite periodic plates have been derived in this paper: • The obtained 2D-model of periodic Reissner-type plates, makes it po- ssible to investigate dynamic (also and stability) problems, in which the constant plate thickness d is of an order of the period length l. • In contrast to the homogenized model, the model obtained in this con- tribution can also be used to determination of higher free vibration fre- quencies caused by the plate periodic structure. • Theproposed 2D-model is a certain complementation for themodel pre- sented by Baron and Woźniak (1995) in which the period lengths were assumed to be much larger than the plate thickness. • The analysis confirms thesis that if the period lengths are small when compared to the plate thickness then the length-scale effect is reduced; in this case the homogenisation approach is used. On modelling of periodic plates... 17 • The analysis of the free vibration problem of a simply supported plate- band leads to the conclusion that the asymptoticmodel gives the lowest values of the basic free vibration frequency. • The calculations for glued laminated timber plates prove that no signi- ficant differences in the values of vibration frequencies can be observed, both for quasi-isotropic and anisotropic timber, if the period l is equal to the plate thickness or smaller. References 1. Baron E., 2002, Onmodelling of medium thickness plates with a uniperiodic structure, Journal of Theoretical and Applied Mechanics, 40, 1, 7-22 2. Baron E., 2005, On a certain model of uniperiodic medium thickness plates subjected to initial stresses, Journal of Theoretical and Applied Mechanics, 43, 1, 93110 3. BaronE.,WoźniakC., 1995,Onmicrodynamicsof compositeplates,Archive of Applied Mechanics, 66, 126-133 4. Bensoussan A., Lions J.L., Papanicolau G., 1978, Asymptotic Analysis for Periodic Structures, Amsterdam, North Holland 5. CaillerieD., 1984,Thin elastic andperiodic plates,Math. Meth. in the Appl. Sci. 6, 159191 6. KohnR.,VogeliusM., 1984,Anewmodel of thinplateswith rapidlyvarying thickness, Int. J. Solids Structures, 20, 333-350 7. Lewiński T., 1991, Effective models of composite periodic plates: I. Asymp- totic solutions, II. Simplifications due to symmetries, III. Two dimensional ap- proaches, Int. J. Solids Structures, 27, 1155-1172, 1173-1184, 1185-1203 8. Lewiński T., 1992, Homogenizing stiffness of plates with periodic structure, Int. J. Solids Structures, 21, 309-326 9. Lewiński T., Telega J.J., 2000, Plates, Laminates and Shells, Singapore, World Scientific Publishing Company 10. Neuhaus H., 2004,Budownictwo drewniane, PWT, Rzeszów 11. Woźniak C., Wierzbicki E., 2000,Averaging Techniques in Thermomecha- nics of Composite Solids, Wydawnictwo Politechniki Częstochowskiej 12. WoźniakM.,Wierzbicki E.,WoźniakC., 2004,Macroscopicmodelling of prestressedmicroperiodic media,Acta Mechanica, 173, 107-117 18 E.Baron Modelowanie periodycznie niejednorodnych płyt o okresie periodyczności rzędu ich grubości Streszczenie W pracy zaproponowano nowy uśrednionymodel płyt typu Reissnera o struktu- rze periodycznie niejednorodnej. Jest to model nieasymptotyczny, otrzymany tech- niką uśredniania tolerancyjnego (tolerance averaging technique, TAT), pozwalający uwzględnić wpływ okresu powtarzalności l na makro-mechaniczne (w sensie mecha- niki kompozytów) własności płyty. Wpływ ten nazywamy efektem skali. Dotychczas metodaminieasymptotycznymimodelowanoperiodycznepłyty średniej grubości speł- niające założenie, że okres l jest dużo większy odmaksymalnej grubości płyty. TAT stosowanowtedy do uśrednionych na grubości, równań 2Dmodelu płyty. Elementem oryginalnym jest zastosowanie TAT bezpośrednio do równań trójwy- miarowej liniowej teorii sprężystości ośrodka o strukturze periodycznej w kierunkach równoległychdopewnej płaszczyzny środkowej.Uwzględniającw tych równaniachhi- potezę kinematyczną Henckey-Bolle’a otrzymano równania 2D-modelu średniej gru- bości płyt o płaskiej strukturze periodycznej i okresie l rzędu grubości płyty. Jak dotądmodelowanow ten sposób tylko periodyczne płyty spełniające założeniaKich- hoffa. Dla przypadku szczególnego, swobodnie podpartego pasma płytowegowyznaczo- no częstości drgań własnych w zależności od parametrów geometrycznych oraz ma- teriałowych i porównano je z częstościami uzyskanymi w ramach modelu asympto- tycznego. Manuscript received July 11, 2005; accepted for print September 30, 2005