Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 52, 2, pp. 485-498, Warsaw 2014 MODELLING OF ANNULAR PLATES STABILITY WITH FUNCTIONALLY GRADED STRUCTURE INTERACTING WITH ELASTIC HETEROGENEOUS SUBSOIL Wojciech Perliński, Michał Gajdzicki, Bohdan Michalak Lodz University of Technology, Department of Structural Mechanics, Łódź, Poland e-mail: wojciech.perlinski@p.lodz.pl; michal.gajdzicki@p.lodz.pl; bohdan.michalak@p.lodz.pl This contribution deals with the modelling and analysis of stability problems for thin com- posite annular plates interacting with elastic heterogeneous subsoil. The object of analysis is an annular platewith a deterministic heterogeneousmicrostructure and the apparentpro- perties smoothly varying along a radial direction. The aim of contribution is to formulate twomacroscopicmathematical models describing stability of this plate. The considerations are based on a tolerance averaging technique. The general results are applied to the analysis of some special stability problems.The obtained results of critical forceswith those obtained from finite elementmethod are compared. Keywords: functionally gradedmaterials, stability, annular thinplates, heterogeneous subsoil 1. Introduction The object of this contribution is a two-phased composite plate interacting with elastic micro- heterogeneous subsoil with two moduli (Fig. 1). The assumed model of elastic foundation is a generalization of the well-known Winkler model. The introduction of an additional modulus of horizontal deformability of the foundation makes it possible to describe the stability of the plate resting on a sufficiently fine net of elastic point supports such as piles or columns. The annular plates under consideration have a space varyingmicrostructure and hence are described by partial differential equations with highly oscillating, non-continuous coefficients, which are not a good tool for application to engineering problems. Hence, various simplified models are proposed, replacing these plates by plates with effective properties described by smooth, slowly varying functions. The plates under consideration aremade of an isotropic homogeneousmatrix and isotropic homogeneous ribs which are situated along the radial direction. Fig. 1. Fragment of the midplane of a plate with longitudinally gradedmicrostructure: (a) microscopic level, (b) macroscopic level 486 W. Perliński et al. Theplate and foundationhave λ-periodic structurealong theangular axis andslowgradation of effective properties in the radial direction. The period λ of inhomogeneity is assumed to be very small when compared to the characteristic length dimension of the plate along the angular axis. The apparent properties of the plate and foundation are constant in the angular direction and slowly graded in space in the radial direction. Hence we deal here with a special case of a functionally gradedmaterial (FGM) and functionally graded foundation properties. Functionally gradedmaterials are a class of composite materials where composition of each material constituent determines continuously varying effective properties of the composite.Many papers have been dedicated to analyse the behaviour of functionally graded (FGM) plates. The analysis of functionally graded plates subjected to in-plane compressive loading can be found in several papers. Javaheri and Eslani (2002) analyzed stability of rectangular FGM plates simply supported on all edges. It is assumed that Young’s modulus varies along the thickness direction. In the paper of Tylikowski (2005) analysis of dynamic stability of FGM rectangular plates subjected to in-plane time dependent forces is presented. Material properties are graded in the thickness direction according to the volume fraction power law distribution. You et al. (2009) developed an analytical solution to determine deformations and stresses in circular disks made of an FGM subjected to internal and/or external pressure. The governing equations are derived frombasic equations of axisymmetric, plane stress problem in elasticity. Themechanical properties of materials are functions of the radial coordinate. In the paper by Tung and Duc (2010) explicit expressions of postbuckling load-deflections curves by the Galerkin method are obtained. Material properties in simply supported rectangular plates are assumed to be graded along the thickness direction according the power law distribution of constituents. However, analyses of FGM plates resting on an elastic foundation are quite limited. In the paper by Benyoucef et al. (2010), the thick rectangular FGMplate withmaterial properties graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of constituents is analyzed. The plates are resting on a homogeneous elastic foundation. The foundation ismodelledasa two-parameterPasternakorone-parameterWinkler-type foundation. In the paper by Naderi and Saidi (2011), the exact solution of the buckling problem for FGM sector plates resting on a homogeneous elastic foundation with one modulus is presented. It is assumed that themodulus of elasticity E in the thickness direction varies according to a power law function. The majority of the above mentioned papers deal with plates where it is assumed that the material properties vary along the plate thickness direction. In contrast to these papers, in the present contribution, we deal with effective properties of the plate material and foundation varying in the midplane of the plates. Thedirect description of the plate under consideration leads to equationswith highly oscilla- ting andnon-continuous coefficients. Hence, the aimof this contribution is to formulate averaged models described by equations with functional but smooth and slowly varying coefficients. Here we can mention these models which are based on the asymptotic homogenization technique for equations with non-uniformly oscillating coefficients, cf. Jikov et al. (1994). However, because the formulation of the averagedmodel byusing the asymptotic homogenization is rather compli- cated from the computational point of view, these asymptoticmethods are restricted to the first approximation. Hence, the averaged model obtained by using this method neglects the effect of the microstructure size on the overall response of the FGM-plate. The formulation of the ave- raged mathematical model for the analysis of stability of the plates under consideration will be based on the tolerance averaging technique. The general modelling procedures of this technique are given byWoźniak et al. (2008, 2010). One should also mention a few papers, where various special problems of microstructured media are presented; e.g. Matysiak (1995), Nagórko and Wągrowska (2002), Wierzbicki (1995). The applications of the tolerance averaging technique for the modelling of stability of various periodic composites were presented in a series papers, Modelling of annular plates stability with functionally graded structure... 487 e.g. Baron (2003), Michalak (1998), Tomczyk (2005), Wierzbicki et al. (1997). The approach, based on the tolerance averaging technique, to formulate macroscopic mathematical models for functionally graded stratified media was proposed by Michalak et al. (2007), Ostrowski and Michalak (2011) for the heat conductions problem, and by Jędrysiak and Michalak (2011) for the stability of thin plates. In the paper by Michalak (2012), shells with functionally graded effective properties are analysed.Michalak andWirowski (2012) analysed dynamic behaviour of thin annular FGMplates with gradation of thematerial properties along the specified direction. Throughout the paper indices i,j,k, . . . run over 1,2,3, indices α,β,γ, . . . run over 1,2. We also introduce non-tensorial indices A,B,C,. . . which run over the sequence 1, . . . ,N. The summation convention holds for all aforementioned sub- and superscripts. 2. Preliminaries Theobject of our considerationsare annular functionally gradedplateswithmicrostructuregiven in Fig. 1 resting on a microheterogeneous foundation. Let us introduce the orthogonal curvili- near coordinate system Oξ1ξ2ξ3 in the physical space occupied by a plate under consideration. Setting x ≡ (ξ1,ξ2) and z = ξ3, it is assumed that the undeformed plate occupies the region Ω ≡{(x,z) : −H/2¬ z ¬H/2, x∈Π}, where Π is the plate midplane and H is the plate thickness.We denote by gαβ ametric tensors and by ǫαβ aRicci tensor. Here and in the sequel, a vertical line before the subscripts stands for the covariant derivative and ∂α = ∂/∂ξ α, ξ1 =ϕ, ξ2 = ρ. The plate rests on the generalized Winkler foundation whose properties are characteri- zed by vertical kz and horizontal kt foundation moduli. The foundation reaction according to Gomuliński (1967) has three components acting in the direction of the coordinates (z,ρ,ϕ) Rz = kzw Rρ = kt H 2 ∂ρw Rϕ = kt H 2 1 ρ ∂ϕw (2.1) Themodel equations for the stability of the considered plate will be obtained in the framework of thewell-known second order non-linear theory for thin plates resting on the elastic foundation (Woźniak, 2001).Denoting thedisplacementfield of theplatemidsurfaceby w(x, t), the external forces by p(x, t) and by µ the mass density related to this midsurface, we obtain: (i) strain-displacement and constitutive equations καβ =−w|αβ m αβ =−Dαβγµκγµ (2.2) where: Dαβγµ =0.5D(gαµgβγ +gαγgβµ+ν(ǫαγǫβµ+ ǫαµǫβγ)), D=Eh3/12(1−ν2). (ii) the strain energy averaged over the plate thickness E(ξλ)= 1 2 Dαβγδw|αβw|γδ + 1 2 nαβw|αw|β + 1 2 kz(w) 2+ 1 2 h2 4 ktδ αβ∂αw∂βw (2.3) (iii) kinetic energy K(ξα)= 1 2 µẇẇ (2.4) The governing equations of the plate under consideration can be described by the well-known principle of stationary action. We introduce the action functional defined by A(w(·)) = ∫ Π t1∫ t0 [L(y,w|αβ(y, t),w|α(y, t), ẇ(y, t),w(y, t))+pw] dt dy (2.5) 488 W. Perliński et al. with the Lagrangian defined by L(·,w|αβ, ẇ,∂αw,w) =K(·, ẇ)−E(·,w|αβ,∂αw,w) = 1 2 ( µẇẇ−nαβw|αw|β −D αβγδw|αβw|γδ −kzww− h2 4 ktδ αβw|αw|β ) (2.6) where nαβ are in-plane forces, and the Kronecker-deltas δαβ will be treated as a tensor; δ11 =1/ρ2, δ22 =1. The principle stationary action applied to the functional Awith the Lagrangian L, defined by Eq. (2.6), leads to the Euler-Lagrange equation ∂ ∂t ∂L ∂ẇ − ∂L ∂w + ( ∂L ∂w|α ) |α − ( ∂L ∂w|αβ ) |αβ = p (2.7) and the equilibrium equations (Dαβγδw|γδ)|αβ − (n αβ |β )|α− h2 4 (ktδ αβw|β)|α+kzw+µẅ= p (2.8) This direct description leads to plate equations with discontinuous and highly oscillating coeffi- cients. These equations are too complicated to be used in the engineering analysis and will be used as the starting point in the tolerance modelling procedure. 3. Averaged models Let us introduce the polar coordinates system Oξ1ξ2, 0 ¬ ξ1 ¬ ϕ, R1 ¬ ξ 2 ¬ R2 so that the undeformed midplane of the annular plate occupies the region Π ≡ [0,ϕ]× [R1,R2]. Let λ, 0 < λ ≪ ϕ, be the known microstructure parameter. Denote Π∆ as a subset of Π of po- ints with coordinates determined by conditions (ξ1,ξ2) ∈ (λ/2,ϕ− λ/2)× (R1,R2). An ar- bitrary cell with a center at the point with coordinates (ξ1ξ2) in Π∆ will be determined by ∆(ξ1,ξ2) = (ξ1 −λ/2,ξ1 +λ/2)×{ξ2}. At the same time, the thickness h of the plate under consideration is supposed to be constant and small compared to themicrostructureparameter λ. In order to derive averaged model equations, we applied the tolerance averaging approach. This technique based on the concept of tolerance and indiscrenibility relations. The general modelling procedures of this technique and basic concepts of this technique, as a tolerance parameter, a tolerance periodic function, a slowly varying function, a highly oscillating function are given byWoźniak et al. (2008, 2010). Wemention here only the averaging operator. For an arbitrary integrable function f(·), the averaging operator over the cell ∆(·) is defined by 〈f〉(ξ1,ξ2)= 1 λ ξ1+λ/2∫ ξ1−λ/2 f(η,ξ2) dη (3.1) for every ξ1 ∈ [λ/2,ϕ−λ/2], ξ2 ∈ [R1,R2]. 3.1. Tolerance model The tolerance averaging technique will be applied to equations (2.1)-(2.7) in order to derive averagedmodel equations. The first assumption in this technique ismicro-macro decomposition of the displacement field Modelling of annular plates stability with functionally graded structure... 489 w(ξ1,ξ2, t)=w0(ξ1,ξ2, t)+hA(ξ1)VA(ξ 1,ξ2, t) A=1, . . . ,N (3.2) for ξα ∈Π and t∈ (t0, t1). The modelling assumption states that the functions w0(·,ξ2, t) ∈ SV 2δ (Ω,∆), VA(·,ξ 2, t) ∈ SV 2δ (Ω,∆) are slowly varying functions together with all partial derivatives. The functions w0(·,ξ2, t), VA(·,ξ 2, t) are the basic unknowns of the modelling problem. The functions hA(·) are known, dependent on themicrostructure length parameter λ, fluctuation shape functions. Let h̃A(·), ∂1h̃ A(·) stand for the periodic approximation of hA(·), ∂1h A(·) in the cell ∆, respectively. Due to the fact that w(·,ξ2, t) are tolerance periodic functions, it can be observed that the periodic approximation of wh(·,ξ 2, t) and ∂αwh(·,ξ 2, t) in ∆(·) have the form wh(y,ξ 2, t)=w0(ξα, t)+hA(y)VA(ξ α, t) ∂αwh(y,ξ 2, t)= ∂αw 0(ξα, t)+∂1h A(y)VA(ξ α, t)+hA(y)∂2VA(ξ α, t) ẇh(y,ξ 2, t)= ẇ0(ξα, t)+hA(y)V̇A(ξ α, t) (3.3) for every ξα ∈Π, almost every y∈∆(ξα) and every t∈ (t0, t1). The tolerance model equations will be obtained by the averaging of the functional A, Eq. (2.5). Substituting decomposition (3.2) of the displacement field into the Lagrangian L(ξα,w,w|αβ,w|α, ẇ), described by equation (2.6), and using the tolerance averaging techni- que, we obtain Ah(w 0,VA)= t1∫ t0 ∫ Π [〈L〉+ 〈p(·)〉w0(·)+ 〈p(·)hA(·)〉VA(·)] dξ α dt (3.4) where averaged Lagrangian (2.6) has the form 〈L〉= 1 2 〈µ〉ẇ0ẇ0+ 〈µhA〉ẇ0V̇A+ 1 2 〈µhAϕB〉V̇AV̇B + 〈p〉w 0+ 〈phA〉VA − 1 2 〈Dαβγµ〉w0|αβw 0 |γµ−〈D 11γµhA|11〉w 0 |γµVA−〈D 22γµhA〉VA|22w 0 |γµ −2〈D12γµhA|1〉w 0 |γµVA|2−〈D 1122hA|11h B〉VAVB|22− 1 2 〈D1111hA|11h B |11〉VAVB −2〈D1212hA|1h B |1〉VA|2VB|2− 1 2 〈D2222hAhB〉VA|22VB|22− 1 2 〈kz〉w 0w0 −〈kzh A〉w0VA− 1 2 〈kzh AhB〉VBVA− H2 8 〈ktg AgB〉δ22VA|2VB|2 − H2 4 〈kth A |1〉δ 1βw0|βVA− H2 4 〈kth A〉δ2βw0|βVA|2+ H2 4 〈kth A |1h B |1〉δ 11VAVB − H2 8 〈kt〉δ αβw0|αw 0 |β − 1 2 〈nαβ〉w0|αw 0 |β −〈n 1βhA|1〉w 0 |βVA−〈n 2βhA〉VA|2w 0 |β − 1 2 〈n11hA|1h B |1〉VAVB −〈n 12hA|1h B |1〉VAVB|2− 1 2 〈n22hAhB〉VA|2VB|2 (3.5) Applying the principle of stationary action to the averaged functional Ah, the Euler-Lagrange equations take the form ∂ ∂t ∂〈L〉 ∂ẇ0 − ( ∂〈L〉 ∂w0 |αβ ) |αβ + (∂〈L〉 ∂w0 |α ) |α − ∂〈L〉 ∂w0 = 〈p〉 ∂ ∂t ∂〈L〉 ∂V̇A − ( ∂〈L〉 ∂VA|22 ) |22 + ( ∂〈L〉 ∂VA|2 ) |2 − ∂〈L〉 ∂VA = 〈phA〉 (3.6) 490 W. Perliński et al. UsingaveragedLagrangian (3.5),we obtain the following systemof equations describing stability of the plate resting on the microheterogeneous foundation (〈Dαβγµ〉w0|γµ)|αβ +(〈D αβ11hA|11〉VA)|αβ +(〈D αβ22hA〉VA|22)|αβ + 〈kz〉w 0+ 〈kzh A〉VA− H2 4 (〈kt〉δ αβw0|β)|α− H2 4 (〈kth A |1〉δ 1βVA)|β − H2 4 (〈kth A〉δ2βVA|2)|β − (N αβw0|β)|α+ 〈µ〉ẅ 0 = 〈p〉 〈D11γµhA|11〉w 0 |γµ+ 〈D 1122hA|11h B〉VB|22+ 〈D 1111hA|11h B |11〉VB +(〈D22γµhA〉w0|γµ)|22+(〈D 1122hAhB|11〉VB)|22+(〈D 2222hAhB〉VB|22)|22 −2(〈D12γµhhA|1〉w 0 |γµ)|2−4(〈D 1212hA|1h B |1〉VB|2)|2+ 〈kzh AhB〉VB + 〈kzh A〉w0− H2 4 (〈kth A〉δ2βw0|β)|2+ H2 4 〈kth A |1〉δ 11w0|β − H2 4 (〈kth AhB〉δ22VB|2)|2+ H2 4 〈kth A |1h B |1〉δ 11VB − (N22〈hAhB〉VB|2)|2+N 11〈hA|1h B |1〉VB + 〈µh AhB〉V̈B = 〈ph A〉 (3.7) We have assumed that the forces nαβ can be represented by a decomposition nαβ(ξγ)=Nαβ(ξγ)+ ñαβ(ξγ) (3.8) where Nαβ = 〈nαβ〉 is a slowly varying function and ñαβ(·) is the fluctuating part of the forces nαβ(·), such that 〈ñαβ〉 = 0. In Eq. (3.5), we have assumed that the fluctuating part ñαβ(·) of the forces nαβ(·) is very small compared to their averaged part Nαβ(·), and hence 〈n22hAhB〉∼=N22〈hAhB〉. 3.2. Asymptotic model For the asymptotic modelling procedure we recall only the concept of highly oscillating function. We shall not deal with the notion of the tolerance periodic function as well as slowly-varying function. For every parameter ε = 1/n, n = 1,2, . . . we define the scaled cell ∆ε ≡ (−εl/2,εl/2) and by ∆ε(x)=x+∆ε the scaled cell with a centre at ξ α ∈Π. Themass density µ(·),moduli of the foundation kz(·), kt(·) and tensor of elasticity D αβγδ(·) are assumed to be highly oscillating discontinuous functions for almost every ξα ∈ Π. If µ(·),kz(·),kt(·),D αβγδ(·) ∈ HO0δ(Π,∆) then for every ξ α ∈ Π there exist functions µ(y,ξ2), kz(y,ξ 2), kt(y,ξ 2), Dαβγδ(y,x2) which are periodic approximations of the functions µ(·), kz(·), kt(·),D αβγδ(·), respectively. The asymptotic modelling procedure begins with decomposition of the displacement as a family of fields wε(y,ξ 2, t)=w0(y,ξ2, t)+ε2h̃A (y ε ,ξ2 ) VA(y,ξ 2, t) y∈∆ε(ξ α) (3.9) where h̃A(y,ξ2) is a periodic approximation of highly oscillating functions hA(·). From formula (3.3) we obtain ∂αwε(y,ξ 2, t)= ∂αw 0(y,ξ2, t)+ε∂1h̃ A (y ε ,ξ2 ) VA(y,ξ 2, t)+ε2h̃A (y ε ,ξ2 ) ∂2VA(y,ξ 2, t) ∂αβwε(y,ξ 2, t)= ∂αβw 0(y,ξ2, t)+∂11h̃ A (y ε ,ξ2 ) VA(y,ξ 2, t) +2ε∂1h̃ A (y ε ,ξ2 ) ∂2VA(y,ξ 2, t)+ε2h̃A (y ε ,ξ2 ) ∂22VA(y,ξ 2, t) ẇε(y,ξ 2, t)= ẇ0(y,ξ2, t)+ε2h̃A (y ε ,ξ2 ) V̇A(y,ξ 2, t) (3.10) Modelling of annular plates stability with functionally graded structure... 491 Under limit passage ε→ 0 for y∈∆ε(ξ α), ξα ∈Π we rewrite formulae (3.9) and (3.10) in the form wε(y,ξ 2, t)=w0(y,ξ2, t)+O(ε) ∂αwε(y,ξ 2, t)= ∂αw 0(y,ξ2, t)+O(ε) ∂αβwε(y,ξ 2, t)= ∂αβw 0(y,ξ2, t)+∂11h̃ A (y ε ,ξ2 ) VA(y,ξ 2, t)+O(ε) ẇε(y,ξ 2, t)= ẇ0(y,ξ2, t)+O(ε) (3.11) For a periodic approximation of the Lagrangian L, we have L̃ε ( y ε , ξ2, w0 (y ε ,ξ2, t ) +O(ε), ∂αw 0 (y ε ,ξ2, t ) +O(ε), ẇ0 (y ε ,ξ2, t ) +O(ε), ∂αβw 0 (y ε ,ξ2, t ) +∂11h̃ A (y ε ,ξ2 ) VA (y ε ,ξ2, t ) +O(ε) ) (3.12) If ε→ 0 then L̃ε bymeans of property of the mean value, see Jikov et al. (1994), weakly tends to L0 ( ξα,w0(ξα, t),∂αw 0(ξα, t), ẇ0(ξα, t),∂αβw 0(ξα, t),VA(ξ α, t) ) = 1 |∆| ∫ ∆(x) L̃ ( y,ξα,w0(ξα, t),∂αw 0(ξα, t), ẇ0(ξα, t),∂αβw 0(ξα, t) +∂11h̃ A(y,ξ2)VA(ξ α, t) ) dy (3.13) The asymptotic action functional has the form A0ε(w 0,VA)= t1∫ t0 ∫ Π L0 ( ξα,w0(·),∂αw 0(·),w0|αβ(·),VA(·), ẇ 0(·) ) dξα dt (3.14) where the Lagrangian is given by L0(ξ α,w0,∂αw 0,w0|αβ,VA, ẇ 0)= 1 2 〈Dαβγµ〉w0|αβw 0 |γµ+ 〈D 11γµhA|11〉VAw 0 |γµ + 1 2 〈D1111hA|11h B |11〉VAVB + 1 2 〈kz〉w 0w0+ H2 8 〈kt〉δ αβ∂αw 0∂βw 0 + 1 2 〈nαβ〉∂αw 0∂βw 0− 1 2 〈µ〉ẇ0ẇ0−〈p〉w0 (3.15) Applying the principle of stationary action, we derive the Euler-Lagrangian equations ∂ ∂t ∂L0 ∂ẇ0 − ( ∂L0 ∂w0 |αβ ) |αβ +∂α ( ∂L0 ∂w0 |α ) − ∂L0 ∂w0 = 〈p〉 ∂L0 ∂VA =0 A=1, . . . ,N (3.16) Substituting formulae (3.15) into equations (3.16), we obtain the following system of equations describing the stability of the plate under consideration (〈Dαβγµ〉w0|γµ)|αβ +(〈D 11αβhA|11〉VA)|αβ + 〈kz〉w 0− H2 4 ∂α(〈kt〉δ αβ∂βw 0) − (Nαβw0|β)|α+ 〈µ〉ẅ 0 = 〈p〉 〈D11αβhA|11〉w 0 |αβ + 〈D 1111hA|11h B |11〉VB =0 (3.17) 492 W. Perliński et al. Eliminating VA from second equation (3.17) VA =− 〈D11γµhB |11 〉 〈D1111hA |11 hB |11 〉 w0|γµ (3.18) and denoting the effective elastic moduli D αβγµ eff = 〈D αβγµ〉− 〈D11γµhB |11 〉 〈D1111hA |11 hB |11 〉 〈D11αβhA|11〉 (3.19) we arrive at the following asymptotic model equation for the averaged displacement of the plate midplane w0(ξα, t) (〈D αβγµ eff 〉w0|γµ)|αβ + 〈kz〉w 0− H2 4 ∂α(〈kt〉δ αβ∂βw 0)− (Nαβw0|β)|α+ 〈µ〉ẅ 0 = 〈p〉 (3.20) Equations (3.18)-(3.20) represent the asymptotic model of the stability of the thin plate inte- racting with microheterogeneous subsoil. The coefficients ofmodel equations (3.7), (3.20) are smooth functions of the radial coordinate ρ ∈ (R0,R1) in contrast to equations in direct description with the discontinuous and highly oscillating coefficients. 4. Applications In order to illustrate themodel equations (3.7) and (3.20), we shall investigate a simple problem of the linear polar-symmetrical stability of the annular plate clamped on its boundary (Fig. 2). The considered composite plate is interacting with heterogeneous elastic subsoil. Fig. 2. The annular plate with a longitudinally graded structure The important point of the tolerance modeling approach is the determination of the fluc- tuation shape functions (FSF). Our analysis we restrict to the case when we have only one fluctuation shape function, hence A,B = 1 and VA(ξ α, t) = V (ξα, t). The calculation of the fluctuation shape functions is usually very difficult. Hence we apply an approximate form of the fluctuation shape function analogous to dynamic analysis. For one-dimensional cell under consideration ∆(ξ1,ξ2), as the fluctuation, shape function we assume h(·) =λ2cos 2πξ1 λ (4.1) Modelling of annular plates stability with functionally graded structure... 493 The comparison of results for the exact and this approximate formof the fluctuation shape func- tion, we can find in papers by Jędrysiak (2001), Jędrysiak andMichalak (2005). For differences between thevalue ofYoung’smodulus 0.25¬Eb/Em ¬ 4and for the ratio 0.25¬ d/(λρ)¬ 0.75 (d – thewidths of the ribs, λρ –microstructure length parameter) the error for the approximate form of FSF is smaller than 10%. 4.1. Tolerance model Let us introduce the polar coordinate system Oξ1ξ2, where ϕ= ξ1 is the angular coordinate and ρ = ξ2 – radial coordinate. Setting w0 = w0(ρ), V = V (ρ), we obtain from equations (3.7) the following system of equations describing the stability of annular plates interacting with heterogeneous subsoil ∂22(〈D̃ 2222〉∂22w 0)+ 1 ρ ∂22(〈D̃ 2211〉∂2w 0)+ 2 ρ ∂2(〈D̃ 2222〉∂22w 0)+ 1 ρ3 〈D̃1111〉∂2w 0 − 1 ρ2 ∂2(〈D̃ 1111〉∂2w 0)− 1 ρ ∂2(〈D̃ 2211〉∂22w 0)+ 2 ρ4 〈D̃2211h|11〉V − 2 ρ3 ∂2(〈D̃ 2211h|11〉V ) + 1 ρ2 ∂22(〈D̃ 2211h|11〉V )+ 2 ρ4 〈D̃1111h|11〉V − 1 ρ3 ∂2(〈D̃ 1111h|11〉V )+∂22(〈D̃ 2222h〉∂22V ) + 2 ρ ∂2(〈D̃ 2222h〉∂22V )− 1 ρ ∂2(〈D̃ 1122h〉∂22V )+ 〈kz〉w 0+ 〈kzh〉V − H2 4 〈kt〉 1 ρ ∂2w 0 − H2 4 ∂2(〈kt〉∂2w 0)− H2 4 ∂2(〈kth〉∂2V )− 1 ρ Nϕ∂2w 0−Nρ∂22u=0 1 ρ3 〈D̃1111h|11〉∂2w 0+ 1 ρ2 〈D̃1122h|11〉∂22w 0+ 1 ρ4 〈D̃1111h|11h|11〉V + 1 ρ2 〈D̃1111h|11h〉∂22V +∂22 (1 ρ 〈D̃2211h〉∂2w 0 ) +∂22(〈D̃ 2222h〉∂22w 0)+∂22 ( 1 ρ2 〈D̃2211h|11h〉V ) +∂22(〈D̃ 2222hh〉∂22V )−4∂2 ( 1 ρ2 〈D̃1212h|1h|1〉∂2V ) + 〈kzhh〉V − H2 4 ∂2(〈kth〉∂2w 0) − H2 4 ∂2(〈kthh〉∂2V )+ H2 4 〈kth|1h|1〉 1 ρ2 V −Nρ〈hh〉∂22V + 1 ρ2 Nϕ〈h|1h|1〉V =0 (4.2) where we have denoted D̃2222 = D2222, D̃1122 = ρ2D1122, D̃1111 = ρ4D1111, Nϕ = ρ 2N11, Nρ = N 22. Equations (4.2) represent a system of two partial differential equations for the averaged deflection w0(·) and the fluctuation amplitude V (·). The boundary conditions for the clamped plate are given by w0(ρ=R1)=w 0(ρ=R2)=0 ∂2w 0(ρ=R1)= ∂2w 0(ρ=R2)= 0 V (ρ=R1)=V (ρ=R2)= 0 ∂2V (ρ=R1)= ∂2V (ρ=R2)= 0 (4.3) Since h(·) ∈ O(λ2), the underlined moduli depend on the microstructure length parameter λ. Hence, the tolerance model equations describe the microstructure length-scale effect on the stability of the plate under consideration. 4.2. Asymptotic model For analysis of the asymptotic model we use equations (3.20). Denoting Dr(ρ) = D 2222 eff , Dϕ(ρ) = ρ 4D1111eff , Drϕ(ρ) = ρ 2D1122eff , Kz = 〈kz〉, Kt = 〈kt〉 we obtain from equation (3.20) a single equation describing stability for the asymptotic model of the plate under consideration 494 W. Perliński et al. ∂22(Dr∂22w 0)+ 2 ρ ∂2 (( Dr − 1 2 Drϕ ) ∂22w 0 ) + 1 ρ ∂22(Drϕ∂2w 0)− 1 ρ2 ∂2(Dϕ∂2w 0) + 1 ρ3 Dϕ∂2w 0+Kzw 0− H2 4 Kz 1 ρ ∂2w 0− H2 4 ∂2(Kt∂2w 0)−Nρ∂22w 0− 1 ρ Nϕ∂2w 0 =0 (4.4) The above equation represents the single partial differential equation for the averaged deflec- tion w0(·) and has the form similar to the equation for buckling of the annular plate with cylindrical orthotropy. 4.3. Numerical results for the asymptotic model In order to derive the critical value of forces for buckling of the plate under consideration we shall use the asymptotic model equation. We look for the solution to equation (4.4), where the problem of the forces Nρ and Nϕ will be solve similarly to plate with cylindrical orthotropy (cf. Mossakowski, 1960) Nρ =−N ( ρ R2 )k−1 Nϕ =−Nk ( ρ R2 )k−1 k= √ Dϕ Dr (4.5) Substituting (4.5) into equation (4.4) we obtain differential operator in the form L(w0)= ∂22(Dr∂22w 0)+ 2 ρ ∂2 (( Dr− 1 2 Drϕ ) ∂22w 0 ) + 1 ρ ∂22(Drϕ∂2w 0) − 1 ρ2 ∂2(Dϕ∂2w 0)+ 1 ρ3 Dϕ∂2w 0+Kzw 0− h2 4 Kt 1 ρ ∂2w 0− h2 4 ∂2(Kt∂2w 0) +N ( ρ R2 )k−1 ∂22w 0+N 1 ρ k ( ρ R2 )k−1 ∂2w 0 =0 (4.6) Operator (4.4) has smoothly varying functional coefficients along the radial direction. Hence, in most cases, solutions to specific problems for the plates under consideration have to be obtained using approximate methods. In order to obtain the approximate solution to equation (4.4) for the annular clamped plate interacting with heterogeneous subsoil, the Galerkin method will be used. The smallest value of critical forces can be obtained from the following equation R2∫ R1 L(f(ρ))f(ρ) dρ=0 (4.7) As the function f(ρ), we assume the first shape function of stability for the isotropic annular clamped plate with the radius R1 =1.0m and R2 =3.0m resting on the elastic homogeneous foundation f(r)=w1 ( J0 ( 1.1875 ρ R1 ) +23.9767Y0 ( 1.1875 ρ R1 ) +16.0964J0 ( 3.8405 ρ R1 ) +11.1116Y0 ( 3.8405 ρ R1 )) (4.8) where J0(·), Y0(·) are Bessel’s functions of the first and second kind, respectively. 4.3.1. Comparison of the test tasks with results from the finite element method In order to verify the correctness of the derived equations, we analysed the obtained results for a test task. We shall investigate the simple problem of polar-symmetrical stability of an annular clamped plate. We compare the value of critical forces from the asymptotic model Modelling of annular plates stability with functionally graded structure... 495 with the results from the finite element method (Abaqus program). The following material and geometrical parameters of the plate were assumed: matrix: Em = E1 = 210, 150 and 69GPa, ν1 = 0.3, ribs: Er = E2 = 210GPa, ν1 = 0.3, number of periodic cells N = 60, thickness of the plate H = 0.05m, internal radius R1 = 1m, external radius R2 = 3m, width of ribs d = 0.75λR1 = 0.75(2π/60)R1 and the foundation parameters: vertical modulus of elasticity of foundation below the matrix kzm = 25MN/m 3, vertical modulus of elasticity of foundation below the ribs kzr =50MN/m 3, horizontalmodulus of elasticity of foundation below thematrix ktm = 0MN/m 3, horizontal modulus of tangent elasticity of the foundation below the ribs ktr =0MN/m 3. The value of critical forces was calculated in two ways: by making use of asymptotic model equations (AS) and through the finite element method (FEM, Abaqus program). These results are summarized in Table 1 Table 1.The comparison of results for the test task calculated by two independentmethods Matrix Ribs Matrix Ribs AM FEM Ratio (asym. model) (Abaqus) Modulus Modulus Elast. found. Elast. found. Ncr Ncr NcrFEM NcrASEm [GPa] Er [GPa] kzm [MN/m 3] kzr [MN/m 3] [kN/m] [kN/m] 210 210 0 0 24303 23843 0.98 150 210 0 0 20321 19533 0.96 69 210 0 0 14239 13181 0.93 210 210 50 50 39212 38756 0.99 150 210 50 50 34990 34054 0.97 69 210 50 50 28090 26615 0.95 210 210 25 50 34720 34275 0.99 150 210 25 50 30510 29712 0.97 69 210 25 50 23790 22690 0.95 The above table shows that the results obtained from equations for the tolerance averaging technique coincide with the results from the well-known finite element method. 4.3.2. Influence of material properties of the plate and foundation on the critical forces The aim of this Subsection is to investigate the influence of material properties of the plate and foundation on the value of the critical forces. The material and geometrical parameter of the plate we assume identical as in the above example. In Fig. 3a, there is shown a diagram of the value of critical forces Nkr [MN] versus k= kz2/kz1, where kz2 is the vertical modulus of the foundation under the matrix and kz1 un- der the ribs. The diagram is derived for the ratio kt1/kz1 = 0.5, kt2/kz2 = 0.5, horizontal and vertical moduli of elasticity of the foundation and kz1 =500.0MN/m 3. In Fig. 3b, there is shown the influence of the ratio p = kt/kz, horizontal kt and verti- cal kz modulus of the elastic foundation. The diagrams in Fig. 3b are derived for the ratio kz2/kz1 = 0.1, kt2/kt1 = 0.1 and the vertical modulus kz1 = 5000.0MN/m 3. Diagram Nkr1(p) shows the smallest value of critical forces for the plate thickness H = 0.05m and Nkr2(p) for the plate thickness H =0.20m. 4.4. Numerical results for the tolerance model We look for an approximate solution to equations (4.2) similarly to the asymptotic model using the Galerkin method. For the tolerancemodel,we obtain twovalues of critical forces, formacro andmicrobuckling. 496 W. Perliński et al. Fig. 3. Diagrams of the value of critical forces Nkr [MN] versus: (a) k= kz2/kz1, (b) p= kt/kz 4.4.1. Influence of the number of cells on the critical forces The aim of this Subsection is to investigate the influence of the microstructure length para- meter λ=2π/α on the value of critical forces. In Fig. 4, diagrams of the value of critical forces versus numbers of the cells α are shown. The diagrams are derived for the annular clamped plate with geometry: H =0.05m,R1 =1m,R2 =3m. Thematerial parameters of the matrix are: Em = E1 = 69GPa, ν1 = 0.3 and of the ribs: Er = E2 = 210GPa, ν2 = 0.3. The width of the ribs is d = 0.75λR1, subsoil moduli: kz2/kz1 = kt2/kt1 = 0.2, kt1/kz1 = kt2/kz2 = 0.2, kz1 =50.0MN/m 3. In Fig. 4a, the diagram of the value of critical forces for macro buckling of the plate versus number of the cells α is shown. The value of critical forces for α> 25-30 is independent of the number of the cells and conforms with the results from the asymptotic model. Let us note that the number of the microstructure cells should be bigger than 30 to provide the correct solution for the tolerance and asymptotic models. Fig. 4. The value of critical forces N [MN/m] for: (a) macro buckling, (b) micro buckling versus the number of the cells α In Fig. 4b, the diagram of the value of critical forces for micro buckling of the plate versus number of the cells α is shown. As one should expect, the value of critical forces for micro buckling grows with the increasing number of the microstructure cells. 5. Conclusions • The composite plate interacting with elastic heterogeneous subsoil having a functionally graded structure is described by model equations involving only smooth coefficients in contrast to the coefficients in equations for direct description, which are non-continuous and highly oscillating. Modelling of annular plates stability with functionally graded structure... 497 • Since theproposedmodel equations have smoothand slowly varying functional coefficients, hence in most cases, solutions to specific problems of stability of the functionally graded plate under consideration have to be obtained using well known numerical methods. • The contribution contains two model equations – tolerance model equations (3.7) with coefficients depending on the microstructure length λ and simplified asymptotic model equations (3.20). • Solutions to the boundary value problems formulated in the framework of the proposed models have the physical sense only if they are slowly varying in the distinguished direc- tions. The number of themicrostructure cells should be bigger than 30. 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