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Cung cấp cho học viên những kiến thức cơ bản khi nghiên cứu tính toán kết cấu tấm, kết cấu vỏ và kết cấu màng. Từ đó có thể áp dụng các bài toán vào trong kết cấu thực silo, bể chứa trụ tròn,…Cung cấp cho học viên những kiến thức cơ bản khi nghiên cứu tính toán kết cấu tấm, kết cấu vỏ và kết cấu màng. Từ đó có thể áp dụng các bài toán vào trong kết cấu thực silo, bể chứa trụ tròn,…
Composite Structures 137 (2016) 85–92 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct Nonlinear free vibration of pre- and post-buckled FGM plates on two-parameter foundation in the thermal environment Maciej Taczała a, Ryszard Buczkowski b,⇑, Michal Kleiber c a West Pomeranian University of Technology, Piastow 41, 71-065 Szczecin, Poland Maritime University of Szczecin, Division of Computer Methods, Poboznego 11, 70-507 Szczecin, Poland c Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawinskiego 5B, 02-106 Warsaw, Poland b a r t i c l e i n f o Article history: Available online 14 November 2015 Keywords: FGM plates Two-parameter elastic foundation Nonlinear free vibration Finite element method a b s t r a c t The geometrically nonlinear free vibration of functionally graded thick plates resting on the elastic Pasternak foundation is investigated The motion equations are derived applying the Hamilton principle We consider the first order shear deformation plate theory (FSDT), in which the modified shear correction factor is required A 16-noded Mindlin plate element of the Lagrange family which is free from shear locking due to small thickness of the plate used The material properties are assumed to be temperature-dependent and expressed as a nonlinear function of temperature Because the FGM plates are not homogeneous, the basic equations are calculated in the equivalent physical neutral surface which differs from the geometric mid-plane In the pre-buckling range natural frequencies decrease ultimately reaching zero for critical stress in the bifurcation point Ó 2015 Elsevier Ltd All rights reserved Introduction Functionally graded materials (FGMs) are new inhomogeneous composite materials in which the volume fraction of two components varies smoothly and continuously across the given direction FGMs are mixtures of ceramics and metal, where external ceramic layers due to large thermal resistance are exposed to high temperatures, while internal metallic constituents, owing to their stronger mechanical performance, are able to reduce the possibility of fracture Manufacturing techniques must guarantee controlled changes in composition and density, so that the product will have a required structure and properties along the given direction, often across plate thickness Because there are many papers dealing with linear and nonlinear thermal stresses and deformations of FGMs beams, plates and shells, we focus only on the studies of the nonlinear vibration of functionally graded plates and shells resting on an elastic foundation Interested readers may wish to review the papers of Liew, Lei and Zhang [2] and Swaminathan and others [1], who have presented comprehensively analytical and numerical solutions for a number of examples concerning the statics, dynamics and stability ⇑ Corresponding author E-mail addresses: maciej.taczala@zut.edu.pl (M Taczała), rbuczkowski@ps.pl (R Buczkowski), mkleiber@ippt.pan.pl (M Kleiber) http://dx.doi.org/10.1016/j.compstruct.2015.11.017 0263-8223/Ó 2015 Elsevier Ltd All rights reserved of functionally graded plates and functionally graded carbon nanotube reinforced composites The first paper investigating the large amplitude of FGM cylindrical shells resting on elastic foundation in thermal environments was presented by Shen [3] The results showed that the differences of the fundamental frequencies between Voigt and Mori–Tanaka micromechanical models are very small, and the differences of the nonlinear to linear frequency ratios of the FGMs shells are negligible Shen and Xiang [4,5] studied the nonlinear vibration of nanocomposite cylindrical beams, plates and shells resting on elastic foundations subjected to thermal loading The paper of Yang, Ke and Kitiporchnai [6], using the differential quadrature method, made the first attempt to study the nonlinear free vibration of single-walled carbon nanotubes (SWCNTs) based on geometric nonlinearity and Eringen’s nonlocal elasticity theory The geometrically nonlinear free vibration of carbon nanotubes/fibre/polymer laminated composite plates with immovable simple supported boundary conditions using the Galerkin procedure was studied by Rafiee and others [7] Zenkour and Sobhy [8] investigated dynamic bending behaviour of FGM plate resting on twoparameter elastic foundation and subjected to time harmonic thermal load An analytical approach was presented by Duc [9] and Duc and Quan [10] to investigate the transient responses of imperfect eccentrically stiffened FGM double curved thin shallow shells using Galerkin method (the Lekhnitsky smeared stiffeners technique is used) The non-linear dynamics, in the sense of strain 86 M Taczała et al / Composite Structures 137 (2016) 85–92 displacement relations, of FGM truncated conical shells on the Pasternak elastic foundation was investigated by Najafov and Sofiyev [11] and the problem was solved using superposition, Galerkin and harmonic balance methods A general third-order deformation theory for functionally graded plates including a modified couple stress effect that accounts for material length scale and geometrical nonlinearity has been recently presented by Kim and Reddy [12] and Neves et al [13] and some time earlier by Ferreira with co-authors [14] We study the non-linear free vibration of FGM plate resting on an elastic foundation using the finite element method The problem has been solved using a 16-noded Mindlin plate element of the Lagrange family which is free from shear locking [15] For the solution of eigenvalues and eigenvectors the subspace iteration technique has been used here All the equations are calculated in reference to the equivalent physical neutral surface which is different from the mid-plate The problem we tackle refers to the formulation and solution of natural vibrations having small amplitudes around the equilibrium of a pre- or post-buckled functionally graded plate subjected to thermal loading It should be noted that the nonlinear analysis concerns the determination of the equilibrium state of the plate, while the vibrations are analysed using the linear eigenproblem formulation Contrary to this formulation, a problem is defined in the literature [16] referring to the nonlinear vibrations with large amplitudes The problem is governed by a nonlinear equation solved analytically or using the finite element procedure combined with the iteration technique The natural frequency is dependent on the amplitude and can also be dependent on initial imperfections, stresses and many other parameters, such as temperature field and material distribution in a FG plate Considering large out-of-plane displacements, we write down the components of the Green–Lagrange strain tensor as follows tþDt w;x tþDt Eyy ¼ tþDt v ;y À ztþDt hx;y þ tþDt w2;y À Á à ÂtþDt tþDt tþDt Exy ¼ u;y þ v ;x þ z tþDt hy;y À tþDt hx;x þ tþDt w;x tþDt w;y à 1 tþDt Exz ¼ tþDt hy þ tþDt w;x à 1 tþDt Eyz ¼ ÀtþDt hx þ tþDt w;y ð5Þ tþDt In the incremental approach, the stress in the actual increment and iteration can be expressed via the value in the previous iteration tþDt 2.1 Derivation of the static equations Si ¼ t Si þ tþDt DSi ð6Þ The increment of the second Piola–Kirchhoff stress tensor written in the vector form is tþDt DSi ¼ t C ij tþDt DEj À tþDt DEðTÞ j ð7Þ where DEðTÞ is the thermal strain being i n o ðTÞ ð0Þ tþDt DEi ¼ DkTEð0Þ Ei ¼ colf 1 i ; tþDt 0 0g ð8Þ and the constitutive matrix C is typical for plane stress analysis with the influence of shear stresses: 2t t Mathematical relations Exx ¼ tþDt u;x þ ztþDt hy;x þ E 6m C ij ¼ 60 40 m E 0 1Àm 0 0 t 7 0 7 bt G bt G 0 ð9Þ The static equation governing the structural response of the FG plate subject to thermal loading will be derived employing the principle of virtual work where b ¼ 5=6 is the shear correction factor Considering dependence of the material parameters on temperature the constitutive matrix is dependent on the loading level Strain components expressed in the incremental form are dtþDt W int ¼ dtþDt W ext tþDt ð1Þ Since it is only the thermal loading which is considered, therefore dtþDt W ext ¼ ð2Þ The virtual work of internal forces for the plate on twoparameter elastic foundation can be expressed as (the total Lagrangian formulation is used here): dtþDt W int ¼ Z tþDt Si dtþDt Ei dV þ Vp Z Af tþDt ½k0 wdtþDt w þ k1 ðtþDt w;x dtþDt w;x þ tþDt w;y dtþDt w;y ÞdA ð3Þ where {Si} are components of the second Piola Kirchhoff stress tensor, {Ei} – components of the Green–Lagrange strain tensor, k0 and k1 – parameters of the elastic foundation Using use the first-order shear deformation theory, displacements of arbitrary point of the plate functions are tþDt u ¼ tþDt um þ ztþDt hy tþDt v ¼ tþDt v m À ztþDt hx tþDt w ¼ tþDt wm ð4Þ where um ; v m ; wm are displacements of the neutral layer in x, y and z directions, respectively, and hx ; hy are rotations with respect to x and y axes tþDt Dw2;x Dt tþDt DEyy ¼ tþDt Dv ;y À ztþDt Dhx;y þ t w;y tþDt Dw;y þ tþ Dw2;y iþ1 1 tþDt DExy ¼ tþDt Du;y þ tþDt Dv ;x þ zðtþDt Dhy;y À tþDt Dhx;x Þ i tþDt Dw;x þ t w;x tþDt Dw;y þ tþDt Dw;x tþDt Dw;y þt w;y iþ1 DExx ¼ tþDt Du;x þ ztþDt Dhy;x þ t w;x tþDt Dw;x þ ð10Þ DExz ¼ ðtþDt Dhy þ tþDt Dw;x Þ tþDt DEyz ¼ ðÀtþDt Dhx þ tþDt Dw;y Þ tþDt With the use of the finite element approximation, the strain increments can be presented in the form tþDt tþDt DEi ¼ t Bijð1Þ tþDt Ddj þ Bð2Þ Ddj tþDt Ddk ijk ð11Þ where t Bð1Þ and t Bð2Þ are strain–displacement matrices t ð1Þ B1j ¼ N1j;x þ zN 5j;x þ N3j;x N3k;x t dk ; t B2j ¼ N2j;y À zN 4j;y þ N3j;y N3k;y t dk ð1Þ t ð1Þ B3j ¼ N1j;y þ N2j;x þ zðN 5j;y À N4j;x Þ þ ðN3k;y N3j;x þ N3k;x N3j;y Þt dk t ð1Þ B4j ¼ N5j þ N3j;x ; t B1j ¼ ÀN4j þ N3j;y ð1Þ 1 ð2Þ ð2Þ ð2Þ B1jk ¼ N3j;x N3k;x ; B2jk ¼ N3j;y N3k;y ; B3jk ¼ N3j;x N3k;y 2 B4km ¼ 0; B5km ¼ 0: ð12Þ 87 M Taczała et al / Composite Structures 137 (2016) 85–92 Shape functions of the 16-node finite element employed in the present analyses and matrices N 1j ; N 2j ; N 3j ; N 4j ; N 5j are defined in Appendix B Note that constructing the vector of strains and strain increments for the finite element analysis, the shear components are doubled, therefore the vector takes the form tþDt DE ¼ col È tþDt tþDt DExx DEyy tþDt tþDt DExy DExz tþDt DEyz É ðintÞ Ddk t t ðTÞ Ddk ¼ t K À1 jk F j t K jk ¼ ðTÞ Rows of the shape function and strain–displacement matrices as well as their derivatives correspond to the degrees of freedom of the finite element Applying Eqs (6–8 and 11) in Eq (3) we obtain t V Z ð2Þ ð2Þ Si Bijk þ Bikj dV þ V t Z ð1Þ ð1Þ C ip t Bpk t Bij dVþ ½k0 N3k N3j ÉtþDt ð14Þ ðdÞ where t i ðdÞ Ddk ¼ Àt F jðintÞ þ tþDt Dkt F ðTÞ j ð1Þ t V Z ð1Þ Â Ã k0 N3k N3j þ k1 ðN3k;x N3j;x þ N 3k;y N3j;y Þ dA ð16Þ Z t V ð2Þ ð2Þ Si Bijk þ Bikj dV ¼ TEð0Þ p Z V ð1Þ t C ip t Bij dV V ð19Þ Af ðincrÞ tþDt Ddk ðincrÞ Ddk ð20Þ Direct application of Eq (20) destroys the symmetry of the stiffness matrix, therefore the procedure is proposed following the original consideration by Crisfield [17] The iterative correction tþDt Ddk appearing while evaluating the displacement increment in the actual step ðincrÞ Ddk ¼ t ðincrÞ dk þ tþDt Ddk ð21Þ is divided into two parts; dependent on internal forces and reference thermal forces tþDt tþDt pi dtþDt ui dA ðintÞ Ddk ¼ t Ddk ðTÞ þ tþDt Dkt Ddk Solving Eq (14), we have tþDt qtþDt u€i dtþDt ui dV ð26Þ Using the displacement functions for FSDT (Eq (5)) and FE approximation, Eq (25) becomes d tþDt Z tþDt W ext ¼ À V N 1k N 1j þ zðN 5k N1j þ N 1k N 5j Þ þ z2 N 5k N 5j þ €ðVÞ ddðVÞ q4 N2k N2j À zðN4k N2j þ N2k N4j Þ þ z N4k N4j þ 5dV d j k N 3k N 3j ð27Þ €ðVÞ ddðVÞ dtþDt W ext ¼ ÀtþDt M jk d j k tþDt Z Mjk ¼ Vp ¼ Dl Z where is the internal force vector The constant arc-length method has been applied to solve the problem The method consists in enhancing the system of equations given by Eq (15) with the constraint equation delimiting increment of displacements in each step tþDt @A which can be also presented in the form ð18Þ is the thermal reference force vector, Z Z  à ð1Þ t ðintÞ F j ¼ t Si t Bij dV þ k0 N 3k N 3j þ k1 ðN 3k;x N 3j;x þ N 3k;y N 3j;y Þ dA tþDt Z ð25Þ ð17Þ is the stiffness matrix dependent on stresses, t ðTÞ Fj € i ÞdtþDt ui dV þ ðtþDt bi À tþDt qtþDt u V is the stiffness matrix dependent on displacements, accounting also for contribution of the elastic foundation, ðSÞ Z dtþDt W ext ¼ À Af K jk ¼ If we assume a possibility of small amplitude vibrations d around the equilibrium positioned of the plate subject to thermal loading, an equation of motion must be derived In this case the virtual work of external forces takes the form ð15Þ C ip t Bpk t Bij dV þ t a quadratic equation for the iterative change of the loading parameter governing the actual temperature of the plate tþDt Dk Once the correct solution is selected, we should follow the rules specified by Crisfield [17] For free vibrations body b and surface p forces vanish, and Eq (24) reduces to Z K jk ¼ ðintÞ ðintÞ t ðincrÞ þ t Ddk dk þ t Dd k ðincrÞ ðintÞ t ðTÞ ðTÞ ðTÞ þ t Ddk Ddk tþDt Dk þ t Ddk t Ddk tþDt Dk2 ¼ Dl ð24Þ þ t dk V ðSÞ tþDt K jk þ t K jk ð23Þ ðSÞ K jk Using Eqs (21), (22), we can transform Eq (20) to dtþDt W ext ¼ Eq (14) can be presented in the following form: t þ t ðVÞ Af h ðdÞ K jk 2.2 Derivation of the dynamic equations Af þ k1 ðN3k;x N3j;x þ N 3k;y N3j;y ÞdA Ddk Z Z ð0Þ t t t ð1Þ t ð1Þ tþDt ¼ DkTEp C ip Bij dV À Si Bij dV V V Z À ½k0 N3k N 3j þ k1 ðN3k;x N3j;x þ N3k;y N3j;y ÞdAt dk t t ðincrÞ dk ð13Þ &Z ðintÞ t ¼ Àt K À1 jk F j t ð22Þ tþDt N1k N1j þ zðN5k N1j þ N1k N5j Þ þ z2 N5k N5j þ ð28Þ q6 N2k N2j À zðN4k N2j þ N2k N4j Þ þ z2 N4k N4j þ 5dV N3k N3j ð29Þ Note that the mass matrix is dependent on the increment of the nonlinear static analysis, as material properties change with the temperature increase The mass matrix is computed using the  Lobatto integration rule 2.3 Modelling material properties To model the distribution of ceramic fraction throughout the plate thickness, the power law is used Vc ¼ n z À zC ; þ h nP0 ð30Þ where zC is a coordinate of the mid-layer The metallic fraction is Vm ¼ À Vc ð31Þ Effective material properties are evaluated depending on the proportion of both fractions 88 M Taczała et al / Composite Structures 137 (2016) 85–92 Rh E ¼ Ec V c þ Em V m a ¼ ac V c þ am V m q ¼ qc V c þ qm V m ð32Þ Ec ¼ Ec0 EcðÀ1Þ T À1 þ þ Ec1 T þ Ec2 T þ Ec3 T Em ¼ Em0 EmðÀ1Þ T À1 þ þ Em1 T þ Em2 T þ Em3 T ac ¼ ac0 acðÀ1Þ T À1 þ þ ac1 T þ ac2 T þ ac3 T am ¼ am0 amðÀ1Þ T À1 þ þ am1 T þ am2 T þ am3 T 2.4 Position of the neutral layer We determine the position of the neutral layer z0 (Fig 1) assuming that the resultant membrane force due to bending stresses SðbÞ xx equals zero tþDt ðbÞ Sxx dz ¼0 ð34Þ The value of stress in the actual iteration can be evaluated in the incremental form: tþDt ðbÞ Sxx ¼ t Sxx þ tþDt DSðbÞ xx ð35Þ The stress increment, based on the constitutive equation, is tþDt ðbÞ ðbÞ DSxx ¼ t C 11 tþDt DExx þ t C 12 tþDt DEðbÞ yy ð36Þ Substituting Eqs (35), (36) to Eq (34) we arrive at Z h t ðbÞ Sxx dz þ Z h " # ÀtþDt Á Ef t tþDt dz ¼ ðz À z Þ D h À m D h y x f À t m2f t ð37Þ Since Z h t ðbÞ Sxx dz ¼0 ð38Þ we can find the position of the neutral layer Rh z0 ¼ Rh ! Á Dhy À t mf tþDt Dhx dz f ! À Á tE f tþDt Dh À t m tþDt Dh dz y x f 1Àt m tE fz 1Àt m2 tE f dz ð40Þ dz Note that in general the neutral layer changes the position along with the temperature-dependent variation of material properties and should be evaluated for each increment/iteration Numerical examples ð33Þ Generally, Poisson ratio can also be modelled using equations using analogous to Eqs (32) and (33) However, in the present paper the Poisson ratio is constant, and density is independent of temperature h z0 ¼ R h 1À m2 f Temperature-dependent variations of material parameters are modelled using the following equations: Z tE z f 1À m2 f ÀtþDt ð39Þ f In the case of Poisson number independent of temperature, Eq (39) reduces to To verify the code used in this study, three examples are discussed First, temperatures at which the plate buckles are computed analytically for isotropic plates (see Appendix A) À Á4 À Á2 4D pb þ k0 þ 2k1 pb DT ¼ ÀpÁ2 Et 1À ma b ð41Þ and compared with the own numerical results using finite element method Due to nonlinear dependency of the thermal expansion coefficient a on temperature, the solution of Eq (41) calls for application of the iterative procedure The critical temperature was calculated for the square plate with b = 300 mm and the thickness of the plate t = mm The simply supported boundary conditions were assumed for all edges, which were also restrained against inplane displacements Material properties are taken from reference [18] A uniform temperature change was applied to the plate, and the assumed reference temperature T was 300 K The results for isotropic plates, made of SUS304 stainless steel and silicon nitride Si3N4, without elastic foundation and positioned on the Winkler elastic foundation are given in Table and plates on the twoparameter foundation in Table For all numerical calculations the mesh (16  16) is taken into account Very good agreement of the numerical and analytical results can be seen.Influence of the exponent in Eq (30) was investigated for the FGM plate having the same dimensions and material properties as in the first example Structural response of the plate is presented in Fig It can be seen that the plate exhibits bifurcation-type behaviour for n ¼ (pure ceramic) and n ¼ 10000 (practically pure metal), as the boundary cases For the intermediate values of the exponent, the curves are similar to those for nonlinear response of isotropic plate with initial imperfection [19] It can be explained by the distribution of material across the thickness; changing fractions of ceramic and metal result in the bending moment causing out-of-plane deformations Note that the structural response of true FGM plates does not fall between the curves for two boundary cases These results are also confirmed by the results of the vibration analysis – Fig For each increment of temperature, the linear eigenvalue problem is solved and natural frequency of free vibration evaluated For the isotropic pure ceramic and metal plates the frequencies decrease to reach zero for critical temperatures, corresponding to the bifurcation points as in Fig For the first time the same observation for isotropic plates without and with foundation was made by Park and Kim [20] and Shen and Xiang [21], respectively The decrease of natural frequencies is the result Table Buckling temperatures in K for isotropic plates on elastic foundation for k1 ¼ Material k0 N=mm ! 0.00 0.01 0.05 0.10 SUS304 Analytically Numerically Analytically Numerically 8.2149 8.2103 16.7286 16.7193 11.5177 11.5960 21.0486 21.0393 24.6136 24.6091 38.1517 38.1426 40.7433 40.7389 59.1485 59.1397 Si3N4 Fig Position of mid-layer and neutral layer M Taczała et al / Composite Structures 137 (2016) 85–92 Table Buckling temperatures in K for isotropic plates on elastic foundation for k1 ¼ 500 N=mm3 Material k0 N=mm ! 0.00 0.01 0.05 0.10 SUS304 Analytically Numerically Analytically Numerically 43.8318 43.8274 63.1590 63.1502 47.0191 47.0147 67.2941 67.2854 59.6793 59.6759 83.6796 83.6710 75.3199 75.3161 103.8254 103.8171 Si3N4 89 diagram is governed by significant increase of deformations in the post-buckling range and the increase of stiffness resulting from nonlinear terms in strain–displacement relationship (Eq (12))) We note that the curves representing the calculated fundamental frequencies follow those presented by Park and Kim [20] The influence of elastic foundation parameters k0 and k1 on structural response and natural frequencies is presented in Figs 4– The behaviour typical of bifurcation-type buckling can be again observed for various values of the first foundation parameter, k0 ranging from to 0.1 N/mm, with the critical temperature increasing with the increase of that value – Fig The effect is replicated for the vibration analysis – Fig as well as the variation of natural frequencies in the pre- and post-buckling range The influence of the shear parameter k1 N=mm3 for constant value k0 ¼ 0:05 N=mm is presented in Figs and The next example refers to the vibration at small amplitudes and the influence of initial imperfection – in the mode corresponding to the eigenvibration mode – on natural frequency is illustrated Here nonlinearity is related to nonlinear terms in the strain–displacement relationships Influence of the imperfections is analysed for various values of the exponents in the power law (Eq (30)), defining the fraction of the ceramic constituent regarding various amplitudes of initial imperfection considering also the sign of the amplitude: positive or and negative, see Table for comparison Calculations were performed for square simply supported plate b = 300 mm, t = 15 mm, material properties for metal fraction: density 8166 kg=m3 , Young modulus 207700 N=m2 , Poisson ratio 0.3177, for ceramic fraction: density 2370 kg=m3 , Young modulus 322200 N=m2 , Poisson ratio 0.24 For isotropic plates (metal or ceramic) the natural frequency for both cases are identical, however, for exponents resulting in varying mixture of ceramic and metallic fractions throughout the plate Fig Curves of equilibrium states for simply supported square Si3N4/SUS304 FG plates Fig Vibration behaviour of Si3N4/SUS304 FG plate of increasing compressive stresses, ultimately resulting in singularity of the total stiffness matrix (sum of stiffness and geometrical matrices) for the critical temperatures of the isotropic plates (308.21 K in the case of SUS304 and 316.73 K for Si3N4), see Fig 3) The increase of the frequencies in the further part of the Fig Curves of the equilibrium states for simply supported square Si3N4/SUS304 FG plate assuming n ¼ for various values of the first foundation parameter k0 in N/mm assuming k1 ¼ 90 M Taczała et al / Composite Structures 137 (2016) 85–92 Fig Vibration behaviour of Si3N4/SUS304 FG plate assuming n ¼ for various values of the first foundation parameter k0 in N/mm assuming k1 ¼ Fig Vibration behaviour of Si3N4/SUS304 FG plate n ¼ for various values of the second foundation parameter k1 in N/mm3 assuming k0 = 0.05 N/mm The volume fraction of the constituent materials has a significant influence on thermal post-buckling behaviour and natural frequencies of FGM plates In the pre-buckling range natural frequencies decrease, following an increase of compressive stress, ultimately reaching zero for critical stress in the bifurcation point Further increase of natural frequencies in the post-buckling range is governed by out-of-plane displacements for a stable equilibrium path Thermal buckling stress and post-buckling response as well as natural frequencies are influenced by both parameters of the Pasternak foundation Acknowledgements The work has been performed under the project Static and dynamic finite element analysis of layered structures on elastic nonhomogeneous foundation, financed by the Polish National Science Centre (NCN) under the contract 2012/05/B/ST6/03086 The support is gratefully acknowledged Appendix A Thermal buckling of plate on elastic foundation – analytical approach Fig Curves of the equilibrium states for a simply supported square Si3N4/SUS304 FG plate n ¼ and b=t ¼ 20 for various values of the second foundation parameter k1 in N/mm3 assuming k0 = 0.05 N/mm Equation for the buckling temperature of the plate on the twoparameter elastic foundation will be derived beginning with the principle of virtual work, due to the absence of external loading dW int ¼ where thickness the frequency depends on the direction of imperfection or the amplitude sign The effect can be explained by asymmetric properties of the plate We note that the natural frequency increases with the increase of the initial imperfection amplitude w0 what can be explained by increasing stiffness of the plate due to nonlinear terms Concluding remarks The formulations were verified against the results of thermal post buckling analysis and free vibration analysis of isotropic and FG plates The numerical results show that structural response and vibration of the FG plates subjected to thermal loading are different from those for the response isotropic plate ðA:1Þ Z À dW int ¼ Á Sx dEx þ Sy dEy þ Sxy dExy dV VP Z  þ Af à k0 wdw þ k1 ðw;x dw;x þ w;y dw;y Þ dA ðA:2Þ Is the virtual work of internal forces of the plate and the elastic foundation Note that Exy is the shear strain and not a component of the strain tensor Constitutive equations including thermal strains are as follows i E h Ex À ExðTÞ þ mðEy À EðTÞ y Þ 1Àm i E h Sy ¼ Ey À EyðTÞ þ mðEx À EðTÞ x Þ 1Àm E Sxy ¼ Exy 2ð1 þ mÞ Sx ¼ ðA:3Þ 91 M Taczała et al / Composite Structures 137 (2016) 85–92 Table Natural frequencies for simply supported FG plate for various exponents of power law, sÀ1 n # w0 =t ! (ceramic) 0.2 Amplitude Positive Negative Positive Negative Positive Negative 10000 (metal) 0.00 0.01 0.02 0.05 0.1 0.2 0.5 11310.70 9226.07 9226.07 6902.22 6902.22 5663.98 5663.98 5005.21 11311.20 9226.16 9226.66 6901.95 6902.67 5663.83 5664.32 5005.43 11312.27 9227.04 9227.85 6902.61 6903.55 5664.35 5665.01 5005.90 11316.85 9230.62 9233.68 6905.06 6907.92 5666.30 5668.40 5007.93 11377.82 9273.56 9301.49 6931.02 6963.17 5687.09 5708.62 5045.58 11677.82 9518.23 9653.62 7110.80 7220.50 5825.59 5908.89 5232.12 13703.90 11183.73 11888.17 8339.31 8886.85 6800.62 7218.24 6089.55 Strains including nonlinear out-of-plane terms are given by Ex ¼ Àzw;xx þ w2;x Ey ¼ Àzw;yy þ w2;y Exy ¼ À2zw;xy þ w;x w;y ðA:4Þ and thermal strains by ¼ ETy ¼ aDT ðA:5Þ Substituting Eqs (A.2–A.4) to Eq (A.1) the virtual work of internal forces becomes Z dW int ¼ D ½ðw;xx þ mw;yy Þdw;xx þ ðw;yy þ mw;xx Þdw;yy AðpÞ Z þ 2ð1 À mÞw;xy dw;xy dA þ ½k0 wdw þ k1 ðw;x dw;x þ w;y dw;y ÞdA À Et aD T 1Àm Af Z AðpÞ ½w;x dw;x þ w;y dw;y dA ðA:6Þ Thermal loading corresponds to the problem of biaxial compression In the case of a simply supported rectangular plate, the displacement function can be taken in the form of one term of the double Fourier series w ¼ wmn sin mpx npy sin a b ðA:7Þ where m and n are integer numbers defining the buckling mode Substituting the assumed displacement function and integrating we arrive at ( dW int ¼ mp2 np2 !2 mp2 np2 ! þ þ k0 þ k1 þ a b a b !' 2 Et mp np aDT þ abwmn dwmn À 1Àm a b D À Á ð5n3 À 5n2 À n þ 1Þ Á 5g3 À 5g2 À g þ 64 N2 ¼ À N3 ¼ dex ¼ Àzdw;xx þ w;x dw;x ETx N1 ¼ À Á ð5n3 þ 5n2 À n À 1Þ Á 5g3 À 5g2 À g þ 64 À Á ð5n3 þ 5n2 À n À 1Þ Á 5g3 þ 5g2 À g À 64 DT ¼ hÀ Á mp a þ hÀ Á À Á2 i þ k0 þ k1 map þ nbp h À Á i Et a ðmapÞ2 þ nbp 1Àm ðB:4Þ N5 ¼ À À pffiffiffi Á pffiffiffi 5n À n2 À 5n þ Á 5g3 À 5g2 À g þ 64 ðB:5Þ N6 ¼ À pffiffiffi Á pffiffiffi 5n þ n2 À 5n À Á 5g3 À 5g2 À g þ 64 ðB:6Þ N7 ¼ pffiffiffi pffiffiffi ð5n3 þ 5n2 À n À 1Þ Á 5g3 À g2 À 5g þ 64 ðB:7Þ N8 ¼ À pffiffiffi pffiffiffi 5g3 þ g2 À 5g À ð5n3 þ 5n2 À n À 1Þ Á 64 ðB:8Þ N9 ¼ À pffiffiffi pffiffiffi 5n þ n2 À 5n À Á ð5g3 þ 5g2 À g À 1Þ 64 ðB:9Þ N10 ¼ pffiffiffi pffiffiffi 5n À n2 À 5n þ Á ð5g3 þ 5g2 À g À 1Þ 64 ðB:10Þ N11 ¼ pffiffiffi pffiffiffi ð5n3 À 5n2 À n þ 1Þ Á 5g3 þ g2 À 5g À 64 ðB:11Þ N12 ¼ À pffiffiffi pffiffiffi ð5n3 À 5n2 À n þ 1Þ Á 5g3 À g2 À 5g þ 64 ðA:8Þ ðA:9Þ Numbers of halfwaves m and n should be adjusted to yield the minimum temperature For square plate a = b and for m = n = we get finally DT ¼ À Á4 À Á2 4D pb þ k0 þ 2k1 pb : ÀpÁ2 Et 1À ma b ðA:10Þ Appendix B Shape functions of the 16-node finite element with Lobatto integration scheme Shape functions of the 16-node plate finite element are as follows (for details we refer the readers to reference [15]) (see Fig B.1): ðB:3Þ À Á ð5n3 À 5n2 À n þ 1Þ Á 5g3 þ 5g2 À g À 64 ÀnpÁ2 i2 b ðB:2Þ N4 ¼ À Hence the temperature increment resulting in buckling is D ðB:1Þ pffiffiffi Fig B.1 16-node finite element of the Lagrange family c ¼ 1= ðB:12Þ 92 M Taczała et al / Composite Structures 137 (2016) 85–92 N13 ¼ pffiffiffi pffiffiffi pffiffiffi 25 pffiffiffi 5n À n2 À 5n þ Á 5g3 À g2 À 5g þ 64 ðB:13Þ N14 ¼ À N15 ¼ pffiffiffi pffiffiffi pffiffiffi 25 pffiffiffi 5n þ n2 À 5n À Á 5g3 À g2 À 5g þ 64 ðB:14Þ pffiffiffi pffiffiffi pffiffiffi 25 pffiffiffi 5n þ n2 À 5n À Á 5g3 þ g2 À 5g À 64 ðB:15Þ N16 ¼ À pffiffiffi pffiffiffi pffiffiffi 25 pffiffiffi 5n À n2 À 5n þ Á 5g3 þ g2 À 5g À 64 ðB:16Þ Displacements are approximated according to u ¼ Ni ui ; 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