Buckling of Laminated and Functionally Graded Plates using Radial Basis Functions

Paper 223 Buckling of Laminated and Functionally Graded Plates using Radial Basis Functions A.M.A. Neves, A.J.M. Ferreira, E. Carrera 3, M. Cinefra 3 C.M.C. Roque 2, C.M. Mota Soares 4 and R.M.N. Jorge Departamento de Engenaria Mecânica, 2 INEGI Faculdade de Engenaria, Universidade do Porto, Portugal 3 Departament of Aeronautics and Aerospace Engineering Politecnico di Torino, Italy 4 Instituto Superior Técnico, Lisbon, Portugal Civil-Comp Press, 22 Proceedings of te Elevent International Conference on Computational Structures Tecnology, B.H.V. Topping, (Editor, Civil-Comp Press, Stirlingsire, Scotland Abstract A yperbolic sine sear deformation teory is used for te buckling analysis of functionally graded plates, accounting for troug-te-tickness deformations. Te linearized buckling equations and boundary conditions are derived using Carrera s Unified Formulation and furter interpolated by collocation wit radial basis functions. Numerical results demonstrate te accuracy of te present approac. Keywords: plates and sells, collocation, mesless metods, functionally graded plates. Introduction A conventional functionally graded plate (FGM considers a continuous variation of material properties over te tickness direction by mixing two different materials []. Te material properties of te FGM plate are assumed to cange continuously trougout te tickness of te plate, according to te volume fraction of te constituent materials. Analysis of vibrations of FGM plates can be found in Batra and Jin [2], Ferreira et al. [3], Vel and Batra [4], Zenkour [5], and Ceng and Batra [6]. Te analysis of mecanical buckling of FGM structures is less common in te literature. It can be found in Najafizade and Eslami [7], Zenkour [5], Ceng and Batra [6], Birman [8], Javaeri and Eslami [9]. Typically, te analysis of FGM plates is performed using te clasical plate teory (CPT [, ], te first-order sear deformation teory (FSDT [2, 2, 3, 3] or iger-order sear deformation teories (HSDT [4, 5, 3, 6, 3]. Te FSDT gives acceptable results but depends on te sear correction factor wic is ard to find as it depends on many parameters. Tere is no need of a sear correction factor wen using a HSDT but linearized buckling equations are more complicated tan

tose of te FSDT. Te Unified Formulation proposed by Carrera [7, 8] made te implementation of suc teories easier. Te use of alternative-to-finite element metods for te analysis of plates, suc as te mesless metods based on collocation wit radial basis functions is atractive due to te absence of a mes and te ease of collocation metods. In recent years, radial basis functions (RBFs sowed excellent accuracy in te interpolation of data and functions. Kansa [9] introduced te concept of solving partial differential equations by an unsymmetric RBF collocation metod based upon te multiquadric interpolation functions. Te autors ave applied successfully te RBF collocation tecnique to te static and dynamic analysis of composite structures [2, 2, 22, 23, 24, 25, 26, 27, 28, 29]. Te present paper considers te tickness stretcing issue on te buckling analysis of FGM plates, by a mesless tecnique based on collocation wit radial basis functions. 2 Problem formulation Te problem consists of a rectangular sandwic plate of plan-form dimensions a and b and uniform tickness. Te co-ordinate system is taken suc tat te x-y plane coincides wit te midplane of te plate (z [, /2]. Te sandwic core is a ceramic material and skins are composed of a functionally graded material across te tickness direction. Te bottom skin varies from a metal-ric surface (z = = to a ceramic-ric surface wile te top skin face varies from a ceramic-ric surface to a metal-ric surface (z = 3 = /2 as illustrated in figure. Te volume fraction of te ceramic pase is obtained from a simple rule of mixtures as: ( p z V c =, z [, ] V c =, z [, 2 ] ( ( p z 3 V c =, z [ 2, 3 ] 2 3 were p is a scalar parameter tat allows te user to define te gradation of material properties across te tickness direction of te skins. Wit tis formulation te interfaces between core and skins disappear. Note tat te core of te present sandwic and any isotropic material can be obtained as a particular case of te power-law function by setting p =. Te volume fraction for te metal pase is given as V m = V c. Te sandwic may be symmetric or non-symmetric about te mid-plane as we may vary te tickness of eac face. Figure 2 sows a non-symmetric sandwic wit volume fraction defined by te power-law ( for various exponents p, in wic top skin tickness is te same as te core tickness and te bottom skin tickness is twice te core tickness. Suc tickness relation is denoted as 2--. A bottom-core-top notation is used. -- means tat skins and core ave te same tickness. Te sandwic plate is subjected to compressive in-plane forces acting on te midplane of te plate. Nxx and N denote te in-plane loads perpendicular to te edges 2

3 = /2 z METAL 2 CERAMIC CERAMIC CERAMIC = -/2 METAL Figure : Sandwic wit isotropic core and FGM skins. /2 5 2.2.2 z coordinate 5 2.2.2 V c Figure 2: Illustration of a 2-- sandwic wit FGM skins for several volume fractions. 3

y N N xx x N N yx Figure 3: Rectangular plate subjected to in-plane forces. x = and y = respectively, and N denotes te distributed sear force parallel to te edges x = and y = respectively (see fig. 3. 3 A quasi-3d yperbolic sine plate sear deformation teory 3. Displacement field Te present teory is based on te following displacement field: u(x,y,z,t = u (x,y,t + zu (x,y,t + sin v(x,y,z,t = v (x,y,t + zv (x,y,t + sin u Z (x,y,t (2 v Z (x,y,t (3 w(x,y,z,t = w (x,y,t + zw (x,y,t + z 2 w 2 (x,y,t (4 were u, v, and w are te displacements in te x, y, and z directions, respectively. u, u, u Z, v, v, v Z, w, w, and w 2 are te unknowns to be determined. A constant term is assumed for te transverse displacement component instead of (4 (w = w to investigate te effect of te tickness stretcing on te buckling load. 4

3.2 Strains Te strains can be related to te displacement field as: ǫ xx ǫ γ = u + ( w 2 x 2( x 2 v + w 2 u + v + w x x w, γ γ ǫ = u + w z x v + w z w z (5 By substitution of te displacement field in (5, te strains are obtained: ǫ xx ǫ γ = ǫ ( xx ǫ ( γ ( + ǫ (nl xx ǫ (nl γ (nl + z ǫ ( xx ǫ ( γ ( + sin ǫ (Z xx ǫ (Z γ (Z (6 γ γ ǫ = γ ( γ ( ǫ ( +z γ ( γ ( ǫ ( +z2 γ (2 γ (2 ǫ (2 +π cos γ (Z γ (Z ǫ (Z (7 being te strain components obtained as ǫ ( xx ǫ ( γ ( ǫ ( xx ǫ ( γ ( = = γ ( γ ( ǫ ( = γ (2 γ (2 ǫ (2 u x v u + v x u x v u + v x = ; ; u + w x v + w w w 2 x w 2 ; ; ǫ (nl xx ǫ (nl γ (nl ǫ (Z xx ǫ (Z γ (Z = = γ ( γ ( ǫ ( γ (Z γ (Z ǫ (Z = = ( w 2 2( x 2 w 2 w w x u Z x v Z u Z + v Z x u Z v Z w x w 2w 2 (8 (9 ( ( are te non-linear terms tat will originate te linearized buckling equa- were ǫ (nl αβ tions. 5

3.3 Elastic stress-strain relations In te case of isotropic functionally graded materials, te 3D constitutive equations can be written as: σ xx σ τ τ τ σ = = C C 2 C 2 C C 44 C 2 C 2 ǫ xx ǫ γ ǫ xx ǫ γ + + C 2 C 2 C 44 C 44 C 33 γ γ ǫ γ γ ǫ (2 Te computation of te elastic constants C ij depends on wic assumption of ǫ we consider. If ǫ =, ten C ij are te plane-stress reduced elastic constants: C = E ν 2; C E 2 = ν ν 2; C 44 = G; C 33 = (3 were E is te modulus of elasticity, ν is te Poisson s ratio, and G is te sear modulus G = E. 2(+ν If ǫ (tickness stretcing, ten te elastic coefficients C ij are tose of te tree-dimensional stress state, given by C = E( ν2 3ν 2 2ν 3, C 2 = E(ν + ν2 3ν 2 2ν 3 (4 C 44 = G, C 33 = E( ν2 3ν 2 2ν 3 (5 3.4 Governing equations and boundary conditions Te governing equations of present teory are derived from te dynamic version of te Principle of Virtual Displacements. Te internal virtual work is δu = Ω { /2 [σ xx ( δǫ ( xx + zδǫ ( xx + sin + σ ( δγ ( + zδγ ( + sin + σ ( δγ ( + zδγ ( + z 2 δγ (2 ( δǫ xx (Z + σ δǫ ( + zδǫ ( + sin ( δγ (Z + σ + π cos δγ ( + zδγ ( + z 2 δγ (2 ( + σ δγ (Z δǫ ( + zδǫ ( δǫ (Z + π cos ] } dz dx dy (6 δγ (Z 6

δu = ( Nxx δǫ ( xx + M xx δǫ ( xx + Rxxδǫ Z xx (Z + N δǫ ( + M δǫ ( + Rδǫ Z (Z Ω + N δγ ( + M δγ ( + Rδγ Z (Z + Q δγ ( + M δγ ( + Rδγ 2 (2 + Q δγ ( + M δγ ( + Rδγ 2 (2 + Rδγ Z (Z + Q δǫ ( + M δǫ ( were Ω is te integration domain on plane (x,y and N xx N N = /2 σ xx σ σ dz, Q Q Q = /2 σ σ σ + Rδγ Z (Z dx dy (7 dz (8 M xx M M = /2 z σ xx σ σ dz, M M M = /2 z σ σ σ dz (9 R Z xx R Z R Z = Ω /2 sin { R 2 R 2 σ xx σ σ } = dz, /2 { R Z R Z z 2 { σ σ } = /2 π { cos σ σ (2 } dz. (2 Te external virtual work due to external loads applied to te plate is given as: δv = (p x δu + p y δv + p z δwdx dy Ω ( ( = (p x δu + zδu + sin δu Z + p y δv + zδv + sin δv Z + p z ( δw + zδw + z 2 δw 2 dx dy Te external virtual work due to in-plane forces and sear forces applied to te plate is given as: [ δv = Ω N xx w x δ w x + N w δ w x + N w δ w yx x + N w (22 } dz ] δ w dx dy (23 7

being N xx and N te in-plane loads perpendicular to te edges x = and y = respectively, and N and N yx te distributed sear forces parallel to te edges x = and y = respectively. Te virtual kinetic energy is given as: δk = Ω = /2 ρ ( uδ u + vδ v + ẇδẇdz dx dy [ I ( u δ u + v δ v + ẇ δẇ Ω + I ( u δ u + u δ u + v δ v + v δ v + ẇ δẇ + ẇ δẇ + I 2 ( u δ u + v δ v + ẇ δẇ 2 + ẇ δẇ + ẇ 2 δẇ (24 + I 3 (ẇ δẇ 2 + ẇ 2 δẇ + I 4 ẇ 2 δẇ 2 + I 5 ( u δ u Z + u Z δ u + v δ v Z + v Z δ v + I 6 ( u Z δ u Z + v Z δ v Z + I 7 ( u δ u Z + u Z δ u + v Z δ v + v δ v Z ] dx dy were te dots denote te derivative wit respect to time t and te inertia terms are defined as I i = /2 ρz i dz i =,, 2, 3, 4 (25 I 5 = /2 ρ sin dz; I 6 = /2 ρ sin 2 dz; I 7 = /2 ρz sin dz (26 Substituting δu, δv, and δk in te virtual work statement, integrating by parts wit respect to x, y, and t and collecting te coefficients of δu, δu, δu Z, δv, δv, δv Z, 8

δw, δw, δw 2, te following governing equations are obtained: δu : N xx x N = δv : N x N = /2 /2 { ( } ρ ü + zü + sin ü Z + p x dz { ( } ρ v + z v + sin v Z + p y dz δw : Q x Q + N xx 2 w x 2 + N 2 w x + N yx 2 w x + N 2 w 2 = /2 δu : M xx x M δv : M x M δw : M x M { ρ (ẅ + zẅ + z 2 ẅ 2 + pz } dz + Q = + Q = + Q = δu Z : RZ xx x RZ + RZ = δv Z : RZ x RZ + RZ = δw 2 : R2 x R2 + 2M = /2 /2 /2 /2 /2 /2 { ( } ρz ü + zü + sin ü Z + zp x dz { ( } ρz v + z v + sin v Z + zp y dz { ρz (ẅ + zẅ + z 2 ẅ 2 + zpz } dz { ρ sin { ρ sin ( ü + zü + sin ( v + z v + sin { ρz 2 ( ẅ + zẅ + z 2 ẅ 2 + z 2 p z } dz ü Z + sin v Z + sin (27 } p x dz } p y dz Te mecanical boundary conditions are: 9

grid 3 2 7 2 2 2 P 4.52 4.57 4.565 Table : Convergence study for te uni-axial buckling load of a simply supported 2-2- sandwic square plate wit FGM skins and p = 5 case using te iger-order teory. δu : n x N xx + n y N = n x Nxx + n y N δv : n x N + n y N = n x N + n y N δw : n x Q + n y Q = n x Q + n y Q δu : n x M xx + n y M = n x Mxx + n y M δv : n x M + n y M = n x M + n y M δw : n x M + n y M = n x M + n y M (28 δu Z : n x Rxx Z + n y R Z = n x RZ xx + n y RZ δv Z : n x R Z + n y R Z = n x RZ + n y RZ δw 2 : n x R 2 + n y R 2 = n x R2 + n y R2 were (n x,n y denotes te unit normal-to-boundary vector. 4 Numerical examples Te iger-order plate teory and collocation wit RBFs are considered for te uniaxial and bi-axial buckling analysis of simply supported functionally graded sandwic square plates (a = b of type C wit side-to-tickness ratio a/ =. Te material properties considered are E m = 7E (aluminum for te metal and E c = 38E (alumina for te ceramic being E = GPa. Te non-dimensional parameter used is P = Pa2 3 E. An initial convergence study wit te iger-order teory was conducted for eac buckling load type using 3 2, 7 2, and 2 2 grids. In table te uniaxial case is sown for te 2-2- sandwic wit p = 5 and in table 2 te bi-axial case is presented for te -2- sandwic wit p =. In te following, results are obtained by considering a grid of 7 2 points, wic seems acceptable by te convergence study. In tables 3 and 4 te critical buckling loads obtained from te present approac wit ǫ and ǫ = are tabulated for various power-law exponents p and tickness ratios. Bot tables include results obtained from classical plate teory (CLPT, firstorder sear deformation plate teory (FSDPT, K = 5/6 as sear correction factor, Reddy s iger-order sear deformation plate teory (TSDPT [6], and Zenkour s

grid 3 2 7 2 2 2 P 3.6628 3.65998 3.65994 Table 2: Convergence study for te bi-axial buckling load of a simply supported -2- sandwic square plate wit FGM skins and p = case using te iger-order teory. sinusoidal sear deformation plate teory (SSDPT [5]. Table 3 refers to te uni-axial buckling load and table 4 refers to te bi-axial buckling load. A good agreement between te present solution and references considered, specially [6] and [5] is obtained. Tis allow us to conclude tat te present iger-order plate teory is good for te modeling of simply supported sandwic FGM plates and tat collocation wit RBFs is a good formulation. Present results wit ǫ = approximates better references [6] and [5] tan ǫ as te autors use te ǫ = approac. Tis study also lead us to conclude tat te tickness stretcing effect as a strong influence on te buckling analysis of sandwic FGM plates as ǫ = gives iger fundamental buckling loads tan ǫ. Te isotropic fully ceramic plate (first line on tables 3 and 4 as te iger fundamental buckling loads. As te core tickness to te total tickness of te plate ratio (( 2 / increases te buckling loads increase as well. We may conclude tat te critical buckling loads decrease as te power-law exponent p increases. By comparing tables 3 and 4 we also conclude tat te bi-axial buckling load of simply supported sandwic square plate wit FGM skins is alf te uni-axial one for te same plate. In figure 4 te first four buckling modes of a simply supported 2--2 sandwic square plate wit FGM skins, p =.5, subjected to a uni-axial in-plane compressive load, using te iger-order plate teory and 7 2 grid is presented. Figure 5 presents te first four buckling modes of a simply supported 2-- sandwic square plate wit FGM skins, p =, subjected to a bi-axial in-plane compressive load. 5 Conclusions An application of a Unified formulation by a mesless discretization is proposed, based on a tickness-stretcing yperbolic sine sear deformation teory tat was implemented for te buckling analysis of functionally graded sandwic plates. Te present formulation was compared wit analytical, mesless or finite element metods and sowed very accurate results. Te effect of ǫ sowed to be significant in suc sandwic problems. Acknowledgments Ana M. A. Neves acknowledges support from FCT grant SFRH/BD/45554/28.

P p Teory -- 2--2 2-- -- 2-2- -2- CLPT 3.7379 3.7379 3.7379 3.7379 3.7379 3.7379 FSDPT 3.449 3.449 3.449 3.449 3.449 3.449 TSDPT [6] 3.495 3.495 3.495 3.495 3.495 3.495 SSDPT [5] 3.66 3.66 3.66 3.66 3.66 3.66 present ǫ 2.95287 2.95287 2.95287 2.95287 2.95287 2.95287 present ǫ = 3.58 3.58 3.58 3.58 3.58 3.58.5 CLPT 7.65398 8.25597 8.56223 8.7863 9.8254 9.6525 FSDPT 7.33732 7.932 8.25 8.434 8.78673 9.957 TSDPT [6] 7.36437 7.9484 8.2247 8.43645 8.8997 9.268 SSDPT [5] 7.36568 7.9495 8.22538 8.4372 8.837 9.267 present ǫ 7.627 7.7627 7.98956 8.9278 8.5572 8.949 present ǫ = 7.8728 7.74326 8.7 8.2233 8.5829 8.973 CLPT 5.33248 6.2733 6.439 6.685 7.9663 7.7846 FSDPT 5.4236 5.8379 6.72 6.43892 6.9257 7.48365 TSDPT [6] 5.673 5.846 6.9394 6.46474 6.94944 7.5656 SSDPT [5] 5.6846 5.849 6.946 6.46539 6.9498 7.5629 present ǫ 5.637 5.735 6.5467 6.35 6.7845 7.3995 present ǫ = 5.7848 5.7322 6.7358 6.33556 6.8547 7.34367 5 CLPT 2.738 3.74 3.4848 3.65732 4.2238 4.8577 FSDPT 2.63842 3.2252 3.38538 3.55958 4.9285 4.7475 TSDPT [6] 2.6582 3.4257 3.435 3.57956 4.29 4.73469 SSDPT [5] 2.666 3.446 3.4449 3.5863 4.288 4.73488 present ǫ 2.63652 3.79 3.36255 3.535 4.57 4.647 present ǫ = 2.6468 3.865 3.3796 3.5448 4.663 4.6659 CLPT 2.56985 2.834 3.6427 3.25924 3.79238 4.3822 FSDPT 2.4694 2.72626 3.7428 3.752 3.6889 4.264 TSDPT [6] 2.48727 2.74632 3.99 3.947 3.7752 4.2799 SSDPT [5] 2.48928 2.74844 3.3443 3.9456 3.4574 4.3875 present ǫ 2.4726 2.7246 3.667 3.576 3.6666 4.255 present ǫ = 2.4829 2.738 3.6943 3.6837 3.6753 4.2792 Table 3: Uni-axial buckling load of simply supported plate of C-type using te igerorder teory and a grid wit 7 2 points. 2

P p Teory -- 2--2 2-- -- 2-2- -2- CLPT 6.86896 6.86896 6.86896 6.86896 6.86896 6.86896 FSDPT 6.5224 6.5224 6.5224 6.5224 6.5224 6.5224 TSDPT [6] 6.5248 6.5248 6.5248 6.5248 6.5248 6.5248 SSDPT [5] 6.533 6.533 6.533 6.533 6.533 6.533 present ǫ 6.47643 6.47643 6.47643 6.47643 6.47643 6.47643 present ǫ = 6.5254 6.5254 6.5254 6.5254 6.5254 6.5254.5 CLPT 3.82699 4.2798 4.282 4.3932 4.5927 4.8762 FSDPT 3.66866 3.9566 4.7 4.257 4.39336 4.59758 TSDPT [6] 3.6829 3.9742 4.235 4.2823 4.4499 4.684 SSDPT [5] 3.68284 3.9797 4.269 4.2856 4.459 4.6835 present ǫ 3.584 3.8583 3.99478 4.9639 4.27586 4.4795 present ǫ = 3.59364 3.8763 4.85 4.67 4.2964 4.48655 CLPT 2.66624 3.366 3.295 3.3475 3.5983 3.8923 FSDPT 2.578 2.969 3.85 3.2946 3.46286 3.7482 TSDPT [6] 2.58357 2.923 3.9697 3.23237 3.47472 3.75328 SSDPT [5] 2.58423 2.926 3.973 3.2327 3.4749 3.7534 present ǫ 2.5369 2.85568 3.2733 3.575 3.3922 3.65998 present ǫ = 2.53924 2.865 3.3679 3.6778 3.4274 3.6783 5 CLPT.3654.55352.7429.82866 2.69 2.42859 FSDPT.392.526.69269.77979 2.4642 2.35737 TSDPT [6].329.5229.776.78978 2.565 2.36734 SSDPT [5].333.5223.7224.7932 2.5644 2.36744 present ǫ.3826.5395.6828.7652 2.2535 2.3235 present ǫ =.3234.5933.68598.7774 2.38 2.3329 CLPT.28493.47.5824.62962.8969 2.9 FSDPT.23452.3633.5374.5876.84445 2.32 TSDPT [6].24363.3736.54595.59736.85376 2.3995 SSDPT [5].24475.37422.5672.59728.57287 2.987 present ǫ.2368.3623.5334.5788.8383 2.275 present ǫ =.249.3654.53472.5849.83576 2.896 Table 4: Bi-axial buckling load of simply supported plate of C-type using te igerorder teory and a grid wit 7 2 points. 3

.5 eig = 7.76265245435.5.5 eig = 8.47436848.5.5 eig =.3592283955.5.5 eig = 26.6565266924.5 Figure 4: First four buckling modes. Uni-axial buckling load of a simply supported 2--2 plate C-type, p =.5, a 7 2 points grid, and using te iger-order teory..5 eig =.5333693836.5 eig = 3.67827255222.5.5.5 eig = 3.67826949879.5.5 eig = 5.6663489369.5 Figure 5: First four buckling modes. Bi-axial buckling load of a simply supported 2-- plate C-type, p =, a 7 2 points grid, and using te iger-order teory. 4

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