Natural frequencies of functionally graded plates by a meshless method

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1 Composite Structures 75 (6) Natural frequencies of functionall graded plates b a meshless method A.J.M. Ferreira a, *, R.C. Batra b, C.M.C. Roque a, L.F. Qian c, R.M.N. Jorge a a Departamento de Engenharia Mecânica e Gestão Industrial, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, Porto, Portugal b Department of Engineering Science and Mechanics, MC, Virginia Poltechnic Institute and State Universit, Blacksburg, VA 46, USA c Department of Mechanical Engineering, Nanjing Universit of Science and Technolog, Nanjing, China Available online 5 June 6 Abstract We use the global collocation method, the first and the third-order shear deformation plate theories, the Mori Tanaka technique to homogenize material properties, and apprimate the trial solution with multiquadric radial basis functions to anale free vibrations of functionall graded plates. Frequencies computed b the present method are found to agree well with those from the analtical solution of Vel and Batra, and the numerical solution of Qian et al. based on the meshless local Petrov Galerkin formulation. Ó 6 Elsevier Ltd. All rights reserved. Kewords: Collocation method Multiquadrics Natural frequencies Functionall graded plates. Introduction Qian et al. [,] recentl empled the meshless local Petrov Galerkin method (MLPG) to anale free and forced vibrations of both homogeneous and functionall graded (FG) thick plates with the higher-order shear and normal deformable plate theor (HOSNDPT) of Batra and Vidoli [3]. Computed frequencies for a simpl supported FG plate were found to match well with those obtained from the analtical solution of the three-dimensional (3D) elasticit equations of Vel and Batra [4]. For a simpl supported plate, Vel and Batra [4] used the classical plate theor, the first-order shear deformation (FSDT), and the third-order shear deformation (TSDT) apprimations [5] for the displacement fields, assumed each displacement component to var sinusoidall in the x- and the -directions, and derived an algebraic equation for the frequencies. The assumed forms of displacements satisf boundar conditions at the plate edges onl when the are simpl supported. * Corresponding author. Tel.: fax: address: ferreira@fe.up.pt (A.J.M. Ferreira). Here we use the asmmetric collocation method with multiquadrics basis functions, and the FSDT and the TSDT to find natural frequencies of square FG plates of various aspect ratios, and under different boundar conditions at the edges. This method has been empled earlier b Ferreira et al. [6] to stud static deformations of FG plates. An advantage of this method over the finite element method (FEM) is that the discretization of the domain into D or 3D elements, and the element connectivit are not needed. The present method, like the MLPG method empled b Qian et al. [,], requires onl coordinates of nodes on the midsurface of the plate. Thus the input required for the present meshless method, and the effort required to prepare the input are considerabl less than that needed for the FEM. Details of the collocation method with multiquadrics and its application to the analsis of plate problems are given in Refs. [6 3].. The finite point multiquadric method Consider the following linear elliptic boundar-value problem defined on a smooth domain X: 63-3/$ - see front matter Ó 6 Elsevier Ltd. All rights reserved. doi:.6/j.compstruct.6.4.

2 54 A.J.M. Ferreira et al. / Composite Structures 75 (6) 53 6 LuðxÞ sðxþ x X ðþ BuðxÞ f ðxþ x ox where ox is the boundar of X, L and B are linear differential operators, and s and f are smooth functions defined on X and ox respectivel. We select N B points (x (j), j =,...,N B )onoxand (N N B ) points (x (j), j = N B +, N B +,...,N) in the interior of X. Let u h ðxþ XN a j gðjjx x ðjþ jj cþ XN a j g j ðxþ ðþ j be an apprimate solution of the boundar-value problem where a,a,...,a N are constants to be determined, kx x (j) k is the Euclidean distance between points x and x (j), c is a constant, and g is a function of kx x (j) k and c. Different forms of functions g and names associated with them are Multiquadrics: g j ðxþ ðjjx x ðjþ jj þ c Þ = Inverse multiquadrics: g j ðxþ ðjjx x ðjþ jj þ c Þ = Gaussian: g j ðxþ e c jjx x ðjþ jj Thin plate splines: g j ðxþ jjx x ðjþ jj log jjx x ðjþ jj: ð3þ Substitution from () into () and evaluating equations resulting from () at the N B points x (j), j =,,...,N B, and from () at (N N B ) points x (j), j = N B +, N B +,...,N give the following N algebraic equations for the determination of a,a,...,a N. X N j X N j j a j Lgðjjx x ðjþ jj cþj xx ðiþ sðx ðiþ Þ i N B þ N B þ... N a j Bgðjjx x ðjþ jj cþj xx ðiþ f ðx ðiþ Þ i... N B ð4þ Depending upon the value of the parameter c and the form of function g, the set of Eq. (4) that determines a,a,...,a N ma become ill-conditioned e.g. see [4]. Also, the computational effort involved in solving (4) for a,a,...,a N varies with the choice of the function g. Once Eqs. (4) have been solved for a s, then the apprimate solution of the problem is given b (). 3. Review of the third-order shear deformation plate theor A schematic sketch of the problem studied, dimensions of the FG plate, and the location of the rectangular Cartesian coordinate axes used to describe deformations of the plate are given in Fig.. The displacement field in the TSDT is given b uðx zþ u ðx Þþz c z 3 þ ow h vðx zþ v ðx Þþz/ c z 3 / þ ow wðx zþ w ðx Þ where c = 4/(3h ), h is the plate thickness, z is the coordinate in the thickness direction, the -plane of the rectangular Cartesian coordinate sstem is located in the midplane of the plate. Functions and / describe, respectivel, rotations about the - and the x-axes of a line that is along the normal to the midsurface of the plate u, v and w give, respectivel, displacements of a point on the midsurface of the plate along the x-, - and z-axes. The constant c is determined b requiring that the transverse shear strain vanishes on the top and the bottom surfaces of the plate. Note that the transverse normal strain identicall vanishes in the TSDT. Batra and Vidoli [3] have proposed a mixed HOSNDPT in which natural boundar conditions prescribed on the top and the bottom surfaces of the plate are exactl satisfied. The displacement field for the FSDT can be obtained from (5) b setting c =. From the strain displacement relations appropriate for infinitesimal deformations, we obtain e e e e e where e ðþ e ðþ O e ðþ e ðþ e ðþ z e ðþ e ðþ e ðþ e ðþ e ðþ þ z ou ov ou þ ov >= a ow þ / ow þ e ðþ e ðþ þ z 3 Fig.. Plate geometr. e ðþ e ðþ x ð5þ >= ð6þ b

3 e ðþ e ðþ e ðþ e ðþ o o/ o þ o/ >= o þ o w o/ þ o w o/ c x þ o/ >= þ o w : ð7þ 3 ow z þ / 3 z ow þ Equations for the plate theor, derived b using the principle of virtual work, are on þ on I u ot þ J ot c ow I 3 ðþ ot on þ on v ot þ J / I ow ot oq x þ oq þ c P þ o P þ o P þ q w I ot c w I 6 ot þ o w ou þ c I 3 ot þ ov o þ o/ J x 4 ot þ o/ om þ om Q x o ow J ot u þ K c J 4 om þ om ot c I 3 Q o ow J ot v þ K / c J 4 ðþ ðþ ðþ ðþ where q is the sum of the distributed normal tractions applied on the top and the bottom surfaces of the plate, and M ab M ab c P ab Q a Q a c R a ð3þ Z I i XN kþ q ðkþ ðzþ i dz ði... 6Þ k k J i I i c I iþ ði 4Þ ð4þ K I c I 4 þ c I 6 c 4 3h c 4 h 3c ð5þ where a, b take the values x,. Furthermore (N,N,N ) denote the in-plane force resultants, (M,M,M ) the moment resultants, (Q x,q ) the shear resultants, and (P,P,P ) and (R x,r ) the higher-order stress resultants. These are defined b A.J.M. Ferreira et al. / Composite Structures 75 (6) N ab M ab P ab Q a R a >= Z h= Z h= h= h= r ab z r az z z 3 >= dz: dz ð6þ ð7þ The FSDT equations are readil obtained from those of the TSDT equations b setting c =. Expressions for M ab, N ab, P ab, Q a and R a in terms of strains can be derived b substituting into (6) and (7) from the following stress strain relation for an isotropic material: 3 r Q Q e r >= Q Q e >= r Q 33 e 6 7 r 4 kq 33 5 e r kq 33 e ðþ where Q E=ð m Þ Q me=ð m Þ Q 33 E=ð þ mþ ðþ E is the effective Young s modulus, and m the effective Poisson s ratio at a point in a FG plate. The shear correction factor, k, is taken as 5/6 for the FSDT and. for the TSDT. Substitution for strains in terms of displacements from (5) into (6), for stresses from () into (6), for M ab, N ab, etc. from (6) into () () ield equations of motion in terms of the generalized displacements u, v, w, and / these are summarized in Appendix I. An apprimate solution of these equations and the pertinent boundar conditions is found b using the meshless method described in Section. That is, we assume that u h ðxþ XN a u j gðjjx xðjþ jj cþ etc: ðþ j These expressions are substituted in equations of motion listed in Appendix I, and also in relevant boundar conditions. Boundar conditions at a simpl supported edge, x = a, are w ða Þ v ða Þ / ða Þ N ða Þ M ða Þ : ðþ Boundar conditions imposed at a rigidl clamped edge, = b, are u ðx bþ v ðx bþ w ðx bþ ðx bþ / ðx bþ : ðþ Boundar conditions imposed at a free edge, x = a, are Q x ða Þ N ða Þ N ða Þ M ða Þ M ða Þ : ð3þ

4 56 A.J.M. Ferreira et al. / Composite Structures 75 (6) Homogenization of material properties We assume that the plate is made of two randoml distributed isotropic constituents, the macroscopic response of the composite is isotropic, and its composition varies onl in the z-direction. Qian and Batra [5] have studied free vibrations of a FG plate with material properties varing smoothl in two directions. The volume fraction of constituent is assumed to be given b V V þðv þ V Þ þ z p : h Thus V V at the bottom surface z = h/, and V V þ at the top surface z = h/ of the plate. Fig. depicts the through-the-thickness distribution of the volume fraction of phase for different values of p. Here we empl the Mori Tanaka [6] homogenization method to find the effective bulk modulus, K, and the effective shear modulus, G, of the composite from Z Volume fraction function Table Fundamental frequenc of a simpl supported square thick Al/ZrO FG plate, TSDT, V c V þ c p h/a =.5 h/a =. h/a =. Present Ref. [] f =. f =. f =.5 f =. f =. f = 3 Fig.. Through-the-thickness distribution of volume fraction. Exact Present Ref. [] Exact Present Ref. [] Exact Table Fundamental frequenc of a simpl supported square thick Al/ZrO FG plate, TSDT, V c, V þ c, h/a =. p = p =3 p =5 Present Ref. [] Exact Present Ref. [] Exact Present Ref. [] Exact Table 3 First natural frequencies of a simpl supported square thick Al/ZrO FG plate, TSDT, V c, V þ c, h/a =. Ceramic p = p = p = 5 Metal N =7 N = N = Ref. [] N =7 N = N = Ref. [] N =7 N = N = Ref. [] N =7 N = N = Ref. [] N =7 N = N = Ref. []

5 A.J.M. Ferreira et al. / Composite Structures 75 (6) K K K K G G G G V þð V Þ K K K þ 4 3 G V þð V Þ G G G þf ð4þ where f G ðk þg Þ 6ðK þg. The effective values of Young s Þ modulus, E, and Poisson s ratio, m, are found from E KG 3K þ G 5. Results 3K G m ð3k þ GÞ : ð5þ We compute results for a FG plate comprised of aluminum and zirconia mainl because analtical results for a plate made of these materials are available for comparison [4]. Material properties of the aluminum (Al) and zirconia (ZrO ) are Al: E m 7 GPa m m :3 q m 7 kg=m 3 ZrO : E z GPa m z :3 q z 57 kg=m 3 : We assume that the volume fraction of the ceramic phase is given b (4) and henceforth replace subscripts and b c and m respectivel. Natural frequencies are non-dimensionalized b rffiffiffiffiffiffi q x xh m : E m We consider simpl supported (SSSS), clamped (CCCC), simpl supported/clamped (SCSC) and clamped/free (CFCF) boundar conditions. Notation SCSC, for example, indicates that edges x = and x = a are simpl supported (S), and edges = and = b are clamped (C). When possible, we compare present results with existing ones. We use multiquadric functions (3) with c =6d, d being the distance between two consecutive nodes. For a square plate, we use nodes equall spaced in the x- and the -directions. In Tables and the fundamental frequenc from the present meshless method, with uniforml spaced Table 4 First natural frequencies of a simpl supported square thick FG plate, TSDT, V c, V þ c, p =. h/a =.5 h/a =. N =7 N = N = Ref. [] N =7 N = N = Ref. [] Table 5 First natural frequencies of a simpl supported square thick Al/ZrO FG plate, FSDT, V c, V þ c, h/a =. Ceramic p = p = p = 5 Metal N =7 N = N = Ref. [] N =7 N = N = Ref. [] N =7 N = N = Ref. [] N =7 N = N = Ref. [] N =7 N = N = Ref. []

6 5 A.J.M. Ferreira et al. / Composite Structures 75 (6) 53 6 Table 6 First natural frequencies of a simpl supported square thick Al/ZrO FG plate, FSDT, V c, V þ c, p =. h/a =.5 h/a =. N =7 N = N = Ref. [] N =7 N = N = Ref. [] collocation points (or nodes), is compared with that from the exact solution of Vel and Batra [4], and the meshless Petrov Galerkin solution of Qian et al. []. It is clear that the presentl computed first frequenc is in excellent agreement with the exact one, particularl for a thick plate. The present method gives closer-to-exact values than Qian et al. s [] solution. However in [] an nodal arrangement was used. In Tables 3 we have listed the first ten frequencies computed with the present method b using 7 7, and collocation points distributed uniforml on the plate s midsurface. Whenever possible, we compare results with those of Qian et al. []. Table 7 First natural frequencies of a clamped square thick Al/ZrO FG plate, TSDT, V c, V þ c, h/a =. Ceramic p = p = p = 5. Metal N =7 N = N = N =7 N = N = N =7 N = N = N =7 N = N = N =7 N = N = Table First natural frequencies of a SCSC thick Al/ZrO FG plate, TSDT, V c, V þ c, h/a =. Ceramic p = p = p = 5. Metal N =7 N = N = N =7 N = N = N =7 N = N = N =7 N = N = N =7 N = N = Table First natural frequencies of a CFCF thick Al/ZrO FG plate, TSDT, V c, V þ c, h/a =. Ceramic p = p = p = 5. Metal N =7 N = N = N =7 N = N = N =7 N = N = N =7 N = N = N =7 N = N =

7 For all cases studied, the computed frequencies are found to be close to those given b Qian et al. []. Whereas integrals over subdomains of X appearing in the MLPG weak formulation need to be evaluated numericall, that is not the case here. Thus the present meshless method is computationall more efficient than the MLPG method. 6. Conclusions The collocation method with multiquadric radial basis functions to apprimate the trial solution and the thirdorder shear deformation theor are found to give frequencies of functionall graded plates that agree ver well with those found b Vel and Batra b solving analticall the three-dimensional elasticit equations. The present method is ver efficient since it neither requires nodal connectivit nor evaluation of an integral over a subdomain of plate s midsurface. The accurac of computed frequencies is controlled b the number of collocation points or nodes, their locations, and the parameter, c, in the multiquadric basis functions. Acknowledgements The financial support of the INTERREG programme, through grant MNAA Materials Network for the Atlantic Arc, is gratefull acknowledged. RCB s work was partiall supported b the ONR grant N4---3 to Virginia Poltechnic Institute and State Universit with Dr. Y.D.S. Rajapakse as the program manager. Views expressed herein are those of authors and not of funding agencies. Appendix I Equations for the determination of the generalized displacements u, v, w, and / of a TSDT are listed below. u A þ A v þ B þ B / 3h E þ o3 w 3 u þ A 33 þ o v þ B 33 3h E 33 3h E þ o / þ o3 w / þ o3 w þ o / u I ot þ J ot c ow I 3 ða:þ ot A.J.M. Ferreira et al. / Composite Structures 75 (6) u A 33 þ o v þ B 33 A 55 3h E 33 þ o / þ o / þ o3 w u þ A þ A v þ B þ B / 3h E þ o3 w 3h E v I ot þ J / ot c ow I 3 ot o þ o w h D o 55 þ o w þ 6 h F o 4 55 þ o w o/ þ A 44 þ o w h D o/ 44 þ o w þ 6 h F o/ 4 44 þ o w / þ o3 w 3 ða:þ þ 4 3 E o 3 u þ E o 3 v 3 þ F o 3 þ F o 3 / 3 3h H o 3 þ o4 w 3 4 3h H o 3 / þ o4 w o 3 u þ E 33 þ o3 v o 3 / þ F x 33 þ o3 / 3h H o 3 33 þ o3 / þ o4 w o 3 u þ E o 3 v þ E þ F o 3 3 þ F 3h H o 3 þ o4 w o 3 / 3 3h H o 3 / þ o4 w 3 4 w I ot c w I 6 ot þ o w ou þ c I 3 ot þ ov o þ o/ J x 4 ot þ o/ A 55 þ ow þ h D 55 þ ow 6 # h ða:3þ h F 4 55 þ ow u þ B þ B v þ D þ D / 3h F þ o3 w 3 3h F / þ o3 w 3 E u þ E v þ F þ F / 3h H # þ o3 w 3 3h H / þ o3 w h

8 6 A.J.M. Ferreira et al. / Composite Structures 75 (6) 53 6 u þ B 33 þ o v þ D 33 3h F 33 3 B 33 3h H 33 o þ o / u þ ov A 44 / þ ow þ o3 w / þ F x 33 # þ o / þ o3 w ow J ot u þ K c J 4 þ h D 44 / þ ow þ o / þ o / h ða:4þ 6 h F 4 44 / þ ow u þ B þ B v þ D þ D / 3h F þ o3 w 3h F / þ o3 w 3 3 E u þ E v þ F þ F / 3h H # þ o3 w 3h H / þ o3 w 3 u þ B 33 þ o v 3h F 33 3 E 33 3h H 33 þ D 33 þ o / þ o3 w u þ o v þ F 33 þ o / þ o3 w o ow J ot v þ K / c J 4 þ o / þ o / # h h ða:5þ where A ij B ij D ij E ij F ij H ij Z h z ðp t P b Þ h h þ p ð z z z 3 z 4 z 6 Þ þ Q b ð z z z 3 z 4 z 6 Þ dz ða:6þ where P t and P b correspond to generic propert at top and bottom surfaces and Q b correspond to the elasticit matrix at the bottom surface. References [] Qian LF, Batra RC, Chen LM. Free and forced vibrations of thick rectangular plates b using higher-order shear and normal deformable plate theor and meshless local Petrov Galerkin (MLPG) method. Comput Model Eng Sci 34:5 34. [] Qian LF, Batra RC, Chen LM. Static and dnamic deformations of thick functionall graded elastic plate b using higher-order shear and normal deformable plate theor and meshless local Petrov Galerkin method. Composites: Part B 435:65 7. [3] Batra RC, Vidoli S. Higher order piezoelectric plate theor derived from a three-dimensional variational principle. AIAA J 4(): 4. [4] Vel SS, Batra RC. Three-dimensional exact solution for the vibration of functionall graded rectangular plates. J. Sound Vibr 4 7:73 3. [5] Redd JN. Mechanics of laminated composite plates. New York: CRC Press 7. [6] Ferreira AJM, Batra RC, Roque CMC, Qian LF, Martins PALS. Static analsis of functionall graded plates using third-order shear deformation theor and a meshless method. Comp Struct 5 6: [7] Hard RL. Multiquadric equations of topograph and other irregular surfaces. Geophs Res 776:5 5. [] Kansa EJ. Multiquadrics a scattered data apprimation scheme with applications to computational fluid dnamics. i. Surface apprimations and partial derivative estimates. Comput Math Appl (/):7 45. [] Kansa EJ. Multiquadrics a scattered data apprimation scheme with applications to computational fluid dnamics. ii. Solutions to parabolic, hperbolic and elliptic partial differential equations. Comput Math Appl (/):47 6. [] Ferreira AJM. A formulation of the multiquadric radial basis function method for the analsis of laminated composite plates. Comp Struct 35:35. [] Ferreira AJM. Thick composite beam analsis using a global meshless apprimation based on radial basis functions. Mech Adv Mater Struct 3:7 4. [] Ferreira AJM. Analsis of composite plates using a laerwise shear deformation theor and multiquadrics discretization. Mech Adv Mater Struct 5():. [3] Ferreira AJM, Roque CMC, Martins PALS. Radial basis functions and higher-order theories in the analsis of laminated composite beams and plates. Comp Struct 466:7 3. [4] Liu GR. Meshfree methods, moving beond the finite element method. CRC Press. [5] Qian LF, Batra RC. Design of bidirectional functionall graded plate for optimal natural frequencies. J Sound Vibr 5:45 4. [6] Mori T, Tanaka K. Average stress in matrix and average elastic energ of materials with misfitting inclusions. Acta Metall 73:57 4.