A Homogenization Procedure for Geometrically. Non-Linear Free Vibration Analysis of. Functionally Graded Circular Plates Involving

Transcription

1 Applied Mathematical Sciences, Vol., 6, no. 4, 3-36 HIKARI Ltd, A Homogenization Procedure for Geometrically Non-Linear Free Vibration Analysis of Functionally Graded Circular Plates Involving the Coupling between Transverse and In-Plane Displacements Rachid El Kk Mohammed V University in Rabat Ecole Normale Supérieure de l'enseignement Technique de Rabat Département de génie mécanique, LaMIPI, CEDoc-STi, B.P. 67 Rabat Instituts, Rabat, Morocco Khalid El Bikri Mohammed V University in Rabat Ecole Normale Supérieure de l'enseignement Technique de Rabat Département de génie mécanique, LaMIPI, B.P. 67 Rabat Instituts, Rabat, Morocco Rhali Benamar Mohammed V University in Rabat Ecole Mohammadia D ingénieurs Avenue Ibn Sina, B.P.765 Agdal Rabat, Morocco Copyright 6 Rachid El Kk, Khalid El Bikri and Rhali Benamar. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 4 Rachid El Kk et al. Abstract A non-linear free axisymmetric vibration of functionally graded thin circular plates denoted by FGCP whose properties vary through the thickness subjected to the coupling between transverse and in-plane displacements is investigated. The equations of motion are derived using the energy method and a multimode approach. A homogenization procedure has been employed to reduce the problem under consideration to that of an isotropic homogeneous circular plate. The inhomogeneity of the plate is characterized by a power law variation of Young s modulus and mass density of the material along the thickness direction whereas Poisson s ratio is assumed to be constant. This variation in material properties of the plate introduces a coupling between the in-plane and transverse displacements. The problem is solved by a numerical iterative method. The formulations are validated by comparing the results with the available solutions in the literature for FG circular plates. The non-linear to linear frequency ratios are presented for various volume fraction index n. The effects of the coupling between the in-plane and transverse displacements on the frequency parameters are proved to be significant. The distributions of the associated bending, membrane and total stresses are also given for various vibration amplitudes with different values of the volume fraction index n and compared with those predicted by the linear theory. Keywords: Non-linear vibration, Circular plate, functionally graded material (FGM), Homogenization procedure Introduction The problem of plates vibration is of a continuing interest, due to their frequent use as structural components, especially in aerospace []. The geometrically non-linear behavior of plates is encountered in many recent applications, in which aircraft panels are subjected to high excitation levels. In a previous series of papers [ 6], a semi-analytical model has been developed for non-linear free vibrations of thin structures such as beams, plates, and shells. The non-linear vibration problem was reduced to iterative solution of a set of non-linear algebraic equations, which allows the amplitude-dependent, non-linear frequencies and mode shapes of the structure considered to be determined. Functionally graded materials, are a new class of materials, have attracted increasing attention in recent years. They are advanced composite materials, realized by smoothly changing the composition of the constituent materials, in a chosen preferred direction. FGMs are designed to withstand high-temperature environments and maintain their structural integrity. Due to these advantages, they are suitable for aerospace applications, such as aircraft, space vehicles, barrier coating and propulsion systems. On the one hand, these materials are typically made from a mixture of ceramic and metal in which the ceramic component provides high-temperature resistance due to its low thermal conductivity; on the

3 Geometrically non-linear free vibration analysis 5 other hand, the ductile metal component prevents fracture caused by thermal or mechanical stresses [7]. The vibration of functionally graded circular plates has had relatively few investigations. Allahverdizadeh et al [8] investigated the nonlinear free and forced vibration of thin circular functionally graded plates by using assumed-time-mode method and Kantorovich time averaging technique. Zhou et al. [9] analyzed the Natural vibration of circular and annular thin plates by a Hamiltonian approach. The nonlinear theories of axisymmetric bending of functionally graded circular plates with modified couple stress have been developed by Reddy and Jessica Berry []. The differential quadrature method, which has been successfully used in solving boundary value problems, has also been extended to solve initial value problems of plates and was used to discretize the time domain [, ]. Zerkane et al [3] used a homogenization procedure to nonlinear free vibration analysis of functionally graded beams resting on nonlinear elastic foundations. Sepahi et al [4] investigated the effects of three-parameter of elastic foundations and thermo-mechanical loading on the axisymmetric large deflection response of a simply supported annular FG plate based on the first-order shear deformation theory (FSDT) in conjunction with nonlinear von Karman assumptions. Talha and Singh [5] performed the large amplitude free flexural vibration analysis of shear deformable functionally graded plates using higher order shear deformation theory. Xia and Shen [6] analyzed the nonlinear vibration and the dynamic response of a shear deformable functionally graded plate with surface-bonded piezoelectric fiber reinforced composite actuators (PFRC) in thermal environments. Yang et al [7] reported a large vibration amplitudes analysis of pre-stressed functionally graded laminated plates that are composed of a shear deformable functionally graded layer and two surface-mounted piezoelectric actuator layers. Using a semi-analytical method based on one dimensional differential quadrature and Galerkin technique to predict the large vibration amplitudes behavior of laminated rectangular plates with two opposite clamped edges, Nie and Zhong [8] studied the free and forced vibration of functionally graded annular sectorial plates with simply supported radial edges and arbitrary circular edges using the state space method (SSM) and a differential quadrature method (DQM). An Analytical investigation of the free vibration behavior of thin circular functionally graded plates integrated with two uniformly distributed actuator layers made of piezoelectric material based on the classical plate theory (CPT) has been presented in [9]. Hosseini-Hashemi et al [3] performed the analytical solutions for free vibration analysis of moderately thick rectangular plates, which are composed of functionally graded materials (FGMs) and supported by either Winkler or Pasternak elastic foundations. The analysis procedure was based on the first-order shear deformation plate theory (FSDT) to derive and to solve exactly the equations of motion and the free vibration characteristics of side-cracked rectangular functionally graded thick plates [3]. Zheng and Zhong [3] investigated axisymmetric bending problem of FG circular plates under two boundary conditions, rigid slipping and elastically supported, subjected to transverse normal and shear loadings. They utilized Fourier

4 6 Rachid El Kk et al. Bessel series as the displacement function. Sahraee and Saidi [33] investigated axisymmetric bending of functionally graded circular plates under uniform transverse loadings using the fourth-order shear deformation plate theory. They studied the effect of various percentages of ceramic metal volume fractions on maximum out-plane displacement and shear stress. Their results were compared with those obtained based on the first-order shear deformation plate theory, the third-order shear deformation plate theory of Reddy and the exact three-dimensional elasticity solution and a good agreement was found between the different approaches. Chen [34] suggested an innovative technique for solving nonlinear differential equations for the bending problem of a circular plate. He used a type of pseudo-linearization to obtain the final solution for large deformations of the circular plate examined. The objective of this paper was the investigation of the geometrically non-linear free axisymmetric vibrations of a thin Functionally Graded circular Plate, using the theoretical model successfully applied to the analysis of large vibration amplitudes of various structures [6, 3, 4]. In the following analysis, a homogenization procedure was developed and used to reduce the problem under consideration to that of an equivalent isotropic homogeneous circular plate. By assuming a harmonic transverse motion, the in-plane and transverse displacements were expanded in the form of finite series of basic functions, namely the linear free vibration modes of the FGCP, obtained in terms of Bessel s functions. The discretized expressions for the total strain and kinetic energies have then been derived. The application of Hamilton s principle reduced the problem of large free vibration amplitudes to the solution of a set of coupled non-linear algebraic equations in terms of the contribution coefficients of the in-plane and transverse basic functions, which has been solved by a numerical iterative method in order to obtain accurate results for vibration amplitudes up to twice the plate thickness. The plate thickness was supposed to be constant. The results are compared with those obtained when the in-plane displacements is neglected [35] and also with the published literature to demonstrate the applicability and the computational efficiency of the proposed method. Material properties and formulation A functionally graded circular plate with thickness h and radius a is considered here. It is assumed that the mechanical and thermal properties of FGM vary through the thickness of plate, and the material properties P can be expressed as PP(zz) = (PP mm PP cc )VV mm + PP cc () Where the subscripts m and c denote the metallic and ceramic constituents, respectively Vm denotes the volume fraction of metal and follows a simple power law as VV mm = zz h + nn. ()

5 Geometrically non-linear free vibration analysis 7 Where z is the thickness coordinate (-h/ z h/), and n is a material constant. According to this distribution, bottom surface (z = - h/) of the functionally graded plate is pure metal, the top surface (z = h/) is pure ceramics, and for different values of n ( n ). This dictates the material variation profile across the plate thickness. The plate is fully metallic and ceramic, when n tends to and to infinity, respectively; whereas the composition of metal and ceramic is linear for n =, as shown in figure. n = n =. n = 5. n = 3. n =. z / h n =. -. n = n =. n = Vm Figure : Volume fraction of metal along the thickness Considering axisymmetric vibrations of the FG circular plate, the displacements are given, in accordance with the classical plate theory, by: uu rr (rr, zz, tt) = UU(rr, tt) zz (rr,tt), uu θθ (rr, tt) =, uu zz (rr, tt) = WW(rr, tt) (3) Where U and W are the in-plane and out-of-plane displacements of the middle plane point (r,θ,) respectively, and ur, uθ and uz are the displacements along ee rr, ee θθ and ee zz directions, respectively. And the force and moment components N and M are as follows, h (NN rr, NN θθ ) = (σσ rr, σσ θθ ) h h dddd, (MM rr, MM θθ ) = (σσ rr, σσ θθ ) h zzzzzz. (4) The constitutive relations for the FGMs are given by σσ rr σσ θθ = EE(zz) νν νν νν εε rr εε + zz kk rr. (5) θθ kk θθ

6 8 Rachid El Kk et al. The Young s Modulus E (z) in Eqs. (5) follow the distribution law of Eqs. () and (), namely, EE(zz) = (EE mm EE cc )VV mm + EE cc, (6) For simplicity, the Poisson s ratio ν in Eqs.(5) is assumed to be constant. The radial and circumferential strain components εε rr and εε θθ in the mid-plane of the plate (i.e. z = ) can be calculated as εε rr = +, εε θθ = UU rr (7 7bb) With U being the displacement in r direction. Variations of the curvature kk rr and kk θθ in the mid-plane of the plate (z = ) can be calculated as, kk rr = WW, kk θθ = rr, (8 8bb) From Eqs. (4) and (5), one obtains NN rr = AA AA εε rr NN θθ AA AA εε + BB BB kk rr. (9) θθ BB BB kk θθ MM rr = BB BB εε rr MM θθ BB BB εε + DD DD kk rr. () θθ DD DD kk θθ Where Aij, Bij and Dij are stiffness coefficients of the plate and can be calculated as h AA iiii, BB iiii, DD iiii = QQ iiii (, zz, zz ) dddd. () h With: QQ = QQ = EE(zz) νν, QQ = νννν From the previous equations, one then obtains the expression for the bending strain energy Vb, the membrane strain energy Vm and the kinetic energy T expressed in terms of the displacements VV bb = ππ BB WW + νν rr rr dddd + +ππ DD WW + rr + νν rr WW rr dddd, ()

7 Geometrically non-linear free vibration analysis 9 VV mm = ππ AA BB WW + UU rr + νν rr νν UU rr + UU rr νν UU + rr νν UU + rr rrrrrr.. WW rrrrrr, (3) And TT = ππ II + rrrrrr. (4) Where I is the inertial term given by: h/ II = ρρ(zz) dddd h/ 3 Homogenization procedure. (5) A new coordinate system is assumed, the midplane is used as the reference surface. If, instead, we select a different reference surface so that zz = zz + δδ (6) Then BB = h/ QQ iiii (zz + δδ) h/ dddd = BB iiii + δδaa iiii (7) And by selecting the distance δ in order to achieve BB = so that δδ = BB AA In the new coordinate system and the extension bending coupling will be eliminated from the equation of motion, and the bending rigidity in the new coordinate system is = DD eeeeee = (DD BB AA ) (8) DD The effective extensional stiffness AA does not change in this new coordinate system. AA = AA eeeeee As the transverse displacement in the new coordinate system is the same as in the

8 Rachid El Kk et al. previous one; equations () and (3) can be expressed in the new systems: VV bb = ππ DDDDDDDD WW + rr + νν rr WW rr dddd, (9) + VV mm = ππ AAAAAAAA UU νν UU + rr rr +.. νν UU rr rrrrrr, () 4 Approximate solution The transverse displacement function W(r,t), which has been assumed in the present paper, which is mainly concerned with the amplitude dependence of the first harmonic component spatial distribution and can be presented as follows: WW(rr, tt) = ww(rr )cccccc(ωωωω), () The in-plane radial displacement function U(r,t) in the following form: UU(rr, tt) = uu(rr )cccccc (ωωωω), () The spatial functions u(r) and w(r) are expanded in the form of finite series of pi and po in-plane ui(r) and out-of-plane wi(r) basic functions, respectively, as follows: ww(rr) = ii ww ii (rr ), uu(rr) = bb ii uu ii (rr ), (3) Where the usual summation convention for repeated indices is used from to po and from to pi for the ai s and bi s coefficients respectively. The discretized forms for the total strain and kinetic energies are respectively given by: VV = ii jj kk ww iiii cccccc (ωωωω) + ii jj kk ll bb ww iiiiiiii + ii jj bb kk CC uuuu iiiiii + bb ii bb jj kk uu iiii cccccc 4 (ωωωω), (4) TT = ωω ii jj mm iiii ww ssssss (ωωωω) + bb ii bb jj mm iiii uu ssssss (ωωωω), (5) In these equations, mm ww iiii, mm uu iiii, kk ww uu iiii, kk iiii are the mass and rigidity tensors associated ww with W and U, respectively, bb iiiiiiii and CC uuuu iiiiii are a fourth and a third order non-linearity and coupling tensors respectively. The general terms of these tensors are given by

9 Geometrically non-linear free vibration analysis mm ww iiii = ππππ ww ii ww jj rr dddd, mm uu iiii = ππππ uu ii uu jj rr dddd, (6, 6bb) kk ww iiii = ππ DD eeeeee dddd ii dddd jj dddd dddd + ddww ii rr dddd dd ww jj dddd rr dddd, (7) kk uu iiii = ππ AA eeeeee dduu ii dddd dd uu jj ddrr + rr uu iiuu jj + νν dd uu ii rr dddd uu jj + νν rr uu dd uu jj ii dddd rr dddd, (8) uuuu = ππ AA eeeeee ddww ii dddd CC iiiiii dd ww jj dddd dd uu kk dddd + νν ddww ii dd ww jj rr dddd dddd uu kk rr dddd, (9) bb ww iiiiiiii = ππ AA eeeeee ddww ii dddd ddww jj dddd ddww kk ddww ll dddd dddd rr dddd, (3) It appears from Eqs. (6) - (3) that the mass and rigidity tensors are symmetric, ww uuuu and the fourth order bb iiiiiiii and the third order tensors CC iiiiii are illustrated below bb ww iiiiiiii = bb ww kkkkkkkk = bb ww jjjjjjjj ww = bb iiiiiiii, CC uuuu uuuu iiiiii = CC jjjjjj (3) The dynamic behavior of the structure is governed by Hamilton s principle, which is symbolically written as δδ ππ ωω (VV TT) dddd = δδ = (3) Replacing T and V by their discretized expressions in the energy condition (3), integrating the time functions and calculating the derivatives with respect to the,, ii ss and bb ii ss, and taking into account the properties of symmetry of the tensors involved, leads to the following set of non-linear algebraic equations: ii kk ww iiii + 3 ii jj kk bb ww iiiiiiii + 3 iibb kk CC uuuu iiiiii ωω ii mm ww iiii =, rr =,, pppp. (33) 3 4 ii jj CC uuuu iiiiii + bb ii kk uu iiii ωω bb ii mm uu iiii =, ss =,, pppp. (33bb) To simplify the analysis and the numerical treatment of the set of non-linear algebraic equations, non-dimensional formulation has been considered by putting the spatial displacement functions as follows: ww ii (rr) = hww ii (rr ), uu ii (rr) = λλhuu ii (rr ), (34)

10 Rachid El Kk et al. Where rr = rr is the non-dimensional radial co-ordinate and λλ = hh is a non-dimensional geometrical parameter representing the ratio of the plate thickness to its radius. Eqs. (33) can be written in a non-dimensional form as ii kk ww iiii + 3 ii jj kk bb ww iiiiiiii + 3 iibb kk CC uuuu iiiiii ωω ii mm ww iiii =, rr =,, pppp. (35) 3 4 ii jj CC uuuu iiiiii + bb ii kk uu iiii λλ ωω bb ii mm uu iiii =, ss =,, pppp. (35bb) Where ωω is the non-dimensional non-linear frequency parameter defined by: ωω = γγωω, wwwwwwh, γγ = II4 DD eeeeee, (36) The non-dimensional terms mm ww iiii, mm uu iiii, kk ww iiii, kk uu iiii, CC uuuu ww iiiiii bb iiiiiiii are given by: mm ww iiii = ww ii ww jj rr dddd, mm uu iiii = uu ii uu jj rr dddd, (37, 37bb) kk ww iiii = dd ww ii dd ww jj dddd dddd + ddww ii dd ww jj rr dddd dddd rr dddd (38) kk uu iiii = ββ dduu ii dd uu jj dddd dddd + rr uu ii uu jj + νν dd uu ii rr dddd uu jj + νν rr uu dd uu ii jj dddd rr dddd (39) CC iiiiii uuuu = ββ ddww ii dd ww jj dddd dddd bb ww iiiiiiii = αα ddww ii dd ww jj dddd dddd with dd uu kk dddd ddww kk dddd + νν ddww ii rr dd ww ll dddd ββ = AA eeeeee h DD eeeeee, αα = AA eeeeee h dd ww jj dddd dddd uu kk rr dddd, (4) rr dddd, (4) 4 DD eeeeee, (4) These non-dimensional tensors are related to the dimensional ones by the following equations: mm iiii uu, mm iiii ww = ππππh λλ mm iiii uu, mm iiii ww, (43)

11 Geometrically non-linear free vibration analysis 3 kk iiii ww = ππ DD eeeeee h kk iiii ww, (44) kk iiii uu = ππ AA eeeeee λλ h kk iiii uu, (45) uuuu = ππ AA eeeeee λλ h 3 CC iiiiii uuuu CC iiiiii, (46) bb ww iiiiiiii = ππ AA eeeeee h 4 ww bb iiiiiiii, (47) In the case of thin plates, for whichλλ is very small; the in-plane inertia term involving the termλλ can be neglected. This is an acceptable assumption in most engineering applications of thin plates [4]. Consequently, equation (33b) can be solved for thebb ii ss leading to: bb ii = jj ll dd jjjjjj, ii =,, pppp, (48) where dd iiiiii = kk iiii uu uuuu CC iiiiii (49) dd iiiiii is a third order tensor expressing the coupling between the in-plane and uu out-of-plane vibrations and kk iiii is the inverse of the tensor kk uu iiii. Substituting equation (48) by equation (33a) leads to an uncoupled set of non-linear algebraic equations in terms of the ii sscoefficients only. ii kk ww iiii + 3 ii jj kk bb iiiiiiii ωω ii mm ww iiii =, rr =,, pppp. (5) bb iiiiiiii is a fourth order tensor given by: bb iiiiiiii = bb ww iiiiiiii + CC iiiiii uuuu dd kkkkkk. (5) Equation (5) will be satisfied in the new coordinate system and it can be seen that the FG circular plate will behave as an isotropic homogeneous circular plate having the equivalent parameters defined above. Eqs. (5) can be written in matrix form as: {AA} TT [KK ww ]{AA} + 3 {AA}TT [BB ]{AA} ωω {AA} TT [MM ww ]{AA} =, (5)

12 4 Rachid El Kk et al. Pre-multiplying Eq. (5) by the vector (AA) TT = [,,, nn ] leads to the following expression for ωω : ωω = ii jj kk ww iiii + 3 ii jj kk ll bb iiiiiiii ww, (53) ii jj mm iiii The basic functions ww ii used in the expansion series of wwin Eq. (34) must satisfy the theoretical clamped boundary conditions, i.e., zero displacement and zero slopes along the circular edge. Since the linear problem of free axisymmetric flexural vibration of a clamped FG circular plate has an exact analytical solution, the chosen basic functions ww ii were taken as the linear mode shapes of fully clamped circular plates given by [4]: ww ii (rr ) = AA ii JJ (ββ ii rr ) JJ (ββ ii ) II (ββ ii ) II (ββ ii rr ), (54) Where ββ ii is the i th real positive root of the transcendental equation. JJ (ββ)ii (ββ) + JJ (ββ)ii (ββ) = (55) In which JJ nn and II nn are, respectively, the Bessel and the modified Bessel functions of the first kind and of order n, The parameter ββ ii is related to the i th non-dimensional linear frequency parameter (ωω ll ) ii of the plate by: ββ ii = (ωω ll ) ii. AA ii was chosen such that: ww ii rr ddrr =. (56) The chosen in-plane basic functions uu ii (rr) for an immovable axisymmetric FG circular plate are given by: uu ii (rr ) = BB ii JJ (αα ii rr ) (57) Where αα ii is the i th real positive root of the equation JJ (αα) =, the functions ww ii (rr)and uu ii (rr) are normalized in such a manner that mm ww iiii = ww ii ww jj rr dddd = δδ iiii, mm uu iiii = uu ii uu jj rr dddd = δδ iiii. 5 Numerical results and discussion (58, 58bb) In order to validate the proposed model, the case of a functionally graded circular plate is treated and the numerical results obtained with iterative solutions are compared to that obtained from using assumed-time-mode method and Kantorovich time averaging technique [8]. The set of non-linear algebraic equations

13 Geometrically non-linear free vibration analysis 5 (5) is solved numerically using the Harwell library routine NSA, which is based on a hybrid iterative method combining the steep descent and Newton's methods. 5. Amplitude frequency dependence The dependence of the non-linear frequency on the non-dimensional vibration amplitude is showed through the so-called backbone curve as plotted in figure for a functionally graded circular plate by considering the in-plane displacements. The comparison made between the results obtained here and those obtained in ref. [35], where in-plane displacements are omitted. It can be seen that the w-formulation leads to higher frequency parameters than those obtained from the u-w-formulation due to the mathematical stiffening that occurs once one neglects the in-plane displacements. In table, the non-linear frequency ratio of a functionally graded circular plate obtained in ref. [8] by considering the in-plane displacements u and w is compared to the results of the present works. As it can be seen, close agreement is found. The same comparison is also made in the table where in-plane displacements are omitted and the same concordance is also noticed. On the other hand, It can be shown from the numerical data in table and that the in-plane displacements have a clear influence on the predicted non-linear resonance frequency, since the discrepancy between the results obtained from w-formulation and u-w-formulation is about 6.8 % for a non-dimensional vibration amplitude W*max=.5. Figure 3 presents the dependence of the frequency ratio of the clamped FG circular plate on the amplitude of vibration for various values of the power law index n. It may be noticed that, by increasing the values of power law index in the range [, ], the frequency increases. For values higher than n =., the frequency decreases when n increases. This may be expected, since when the power law index n =. or n =., the material is pure metallic or pure ceramic respectively and the non-dimensional frequency corresponds the isotropic homogeneous materiel case..7 Non-linear to linear frequency ratio Tm = Tc = 3(K) n =.5 () : Uncoupled model [35] () : Coupled model () () Non-dimensional amplitude Figure : Effect of coupling on the non-linear frequency ratios for FG circular plate, (n =.5): ( ) Uncoupled model [35], (- - -) Coupled model (Present work)

14 6 Rachid El Kk et al. Table : Frequency ratios (ωω nnnn ) of the clamped coupled FGCP ωω ll Method n ww mmmmmm Present work Coupled FGCP. (Model with u-w formulation) Ref.[8] Coupled FGCP. (Model with u-w formulation) Deviation (%) Table : Comparison of the non linear frequency ratios (ωω nnnn ωω ll ) obtained via the coupledisotropic homogeneous plate and uncoupled models for isotropic homogeneous and functionally graded circular plates Method n ww mmmmmm Present work Coupled isotropic plate. (Model with u-w formulation) Ref.[35] Uncoupled isotropic plate. (Model with w formulation only) Present work Uncoupled FGCP (Model with w formulation only)

15 Geometrically non-linear free vibration analysis Non-linear to linear frequency ratio Tm = Tc = 3(K) n =. n =.5 n =. n =. n = Non- dimensional amplitude Figure 3: Effect of the power law index (n) on the variation of the non-linear frequency ratios (ωω nnnn ωω ll ) of the clamped coupled FG circular plate, with the amplitude of vibration 5. Stress analysis As mentioned above, the present multimodal model enables not only determination of the amplitude frequency dependence, but also the deformation of the mode shape due to the geometrical non-linearity. From this last result, it was expected that the effect of the amplitude of vibration on the distribution of the associated bending stress would be greater significance; since the bending stress is related to the derivatives of the amplitude-dependent transverse mode shape. Figure 4, in which the radial bending stress distributions associated with the first non-linear axisymmetric mode shape is plotted, for the power index n =.5 and various values of the vibration amplitude, show the amplitude dependence of the bending stress distribution. It can be seen also from figure 5 that the non linear bending stress exhibits a higher increase near to the clamped edge, but behaves in an opposite manner near to the plate centre. Figure 6 exhibit the radial membrane stress results associated with the first non-linear axisymmetric mode shape at the center and the edge of functionally graded circular plate. Review of this figure show a rapid increase of the membrane stress with increasing amplitude of vibration, especially at the centre of the plate. The non-dimensional radial membrane and total stress distributions associated with the first axisymmetric non-linear mode shape are plotted in figures 7 and 8, respectively, for the power law index n =.5 and various values of the non-dimensional vibration amplitude. It can be seen that the membrane stress can be neglected at small vibration amplitudes. The result found in the present work for n =. is coherent with that given in ref. [4] for the stress distributions of a clamped homogenous circular plate. The radial bending stresses, the membrane stresses and the radial total stresses associated to the first non-linear axisymmetric

16 8 Rachid El Kk et al. mode shape of a clamped FG circular plate for n =.5 and W*max =.5 are plotted in figure 9. It can be shown that the membrane stresses contribute significantly to the total radial stresses. 4 x -3 Non-dimensional radial bending stress (4) (3) () () Tm = Tc = 3(K) n =.5 () : w*max =.3 () : w*max =.5 (3) : w*max =. (4) : w*max = r * Figure 4: Non-dimensional radial bending stress distribution associated to the first non-linear axisymmetric mode shape of a clamped FG circular plate for n =.5 and various non-dimensional vibration amplitudes Non-dimensional radial bending stress Edge Centre Tm = Tc = 3(K) n = Non-dimensional amplitude Figure 5: Effect of large vibration amplitudes on the non dimensional radial bending stress associated with the first non linear axisymmetric mode shapes at the centre and the Edge of FG circular plate

17 Geometrically non-linear free vibration analysis 9 3 Non-dimensional radial membrane stress Centre Edge Tm = Tc = 3(K) n = Non-dimensional amplitude Figure 6: Effect of large vibration amplitudes on the non dimensional radial membrane stress associated with the first non linear axisymmetric mode shapes at the centre and the Edge of FG circular plate Non-dimensional radial membrane stress x -4 (4) (3) () : w*max =.3 () : w*max =.5 (3) : w*max =. (4) : w*max =.5 Tm = Tc = 3(K) n =.5 () () r * Figure 7: Non-dimensional radial membrane stress distribution associated to the first non-linear axisymmetric mode shape of a clamped FG circular plate for n =.5 and various non-dimensional vibration amplitudes

18 3 Rachid El Kk et al. 6 x -3 Non-dimensional radial total stress Tm = Tc = 3(K) n =.5 () : w*max =.3 () : w*max =.5 (3) : w*max =. (4) : w*max =.5 (4) (3) () () r * Figure 8: Non-dimensional radial total stress distribution associated to the first non-linear axisymmetric mode shape of a clamped FG circular plate for n =.5 and various non-dimensional vibration amplitudes 6 x -3 Non-dimensional radial stress 4 - (3) () () Tm = Tc = 3(K) W*max =.5 n =.5 () : Bending stress () : Membrane stress (3) : Total stress r * Figure 9: Non-dimensional radial Bending, membrane and total stress distribution associated to the first non-linear axisymmetric mode shape of a clamped FG circular plate for n =.5 and W*max =.5 The dimensionless radial bending and membrane stresses with dimensionless thickness of the plate for different values of n at T = 3 (K) and W* max =.5 are depicted in figures (a-b) and (-b). It is obvious that with increasing n, the stresses decrease in any arbitrary transverse section of the plate. For ceramic-rich and metal rich plates (n = and n = ), the stress distribution is linear, where for the functionally graded (Si3N4/SUS34) circular plate, the behavior is nonlinear and is governed by the variation of the properties in the thickness

19 Geometrically non-linear free vibration analysis 3 direction. The material properties of the functionally graded plate are assumed to vary through the thickness of the plate. Figures (a) and (b) demonstrate how the radial bending and membrane stress distributions exhibit a regular and continuous change from one surface to another (5) (4) (3) () (5) (4) (3) ().. Z * () Tc = Tm = 3(K) W*max =.5 () : n =. () : n =.5 (3) : n = 5. (4) : n =. (5) : n = Z * Tc = Tm = 3(K) W*max =.5 () : n =. () : n =.5 (3) : n = 5. (4) : n =. (5) : n =. () -.4 ( a ) -.4 ( b ) Radial bending stress Radial bending stress Figure : Variation of the radial bending stress of FG circular plate through the dimensionless thickness for different values of the power law index n for W*max =.5, case (a): Centre, case (b): Edge (3) (4) (5) (3) (4) (5).. ().. () Z * () Tc = Tm = 3(k) W*max =.5 () : n =. () : n =.5 (3) : n = 5. (4) : n =. (5) : n =. -.4 ( a ) Radial membrane stress -.4 ( b ) Radial membrane stress Figure : Variation of the radial membrane stress of FG circular plate through the dimensionless thickness for different values of the power law index n for W*max =.5, case (a): Centre, case (b): Edge Z * () Tc = Tm = 3(K) W*max =.5 () : n =. () : n =.5 (3) : n = 5. (4) : n =. (5) : n = (5) (4).5. z = h / Tc = Tm = 3(K) W*max =.5 n =.5 Radial bending stress (3) () () Tc = Tm = 3(K) W*max =.5 n =.5 () : z = - h/ () : z = - h/4 (3) : z =. (4) : z = h/4 (5) : z = h/ r * ( a ) z = - h / ( b ) r * Figure : Variation of the radial bending stress (a) and radial membrane stress (b) of clamped FG circular plate on different thickness distribution for n =.5 and W*max =.5 Radial membrane stress z = h / 4 z =. z = - h / 4

20 3 Rachid El Kk et al. 6 Conclusion In this work, it has been shown that the model for non-linear free vibrations of a clamped thin isotropic circular plate developed previously in [4] can be generalized and extended to examine qualitatively and quantitatively the coupling between the membrane and transverse displacements for the functionally graded circular plate in the non-linear range. A homogenization procedure has been developed to reduce the problem under consideration to that of an equivalent isotropic homogeneous circular plate which means that no special software needs to be developed for their analysis. Judicious choice of admissible and compatible basic functions for a clamped functionally graded circular plate has been made and iterative method has been employed to solve the amplitude equations of motion in order to establish the validity of the present u-w coupled formulation through comparisons of the numerical analytical results obtained here with those found in the published literature. In the results, parametric studies were devoted to the effects of the coupling between in-plane and transverse displacements, the influences of vibration amplitude and volume fraction index n have been examined. Several conclusions may be drawn from this study. It is considered that the effects of coupling between in-plane and transverse displacements on the frequency parameters are proved to be significant. It is also concluded that variation of volume fraction index is influential in FGM properties, dynamic treatment and the amount of stresses. The vibration frequencies are dependent on the large vibration amplitudes. Also, the results show that non-linear stresses distribution exhibits a high decrease with increasing the volume fraction index n, in contrast with the non-linear frequencies which exhibit only a slight change with the variation of the power index n. The latter results showed that the in-plane membrane stresses have a large contribution to the total radial stresses when large vibration amplitudes occur. Consequently, they cannot be neglected in the engineering design of large deflected structures. References [] R.G. White, Developments in the acoustic fatigue design process for composite aircraft structures, Composite Structures, 6 (99), [] R. Benamar, Non-linear Dynamic Behaviour of Fully Clamped Beams and Rectangular Isotropic and Laminated Plates, Ph.D. Thesis, University of Southampton, 99. [3] R. Benamar, M.M.K. Bennouna, R.G. White, The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures. Part I: simply supported and clamped clamped beams, Journal of

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24 36 Rachid El Kk et al. [3] Sh. Hosseini-Hashemi, H. Rokni Damavandi Taher, H. Akhavan, M. Omidi, Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory, Applied Mathematical Modelling, 34 (), [3] C.S. Huang, O.G. McGee III, M.J. Chang, Vibrations of cracked rectangular FGM thick plates, Composite Structures, 93 (), [3] Lei Zheng and Zheng Zhong, Exact solution for axisymmetric bending of functionally graded circular plate, Tsinghua Science and Technology, 4 (9), no. S, [33] S. Sahraee, A.R. Saidi, Axisymmetric bending analysis of thick functionally graded circular plates using fourth-order shear deformation theory, European Journal of Mechanics-A/ Solids, 8 (9), [34] Y.Z. Chen, Innovative iteration technique for nonlinear ordinary differential equations of large deflection problem of circular plates, Mechanics Research Communications, 43 (), [35] El Kk Rachid, El Bikri Khalid, Benamar Rhali, Geometrically Nonlinear Free Axisymmetric Vibrations Analysis of Thin Circular Functionally Graded Plates, Advanced Materials Research, (4), Received: April, 6; Published: June 5, 6