Applied Mathematical Sciences, Vol. 10, 2016, no. 41, 2013 - 2036 HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2016.63122

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 Kaak

Mohammed V University in Rabat Ecole Normale Supérieure de l'Enseignement Technique de Rabat

Département de génie mécanique, LaMIPI, CEDoc-ST2i, B.P. 6207 Rabat Instituts, 10100 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. 6207

Rabat Instituts, 10100 Rabat, Morocco

Rhali Benamar

Mohammed V University in Rabat Ecole Mohammadia D’ingénieurs Avenue Ibn Sina, B.P.765 Agdal

Rabat, Morocco Copyright © 2016 Rachid El Kaak, 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.

2014 Rachid El Kaak 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 1 Introduction The problem of plates vibration is of a continuing interest, due to their frequent use as structural components, especially in aerospace [1]. 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 [2–16], 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

Geometrically non-linear free vibration analysis 2015 other hand, the ductile metal component prevents fracture caused by thermal or mechanical stresses [17]. The vibration of functionally graded circular plates has had relatively few investigations. Allahverdizadeh et al [18] 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. [19] 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 [20]. 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 [21, 22]. Zerkane et al [23] used a homogenization procedure to nonlinear free vibration analysis of functionally graded beams resting on nonlinear elastic foundations. Sepahi et al [24] 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 [25] performed the large amplitude free flexural vibration analysis of shear deformable functionally graded plates using higher order shear deformation theory. Xia and Shen [26] 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 [27] 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 [28] 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 [29]. Hosseini-Hashemi et al [30] 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 [31]. Zheng and Zhong [32] 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–

2016 Rachid El Kaak 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, 13, 14]. 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. 2 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) = (𝑃𝑃𝑚𝑚 − 𝑃𝑃𝑐𝑐)𝑉𝑉𝑚𝑚 + 𝑃𝑃𝑐𝑐(1)

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

𝑉𝑉𝑚𝑚 = �𝑧𝑧ℎ +

12�

𝑛𝑛

. (2)

Geometrically non-linear free vibration analysis 2017 Where z is the thickness coordinate (-h/2 ≤ z ≤ h/2), and n is a material constant. According to this distribution, bottom surface (z = - h/2) of the functionally graded plate is pure metal, the top surface (z = h/2) is pure ceramics, and for different values of n (0 ≤ n ≤ ∞). This dictates the material variation profile across the plate thickness. The plate is fully metallic and ceramic, when n tends to 0 and to infinity, respectively; whereas the composition of metal and ceramic is linear for n = 1, as shown in figure 1.

Figure 1: 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:

𝑢𝑢𝑟𝑟(𝑟𝑟, 𝑧𝑧, 𝑡𝑡) = 𝑈𝑈(𝑟𝑟, 𝑡𝑡) − 𝑧𝑧𝜕𝜕𝜕𝜕(𝑟𝑟,𝑡𝑡)

𝜕𝜕𝑟𝑟 ,𝑢𝑢𝜃𝜃(𝑟𝑟, 𝑡𝑡) = 0 , 𝑢𝑢𝑧𝑧(𝑟𝑟, 𝑡𝑡) = 𝜕𝜕(𝑟𝑟, 𝑡𝑡) (3)

Where U and W are the in-plane and out-of-plane displacements of the middle plane point (r,θ,0) respectively, and ur, uθ and uz are the displacements along 𝑒𝑒𝑟𝑟 , 𝑒𝑒𝜃𝜃 and 𝑒𝑒𝑧𝑧 directions, respectively. And the force and moment components N and M are as follows,

(𝑁𝑁𝑟𝑟 ,𝑁𝑁𝜃𝜃) = � (𝜎𝜎𝑟𝑟 ,𝜎𝜎𝜃𝜃)ℎ 2⁄

−ℎ 2⁄ 𝑑𝑑𝑧𝑧, (𝑀𝑀𝑟𝑟,𝑀𝑀𝜃𝜃) = � (𝜎𝜎𝑟𝑟 ,𝜎𝜎𝜃𝜃)

ℎ 2⁄

−ℎ 2⁄ 𝑧𝑧𝑑𝑑𝑧𝑧. (4)

The constitutive relations for the FGMs are given by

�𝜎𝜎𝑟𝑟𝜎𝜎𝜃𝜃� =

𝐸𝐸(𝑧𝑧)1 − 𝜈𝜈2 �

1𝜈𝜈𝜈𝜈1� ��

𝜀𝜀𝑟𝑟0

𝜀𝜀𝜃𝜃0� + 𝑧𝑧 �

𝑘𝑘𝑟𝑟𝑘𝑘𝜃𝜃�� . (5)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Vm

z / h

n = 10.0

n = 5.0

n = 100.0

n = 3.0

n = 2.0

n =1.0

n = 0.5

n = 0.2

n = 0.1

2018 Rachid El Kaak et al. The Young’s Modulus E (z) in Eqs. (5) follow the distribution law of Eqs. (1) and (2), namely, 𝐸𝐸(𝑧𝑧) = (𝐸𝐸𝑚𝑚 − 𝐸𝐸𝑐𝑐)𝑉𝑉𝑚𝑚 + 𝐸𝐸𝑐𝑐 , (6)

For simplicity, the Poisson’s ratio ν in Eqs.(5) is assumed to be constant. The radial and circumferential strain components 𝜀𝜀𝑟𝑟0 and 𝜀𝜀𝜃𝜃0 in the mid-plane of the plate (i.e. z = 0) can be calculated as

𝜀𝜀𝑟𝑟0 = 𝜕𝜕𝑈𝑈𝜕𝜕𝑟𝑟 +

12 �𝜕𝜕𝜕𝜕𝜕𝜕𝑟𝑟 �

2

, 𝜀𝜀𝜃𝜃0 = 𝑈𝑈𝑟𝑟 (7𝑎𝑎 − 7𝑏𝑏)

With U being the displacement in r direction. Variations of the curvature 𝑘𝑘𝑟𝑟 and 𝑘𝑘𝜃𝜃 in the mid-plane of the plate (z = 0) can be calculated as,

𝑘𝑘𝑟𝑟 = −𝜕𝜕2𝜕𝜕𝜕𝜕𝑟𝑟2 , 𝑘𝑘𝜃𝜃 = −

1𝑟𝑟𝜕𝜕𝜕𝜕𝜕𝜕𝑟𝑟

, (8𝑎𝑎 − 8𝑏𝑏)

From Eqs. (4) and (5), one obtains

�𝑁𝑁𝑟𝑟𝑁𝑁𝜃𝜃� = �𝐴𝐴11𝐴𝐴12𝐴𝐴12𝐴𝐴22

� �𝜀𝜀𝑟𝑟0

𝜀𝜀𝜃𝜃0� + �𝐵𝐵11𝐵𝐵12𝐵𝐵12𝐵𝐵22

� �𝑘𝑘𝑟𝑟𝑘𝑘𝜃𝜃� . (9)

�𝑀𝑀𝑟𝑟

𝑀𝑀𝜃𝜃� = �𝐵𝐵11𝐵𝐵12𝐵𝐵12𝐵𝐵22

� �𝜀𝜀𝑟𝑟0

𝜀𝜀𝜃𝜃0� + �𝐷𝐷11𝐷𝐷12𝐷𝐷12𝐷𝐷22

� �𝑘𝑘𝑟𝑟𝑘𝑘𝜃𝜃� . (10)

Where Aij, Bij and Dij are stiffness coefficients of the plate and can be calculated as

�𝐴𝐴𝑖𝑖𝑖𝑖,𝐵𝐵𝑖𝑖𝑖𝑖 ,𝐷𝐷𝑖𝑖𝑖𝑖� = � 𝑄𝑄𝑖𝑖𝑖𝑖(1, 𝑧𝑧, 𝑧𝑧2)ℎ 2⁄

−ℎ 2⁄𝑑𝑑𝑧𝑧 . (11)

With: 𝑄𝑄11 = 𝑄𝑄22 = 𝐸𝐸(𝑧𝑧)1− 𝜈𝜈2

, 𝑄𝑄12 = 𝜈𝜈𝑄𝑄11 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

𝑉𝑉𝑏𝑏 = 𝜋𝜋 � −𝐵𝐵11 �𝜕𝜕2𝜕𝜕𝜕𝜕𝑟𝑟2 �

𝜕𝜕𝜕𝜕𝜕𝜕𝑟𝑟 �

2

+ 𝜈𝜈

𝑟𝑟 𝜕𝜕𝜕𝜕𝜕𝜕𝑟𝑟 �

𝜕𝜕𝜕𝜕𝜕𝜕𝑟𝑟 �

2

� 𝑟𝑟 𝑑𝑑𝑟𝑟𝑎𝑎

0+

+𝜋𝜋� 𝐷𝐷11 ��𝜕𝜕2𝜕𝜕𝜕𝜕𝑟𝑟2 �

2

+ 1𝑟𝑟2 �

𝜕𝜕𝜕𝜕𝜕𝜕𝑟𝑟 �

2

+ 2 𝜈𝜈𝑟𝑟𝜕𝜕𝜕𝜕𝜕𝜕𝑟𝑟

𝜕𝜕2𝜕𝜕𝜕𝜕𝑟𝑟2 � 𝑟𝑟 𝑑𝑑𝑟𝑟

𝑎𝑎

0, (12)

Geometrically non-linear free vibration analysis 2019

𝑉𝑉𝑚𝑚 = 𝜋𝜋� 𝐴𝐴11 ��𝜕𝜕𝑈𝑈𝜕𝜕𝑟𝑟�2

+ 𝜕𝜕𝑈𝑈𝜕𝜕𝑟𝑟

�𝜕𝜕𝜕𝜕𝜕𝜕𝑟𝑟

�2

+ 14�𝜕𝜕𝜕𝜕𝜕𝜕𝑟𝑟

�4

+ 𝑈𝑈2

𝑟𝑟2+

2𝜈𝜈 𝑈𝑈𝑟𝑟

𝜕𝜕𝑈𝑈𝜕𝜕𝑟𝑟

+ 𝜈𝜈 𝑈𝑈𝑟𝑟�𝜕𝜕𝜕𝜕𝜕𝜕𝑟𝑟

�2

�𝑎𝑎

0𝑟𝑟𝑑𝑑𝑟𝑟− ..

−2𝐵𝐵11 �𝜕𝜕𝑈𝑈𝜕𝜕𝑟𝑟

𝜕𝜕2𝜕𝜕𝜕𝜕𝑟𝑟2

+𝑈𝑈𝑟𝑟2𝜕𝜕𝜕𝜕𝜕𝜕𝑟𝑟

+𝜈𝜈𝑟𝑟𝜕𝜕𝑈𝑈𝜕𝜕𝑟𝑟

𝜕𝜕𝜕𝜕𝜕𝜕𝑟𝑟

+𝜈𝜈 𝑈𝑈𝑟𝑟𝜕𝜕2𝜕𝜕𝜕𝜕𝑟𝑟2

� 𝑟𝑟𝑑𝑑𝑟𝑟, (13)

And

𝑇𝑇 = 𝜋𝜋 𝐼𝐼0 � ��𝜕𝜕𝜕𝜕𝜕𝜕𝑡𝑡 �

2

+ �𝜕𝜕𝑈𝑈𝜕𝜕𝑡𝑡 �

2

�𝑎𝑎

0𝑟𝑟𝑑𝑑𝑟𝑟. (14)

Where I0 is the inertial term given by:

𝐼𝐼0 = � 𝜌𝜌(𝑧𝑧)𝑑𝑑𝑧𝑧 ℎ/2

−ℎ/2. (15)

3 Homogenization procedure 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 (16) Then

𝐵𝐵11 = � 𝑄𝑄𝑖𝑖𝑖𝑖(𝑧𝑧′ + 𝛿𝛿)ℎ/2

−ℎ/2𝑑𝑑𝑧𝑧′ = 𝐵𝐵𝑖𝑖𝑖𝑖′ + 𝛿𝛿𝐴𝐴𝑖𝑖𝑖𝑖 (17)

And by selecting the distance δ in order to achieve 𝐵𝐵11′ = 0 so that

𝛿𝛿 = 𝐵𝐵11 𝐴𝐴11⁄

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

𝐷𝐷11′ = 𝐷𝐷𝑒𝑒𝑒𝑒𝑒𝑒 = (𝐷𝐷11 − 𝐵𝐵112 𝐴𝐴11⁄ ) (18)

The effective extensional stiffness 𝐴𝐴11does not change in this new coordinate system. 𝐴𝐴11 = 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒 As the transverse displacement in the new coordinate system is the same as in the

2020 Rachid El Kaak et al. previous one; equations (12) and (13) can be expressed in the new systems:

𝑉𝑉𝑏𝑏 = 𝜋𝜋 𝐷𝐷𝑒𝑒𝐷𝐷𝐷𝐷 � ��𝜕𝜕2𝜕𝜕𝜕𝜕𝑟𝑟2 �

2

+ 1𝑟𝑟2 �

𝜕𝜕𝜕𝜕𝜕𝜕𝑟𝑟 �

2

+ 2 𝜈𝜈𝑟𝑟𝜕𝜕𝜕𝜕𝜕𝜕𝑟𝑟

𝜕𝜕2𝜕𝜕𝜕𝜕𝑟𝑟2 �

𝑎𝑎

0 𝑟𝑟 𝑑𝑑𝑟𝑟, (19)

𝑉𝑉𝑚𝑚 = 𝜋𝜋 𝐴𝐴𝑒𝑒𝐷𝐷𝐷𝐷� ��𝜕𝜕𝑈𝑈𝜕𝜕𝑟𝑟�

2

+ 𝜕𝜕𝑈𝑈𝜕𝜕𝑟𝑟 �

𝜕𝜕𝜕𝜕𝜕𝜕𝑟𝑟 �

2

+ 14 �𝜕𝜕𝜕𝜕𝜕𝜕𝑟𝑟 �

4

+ 𝑈𝑈2

𝑟𝑟2 +2𝜈𝜈 𝑈𝑈𝑟𝑟

𝜕𝜕𝑈𝑈𝜕𝜕𝑟𝑟 +

𝑎𝑎

0… ..

+ 𝜈𝜈 𝑈𝑈𝑟𝑟 �

𝜕𝜕𝜕𝜕𝜕𝜕𝑟𝑟 �

2

� 𝑟𝑟𝑑𝑑𝑟𝑟, (20)

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: 𝜕𝜕(𝑟𝑟, 𝑡𝑡) = 𝑤𝑤(𝑟𝑟 )𝑐𝑐𝑐𝑐𝑐𝑐(𝜔𝜔𝑡𝑡), (21) The in-plane radial displacement function U(r,t) in the following form: 𝑈𝑈(𝑟𝑟, 𝑡𝑡) = 𝑢𝑢(𝑟𝑟 )𝑐𝑐𝑐𝑐𝑐𝑐2(𝜔𝜔𝑡𝑡), (22) 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) = 𝑎𝑎𝑖𝑖𝑤𝑤𝑖𝑖(𝑟𝑟 ), 𝑢𝑢(𝑟𝑟) = 𝑏𝑏𝑖𝑖𝑢𝑢𝑖𝑖(𝑟𝑟 ), (23) Where the usual summation convention for repeated indices is used from 1 to po and from1 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 =

12𝑎𝑎𝑖𝑖𝑎𝑎𝑖𝑖𝑘𝑘𝑖𝑖𝑖𝑖𝑤𝑤𝑐𝑐𝑐𝑐𝑐𝑐2(𝜔𝜔𝑡𝑡) +

12�𝑎𝑎𝑖𝑖𝑎𝑎𝑖𝑖𝑎𝑎𝑘𝑘𝑎𝑎𝑙𝑙𝑏𝑏𝑖𝑖𝑖𝑖𝑘𝑘𝑙𝑙𝑤𝑤 + 𝑎𝑎𝑖𝑖𝑎𝑎𝑖𝑖𝑏𝑏𝑘𝑘𝐶𝐶𝑖𝑖𝑖𝑖𝑘𝑘𝑢𝑢𝑤𝑤 + 𝑏𝑏𝑖𝑖𝑏𝑏𝑖𝑖𝑘𝑘𝑖𝑖𝑖𝑖𝑢𝑢 �𝑐𝑐𝑐𝑐𝑐𝑐4(𝜔𝜔𝑡𝑡), (24)

𝑇𝑇 = 12𝜔𝜔2�𝑎𝑎𝑖𝑖𝑎𝑎𝑖𝑖𝑚𝑚𝑖𝑖𝑖𝑖

𝑤𝑤𝑐𝑐𝑠𝑠𝑠𝑠2(𝜔𝜔𝑡𝑡) + 𝑏𝑏𝑖𝑖𝑏𝑏𝑖𝑖𝑚𝑚𝑖𝑖𝑖𝑖𝑢𝑢 𝑐𝑐𝑠𝑠𝑠𝑠2(2𝜔𝜔𝑡𝑡)�, (25)

In these equations, 𝒎𝒎𝒊𝒊𝒊𝒊

𝒘𝒘,𝒎𝒎𝒊𝒊𝒊𝒊𝒖𝒖 ,𝒌𝒌𝒊𝒊𝒊𝒊𝒘𝒘,𝒌𝒌𝒊𝒊𝒊𝒊𝒖𝒖 are the mass and rigidity tensors associated

with W and U, respectively,𝒃𝒃𝒊𝒊𝒊𝒊𝒌𝒌𝒊𝒊𝒘𝒘 and 𝑪𝑪𝒊𝒊𝒊𝒊𝒌𝒌𝒖𝒖𝒘𝒘 are a fourth and a third order non-linearity and coupling tensors respectively. The general terms of these tensors are given by

Geometrically non-linear free vibration analysis 2021

𝑚𝑚𝑖𝑖𝑖𝑖𝑤𝑤 = 2𝜋𝜋𝐼𝐼 � 𝑤𝑤𝑖𝑖𝑤𝑤𝑖𝑖

𝑎𝑎

0𝑟𝑟 𝑑𝑑𝑟𝑟 , 𝑚𝑚𝑖𝑖𝑖𝑖

𝑢𝑢 = 2𝜋𝜋𝐼𝐼 � 𝑢𝑢𝑖𝑖𝑢𝑢𝑖𝑖𝑎𝑎

0𝑟𝑟 𝑑𝑑𝑟𝑟 , (26𝑎𝑎, 26𝑏𝑏)

𝑘𝑘𝑖𝑖𝑖𝑖𝑤𝑤 = 2𝜋𝜋 𝐷𝐷𝑒𝑒𝑒𝑒𝑒𝑒 � �𝑑𝑑𝑤𝑤𝑖𝑖

2

𝑑𝑑𝑟𝑟2𝑑𝑑𝑤𝑤𝑖𝑖2

𝑑𝑑𝑟𝑟2 + 1𝑟𝑟2𝑑𝑑𝑤𝑤𝑖𝑖

𝑑𝑑𝑟𝑟𝑑𝑑 𝑤𝑤𝑖𝑖𝑑𝑑𝑟𝑟 �

𝑎𝑎

0 𝑟𝑟 𝑑𝑑𝑟𝑟, (27)

𝑘𝑘𝑖𝑖𝑖𝑖𝑢𝑢 = 2𝜋𝜋 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒 � �𝑑𝑑𝑢𝑢𝑖𝑖𝑑𝑑𝑟𝑟

𝑑𝑑 𝑢𝑢𝑖𝑖𝑑𝑑𝑟𝑟 +

1𝑟𝑟2 𝑢𝑢𝑖𝑖𝑢𝑢𝑖𝑖 +

𝜈𝜈𝑟𝑟𝑑𝑑 𝑢𝑢𝑖𝑖𝑑𝑑𝑟𝑟 𝑢𝑢𝑖𝑖 +

𝜈𝜈𝑟𝑟 𝑢𝑢𝑖𝑖

𝑑𝑑 𝑢𝑢𝑖𝑖𝑑𝑑𝑟𝑟 �

𝑎𝑎

0 𝑟𝑟 𝑑𝑑𝑟𝑟 , (28)

𝐶𝐶𝑖𝑖𝑖𝑖𝑘𝑘𝑢𝑢𝑤𝑤 = 2𝜋𝜋 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒 � �𝑑𝑑𝑤𝑤𝑖𝑖

𝑑𝑑𝑟𝑟𝑑𝑑 𝑤𝑤𝑖𝑖𝑑𝑑𝑟𝑟

𝑑𝑑 𝑢𝑢𝑘𝑘𝑑𝑑𝑟𝑟 +

𝜈𝜈𝑟𝑟𝑑𝑑𝑤𝑤𝑖𝑖

𝑑𝑑𝑟𝑟𝑑𝑑 𝑤𝑤𝑖𝑖𝑑𝑑𝑟𝑟 𝑢𝑢𝑘𝑘�

𝑎𝑎

0 𝑟𝑟 𝑑𝑑𝑟𝑟 , (29)

𝑏𝑏𝑖𝑖𝑖𝑖𝑘𝑘𝑙𝑙𝑤𝑤 = 𝜋𝜋 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒

2 � �𝑑𝑑𝑤𝑤𝑖𝑖

𝑑𝑑𝑟𝑟𝑑𝑑𝑤𝑤𝑖𝑖𝑑𝑑𝑟𝑟

𝑑𝑑𝑤𝑤𝑘𝑘

𝑑𝑑𝑟𝑟𝑑𝑑𝑤𝑤𝑙𝑙

𝑑𝑑𝑟𝑟 �𝑎𝑎

0 𝑟𝑟 𝑑𝑑𝑟𝑟 , (30)

It appears from Eqs. (26) - (30) that the mass and rigidity tensors are symmetric, and the fourth order 𝒃𝒃𝒊𝒊𝒊𝒊𝒌𝒌𝒊𝒊𝒘𝒘 and the third order tensors 𝑪𝑪𝒊𝒊𝒊𝒊𝒌𝒌𝒖𝒖𝒘𝒘 are illustrated below 𝑏𝑏𝑖𝑖𝑖𝑖𝑘𝑘𝑙𝑙𝑤𝑤 = 𝑏𝑏𝑘𝑘𝑙𝑙𝑖𝑖𝑖𝑖𝑤𝑤 = 𝑏𝑏𝑖𝑖𝑖𝑖𝑙𝑙𝑘𝑘𝑤𝑤 = 𝑏𝑏𝑖𝑖𝑘𝑘𝑖𝑖𝑙𝑙𝑤𝑤 , 𝐶𝐶𝑖𝑖𝑖𝑖𝑘𝑘𝑢𝑢𝑤𝑤 = 𝐶𝐶𝑖𝑖𝑖𝑖𝑘𝑘𝑢𝑢𝑤𝑤 (31) The dynamic behavior of the structure is governed by Hamilton’s principle, which is symbolically written as

𝛿𝛿 � (𝑉𝑉 − 𝑇𝑇) 𝑑𝑑𝑡𝑡 𝜋𝜋2𝜔𝜔

0= 𝛿𝛿∅ = 0 (32)

Replacing T and V by their discretized expressions in the energy condition (32), integrating the time functions and calculating the derivatives with respect to the 𝒂𝒂𝒊𝒊

, 𝒔𝒔 and 𝒃𝒃𝒊𝒊, 𝒔𝒔 , and taking into account the properties of symmetry of the tensors

involved, leads to the following set of non-linear algebraic equations:

2 𝑎𝑎𝑖𝑖𝑘𝑘𝑖𝑖𝑟𝑟𝑤𝑤 + 3𝑎𝑎𝑖𝑖𝑎𝑎𝑖𝑖𝑎𝑎𝑘𝑘𝑏𝑏𝑖𝑖𝑖𝑖𝑘𝑘𝑟𝑟𝑤𝑤 +32

𝑎𝑎𝑖𝑖𝑏𝑏𝑘𝑘𝐶𝐶𝑖𝑖𝑟𝑟𝑘𝑘𝑢𝑢𝑤𝑤 − 2𝜔𝜔2𝑎𝑎𝑖𝑖𝑚𝑚𝑖𝑖𝑟𝑟𝑤𝑤 = 0, 𝑟𝑟 = 1, … ,𝑝𝑝𝑐𝑐. (33𝑎𝑎)

34 �

𝑎𝑎𝑖𝑖𝑎𝑎𝑖𝑖𝐶𝐶𝑖𝑖𝑖𝑖𝑖𝑖𝑢𝑢𝑤𝑤 + 2𝑏𝑏𝑖𝑖𝑘𝑘𝑖𝑖𝑖𝑖𝑢𝑢 � − 2𝜔𝜔2𝑏𝑏𝑖𝑖𝑚𝑚𝑖𝑖𝑖𝑖𝑢𝑢 = 0, 𝑐𝑐 = 1, … ,𝑝𝑝𝑠𝑠. (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: 𝑤𝑤𝑖𝑖(𝑟𝑟) = ℎ𝑤𝑤𝑖𝑖

∗(𝑟𝑟∗), 𝑢𝑢𝑖𝑖(𝑟𝑟) = 𝜆𝜆ℎ𝑢𝑢𝑖𝑖∗(𝑟𝑟∗), (34)

2022 Rachid El Kaak et al. Where 𝒓𝒓∗ = 𝒓𝒓 𝒂𝒂⁄ is the non-dimensional radial co-ordinate and 𝛌𝛌 = 𝒉𝒉 𝒂𝒂⁄ 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

2 𝑎𝑎𝑖𝑖𝑘𝑘𝑖𝑖𝑟𝑟𝑤𝑤∗ + 3𝑎𝑎𝑖𝑖𝑎𝑎𝑖𝑖𝑎𝑎𝑘𝑘𝑏𝑏𝑖𝑖𝑖𝑖𝑘𝑘𝑟𝑟𝑤𝑤∗ +32

𝑎𝑎𝑖𝑖𝑏𝑏𝑘𝑘𝐶𝐶𝑖𝑖𝑟𝑟𝑘𝑘𝑢𝑢𝑤𝑤∗ − 2𝜔𝜔∗2𝑎𝑎𝑖𝑖𝑚𝑚𝑖𝑖𝑟𝑟𝑤𝑤∗ = 0, 𝑟𝑟 = 1, … ,𝑝𝑝𝑐𝑐. (35𝑎𝑎)

34 � 𝑎𝑎𝑖𝑖𝑎𝑎𝑖𝑖𝐶𝐶𝑖𝑖𝑟𝑟𝑖𝑖𝑢𝑢𝑤𝑤∗ + 2𝑏𝑏𝑖𝑖𝑘𝑘𝑖𝑖𝑖𝑖𝑢𝑢∗� − 2𝜆𝜆2𝜔𝜔∗2𝑏𝑏𝑖𝑖𝑚𝑚𝑖𝑖𝑖𝑖

𝑢𝑢∗ = 0, 𝑐𝑐 = 1, … ,𝑝𝑝𝑠𝑠. (35𝑏𝑏) Where 𝛚𝛚∗is the non-dimensional non-linear frequency parameter defined by:

𝜔𝜔∗2 = 𝛾𝛾𝜔𝜔2, 𝑤𝑤𝑠𝑠𝑡𝑡ℎ, 𝛾𝛾 =𝐼𝐼𝑎𝑎4

𝐷𝐷𝑒𝑒𝑒𝑒𝑒𝑒, (36)

The non-dimensional terms 𝒎𝒎𝒊𝒊𝒊𝒊

𝒘𝒘∗,𝒎𝒎𝒊𝒊𝒊𝒊𝒖𝒖∗,𝒌𝒌𝒊𝒊𝒊𝒊𝒘𝒘∗,𝒌𝒌𝒊𝒊𝒊𝒊𝒖𝒖∗,𝑪𝑪𝒊𝒊𝒊𝒊𝒌𝒌𝒖𝒖𝒘𝒘∗𝒂𝒂𝒂𝒂𝒂𝒂𝒃𝒃𝒊𝒊𝒊𝒊𝒌𝒌𝒓𝒓𝒘𝒘∗ are given by:

𝑚𝑚𝑖𝑖𝑖𝑖𝑤𝑤∗ = � 𝑤𝑤𝑖𝑖

∗𝑤𝑤𝑖𝑖∗1

0𝑟𝑟∗𝑑𝑑𝑟𝑟∗ , 𝑚𝑚𝑖𝑖𝑖𝑖

𝑢𝑢∗ = � 𝑢𝑢𝑖𝑖∗𝑢𝑢𝑖𝑖∗1

0𝑟𝑟∗𝑑𝑑𝑟𝑟∗ , (37𝑎𝑎, 37𝑏𝑏)

𝑘𝑘𝑖𝑖𝑖𝑖𝑤𝑤∗ = � �𝑑𝑑2𝑤𝑤𝑖𝑖

∗

𝑑𝑑𝑟𝑟∗2𝑑𝑑2𝑤𝑤𝑖𝑖∗

𝑑𝑑𝑟𝑟∗2 + 1𝑟𝑟∗2

𝑑𝑑𝑤𝑤𝑖𝑖∗

𝑑𝑑𝑟𝑟∗𝑑𝑑 𝑤𝑤𝑖𝑖

∗

𝑑𝑑𝑟𝑟∗ �1

0𝑟𝑟∗𝑑𝑑𝑟𝑟∗ (38)

𝑘𝑘𝑖𝑖𝑖𝑖𝑢𝑢∗ = 𝛽𝛽� �𝑑𝑑𝑢𝑢𝑖𝑖

∗

𝑑𝑑𝑟𝑟∗𝑑𝑑 𝑢𝑢𝑖𝑖

∗

𝑑𝑑𝑟𝑟∗ +1𝑟𝑟∗2 𝑢𝑢𝑖𝑖

∗𝑢𝑢𝑖𝑖∗ +𝜈𝜈𝑟𝑟∗𝑑𝑑 𝑢𝑢𝑖𝑖

∗

𝑑𝑑𝑟𝑟∗ 𝑢𝑢𝑖𝑖∗ +

𝜈𝜈𝑟𝑟∗ 𝑢𝑢𝑖𝑖

∗ 𝑑𝑑 𝑢𝑢𝑖𝑖∗

𝑑𝑑𝑟𝑟∗ �1

0𝑟𝑟∗𝑑𝑑𝑟𝑟∗ (39)

𝐶𝐶𝑖𝑖𝑖𝑖𝑘𝑘𝑢𝑢𝑤𝑤∗ = 𝛽𝛽� �𝑑𝑑𝑤𝑤𝑖𝑖

∗

𝑑𝑑𝑟𝑟∗𝑑𝑑 𝑤𝑤𝑖𝑖

∗

𝑑𝑑𝑟𝑟∗𝑑𝑑 𝑢𝑢𝑘𝑘

∗

𝑑𝑑𝑟𝑟∗ +𝜈𝜈𝑟𝑟∗𝑑𝑑𝑤𝑤𝑖𝑖

∗

𝑑𝑑𝑟𝑟∗𝑑𝑑 𝑤𝑤𝑖𝑖

∗

𝑑𝑑𝑟𝑟∗ 𝑢𝑢𝑘𝑘∗�1

0𝑟𝑟∗𝑑𝑑𝑟𝑟∗, (40)

𝑏𝑏𝑖𝑖𝑖𝑖𝑘𝑘𝑙𝑙𝑤𝑤∗ = 𝛼𝛼� �𝑑𝑑𝑤𝑤𝑖𝑖

∗

𝑑𝑑𝑟𝑟∗𝑑𝑑 𝑤𝑤𝑖𝑖

∗

𝑑𝑑𝑟𝑟∗𝑑𝑑𝑤𝑤𝑘𝑘

∗

𝑑𝑑𝑟𝑟∗𝑑𝑑 𝑤𝑤𝑙𝑙

∗

𝑑𝑑𝑟𝑟∗ �1

0𝑟𝑟∗𝑑𝑑𝑟𝑟∗ , (41)

with

𝛽𝛽 = 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒 ℎ2

𝐷𝐷𝑒𝑒𝑒𝑒𝑒𝑒 , 𝛼𝛼 =

𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒 ℎ2

4 𝐷𝐷𝑒𝑒𝑒𝑒𝑒𝑒, (42)

These non-dimensional tensors are related to the dimensional ones by the following equations:

�𝑚𝑚𝑖𝑖𝑖𝑖

𝑢𝑢 ,𝑚𝑚𝑖𝑖𝑖𝑖𝑤𝑤� = 2𝜋𝜋𝐼𝐼ℎ2𝑎𝑎2�𝜆𝜆2𝑚𝑚𝑖𝑖𝑖𝑖

𝑢𝑢∗,𝑚𝑚𝑖𝑖𝑖𝑖𝑤𝑤∗�, (43)

Geometrically non-linear free vibration analysis 2023

𝑘𝑘𝑖𝑖𝑖𝑖𝑤𝑤 = 2 𝜋𝜋 𝐷𝐷𝑒𝑒𝑒𝑒𝑒𝑒 ℎ2

𝑎𝑎2 𝑘𝑘𝑖𝑖𝑖𝑖𝑤𝑤∗ , (44) 𝑘𝑘𝑖𝑖𝑖𝑖𝑢𝑢 = 2 𝜋𝜋 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒 𝜆𝜆2ℎ2𝑘𝑘𝑖𝑖𝑖𝑖𝑢𝑢∗, (45)

𝐶𝐶𝑖𝑖𝑖𝑖𝑘𝑘𝑢𝑢𝑤𝑤 = 2 𝜋𝜋 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒 𝜆𝜆 ℎ3

𝑎𝑎 𝐶𝐶𝑖𝑖𝑖𝑖𝑘𝑘𝑢𝑢𝑤𝑤∗ , (46)

𝑏𝑏𝑖𝑖𝑖𝑖𝑘𝑘𝑙𝑙𝑤𝑤 = 𝜋𝜋 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒 ℎ4

2 𝑎𝑎2 𝑏𝑏𝑖𝑖𝑖𝑖𝑘𝑘𝑙𝑙𝑤𝑤∗ , (47) In the case of thin plates, for which𝛌𝛌 is very small; the in-plane inertia term involving the term𝛌𝛌𝟐𝟐 can be neglected. This is an acceptable assumption in most engineering applications of thin plates [14]. Consequently, equation (33b) can be solved for the𝒃𝒃𝒊𝒊′𝒔𝒔 leading to: 𝑏𝑏𝑖𝑖 = 𝑎𝑎𝑖𝑖𝑎𝑎𝑙𝑙𝑑𝑑𝑖𝑖𝑙𝑙𝑖𝑖∗ , 𝑠𝑠 = 1, … , 𝑝𝑝𝑠𝑠, (48) where

𝑑𝑑𝑖𝑖𝑖𝑖𝑘𝑘∗ = −12𝑘𝑘𝑖𝑖𝑖𝑖

𝑢𝑢∗−1𝐶𝐶𝑖𝑖𝑖𝑖𝑘𝑘𝑢𝑢𝑤𝑤∗ (49) 𝒂𝒂𝒊𝒊𝒊𝒊𝒌𝒌∗ is a third order tensor expressing the coupling between the in-plane and out-of-plane vibrations and 𝒌𝒌𝒊𝒊𝒊𝒊𝒖𝒖∗−𝟏𝟏 is the inverse of the tensor 𝒌𝒌𝒊𝒊𝒊𝒊𝒖𝒖∗. Substituting equation (48) by equation (33a) leads to an uncoupled set of non-linear algebraic equations in terms of the𝒂𝒂𝒊𝒊′𝒔𝒔coefficients only.

𝑎𝑎𝑖𝑖𝑘𝑘𝑖𝑖𝑟𝑟𝑤𝑤∗ + 32𝑎𝑎𝑖𝑖𝑎𝑎𝑖𝑖𝑎𝑎𝑘𝑘𝑏𝑏𝑖𝑖𝑖𝑖𝑘𝑘𝑟𝑟

∗ − 𝜔𝜔∗2𝑎𝑎𝑖𝑖𝑚𝑚𝑖𝑖𝑟𝑟𝑤𝑤∗ = 0, 𝑟𝑟 = 1, … , 𝑝𝑝𝑐𝑐. (50)

𝒃𝒃𝒊𝒊𝒊𝒊𝒌𝒌𝒊𝒊∗ is a fourth order tensor given by:

𝑏𝑏𝑖𝑖𝑖𝑖𝑘𝑘𝑙𝑙∗ = 𝑏𝑏𝑖𝑖𝑖𝑖𝑘𝑘𝑙𝑙𝑤𝑤∗ + 12𝐶𝐶𝑖𝑖𝑖𝑖𝑛𝑛

𝑢𝑢𝑤𝑤∗𝑑𝑑𝑘𝑘𝑙𝑙𝑛𝑛∗ . (51) Equation (50) 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. (50) can be written in matrix form as:

{𝐴𝐴}𝑇𝑇[𝐾𝐾𝑤𝑤∗]{𝐴𝐴} +32

{𝐴𝐴}𝑇𝑇[𝐵𝐵∗]{𝐴𝐴}− 𝜔𝜔∗2{𝐴𝐴}𝑇𝑇[𝑀𝑀𝑤𝑤∗]{𝐴𝐴} = 0, (52)

2024 Rachid El Kaak et al. Pre-multiplying Eq. (50) by the vector (𝑨𝑨)𝑻𝑻 = [𝒂𝒂𝟏𝟏 ,𝒂𝒂𝟐𝟐, … … ,𝒂𝒂𝒂𝒂] leads to the following expression for 𝝎𝝎∗𝟐𝟐 :

𝜔𝜔∗2 =𝑎𝑎𝑖𝑖𝑎𝑎𝑖𝑖𝑘𝑘𝑖𝑖𝑖𝑖𝑤𝑤∗ + 3

2𝑎𝑎𝑖𝑖𝑎𝑎𝑖𝑖𝑎𝑎𝑘𝑘𝑎𝑎𝑙𝑙𝑏𝑏𝑖𝑖𝑖𝑖𝑘𝑘𝑙𝑙∗

𝑎𝑎𝑖𝑖𝑎𝑎𝑖𝑖𝑚𝑚𝑖𝑖𝑟𝑟𝑤𝑤∗ , (53)

The basic functions 𝒘𝒘𝒊𝒊∗ 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 𝒘𝒘𝒊𝒊

∗were taken as the linear mode shapes of fully clamped circular plates given by [14]:

𝑤𝑤𝑖𝑖∗(𝑟𝑟∗) = 𝐴𝐴𝑖𝑖 �𝐽𝐽0(𝛽𝛽𝑖𝑖𝑟𝑟∗)−

𝐽𝐽0(𝛽𝛽𝑖𝑖)𝐼𝐼0(𝛽𝛽𝑖𝑖)

𝐼𝐼0(𝛽𝛽𝑖𝑖𝑟𝑟∗)� , (54)

Where 𝜷𝜷𝒊𝒊 is the ith real positive root of the transcendental equation. 𝐽𝐽1(𝛽𝛽)𝐼𝐼0(𝛽𝛽) + 𝐽𝐽0(𝛽𝛽)𝐼𝐼1(𝛽𝛽) = 0 (55) In which 𝑱𝑱𝒂𝒂 and 𝑰𝑰𝒂𝒂 are, respectively, the Bessel and the modified Bessel functions of the first kind and of order n, The parameter 𝜷𝜷𝒊𝒊 is related to the ith non-dimensional linear frequency parameter (𝝎𝝎𝒊𝒊

∗)𝒊𝒊 of the plate by: 𝜷𝜷𝒊𝒊𝟐𝟐 = (𝝎𝝎𝒊𝒊

∗)𝒊𝒊 .𝑨𝑨𝒊𝒊 was chosen such that:

� 𝑤𝑤𝑖𝑖∗2

1

0𝑟𝑟∗𝑑𝑑𝑟𝑟∗ = 1. (56)

The chosen in-plane basic functions 𝒖𝒖𝒊𝒊∗(𝒓𝒓) for an immovable axisymmetric FG circular plate are given by: 𝑢𝑢𝑖𝑖∗(𝑟𝑟∗) = 𝐵𝐵𝑖𝑖𝐽𝐽1(𝛼𝛼𝑖𝑖𝑟𝑟∗) (57) Where 𝜶𝜶𝒊𝒊 is the ith real positive root of the equation 𝑱𝑱𝟏𝟏(𝜶𝜶) = 𝟎𝟎, the functions 𝒘𝒘𝒊𝒊∗(𝒓𝒓)and 𝒖𝒖𝒊𝒊∗(𝒓𝒓) are normalized in such a manner that

𝑚𝑚𝑖𝑖𝑖𝑖𝑤𝑤∗ = � 𝑤𝑤𝑖𝑖

∗𝑤𝑤𝑖𝑖∗1

0𝑟𝑟∗𝑑𝑑𝑟𝑟∗ = 𝛿𝛿𝑖𝑖𝑖𝑖 , 𝑚𝑚𝑖𝑖𝑖𝑖

𝑢𝑢∗ = � 𝑢𝑢𝑖𝑖∗𝑢𝑢𝑖𝑖∗1

0𝑟𝑟∗𝑑𝑑𝑟𝑟∗ = 𝛿𝛿𝑖𝑖𝑖𝑖 . (58𝑎𝑎, 58𝑏𝑏)

5 Numerical results and discussion 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 [18]. The set of non-linear algebraic equations

Geometrically non-linear free vibration analysis 2025 (50) is solved numerically using the Harwell library routine NS01A, which is based on a hybrid iterative method combining the steep descent and Newton's methods. 5.1 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 2 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 1, the non-linear frequency ratio of a functionally graded circular plate obtained in ref. [18] 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 2 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 1 and 2 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=1.5. 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 [0, 2], the frequency increases. For values higher than n = 2.0, the frequency decreases when n increases. This may be expected, since when the power law index n = 0.0 or n = 1000.0, the material is pure metallic or pure ceramic respectively and the non-dimensional frequency corresponds the isotropic homogeneous materiel case.

Figure 2: Effect of coupling on the non-linear frequency ratios for FG circular

plate, (n = 0.5): (—) Uncoupled model [35], (- - -) Coupled model (Present work)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.81

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Non-dimensional amplitude

Non

-line

ar to

line

ar fr

eque

ncy

ratio

(1) : Uncoupled model [35](2) : Coupled model

(2)

(1)

Tm = Tc = 300(K)n = 0.5

2026 Rachid El Kaak et al.

Table 1: Frequency ratios (𝜔𝜔𝑛𝑛𝑙𝑙∗ 𝜔𝜔𝑙𝑙

∗⁄ ) of the clamped coupled FGCP

Method n 𝒘𝒘𝒎𝒎𝒂𝒂𝒎𝒎∗

0.2 0.4 0.8 1.0 1.5

Present work 0.5 1.0076 1.0301 1.1153 1.1752 1.3605

Coupled FGCP.

(Model with u-w formulation)

Ref.[18] 1.0074 1.0259 1.1037 1.1629 1.3370

Coupled FGCP.

(Model with u-w formulation)

Deviation (%) 0.0198 0.4077 1.0400 1.0466 1.7273

Table 2: Comparison of the non linear frequency ratios (𝜔𝜔𝑛𝑛𝑙𝑙∗ 𝜔𝜔𝑙𝑙

∗⁄ ) obtained via the coupledisotropic homogeneous plate and uncoupled models for isotropic

homogeneous and functionally graded circular plates

Method n 𝒘𝒘𝒎𝒎𝒂𝒂𝒎𝒎∗

0.2 0.4 0.8 1.0 1.5

Present work 0.0 1.0075 1.0296 1.1135 1.1724 1.3548

Coupled isotropic plate.

(Model with u-w formulation)

Ref.[35] 1.0108 1.0421 1.1560 1.2318 1.4542

Uncoupled isotropic plate.

(Model with w formulation only)

Present work 0.5 1.0110 1.0429 1.1587 1.2393 1.4610

Uncoupled FGCP

(Model with w formulation only)

Geometrically non-linear free vibration analysis 2027

Figure 3: Effect of the power law index (n) on the variation of the non-linear frequency ratios (𝜔𝜔𝑛𝑛𝑙𝑙

∗ 𝜔𝜔𝑙𝑙∗⁄ ) of the clamped coupled FG circular plate, with the

amplitude of vibration 5.2 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 = 0.5 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 = 0.5 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 = 0.0 is coherent with that given in ref. [14] 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

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.61

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

Non- dimensional amplitude

Non

-line

ar to

line

ar fr

eque

ncy

ratio

n = 0.0n = 0.5n = 2.0n = 10.0n = 1000.0

1.02 1.041.17

1.175

1.18

Tm = Tc = 300(K)

2028 Rachid El Kaak et al. mode shape of a clamped FG circular plate for n = 0.5 and W*max = 1.5 are plotted in figure 9. It can be shown that the membrane stresses contribute significantly to the total radial stresses.

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 = 0.5

and various non-dimensional vibration amplitudes

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

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-6

-5

-4

-3

-2

-1

0

1

2

3

4x 10

-3

r *

Non

-dim

ensi

onal

radi

al b

endi

ng s

tress

(1) : w*max = 0.03(2) : w*max = 0.5(3) : w*max = 1.0(4) : w*max = 1.5

(4)

(3)

(2)

(1)

Tm = Tc = 300(K)n = 0.5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

1

2

3

4

5

6

Non-dimensional amplitude

Non

-dim

ensi

onal

radi

al b

endi

ng st

ress

Edge

Centre

Tm = Tc = 300(K)n = 0.5

Geometrically non-linear free vibration analysis 2029

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

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 =

0.5 and various non-dimensional vibration amplitudes

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.5

1

1.5

2

2.5

3

Non-dimensional amplitude

Non

-dim

ensi

onal

radi

al m

embr

ane

stre

ss

Centre

Edge

Tm = Tc = 300(K)n = 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

1

2

3

4

5

6

7

8

9

x 10-4

r *

Non

-dim

ensi

onal

radi

al m

embr

ane

stre

ss

(1) : w*max = 0.03(2) : w*max = 0.5(3) : w*max = 1.0(4) : w*max = 1.5

(3)

(2)

(1)

(4)

Tm = Tc = 300(K)n = 0.5

2030 Rachid El Kaak et al.

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 = 0.5

and various non-dimensional vibration amplitudes

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 = 0.5 and W*max = 1.5 The dimensionless radial bending and membrane stresses with dimensionless thickness of the plate for different values of n at T = 300 (K) and W* max = 1.5 are depicted in figures (10a-10b) and (11-11b). 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 = 0 and n = ∞), the stress distribution is linear, where for the functionally graded (Si3N4/SUS304) circular plate, the behavior is nonlinear and is governed by the variation of the properties in the thickness

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3

-2

-1

0

1

2

3

4

5

6x 10-3

r *

Non

-dim

ensi

onal

radi

al to

tal s

tress

(1) : w*max = 0.03(2) : w*max = 0.5(3) : w*max = 1.0(4) : w*max = 1.5

(1)

(2)

(3)

(4)Tm = Tc = 300(K)n = 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-6

-4

-2

0

2

4

6x 10-3

r *

Non

-dim

ensi

onal

radi

al st

ress

(1) : Bending stress (2) : Membrane stress(3) : Total stress(1)

(3)

(2)

Tm = Tc = 300(K)W*max = 1.5n = 0.5

Geometrically non-linear free vibration analysis 2031 direction. The material properties of the functionally graded plate are assumed to vary through the thickness of the plate. Figures 12(a) and 12(b) demonstrate how the radial bending and membrane stress distributions exhibit a regular and continuous change from one surface to another.

Figure 10: 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

= 1.5, case (a): Centre, case (b): Edge

Figure 11: 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 = 1.5, case (a): Centre, case (b): Edge

Figure 12: Variation of the radial bending stress (a) and radial membrane stress (b)

of clamped FG circular plate on different thickness distribution for n = 0.5 and W*max = 1.5

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Radial bending stress

Z *

(1) : n = 0.0(2) : n = 0.5(3) : n = 5.0(4) : n = 10.0(5) : n = 1000

(2)

(3)

(4)(5)

(1)

Tc = Tm = 300(K)W*max = 1.5

( a )

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Radial bending stressZ

*

(1) : n = 0.0(2) : n = 0.5(3) : n = 5.0(4) : n = 10.0(5) : n = 1000.0

(2)

(4)

(5)

(3)

(1)

( b )

Tc = Tm = 300(K)W*max = 1.5

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Radial membrane stress

Z *

(1) : n = 0.0(2) : n = 0.5(3) : n = 5.0(4) : n = 10.0(5) : n = 1000.0

( a )

(1)

(2)

(3)

(4)

(5)

Tc = Tm = 300(k)W*max = 1.5

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Radial membrane stress

Z *

(1) : n = 0.0(2) : n = 0.5(3) : n = 5.0(4) : n = 10.0(5) : n = 1000.0

( b )

(2)

(3)

(4)

(5)

(1)Tc = Tm = 300(K)W*max = 1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

r *

Radi

al b

endi

ng st

ress

(1) : z = - h/2(2) : z = - h/4(3) : z = 0.0(4) : z = h/4(5) : z = h/2

(2)

(3)

(4)

(5)

Tc = Tm = 300(K)W*max = 1.5n = 0.5

(1)

( a )

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1

-0.05

0

0.05

0.1

0.15

r *

Radi

al m

embr

ane

stre

ss

z = - h / 2

z = - h / 4

z = h / 4

z = h / 2

z = 0.0

( b )

Tc = Tm = 300(K)W*max = 1.5n = 0.5

2032 Rachid El Kaak 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 [14] 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 [1] R.G. White, Developments in the acoustic fatigue design process for

composite aircraft structures, Composite Structures, 16 (1990), 171-192. http://dx.doi.org/10.1016/0263-8223(90)90071-l

[2] R. Benamar, Non-linear Dynamic Behaviour of Fully Clamped Beams and

Rectangular Isotropic and Laminated Plates, Ph.D. Thesis, University of Southampton, 1990.

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http://dx.doi.org/10.4028/www.scientific.net/amr.971-973.489 Received: April 10, 2016; Published: June 15, 2016