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Linear and non-linear vibration analysis of moderately thick isosceles triangular FGPs using a triangular finite p-element

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Abstract

Background

The geometrically non-linear formulation based on Von-Karman’s hypothesis is used to study the free vibration isosceles triangular plates by using four types of mixtures of functionally graded materials (FGMs - AL/AL2O3, SUS304/Si3N4, Ti- AL-4V/Aluminum oxide, AL/ZrO2). Material properties are assumed to be temperature dependent and graded in the thickness direction according to power law distribution.

Methods

A hierarchical finite element based on triangular p-element is employed to define the model, taking into account the hypotheses of first-order shear deformation theory. The equations of non-linear free motion are derived from Lagrange's equation in combination with the harmonic balance method and solved iteratively using the linearized updated mode method.

Results

Results for the linear and nonlinear frequencies parameters of clamped isosceles triangular plates are obtained. The accuracy of the present results are established through convergence studies and comparison with results of literature for metallic plates. The results of the linear vibration of clamped FGMs isosceles triangular plates are also presented in this study.

Conclusion

The effects of apex angle, thickness ratio, volume fraction exponent and mixtures of FGMs on the backbone curves and mode shape of clamped isosceles triangular plates are studied. The results obtained in this work reveal that the physical and geometrical parameters have a important effect on the non-linear vibration of FGMs triangular plates.

Background

In recent years, the geometrically non-linear vibration of functionally graded Materials (FGMs) for different structures has acquired great interest in many researches. In 1984, The concept of the FGMs was introduced in Japan by scientific researchers (Koizumi 1993; Koizumi 1997). FGMs are composite materials which are microscopically inhomogeneous. The mechanical properties of FGMs are expressed with mathematical functions, and assumed to vary continuously from one surface to the other.

Since the variation of mechanical properties of FGM is nonlinear, therefore, studies based on the nonlinear deformation theory is required for these type of materials. Many works have studied the static and dynamic nonlinear behavior of functionally graded plates with various shapes. The group of researchers headed by (Reddy and Chin 1998; Reddy et al. 1999; Reddy 2000) have done a lot of numerical and theoretical work on FG plates under several effects (thermoelastic response, axisymmetric bending and stretching, finite element models, FSDT-plate and TSDT-plate). Woo & Meguid (2001) analyzed the nonlinear behavior of functionally graded shallow shells and thin plates under temperature effects and mechanical loads. The analysis of nonlinear bending of FG simply supported rectangular plates submissive to thermal and mechanical loading was studied by (Shen 2002). (Huang & Shen 2004) applied the perturbation technique to nonlinear vibration and dynamic response of FG plates in a thermal environment. Chen (2005) investigated the large amplitude vibration of FG plate with arbitrary initial stresses based on FSDT. An analytical solution was proposed by Woo et al. 2006 to analyzed the nonlinear vibration of functionally graded plates using classic plate theory. Allahverdizadeh et al. (2008a, 2008b) have studied the non-linear forced and free vibration analysis of circular functionally graded plate in thermal environment. The p-version of the FEM has been applied to investigate the non-linear free vibration of elliptic sector plates and functionally graded sector plates by (Belalia & Houmat 2010; 2012). Hao et al. 2011 analyzed the non-linear vibration of a cantilever functionally graded plate based on TSDT of plate and asymptotic analysis and perturbation method. Duc & Cong 2013 analyzed the non-linear dynamic response of imperfect symmetric thin sandwich FGM plate on elastic foundation. Yin et al. 2015 proposed a novel approach based on isogeometric analysis (IGA) for the geometrically nonlinear analysis of functionally graded plates (FGPs). the same approach (IGA) and a simple first-order shear deformation plate theory (S-FSDT) are used by Yu et al. 2015 to investigated geometrically nonlinear analysis of homogeneous and non-homogeneous functionally graded plates. Alinaghizadeh & Shariati 2016, investigated the non-linear bending analysis of variable thickness two-directional FG circular and annular sector plates resting on the non-linear elastic foundation using the generalized differential quadrature (GDQ) and the Newton–Raphson iterative methods.

The p-version FEM has many advantages over the classic finite element method (h-version), which includes the ability to increase the accuracy of the solution without re-defining the mesh (Han & Petyt 1997; Ribeiro 2003). This advantage is suitable in non-linear study because the problem is solved iteratively and the non-linear stiffness matrices are reconstructed throughout each iteration. Using the p-version with higher order polynomials, the structure is modeled by one element while satisfying the exactitude requirement. In p-version, the point where the maximum amplitude is easy to find it as there is a single element, contrary to the h-version this point must be sought in every element of the mesh which is very difficult. The advantages of the p-version mentioned previously, make it more powerful to the nonlinear vibration analysis of plates. So far, no work has been published to the study of linear and nonlinear vibration of FGMs isosceles triangular plate by using the p-version of FEM.

In the present work, the non-linear vibration analysis of moderately thick FGMs isosceles triangular plates was investigated by a triangular finite p-element. The shape functions of triangular finite p-element are obtained by the shifted orthogonal polynomials of Legendre. The effects of rotatory inertia and transverse shear deformations are taken into account (Mindlin 1951). The Von-Karman hypothesis are used in combination with the harmonic balance method (HBM) to obtained the motion equations. The resultant equations of motion are solved iteratively using the linearized updated mode method. The exactitude of the p-element is investigated with a clamped metallic triangular plate. Comparisons are made between current results and those from published results. The effects of thickness ratio, apex angle, exponent of volume fraction and mixtures of FGMs on the backbone curves and mode shape of clamped isosceles triangular plates are also studied.

Methods

Consider a moderately thick isosceles triangular plate with the following geometrical parameters thickness h, base b, height a and apex angle β (Fig. 1). The triangular p-element is mapped to global coordinates from the local coordinates ξ and η. The differential relationship between the two coordinates systems is given as a function of the Jacobian matrix ( J ) by

Fig. 1
figure1

Geometry of isosceles triangular plate

$$ \left\{\begin{array}{c}\hfill \frac{\partial }{\partial \xi}\hfill \\ {}\hfill \frac{\partial }{\partial \eta}\hfill \end{array}\right\}=\boldsymbol{J}\left\{\begin{array}{c}\hfill \frac{\partial }{\partial x}\hfill \\ {}\hfill \frac{\partial }{\partial y}\hfill \end{array}\right\} $$
(1)

where J is given by

$$ \boldsymbol{J}=\left[\begin{array}{cc}\hfill \frac{\partial x}{\partial \xi}\hfill & \hfill \frac{\partial y}{\partial \xi}\hfill \\ {}\hfill \frac{\partial x}{\partial \eta}\hfill & \hfill \frac{\partial y}{\partial \eta}\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill b\hfill & \hfill 0\hfill \\ {}\hfill b/2\hfill & \hfill b/2 tg\left(\frac{\beta}{2}\right)\hfill \end{array}\right] $$
(2)

In first-order shear deformation plate theory, the displacements (u, v and w) at a point with coordinate (x, y, z) from the median surface are given as functions of midplane displacements (u 0 , v 0 , w) and independent rotations (θ x and θ y ) about the x and y axes as

$$ \begin{array}{c}\hfill u\;\left( x, y, z, t\right)={u}_0\left( x, y, t\right)+ z{\theta}_y\left( x, y, t\right)\hfill \\ {}\hfill v\;\left( x, y, z, t\right)={v}_0\left( x, y, t\right)- z{\theta}_x\left( x, y, t\right)\hfill \\ {}\hfill w\;\left( x, y, z, t\right)= w\;\left( x, y, t\right)\hfill \end{array} $$
(3)

The in-plane displacements (u, v) and out-of-plane displacements (w, θ x and θ y ) will be expressed using the p-version FEM as

$$ \left\{\begin{array}{c}\hfill u\hfill \\ {}\hfill v\hfill \end{array}\right\}=\left[\begin{array}{cc}\hfill N\left(\xi, \eta \right)\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill N\left(\xi, \eta \right)\hfill \end{array}\right]\left\{\begin{array}{c}\hfill {q}_u\hfill \\ {}\hfill {q}_v\hfill \end{array}\right\} $$
(4)
$$ \left\{\begin{array}{c}\hfill \begin{array}{c}\hfill w\hfill \\ {}\hfill {\theta}_y\hfill \end{array}\hfill \\ {}\hfill {\theta}_x\hfill \end{array}\right\}=\left[\begin{array}{ccc}\hfill N\left(\xi, \eta \right)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill N\left(\xi, \eta \right)\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill N\left(\xi, \eta \right)\hfill \end{array}\right]\left\{\begin{array}{c}\hfill \begin{array}{c}\hfill {q}_w\hfill \\ {}\hfill {q}_{\theta_y}\hfill \end{array}\hfill \\ {}\hfill {q}_{\theta_x}\hfill \end{array}\right\} $$
(5)

where q u , q v are the vectors of generalized in-plane displacements, q w , \( {q}_{\theta_y} \) and \( {q}_{\theta_x} \) are the vectors of generalized transverse displacement and rotations, respectively, N(ξ, η) are the hierarchical shape functions of triangular p-element (Belalia & Houmat 2010).

Using FSDT of plate in combination with Von-Karman hypothesis, the nonlinear strain–displacement relationships are expressed as

$$ \left\{\varepsilon \right\}=\left\{{\varepsilon}^L\right\}+\left\{{\varepsilon}^{NL}\right\} $$
(6)

where the linear and the non-linear strains can be expressed as,

$$ \left\{{\varepsilon}^L\right\}=\left\{\begin{array}{c}\hfill {\varepsilon}_P^L\hfill \\ {}\hfill 0\hfill \end{array}\right\}+\left\{\begin{array}{c}\hfill z{\varepsilon}_b\hfill \\ {}\hfill {\varepsilon}_s\hfill \end{array}\right\}\kern1em \mathrm{and}\kern1em \left\{{\varepsilon}^{NL}\right\}=\left\{\begin{array}{c}\hfill {\varepsilon}_P^{NL}\hfill \\ {}\hfill 0\hfill \end{array}\right\} $$
(7)

the components of linear and the non-linear strains given in Eq. (7) are defined as

$$ \left\{{\varepsilon}_P^L\right\}=\left\{\begin{array}{c}\hfill \frac{\partial u}{\partial x}\hfill \\ {}\hfill \frac{\partial v}{\partial y}\hfill \\ {}\hfill \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\hfill \end{array}\right\},\kern1em \left\{{\varepsilon}_b\right\}=\left\{\begin{array}{c}\hfill \frac{\partial {\theta}_y}{\partial x}\hfill \\ {}\hfill -\frac{\partial {\theta}_x}{\partial y}\hfill \\ {}\hfill \frac{\partial {\theta}_y}{\partial y}-\frac{\partial {\theta}_x}{\partial x}\hfill \end{array}\right\} $$
(8)
$$ \left\{{\varepsilon}_s\right\}=\left\{\begin{array}{c}\hfill \frac{\partial w}{\partial x}+{\theta}_y\hfill \\ {}\hfill \frac{\partial w}{\partial y}-{\theta}_x\hfill \end{array}\right\}\kern1em \left\{{\varepsilon}_P^{NL}\right\}=\left\{\begin{array}{c}\hfill \frac{1}{2}{\left(\frac{\partial w}{\partial x}\right)}^2\hfill \\ {}\hfill \frac{1}{2}{\left(\frac{\partial w}{\partial y}\right)}^2\hfill \\ {}\hfill \frac{\partial w}{\partial x}\frac{\partial w}{\partial y}\hfill \end{array}\right\} $$
(9)

The differential relationship used in Eqs. 89 is obtained by inversing Eq. 1 as

$$ \left\{\begin{array}{c}\hfill \frac{\partial }{\partial x}\hfill \\ {}\hfill \frac{\partial }{\partial y}\hfill \end{array}\right\}={\boldsymbol{J}}^{-1}\left\{\begin{array}{c}\hfill \frac{\partial }{\partial \xi}\hfill \\ {}\hfill \frac{\partial }{\partial \eta}\hfill \end{array}\right\} $$
(10)

The strain energy E S and kinetic energy E K of the functionally graded moderately thick plate can expressed as

$$ {E}_S=\frac{1}{2}{\displaystyle \iint}\left[{\left\{{\varepsilon}_p\right\}}^{\mathrm{T}}\left[{\mathrm{A}}_{\mathrm{ij}}\right]\left\{{\varepsilon}_p\right\}+{\left\{{\varepsilon}_p\right\}}^{\mathrm{T}}\left[{\mathrm{B}}_{\mathrm{ij}}\right]\left\{{\varepsilon}_b\right\}\right.+{\left\{{\varepsilon}_b\right\}}^T\left[{B}_{ij}\right]\left\{{\varepsilon}_p\right\}+{\left\{{\varepsilon}_b\right\}}^T\left[{D}_{ij}\right]\left\{{\varepsilon}_b\right\}+\left.{\left\{{\varepsilon}_s\right\}}^T\left[{S}_{ij}\right]\left\{{\varepsilon}_s\right\}\right] dxdy $$
(11)
$$ {E}_K=\frac{1}{2}{\displaystyle \iint}\left[{I}_1\left({\left(\frac{\partial u}{\partial t}\right)}^2+{\left(\frac{\partial v}{\partial t}\right)}^2+{\left(\frac{\partial w}{\partial t}\right)}^2\right)+{I}_3\left({\left(\frac{\partial {\theta}_x}{\partial t}\right)}^2+{\left(\frac{\partial {\theta}_y}{\partial t}\right)}^2\right)\right] dxdy $$
(12)

where [A ij ], [B ij ] and [D ij ], are extensional, bending-extensional and bending stiffness constants of the FG plate and are given by

$$ \left[{A}_{ij},{B}_{ij},{D}_{ij}\right]={\displaystyle \underset{-\frac{h}{2}}{\overset{+\frac{h}{2}}{\int }}}{Q}_{ij}\left(1, z,{z}^2\right) d z\kern0.75em \left( i, j=1,2,6\right) $$
(13)
$$ \left[{S}_{ij}\right]= k{\displaystyle \underset{-\frac{h}{2}}{\overset{+\frac{h}{2}}{\int }}}{Q}_{ij} d z\kern0.75em \left( i, j=4,5\right) $$
(14)

where k is a shear correction factor and is equal to π2/12

$$ {Q}_{11}={Q}_{22}=\frac{E(z)}{1-{\nu}^2(z)}\kern1.12em {Q}_{12}=\nu (z){Q}_{11}\kern0.62em {Q}_{44}={Q}_{55}={Q}_{66}=\frac{E(z)}{2\left(1+\nu (z)\right)} $$
(15)
$$ \left({I}_1,{I}_3\right)={\displaystyle \underset{- h/2}{\overset{+ h/2}{\int }}}\rho (z)\left(1, z \mathit{^2}\right) d z $$
(16)

The material properties E(z),ν(z), and ρ(z) of the functionally graded triangular plate assumed to be graded only in the thickness direction according to a simple power law distribution in terms of the volume fraction of the constituents which is expressed a

$$ E(z)=\kern0.75em \left({E}_c-{E}_m\right){\left(\frac{1}{2}+\frac{z}{h}\right)}^n+{E}_m $$
(17)
$$ \nu (z)=\kern0.75em \left({\nu}_c-{\nu}_m\right){\left(\frac{1}{2}+\frac{z}{h}\right)}^n+{\nu}_m $$
(18)
$$ \rho (z)=\kern0.75em \left({\rho}_c-{\rho}_m\right){\left(\frac{1}{2}+\frac{z}{h}\right)}^n+{\rho}_m $$
(19)

where c and m index designate the ceramic and the metal, respectively, n is the exponent of the volume fraction (n ≥ 0), z is the thickness coordinate variable, E elastic modulus, ρ mass density, h is the thickness of the plate and ν is the Poisson’s ratio. The bottom layer of the functionally graded triangular plate is fully metallic material and the top layer is fully ceramic material. The constants of material for four types of FGMs considered in this study (AL/AL2O3,SUS304/Si3N4, Ti-6AL-4 V/Aluminum oxide, AL/ZrO2) are shown in Table 1.

Table 1 Mechanical properties of FGMs components Yang et al. (2003) and Zhao et al. (2009)

Inserting Eqs. (1112) in Lagrange’s equations the equations of free motion are obtained as:

$$ \left[\overline{\boldsymbol{M}}\right]\left\{\begin{array}{c}\hfill {\ddot{q}}_u\hfill \\ {}\hfill {\ddot{q}}_v\hfill \end{array}\right\}+\left[\overline{\boldsymbol{K}}\right]\left\{\begin{array}{c}\hfill {q}_u\hfill \\ {}\hfill {q}_v\hfill \end{array}\right\}+\left[\overset{\smile }{\boldsymbol{K}}+\widehat{\boldsymbol{K}}\right]\left\{\begin{array}{c}\hfill \begin{array}{c}\hfill {q}_w\hfill \\ {}\hfill {q}_{\theta_y}\hfill \end{array}\hfill \\ {}\hfill {q}_{\theta_x}\hfill \end{array}\right\}=0 $$
(20)
$$ \left[\boldsymbol{M}\right]\left\{\begin{array}{c}\hfill \begin{array}{c}\hfill {\ddot{q}}_w\hfill \\ {}\hfill {\ddot{q}}_{\theta_y}\hfill \end{array}\hfill \\ {}\hfill {\ddot{q}}_{\theta_x}\hfill \end{array}\right\}+\left[\tilde{\boldsymbol{K}}+\boldsymbol{K}\right]\left\{\begin{array}{c}\hfill \begin{array}{c}\hfill {q}_w\hfill \\ {}\hfill {q}_{\theta_y}\hfill \end{array}\hfill \\ {}\hfill {q}_{\theta_x}\hfill \end{array}\right\}+\left[2\widehat{\boldsymbol{K}}+\overset{\smile }{\boldsymbol{K}}\right]\left\{\begin{array}{c}\hfill {q}_u\hfill \\ {}\hfill {q}_v\hfill \end{array}\right\}=0 $$
(21)

The vector of generalized displacement in free motion will be given as

$$ \left\{\begin{array}{c}\hfill \begin{array}{c}\hfill {q}_w\hfill \\ {}\hfill {q}_{\theta_y}\hfill \end{array}\hfill \\ {}\hfill {q}_{\theta_x}\hfill \end{array}\right\}=\left\{\begin{array}{c}\hfill \begin{array}{c}\hfill {Q}_w\hfill \\ {}\hfill {Q}_{\theta_y}\hfill \end{array}\hfill \\ {}\hfill {Q}_{\theta_x}\hfill \end{array}\right\} cos\left(\omega t\right)= Qcos\left(\omega t\right) $$
(22)

By neglecting the in-plane inertia, and taking into account the effects of the transverse shear deformation and inertia of rotation. Inserting Eqs. (20) and (22) into Eq. (21) and applying the HB-method, the final equation of free motion are of the form

$$ \left[-\omega {}^2\boldsymbol{M}+\boldsymbol{K}-{\overset{\smile }{\boldsymbol{K}}}^{T}{\overline{\boldsymbol{K}}}^{-1}\overset{\smile}{\boldsymbol{K}}\right] \left\{\begin{array}{c}\hfill \begin{array}{c}\hfill {Q}_w\hfill \\ {}\hfill {Q}_{\theta_y}\hfill \end{array}\hfill \\ {}\hfill {Q}_{\theta_x}\hfill \end{array}\right\} +\frac{3}{4}\left[\begin{array}{ccc}\hfill \widetilde{\boldsymbol{K}}-2{\widehat{\boldsymbol{K}}}^T{\overline{\boldsymbol{K}}}^{-1}\widehat{\boldsymbol{K}}\hfill & \hfill \kern.5em 0\hfill & \hfill \kern.5em 0\hfill \\ {}\hfill 0\hfill & \hfill \kern2.7em 0\hfill & \hfill \kern.5em 0\hfill \\ {}\hfill 0\hfill & \hfill \kern2.7em 0\hfill & \hfill \kern.5em 0\hfill \end{array}\right] \left\{\begin{array}{c}\hfill \begin{array}{c}\hfill {Q}_w\hfill \\ {}\hfill {Q}_{\theta_y}\hfill \end{array}\hfill \\ {}\hfill {Q}_{\theta_x}\hfill \end{array}\right\}=0 $$
(23)

Where M is the out-of-plane inertia matrices, \( \overline{\boldsymbol{K}} \), K and \( \overset{\smile }{\boldsymbol{K}} \) are the extension, bending and coupled extension-rotation linear stiffness matrices, \( \tilde{\boldsymbol{K}} \) and \( \widehat{\boldsymbol{K}} \) represent the nonlinear stiffness matrices. These matrices are given in Appendix A.

The system of equations given in Eq. (23) are solved iteratively using the linearized updated mode method. This method needs two type of amplitudes, the first is the specific amplitude which depends on the plate thickness, the second is the maximum amplitude to be calculated for each iteration. The new system of equations is solved using any known technique with an accuracy of around (e.g.10−5).

The maximum amplitude w max is evaluated as

$$ {w}_{max}=\left[\begin{array}{ccc}\hfill N\left({\xi}_0,{\eta}_0\right)\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\left\{\begin{array}{c}\hfill \begin{array}{c}\hfill {Q}_w\hfill \\ {}\hfill {Q}_{\theta_y}\hfill \end{array}\hfill \\ {}\hfill {Q}_{\theta_x}\hfill \end{array}\right\}\kern2.5em \left( i=1,2,\dots \left( p+1\right)\left( p+2\right)/2\right) $$
(24)

Results

Study of convergence and comparison for linear vibration

In this part a convergence and comparison study is made for the linear vibration of clamped metallic isosceles triangular plates to validate the current formulation and methods proposed.

Table 2 shows the convergence of the first three frequencies parameter \( \Omega =\omega b{}^2\sqrt{\rho h/ D} \) of metallic clamped isosceles triangular plate (β = 90°) for the three following different thickness ratio (h/b = 0.05, 0.1 and 0.15). The convergence of results can be accelerated by increasing the polynomial order p from 6 to 11. To validate the accuracy of the present solution, a comparison, listed in Table 3, is made between the present results and the results of p-version Ritz method (Liew et al. 1998) of first three linear frequency parameters for metallic clamped isosceles triangular plate, the geometric parameters of this plate are taken (β = 30°, 60° and 90°) for apex angle and (h/b = 0.05, 0.1 and 0.15) for thickness ratio. From this table, it can be found that the present results are in good agreement with the published results. From this table, it can be found that the present results are in good agreement with the published results.

Table 2 Convergence of the first three linear frequency parameters for clamped metallic isosceles triangular plate (β = 90°)
Table 3 Comparison of the first three linear frequency parameters for clamped metallic isosceles triangular plate

Linear vibration of FGMs isosceles triangular plate

This part of study present the linear free vibration of thick FGMs isosceles triangular plates designed by four different mixtures (FGM 1: AL/AL2O3, FGM 2: SUS304/Si3N4, FGM 3: Ti-6AL-4 V/Aluminum oxide and FGM 4: AL/ZrO2). Tables 4, 5, 6, 7 display the first three linear frequency parameters \( {\Omega}_{\mathrm{L}}=\omega b{}^2\sqrt{12{\rho}_m\left(1-\nu \mathit{^2}\right)/{E}_m} \) for a clamped FGMs isosceles triangular plate, three apex angles (β = 30°, 60° and 90°) and three thickness ratio (h/b = 0.05, 0.1 and 0.15) are considered. The exponent of volume fraction vary from 0 to ∞ and it takes the values presented in tables. The results presented in this section comes to enrich the results of literatures. The tables visibly show that the linear frequency parameters is proportional to the angle and thickness and inversely proportional to the volume fraction exponent. For the triangular plate with apex angle (β = 60°), it is noted that the second and third modes are double modes for cases purely metal or purely ceramic, but varied the volume fraction exponent there is a small spacing between the two modes, the maximum spacing is the round of n = 1.

Table 4 The first three linear frequency parameters of clamped FG AL/AL2O3 isosceles triangular plate
Table 5 The first three linear frequency parameters of clamped FG SUS304/Si3N4 isosceles triangular plate
Table 6 The first three linear frequency parameters of clamped FG Ti-6AL-4 V/Aluminum oxide isosceles triangular plate
Table 7 The first three linear frequency parameters of clamped FG AL/ZrO2 isosceles triangular plate

Non-linear vibration of isosceles triangular FG-plate

The investigation of the effects of the FGM mixtures, volume fraction exponent, thickness ratio, apex angle and boundary conditions on the hardening behavior are investigated in this part. The resultant backbone curves which shows the change in the nonlinear-to-linear frequency ratio ΩNLL according to maximum amplitude-to-thickness ratios |w max |/h are plotted in Figs. 2, 3, 4, 5 for clamped FG isosceles triangular plate. In Fig. 2, four different mixtures of FGM (FGM 1: AL/AL2O3, FGM 2: SUS304/Si3N4, FGM 3: Ti-6AL-4 V/Aluminum oxide and FGM 4: AL/ZrO2) are considered for volume fraction exponent n = 0.5. The thickness ratio and the apex angle of FG isosceles triangular plate are taken respectively as h/b = 0.1, β = 60°. The effect of apex angle and thickness on the backbone curve for the first mode of the functionally garded AL/AL2O3 clamped triangular plate with (β = 60°) and n =1 are presented in Figs. 3, 4. The effects of mixtures, thickness ratio and apex angle are clearly shown on the plot of these figures. The plots clearly show that if the thickness and angle increases the effects of the hardening behavior increases automatically. Also, the nonlinear vibration of the triangular plate with mixture FGM 4 presents the greatest hardening behavior compared to others mixtures of FGM.

Fig. 2
figure2

Material mixtures effects on the fundamental backbone curves for clamped FG triangular plate (β = 60°, h/b = 0.1, n = 0.5)

Fig. 3
figure3

The thickness effects on the fundamental backbone curves for clamped FG AL/AL2O3 isosceles triangular plate (β = 60° and n =1)

Fig. 4
figure4

The apex angle effects on the fundamental backbone curves for clamped FG AL/AL2O3 isosceles triangular plate (h/b = 0.1and n =1)

Fig. 5
figure5

The boundary conditions effects on the fundamental backbone curves for FG AL/AL2O3 isosceles triangular plate (β = 90°, h/b = 0.05 and n =1)

The boundary conditions effects on the fundamental backbone curves for FG AL/AL2O3 isosceles triangular plate are investigated in Fig. 5. Four different boundary conditions are considered in this part of study SSS, CSS, SCC and CCC (S: simply supported edge and C : clamped edge). The volume fraction exponent, thickness ratio and the apex angle of FG isosceles triangular plate are taken respectively as n =1, h/b = 0.05 and β = 90°. The figure clearly show that the FG plate with simply supported boundary conditions presents a more accentuated hardening behavior than the other boundary conditions. It is noted that the hardening effect increases when the plate becomes more free (SSS) and decreases as the plate becomes more fixed (CCC), this difference in the results is due to the rotation of the edges.

The variation of frequency ratio ΩNLL according to volume fraction exponent for clamped isosceles triangular plate with four different mixtures of FGMs is shown in Fig. 6. The exponent of volume fraction take values from 0 to 20 and maximum amplitude-to-thickness ratios take three values |w max |/h = 0.6, 0.8 and 1. The geometric parameters of the plate are (β = 90°) and h/b =0.1. Noted that the shape of the graph is similar for three values of the maximum amplitude-to-thickness ratios of this fact and to understand the phenomenon and good interpretation, Fig. 7 plot only the results of the largest value of the maximum amplitude |w max |/h = 1. It can be seen for volume fraction exponent which varied between n = 0 to n = 4 the hardening effect is maximum for the first mixture (AL/AL2O3), for values n ≥ 4 the second mixture (which SUS304/Si3N4) presents the greatest hardening effect. For third and fourth mixtures (Ti-6AL-4 V/Aluminum oxide and AL/ZrO2) the shape of the two curves are parallel with superiority of the values obtained for the fourth mixture FGM 4. Note that the peak of the hardening behavior for four curves is obtained for volume fraction exponent n = 1, at which corresponds to a linear variation of constituent materials of the mixture. By comparing the spacing between curves FGM1 (Al/Al2O3) and FGM4 (Al/ZrO2) we see clearly the influence of physical properties of the two ceramic (Al2O3 and ZrO2) on hardening behavior. This influence is not due to metal (Al) since the same metal is used in both mixtures.

Fig. 6
figure6

Material mixtures effects on the variation of the nonlinear-to-linear fundamental frequency ratio with the volume fraction exponent for clamped FG isosceles triangular plate (h/b = 0.1, β = 90°)

Fig. 7
figure7

Material mixtures effects on the variation of the nonlinear-to-linear fundamental frequency ratio with the volume fraction exponent for clamped FG isosceles triangular plate (|w max |/h = 1, h/b = 0.1, β = 90°)

Figures 8, 9, 10 shows the normalized non-linear fundamental mode shape of isosceles triangular plate for four different mixtures of FGM along the line passes through the point of maximum amplitude (ξ 0, η 0). The mode shape are normalized by dividing by their own maximum displacement. Three apex angles and thickness ratio of FG plate are considered (β = 30°, 60° and 90°), (h/b = 0.05) respectively, volume fraction exponent n = 1 and the maximum amplitude |w max |/h = 1. It can see from these graphs that the displacement is maximum for the FGM 2 (SUS304/Si3N4) then comes FGM3 (Ti-6Al-4 V/Aluminum oxide) with a percentage of displacement 83% of maximum displacement, FGM 1 (AL/AL2O3) with 72% and lastly FGM 4 (AL/ZrO2) with 64%. The normalized non-linear of second and third modes shape of isosceles triangular plates for the same mixtures used early are plotted in Figs. 11, 12, respectively. The geometric parameters used are h/b = 0.05, β = 90° and |w max |/h = 0.8. It can be seen from this plot the effect of mixtures on normalized non-linear first three fundamental mode shape of isosceles triangular plate. This is due to fact that the composition of mixtures contribute to various in-plane forces in the isosceles triangular plate.

Fig. 8
figure8

Section of normalized non-linear fundamental mode shapes of FG isosceles triangular plate : a) along of ξ; b) alone of η (β = 30°, n = 1, h/b = 0.05)

Fig. 9
figure9

Section of normalized non-linear fundamental mode shapes of FG isosceles triangular plate: a) along of ξ; b) alone of η (β = 60°, n = 1, h/b = 0.05)

Fig. 10
figure10

Section of normalized non-linear fundamental mode shapes of FG isosceles triangular plate: a) along of ξ; b) alone of η (β = 90°, n = 1, h/b = 0.05)

Fig. 11
figure11

Section of normalized non-linear second mode shapes of FG isosceles triangular plate: a) along of ξ; b) alone of η (β = 90°, n = 1, h/b = 0.05)

Fig. 12
figure12

Section of normalized non-linear third mode shapes of FG isosceles triangular plate: a) along of ξ; b) alone of η (β = 90°, n = 1, h/b = 0.05)

Conclusions

The non-linear free vibration of moderately thick FGMs clamped isosceles triangular plates was analyzed by a triangular p-element. The material properties of the functionally graded triangular plate assumed to be graded only in the thickness direction according to a simple power law distribution in terms of the volume fraction of the constituents. The shape functions of triangular finite p-element are obtained by the shifted orthogonal polynomials of Legendre. The components of stiffness and mass matrices were calculated using numerical integration of Gauss-Legendre. The equations of motion are obtained from Lagrange's equation in combination with the harmonic balance method (HBM). Results for linear and non-linear frequency for the lowest three modes of FGMs clamped isosceles triangular plates were obtained. The parametric studies show that the boundary conditions have a great influence on the shape of the backbone curves, the hardening spring effect decreases for clamped FG plate. For simply supported FG plate and by increasing thickness ratio and sector angle of FG plates the hardening spring effect increases. A increase in the volume fraction exponent produces a variation in the hardening spring effect with an increasing part and another decreasing part, the peak in the curves of the nonlinear-to-linear fundamental frequency ratio FG triangular plate is obtained around of n = 1 at which the hardening behavior is maximum, and is obtained for AL/AL2O3 FG plate. This value of volume fraction exponent corresponds to equal mixtures of metal and ceramic in the composition of the FG plate. Not only the hardening behavior is influenced by this mixture but the non-linear mode shape of FG isosceles triangular plate is also influenced.

Appendix A

$$ {\overline{\mathbf{K}}}_{\alpha, \beta}=\left[\begin{array}{cc}\hfill {\overline{K}}_{2\alpha -1,2\beta -1}\hfill & \hfill {\overline{K}}_{2\alpha -1,2\beta}\hfill \\ {}\hfill {\overline{K}}_{2\alpha, 2\beta -1}\kern1em \hfill & \hfill {\overline{K}}_{2\alpha, 2\beta}\kern1em \hfill \end{array}\right] $$
(A.1)
$$ {\mathbf{K}}_{\alpha, \beta}=\left[\begin{array}{ccc}\hfill {K}_{3\alpha -2,3\beta -2}\kern1em \hfill & \hfill {K}_{3\alpha -2,3\beta -1}\kern1em \hfill & \hfill {K}_{3\alpha -2,3\beta}\hfill \\ {}\hfill {K}_{3\alpha -1,3\beta -2}\kern1em \hfill & \hfill {K}_{3\alpha -1,3\beta -1}\kern1em \hfill & \hfill {K}_{3\alpha -1,3\beta}\hfill \\ {}\hfill {K}_{3\alpha, 3\beta -2}\kern1em \hfill & \hfill {K}_{3\alpha, 3\beta -1}\kern1em \hfill & \hfill {K}_{3\alpha, 3\beta}\kern1em \hfill \end{array}\right] $$
(A.2)
$$ {\mathbf{M}}_{\alpha, \beta}=\left[\begin{array}{ccc}\hfill {M}_{3\alpha -2,3\beta -2}\kern1em \hfill & \hfill {M}_{3\alpha -2,3\beta -1}\kern1em \hfill & \hfill {M}_{3\alpha -2,3\beta}\hfill \\ {}\hfill {M}_{3\alpha -1,3\beta -2}\kern1em \hfill & \hfill\ {M}_{3\alpha -1,3\beta -1}\kern1em \hfill & \hfill {M}_{3\alpha -1,3\beta}\hfill \\ {}\hfill {M}_{3\alpha, 3\beta -2}\kern1em \hfill & \hfill {M}_{3\alpha, 3\beta -1}\kern1em \hfill & \hfill {M}_{3\alpha, 3\beta}\kern1em \hfill \end{array}\right] $$
(A.3)
$$ {\widehat{\mathbf{K}}}_{\alpha, \beta}=\left[\begin{array}{ccc}\hfill {\widehat{K}}_{2\alpha -1,3\beta -2}\kern1em \hfill & \hfill {\widehat{K}}_{2\alpha -1,3\beta -1}\kern1em \hfill & \hfill {\widehat{K}}_{2\alpha -1,3\beta}\hfill \\ {}\hfill {\widehat{K}}_{2\alpha, 3\beta -2}\kern1em \hfill & \hfill {\widehat{K}}_{2\alpha, 3\beta -1}\kern1em \hfill & \hfill {\widehat{K}}_{2\alpha, 3\beta}\kern1em \hfill \end{array}\right] $$
(A.4)
$$ {\overset{\smile }{\mathbf{K}}}_{\alpha, \beta}=\left[\begin{array}{ccc}\hfill {\overset{\smile }{K}}_{2\alpha -1,3\beta -2}\kern1em \hfill & \hfill {\overset{\smile }{K}}_{2\alpha -1,3\beta -1}\kern1em \hfill & \hfill {\overset{\smile }{K}}_{2\alpha -1,3\beta}\hfill \\ {}\hfill {\overset{\smile }{K}}_{2\alpha, 3\beta -2}\kern1em \hfill & \hfill {\overset{\smile }{K}}_{2\alpha, 3\beta -1}\kern1em \hfill & \hfill {\overset{\smile }{K}}_{2\alpha, 3\beta}\kern1em \hfill \end{array}\right] $$
(A.5)
$$ {\tilde{\mathbf{K}}}_{\alpha, \beta}=\left[\begin{array}{ccc}\hfill {\tilde{K}}_{3\alpha -2,3\beta -2}\kern1em \hfill & \hfill {\tilde{K}}_{3\alpha -2,3\beta -1}\kern1em \hfill & \hfill {\tilde{K}}_{3\alpha -2,3\beta}\hfill \\ {}\hfill {\tilde{K}}_{3\alpha -1,3\beta -2}\kern1em \hfill & \hfill {\tilde{K}}_{3\alpha -1,3\beta -1}\kern1em \hfill & \hfill {\tilde{K}}_{3\alpha -1,3\beta}\hfill \\ {}\hfill {\tilde{K}}_{3\alpha, 3\beta -2}\kern1em \hfill & \hfill {\tilde{K}}_{3\alpha, 3\beta -1}\kern1em \hfill & \hfill {\tilde{K}}_{3\alpha, 3\beta}\kern1em \hfill \end{array}\right] $$
(A.6)

The non-zero elements of the matrices M, K, \( \overline{\mathrm{K}} \), \( \widehat{\mathrm{K}} \), \( \overset{\smile }{\mathrm{K}} \) and \( \tilde{\mathrm{K}} \) are expressed as

$$ {M}_{3\;\alpha -2,3\beta -2}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}{I}_1{N}_{\alpha}{N}_{\beta}\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.7)
$$ {M}_{3\;\alpha -1,3\beta -1}={M}_{3\;\alpha, 3\beta}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}{I}_3{N}_{\alpha}{N}_{\beta}\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.8)
$$ {K}_{3\alpha -2,3\beta -2}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }} k\;\left({A}_{44}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}+{A}_{55}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \eta}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.9)
$$ {K}_{3\alpha -2,3\beta -1}=-{\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }} k\;{A}_{44}\frac{\partial {N}_{\alpha}}{\partial \eta}{N}_{\beta}\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.10)
$$ {K}_{3\alpha -2,3\beta}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }} k\;{A}_{55}\frac{\partial {N}_{\alpha}}{\partial \xi}{N}_{\beta}\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.11)
$$ {K}_{3\alpha -1,3\beta -2}=-{\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }} k\;{A}_{44}{N}_{\alpha}\frac{\partial {N}_{\beta}}{\partial \eta}\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.12)
$$ {K}_{3\alpha -1,3\beta -1}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\;\left({D}_{22}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \eta}+{D}_{66}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}+ k\;{A}_{44}{N}_{\alpha}{N}_{\beta}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.13)
$$ {K}_{3\alpha -1,3\beta}=-{\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\;\left({D}_{12}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \xi}+{D}_{66}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \eta}\right)\;}}\left|\mathrm{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.14)
$$ {K}_{3\alpha, 3\beta -2}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }} k\;{A}_{55}{N}_{\alpha}\frac{\partial {N}_{\beta}}{\partial \eta}\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.15)
$$ {K}_{3\alpha, 3\beta -1}=-{\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\;\left({D}_{12}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \eta}+{D}_{66}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \xi}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.16)
$$ {K}_{3\alpha, 3\beta}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\;\left({D}_{11}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}+{D}_{66}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \eta}+ k\;{A}_{55}{N}_{\alpha}{N}_{\beta}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.17)
$$ {\overline{K}}_{2\alpha -1,2\beta -1}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\left({A}_{11}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}+{A}_{66}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \eta}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.18)
$$ {\overline{K}}_{2\alpha -1,2\beta}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\left({A}_{12}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \eta}+{A}_{66}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \xi}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.19)
$$ {\overline{K}}_{2\alpha, 2\beta -1}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\left({A}_{12}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \xi}+{A}_{66}\frac{\partial {N}_{\alpha}}{\partial y}\frac{\partial {N}_{\beta}}{\partial \eta}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.20)
$$ {\overline{K}}_{2\alpha, 2\beta}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\left({A}_{22}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \eta}+{A}_{66}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.21)
$$ \begin{array}{l}{\widehat{K}}_{2\alpha -1,3\beta -2}=\frac{1}{2}{\displaystyle \sum_{\delta =1}^r\left({\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\;\left({A}_{11}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}\frac{\partial {N}_{\delta}}{\partial \xi}+{A}_{12}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \eta}\frac{\partial {N}_{\delta}}{\partial \eta}\right.}}\right.}\\ {}\kern4em +\left.2{A}_{66}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \xi}\frac{\partial {N}_{\delta}}{\partial \eta}\right)\;\left.\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta \right)\kern0.24em {Q}_{3\delta -2}\end{array} $$
(A.22)
$$ \begin{array}{l}{\widehat{K}}_{2\alpha, 3\beta -2}=\frac{1}{2}{\displaystyle \sum_{\delta =1}^r\left({\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\;}}\left({A}_{22}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \eta}\frac{\partial {N}_{\delta}}{\partial \eta}+{A}_{12}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \xi}\frac{\partial {N}_{\delta}}{\partial \xi}\right.\right.}\\ {}\kern3em +\left.2{A}_{66}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}\frac{\partial {N}_{\delta}}{\partial \eta}\right)\;\left.\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta\;\right)\kern0.24em {Q}_{3\delta -2}\end{array} $$
(A.23)
$$ {\overset{\smile }{K}}_{2\alpha -1,3\beta -1}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\left({B}_{12}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \eta}-{B}_{66}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \xi}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.24)
$$ {\overset{\smile }{K}}_{2\alpha -1,3\beta}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\left({B}_{11}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \eta}+{B}_{66}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \xi}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.25)
$$ {\overset{\smile }{K}}_{2\alpha, 3\beta -1}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\left({B}_{22}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \eta}-{B}_{66}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.26)
$$ {\overset{\smile }{K}}_{2\alpha, 3\beta}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\left({B}_{12}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \xi}+{B}_{66}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \eta}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.27)
$$ \begin{array}{l}{\tilde{K}}_{3\alpha -2,3\beta -2}=\frac{1}{2}{\displaystyle \sum_{\delta =1}^r{\displaystyle \sum_{\gamma =1}^r\left({\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\;}}\left({A}_{11}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}\frac{\partial {N}_{\delta}}{\partial \xi}\frac{\partial {N}_{\gamma}}{\partial \xi}\right.\right.}}\\ {}\kern10em +{A}_{22}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \eta}\frac{\partial {N}_{\delta}}{\partial \eta}\frac{\partial {N}_{\gamma}}{\partial \eta}\kern5em \\ {}\kern10em +\left({A}_{12}\right.+\left.2{A}_{66}\right)\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}\frac{\partial {N}_{\delta}}{\partial \eta}\frac{\partial {N}_{\gamma}}{\partial \eta}\\ {}\kern10em \left.+\left({A}_{12}\right.+\left.2{A}_{66}\right)\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \eta}\frac{\partial {N}_{\delta}}{\partial \xi}\frac{\partial {N}_{\gamma}}{\partial \xi}\right)\;\left.\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta\;\right)\kern0.24em {Q}_{3\delta -2}{Q}_{3\gamma -2}\end{array} $$
(A.28)

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Laboratory of Computational Mechanics, Faculty of Technology, Department of Mechanical Engineering, University of Tlemcen, B.P. 230, Tlemcen 13000, Algeria.

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Correspondence to SA. Belalia.

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Keywords

  • The mixtures effect of Ceramic-Metal
  • Linear and Non-linear vibration
  • Moderately thick FGM plates
  • p-version of finite element method