Open Access

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

Mechanics of Advanced Materials and Modern Processes20173:4

DOI: 10.1186/s40759-017-0018-0

Received: 2 September 2016

Accepted: 12 January 2017

Published: 1 February 2017

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.

Keywords

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

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

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)

Material

Properties

E (109 N/m2)

ν

ρ (kg/m3)

Aluminium (Al)

70.00

0.30

2707

Alumina (Al2O3)

380.00

0.30

3800

Stainless steel SUS304

207.78

0.3177

8166

Silicon nitride Si3N4

322.27

0.24

2370

Ti-6AL-4 V

105.7

0.2981

4429

Aluminum oxide

320.24

0.26

3750

Zirconia (ZrO2)

151.00

0.30

3000

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°)

h/b

Mode

p

6

7

8

9

10

11

0.05

Ω1

166.3

164.6

164.4

164.3

164.3

164.3

Ω2

277.2

265.7

261.9

261.1

260.9

260.9

Ω3

330.4

321.2

316.3

314.3

313.8

313.7

0.1

Ω1

128.2

127.9

127.9

127.9

127.9

127.9

Ω2

195.1

191.2

190.5

190.2

190.2

190.2

Ω3

227.9

225.0

223.6

223.3

223.2

223.2

0.15

Ω1

100.3

100.2

100.2

100.2

100.2

100.2

Ω2

146.1

144.3

144.1

144.0

144.0

144.0

Ω3

168.7

167.4

166.8

166.7

166.7

166.7

Table 3

Comparison of the first three linear frequency parameters for clamped metallic isosceles triangular plate

h/b

Mode

β

30°

60°

90°

Present

Liew et al. (1998)

Present

Liew et al. (1998)

Present

Liew et al. (1998)

0.05

Ω1

51.55

51.55

91.86

91.86

164.3

164.4

Ω2

80.60

80.61

167.5

167.5

260.9

260.9

Ω3

109.5

109.5

167.5

167.5

313.7

313.7

0.1

Ω1

46.35

46.35

77.76

77.79

127.9

127.9

Ω2

69.82

69.81

132.3

132.3

190.2

190.3

Ω3

92.16

92.17

132.3

132.3

223.1

223.2

0.15

Ω1

40.55

40.55

64.58

64.59

100.2

100.2

Ω2

59.07

59.07

104.7

104.7

143.0

144.0

Ω3

76.12

76.15

104.7

104.7

166.7

166.7

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

β

h/b

Mode

n

ceramic

0.1

0.5

1

2

5

10

metal

30°

0.05

ΩL1

5.0681

4.8827

4.3399

3.9585

3.6367

3.4112

3.2521

2.5773

ΩL2

7.9247

7.6385

6.8050

6.2214

5.7244

5.3517

5.0840

4.0300

ΩL3

10.768

10.382

9.2453

8.4385

7.7440

7.2297

6.8741

5.4761

0.1

ΩL1

9.1144

8.7992

7.8656

7.1908

6.5871

6.0926

5.7606

4.6349

ΩL2

13.729

13.267

11.904

10.916

10.009

9.1978

8.6491

6.9818

ΩL3

18.122

17.522

15.727

14.401

13.158

12.049

11.330

9.2160

0.15

ΩL1

11.960

11.571

10.404

9.5364

8.7114

7.9458

7.4521

6.0824

ΩL2

17.422

16.873

15.237

14.013

12.812

11.599

10.814

8.8598

ΩL3

22.452

21.759

19.656

18.046

16.436

14.823

13.822

11.418

60°

0.05

ΩL1

9.0322

8.7089

7.7700

7.1137

6.5499

6.1081

5.7894

4.5931

ΩL2

16.471

15.893

14.197

12.987

11.922

11.066

10.472

8.3760

ΩL3

16.471

15.895

14.220

13.039

11.998

11.121

10.501

8.3760

0.1

ΩL1

15.290

14.785

13.290

12.200

11.179

10.225

9.5914

7.7757

ΩL2

26.023

25.197

22.727

20.880

19.081

17.304

16.159

13.233

ΩL3

26.023

25.197

22.734

20.895

19.098

17.307

16.159

13.233

0.15

ΩL1

19.048

18.460

16.696

15.361

14.020

12.629

11.751

9.6867

ΩL2

30.889

29.973

27.193

25.036

22.785

20.357

18.876

15.708

ΩL3

30.889

29.974

27.208

25.070

22.840

20.408

18.904

15.708

90°

0.05

ΩL1

16.158

15.597

13.957

12.793

11.760

10.885

10.273

8.2171

ΩL2

25.654

24.788

22.260

20.454

18.802

17.278

16.227

13.046

ΩL3

30.842

29.806

26.705

24.420

22.326

20.519

19.338

15.684

0.1

ΩL1

25.141

24.354

22.002

20.236

18.487

16.701

15.561

12.785

ΩL2

37.407

36.275

32.868

30.265

27.597

24.749

22.981

19.022

ΩL3

43.879

42.565

38.556

35.441

32.237

28.877

26.830

22.314

0.15

ΩL1

29.543

28.682

26.067

24.026

21.861

19.453

17.991

15.023

ΩL2

42.470

41.260

37.557

34.620

31.440

27.861

25.732

21.597

ΩL3

49.166

47.778

43.507

40.090

36.371

32.198

29.726

25.002

Table 5

The first three linear frequency parameters of clamped FG SUS304/Si3N4 isosceles triangular plate

β

h/b

Mode

n

ceramic

0.1

0.5

1

2

5

10

metal

30°

0.05

ΩL1

5.8366

5.1789

4.0506

3.5610

3.2026

2.9134

2.7785

2.5747

ΩL2

9.1371

8.1064

6.3395

5.5721

5.0090

4.5535

4.3414

4.0244

ΩL3

12.427

11.024

8.6199

7.5746

6.8066

6.1846

5.8959

5.4669

0.1

ΩL1

10.564

9.3705

7.3225

6.4269

5.7626

5.2220

4.9759

4.6207

ΩL2

15.956

14.149

11.053

9.6966

8.6850

7.8582

7.4845

6.9544

ΩL3

21.102

18.708

14.609

12.811

11.465

10.363

9.8682

9.1743

0.15

ΩL1

13.954

12.374

9.6609

8.4647

7.5657

6.8293

6.5019

6.0512

ΩL2

20.390

18.075

14.106

12.353

11.028

9.9374

9.4558

8.8055

ΩL3

26.332

23.336

18.2050

15.934

14.212

12.793

12.170

11.341

60°

0.05

ΩL1

10.423

9.2477

7.2314

6.3544

5.7096

5.1871

4.9454

4.5855

ΩL2

19.056

16.902

13.210

11.601

10.413

9.4482

9.0049

8.3552

ΩL3

19.056

16.902

13.211

11.603

10.415

9.4492

9.0053

8.3552

0.1

ΩL1

17.805

15.790

12.331

10.810

9.6719

8.7402

8.3229

7.7404

ΩL2

30.433

26.975

21.052

18.440

16.469

14.849

14.131

13.156

ΩL3

30.433

26.975

21.053

18.441

16.470

14.850

14.132

13.156

0.15

ΩL1

22.345

19.812

15.456

13.522

12.053

10.845

10.318

9.6205

ΩL2

36.377

32.233

25.130

21.971

19.558

17.564

16.700

15.581

ΩL3

36.377

32.234

25.132

21.975

19.562

17.566

16.701

15.581

90°

0.05

ΩL1

18.711

16.598

12.972

11.388

10.216

9.2627

8.8272

8.1943

ΩL2

29.792

26.422

20.641

18.112

16.227

14.689

13.992

12.998

ΩL3

35.858

31.808

24.843

21.792

19.508

17.650

16.8101

15.621

0.1

ΩL1

29.448

26.111

20.374

17.831

15.906

14.324

13.631

12.704

ΩL2

43.964

38.964

30.385

26.577

23.679

21.289

20.248

18.882

ΩL3

51.630

45.755

35.676

31.196

27.781

24.961

23.736

22.141

0.15

ΩL1

34.858

30.906

24.089

21.036

18.693

16.762

15.937

14.894

ΩL2

50.223

44.496

34.666

30.273

26.895

24.098

22.903

21.397

ΩL3

58.180

51.535

40.150

35.064

31.148

27.898

26.509

24.766

Table 6

The first three linear frequency parameters of clamped FG Ti-6AL-4 V/Aluminum oxide isosceles triangular plate

β

h/b

Mode

n

ceramic

0.1

0.5

1

2

5

10

metal

30°

0.05

ΩL1

4.8290

4.6023

4.0178

3.6640

3.3821

3.1477

2.9915

2.5775

ΩL2

7.5568

7.2037

6.2958

5.7453

5.3021

4.9231

4.6728

4.0305

ΩL3

10.275

9.7962

8.5595

7.8050

7.1942

6.6742

6.3363

5.4770

0.1

ΩL1

8.7224

8.3224

7.2851

6.6421

6.1042

5.6259

5.3293

4.6364

ΩL2

13.162

12.565

11.018

10.053

9.2288

8.4668

8.0039

6.9847

ΩL3

17.397

16.613

14.569

13.281

12.167

11.137

10.527

9.2205

0.15

ΩL1

11.497

10.982

9.6405

8.7886

8.0410

7.3389

6.9297

6.0858

ΩL2

16.783

16.041

14.109

12.873

11.761

10.679

10.060

8.8657

ΩL3

21.659

20.709

18.216

16.602

15.134

13.708

12.913

11.426

60°

0.05

ΩL1

8.6182

8.2172

7.1865

6.5606

6.0523

5.6101

5.3205

4.5939

ΩL2

15.743

15.016

13.139

11.986

11.033

10.193

9.6614

8.3781

ΩL3

15.743

15.017

13.149

12.006

11.058

10.209

9.6685

8.3781

0.1

ΩL1

14.679

14.018

12.303

11.225

10.290

9.4121

8.8890

7.7795

ΩL2

25.055

23.944

21.047

19.198

17.544

15.951

15.037

13.242

ΩL3

25.055

23.944

21.049

19.201

17.547

15.951

15.037

13.242

0.15

ΩL1

18.378

17.573

15.468

14.107

12.863

11.641

10.959

9.6939

ΩL2

29.880

28.591

25.201

22.974

20.887

18.802

17.675

15.721

ΩL3

29.880

28.591

25.209

22.991

20.911

18.818

17.682

15.721

90°

0.05

ΩL1

15.453

14.743

12.911

11.784

10.845

10.002

9.4702

8.2194

ΩL2

24.582

23.465

20.582

18.795

17.269

15.851

14.981

13.051

ΩL3

29.577

28.236

24.742

22.547

20.676

18.974

17.953

15.691

0.1

ΩL1

24.232

23.165

20.377

18.586

16.965

15.385

14.493

12.794

ΩL2

36.137

34.565

30.445

27.765

25.284

22.822

21.468

19.038

ΩL3

42.422

40.584

35.747

32.574

29.626

26.711

25.127

22.333

0.15

ΩL1

28.615

27.392

24.165

22.028

19.998

17.948

16.857

15.037

ΩL2

41.198

39.447

34.821

31.734

28.767

25.752

24.171

21.619

ΩL3

47.715

45.694

40.348

36.768

33.310

29.790

27.951

25.028

Table 7

The first three linear frequency parameters of clamped FG AL/ZrO2 isosceles triangular plate

β

h/b

Mode

n

ceramic

0.1

0.5

1

2

5

10

metal

30°

0.05

ΩL1

3.0021

2.9572

2.8416

2.7888

2.7883

2.8317

2.8112

2.5773

ΩL2

4.6942

4.6257

4.4505

4.3711

4.3692

4.4266

4.3896

4.0299

ΩL3

6.3787

6.2869

6.0482

5.9351

5.9251

5.9981

5.9503

5.4760

0.1

ΩL1

5.3989

5.3268

5.1359

5.0398

5.0174

5.0496

5.0015

4.6348

ΩL2

8.1325

8.0301

7.7590

7.6194

7.5751

7.5901

7.5050

6.9816

ΩL3

10.735

10.604

10.250

10.056

9.9768

9.9757

9.8654

9.2158

0.15

ΩL1

7.0850

7.0016

6.7739

6.6473

6.5896

6.5742

6.4964

6.0823

ΩL2

10.320

10.207

9.8999

9.7224

9.6219

9.5516

9.4211

8.8596

ΩL3

13.300

13.161

12.769

12.526

12.368

12.251

12.086

11.418

60°

0.05

ΩL1

5.3502

5.2734

5.0781

4.9894

4.9857

5.0425

4.9975

4.5931

ΩL2

9.7566

9.6227

9.2739

9.1052

9.0780

9.1532

9.0675

8.3758

ΩL3

9.7566

9.6232

9.2816

9.1224

9.0985

9.1668

9.0741

8.3758

0.1

ΩL1

9.0573

8.9471

8.6541

8.4990

8.4407

8.4359

8.3353

7.7755

ΩL2

15.415

15.244

14.774

14.500

14.358

14.268

14.081

13.233

ΩL3

15.415

15.244

14.777

14.509

14.362

14.270

14.082

13.233

0.15

ΩL1

11.283

11.165

10.838

10.642

10.517

10.414

10.266

9.6865

ΩL2

18.297

18.125

17.627

17.299

17.040

16.784

16.532

15.707

ΩL3

18.297

18.125

17.632

17.310

17.056

16.797

16.538

15.707

90°

0.05

ΩL1

9.5715

9.4422

9.1085

8.9487

8.9212

8.9816

8.8897

8.2169

ΩL2

15.196

15.003

14.502

14.254

14.185

14.215

14.050

13.045

ΩL3

18.269

18.040

17.419

17.086

16.970

17.004

16.829

15.684

0.1

ΩL1

14.892

14.732

14.291

14.033

13.882

13.769

13.580

12.785

ΩL2

22.158

21.938

21.319

20.930

20.650

20.390

20.091

19.022

ΩL3

25.992

25.741

25.014

24.537

24.177

23.847

23.505

22.313

0.15

ΩL1

17.500

17.340

16.880

16.571

16.314

16.039

15.785

15.023

ΩL2

25.157

24.943

24.305

23.848

23.428

22.966

22.596

21.596

ΩL3

29.123

28.883

28.152

27.614

27.104

26.542

26.113

25.002

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

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

Fig. 3

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

Fig. 4

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

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

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

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

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

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

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

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

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)

Declarations

Competing interests

The author declare no significant competing financial, professional or personal interests that might have influenced the performance or presentation of the work described in this manuscript.

Author details

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

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Faculty of Technology, Department of Mechanical Engineering, University of Tlemcen

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Copyright

© The Author(s). 2017