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Уточненная квазитрехмерная теория функционально-градиентных пористых пластин: исследование колебаний
A.M. Zenkour1,2, M.H. Aljadani3
1 Университет короля Абдулазиза, Джидда, 21589, Саудовская Аравия
2 Университет Кафр-эш-Шейха, Кафр-эш-Шейх, 33516, Египет
3 Университет Умм аль-Кура, Мекка, 21421, Саудовская Аравия
В статье обсуждаются свободные колебания пористых функционально-градиентных прямоугольных пластин большой толщины в рамках уточненной квазитрехмерной теории, которая учитывает влияние растяжения по толщине при анализе колебаний пористых пластин. Предполагается, что свойства материала пористой пластины меняются по толщине пластины в соответствии с модифицированным полиномиальным законом материала. С помощью принципа Гамильтона получены уравнения движения пористой пластины. На основе метода Навье получены решения в замкнутой форме для свободно опертых пористых пластин из функционально-градиентных материалов. Представлены численные результаты, подтверждающие точность прогноза, предложенного квазитрехмерной теорией отклика пористых пластин на свободные колебания. Обсуждается влияние коэффициента пористости, градиентного параметра в экспоненциальной форме, соотношения сторон, а также отношения сторон к толщине пластин.
Ключевые слова: функционально-градиентные материалы, пористые пластины, уточненная квазитрехмерная теория, свободные колебания
DOI 10.24412/1683-805X-2021-2-56-70
Quasi-3D refined theory for functionally graded porous plates:
Vibration analysis
A.M. Zenkour1,2 and M.H. Aljadani3
1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh, 33516, Egypt 3 Department of Mathematics, Jamoum University College, Umm Al-Qura University, Makkah, 21421, Saudi Arabia
This paper aims to present free vibration of porous functionally graded thick rectangular plates using a quasi-3D refined theory. This theory considers the thickness stretching effect for vibration analysis of porous plates. It is assumed that the material properties of the porous plate are varying across the plate thickness according to a modified polynomial material law. The equations of motion of the porous plate are obtained via the Hamilton principle. Navier's technique is applied to obtain the closed-form solution for simply-supported functionally graded materials porous plates. Some numerical validations are presented to prove the accuracy of the present quasi-3D refined theory in predicting the free vibration response of porous plates. The influence of porosity parameter, aspect ratio, side-to-thickness ratio, and exponent graded factor are discussed.
Keywords: functionally graded materials, porous plates, quasi-3D refined theory, free vibration
1. Introduction
With the increasing demand for low thermal con-
ductivity and high fracture toughness materials, a new advanced material called functionally graded
materials (FGMs) are introduced. These materials exhibit continuous smooth variation of material properties from one plane to another and thus reduce inplane transverse stresses and give high thermal resis-
© Zenkour A.M., Aljadani M.H., 2021
tance encountered in traditional composites structures. Recently, functionally graded materials have found various applications in aerospace, electrical devices, energy transformation, civil engineering, biomedical engineering, automotive industry, and optics.
Various studies have been performed to investigate the vibration of functionally graded plates. Hos-seini-Hashemi et al. [1] examined the free vibration response of moderately thick rectangular FGM plates using first-order shear deformation plate theory. Fares et al. [2] analyzed the free vibration response of FGM plates by applying a mixed variational approach with the framework of refined two-dimensional plate theory. Thai and Choi [3] used a refined shear deformation plate theory to investigate the free vibration of FGM plates laying on the elastic base. Neves et al. [4] obtained the bending and free vibration of FGM plates based on Carrera's unified formulation. A sinusoidal shear deformation plate theory is used for bending, buckling, and vibration analysis of FGM plates by Thai and Vo [5]. By using the mesh-less technique, Neves et al. [6] examined the static and free vibration of FGM plates. Thai et al. [7] utilized an efficient shear deformation theory to analyze the free vibration of FGM plates. Zaoui et al. [8] presented the free vibration response of FGM plates laying on elastic foundations by employing a quasi-3D hybrid-type higher-order shear deformation plate theory. A two-dimensional trigonometric shear deformation plate theory is used to obtain the free vibration of FGM plates by Zenkour [9]. Hebali et al. [10] examined the static, buckling, and vibration response of FGM plates by using a novel higher-order shear deformation plate theory. By considering the thickness stretching effect, Jha et al. [11] reported the free vibration of FGM plates. By using a four-variable shear deformation shell theory, Djedid et al. [12] applied a simple n-order four variable refined theory to investigate the bending and free vibration response of FGM plates.
The inclusion of porosity during the manufacturing process of FGM structures reduces the stiffness of the structure and gives exceptional energy-absorbing capability. These materials are mainly used in enhanced filtration, energy-absorbing systems, medical implants, sound absorbers, insulating materials, automotive industry, and electromagnetic shielding. A large number of research efforts have been made to investigate the vibration response of FGM beams [13-17] and shell [18-20]. Akba§ [21] presented the free vibration and bending response of the FGM po-
rous plate based on the first-order shear deformation plate theory. Rezaei et al. [22] investigated the free vibration of FGM plates with porosity according to a simple first-order shear deformation plate theory. The Rayleigh-Ritz procedure is utilized for the vibration response of thick FGM porous plates by Zhao et al. [23]. Using a state-space approach with a four variable plate theory, Demirhan and Taskin [24] presented the free vibration and static response of FGM porous plates. Based on a refined higher-order theory, Daikh and Zenkour [25] reported the free vibration and buckling of FGM sandwich plates with porosity.
The thickness stretching effect is considered in various plate theories such as Carrera's unified formulation [26] and quasi-3D plate theory. Among them, Zenkour [27] used a quasi-3D refined plate theory to investigate the static of FGM porous plates. Shear and normal deformations plate theory with the finite element method are applied for the vibration response of FGM microplates by Karamanli and Ay-dogdu [28]. Matsunaga [29] reported the free vibration and buckling of FGM plates via higher-order theory shear and normal deformations plate theory.
Based on open literature, there has been no study on free vibration of the FGM porous plates based on a quasi-3D refined theory. Therefore, this article aims to propose a quasi-3D refined plate theory for the free vibration response of FGM porous plates. The quasi-3D refined theory takes into consideration the thickness stretching effect. The material properties of the porous plate are varying across the plate thickness. The porous plate is graded according to modified polynomial material law. Hamilton's principle is used to obtain the motion equations according to the quasi-3D refined theory. Navier's technique is applied to obtain the closed-form solution for simply-supported FGM porous plates. The influence of porosity parameter, aspect ratio, side thickness, and exponent graded factor are discussed in detail.
2. Formulation of the problem
2.1. Structural model
Consider an isotropic rectangular porous plate with uniform thickness h and side dimensions a, b made of functionally graded materials, as illustrated in Fig. 1. A Cartesian coordinate system (x,y, z) is used to describe the plate geometry whereas 0 < x < a, 0 <y < b and -h/2 < z < h/2. The porous plate is composed of ceramics at the upper plane (z = h/2), and it
Fig. 1. The geometry and coordinates of the FGM porous plate
is continuously varying to metal at the lower plane
(z = -h/2).
Assuming even pores scatting across the FGM plate, the effective material properties, as E(z) is Young's modulus, v(z) is Poisson's ratio, and p(z) is the material density, are assumed to vary continuously across the plate thickness based on a modified polynomial material law as [30]
+ P
P ( * ) = P
z 1 h + 2
1 -I z+if-^
h2
, p > 0,0 <a « 1, (1)
where p is an exponent graded factor that presents the variation of material properties along with the thickness, and a is the porosity volume fraction that describes the fraction of pore volume to the total volume of the structure. Besides, subscripts m and c indicate the metal and ceramic materials of the FGM plate, respectively.
2.2. The quasi-3D theory
According to the quasi-3D theory [31], the displacements u, v, and w of any point in the Cartesian coordinate system (x, y, z) could be assumed as
u(x, y, z) = u0(x, y) - z + y(z)u1(x, y), dx
dw0
v(x y, z) = vo(x, y) - z—0 + v(z) V1(x, yx (2) ox
w(x, y, z) = Wo(x, y) + z)wi(x, y),
where u0 and v0 are in-plane displacements of the midplane, w0 is the deflection in the normal transverse direction, u1 and v1 are rotations of the yz and xz planes caused by the bending, and w1 denotes an additional displacement accounts for thickness stretching effect. The superscript notation (') represents differentiation concerning z. The shape function v(z) of the present investigation is given as
V (*) = (3)
The advantage of the present theory over the other higher-order shear deformation theories is this quasi-theory takes into consideration the effect of normal transverse strain through the plate thickness. The current formulation does not require any shear correction factor.
2.3. Strain-displacement relation
Based on the displacement filed of the quasi-3D deformation plate theory, the linear strain relations of the porous FGM plate e, (ei = en, e2 = e22, e3 = e33, e4 = e23, e5 = e13, e6 = e12) can be expressed as
(4)
e1 [ei0)' [e(1)' [e(2)"
e 2 > = < e 20) > + z< e21) • + V( z )- e22)
.e6. e60) e61) e62)
(e 4, e5} = y'( z ){ei0), e50)}, e3 = y"( z )e£
where the linear strain-displacement geometrical relationships are given as
,(0)
e(0) =
du,
0 c(0)
dv,
dx
, e2 =
0 c(0)
(0)
(0) dw1
e50) = U1 +-d1, e r =
dx
dy (0) dv,
e3 = w1, e 4 = v1 +
dw1 dy
0 , uu0 „(1) _ "T , &i —
dx dy
d 2 w0
dx2
e 21) =
2
d w,
0 c(1)
dy2 D
,(2) = dv1 '2 —
dy
= -2 d2 W0 e12) = du1 dxdy ' 1 dx
e(2) = dv1 + du1 6 _
dx dy
(5)
The constitutive relations of stress-strain components for the FGM porous plate are stated as
a/ = z )(si + s2 + S3)0/ + 2I(z )s/, (6)
where ôi = 1 for i = 1, 2, 3 and ôi = 0 for i = 4, 5, 6. Also, X(z) and |(z) indicate Lamé's coefficients defined as
vE(z) , , E(z)
K * ) =
(1 + v)(1 - 2 v)
, K z ) =
2(1 + v)'
(7)
3. The equations of motion
The equations of motion can be obtained by using the Hamilton's principle. The principle can be presented as
} (SU -ÔK )dt = 0,
(8)
where SU, SK indicate the virtual of strain energy, and the virtual of the kinetic energy of FGM plates, respectively, and can be presented as
h 2
5U = jj J (c,Ss, +Tj.Syij)dzdQ = 0, (9)
Q-h/ 2 h/ 2
SK = JJ J p(z)(uSu + vSv + wSw)dzdQ = 0, (10)
Q- h 2
in which () represents the derivative concerning time, i, j = 1, 2, 3. Substituting Eqs. (9) and (10) into Eq. (8), then integrating the variation by parts the coefficients of Su0, Sv0, Sw0, Sui, Sv and Sw to zero, separately, the following equations of motion are obtained
Su0: + °N6 = I1u0 -12^^+14u1,
cx cy
dx
Sv0: + = iv -+ i4v
Cx cy
Cy
C 2M1 „ c2M6 c 2m2
4,:-^ + 2—6- +
= I1WW П +12
ax2 CUn
dxdy cy
CVn
( c 2 w n c 2 w n
dx cy
di1 ctVj
- I-
cx2
cy2
+^1+^
(11)
„ SS1 dS6 dw>0
SuV —1 + —6 - Q = /4% -15 ~~~ + ,
ox cy ox
0S6 0S2 Ow0 ..
^ 1*- + ~CT - Q4 =17V0 -15 ^ +17^
ox oy oy
S.',:f + Q - N3. = /s»>1,
ox2 oy
where Nj and Mj (j = 1, 2, 6) indicate stress resultants and stress couples, Sj indicate additional stress couples, Ql (l = 4, 5) and N3 indicate transverse and normal shear stress resultants. They are presented as
h 2
{Nj, Mj, Sj} = J {1, z V(z)}cjdz, j = 1,2,6,
-h/ 2
h/2 h/2 Ql = J V(z)cdz, N3 = J V"(z)c3dz, (12)
-h/2 -h/2
h 2
^ ^ ^ 14, /5,16,17, 78} = J P(z)
-h/ 2
X {1, z, z2, v(z), zy(z), V(z)[y(z)]2, [y'(z)]2}dz. The stress and moment resultants could be presented in terms of total strains by inserting Eq. (6) into Eq. (12):
( N N2 N6
M1 M 2
M6
N3
A11 A12
A12 A11
n n
B11 B12
B12 B11
n n
$1
B12
B12 B11 n n
n n
A66 n n
B66 n n
B^6
B11 B1 Br
Gi, Gi, Hi
A1
ft 2
D11
Ds D12
n
H1,
12 B11
n -12 -11 n
D12 D11 n HS
n n
B66 n
B11 B1
BS
12 B1S1 n
D11 D12
n DS
-66 n n
Ds
D66
Ds D11
n
F11 F12 F12 F11
n n
B66 n
n Ds
D66 n n
n n F6
n Jt
GI,
HS,
Js
J13 J
J13
n
L-,
(13)
1,
cu0/cx cvq/ cy Cv0I cx + CU0I cy -C2w0/cx2
-c4/ cy2
- 2d2w0/(dxdy)
cuj cx cvj cy cvj cx + cu1/ cy
{Q4,65} = A44 W2 (14)
in which
hj 2
{An, Bn, B/l5 Dn, Dh, F\} = J [X(z) + 2ц(z)]
-h/ 2
X {1, z, y(z), z2, zy(z), [y(z)]2}dz,
h 2
{A12, B12, Br2, D12, Dr2, F12}
= J X(z){1, z, v(z), z2, zV(z), [y(z)]2}dz,
-h) 2
{Л56, B66, B66, D66, D66, F66 }
h2 (15)
= J Ц(z){1, z, v(z), z2, zy(z),[y(z)]2}dz,
-h 2
h 2
A44 = J z)[V'( z)]2dz,
-h/ 2
h 2
¿33 = J [X(z) + 2^(z)][V"(z)]2dz,
-h/ 2
h/ 2
{G/3, H/3, J/3} = J X(z){1, z, v(z)}V"(z)dz.
-h/ 2
Substituting Eqs. (13) and (14) into the motion equations, the following system of algebraic equations is given:
([k ]-ш>]){Л} = {0},
(16)
X
in which [k] and [m] represent the symmetric matrices of differential operators and {A} = {u0, v0, w0, u1, v1, wi}t. The components of the symmetric matrices [k] and [m] are given in Appendix A.
4. Closed-form solution
This analysis is concerned with the exact solution for a simply-supported FGM porous plate. The following boundary conditions are imposed at the side ends of porous plate:
v0 = w0 = v1 = w1 = N1 = M1 = S1 = 0
at x = 0, a, u0 = w0 = u1 = w1 = N2 = M 2 = S2 = 0
(17)
(u0 U1)"
(w0 =
,( v0 v1).
at x = 0, b,
where = mi/a, ^ = ni/b, n and m are vibration mode. Based on Navier's technique, the following forms for (u0, v0, w0, u1, v1, w1) are assumed to satisfy the above boundary conditions:
(U, X )cos(^x)sin(ny ) (W, Z)sin(^x)sin(ny) |, (18) (V, Y)sin(^x)cos(^j) ^
where U, V, W, X, Y and Z are arbitrary parameters. By combining Eqs. (16) and (18), the following system of equations is obtained:
([K] -ra2[M]){A} = {0}, (19)
where [K] and [M] are stiffness and mass matrices, respectively, {A} is expressed as
{A} = {U, V, W, X, Y, Z}l, (20)
and the components of the symmetric matrices [K] and [M] are given in Appendix B.
5. Numerical results and discussions
5.1. Comparision studies
The fundamental natural frequency response of simply-supported porous FGM plates is investigated. In the present analysis of the quasi-3D refined theory, the effect of thickness stretching s3 in porous FGM plates is taken into consideration. The plate is made of a mixture of aluminum (Al) and alumina (Al2O3). The bottom surface of the plate structure is metal, while the top surface is ceramic. The following material properties are considered unless stated otherwise:
Em = 70 GPa, vm = 0.3, pm = 2707 kg/m3,
(21)
Ec = 380GPa, vc = 0.3, pc = 3800kg/m3. Various numerical examples are introduced to present the influence of porosity volume fraction, in-
homogeneity, and geometric parameters. The present investigations (s3 = 0 and s3 ^ 0) are compared with the existing data in the literature.
Example 1. The fundamental natural frequency of simply supported FGM square plates for different side-to-thickness ratios a/h, vibration modes, and exponent graded factors are presented in Table 1. In this example, the fundamental frequency is reported by using the nondimensional form ra = rahyj pc/ Ec [32]. The present fundamental frequency ra results are compared the first-order shear deformation theory (FSDT) with shear correction factor K = 5/6 in [33], FSDT that used the element-free KP-Ritz method [1], higher-order shear deformation plate theory (HSDT) used in [29, 34], Reddy's third-order shear deformation plate theory (TSDT) [34]; and eight unknown higher-order shear deformation plate theory (HSDT-8) that accounts for s3 ^ 0 [32]. It can be seen that the current obtained results when s3 ^ 0 are in excellent agreement with one obtained by (HSDT-8) [32]. Moreover, the present results in the case of s3 = 0 are in good agreement with those obtained by TSDT [34]. It worth mentioning that the results of FSDT are underestimated since the theory assumed constant shear strain across the structure.
Example 2. The fundamental natural frequency of simply supported FGM rectangular plates for different side-to-thickness ratios a/h, vibration modes (m, n) and exponent graded factors p are shown in Table 2. The nondimensional fundamental frequency is given as ra = raa2 / (h^j pc/ Ec). The obtained results according to the first-order shear deformation theory [35] and sinusoidal shear deformation plate theory (SSDT) are compared with the present quasi-3D refined theory (s3 ^ 0). As observed, the present results when s3 = 0 are in excellent agreement with one obtained by SSDT [5]. However, the present results in the case of s3 ^ 0 are given more accurate results since the thickness stretching effect is considered in the analysis.
Example 3. Table 3 demonstrates the fundamental natural frequency of simply supported nonporous and porous FGM plates for different porosity parameters a, side-to-thickness ratios a/h, aspect ratios a/b and exponent graded factors p (pm = 2707 kg/m3). In this table, the nondimensional fundamental frequency is obtained based on the form ra = rah^j pm/ Em. The obtained results according to the first-order shear de-
Table 1. Comparison of the nondimensional fundamental frequency ra = roh^/pc/Ec of square FGM plates
a/h Mode (m, n) Method a S3 p
0 0.5 1 4 10 Metal
5 (1, 1) Present 0.1 * 0 0.2150 0.1820 0.1621 0.1306 0.1210 0.0952
0.0 * 0 0.2136 0.1836 0.1671 0.1417 0.1325 0.1091
0.0 = 0 0.2112 0.1809 0.1632 0.1377 0.1301 0.1080
HSDT-8 [32] 0.0 * 0 0.2122 0.1816 0.1640 0.1386 0.1307 -
TSDT [34] 0.0 = 0 0.2113 0.1807 0.1631 0.1378 0.1301 0.1076
HSDT [29] 0.0 = 0 0.2121 0.1819 0.1640 0.1383 0.1306 0.1077
FSDT [1] 0.0 = 0 0.2112 0.1806 0.1650 0.1371 0.1304 0.1075
FSDT [33] 0.0 = 0 0.2055 0.1757 0.1587 0.1356 0.1284 -
(1, 2) Present 0.1 * 0 0.4738 0.4050 0.3612 0.2836 0.2572 0.2100
0.0 * 0 0.4696 0.4076 0.3704 0.3064 0.2830 0.2398
0.0 = 0 0.4626 0.3992 0.3614 0.2976 0.2772 0.2362
TSDT [34] 0.0 = 0 0.4623 0.3989 0.3607 0.2980 0.2771 0.2355
HSDT [29] 0.0 = 0 0.4658 0.4040 0.3644 0.3000 0.2790 0.2365
(2, 2) Present 0.1 * 0 0.6886 0.5914 0.5292 0.4090 0.3666 0.3054
0.0 * 0 0.6824 0.5924 0.5410 0.4408 0.4038 0.3486
0.0 = 0 0.6704 0.5814 0.5266 0.4284 0.3956 0.3420
TSDT [34] 0.0 = 0 0.6688 0.5803 0.5254 0.4284 0.3948 0.3407
HSDT [29] 0.0 = 0 0.6753 0.5891 0.5444 0.4362 0.3981 0.3429
10 (1, 1) Present 0.1 * 0 0.0583 0.0490 0.0436 0.0359 0.0339 0.0258
0.0 * 0 0.0581 0.0496 0.0451 0.0391 0.0370 0.0296
0.0 = 0 0.0576 0.0490 0.0442 0.0380 0.0364 0.0294
HSDT-8 [32] 0.0 * 0 0.0578 0.0491 0.0443 0.0381 0.0364 -
TSDT [34] 0.0 = 0 0.0577 0.0490 0.0442 0.0381 0.0364 0.0293
HSDT [29] 0.0 = 0 0.0577 0.0492 0.0443 0.0381 0.0364 0.0293
FSDT [1] 0.0 = 0 0.0577 0.0492 0.0445 0.0383 0.0363 0.0294
FSDT [33] 0.0 = 0 0.0568 0.0482 0.0435 0.0376 0.3592 -
(1, 2) Present 0.1 * 0 0.1397 0.1179 0.1050 0.0853 0.0796 0.0619
0.0 * 0 0.1389 0.1192 0.1083 0.0927 0.0871 0.0709
0.0 = 0 0.1376 0.1174 0.1060 0.0902 0.0856 0.0702
TSDT [34] 0.0 = 0 0.1377 0.1174 0.1059 0.0903 0.0856 0.0701
HSDT [29] 0.0 = 0 0.1381 0.1180 0.1063 0.0904 0.0859 0.0701
FSDT [33] 0.0 = 0 0.1354 0.1154 0.1042 - 0.0850 -
(2, 2) Present 0.1 * 0 0.2150 0.1820 0.1622 0.1306 0.1209 0.0952
0.0 * 0 0.2137 0.1837 0.1671 0.1417 0.1325 0.1092
0.0 = 0 0.2113 0.1808 0.1632 0.1378 0.1301 0.1079
TSDT [34] 0.0 = 0 0.2113 0.1807 0.1631 0.1378 0.1301 0.1076
HSDT [29] 0.0 = 0 0.2121 0.1819 0.1640 0.1383 0.1306 0.1077
FSDT [33] 0.0 = 0 0.2063 0.1764 0.1594 - 0.1289 -
20 (1, 1) Present 0.1 * 0 0.0150 0.0125 0.0111 0.0092 0.0088 0.0066
0.0 * 0 0.0149 0.0127 0.0115 0.0100 0.0096 0.0076
0.0 = 0 0.0148 0.0125 0.0113 0.0098 0.0094 0.0075
TSDT [34] 0.0 = 0 0.0148 0.0125 0.0113 0.0098 0.0094 -
FSDT [1] 0.0 = 0 0.0148 0.0128 0.0115 0.0101 0.0096 -
FSDT [33] 0.0 = 0 0.0146 0.0124 0.0112 0.0097 0.0093 -
Table 2. Comparison of the first four nondimensional frequencies co = <»a2/hyjpc/Ec of a rectangular FGM plate (b = 2a)
p a/h Method a e, Mode (m, n)
1 (1,1) 2(1, 2) , (1, ,) 4 (2,1)
n 5 Present 0.1 * 0 ,.4928 5. ,745 8.2455 10.,544
= 0 * 0 ,.4756 5. ,4,0 8.1890 10.2744
= 0 = 0 ,.4414 5.2810 8.0745 10.1282
SSDT [5] = 0 = 0 ,.4416 5.2822 8.0772 10.1201
FSDT [35] = 0 = 0 ,.4409 5.2802 8.0710 9.7416
10 Present 0.1 * 0 ,.6920 5.8,76 9.,089 12.000,
= 0 * 0 ,.6787 5.8155 9.2696 11.94,6
= 0 = 0 ,.6518 5.769, 9.1874 11.8,51
SSDT [5] = 0 = 0 ,.6519 5.7697 9.1887 11.8,26
FSDT [35] = 0 = 0 ,.6518 5.769, 9.1876 11.8,10
20 Present 0.1 * 0 ,.7490 5.9792 9.6672 12.5901
= 0 * 0 ,.7,70 5.9602 9.5666 12.5461
= 0 = 0 ,.7124 5.9196 9.5671 12.4569
SSDT [5] = 0 = 0 ,.7Ш 5.9199 9.5668 12.4565
FSDT [35] = 0 = 0 ,.7Ш 5.9198 9.6,42 12.4560
1 5 Present 0.1 * 0 2.6244 4.05,0 6.2505 7.8770
= 0 * 0 2.7079 4.1760 6.4260 8.0845
= 0 = 0 2.6484 4.080, 6.2710 7.8845
SSDT [5] = 0 = 0 2.6478 4.0787 6.2678 7.8787
FSDT [35] = 0 = 0 2.647, 4.077, 6.26,6 7.8711
10 Present 0.1 * 0 2.7580 4.,662 6.9764 9.0069
= 0 * 0 2.8527 4.5145 7.2066 9.2984
= 0 = 0 2.7940 4.4199 7.05,2 9.0960
SSDT [5] = 0 = 0 2.79,7 4.4194 7.0519 9.0940
FSDT [35] = 0 = 0 2.79,7 4.4192 7.0512 9.0928
20 Present 0.1 * 0 2.7956 4.4602 7.2144 9.4016
= 0 * 0 2.89,8 4.6168 7.4662 9.7272
= 0 = 0 2.8,54 4.5228 7.,126 9.5274
SSDT [5] = 0 = 0 2.8,5, 4.5228 7.3133 9.526,
FSDT [35] = 0 = 0 2.8,52 4.5228 7.,1,2 9.5261
5 5 Present 0.1 * 0 2.0948 ,.1946 4.8474 6.0450
= 0 * 0 2.2828 ,.4788 5.2725 6.5710
= 0 = 0 2.225, ,.,902 5.1,60 6.,990
SSDT [5] = 0 = 0 2.2260 ,.,914 5.1,78 6.4010
FSDT [35] = 0 = 0 2.2528 ,.4492 5.2579 6.5749
10 Present 0.1 * 0 2.2468 ,.5411 5.6178 7.2161
= 0 * 0 2.4514 ,.8624 6.1257 7.8656
= 0 = 0 2.,909 ,.7669 5.97,0 7.6676
SSDT [5] = 0 = 0 2.,912 ,.767, 5.9742 7.6696
FSDT [35] = 0 = 0 2.,998 ,.7881 6.0247 7.7505
20 Present * 0 * 0 2.2914 ,.6524 5.8950 7.6692
= 0 * 0 2.5014 ,.9860 6.4—4 8.,680
= 0 = 0 2.4400 ,.8880 6.2748 8.1618
SSDT [5] = 0 = 0 2.4401 ,.8881 6.275, 8.1624
FSDT [35] = 0 = 0 2.4425 ,.89,9 6.290, 8.1875
Table 3. Nondimensional natural frequency rô = mh^pm/Em of nonporous and porous FGM plates
a/h
a/b a Method S3 p
0.0 0.1 0.5 1.0
Present * 0 0.2732870 0.2636090 0.2342840 0.2129170
0.0 = 0 0.2706930 0.2610400 0.2308640 0.2082430
FSDT [22] = 0 0.2719460 0.2620170 0.2315730 0.2089970
Present * 0 0.2771090 0.2660040 0.2292760 0.1982500
0.5 0.2 = 0 0.2755850 0.2644900 0.2271030 0.1950560
FSDT [22] = 0 0.2804780 0.2689600 0.2308750 0.1984520
Present * 0 0.2860830 0.2729680 0.2238090 0.1700280
0.4 = 0 0.2851950 0.2720880 0.2224540 0.1679680
FSDT [22] = 0 0.2929770 0.2792940 0.2284530 0.1727790
Present * 0 0.4201340 0.4056140 0.3612050 0.3283230
0.0 = 0 0.4155280 0.4010710 0.3554850 0.3208230
FSDT [22] = 0 0.4182100 0.4032200 0.3570500 0.3224900
HSDT [24] = 0 0.4362400 0.4203500 0.3711200 0.3342500
Present * 0 0.4267310 0.4100310 0.3543430 0.3067510
1.0 0.2 = 0 0.4239140 0.4072480 0.3506540 0.3016170
FSDT [22] = 0 0.4313300 0.4139600 0.3562900 0.3068100
HSDT [24] = 0 0.4499300 0.4315000 0.3698100 0.3168400
Present * 0 0.4411880 0.4214400 0.3468170 0.2643920
0.4 = 0 0.4394850 0.4197600 0.3445030 0.2610960
FSDT [22] = 0 0.4505500 0.4299400 0.3531000 0.2682500
HSDT [24] = 0 0.4699800 0.4480800 0.3655800 0.2748200
Present * 0 0.0723151 0.0696561 0.0617164 0.0560759
0.0 = 0 0.0717915 0.0691331 0.0609327 0.0549209
FSDT [22] = 0 0.0719090 0.0692080 0.0609870 0.0549790
Present * 0 0.0731311 0.0700914 0.0601708 0.0519398
0.5 0.2 = 0 0.0728524 0.0698102 0.0596852 0.0511519
FSDT [22] = 0 0.0741650 0.0710290 0.0607210 0.0520520
Present * 0 0.0753255 0.0717474 0.0584964 0.0442117
0.4 = 0 0.0751819 0.0715999 0.0581979 0.0436978
FSDT [22] = 0 0.0774700 0.0737370 0.0599430 0.0450290
Present * 0 0.1143150 0.1101460 0.0976585 0.0887366
0.0 = 0 0.1134300 0.1092650 0.0963775 0.0868822
FSDT [22] = 0 0.1136900 0.1094500 0.0965100 0.0870200
HSDT [24] = 0 0.1151400 0.1108200 0.0976300 0.0879600
Present * 0 0.1156730 0.1109050 0.0952916 0.0822855
1.0 0.2 = 0 0.1151890 0.1104180 0.0944932 0.0810212
FSDT [22] = 0 0.1172600 0.1123300 0.0961200 0.0824400
HSDT [24] = 0 0.1187500 0.1137300 0.0971900 0.0832400
Present * 0 0.1192050 0.1135870 0.0927233 0.0701554
0.4 = 0 0.1189470 0.1133240 0.0922309 0.0693358
FSDT [22] = 0 0.1224900 0.1166200 0.0949400 0.0714200
HSDT [24] = 0 0.1240500 0.1180700 0.0959200 0.0719400
10
5
Table 3 (continued)
a/h a/b a Method S3 p
0.0 0.1 0.5 1.0
20 0.5 0.0 Present * 0 0.0183647 0.0176819 0.0156521 0.0142209
= 0 0.0182440 0.0175608 0.0154620 0.0139337
FSDT [22] = 0 0.0182570 0.0175650 0.0154650 0.0139370
0.2 Present * 0 0.0185571 0.0177776 0.0152433 0.0131517
= 0 0.0184955 0.0177150 0.0151264 0.0129558
FSDT [22] = 0 0.0188290 0.0180260 0.0153920 0.0131840
0.4 Present * 0 0.0191010 0.0181841 0.0148014 0.0111710
= 0 0.0190712 0.0181530 0.0147300 0.0110421
FSDT [22] = 0 0.0196690 0.0187120 0.0151840 0.0113840
1.0 0.0 Present * 0 0.0292899 0.0282034 0.0249707 0.0226875
= 0 0.0290933 0.0280064 0.0246644 0.0222274
FSDT [22] = 0 0.0291200 0.0280200 0.0246700 0.0222400
HSDT [24] = 0 0.0292200 0.0281100 0.0247500 0.0223000
0.2 Present * 0 0.0296015 0.0283609 0.0243238 0.0209883
= 0 0.0295003 0.0282581 0.0241354 0.0206745
FSDT [22] = 0 0.0300300 0.0287500 0.0245600 0.0210400
HSDT [24] = 0 0.0301400 0.0288500 0.0246300 0.0211000
0.4 Present * 0 0.0304734 0.0290138 0.0236244 0.0143570
= 0 0.0304236 0.0289622 0.0235091 0.0176292
FSDT [22] = 0 0.0313700 0.0298500 0.0242300 0.0181700
HSDT [24] = 0 0.0314800 0.0299500 0.0242900 0.0182100
formation theory; that uses a shear correction factor K = (tc2/12)V2 [22] and sinusoidal shear deformation plate theory are compared with the present quasi-3D refined theory (s3 ^ 0). As observed, the present results when s3 = 0 are in excellent agreement with one obtained by HSDT [24]. However, both of the results in [22, 24] do not consider the thickness stretching effect and therefore the present results s3 ^ 0 give more accurate results for the fundamental natural frequency.
In Table 3, the nondimensional natural frequency of FGM plates is decreasing with the increase of exponent graded factor. Furthermore, the rising of aspect ratio value a/b increases the fundamental frequency. The fundamental frequency decreases as a/h increases. As an observation note, no specific trend concerning the impact of a on the natural frequency of porous FGM plates for several exponents graded factors p was noticed. For example, the natural frequency co when p = 0.1 increases as a increases,
meanwhile the response is the opposite when p increases.
Example 4. The first four nondimensional natural frequencies co of porous FGM plates (a/h = 20) are presented in Table 4 (pm = 2707 kg/m3). As observed, the obtained results are in excellent agreement with those in [22]. The natural frequency of the mode (1, 3) has the highest natural frequency response and (1, 1) has the lowest. As mentioned in Table 3, co rises as porosity increases when p = 0.1 and << reduces as porosity increases when p = 1. The impact of a on co is more noticeable for the 3rd and 4th vibration modes.
5.2. Parametric studies
After clarifying the reliability and accuracy of the present quasi-3D refined plate theory, the following illustrations for the nondimensional natural frequency response of FGM porous plates can be used as a benchmark for future research works. The influence of porosity volume fraction, inhomogeneity, and geo-
Table 4. The first four nondimensional natural frequencies co = chyjpm/Em of FGM porous square plates with a/h = 20
p a Method S3 Mode
1 (1, 1) 2 (1, 2) 3 (2, 2) 4 (1, 3)
0.1 0.1 Present * 0 0.0282 0.0695 0.1103 0.1369
= 0 0.0281 0.0693 0.1096 0.1360
FSDT [22] = 0 0.0283 0.0700 0.1108 0.1374
0.2 Present * 0 0.0284 0.0701 0.1109 0.1376
= 0 0.0282 0.0699 0.1104 0.1370
FSDT [22] = 0 0.0287 0.0710 0.1123 0.1394
1.0 0.1 Present * 0 0.0218 0.0538 0.0857 0.1065
= 0 0.0215 0.0543 0.0850 0.1046
FSDT [22] = 0 0.0217 0.0538 0.0851 0.1057
0.2 Present * 0 0.0208 0.0506 0.0828 0.1021
= 0 0.0206 0.0525 0.0828 0.1004
FSDT [22] = 0 0.0210 0.0520 0.0824 0.1024
metric parameters are discussed in these figures. In this section, the nondimensional fundamental frequency is obtained based on the form
co = rahyj pm/ £m,
and the following values are considered a/h = 10, a/b = 1, p = 1, a = 0.1, and vibration mode (1, 1).
Figure 2 reveals the effect of the porosity parameter a and exponent graded factor p on the nondimen-sional natural frequency co of FGM porous plates. As observed, the nondimensional fundamental natural frequency increases when the exponent graded factor p is small. However, the nondimensional fundamental natural frequency decreases as the exponent graded factor p increases. The dropping of the fundamental natural frequency because of the impact of reducing structural stiffness dominates the reducing inertia at this point.
The influence of the porosity on the first four non-dimensional natural frequencies of FGM porous
Fig. 2. Effect of the porosity and exponent graded factor on the nondimensional natural frequency of FGM porous plates. p = 0.0 (7), 0.1 (2), 0.5 (3), 1.0 (4)
plates for two exponent graded factors is given in Fig. 3. It can be seen that the first four nondimen-sional natural frequencies increase with the increase of the porosity parameter when p = 0.1. On the contrary, the behavior of the first four nondimensional natural frequencies changes to decreasing when p = 1.
Figure 4 exhibits the effect of exponent graded factor p and porosity parameter a on the nondimen-
0.0
0.
0
0.3
Fig. 3. Effect of the porosity on the first four nondimensional natural frequencies of FGM porous plates for two exponent graded factors p = 0.1 (a), 1.0 (b), a/b = 0.5, a/h = 10: 7—mode 1 (1, 1), 2—mode 2 (1, 2), 3—mode 3 (2, 2), 4—mode 4 (1, 3)
\
\ K ------
\ -------
\ 2 3
0 2 4 6 8 p
Fig. 4. Effect of exponent graded factor and porosity parameter on the nondimensional natural frequency of FGM porous plates. a = 0.0 (7), 0.1 (2), 0.2 (5)
sional natural frequency co of FGM porous plates. As p rises, the fundamental natural frequency co of FGM porous plates decreases. For p < 0.2, the fundamental natural frequency co increases as a increases. However, for p > 0.2 the fundamental natural frequency co decreases as the porosity parameter a increases.
The effect of exponent graded factor p and aspect ratio a/b on the nondimensional natural frequency co of FGM porous plates is illustrated in Fig. 5. As the aspect ratio a/b increases, the fundamental natural frequency co of FGM porous plates increases. It can be observed that the impact of aspect ratio on the fundamental natural frequency co is significant when the exponent graded factor p is less than 1. Furthermore, the higher values of aspect ratio a/b have a more substantial impact on the fundamental natural frequency co of FGM porous plates.
0 2 4 6 8 p
Fig. 6. Effect of exponent graded factor and side-to-thickness ratio on the nondimensional natural frequency of FGM porous plates. a/h = 5 (7), 10 (2), 15 (5), 20 (4)
Figure 6 illustrates the effect of exponent graded factor p and side-to-thickness ratio a/h on the nondi-mensional natural frequency co of FGM porous plates. The fundamental natural frequency co decreases as the side-to-thickness ratio a/h increases. As observed, the nonlinear trend of the fundamental natural frequency co is due to the increase of the plate stiffness.
The effect of side-to-thickness ratio a/h and exponent graded factor p on the nondimensional natural frequency co of FGM porous plates are presented in Fig. 7. It is evident that the natural frequency co reduces as the side-to-thickness ratio a/h and the exponent graded factorp increase.
Figure 8 displays the impact of porosity parameter a and side-to-thickness ratio a/h on the nondimensional natural frequency co of FGM porous plates. It can be seen that the fundamental natural frequency
Fig. 5. Effect of exponent graded factor and aspect ratio on the nondimensional natural frequency of FGM porous plates. a/b = 0.5 (7), 1.0 (2), 1.5 (5), 2.0 (4)
Fig. 7. Effect of side-to-thickness ratio and exponent graded factor on the nondimensional natural frequency of FGM porous plates. p = 0 (1), 1 (2), 2 (3), 5 (4)
Fig. 8. Effect of porosity parameter and side-to-thickness ratio on the nondimensional natural frequency of FGM porous plates. a = 0.0 (1, 1'), 0.1 (2, 2'), 0.4 (3, 3'); p = 0 (1-3), 2 (1'-3')
co decreases with the increase of side-to-thickness ratio a/h; this is a result of the reduction of the plates bending stiffness as the plate becomes thinner, which accordingly causes a decrease of the fundamental natural frequency. The influence of porosity inclusion is more pronounced for thick plates and depends on the exponent graded factor p.
The impact of aspect ratio a/b and exponent graded factor p on the nondimensional natural frequency co of FGM porous plates are given in Fig. 9. The fundamental natural frequency rises as aspect ratio a/b increases. Furthermore, the fundamental natural frequency reduces as the exponent graded factor increases p.
6. Conclusions
A quasi-3D refined shear and normal deformations plate theory is developed and implemented in the present work for vibration analysis of FGM porous plates. The thickness stretching effect is consi-
dered in the vibration analysis of FGM porous plates. Hamilton's principle is used to derive the equations
of motion, and closed-form solutions via Navier technique are obtained.
Various numerical examples are reported to ensure the accuracy of the present quasi-3D theory. The influence of porosity fraction parameters on the natural frequencies of FGM plates was reported for various values of vibration modes, aspect ratios, graded factors, and side-to thickness ratios.
The present analysis results can be summarized as follows. The natural frequency of nonporous and po-
Fig. 9. Effect of aspect ratio and exponent graded factor on the nondimensional natural frequency of FGM porous plates. p = 0 (1), 1 (2), 2 (3), 5 (4)
rous FGM plates decreases with the rise of the exponent graded factor. The response of the natural frequency with respect to porosity is affected by the value of the exponent graded factor. The impact of porosity on the natural frequency is more noticeable for the 3rd and 4th vibration modes.
Appendix A
The elements of the symmetric matrix [k] presented in Eq. (18) are given by
a2 a2 a2
k11 - A11^T + A66~2, k12 - (A12 + A66) a ^ ,
ax2 dy2 axay
a3 a3
k13 =-511^TT"(B12 + 2B66^ . 2 ,
ax axay
k - Bs a2 + Bs a2
k14 - B1^7T + B6^7T'
dx2
dy2
ki5 = (B2+Bs6)
a a , k16 -dxdy dx
k22 -
d2 a d2
dx2 + Al1 dy2
k23 --(Bi2 +2B66)
By
dx2 dy 11 dy3
d 2 d 2 d
k24 - k15, k25 - B66^T2 + B11^TT, k26 -
dx2 dy2 dy
k33 - B11
'd4
5,4 A
dx4 +dy4
+ 2( B2 + 2 2&)
dx dy
k34 =-A'1 -(^2 +2 D66) T^=
Ox Oxoy
k35 =-( D2 +2 D66) -D/1
ox Oy Oy
k - -Hs "-36 - -"13
fTL+_Ol ^
Ox2 Oy2
k - + ^ - A
«-44-^11 2 +i66 . 2 44, Ox Oy
2
k45 - (F\2 + F6)
k46 - (J13 A44)
OxOy' O
Ox
k55 - F66~2 + F1S1TT A44,
Ox Oy
k56 - (J13 A44) -
k66 - F33 A44
Oy
^il+JOL ^
Ox2 Oy2
The elements of the symmetric matrix [m] presented in Eq. (18) are given by
m11 - ^ m13 --I2^-, m14 - ^
Ox
m12 - m15 - m16 - 0, m22 - I1,
dy'
m23 - ^2 m25 - I4, m24 - m26 - 0,
m33 - I1 +13
vOx2 Oy ,
, m34 - -I5
Ox
A
Oy
m35 --I5T-, m36 -I6,
m44 - m55 - I7, m66 - I8, m45 - m46 - m56 - 0.
Appendix B
The elements of the symmetric matrix [K] presented in Eq. (21) are given by
K11 - f2 A11 + n2 A56, K12 - fn( A12 + A;6 ),
K13 --f[f2£n +n2(^12 + 2^66)],
K14-f2 5/1 +n2 B^,
K15-fn(B1S2 + B^), K16--1^3,
K22 -f2A<56 +n2A11, K23--n[f2(B12 +2B66) + n2 B11], K24 - K15, K25-fB + ^2, K26--^3, K33 -(f4 +n4)BS1 + 2fY(B1s2 + 2B^), K34 --f[f2D11 +n2(A2 +2D^)], K35--n[f2( D2 +2D66) + n2 AU K36 - (f2 +n2)HI3,
K44-f2 F1 +n2 F6 + A44, K45 -fn(F2 + F66 ),
K46 -f(A44 - J3), K55 -fF +n2F1S1 + A44, K56 - n( A44 -Jo), K66-(f2 +n2)A44 +L33.
The elements of the symmetric matrix [M] presented in Eq. (21) are given by
M11 - ^ M13 --I2f M14 - 14, M12 - Mu - M16 - 0,
M22 - I1, M23 - -12n, M25 - I4, M24 - M26 - 0, M33 -11 +13(f2 +n2),
M34 - -I5f, M35 - -I5n M36 -I6,
M44 - M55 - I7, M66 - I8, M45 - M46 - M56 - 0.
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Received 31.08.2020, revised 31.08.2020, accepted 30.09.2020
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Ashraf M. Zenkour, Ph.D., Prof., King Abdulaziz University, Saudi Arabia, Kafrelsheikh University, Egypt, zenkour@kau.edu.sa, zenkour@sci.kfs.edu.eg
Maryam H. Aljadani, Ph.D., Ass. Prof., Umm Al-Qura University, Saudi Arabia, Mhjadani@uqu.edu.sa
