EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 4, 2020, 766-778
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Free vibration analysis of thick FGM spherical shells
based on a third-order shear deformation theory

Mohammad Zannon1,∗,Abdullah Abu-Rqayiq2,Ammar Al-bdour1

1 Department of Mathematics, Tafila Technical University, Tafila, Jordan
2 Department of Mathematics and Statistics, Texas AM University , Corpus Christi, USA

Abstract. In this paper, we consider problems associated with free vibration of functionally
graded thick spherical shells. We perform our analysis by collecting the radial basis functions,
according to the third-order shear deformation theory that accounts the thickness deformation
consuming the principle of virtual work. Numerical results which include spherical shell panels
with all edges simply supported or clamped are presented to validate the accuracy of the present
approach.

2020 Mathematics Subject Classifications: 74H15, 74H45, 74S30

Key Words and Phrases: Free vibration, Spherical shells, functionally graded materials, radial
basis functions

1. Introduction

Functionally graded materials (FGM) have enormous application potential in modern
technology such as aerospace. The use of FGM provides means to solve the problem of
high transverse shear stresses, which are induced when two similar materials with different
properties are bonded. The material properties are varied over the thickness of the bond
by mixing the different materials. FGM can minimize the thermal stress concentration
produced by the high temperature gradient. The vibration of the response of FGM beams
has been studied in different ways, including analytical studies, numerical approaches,
and the finite element method. The latter method is the most common numerical pro-
cedure for the analysis of the shells. Continuous development has resulted in complex
structural designs of the materials; we therefore require a careful analysis in addition to
the numerical methods to solve the mathematical shell models of these complex struc-
tures. Spherical shells are used in many applications, for example, in circular forms such
as the nose of a plane or the caps of pressurized cylindrical tanks. The geometrical prop-
erty of functionally graded (FG) spherical shells also leads to applications such as various

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i4.3826

Email addresses: zanno1ms@gmail.com (M. Zannon),
Abdullah.aburqayiq@tamucc.edu (A. Abu-Rqayiq), ammarbdoor@gmail.com (A. Al-bdour)

https://www.ejpam.com 766 c© 2020 EJPAM All rights reserved.



M. Zannon, A. Abu-Rqayiq, A. Al-bdour / Eur. J. Pure Appl. Math, 13 (4) (2020), 766-778 767

reservoirs, interstage and spherical caps [10],[12]. Tomabene and Viola [21] carried out
several studies on vibration solutions for FG spherical shell structures by applying the
GDQ method. Considering the stretching and predominantly flexural vibration, Reddy
and Cheng [17] discussed the properties of FG spherical, shallow shells by applying the
classical theory and the first-order and third-order shear deformation theories. In the
structure of the higher-order shear deformation theory (HSDT), Neves et al. [12] inves-
tigated the free vibration behaviours of FG shell structures, in which the expressions of
motion and the boundary restraints were acquired by Carreras unified formulation. Kar
and Panda [10] analyzed the vibration characteristics of the FG spherical shell structure
based on the HSDT, and the analytical model is discretised by using quadrilateral La-
grangian element [1],[11] Meshfree techniques are often used to solve this problem. One
such method is the radial base function (RBF) collocation method [2],[6],[13]. The use
of radial basis function for the analysis of structures and materials has been previously
studied [1],[7],[8],[9],[10],[11],[12],[13],[14],[15],[16],[17],[18]. Brischetto and Carrera (2010)
applied the unified formulation (UF) in several finite element analyses of shells by using the
Riessners mixed vibrational theorem or by using the principle of virtual Displacements.
UF can help obtain the external force terms, the stiffness matrix components, and the
inertia terms. An approach proposed by [20],[27],[28] was employed in this work which
uses a third-order shear deformation theory (TSDT), allowing for through the thickness
deformations . We also study the impact of εzz 6= 0 in these problems. The numerical ex-
amples given in this study demonstrate how effective the current approach is in predicting
the free vibrations of thick spherical FG shells. The advantage of the current technique
in expecting free vibrations of thick FG spherical shells are established thru numerical
examples.The accuracy and reliability of the present method are verified by comparing
our results to other available numerical results in literature [20].

2. Theoretical Formulation

2.1. Functionally Grade Material properties

FGM is formed by a mixture of ceramic and metal as shown in Figure 1. The material
properties change continuously from one surface to the other according to a powerlaw of
volume fraction [4],[15],[16]:

P (z) = (Pc − Pm)Vc + Pm
Vc(z) =

(
1
2 + z

h

)n (1)

where P (z) represents the effective material properties: Young’s modulus E, density ρ
and Poisson ratio v ; Pc and Pm denote the properties of the Aluminum and Alumina,
respectively; Vc(z) is the volume fractions of the Aluminum; h the thickness of structure;
is the z ∈

[−h
2 ,

h
2

]
the volume fraction exponent and n ≥ 0 thickness coordinate of the

structure.



M. Zannon, A. Abu-Rqayiq, A. Al-bdour / Eur. J. Pure Appl. Math, 13 (4) (2020), 766-778 768

2.2. Formulation

The shell structure is typically used in a typical vehicle’s outer body. Some of these
shell structures are flat, but most of them are curved in nature, so they are called shells
[28],[29]. Plates are a special type of shells, so the focus in modelling and analyses has
been on shells. Shells bodies are in three dimensions, one of which is thickness, which
is small compared to the other two dimensions. Shell structure theories are based upon
the three-dimensional (3D) theory of elasticity, which is costly and time-consuming. A
schematic diagram of this geometry is shown in Figure 2.
On the basis of the Third-order shear deformation shell theory and applying the displace-
ment field [29] we have

u(α, β, z) = u0(α, β) + zψα(α, β) + z3ϕα(α, β)
v(α, β, z) = v0(α, β) + zψβ(α, β) + z3ϕβ(α, β)
w(α, β, z) = w0(α, β) + zψz(α, β)

(2)

where −h
2 ≤ z ≤

h
2 , h is the shell thickness, u0, v0 and w0 are middle surface displacements

of the shell, ψα, ψβ, ψz are middle surface rotations and ϕα, ϕβ are higher order terms
rotation of transverse normal. Here, we use the method as the basis for a free vibration
study of FG spherical shells [1],[11],[12],[22].
The fundamental nucleo kκτsuu is reported for functionally graded doubly curved spherical
shells (radii of curvature in both α and β directions (see Fig. 2) [1],[5],[6],[7]:(

Kτsk
uu

)
11

= − Eκ(1−(vκ)2)
1−3(vκ)2−2(vκ)3

Jkτsβ/α∂
s
α∂

τ
α − Eκ

2(1+vκ)J
kτs
α/β∂

s
β∂

τ
β

+ Eκ

2(1+vκ)

(
JkτZszαβ − 1

RJ
kτzs
β − 1

RJ
kτsz
β + 1

R2J
kτs
β/α

) (3)

(
KτSk
uu

)
12

= − Eκ(vκ+(vκ)2)
1−3(vκ)2−2(vκ)3

Jkτs∂τα∂
s
β −

Eκ

2(1+vκ)J
kτs∂sα∂

τ
β

(4)

(
Kτsk
uu

)
13

= − Eκ(1−(vκ)2)
1−3(vκ)2−2(vκ)3

1
RJ

kτs
β/α∂

τ
α −

Eκ(vκ+(vκ)2)
1−3(vκ)2−2(vκ)3

1
RJ

kτs
αα ∂

τ
α

− Eκ(vκ+(vκ)2)
1−3(vκ)2−2(vκ)3

JkτSβ ∂τα + Eκ

2(1+vκ)

(
Jkτzsβ ∂sα − 1

RJ
kτS
β/α∂

s
α

) (5)

(
Kτsk
uu

)
21

= − Eκ(vκ+(vκ)2)
1−3(vκ)2−2(vκ)3

Jkτs∂sα∂
τ
β −

Eκ

2(1+vκ)J
kτs∂τα∂

s
β

(6)

(
Kτsk
uu

)
22

= − Eκ

2(1+vκ)J
kτs
β/α∂

S
α∂

τ
α + Eκ

2(1+vκ)

(
UkτZSzαβ − 1

RJ
kτZS
α − 1

RJ
kτsZ
α + 1

R2J
kτS
α/β

)
(7)



M. Zannon, A. Abu-Rqayiq, A. Al-bdour / Eur. J. Pure Appl. Math, 13 (4) (2020), 766-778 769

(
Kτsk
uu

)
23

= − Eκ(vκ+(vκ)2)
1−3(vκ)2−2(vκ)3

1
RJ

kτs
ββ ∂

τ
β −

Eκ(vκ+(vκ)2)
1−3(vκ)2−2(vκ)3

JkτSα ∂τβ

+ Eκ

2(1+vκ)

(
JkτZsα ∂sβ −

1
RJ

kτs
α/β∂

s
β

) (8)

(
Kτsk
uu

)
31

=
Eκ(1−(vκ)2)

1−3(vκ)2−2(vκ)3
1
RJ

kτ
β/α∂

s
α +

Eκ(vκ+(vκ)2)
1−3(vκ)2−2(vκ)3

1
RJ

kτ
α ∂sα

+
Eκ(vκ+(vκ)2)

1−3(vκ)2−2(vκ)3
JkτZsβ δsα − Eκ

2(1+vκ)

(
JkτsZβ ∂τα − 1

RJ
kτS
β/α∂

τ
α

) (9)

(KτSk
uu )32 =

Eκ(vκ+(vκ)2)
1−3(vκ)2−2(vκ)3

1
RJ

kτs
βk ∂

s
β +

Eκ(vκ+(vκ)2)
1−3(vκ)2−2(vκ)3

Jkτzsα ∂sβ

− Eκ

2(1+vκ)

(
Jkτszα ∂τβ −

1
RJ

kτs
α/β∂

τ
β

) (10)

(
Kτsk
uu

)
33

=
Eκ(1−(vκ)2)

1−3(vκ)2−2(vκ)3
1
R2J

kτS
β/α +

Eκ(1−(vκ)2)
1−3(vκ)2−2(vκ)3

3kττSzαβ

+2
Eκ(vκ+(vκ)2)

1−3(vκ)2−2(vκ)3
1
RJ

kτs
1 +

Eκ(vκ+(vκ)2)
1−3(vκ)2−2(vκ)3

1
R

(
Jkτzsβ + Jkτszβ

)
+

Eκ(vκ+(vκ)2)
1−3(vx)2−2(vx)3

1
R

(
Ukτzs
α + Jkτzα

)
− Eκ

2(1+vκ)J
kτs
α/β∂

s
β∂

τ
β

− Eκ

2(1+vκ)J
kτs
β/α∂

s
αα

τ
α

(11)

where(
JkτS , JkτSα , JkτSβ , Jkτsα

β
, Jkτsβ

α

, , JkτSαβ

)
=

∫
Ak

FτFS

(
1, 1 +

z

R
, 1 +

z

R
, 1, 1, 1

)
dz (12)

(
JkτZs, JkτZSα , JkτZsβ , JkτZSα

β
, JkτZSβ

α

, JkτZSαβ

)
=

∫
Ak

∂Fτ
∂z

FS

(
1, 1 +

z

R
, 1 +

z

R
, 1, 1, 1

)
dz

(13)

(
JkτSZ , JkτSZα , JkτSZβ , JkτSZα

β
, JkτSZβ

α

, JkτSZαβ

)
=

∫
Ak

Fτ
∂FS
∂z

(
1, 1 +

z

R
, 1 +

z

R
, 1, 1, 1

)
dz

(14)



M. Zannon, A. Abu-Rqayiq, A. Al-bdour / Eur. J. Pure Appl. Math, 13 (4) (2020), 766-778 770

(
Jkτ2s2 , Jkτ2Szα , Jkτ2szβ , Jkτzszα

β
, Jkτzszβ , Jkτzszαβ

)
=

∫
Ak

∂Fτ
∂z

∂Fs
∂z

(
1, 1 +

z

R
, 1 +

z

R
, 1, 1, 1

)
dz

(15)

The application of boundary conditions makes use of the fundamental nucleo Πd in the
form [13],[14],[15],[17]:

(ΠτSK
uu )11 = cos (ϕα)

Eκ
(

1− (vκ)2
)

1− 3 (vκ)2 − 2 (vκ)3
JkτSβ/α∂

S
α + cos (ϕβ)

Eκ

2 (1 + vκ)
Jkτα/β∂

S
β (16)

(
ΠτSk
uu

)
12

= cos (ϕα)
Eκ
(
νκ + (νκ)2

)
1− 3 (νκ)2 − 2 (νκ)3

Jkτs∂Sβ + cos (ϕβ)
Eκ

2 (1 + νκ)
Jkτs∂Sα (17)

(
Πτsk
uu

)
13

= cos (ϕα) 1
R

Eκ(1−(vκ)2)
1−3(vκ)2−2(vκ)3

JkτSβ/α + cos (ϕα) 1
R

Eκ(vκ+(vκ)2)
1−3(vκ)2−2(vκ)3

Jkτs

+ cos (ϕα)
Eκ(νκ+(νκ)2)

1−3(νκ)2−2(νκ)3
Jkτ2β

(18)

(
Πτsk
uu

)
21

= cos (ϕβ)
Eκ
(
νκ +

(
νK
)2)

1− 3 (νκ)2 − 2 (νκ)3
Jkτs∂Sα + cos (ϕα)

Eκ

2 (1 + νκ)
JkτSδSβ (19)

(ΠτSK
uu )22 = cos (ϕα)

Eκ

2 (1 + vκ)
JkτSβ/α∂

S
α (20)

(
ΠτSk
uu

)
23

= cos (ϕβ)
1

R

Eκ
(
vκ + (vκ)2

)
1− 3 (vκ)2 − 2 (vκ)3

Jkts + cos (ϕβ)
Eκ
(
vκ + (vκ)2

)
1− 3 (vκ)2 − 2 (vκ)3

Jkτεzzα

(21)(
Πτsk
uu

)
31

= − cos (ϕα)
1

R

Eκ

2 (1 + vκ)
Jkτsβ/α + cos (ϕα)

Eκ

2 (1 + vκ)
Jkτszβ (22)

(
Πτsk
uu

)
32

= − cos (ϕβ)
1

R

Eκ

2 (1 + vκ)
Jkτsα/β + cos (ϕβ)

Eκ

2 (1 + vκ)
Jkτszα (23)

(
Πτsk
uu

)
33

= cos (ϕα)
Eκ

2 (1 + vκ)
Jkτsβ/α∂

S
α + cos (ϕβ)

Eκ

2 (1 + vκ)
JkτSα/β ∂

S
β (24)

Different studies describe the radial base function s(x) as a univariate continuous real-
valued function that mainly depends on the distance between the origin or other fixed



M. Zannon, A. Abu-Rqayiq, A. Al-bdour / Eur. J. Pure Appl. Math, 13 (4) (2020), 766-778 771

center point [2],[9],[28].
Definition: [2],[9],[12] Let R+ = {x ∈ R, x ≥ 0} be the non-negative half line and let
φ : R+ → R be a continuous function with φ(0) ≥ 0 A radial basis function on Rd
is a function of the form φ (‖x− xi‖), where x, xi ∈ Rd and r = ‖x− xi‖ denotes the
Euclidean distance between x and xi. If N points {xi}Ni=1 in Rd are chosen, then by

custom s(x) =
∑N

i=1 λiφ (‖x− xi‖) ;λi ∈ R is called a radial basis functions as well.
The Euclidean distance is therefore real and non-negative and is a parameter for controlling
the shape of the functions with effects on the convergence rate [1],[9],[12]. We consider
the elliptic partial differential operator D acting in a bounded region Ω in Rn, and ∂Ω is
the boundary of the domain for purposes of the mathematical formulation.
Consider the following elliptic PDE problem presented in [2],[3],[8],[13],[23]:

Du(x) = f(x) in domain Ω

u(x) = g(x) in domain Ω

For scattered data (xi, u(xi)) ∈ Rd+1 the approximation s(x) for a real function u(x) can
be constructed by linear combinations of translations of one function φ(‖·‖) of one variable
which is centred at

{xi}Ni=1 ⊆ Rd, u(x) ≈ s(x) =
N∑
i=1

γiφ (‖x− xi‖) (25)

The most attractive feature of the RBF methods is that the location of centres can be
chosen arbitrarily in the domain of interest. To determine the unknown coefficients γi, i =
1, 2, 3, . . . .N , we impose the interpolation conditions on s(x). This gives the N ×N linear
system

u (xj) =
N∑
i=1

γiφ (‖xj − xi‖) (26)

where the γi‘s are unknown coefficients to be determined, N is the total number of nodes
for which their NL point are the interior points and φ is the radial basis function approx-
imation of the function u [2],[11],[24]. Collocation with the PDE at the inner points and
boundary conditions, there are the following observations at the boundary points:{

F (xi) = Du (xj) ≈
∑N

i=1 γiDφ (‖xj − xi‖) ,∀j = 1, 2, . . . . . . , N

G (xi) = u (xj) ≈
∑N

i=1 γiφ (‖xj − xi‖) , ∀j = NL+1, . . . . . . . . . , N
(27)

The coefficients {γj}Nj=1 can be solved from the corresponding system of equations with
the coefficient matrix structured as:[

F
G

]
=

[
Dφ
φ

]
[γ] (28)



M. Zannon, A. Abu-Rqayiq, A. Al-bdour / Eur. J. Pure Appl. Math, 13 (4) (2020), 766-778 772

For free vibration problems, a harmonic solution is assumed for the displacements and it
is given in [2],[9],[24]. Substituting the harmonic expansion into equations (28), which can
be written as an eigenvalue problem. Therefore, the resulting equations can be written in
the following matrix form:

{[D]− λ[ζ]}{[∆]} = 0 (29)

where λ = ω2, ω is the natural frequency, {∆} is the displacement vector and {ζ} collects
all terms related to the inertial terms [26],[29].

2.3. Numerical Results

In this section, both theories (the third-order shear deformation theory and the ra-
dial base functions collocation for the free vibration study of functionally graded shell
panels) are combined [1],[19],[20]. Examples below show spherical shell panels with all
edges clamped or simply supported. Particular cases are also considered, namely isotropic
materials (fully ceramic and metal) [24],[25],[26],[27],[28],[29]. Comparing the results with
Pradyumna and Bandyopadhyay [11],[16] there is a marked improvement in HSDT and
finite elements formulation. In the following example, consideration is given to the free
vibration of simply supported and clamped FG spherical shell panels. The fundamental
frequencies of a simply supported square and clamped FG spherical shell panels composed
of aluminium and alumina, with the side-to-thickness ratio a/h = 10 are presented in
tables 1, 2, 3, 4, 5 and 6 considering various side-to-radius ratios a/R as well as power-law
exponent p. This technique possess similar properties as : Aluminumi:

Em = 70GPa, vm = 0.3, ρm = 270Kg/m3

Alumina:
Ec = 380GPa, vc = 0.3, ρc = 3000Kg/m3



M. Zannon, A. Abu-Rqayiq, A. Al-bdour / Eur. J. Pure Appl. Math, 13 (4) (2020), 766-778 773

Figure 1: A functionally graded shell [15]

Figure 2: Geometry and notations for a multilayered shell [12]

Table 1: Fundamental frequencies of CCCC square spherical shell panels collected of aluminum and alumina,
R/a = 0.5 and a/h = 10, for various p

P Ref [3] Present Ref [16]
εzz 6= 0

0 173.9595 176.8240 176.8356

0.2 161.3704 162.2278 163.0460

0.5 147.4598 148.2765 149.0095

1 132.3396 133.1074 133.7710

2 116.4386 117.2276 117.9317

∞ 80.7722 80.3306 79.8994



M. Zannon, A. Abu-Rqayiq, A. Al-bdour / Eur. J. Pure Appl. Math, 13 (4) (2020), 766-778 774

Table 2: Fundamental frequencies of CCCC square spherical shell panels collected of aluminum and alumina,
R/a = 1 and a/h = 10, for various p

P Ref [3] Present Ref [16]
εzz 6= 0

0 120.9210 122.2234 122.3533

0.2 112.2017 112.4581 112.8132

0.5 102.5983 102.8894 103.1490

1 92.2147 92.5215 92.6962

2 81.3963 81.7456 81.9179

∞ 56.2999 55.7926 55.2827

Table 3: Fundamental frequencies of CCCC square spherical shell panels collected of aluminum and alumina,
R/a = 5 and a/h = 10, for various p

P Ref [3] Present Ref [16]
εzz 6= 0

0 73.5550 74.1879 75.2810

0.2 69.6597 68.9369 68.6329

0.5 64.6114 63.1508 62.0789

1 57.8619 56.3608 55.2302

2 37.3914 43.0285 49.0328

∞ 33.2343 33.5202 34.0141

Table 4: Fundamental frequencies of SSSS square spherical shell panels collected of aluminum and alumina,
R/a = 0.5 and a/h = 10, for various p

P Ref [3] Present Ref [16]
εzz 6= 0

0 124.1581 125.2288 126.0882

0.2 115.7499 116.5276 117.0197

0.5 106.5014 107.2529 107.6572

1 96.2587 96.9763 97.2968

2 84.8206 85.5247 85.8028

∞ 57.2005 7.1331 56.9702



M. Zannon, A. Abu-Rqayiq, A. Al-bdour / Eur. J. Pure Appl. Math, 13 (4) (2020), 766-778 775

Table 5: Fundamental frequencies of SSSS square spherical shell panels collected of aluminum and alumina,
R/a = 1 and a/h = 10, for various p

P Ref [3] Present Ref [16]
εzz 6= 0

0 44.0073 44.2264 44.4697

0.2 41.7782 41.0859 40.4211

0.5 38.7731 37.6092 36.4782

1 34.6009 33.4847 32.4101

2 28.7459 28.7646 28.8329

∞ 19.8838 19.9828 20.0927

Table 6: Fundamental frequencies of SSSS square spherical shell panels collected of aluminum and alumina,
R/a = 5 and a/h = 10, for various p

P Ref [3] Present Ref [16]
εzz 6= 0

0 124.1581 125.2288 126.0882

0.2 115.7499 116.5276 117.0197

0.5 106.5014 107.2529 107.6572

1 96.2587 96.9763 97.2968

2 84.8206 85.5247 85.8028

∞ 57.2005 7.1331 56.9702



REFERENCES 776

3. Conclusions

Here we examine the previous work and continue to propose our own methods. Free
vibration analyzes of FG spherical shells is examined. A third-order shear deformation
theory that allows extensibility in the thickness direction was executed and the influence
of εzz 6= 0 was studied. The main conclusion that can be drawn from this work is The
fundamental frequency decreases as the ratio R/a increases, clamped FG shell panels
present higher frequency values than simply supported ones and the fundamental frequency
of FG spherical shell panels decreases as the exponent p in power-law increases. The effect
of εzz 6= 0 shows significance in thicker shells and seems independent of the radius of
curvature. The accuracy and reliability of the present method is carried out by comparing
our present findings with those of available numerical results.

Acknowledgements

The authors thank the fund by the College of Science and Engineering at Texas A&M
University-Corpus Christi.

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