72
Abstract
In this paper, non-linear dynamics analysis of functionally graded
material (FGM) shell structures is investigated using the higher
order solid-shell element based on the Enhanced Assumed Strain
(EAS). With this element, a quadratic distribution of the shear
stress through the thickness is considered in an enhancing part.
Material properties of the shell structure are varied continuously
in the thickness direction according to the general four-parameter
power-law distribution in terms of the volume fractions of the
constituents. Performance and accuracy of the present higher
order solid-shell element are confirmed by comparing the numeri-
cal results obtained from finite element analyses with results from
the literature.
Keywords
Non-linear dynamics, Higher Order Shear Deformation, Shear
locking, Solid-shell element, FGM.
Nonlinear Dynamics Analysis of FGM Shell Structures with
a Higher Order Shear Strain Enhanced Solid-Shell Element
1 INTRODUCTION
In the last decades, functionally graded materials (FGM) are becoming more widely used since they
provide many advantages to structural designers. FGMs shell structures are widely used in aircraft
and space systems due to their advantages of high stiffness and their high heat-resistance. In an
FGM, the structure is prepared from a mixture of ceramic and metal, or a combination of other
metals or other ceramics that are appropriate to achieve the desired objective. Material properties of
A. Hajlaoui
a
E. Triki
a
A. Frikha
a
M. Wali
a
F. Dammak
a
a
Mechanical Modeling and Manufactur-
ing Laboratory (LA2MP), National En-
gineering School of Sfax, B.P W3038,
Sfax, University of Sfax, Tunisia
abdhajlaoui@gmail.com; em-
na_triki@yahoo.fr;
frikhaahmed@yahoo.fr; mondherwa-
li@yahoo.fr; fakhred-
dine.dammak@enis.rnu.tn
http://dx.doi.org/10.1590/1679-78253323
Received 26.08.2016
In revised form 26.10.2016
Accepted 06.11.2016
Available online 07.11.2016
A. Hajlaoui et al / Nonlinear Dynamics Analysis of FGM Shell Structures with a Higher Order Shear Strain Enhanced Solid-Shell Element 73
Latin American Journal of Solids and Structures 14 (2017) 72-91
FGMs are varied continuously in one or more directions. Therefore, the stress distributions are
smooth hence interface problems are eliminated when compared to laminate structure
In order to avoid structural failure caused by dynamic loadings, linear dynamic characteristics
of FGMs shell structures have been considered by many researchers. Finding analytic solutions for
linear and non-linear dynamics analysis of functionally graded material is a very difficult task, and
they only exist for particular cases. Meanwhile, the finite-element (FE) method has become the
most widely used technique to model the processes of dynamics analysis.
There are several solution formulations for vibration and linear dynamic analysis of FGM shells
structures. They can be classified into two groups: the 2D plate and shell models and the full 3D
elasticity model. Zhi-Yuan and Hua-Ning (2007) studied free vibration characteristics of functionally
graded cylindrical shells with holes using Kirchhoff Classical thin Plate Theory (CPT). On the other
hand, Kadoli and Ganesan (2006) studied the buckling and free vibration analysis of functionally
graded cylindrical shells subjected to a temperature-specified boundary condition. In their analysis,
the finite element equations based on the First order Shear Deformation Theory (FSDT) and the
Plane Stress Assumption (PSA) were formulated. Recently, Ansari et al. (2016) developed a non-
classical size-dependent plate model based on the modified strain gradient and the FSDT for the
bending, buckling and free vibration analyses of microscale FG plates.
Based on the Third order Shear Deformation Theory (TSDT) and PSA, Reddy (2000) present-
ed a theoretical formulation, Navier's solutions of rectangular plates, and finite element models to
study the nonlinear dynamic response of FGM plates subjected to a suddenly applied uniform pres-
sure. Yang and Shen (2002) analyzed free and forced vibration for initially stressed FGM plates. In
this contribution, theoretical formulations are based on TSDT and PSA including the thermal ef-
fects. One-dimensional differential quadrature technique and Galerkin approach are used to deter-
mine the transient response of the plate subjected to dynamic loads. Gharooni and Ghannad (2015)
investigated displacements and stresses in pressurized thick FGM cylinders with exponential varia-
tion of material properties based on TSDT. Furthermore, based on a TSDT of shells and PSA, Wali
et al. (2015) Frikha et al. (2016) and Frikha et al. (2017) studied respectively the free vibration,
linear dynamic response and fully geometrical nonlinear mechanical response of FGM shell struc-
tures using an efficient double directors shell element proposed in Wali et al. (2014). The shear
stress boundary conditions on top and bottom faces are considered in a discrete form as given in
(2005).
For full 3D elasticity formulations for vibration and linear dynamic analysis of FGM shells, one
find the work of Vel and Batra (2004) based on three dimensional exact solutions for free and forced
vibrations of simply supported FGM rectangular plates. In Asemi et al. (2014) the static and dy-
namic analyses of FGM skew plates are obtained based on the three-dimensional theory of elastici-
ty. Graded elements, the principle of minimum energy and Rayleigh-Ritz energy method are used.
Using 3D elasticity model, Nguyen and Nguyen-Xuan (2015) proposed an efficiently computational
tool based on an isogeometric finite element formulation for static and dynamic response analysis of
FGM plates.
Regarding the nonlinear dynamics analysis of FGM shells, three kinematics assumption are
used, CPT, FSDT and TSDT.
74 A. Hajlaoui et al / Nonlinear Dynamics Analysis of FGM Shell Structures with a Higher Order Shear Strain Enhanced Solid-Shell Element
Latin American Journal of Solids and Structures 14 (2017) 72-91
Based on the CPT assumption and von-Karman geometrical nonlinearity, Woo et al. (2006) in-
vestigated the nonlinear free vibration behavior of FGM plates. Allahverdizadeh et al. (2008) evalu-
ated the material properties of an FGM thin circular plate and investigated the nonlinear free and
forced vibrations using the shooting technique. The formulation is based on CPT and von-Karman
geometrical nonlinearity. Alijani et al. (2011) studied the nonlinear vibrations of FGM doubly
curved shallow shells. They considered the thermal effect, the CPT assumption and Donnell nonlin-
earity. Using the CPT with an improved Donnell equations and PSA, Bich and Nguyen (2012) ex-
amined the nonlinear vibration of functionally graded circular cylindrical shells. Considering the
CPT with PSA and von-Karman geometrical nonlinearity, Duc (2013) presented an analytic inves-
tigation on the nonlinear dynamic response of eccentrically stiffened functionally graded double
curved shallow shells resting on elastic foundations and being subjected to axial compressive load
and transverse load. Using the same considerations, Duc and Cong (2015) investigated the nonlinear
dynamic response of imperfect symmetrical thin FGM plate on elastic foundation.
Using the FSDT plate finite element method with PSA employing von-Karman nonlinearity,
Praveen and Reddy (1998) investigated the nonlinear transient thermo-elastic response of FGM
plates. Liew et al. (2006) studied the nonlinear vibration of a coating-FGM-substrate cylindrical
panel subjected to a temperature gradient. The FSDT is considered with PSA and von-Karman
geometrical nonlinearity. Also based on the FSDT with PSA and von-Karman geometrical nonline-
arity, Zhang et al. (2012) analyzed the nonlinear dynamics of a clamped–clamped FGM circular
cylindrical shell subjected to an external excitation and uniform temperature change.
Based on TSDT and PSA, Hao et al. (2008) presented an analysis of the nonlinear dynamics of
a simply supported FGM rectangular plate subjected to the transversal and in-plane excitations in a
thermal environment. The von-Karman geometrical nonlinearity assumption is used. Using the
same von-Karman geometrical nonlinearity with TSDT and PSA, Duc et al. (2015) presented an
analytical approach to investigate the nonlinear dynamic response and vibration of imperfect FGM
thick circular cylindrical shells surrounded on elastic foundation. Based on TSDT with PSA and
von-Karman type nonlinear kinematics, Liu et al. (2015) presented a nonlinear dynamic analysis of
a slightly initial imperfect FGM circular cylindrical shell subjected to complex loads including aero-
dynamic pressure and thermal loading.
It can be seen from the previous literature that in most of the studies, full 3D elasticity formu-
lations are only limited to linear dynamic and free vibration of FGM structures. Also, the von-
Karman or Donell geometrical nonlinearity are the only kinematic that has been basically used for
the nonlinear dynamic analysis of shell’s structures where the PSA is largely considered.
However, in this paper we focus on the fully 3D non-linear dynamics analysis of FGM shell
structures by using higher order shear strain enhanced solid-shell element. The present formulation
constitute an extension of the higher order shear deformation solid-shell finite element, developed
by Hajlaoui et al. (2016), to the full nonlinear dynamics. The present solid-shell element formula-
tion is based on the partition of shear strain: one of the parts is independent of the thickness coor-
dinate and formulated by the Assumed Natural Strain (ANS) method, which avoids the shear lock-
ing in thin limit structure; the other enhancing part ensures a quadratic distribution across the
thickness. With this shear strain enhancement, the accuracy of transverse shear stresses will be
improved and the shear correction factors will be avoided.
A. Hajlaoui et al / Nonlinear Dynamics Analysis of FGM Shell Structures with a Higher Order Shear Strain Enhanced Solid-Shell Element 75
Latin American Journal of Solids and Structures 14 (2017) 72-91
The remainder of this paper is organized as follows. Functionally graded materials are described
in section two. After that, solid-shell finite element formulation and a transient analysis of the non-
linear formulation is described in section three and four respectively. Numerical results and discus-
sions of the finite element model are investigated in detail in section five. Finally, some concluding
remarks are analyzed and presented in section six.
2 FUNCTIONALLY GRADED MATERIALS
In this paper, we consider an FGM shell structures made from a mixture of metal and ceramics and
the composition varies continuously in the thickness direction. In fact, the Young’s modulus
()Ez
,
density
()zr
and Poisson’s ratio
(
)
zn
are assumed to vary through the shell thicknesses according
to a power-law distribution as
() ( )
cmcm
zEEVEE =- +
,
() ( )
cmcm
zVrrrr=- +
,
() ( )
cmcm
zVnnn n=- +
(1)
in which the subscripts
m
and
c
refer to metal and ceramic components, respectively. In addition,
the volume fraction
c
V follows two general four-parameter power-law distributions, Su et al. (2014).
()
,,,p
I
FGM a b c
:
()
11
1
22
p
c
c
zz
za b
hh
V
éù
æöæö
êú
÷÷
çç
=- +÷+ +÷
çç
êú
÷÷
çç
÷÷
èøèø
êú
ëû
(2)
(
)
,,,p
II
FGM a b c
:
()
11
1
22
p
c
c
zz
za b
hh
V
éù
æöæö
êú
÷÷
çç
=- -÷+ -÷
çç
êú
÷÷
çç
÷÷
èøèø
êú
ëû
(3)
where a, b and c are the parameters which determine the material variation profile through the
FGM shell thickness and p is the power-law index.
3 SOLID-SHELL FINITE ELEMENT FORMULATION
The developed solid-shell element is an eight nodes hexahedral element with three degrees of free-
dom per node. The transverse shear strain is composed of two parts. The first one is independent of
the thickness coordinate and formulated by the assumed natural strain method (ANS). The second
part is an enhancing part, which ensures a quadratic distribution through the thickness.
The enhanced assumed strain method consists in the enhancement of the compatible part of the
Green Lagrange strain tensor,
c
E
, with an enhanced part
E
to have a total strain as follows
c
=+EE E
(4)
3.1 Variational Formulation
The point of departure is the well-known three-field variational functional in Lagrangean formula-
tion. This variational functional is as follows
76 A. Hajlaoui et al / Nonlinear Dynamics Analysis of FGM Shell Structures with a Higher Order Shear Strain Enhanced Solid-Shell Element
Latin American Journal of Solids and Structures 14 (2017) 72-91
(
)
(
)
,, : . . 0
f
c
VS
VVV
dV dV dAy
¶
éù
P= +- - - =
êú
ëû
ò
òò
uES E E S E F u F u
(5)
where
y
is the strain energy function and u,
E
and S
are the independent tonsorial quantities
which are: displacement, enhanced assumed strain and assumed stress fields respectively. Also in
Eq. (5) appear the prescribed body force
V
F and surface traction
S
F . The orthogonality between
the enhanced strain and stress fields leads to
:0
V
dV =
ò
SE
(6)
This orthogonality condition reduce the number of independent variables in the original func-
tional to just two
()
,uE
. The weak form of this modified reduced functional may be obtained with
the direction derivative leading to
(
)
(
)
,: ..0
f
c
VS
VVV
WdVdVdAddd d d
¶
=P= + - - =
òòò
uE S E E F u F u
(7)
where S is the Piola-Kirchhoff stress tensor
y¶
=
¶
S
E
(8)
3.2 Finite Element Formulation
In each finite element domain, an eight-node hexahedral solid-shell element is considered. The posi-
tion vectors in reference and current configurations respectively are
n
=XNX
,
n
=xNx
(9)
where N is the tri-linear shape functions matrix given by
13 23 33 43 53 63 73 83
NNNNNNNN
é
ù
=
ê
ú
ë
û
NIIIIIIII
(10)
n
x and
n
X are nodal coordinates. The displacement field, with the corresponding variation and
increment, is interpolated in the same manner as follows
n
=uNU
,
n
dd=uNU
,
n
D= DuNU
(11)
where
11 1 88 8
, , ,... , ,
T
n
uvw uvw
éù
=
ëû
U is the nodal displacements vector at the element level. The co-
variant base vectors obtained by partial derivative of the position vectors with respect to convective
coordinate
()()
ξ
123
,, ,,xxx xhz==
in reference and current configuration are given by
k
k
x
¶
=
¶
X
G
,
k
k
x
¶
=
¶
x
g
,
1, 2, 3k =
(12)