nonlinear dynamic response and vibration imperfect shear deformable functionally graded plates subjected to blast and thermal load

Composite Structures 137 (2016) 85–92

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Composite Structures
journal homepage: www.elsevier.com/locate/compstruct

Nonlinear free vibration of pre- and post-buckled FGM plates on
two-parameter foundation in the thermal environment
Maciej Taczała a, Ryszard Buczkowski b,⇑, Michal Kleiber c
a

West Pomeranian University of Technology, Piastow 41, 71-065 Szczecin, Poland
Maritime University of Szczecin, Division of Computer Methods, Poboznego 11, 70-507 Szczecin, Poland
c
Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawinskiego 5B, 02-106 Warsaw, Poland
b

a r t i c l e

i n f o

Article history:
Available online 14 November 2015
Keywords:
FGM plates
Two-parameter elastic foundation
Nonlinear free vibration
Finite element method

a b s t r a c t

The geometrically nonlinear free vibration of functionally graded thick plates resting on the elastic
Pasternak foundation is investigated. The motion equations are derived applying the Hamilton principle.
We consider the first order shear deformation plate theory (FSDT), in which the modified shear correction
factor is required. A 16-noded Mindlin plate element of the Lagrange family which is free from shear
locking due to small thickness of the plate used. The material properties are assumed to be
temperature-dependent and expressed as a nonlinear function of temperature. Because the FGM plates
are not homogeneous, the basic equations are calculated in the equivalent physical neutral surface which
differs from the geometric mid-plane. In the pre-buckling range natural frequencies decrease ultimately
reaching zero for critical stress in the bifurcation point.
Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction
Functionally graded materials (FGMs) are new inhomogeneous
composite materials in which the volume fraction of two components varies smoothly and continuously across the given direction.
FGMs are mixtures of ceramics and metal, where external ceramic
layers due to large thermal resistance are exposed to high temperatures, while internal metallic constituents, owing to their stronger
mechanical performance, are able to reduce the possibility of fracture. Manufacturing techniques must guarantee controlled
changes in composition and density, so that the product will have
a required structure and properties along the given direction, often
across plate thickness.
Because there are many papers dealing with linear and nonlinear thermal stresses and deformations of FGMs beams, plates and
shells, we focus only on the studies of the nonlinear vibration of
functionally graded plates and shells resting on an elastic foundation. Interested readers may wish to review the papers of Liew, Lei
and Zhang [2] and Swaminathan and others [1], who have presented comprehensively analytical and numerical solutions for a
number of examples concerning the statics, dynamics and stability

⇑ Corresponding author.
E-mail addresses: (M. Taczała),
(R. Buczkowski), (M. Kleiber).
/>0263-8223/Ó 2015 Elsevier Ltd. All rights reserved.

of functionally graded plates and functionally graded carbon nanotube reinforced composites.
The first paper investigating the large amplitude of FGM cylindrical shells resting on elastic foundation in thermal environments
was presented by Shen [3]. The results showed that the differences
of the fundamental frequencies between Voigt and Mori–Tanaka
micromechanical models are very small, and the differences of
the nonlinear to linear frequency ratios of the FGMs shells are negligible. Shen and Xiang [4,5] studied the nonlinear vibration of
nanocomposite cylindrical beams, plates and shells resting on elastic foundations subjected to thermal loading. The paper of Yang, Ke
and Kitiporchnai [6], using the differential quadrature method,
made the first attempt to study the nonlinear free vibration of
single-walled carbon nanotubes (SWCNTs) based on geometric
nonlinearity and Eringen’s nonlocal elasticity theory. The geometrically nonlinear free vibration of carbon nanotubes/fibre/polymer
laminated composite plates with immovable simple supported
boundary conditions using the Galerkin procedure was studied
by Rafiee and others [7]. Zenkour and Sobhy [8] investigated
dynamic bending behaviour of FGM plate resting on twoparameter elastic foundation and subjected to time harmonic thermal load. An analytical approach was presented by Duc [9] and Duc
and Quan [10] to investigate the transient responses of imperfect
eccentrically stiffened FGM double curved thin shallow shells
using Galerkin method (the Lekhnitsky smeared stiffeners technique is used). The non-linear dynamics, in the sense of strain


86

M. Taczała et al. / Composite Structures 137 (2016) 85–92

displacement relations, of FGM truncated conical shells on the
Pasternak elastic foundation was investigated by Najafov and
Sofiyev [11] and the problem was solved using superposition,
Galerkin and harmonic balance methods. A general third-order
deformation theory for functionally graded plates including a modified couple stress effect that accounts for material length scale and

geometrical nonlinearity has been recently presented by Kim and
Reddy [12] and Neves et al. [13] and some time earlier by Ferreira
with co-authors [14].
We study the non-linear free vibration of FGM plate resting on
an elastic foundation using the finite element method. The problem has been solved using a 16-noded Mindlin plate element of
the Lagrange family which is free from shear locking [15]. For the
solution of eigenvalues and eigenvectors the subspace iteration
technique has been used here. All the equations are calculated in
reference to the equivalent physical neutral surface which is different from the mid-plate.
The problem we tackle refers to the formulation and solution of
natural vibrations having small amplitudes around the equilibrium
of a pre- or post-buckled functionally graded plate subjected to
thermal loading. It should be noted that the nonlinear analysis concerns the determination of the equilibrium state of the plate, while
the vibrations are analysed using the linear eigenproblem formulation. Contrary to this formulation, a problem is defined in the literature [16] referring to the nonlinear vibrations with large
amplitudes. The problem is governed by a nonlinear equation
solved analytically or using the finite element procedure combined
with the iteration technique. The natural frequency is dependent
on the amplitude and can also be dependent on initial imperfections, stresses and many other parameters, such as temperature
field and material distribution in a FG plate.

Considering large out-of-plane displacements, we write down
the components of the Green–Lagrange strain tensor as follows

1 tþDt 2
w;x
2
1
tþDt
Eyy ¼ tþDt v ;y À ztþDt hx;y þ tþDt w2;y
2

À
Á
Ã
1 ÂtþDt
tþDt
tþDt
Exy ¼
u;y þ
v ;x þ z tþDt hy;y À tþDt hx;x þ tþDt w;x tþDt w;y
2
Ã
1Â
tþDt
Exz ¼ tþDt hy þ tþDt w;x
2
Ã
1Â
tþDt
Eyz ¼ ÀtþDt hx þ tþDt w;y
2
ð5Þ
tþDt

In the incremental approach, the stress in the actual increment
and iteration can be expressed via the value in the previous
iteration
tþDt

2.1. Derivation of the static equations

Si ¼ t Si þ tþDt DSi

ð6Þ

The increment of the second Piola–Kirchhoff stress tensor written in the vector form is
tþDt

DSi ¼ t C ij



tþDt

DEj À tþDt DEðTÞ
j



ð7Þ

where

DEðTÞ
is the thermal strain being
i
n o
ðTÞ
ð0Þ
tþDt
DEi ¼ DkTEð0Þ

Ei
¼ colf 1 1
i ;
tþDt

0 0 0g

ð8Þ

and the constitutive matrix C is typical for plane stress analysis with
the influence of shear stresses:

2t

t

2. Mathematical relations

Exx ¼ tþDt u;x þ ztþDt hy;x þ

E

6
6m
6
C ij ¼ 6
60
6
40
0

m
E

0
0

0

1Àm
2

0

0

0

0

t

3
0
7
0 7
7
0
0 7
7

7
bt G 0 5
0 bt G
0
0

ð9Þ

The static equation governing the structural response of the FG
plate subject to thermal loading will be derived employing the
principle of virtual work.

where b ¼ 5=6 is the shear correction factor. Considering dependence of the material parameters on temperature the constitutive
matrix is dependent on the loading level.
Strain components expressed in the incremental form are

dtþDt W int ¼ dtþDt W ext

tþDt

ð1Þ

Since it is only the thermal loading which is considered,
therefore

dtþDt W ext ¼ 0

ð2Þ

The virtual work of internal forces for the plate on twoparameter elastic foundation can be expressed as (the total Lagrangian formulation is used here):

dtþDt W int ¼

Z

tþDt

Si dtþDt Ei dV þ

Vp

Z
Af

tþDt

½k0

wdtþDt w

þ k1 ðtþDt w;x dtþDt w;x þ tþDt w;y dtþDt w;y ފdA

ð3Þ

where {Si} are components of the second Piola Kirchhoff stress tensor, {Ei} – components of the Green–Lagrange strain tensor, k0 and
k1 – parameters of the elastic foundation. Using use the first-order
shear deformation theory, displacements of arbitrary point of the
plate functions are
tþDt

u ¼ tþDt um þ ztþDt hy

tþDt

v ¼ tþDt v m À ztþDt hx

tþDt

w ¼ tþDt wm

ð4Þ

where um ; v m ; wm are displacements of the neutral layer in x, y and z
directions, respectively, and hx ; hy are rotations with respect to x and
y axes.

1 tþDt
Dw2;x
2
1 Dt
tþDt
DEyy ¼ tþDt Dv ;y À ztþDt Dhx;y þ t w;y tþDt Dw;y þ tþ
Dw2;y
2 iþ1
1Â
tþDt
DExy ¼ tþDt Du;y þ tþDt Dv ;x þ zðtþDt Dhy;y À tþDt Dhx;x Þ
2
i
tþDt

Dw;x þ t w;x tþDt Dw;y þ tþDt Dw;x tþDt Dw;y
þt w;y iþ1

DExx ¼ tþDt Du;x þ ztþDt Dhy;x þ t w;x tþDt Dw;x þ

ð10Þ

1
DExz ¼ ðtþDt Dhy þ tþDt Dw;x Þ
2
1
tþDt
DEyz ¼ ðÀtþDt Dhx þ tþDt Dw;y Þ
2
tþDt

With the use of the finite element approximation, the strain
increments can be presented in the form
tþDt

tþDt
DEi ¼ t Bijð1Þ tþDt Ddj þ Bð2Þ
Ddj tþDt Ddk
ijk

ð11Þ

where t Bð1Þ and t Bð2Þ are strain–displacement matrices
t ð1Þ
B1j

¼ N1j;x þ zN 5j;x þ N3j;x N3k;x t dk ; t B2j ¼ N2j;y À zN 4j;y þ N3j;y N3k;y t dk

ð1Þ

t ð1Þ
B3j

¼ N1j;y þ N2j;x þ zðN 5j;y À N4j;x Þ þ ðN3k;y N3j;x þ N3k;x N3j;y Þt dk

t ð1Þ
B4j

¼ N5j þ N3j;x ; t B1j ¼ ÀN4j þ N3j;y

ð1Þ

1
1
ð2Þ
ð2Þ
ð2Þ
B1jk ¼ N3j;x N3k;x ; B2jk ¼ N3j;y N3k;y ; B3jk ¼ N3j;x N3k;y
2
2
B4km ¼ 0; B5km ¼ 0:
ð12Þ


87

M. Taczała et al. / Composite Structures 137 (2016) 85–92

Shape functions of the 16-node finite element employed in the
present analyses and matrices N 1j ; N 2j ; N 3j ; N 4j ; N 5j are defined in
Appendix B.
Note that constructing the vector of strains and strain increments for the finite element analysis, the shear components are
doubled, therefore the vector takes the form
tþDt

DE ¼ col

È tþDt

tþDt

DExx

DEyy 2

tþDt

tþDt

DExy 2

DExz 2

tþDt

DEyz

É

ðintÞ

Ddk

t

t ðTÞ
Ddk ¼ t K À1
jk F j

t

K jk ¼

ðTÞ



Rows of the shape function and strain–displacement matrices
as well as their derivatives correspond to the degrees of freedom
of the finite element. Applying Eqs. (6–8 and 11) in Eq. (3) we
obtain
t
V

Z



ð2Þ
ð2Þ
Si Bijk þ Bikj dV þ

V

t

Z
ð1Þ ð1Þ
C ip t Bpk t Bij dVþ ½k0 N3k N3j
ÉtþDt

ð14Þ

ðdÞ

where
t

i

ðdÞ

Ddk ¼ Àt F jðintÞ þ tþDt Dkt F ðTÞ
j

ð1Þ

t
V

Z

ð1Þ

Â
Ã
k0 N3k N3j þ k1 ðN3k;x N3j;x þ N 3k;y N3j;y Þ dA

ð16Þ

Z

t
V



ð2Þ
ð2Þ
Si Bijk þ Bikj dV

¼ TEð0Þ
p

Z

V

ð1Þ
t
C ip t Bij dV

V

ð19Þ

Af

ðincrÞ tþDt

Ddk

ðincrÞ

Ddk

2

ð20Þ

Direct application of Eq. (20) destroys the symmetry of the stiffness
matrix, therefore the procedure is proposed following the original
consideration by Crisfield [17]. The iterative correction tþDt Ddk
appearing while evaluating the displacement increment in the
actual step
ðincrÞ

Ddk

¼

t ðincrÞ
dk

þ

tþDt

Ddk

ð21Þ

is divided into two parts; dependent on internal forces and reference thermal forces
tþDt

tþDt

pi dtþDt ui dA

ðintÞ

Ddk ¼ t Ddk

ðTÞ

þ tþDt Dkt Ddk

Solving Eq. (14), we have

tþDt

qtþDt u€i dtþDt ui dV

ð26Þ

Using the displacement functions for FSDT (Eq. (5)) and FE
approximation, Eq. (25) becomes
d

tþDt

2

Z

tþDt

W ext ¼ À
V

3
N 1k N 1j þ zðN 5k N1j þ N 1k N 5j Þ þ z2 N 5k N 5j þ
6
€ðVÞ ddðVÞ
2
q4 N2k N2j À zðN4k N2j þ N2k N4j Þ þ z N4k N4j þ 7
5dV d

j
k
N 3k N 3j
ð27Þ

€ðVÞ ddðVÞ
dtþDt W ext ¼ ÀtþDt M jk d
j
k

tþDt

Z
Mjk ¼
Vp

¼ Dl

Z

where

is the internal force vector.
The constant arc-length method has been applied to solve the
problem. The method consists in enhancing the system of equations given by Eq. (15) with the constraint equation delimiting
increment of displacements in each step

tþDt

@A

which can be also presented in the form

ð18Þ

is the thermal reference force vector,
Z
Z
Â
Ã
ð1Þ
t ðintÞ
F j ¼ t Si t Bij dV þ
k0 N 3k N 3j þ k1 ðN 3k;x N 3j;x þ N 3k;y N 3j;y Þ dA

tþDt

Z

ð25Þ

ð17Þ

is the stiffness matrix dependent on stresses,
t ðTÞ
Fj

€ i ÞdtþDt ui dV þ
ðtþDt bi À tþDt qtþDt u

V

is the stiffness matrix dependent on displacements, accounting also
for contribution of the elastic foundation,
ðSÞ

Z

dtþDt W ext ¼ À

Af

K jk ¼

If we assume a possibility of small amplitude vibrations d
around the equilibrium positioned of the plate subject to thermal
loading, an equation of motion must be derived. In this case the
virtual work of external forces takes the form

ð15Þ

C ip t Bpk t Bij dV

þ

t

a quadratic equation for the iterative change of the loading parameter governing the actual temperature of the plate tþDt Dk. Once the
correct solution is selected, we should follow the rules specified by
Crisfield [17].

For free vibrations body b and surface p forces vanish, and Eq.
(24) reduces to

Z

K jk ¼



ðintÞ
ðintÞ
t ðincrÞ
þ t Ddk
dk
þ t Dd k


ðincrÞ
ðintÞ t
ðTÞ
ðTÞ
ðTÞ
2
þ t Ddk
Ddk tþDt Dk þ t Ddk t Ddk tþDt Dk2 ¼ Dl ð24Þ
þ 2 t dk

V

ðSÞ tþDt

K jk þ t K jk

ð23Þ

ðSÞ
K jk

Using Eqs. (21), (22), we can transform Eq. (20) to

dtþDt W ext ¼

Eq. (14) can be presented in the following form:
t

þ

t

ðVÞ

Af

h

ðdÞ
K jk

2.2. Derivation of the dynamic equations

Af

þ k1 ðN3k;x N3j;x þ N 3k;y N3j;y ފdA
Ddk
Z
Z
ð0Þ
t
t t ð1Þ
t ð1Þ
tþDt
¼
DkTEp
C ip Bij dV À
Si Bij dV
V
V
Z
À ½k0 N3k N 3j þ k1 ðN3k;x N3j;x þ N3k;y N3j;y ފdAt dk

t

t ðincrÞ
dk

ð13Þ

&Z

ðintÞ

t
¼ Àt K À1
jk F j

t

ð22Þ

2
tþDt

N1k N1j þ zðN5k N1j þ N1k N5j Þ þ z2 N5k N5j þ

ð28Þ
3

7
q6
4 N2k N2j À zðN4k N2j þ N2k N4j Þ þ z2 N4k N4j þ 5dV
N3k N3j
ð29Þ

Note that the mass matrix is dependent on the increment
of the nonlinear static analysis, as material properties change
with the temperature increase. The mass matrix is computed
using the 4 Â 4 Lobatto integration rule.
2.3. Modelling material properties
To model the distribution of ceramic fraction throughout the

plate thickness, the power law is used

Vc ¼


n
z À zC 1
;
þ
2
h

nP0

ð30Þ

where zC is a coordinate of the mid-layer. The metallic fraction is

Vm ¼ 1 À Vc

ð31Þ

Effective material properties are evaluated depending on the
proportion of both fractions


88

M. Taczała et al. / Composite Structures 137 (2016) 85–92

Rh

E ¼ Ec V c þ Em V m

a ¼ ac V c þ am V m
q ¼ qc V c þ qm V m

ð32Þ



Ec ¼ Ec0 EcðÀ1Þ T À1 þ 1 þ Ec1 T þ Ec2 T 2 þ Ec3 T 3


Em ¼ Em0 EmðÀ1Þ T À1 þ 1 þ Em1 T þ Em2 T 2 þ Em3 T 3


ac ¼ ac0 acðÀ1Þ T À1 þ 1 þ ac1 T þ ac2 T 2 þ ac3 T 3


am ¼ am0 amðÀ1Þ T À1 þ 1 þ am1 T þ am2 T 2 þ am3 T 3

2.4. Position of the neutral layer
We determine the position of the neutral layer z0 (Fig. 1) assuming that the resultant membrane force due to bending stresses SðbÞ
xx
equals zero.
tþDt ðbÞ
Sxx dz

0

¼0

ð34Þ

The value of stress in the actual iteration can be evaluated in the
incremental form:
tþDt

ðbÞ
Sxx ¼ t Sxx
þ tþDt DSðbÞ
xx

ð35Þ

The stress increment, based on the constitutive equation, is
tþDt

ðbÞ
ðbÞ
DSxx
¼ t C 11 tþDt DExx
þ t C 12 tþDt DEðbÞ
yy

ð36Þ

Substituting Eqs. (35), (36) to Eq. (34) we arrive at

Z

h

0

t ðbÞ
Sxx dz þ

Z
0

h

"

#
ÀtþDt
Á
Ef
t tþDt
dz ¼ 0
ðz
À
z
Þ
D
h
À
m

D
h
0
y
x
f
1 À t m2f
t

ð37Þ

Since

Z
0

h

t ðbÞ
Sxx dz

¼0

ð38Þ

we can find the position of the neutral layer

Rh
z0 ¼

0

Rh
0

!
Á
Dhy À t mf tþDt Dhx dz
f
!
À
Á
tE
f
tþDt Dh À t m tþDt Dh
dz
y
x
f
2
1Àt m
tE

fz

1Àt m2

tE
f

dz
ð40Þ

dz

Note that in general the neutral layer changes the position
along with the temperature-dependent variation of material properties and should be evaluated for each increment/iteration.
3. Numerical examples

ð33Þ

Generally, Poisson ratio can also be modelled using equations
using analogous to Eqs. (32) and (33). However, in the present
paper the Poisson ratio is constant, and density is independent of
temperature.

h

z0 ¼ R h

0 1À m2
f

Temperature-dependent variations of material parameters are
modelled using the following equations:

Z

tE z
f

0 1À m2
f

ÀtþDt

ð39Þ

f

In the case of Poisson number independent of temperature, Eq.
(39) reduces to

To verify the code used in this study, three examples are discussed. First, temperatures at which the plate buckles are computed analytically for isotropic plates (see Appendix A)

À Á4
À Á2
4D pb þ k0 þ 2k1 pb
DT ¼
ÀpÁ2
Et
2 1À
ma b

ð41Þ

and compared with the own numerical results using finite element
method.
Due to nonlinear dependency of the thermal expansion coefficient a on temperature, the solution of Eq. (41) calls for application
of the iterative procedure. The critical temperature was calculated
for the square plate with b = 300 mm and the thickness of the plate

t = 3 mm. The simply supported boundary conditions were
assumed for all edges, which were also restrained against inplane displacements. Material properties are taken from reference
[18].
A uniform temperature change was applied to the plate, and the
assumed reference temperature T 0 was 300 K. The results for isotropic plates, made of SUS304 stainless steel and silicon nitride
Si3N4, without elastic foundation and positioned on the Winkler
elastic foundation are given in Table 1 and plates on the twoparameter foundation in Table 2. For all numerical calculations
the mesh (16 Â 16) is taken into account.
Very good agreement of the numerical and analytical results
can be seen.Influence of the exponent in Eq. (30) was investigated
for the FGM plate having the same dimensions and material properties as in the first example. Structural response of the plate is
presented in Fig. 2. It can be seen that the plate exhibits
bifurcation-type behaviour for n ¼ 0 (pure ceramic) and
n ¼ 10000 (practically pure metal), as the boundary cases. For
the intermediate values of the exponent, the curves are similar to
those for nonlinear response of isotropic plate with initial imperfection [19]. It can be explained by the distribution of material
across the thickness; changing fractions of ceramic and metal
result in the bending moment causing out-of-plane deformations.
Note that the structural response of true FGM plates does not fall
between the curves for two boundary cases.
These results are also confirmed by the results of the vibration
analysis – Fig. 3. For each increment of temperature, the linear
eigenvalue problem is solved and natural frequency of free vibration evaluated. For the isotropic pure ceramic and metal plates
the frequencies decrease to reach zero for critical temperatures,
corresponding to the bifurcation points as in Fig. 2. For the first
time the same observation for isotropic plates without and with
foundation was made by Park and Kim [20] and Shen and Xiang
[21], respectively. The decrease of natural frequencies is the result

Table 1

Buckling temperatures in K for isotropic plates on elastic foundation for k1 ¼ 0.
Material

k0 N=mm !

0.00

0.01

0.05

0.10

SUS304

Analytically
Numerically
Analytically
Numerically

8.2149
8.2103
16.7286
16.7193

11.5177
11.5960
21.0486
21.0393

24.6136
24.6091
38.1517
38.1426

40.7433
40.7389
59.1485
59.1397

Si3N4
Fig. 1. Position of mid-layer and neutral layer.


M. Taczała et al. / Composite Structures 137 (2016) 85–92
Table 2
Buckling temperatures in K for isotropic plates on elastic foundation for
k1 ¼ 500 N=mm3 .
Material

k0 N=mm !

0.00

0.01

0.05

0.10

SUS304

Analytically
Numerically
Analytically
Numerically

43.8318
43.8274
63.1590
63.1502

47.0191
47.0147
67.2941
67.2854

59.6793
59.6759
83.6796
83.6710

75.3199
75.3161
103.8254
103.8171

Si3N4

89

diagram is governed by significant increase of deformations in
the post-buckling range and the increase of stiffness resulting from
nonlinear terms in strain–displacement relationship (Eq. (12))).
We note that the curves representing the calculated fundamental
frequencies follow those presented by Park and Kim [20].
The influence of elastic foundation parameters k0 and k1 on
structural response and natural frequencies is presented in Figs. 4–
7. The behaviour typical of bifurcation-type buckling can be again
observed for various values of the first foundation parameter, k0
ranging from 0 to 0.1 N/mm, with the critical temperature increasing with the increase of that value – Fig. 4. The effect is replicated
for the vibration analysis – Fig. 5 as well as the variation of natural
frequencies in the pre- and post-buckling range.
The influence of the shear parameter k1 N=mm3 for constant
value k0 ¼ 0:05 N=mm is presented in Figs. 6 and 7.
The next example refers to the vibration at small amplitudes
and the influence of initial imperfection – in the mode corresponding to the eigenvibration mode – on natural frequency is illustrated. Here nonlinearity is related to nonlinear terms in the
strain–displacement relationships. Influence of the imperfections
is analysed for various values of the exponents in the power law
(Eq. (30)), defining the fraction of the ceramic constituent regarding various amplitudes of initial imperfection considering also
the sign of the amplitude: positive or and negative, see Table 3
for comparison. Calculations were performed for square simply
supported plate b = 300 mm, t = 15 mm, material properties for
metal
fraction:
density
8166 kg=m3 ,
Young
modulus
207700 N=m2 , Poisson ratio 0.3177, for ceramic fraction: density

2370 kg=m3 , Young modulus 322200 N=m2 , Poisson ratio 0.24.
For isotropic plates (metal or ceramic) the natural frequency for
both cases are identical, however, for exponents resulting in varying mixture of ceramic and metallic fractions throughout the plate

Fig. 2. Curves of equilibrium states for simply supported square Si3N4/SUS304 FG
plates.

Fig. 3. Vibration behaviour of Si3N4/SUS304 FG plate.

of increasing compressive stresses, ultimately resulting in singularity of the total stiffness matrix (sum of stiffness and geometrical
matrices) for the critical temperatures of the isotropic plates
(308.21 K in the case of SUS304 and 316.73 K for Si3N4), see
Fig. 3). The increase of the frequencies in the further part of the

Fig. 4. Curves of the equilibrium states for simply supported square Si3N4/SUS304
FG plate assuming n ¼ 0 for various values of the first foundation parameter k0 in
N/mm assuming k1 ¼ 0.


90

M. Taczała et al. / Composite Structures 137 (2016) 85–92

Fig. 5. Vibration behaviour of Si3N4/SUS304 FG plate assuming n ¼ 0 for various
values of the first foundation parameter k0 in N/mm assuming k1 ¼ 0.

Fig. 7. Vibration behaviour of Si3N4/SUS304 FG plate n ¼ 1 for various values of the
second foundation parameter k1 in N/mm3 assuming k0 = 0.05 N/mm.

3. The volume fraction of the constituent materials has a significant influence on thermal post-buckling behaviour and natural

frequencies of FGM plates.
4. In the pre-buckling range natural frequencies decrease, following an increase of compressive stress, ultimately reaching zero
for critical stress in the bifurcation point.
5. Further increase of natural frequencies in the post-buckling
range is governed by out-of-plane displacements for a stable
equilibrium path.
6. Thermal buckling stress and post-buckling response as well as
natural frequencies are influenced by both parameters of the
Pasternak foundation.

Acknowledgements
The work has been performed under the project Static and
dynamic finite element analysis of layered structures on elastic nonhomogeneous foundation, financed by the Polish National Science Centre (NCN) under the contract 2012/05/B/ST6/03086. The support is
gratefully acknowledged.
Appendix A. Thermal buckling of plate on elastic foundation –
analytical approach

Fig. 6. Curves of the equilibrium states for a simply supported square Si3N4/SUS304
FG plate n ¼ 1 and b=t ¼ 20 for various values of the second foundation parameter
k1 in N/mm3 assuming k0 = 0.05 N/mm.

Equation for the buckling temperature of the plate on the twoparameter elastic foundation will be derived beginning with the
principle of virtual work, due to the absence of external loading

dW int ¼ 0
where

thickness the frequency depends on the direction of imperfection
or the amplitude sign. The effect can be explained by asymmetric
properties of the plate. We note that the natural frequency

increases with the increase of the initial imperfection amplitude
w0 what can be explained by increasing stiffness of the plate due
to nonlinear terms.
4. Concluding remarks
1. The formulations were verified against the results of thermal
post buckling analysis and free vibration analysis of isotropic
and FG plates.
2. The numerical results show that structural response and vibration of the FG plates subjected to thermal loading are different
from those for the response isotropic plate.

ðA:1Þ

Z

À

dW int ¼

Á
Sx dEx þ Sy dEy þ Sxy dExy dV

VP

Z

Â

þ
Af

Ã
k0 wdw þ k1 ðw;x dw;x þ w;y dw;y Þ dA

ðA:2Þ

Is the virtual work of internal forces of the plate and the elastic
foundation. Note that Exy is the shear strain and not a component of
the strain tensor. Constitutive equations including thermal strains
are as follows

i
E h
Ex À ExðTÞ þ mðEy À EðTÞ
y Þ
2
1Àm
i
E h
Sy ¼
Ey À EyðTÞ þ mðEx À EðTÞ
x Þ
2
1Àm
E
Sxy ¼
Exy
2ð1 þ mÞ

Sx ¼

ðA:3Þ


91

M. Taczała et al. / Composite Structures 137 (2016) 85–92
Table 3
Natural frequencies for simply supported FG plate for various exponents of power law, sÀ1 .
n # w0 =t !
0 (ceramic)
0.2

Amplitude
Positive
Negative
Positive
Negative
Positive
Negative

1
5
10000 (metal)

0.00

0.01

0.02

0.05

0.1

0.2

0.5

11310.70
9226.07
9226.07
6902.22
6902.22
5663.98
5663.98
5005.21

11311.20
9226.16
9226.66
6901.95
6902.67
5663.83
5664.32
5005.43

11312.27
9227.04
9227.85
6902.61

6903.55
5664.35
5665.01
5005.90

11316.85
9230.62
9233.68
6905.06
6907.92
5666.30
5668.40
5007.93

11377.82
9273.56
9301.49
6931.02
6963.17
5687.09
5708.62
5045.58

11677.82
9518.23
9653.62
7110.80
7220.50
5825.59
5908.89

5232.12

13703.90
11183.73
11888.17
8339.31
8886.85
6800.62
7218.24
6089.55

Strains including nonlinear out-of-plane terms are given by

1
Ex ¼ Àzw;xx þ w2;x
2
1
Ey ¼ Àzw;yy þ w2;y
2
Exy ¼ À2zw;xy þ w;x w;y

ðA:4Þ

and thermal strains by

¼

ETy

¼ aDT

ðA:5Þ

Substituting Eqs. (A.2–A.4) to Eq. (A.1) the virtual work of internal forces becomes
Z
dW int ¼ D
½ðw;xx þ mw;yy Þdw;xx þ ðw;yy þ mw;xx Þdw;yy
AðpÞ
Z
þ 2ð1 À mÞw;xy dw;xy ŠdA þ
½k0 wdw þ k1 ðw;x dw;x þ w;y dw;y ފdA
À

Et
aD T
1Àm

Af

Z
AðpÞ

½w;x dw;x þ w;y dw;y ŠdA

ðA:6Þ

Thermal loading corresponds to the problem of biaxial compression. In the case of a simply supported rectangular plate, the
displacement function can be taken in the form of one term of
the double Fourier series

w ¼ wmn sin

mpx
npy
sin
a
b

ðA:7Þ

where m and n are integer numbers defining the buckling mode.
Substituting the assumed displacement function and integrating
we arrive at

(

dW int ¼

mp2 np2 !2
mp2 np2 !
þ
þ k0 þ k1
þ
a
b
a
b
!'





2
2
Et
mp
np
1
aDT
þ
abwmn dwmn
À
1Àm
4
a
b
D

À
Á
1
ð5n3 À 5n2 À n þ 1Þ Á 5g3 À 5g2 À g þ 1
64

N2 ¼ À
N3 ¼

dex ¼ Àzdw;xx þ w;x dw;x

ETx

N1 ¼

À
Á
1
ð5n3 þ 5n2 À n À 1Þ Á 5g3 À 5g2 À g þ 1
64

À
Á
1
ð5n3 þ 5n2 À n À 1Þ Á 5g3 þ 5g2 À g À 1
64

DT ¼

hÀ Á
mp 2
a

þ

hÀ Á
À Á2 i
2
þ k0 þ k1 map þ nbp
h
À Á i
Et

a ðmapÞ2 þ nbp 2
1Àm

ðB:4Þ

N5 ¼ À

 À
pffiffiffi
Á
5 pffiffiffi 3
5n À n2 À 5n þ 1 Á 5g3 À 5g2 À g þ 1
64

ðB:5Þ

N6 ¼

 À
pffiffiffi
Á
5 pffiffiffi 3
5n þ n2 À 5n À 1 Á 5g3 À 5g2 À g þ 1
64

ðB:6Þ

N7 ¼

pffiffiffi


pffiffiffi
5
ð5n3 þ 5n2 À n À 1Þ Á
5g3 À g2 À 5g þ 1
64

ðB:7Þ

N8 ¼ À

pffiffiffi

pffiffiffi
5
5g3 þ g2 À 5g À 1
ð5n3 þ 5n2 À n À 1Þ Á
64

ðB:8Þ

N9 ¼ À


pffiffiffi
5 pffiffiffi 3
5n þ n2 À 5n À 1 Á ð5g3 þ 5g2 À g À 1Þ
64

ðB:9Þ

N10 ¼


pffiffiffi
5 pffiffiffi 3
5n À n2 À 5n þ 1 Á ð5g3 þ 5g2 À g À 1Þ
64

ðB:10Þ

N11 ¼

pffiffiffi

pffiffiffi
5
ð5n3 À 5n2 À n þ 1Þ Á
5g3 þ g2 À 5g À 1
64

ðB:11Þ

N12 ¼ À

pffiffiffi

pffiffiffi
5
ð5n3 À 5n2 À n þ 1Þ Á

5g3 À g2 À 5g þ 1
64

ðA:8Þ

ðA:9Þ

Numbers of halfwaves m and n should be adjusted to yield the minimum temperature. For square plate a = b and for m = n = 1 we get
finally

DT ¼

À Á4
À Á2
4D pb þ k0 þ 2k1 pb
:
ÀpÁ2
Et
2 1À
ma b

ðA:10Þ

Appendix B. Shape functions of the 16-node finite element with
Lobatto integration scheme
Shape functions of the 16-node plate finite element are as follows (for details we refer the readers to reference [15]) (see
Fig. B.1):

ðB:3Þ

À
Á
1
ð5n3 À 5n2 À n þ 1Þ Á 5g3 þ 5g2 À g À 1
64

ÀnpÁ2 i2
b

ðB:2Þ

N4 ¼ À

Hence the temperature increment resulting in buckling is

D

ðB:1Þ

pffiffiffi
Fig. B.1. 16-node finite element of the Lagrange family c ¼ 1= 5.

ðB:12Þ


92

M. Taczała et al. / Composite Structures 137 (2016) 85–92

N13 ¼

 pffiffiffi

pffiffiffi
pffiffiffi
25 pffiffiffi 3
5n À n2 À 5n þ 1 Á
5g3 À g2 À 5g þ 1
64
ðB:13Þ

N14 ¼ À

N15 ¼

 pffiffiffi

pffiffiffi
pffiffiffi
25 pffiffiffi 3
5n þ n2 À 5n À 1 Á
5g3 À g2 À 5g þ 1
64
ðB:14Þ

 pffiffiffi

pffiffiffi
pffiffiffi
25 pffiffiffi 3

5n þ n2 À 5n À 1 Á
5g3 þ g2 À 5g À 1
64
ðB:15Þ

N16 ¼ À

 pffiffiffi

pffiffiffi
pffiffiffi
25 pffiffiffi 3
5n À n2 À 5n þ 1 Á
5g3 þ g2 À 5g À 1
64
ðB:16Þ

Displacements are approximated according to

u ¼ Ni ui ;

v ¼ Ni v i ; . . . ;

hy ¼ Ni hyi

ðB:17Þ

Forming the vector of displacement functions, the relationship
between them and nodal displacements can be given introducing
two-dimensional shape function matrix

8 9 2
u>
N1
>
>
>
>
>
6
>
>
>
>
v
>
>
6
< = 6 0
w ¼6
0
>
> 6
>
> hx >
> 6
0
>
>
4

>
>
>
: >
;
hy
0

0

0

0

0

N2

N1
0

0
N1

0
0

0
0

0
0

0

0

N1

0

0

0

0

0

N1

0

8
9
> u1 >
>
>
>
3>

>
>
::: >
v1 >
>
>
>
>
>
>
7>
>
>
>
::: 7>
w
>
1
=
7<
::: 7
h
x1
7>
>
7>
> hy1 >
>
::: 5>
>

>
>
>
>
>
> u2 >
>
>
::: >
>
>
>
>
:
;
:::

ðB:18Þ

what can be expressed also as

u ¼ N1i di ;

v ¼ N2i di ; . . . ;

hy ¼ N5i di

ðB:19Þ

where

d ¼ colf u1

v1

w1

hx1

hy1

u2

::: g

ðB:20Þ

References
[1] Swaminathan K, Naveenkumar DT, Zenkour AM, Carrera E. Stress, vibration
and buckling analyses of FGM plates – a state-of-art review. Compos Struct
2015;120:10–31.

[2] Liew KM, Lei ZX, Zhang LW. Mechanical analysis of functionally graded carbon
nanotube reinforced composites: a review. Compos Struct 2015;120:90–7.
[3] Shen H-S. Nonlinear vibration of shear deformable FGM cylindrical shells
surrounded by an elastic medium. Compos Struct 2012;94:1144–54.
[4] Shen H-S, Xiang Y. Nonlinear analysis of nanotube-reinforced composite
beams resting on elastic foundations in thermal environments. Eng Struct
2013;56:698–708.
[5] Shen H-S, Xiang Y. Nonlinear vibration of nanotube-reinforced composite

cylindrical panels resting on elastic foundations in thermal environments.
Compos Struct 2014;111:291–300.
[6] Yang J, Lee LL, Kitipornchai S. Nonlinear free vibration of single nanotubes
using nonlocal Timoshenko beam theory. Physica E 2010;42:1727–35.
[7] Rafiee M, Liu XF, He XQ, Kitipornchai S. Geometrically nonlinear free vibration
of shear deformable piezoelectric carbon nanotube/fiber/polymer multiscale
laminated composite plates. J Sound Vib 2014;333:3236–51.
[8] Zenkour AM, Sobhy S. Dynamic bending response of thermoelastic functionally
graded plates resting on elastic foundations. Aeoresp Sci Technol
2013;29:7–17.
[9] Duc ND. Nonlinear dynamic response of imperfect eccentrically stiffened FGM
double shallow shells on elastic foundation. Compos Struct 2012;102:306–14.
[10] Duc ND, Quan TQ. Transient responses of functionally graded double shells
with temperature-dependent material properties in thermal environment. Eur
J Mech A/Solids 2012;47:101–23.
[11] Najafov AM, Sofiyev AH. The non-linear dynamics of FGM truncated conical
shells surrounded by an elastic medium. Int J Mech Sci 2013;66:33–44.
[12] Kim J, Reddy JN. A general third-order theory of functionally graded plates
with modified couple stress effect and von the Kármán nonlinearity: theory
and finite element analysis. Acta Mech 2015;226:2973–98.
[13] Neves AMA, Ferreira AJM, Carrera E, Cinefra M, Roque CMC, Jorge RMN, et al.
Static, free vibration and buckling analysis of isotropic and sandwich
functionally graded plates using a quasi-3D higher-order shear deformation
theory and a meshless technique. Compos Part B 2013;44:657–74.
[14] Ferreira AJM, Batra RC, Roque RC, Qian LF, Martins PALS. Static analysis of
functionally graded plates using third-order shear deformation theory and a
meshless method. Compos Struct 2005;69:449–57.
[15] Taczala M, Buczkowski R, Kleiber M. A 16-noded locking-free Mindlin plate
resting on two-parameter elastic foundation – static and eigenvalue analysis.
Comput Assisted Mech Eng Sci; accepted for publication.

[16] Chauhan AP, Ashwell DG. Small- and large-amplitude free vibrations of square
shallow shells. Int J Mech Sci 1969;11:337–49.
[17] Crisfield MA. A fast incremental/iterative solution procedure that handles
‘‘snap-through”. Comput Struct 1981;13:55–62.
[18] Reddy JN, Chin CD. Thermoelastical analysis of functionally graded cylinders
and plates. J Therm Stress 1998;21:593–626.
[19] Taczala M, Buczkowski R, Kleiber M. Postbuckling analysis of functionally
graded plates on an elastic foundation. Compos Struct 2015;132:842–7.
[20] Park J-S, Kim J-H. Thermal postbuckling and vibration analyses of functionally
graded plates. J Sound Vib 2006;289:77–93.
[21] Shen H-S, Xiang Y. Nonlinear vibration of nanotube-reinforced composite
cylindrical panels resting on elastic foundations in thermal environment.
Compos Struct 2014;111:291–300.