Journal of Science and Technology in Civil Engineering, HUCE, 2025, 19 (1): 119–130
ISOGEOMETRIC FREE VIBRATION OF THE POROUS METAL
FOAM PLATES RESTING ON AN ELASTIC FOUNDATION USING
A QUASI-3D REFINED THEORY
Trang Tan Triena, Le Thanh Phonga, Pham Tan Hunga,∗
aFaculty of Civil Engineering, Ho Chi Minh City University of Technology and Education (HCMUTE),
No. 1 Vo Van Ngan street, Linh Chieu ward, Thu Duc city, Ho Chi Minh city, Vietnam
Article history:
Received 13/12/2024, Revised 17/01/2025, Accepted 18/3/2025
Abstract
This study investigates the free vibration behavior of porous metal foam plates using the Quasi-3D refined
plate theory. We consider three types of pores across the plate thickness: uniform, symmetric, and asymmetric
distributions. Besides, the metal foam plate is reinforced by a Winkler-Pasternak foundation. By employing
the variational principle and Quasi-3D refined theory, we derive the weak form for free vibration analysis.
The Quasi-3D theory is essential for analyzing plates, as it accurately captures transverse shear and normal
deformations, which are vital for understanding the behavior of thick and moderately thick plates. Unlike sim-
pler models, it provides a detailed representation of stress and strain distributions across the plate’s thickness,
enabling precise modeling of complex structural behaviors. The natural frequency of the porous metal foam
plates is determined by solving the explicit governing equation using the isogeometric approach. Additionally,
we examine how the porous coefficient, porous distribution, and geometry impact the vibrational frequency of
the porous metal foam plate.
Keywords: quasi-3D refined theory; isogeometric approach; porous metal foam plates; porous distribution.
https://doi.org/10.31814/stce.huce2025-19(1)-10 ©2025 Hanoi University of Civil Engineering (HUCE)
1. Introduction
Porous structures have been the focus of intensive research in recent years due to their excep-
tional mechanical properties. Chen et al. [1] presented the nonlinear vibration of sandwich beams
with a functionally graded (FG) porous metal foam core according to the Timoshenko beam theory.
Jabbari et al. [2,3] used the classical plate theory (CPT) and analytical methods to investigate porous
metal foam plates’ mechanical and thermal buckling. Besides, Barati and colleagues [4] conducted
analytical free vibration and buckling behaviors of the FG piezoelectric porous plates. Keddouri [5]
employed the refined plate theory (RPT) and analytical method to examine the impact of porous coef-
ficient and porous distribution on the deflection and stresses of FG sandwich plates with porosities. In
the study [6], the free vibration of the metal foam cylindrical shell was investigated using the analyt-
ical approach and FSDT. Rezaei et al. [7] determined the vibrational frequency of the FG plate made
of porous materials based on the first-order shear deformation plate theory (FSDT) and analytical ap-
proach. The analytical nonlinear vibration of the metal foam circular cylindrical shells with graphene
platelets (GPL) reinforcement was examined by Wang et al. [8] using Donnell nonlinear shell theory.
Ebrahimi et al. [9] used the analytical method combined with the RPT to introduce the free vibration
of the porous metal foam plate supported in an elastic foundation. Li et al. [10] employed the FSDT
and generalized differential quadrature (GDQ) method to explore the free vibration behavior of the
porous metal foam truncated conical shell. In addition, using the quasi-3D theory, Zenkour et al.
∗Corresponding author. E-mail address: hungpht@hcmute.edu.vn (Hung, P. T.)
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Hung, P. T., et al. /Journal of Science and Technology in Civil Engineering
[11,12] analytically solved the bending of FG and FG sandwich porous plates. Nguyen et al. [13]
utilized the polygonal finite element formulation to study the active vibration control of FG porous
metal foam plates with GPL reinforcement.
From the literature reviews above, it is evident that numerous studies have focused on the behav-
iors of metal foam structures using analytical methods. Nonetheless, these techniques are limited to
addressing straightforward problems with uncomplicated boundaries. For practical real-world struc-
tures, numerical methods like FEM, isogeometric analysis, and meshfree methods are preferred. In
addition, Hung et al. [14] studied the buckling and dynamic of the porous metal foam plates according
to the higher-order shear deformation theory (HSDT) with seven variables and moving Kriging mesh-
free method. Hughes et al. [15] pioneered the introduction of isogeometric analysis (IGA) utilizing
NURBS functions. Throughout this decade, numerous researchers have effectively employed IGA
for both plates and microplates. Phung-Van et al. [16] studied the nonlinear transient of the porous
FGM plates using IGA. Besides, the free vibration, bending and dynamic control of the piezoelectric
plates using the combination of HSDT and IGA were presented in ref. [17]. Lieu et al. [18] em-
ployed IGA to examine the free vibration of the FG porous plate with GPL reinforcement. The free
vibration of the mutidirectional FG plates using IGA was investigated by Son et al. [19,20]. Based
on the Quasi-3D theory and modified couple stress theory, Thai et al. [21] investigated the buckling
and free vibration of the multilayer FG plates reinforced with graphene platelets (GPLRC). Thai et
al. [22] used the MSGT, HSDT and IGA to study the free vibration of the multilayer FG GPLRC
microplates. Currently, there is a lack of research utilizing IGA based on Quasi-3D refined theory
to investigate the effects of porosity properties on the frequency of porous metal foam plates resting
on a Winkler-Pasternak foundation. This article address this gap by constructing a numerical model
for metal foam plates characterized by symmetric, asymmetric and uniform porous distributions. The
impact of the porous coefficient, porous distribution and geometry on the behavior of the metal foam
plate presented.
2. The fundamental equations
2.1. The effective material properties
Let’s contemplate porous metal foam plates with pores distributed throughout thickness of the
plate in three manners: uniform (P-I), symmetric (P-II), and asymmetric (P-III). The porous metal
foam plate and porosity distributions are shown in Fig. 1. The material properties of these microplates
are outlined as follows [23]
P-I
E(z)=E1(1−e0ζ)
G(z)=G1(1−e0ζ)
ρ(z)=ρ11−p1−e0ζ
ζ=1
e0
−1
e0 2
πp1−e0−2
π+1!2
P-II
E(z)=E11−e0cos πz
h
G(z)=G11−e0cos πz
h
ρ(z)=ρ11−emcos πz
h
(1)
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P-III
E(z)=E1"1−e0cosπ
4 2z
2h+1!#
G(z)=G1"1−e0cosπ
4 2z
h+1!#
ρ(z)=ρ1"1−emcosπ
4 2z
h+1!#
in which
em=1−p1−e0(2)
where emand e0are respectively denote the density porosity and porous coefficients. It is important
to note that this study does not consider the local effects of pores. The description of emand e0is
provided bellow
e0=1−E2
E1
=1−G2
G1
,0<e0<1
em=1−ρ2
ρ1
,0<em<1
(3)
where E1,G1, ρ1and E2,G2, ρ2are respectively represent the maximum and minimum values of elas-
tic modulus, shear modulus, and mass density of the plate.
Besides, based on [23], the Poisson’s ratio νis considered constant throughout the plate thickness.
Figure 1. The porous metal foam plate
2.2. The effective material properties
Based on the Quasi-3D refined plate theory [24], the displacement fields are described as follows
u=u0−zwb,x+f(z)ws,x
v=v0−zwb,y+f(z)ws,y
w=wb+ϕ(z)ws
with f(z)=z−4z3
3h2;ϕ(z)=1
6f′(z)(4)
where u0and v0denote displacement of the middle surface, while wb and ws are respectively repre-
sent bending and shear deflection; the index “,” denotes a differential operator.
The variational principle for free vibration of the porous plate resting on an elastic foundation is
defined as follows
δU−δK−δW=0 (5)
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where U,Wand Krepresent the strain energy, work and kinetic energies, respectively.
The strain energy of the metal foam plate is described by
U=Z
V
εTσdV (6)
where εand σrepresent the strain and Cauchy stress tensors, respectively.
The linear strain tensor of the porous plate is presented as follow
ε=1
2h∇u+(∇u)Tiwith ∇=(∂
∂x
∂
∂y
∂
∂z)(7)
where
u=
u
v
w
=u1+zu2+f(z)u3+ϕ(z)u4
u1=
u0
v0
wb
;u2=−
wb,x
wb,y
0
;u3=
ws,x
ws,y
0
;u4=
0
0
ws
(8)
After the substitution Eq. (8) into Eq. (7), the linear strain tensor is reformed as follows
ε=(εb
εs);εb=
εx
εy
γxy
εz
=ε1b+zε2b+f(z)ε3b+ϕ′(z)ε4b
εs=(γxz
γyz )=f′(z)+ϕ(z)γs
(9)
where
ε1b=
u0,x
v0,y
u0,y+v0,x
0
;ε2b=
−wb,xx
−wb,yy
−2wb,xy
0
;ε3b=
ws,xx
ws,yy
2ws,xy
0
;ε4b=
0
0
0
ws
;γs=(ws,x
ws,y)
(10)
The stress-train relation based on the Hook law is presented by
σb=
σx
σy
τxy
σz
Q11 Q12 0Q13
Q12 Q22 0Q23
0 0 Q66 0
Q13 Q23 0Q33
εx
εy
γxy
εz
=Cbεb
σs=(τxz
τyz )="Q55 0
0Q44 #( γxz
γyz )=Csεs
(11)
in which
Q11 =Q22 =Q33 =(1−ν)E(z)
(1−2ν) (1+ν);Q13 =Q23 =Q12 =νE(z)
(1−2ν) (1+ν)
Q66 =Q55 =Q44 =E(z)
2(1 +ν)
(12)
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Substituting σband σsin Eq. (9), into Eq. (6), the virtual strain energy is reformed by following
δU=ZΩ
δ¯
εT
b¯
Db¯
εbdΩ + ZΩ
δγT
s¯
DsγsdΩ(13)
where
¯
εb=nεb1εb2εb3εb4oT;¯
Db=
A B C E
B D F L
C F H O
E L O P
(A,B,C,E,D,F,L,H,O,P)=Zh/2
−h/21,z,f, ϕ′,z2,z f,zϕ′,f2,fϕ′, ϕ′2Cbdz
¯
Ds=Zh/2
−h/2f′+ϕ2Csdz
(14)
The expression for virtual kinetic energy can be expressed as follow
δK=ZΩ
δ¯
uTIm¨
¯
udΩ(15)
in which
¯
u=
u1
u2
u3
u4
;Im=
I00 0
0I00
0 0 I0
;I0=
AmBmCmEm
BmDmFmLm
CmFmHmOm
EmLmOmPm
(Am,Bm,Cm,Em,Dm,Fm,Lm,Hm,Om,Pm)=Zh/2
−h/2
ρ1,z,f, ϕ, z2,z f,zϕ, f2,fϕ, ϕ2dz
(16)
The virtual work performed by an elastic foundation can be represented as follows
δW=−ZΩkww−ks∇2wδwdΩ(17)
in which kwand ksare respectively represent the spring and shear coefficients of the foundation.
Substituting Eqs. (13), (15) and (17) into Eq. (5), the weak form of the porous metal foam plates
is reformed by
ZΩ
δ¯
εT
b¯
Db¯
εbdΩ + ZΩ
δγT
s¯
DsγsdΩ + ZΩ
δ¯
uTIm¨
¯
udΩ
−ZΩhkww−ks∇2wiδwdΩ = 0
(18)
2.3. NURBs approximation
In two dimensions (2D), NURBS basis functions [15] are defined using two knot vectors, ¯
U=
nη1, η2, . . . , ηn+p+1oand ¯
V=nζ1, ζ2, . . . , ζm+q+1o. Here, nand mrepresent the number of control points
in the respective directions, while pand qcorrespond to the polynomial orders. The computation of
the basis functions for 2D B-splines includes:
Ni,j(η, ζ)=⌢
Ni,p(η)⌢
Mj,q(ζ)(19)
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