Mechanics of Advanced Materials and Structures

ISSN: (Print) (Online) Journal homepage: www.tandfonline.com/journals/umcm20

Quasi-3D plate theory for size-dependent static and free

vibration analysis of FG microplate with porosities based
on a modified couple stress theory

Abdeldjebbar Tounsi, Abdelhakim Kaci, Abdelouahed Tounsi, Mohammed A.

Al-Osta, Murat Yaylacı, Sherain M. Y. Mohamed, Saad Althobaiti & Mahmoud
M. Selim

To cite this article: Abdeldjebbar Tounsi, Abdelhakim Kaci, Abdelouahed Tounsi, Mohammed
A. Al-Osta, Murat Yaylacı, Sherain M. Y. Mohamed, Saad Althobaiti & Mahmoud M. Selim (18
Feb 2025): Quasi-3D plate theory for size-dependent static and free vibration analysis of FG
microplate with porosities based on a modified couple stress theory, Mechanics of Advanced
Materials and Structures, DOI: 10.1080/15376494.2025.2463687

To link to this article: https://doi.org/10.1080/15376494.2025.2463687

Published online: 18 Feb 2025.

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MECHANICS OF ADVANCED MATERIALS AND STRUCTURES
https://doi.org/10.1080/15376494.2025.2463687

ORIGINAL ARTICLE

Quasi-3D plate theory for size-dependent static and free vibration analysis of FG
microplate with porosities based on a modified couple stress theory
Abdeldjebbar Tounsia, Abdelhakim Kacib,c, Abdelouahed Tounsib,d,e, Mohammed A. Al-Ostad,e, Murat Yaylacıf,g,
Sherain M. Y. Mohamedh, Saad Althobaitii, and Mahmoud M. Selimh
a
Mechanical Engineering Department, Faculty of Science and Technology, University of R�elizane, R�elizane, Algeria; bFaculty of Technology,
Civil Engineering Department, Material and Hydrology Laboratory, University of Sidi Bel Abbes, Sidi Bel Abbes, Algeria; cFacult�e de
Technologie, D�epartement de G�enie Civil et Hydraulique, Universit�e Dr. Tahar Moulay, Saida, Alg�erie; dDepartment of Civil and
Environmental Engineering, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia; eInterdisciplinary Research Center for
Construction and Building Materials, KFUPM, Dhahran, Saudi Arabia; fDepartment of Civil Engineering, Recep Tayyip Erdogan University,
Rize, Turkey; gFaculty of Turgut Kıran Maritime, Recep Tayyip Erdogan University, Rize, Turkey; hDepartment of Mathematics, College of
Science and Humanities, Prince Sattam bin Abdulaziz University, Al-Kharj, Saudi Arabia; iDepartment of Sciences and Technology, Ranyah
University Collage, Taif University, Taif, Saudi Arabia

ABSTRACT ARTICLE HISTORY

This work presents a static and vibration analysis of a porous functionally graded (FG) microplate Received 10 January 2025
using a simplified quasi-3D plate formulation and a modified couple stress theory that requires Accepted 2 February 2025
only one material length scale parameter. The theory contains only five unknowns, incorporating
KEYWORDS
undetermined integral variables in accordance with the first-order shear deformation theory
Microplate; simplified quasi-
(FSDT). The model satisfies the zero traction boundary conditions on the upper and lower surfaces 3D plate theory; modified
of the microplate. The governing equations are obtained via Hamilton’s principle and subse­ couple stress theory;
quently solved using Navier-type closed-form solutions. The proposed theoretical model is vali­ porosity; material scale
dated by comparing it with existing models in the literature to demonstrate its efficacy. The parameter
influence of the length scale parameter, the power law index, shear and normal deformation
effects, porosity factors, and plate thickness on the responses of microplates is examined through
the presentation of numerical examples. The findings show that taking size effects into account
increases the stiffness of the microplate, which in turn reduces deflections and raises natural fre­
quencies. Conversely, the normal and shear deformation influences have the opposite impact.

1. Introduction differs significantly from that predicted by classical con­

tinuum mechanics, which is no longer suitable for accurately
Micro and nano-electromechanical systems (MEMS / NEMS)
are rapidly evolving technologies with diverse applications in describing these small-scale phenomena.
fields such as materials science, electronics, biotechnology, The emergence of size and surface effects of the micro/
medicine, industry, manufacturing, information technology, nano scale necessitates the development of new theoretical
automotive engineering, and aerospace [1–5]. The fabrication approaches and specific models to account for these unique
of MEMS and NEMS devices requires extreme precision, and behaviors. Contemporary microscopic theories encompass
contamination and defects can significantly affect their per­ nonlocal, strain gradient, couple stress, and surface elasticity
formance. Multi-physical interactions, such as electromechan­ theories. For instance, the nonlocal theory evaluates the
ical and thermoelectric effects, further complicate the process, overall influence of long-range interactions within the
necessitating sophisticated models to predict behaviors at dif­ microstructure [13–18]. The strain gradient theory [19,20]
ferent scales. Long-term reliability remains a major challenge, analyses the effect of higher-order deformation tensors on
with material fatigue and degradation issues due to thermal the material structure. Couple stress theory [21] incorporates
and mechanical cycling.
materials’ rotational degrees of freedom during deformation.
The coupling of phenomena across different scales, from
Finally, surface elasticity theory effectively characterizes the
nano to micro, further complicates the design and integra­
tion of MEMS/NEMS systems [6–12]. These phenomena influence of surface free energy on materials’ mechanical
include changes in mechanical properties, such as stiffness response. These theories partially address the limitations of
and strength, as well as size effects that influence inter- classical continuum theory by exploiting various microme­
atomic interactions and surface forces, thereby altering the chanical mechanisms. Numerous researchers have success­
overall material response to mechanical loads. Consequently, fully employed them for various analyses and experimental
the mechanical response observed at the micro or nanoscale investigations.

CONTACT Abdeldjebbar Tounsi abdeldjebbar.tounsi@univ-relizane.dz Mechanical Engineering Department, Faculty of Science and Technology, University
of R�elizane, R�elizane, Algeria.
� 2025 Taylor & Francis Group, LLC
2 A. TOUNSI ET AL.

Recent developments in this field include in-depth studies approach contributes to the design of micro-devices such as
on the application of nonlocal and strain gradient theories MEMS/NEMS. Van Hieu et al. (2023) [33] utilized Mindlin’s
to nanostructures. For instance, using nonlocal theory, plate theory and Modified Couple Stress Theory to model
Moheimani and Dalir (2020) [22] explored the static, microplates. Analytical solutions are obtained for simply sup­
dynamic, and vibrational behaviors of functional micro- and ported functionally graded (FG) microplates, considering two
nanobeams. They developed governing linear equations models of porosity distribution. The results show the influ­
based on this theory and solved them using analytical meth­ ence of key parameters on the linear and nonlinear behavior
ods under different boundary conditions. The study eval­ of the microplate. Recent studies have employed CCST-based
uated the influence of axial load, nonlocal parameters, and unified shear deformation theories to develop more compre­
power index on natural frequencies under various boundary hensive models for static and dynamic analyses of micro and
conditions. A comparative analysis with classical theory nanoscale plates [34–36]. For instance, researchers have suc­
results highlighted significant nonlocal effects, particularly cessfully implemented finite element methods based on CCST
pronounced at the nanoscale, which substantially impact the to investigate size-dependent bending and vibration behaviors
behaviors of the beams studied. Similarly, Li et al. (2023) of functionally graded materials [37–39]. These analyses dem­
[23] studied the free vibrations of circular porous and onstrate the effectiveness of CCST in predicting mechanical
graded nanoplates under various boundary conditions, con­ responses that align closely with experimental observations,
sidering the continuous variation of material properties thereby enhancing the understanding of size effects in engin­
across the thickness. Eringen’s nonlocal elastic theory was eering applications. Nguyen et al. (2022) [40] presented the
applied to capture the size effect, and the equations of analysis of nonlinear static bending of microplates with vari­
motion were derived from Mindlin plate theory using able thickness by utilizing the finite element method and
Hamilton’s principle. The study numerically solved these modified stresses. The proposed theory and mathematical
equations using the shooting technique to analyze the model are validated by comparing numerical data with exist­
impacts of porosity distribution, porosity coefficient, nonlo­ ing literature, and a parametric study is conducted to exam­
cal scale effect, thickness-to-diameter ratio, and boundary ine the mechanical behavior of the structure, particularly
conditions on the natural frequencies of nanoplates. Phung- highlighting nonlinear effects. The calculated results are an
Van et al. (2021) [24] investigated the nonlinear bending invaluable reference for such engineering structures’ practical
behavior of nanoporous metallic foam plates using nonlocal use and design. Thai et al. (2020) [41] introduced a size
strain gradient theory, showing that the stiffness of these dependent model using NURBS basis functions integrated
structures softens with increasing nonlocal parameters. The with MCST and quasi-3D shear deformation theory to study
results demonstrated that introducing nonlocality signifi­ vibrations and buckling of multilayer composite microplates
cantly alters the rupture and deformation behaviors of nano­ reinforced with graphene nanoplatelets. Numerical simula­
composites, highlighting the importance of this theory for tions demonstrate how geometric parameters, boundary con­
the design of advanced materials. ditions, and material length scale parameters influence natural
Recent efforts have focused on the development of modi­ frequencies and buckling loads. Yang et al. (2021) [42] exam­
fied stress couple theories requiring fewer material parame­ ined the nonlinear geometric-dependent bending response of
ters. The Consistent Couple Stress Theory (CCST) is an arbitrarily shaped microplates with variable thickness in func­
advanced continuum mechanics framework that accounts for tionally graded (FG) composites. They employed continuum
size effects in materials, particularly at micro- and nanoscales elasticity theory with modified stresses incorporating von
[25,26]. Unlike classical theories, CCSTincorporates both Karman’s hypothesis of large deformations in a quasi-three-
stress and couple stress tensors, allowing for a more accurate dimensional (quasi-3D) plate framework, where transverse
representation of material behavior under various loading shear deformations and normal deflection are distributed
conditions [27]. This theory is particularly relevant for analyz­ according to trigonometric patterns. The study analyses varia­
ing structures where size-dependent effects are significant, tions in the thickness of microplates in linear, convex, and
such as in microplates and nanostructures [28]. In contrast, concave patterns using isogeometric techniques with non-uni­
the Modified Couple Stress Theory (MCST) simplifies some form B-spline functions. It highlights how modified stresses
of the assumptions made in CCST by reducing the number induce stiffening effects in linear and nonlinear bending
of parameters involved and focusing primarily on stress responses, especially with thickness variations from convex to
rather than couple stress [29–31]. While MCST is effective linear and then concave. Moreover, transitioning from fixed
for many applications, it may not capture all the nuances of to simply supported boundary conditions accentuates these
material behavior that CCST addresses, particularly in cases effects on FG composite microplates. Afshari and Adab
involving complex loading scenarios or geometries. Tang (2022) [43] investigated the mechanical analysis of buckling
et al. (2022) [32] examined the microscopic scale impacts on and free vibrations of microplates reinforced with FG gra­
thin plates’ bending, buckling, and vibration behavior by cou­ phene nanoplatelets (FG-GNPs), using MCST and sinusoidal
pling Kirchhoff’s plate theory with the Modified Couple shear deformation quasi-3D theory (SSDT). Karami et al.
Stress Theory (MCST). The results demonstrate that scale (2024) [44] studied the vibrational behavior of FG thick
effects increase the equivalent stiffness of the plate, reducing microplates with material imperfections. They employed a
deflection while increasing critical buckling load and natural quasi-3D model and the MCST method to rigorously analyze
frequency without affecting the buckling topology. This both free and forced vibrations. The study emphasized the
 MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 3

significant influence of thickness and imperfections on the edges under transverse load. They employed an inverse
vibrational characteristics of microplates. The design, fabrica­ hyperbolic shear deformation theory (IHSDT) to approxi­
tion, and application of unit structural devices such as micro- mate the displacement field. The study considered five types
cantilevers and microplates can be guided by integrating of porosity distribution functions and evaluated their effects
materials, structures, and micro-scale theories. Research on the plate’s performance. Kumar et al. (2021) [53] investi­
addressing scale effects is crucial. A literature review synthe­ gated the effects of porosity distributions on the nonlinear
sizes several significant works on size effect theories, includ­ free vibration and transient analysis of porous functionally
ing nonlocal strain gradient theory, stress couple theory, and graded skew (PFGS) plates. The effective material properties
modified stress couple theory, among others [45]. were derived from modified power-law equations, using a
While these research endeavors focus on the latest nonlinear finite element formulation based on first-order
advancements in structural engineering, the increasing use shear deformation theory (FSDT) and von Karman’s strain
of functionally graded materials (FGMs) requires the devel­ relations. The study examined how porosity distributions
opment of suitable theoretical and numerical formulations. and parameters influence the nonlinear frequency responses
In classical plate theory (CPT), which is based on of PFGS plates at various skew angles, detailing the effects
Kirchhoff’s hypothesis, only satisfactory results can be of volume fraction grading index and skew angle on
obtained for the analysis of thin plates. Love developed the dynamic responses. Hu and Fu (2023) [54] presented the
theory in 1888, and it utilized Kirchhoff’s assumptions. effects of porosity distribution and grading on the free
Models developed in accordance with the First-Order Shear vibration of FG plates using first-order shear deformation
Deformation Theory (FSDT) are founded upon the assump­ theory. They found that uneven porosity in E-FGM
tions espoused in the Mindlin-Reissner plate theory, wherein increases frequency by 67.3% compared to even porosity,
transverse shear deformations are presumed to be constant while for even porosity, S-FGM increases frequency by 54%
in the thickness direction. This necessitates the incorpor­ and 70% over P-FGM and E-FGM, respectively. This under­
ation of shear correction factors to ensure the precision of scores the substantial impact of porosity and grading on the
the resulting data. Several higher-order shear deformation vibration characteristics of FG plates. Farzam and Hassani
theories have been proposed to enhance the accuracy of pre­ (2019) [55] studied the bending, buckling, and vibration
dicting transverse shear. These theories assume a transverse responses of FG microplates with porosities based on modi­
distribution in the form of a polynomial or higher degree, fied couple stress theory (MCST) and hyperbolic shear
which results in the transverse stress vanishing at the top deformation theory (HSDT). Additionally, Liu et al. (2022)
and bottom of the plate, eliminating the need for shear cor­ [56] conducted an analytical study on the impact response
rection factors. However, since all these theories assume uni­ of shear deformable sandwich cylindrical shells with a func­
form transverse displacements along the thickness direction, tionally graded porous core. Zhou et al. (2022) [57] intro­
they ignore the stretching effect in the transverse direction duced a novel similitude method for predicting natural
(i.e. ez ¼ 0). To address the impact of this stretching effect, frequencies of FG porous plates under thermal environ­
numerous quasi-3D theories have been developed in previ­ ments. Beitollahi et al. (2024) [58] explored variable length
ous studies for analyzing the mechanical behavior of thick scale parameters in functionally graded non-porous and por­
plates [46–50]. These quasi-3D theories are computationally ous microplates/nanoplates. Pham et al. (2022) [59] analyzed
intensive and costly, involving a significant number of bending and hygro-thermo-mechanical vibrations of func­
unknown variables. For instance, some theories deal with tionally graded porous sandwich nanoshells resting on elas­
twelve, eleven, nine, or fewer unknowns. Mantari and Soares tic foundations. Lastly, Do and Pham (2024) [60] focused
(2015) [51] introduced a quasi-3D formulation with six on nonlinear static analysis of functionally graded porous
derivable unknowns. Recently, a simplified quasi-3D theory sandwich plates resting on Kerr foundations, while Wang
using only five unknowns has been proposed for analyzing et al. (2024) [61] developed a refined plate theory for analyz­
the mechanical behavior of plates. Hence, a straightforward ing bending, buckling, and free vibrations of functionally
quasi-3D theory, as proposed in this study, is essential. graded porous plates reinforced by graphene platelets.
The manufacture of functional gradient (FG) materials is The objective of this article is to utilize a simplified quasi-
a delicate process, often marked by the presence of pores 3D plate theory with only five unknowns for the analysis of
and voids. However, rather than simply being a nuisance, size-dependent static and free vibration behaviors of FG
these imperfections can be exploited to our advantage. By microplates with porosities based on modified couple stress
carefully controlling the distribution of porosity, we can theory. A key novelty of this work lies in its ability to achieve
reduce the weight of FG materials and enhance their energy a size range in microscale theory that aligns closely with
absorption capabilities. This represents a fascinating oppor­ experimental observations, addressing a significant gap in cur­
tunity to design structures that respond effectively to both rent literature. Experimental evidence demonstrates that the
static and dynamic loads. Exploring this uncharted territory mechanical response of materials at the micron scale depends
poses a challenge that promises to open new frontiers in on size, which this study effectively incorporates. The plate is
materials engineering and structural optimization. For modeled as either perfectly porous and homogeneous or as
example, Dhuria et al. (2021) [52] studied the effect of por­ having a perfectly homogeneous shape, depending on the vol­
osity distribution on the static and buckling responses of a ume fraction of porosity. The governing equations are derived
functionally graded (FG) porous plate with simply supported using Hamilton’s principle and solved employing the Navier
4 A. TOUNSI ET AL.

solution, involving only five unknown functions—this simpli­ The coefficients k1 and k2 depend on the geometry.where
fication enhances computational efficiency without sacrificing ðu0 , v0 , w0 , h, /z Þ are five unknown displacements of the
accuracy. Comparative studies have been conducted to con­ mid-plane of the plate. In this study, f ðzÞ is a shape function
firm the accuracy and effectiveness of the current theory, chosen in the form.
showcasing its potential for practical applications. The devel­ " � �#
oped model not only provides a robust framework for analyz­ 4 z 2 0
f ðz Þ ¼ z 1 − and g ðzÞ ¼ f ðzÞ (7)
ing MEMS/NEMS but also sets a precedent for future 3 h
research in the size-dependent behavior of porous structures,
Applying Eq. (2a) and Eq. (6), the non-zero strains of the
making it a significant contribution to the field.
present simplified plate theory can be written as:
2. Theoretical formulations @u0 @ 2 w0
exx ¼ −z − f ðzÞk1 h (8a)
@x @x2
2.1. Modified couple stress theory
Based on the modified couple stress model developed by @v0 @ 2 w0
eyy ¼ −z − f ðzÞk2 h (8b)
Yang et al. (2002) [62], the strain energy U for a linear elas­ @y @y2
tic material using the region X is related to curvature ten­
sors and strain and is written as follows: ezz ¼ g 0 ðzÞ/z (8c)
ð
1
U¼ ðrij eij þ mij vij ÞdXdz, ði, j ¼ 1, 2, 3Þ (1) � �
2 X @u0 @v0 @ 2 w0 @2h
cxy ¼ þ − 2z − f ðzÞðk1 A0 þ k2 B0 Þ
In which e; r; v and m represent classical strain tensor, @y @x @x@y @x@y
Cauchy stress tensor, symmetric curvature tensor, and devia­ (8d)
toric part of the couple stress tensor, respectively. The strain
� �
and the curvature tensors are obtained as follows: @/z @h
1 cxz ¼ g ðzÞ − k1 A0 (8e)
eij ¼ ðui, j þ ui, i Þ (2a) @x @x
2
� �
1 � @/z 0 @h
vij ¼ hi, j þ hi, i (2b) cyz ¼ g ðzÞ − k2 B (8f)
2 @y @y
where ui denote the parts of the displacement vector and hi The integrals utilized in the equations can be solved
indicate the parts of the rotation vector and are determined using a Navier-type approach and are given as follows:
as follows: ð 2 ð
@ 0 @ h @ @2h
1 h dx ¼ A , h dy ¼ B0 ,
hi ¼ eijk uk, j (3) @y @x@y @x @x@y
2 ð ð (9)
where eijk represents the permutation symbol. The relations 0 @h 0 @h
h dx ¼ A , h dy ¼ B
of constitutive are given as follows: @x @y
rij ¼ kðzÞekk dij þ 2lðzÞeij (4a) where the coefficients A0 and B0 are determined in accord­
ance with the solution method employed, which, in this
mij ¼ 2lðzÞl2 vij (4b) instance, utilizes the Navier approach. Consequently, the
expressions for A0 and B0 are as follows:
where dij denotes the Kronecker delta, l represents the
1 1
material length scale parameter, which indicates the influ­ A0 ¼ − ~ 2 , k2 ¼ n
, B0 ¼ − 2 , k1 ¼ m ~2 (10)
ence of couple stress, k and l are Lame’s constants and can ~2
m ~
n
be obtained as follows: where m~ and n~ are defined in expression (28).
EðzÞ� ðzÞ EðzÞ Using Eqs. (2b), (3) and (6), the parts of the curvature
kðzÞ ¼ and lðzÞ ¼ (5)
½1 þ � ðzÞ�½1 − 2� ðzÞ� 2½1 þ �ðzÞ� tensor are found as follows:
!
@hx @ 2 w0 1 2
0 @ h @ 2 /z
vx ¼ ¼ þ g ðz Þ k2 B þ (11a)
2.2. Kinematics @x @x@y 2 @x@y @x@y
The displacement field of the new theory can be specified as
!
follows:
ð @hy @ 2 w0 1 @2h @ 2 /z
@w0 vy ¼ ¼− − g ðz Þ k1 A0 þ (11b)
uðx, y, z, tÞ ¼ u0 ðx, y, tÞ − z − f ðzÞk1 hðx, y, t Þdx @y @x@y 2 @x@y @x@y
@x ð
@w0 (6)
vðx, y, z, tÞ ¼ v0 ðx, y, tÞ − z − f ðzÞk2 hðx, y, t Þdy
@y @hz 1 @2h
vz ¼ ¼ g ðzÞðk1 A0 − k2 B0 Þ (11c)
wðx, y, z, tÞ ¼ w0 ðx, y, tÞ þ gðzÞ/z @z 2 @x@y
 MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 5

� � !
1 @hx @hy 1 @ 2 w0 @ 2 w0 Putting Eq. (11) into Eq. (4b), the deviatoric components
vxy ¼ þ ¼ − of the couple stress tensor are derived as follows.
2 @y @x 2 @y2 @x2 " !#
! (11d) 2 2
@ 2 /z
2 @ w0 1 0 @ h
1 @2h @ 2 h @ 2 /z @ 2 /z mx ¼ 2lðzÞl þ g ðz Þ k2 B þ (13a)
þ g ðzÞ k2 B0 2 − k1 A0 þ − @x@y 2 @x@y @x@y
4 @y @x2 @y2 @x2
! " !#
� � 2 2 @ 2 w0 1 @2h @ 2 /z
1 @hx @hz 1 @ v0 @ u0 my ¼ 2lðzÞl − 2
− g ðz Þ k1 A 0 þ
vxz ¼ þ ¼ − @x@y 2 @x@y @x@y
2 @z @x 4 @x2 @x@y
" � �# (13b)
1 0 0 @ 3 w0 0 0 @h @/z
þ f ðz Þðk 1 A − k 2 B Þ 2 þ g ðz Þ k 2 B þ " #
4 @x @y @y @y
1 2 @2h
(11e) mz ¼ 2lðzÞl g ðzÞðk1 A0 − k2 B0 Þ (13c)
2 @x@y
� � !
1 @hy @hz 1 @ 2 v0 @ 2 u0 " !
vyz ¼ þ ¼ − @ 2 w0 @ 2 w0
2 @z @y 4 @x@y @y2 mxy ¼ lðzÞl 2
−
" � �# @y2 @x2
1 0 0 @ 3 w0 0 0 @h @/z !#
þ f ðz Þðk1 A − k2 B Þ − g ðz Þ k 1 A þ 1 @2h @ 2 h @ 2 /z @ 2 /z
4 @x@y2 @x @x þ g ðz Þ k2 B 0 2 − k 1 A0 þ −
2 @y @x2 @y2 @x2
(11f)
Putting Eq. (8) into Eq. (4a) results in the following parts (13d)
of the stress tensor " !
! 1 @ 2 v0 @ 2 u0
2
@u0 @v0 @ 2 w0 @ 2 w0 myz ¼ lðzÞl −
rx ¼ kðzÞ þ − z þz − g 0 ðzÞ/z 2 @x@y @y2
@x @y @x2 @y2
! � � 1 @ 3 w0
@u0 @ 2 w0 þ f ðzÞðk1 A0 − k2 B0 Þ
ð Þ
− f ðzÞh k1 þ k2 þ 2lðzÞ −z − f ðzÞk1 h 2 @x@y2
@x @x2
� �!#
(12a) 0 0 @h @/z
− g ðz Þ k1 A þ (13e)
! @x @x
@u0 @v0 @ 2 w0 @ 2 w0
ry ¼ kðzÞ þ − z þ z − g 0 ðzÞ/z " !
@x @y @x2 @y2 1 @ 2 v0 @ 2 u0
2
! ! mxz ¼ lðzÞl −
@v0 @ 2 w0 2 @x2 @x@y
− f ðzÞhðk1 þ k2 Þ þ 2lðzÞ −z − f ðzÞk2 h
@y @y2 1 @ 3 w0
þ f ðzÞðk1 A0 − k2 B0 Þ 2
(12b) 2 @x @y
! � �!#
@u0 @v0 @ 2 w0 @ 2 w0 0 0 @h @/z
þ g ðz Þ k 2 B þ (13f)
rz ¼ kðzÞ þ − z 2
þz − g 0 ðzÞ/z @y @y
@x @y @x @y2
!
− f ðzÞhðk1 þ k2 Þ þ 2lðzÞg 0 ðzÞ/z
2.3. Material model
(12c) Dealing with an FG microplate consisting of two different
FGMs is depicted in Figure 1. The microplate is designed to
sxy ¼ 2lðzÞexy be composed of metal and ceramic in such a way that the
� � ! material at the top surface ðz ¼ h=2Þ is ceramic-rich, while
@u0 @v0 @ 2 w0 @ 2
h
¼ lðzÞ þ − 2z − f ðzÞðk1 A0 þ k2 B0 Þ the material at the bottom surface ðz ¼ −h=2Þ is metal-rich.
@y @x @x@y @x@y
However, the effect of porosities inside the FG plate materi­
(12d) als during production is included. Firstly, a non-homogen­
� � eity material with a porosity volume function, a (0 � a � 1)
@/z @h
syz ¼ 2lðzÞeyz ¼ lðzÞg ðzÞ − k2 B0 (12e) will be considered.
@y @y The effective Young’s modulus for the power law distri­
� � bution is defined by Benferhat and others [63–66]:
@/z 0 @h a
sxz ¼ 2lðzÞexz ¼ lðzÞg ðzÞ − k1 A (12f) EðzÞ ¼ Em þ ðEc − Em ÞVc ðzÞ − ðEc þ Em Þ (14)
@x @x 2
6 A. TOUNSI ET AL.

�
1 @ 3 dw0 1 3
0 Þ @ dw0
þ Yyz ðk1 A0 − k2 B0 Þ þ Y xz ð k 1 A 0
− k 2 B
2 @x@y2 2 @x2 @y
( !
2 2
0 @ dh 0 @ dh
þ − Sx k1 dh − Sy k2 dh − Sxy k1 A þ k2 B
@x@y @x@y
@dh @dh 1 @ 2 dh
− Qyz k2 B0 − Qxz k1 A0 þ Z x k2 B 0
@y @x 2 @x@y
1 @ 2 dh 1 @ 2 dh
− Z y k1 A0 þ Zz ðk1 A0 − k2 B0 Þ
2 @x@y 2 @x@y
1 @ 2 dh 1 2
0 @ dh 1 @dh
þ Zxy k2 B0 − Zxy k1 A − Wyz k1 A0
2 @y2 2 @x2 2 @x
Figure 1. FGM plate’s geometry. )
1 @dh @d/z @d/z
þ Wxz k2 B0 þ Nz d/z þ Qyz þ Qxz
2 @y @y @x
1 @ 2 d/z 1 @ 2 d/z 1 @ 2 d/z 1 @ 2 d/z
where Vc ðzÞ ¼ ð0:5 þ z=hÞk is the ceramic volume fraction, þ Zx − Zy þ Zxy − Zxy
and the subscripts c and m denote the metallic and ceramic 2 @x@y 2 @x@y 2 @y2 2 @x2
!
constituents. 1 @d/z 1 @d/z
The rule of mixture gives the effectual mass density of þ − Wyz þ Wxz dXdt
2 @x 2 @y
the FG microplate as
a (17)
qðzÞ ¼ qm þ ðqc − qm ÞVc ðzÞ − ðqc þ qm Þ (15)
2 where X indicate the top surface and the stress resultants N;
For convenience, Poisson’s ratio t is considered to M; S; Q; Nz and X, Y, Z, W are defined by
remain constant, as its effects on the response of FG plates Ð h=2
ðNi , Mi , Si Þ ¼ −h=2 ð1, z, f Þðri Þdz, ði ¼ x, y, xyÞ,
are minimal [67–69]. Ð h=2
Qi ¼ −h=2 ðsi Þg ðzÞdz, ði ¼ xz, yzÞ (18)
Ð h=2 Ð
Nz ¼ −h=2 rz g 0 ðzÞdz, ðXi , Yi , Zi , Wi Þ ¼ A ð1, f , g, g 0 Þmi dA
2.4. The governing equations
The work done by external applied forces can be written
This study employs Hamilton’s principle to obtain the as the variation in work.
motion equations. This fundamental principle is formulated ð ð � �
in analytical terms as: d V ¼ qdwdX ¼ qdw0 þ gðzÞjz¼h=2 d/z dX (19)
ð t2 X X

ðdU − dV − dK Þdt ¼ 0 (16) where q denotes the transverse load.

t1 The kinetic energy variation can be obtained as
The strain energy virtual variation is represented by d U; ð ðh
2

the work done virtual variation by the external applied dK ¼ qðzÞ½ud _ u_ þ v_ d v_ þ wd _ w_ �dzdX
X −h2
forces is represented by d V; and the kinetic energy virtual ð (
variation is represented by d K: The symbols t1 and t2 indi­ ¼ I0 ½u_ 0 du_ 0 þ v_ 0 d_v 0 þ w_ 0 dw_ 0 �
cate the initial and final times. The strain energy variation X
� �
of the plate is expressed as: @dw_ 0 @ w_ 0 @dw_ 0 @ w_ 0
− I1 u_ 0 þ du_ 0 þ v_ 0 þ d_v 0
ð @x @x @y @y
" � � !#
U¼ rx dex þ ry dey þ rz dez þ sxy dcxy þ syz dcyz þ sxz dcxz þ mx dvx @d h_ @ h_ @dh_ @ h_
V 0 0
� − J1 k1 A u_ 0 þ du_ 0 þ k2 B v_ 0 þ d_v 0
þ my dvy þ mz dvz þ 2mxy dvxy þ 2myz dvyz þ 2mxz dvxz dXdz @x @x @y @y
� �
@ w_ 0 @dw_ 0 @ w_ 0 @dw_ 0
þ I2 þ
! @x @x @y @y
ð " #
@du0 @du0 1 @ 2 du0 1 @ 2 du0 @ h_ @dh_ _
2 0 @ h @dh
_
d U¼ Nx þ Nxy − Xyz − Xxz þ K2 k1 A 22 0
þ k2 B 2
X @x @y 2 @y2 2 @x@y @x @x @y @y
! " � �
0 @w _ 0 @dh_ @dw_ 0 @ h_
@dv0 @dv0 1 @ 2 dv0 1 @ 2 dv0 þ J2 k 1 A þ
þ Ny þ Nxy þ Xyz þ Xxz @x @x @x @x
@y @x 2 @x@y 2 @x2 !#
� 0 @w _ 0 @dh_ @dw_ 0 @ h_
@ 2 dw0 @ 2 dw0 @ 2 dw0 þ k2 B þ
þ − Mx − My − 2Mxy
@y @y @y @y
@x2 @y2 @x@y )
h i
@ 2 dw0 @ 2 dw0 @ 2 dw0 @ 2 dw0 þ J1s w_ 0 d/_ z þ /_ z dw_ 0 þ J3s /_ z d/_ z dX
þ Xx − Xy þ Xxy − X xy
@x@y @x@y @y2 @x2
(20)
 MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 7

! !
where dot-superscript convention implies the differentiation @ 4 w0 @ 4 w0 @4h @4h
for the time variable t;qðzÞ is the mass density; and (I0 ; I1 ; −J2 k1 A 2 2 þ k2 B0 2 2
0
− J3 k21 A0 2 2 2
þ k22 B0 2 2 2
@x @t @y @t @x @t @y @t
J1 ; J2S I2 ; J2 ; K2 ; K2s ) are the mass inertias described as
(22d)
ðh
� 2 �
I0 , I1 , J1 , J1s , I2 , J2 , J3 , J3s ¼ 1, z, f , g, z2 , zf , f 2 , g qðzÞdz
2
@Qxz @Qyz
−h2 d/z : þ − Nz
@x @y
(21) !
1 @ 2 Zy @ 2 Zx @ 2 Zxy @ 2 Zxy @Wxz @Wyz
After substituting the expressions for dU; dV; and dK þ − − þ þ −
2 @x@y @x@y @y2 @x2 @y @x
from Eqs. (17), (19), and (20) into Eq. (16), the next step
involves integrating by parts with respect to both the space @ 2 w0 @ 2 /z
and time variables. Then, collecting the coefficients of du0 ; ¼ J1s 2
þ J3s
@t @t2
dw0 ; dh and d/z to derive the equations of motion for the (22e)
FG microplate as follows: The natural boundary conditions are of the form:
! � �
1 @Xxz @Xyz
@Nx @Nxy 1 @ 2 Xxz @ 2 Xyz du0 : Nx nx þ Nxy ny þ þ ny (23a)
du0 : þ þ þ 2 @x @y
@x @y 2 @x@y @y2
� �
@ 2 u0 @ 3 w0 3
0 @ h
1 @Xxz @Xyz
¼ I0 2 − I1 − J 1 k 1 A (22a) dv0 : Nxy nx þ Ny ny − þ nx (23b)
@t @x@t 2 @x@t2 2 @x @y
� � � �
! @Mx @Mxy @Mxy @My @Mns
@Nxy @Ny 1 @ Xxz @ Xyz 2 2 dw0 : þ nx þ þ ny þ
dv0 : þ − þ @x @y @x @y @s
@x @y 2 @x2 @x@y
� � � �
@Xxy @Xx @Xy @Xxy
@ 2 v0 @ 3 w0 3
0 @ h þ − nx þ − ny
¼ I0 2 − I1 − J 1 k 2 B (22b) @x @y @y @y
@t @y@t2 @y@t2 !
1 @ 2 Yxz @ 2 Yyz
þ ðk1 A0 − k2 B0 Þ ny þ nx (23c)
@ 2 Mx @ 2 Mxy @ 2 My 2 @x@y @x@y
dw0 : þ 2 þ
@x2 @x@y @y2 � � � �
! 0 @Sxy 0 @Sxy
1 0
� @ Yxz @ Yyz 3 3 dh : k1 A Qxz − nx þ k2 B Qyz − ny
þ k1 A0 − k2 B þ @y @x
2 @x2 @y @x@y2 �
0 0 @Sns
! þ k1 A þ k2 B
@ 2 Xxy @ 2 Xxy @ 2 Xy @ 2 Xx @s
þ − þ − þq � �
@x2 @y2 @x@y @x@y
! ! 1 0 @Zx @Zz @Zxy
− k1 A − þ − Wxz nx
@ 2 w0 @ 3 u0 @ 3 v0 @ 4 w0 @ 4 w0 2 @x @x @x
¼ I0 2 þ I1 þ − I2 þ � �
@t @x@t 2 @y@t2 @x2 @t2 @y2 @t2 1 0 @Zy @Zz @Zxy
þ k2 B − þ − Wyz ny (23d)
! 2 @y @y @y
4 4
@ h @ h @ 2 /z
− J2 k1 A0 2 2 þ k2 B0 2 2 þ J1s � � ��
@x @t @y @t @t2 1 @Zy @Zxy
duz : Qxz þ þ − Wyz nx
(22c) 2 @y @x
� � ��
1 @Zx @Zxy
þ Qyz − þ − Wxz ny
@ 2 Sxy @Qxz 2 @y @y
dh : k1 Sx þ k2 Sy þ ðk1 A0 þ k2 B0 Þ − k1 A0 � �
@x@y @x 1 @d/z @d/z
! − Zy þ Zxy nx
2 2 2 @x @y
@Q yz 1 @ Z y @ Zxy @W yz � �
− k2 B0 þ k1 A0 þ − 1 @d/z @d/z
@y 2 @x@y @x2 @x þ Zx þ Zxy ny (23e)
! 2 @y @x
1 @ 2 Zx @ 2 Zxy @Wxz 1 2
0 Þ @ Zz
− k2 B 0 þ − þ ð k 2 B 0
− k 1 A @dw0
2 @x@y @y2 @y 2 @x@y : Mn (23f)
! @n
@ 3 u0 @ 3 v0
¼ þJ1 k1 A0 2
þ k2 B0 @dh
@x@t @y@t 2 : Sn (23g)
@n
8 A. TOUNSI ET AL.

where @ 3 u0 @ 3 u0 @ 3 v0
dw0 : B11 þ ð B12 þ 2B 66 Þ þ ð B12 þ 2B 66 Þ
� � @x3 @x@y2 @x2 @y
Mns ¼ ðMy − Mx Þnx ny þ Mxy n2x − n2y , Mn
@ 3 v0 @ 4 w0 @ 4 w0
þ B22 − D 11 − 2 ð D 12 þ 2D 66 Þ
¼ Mx n2x þ My n2y þ 2Mxy nx ny (24a) @y3 @x4 @x2 @y2
!
� � Bn @ 3 u0 @ 3 v0 @ 4 w0
− ðk1 A0 − k2 B0 Þr2 − − D 22
Sns ¼ Sxy n2x − n2y , Sn ¼ 2Sxy nx ny (24b) 4 @x@y2 @x2 @y @y4
!2
4
� @ 2 h Cn @ 2 / @ 2 /
@ @ @ @ 0 @ 0 @
− An r w0 − k1 Ds11 þ k2 Ds12 − þ
¼ nx þ ny , ¼ n x k1 A þny k2 B (24c) @x2 2 @y2 @x2
@n @x @y @s @y @x
@4h � @2h
where (nx ; ny ) denote the direction cosines of the outward − 2ðk1 A0 − 2k2 B0 ÞDs66 2 2
− k1 Ds12 þ k2 Ds22
@x @y @y2
unit normal to the boundary of the mid-plane, N; M; S; Q; !
Nz and X, Y, Z, W represent the stress and couple stress Cn @2 @2
− k1 A0 2 þ k2 B0 2 r2 h
resultants through the plate thickness respectively. The spe­ 2 @x @y
cific expressions of them are given in Appendix A. !
Putting Eq. (23) into Eq. (22), the motion equations are Fn 0 0 2 @4 @2h @2h
þ ðk1 A − k2 B Þ þ
obtained in terms of generalized displacements 4 @x2 @y2 @x2 @y2
(u0 , v0 , w0 , h, /z ) as !
4
En 0 0 2 @ h @2/ @2/
− ðk1 A − k2 B Þ þ La þ þq
@ 2 u0 @ 2 u0 @ 2 v0 @ 3 w0 4 @x2 @y2 @x2 @y2
du0 : A11 þ A66 þ ð A12 þ A66 Þ − B11 ! !
@x2 @y2 @x@y @x3
@ 2 w0 @ 3 u0 @ 3 v0 @ 4 w0 @ 4 w0
@ 3 w0 � @h ¼ I0 2 þ I1 þ − I2 þ
− ðB12 þ 2B66 Þ 2
− k1 Bs11 þ k2 Bs12 @t @x@t 2 @y@t2 @x2 @t2 @y2 @t2
@x@y @x !
4 4
@3h @/ 0 @ h 0 @ h @ 2 /z
− Bs66 ðk1 A0 þ k2 B0 Þ þL z − J2 k1 A 2 2 þ k2 B 2 2 þ J1s
@x@y2 @x @x @t @y @t @t2
" ! !#
1 @ 2 @ 2 u0 @ 2 u0 @2 @ 2 v0 @ 2 v0 (25c)
− An þ 2 − þ 2
4 @y2 @x2 @y @x@y @x2 @y
� @u0 � @v0
" ! # dh : k1 Bs11 þ k2 Bs12 þ k1 Bs12 þ k2 Bs22
1 @ 3
@ 2
h @ 2
h @ 3
h @x @y
þ ðk1 A0 − k2 B0 Þ Bn þ − Dn !
4 @x@y2 @x2 @y2 @x@y2 3
@ u0 3
@ v0
þ Bs66 ðk1 A0 þ k2 B0 Þ 2
þ 2
@ 2 u0 @ 3 w0 3
0 @ h
@x@y @x @y
¼ I0 − I1 − J1 k 1 A !
@t2 @x@t2 @x@t 2
Dn 0 0 @ 3 v0 @ 3 u0
(25a) þ ðk2 B − k1 A Þ −
4 @x2 @y @x@y2

@ 2 v0 @ 2 v0 @ 2 u0 @ 3 w0 � @ 2 w0 � @ 2 w0
dv0 : A22 þ A66 2 þ ðA12 þ A66 Þ − B22 − k1 Ds11 þ k2 Ds12 2
− k1 Ds12 þ k2 Ds22
@y 2 @x @x@y @y3 @x @y2
@ 3 w0 � @h @ 4 w0
− ðB12 þ 2B66 Þ − k1 Bs12 þ k2 Bs22 − 2Ds66 ðk1 A0 þ k2 B0 Þ
2
@x @y @y @x2 @y2
" #
@3h @/ Cn @ 2
w @ 2
w
− Bs66 ðk1 A0
þ k2 B þL z
0Þ
− k1 A0 r 2
0
þ k2 B0 r2
0
@x2 @y @y 2 @x2 @y2
" ! !# �
1 @ 2 @ 2 v0 @ 2 v0 @2 @ 2 u0 @ 2 u0 − H11s 2
k1 þ 2H12 s s 2
k1 k2 þ H22 k2 h
− An þ 2 − þ 2
4 @x2 @y2 @x @x@y @x2 @y
s ð @4h
2
" ! # − H66 k1 A0 þ k2 B0 Þ þ Rðk1 þ k2 Þ/z
1 @ 3
@ 2
h @ 2
h @ 3
h @x2 @y2
− ðk1 A0 − k2 B0 Þ Bn 2 þ − Dn 2 � �
4 @x @y @x2 @y2 @x @y 0 s @ 2 /z 2
0@ h
− k1 A A55 − k1 A 2
@ 2 u0 @ 3 w0 3 @x2 @x
0 @ h !
¼ I0 − I1 − J1 k 1 A 2 2
@t2 @x@t2 @x@t 2 0 s @ /z 0@ h
− k2 B A44 − k2 B 2
(25b) @y2 @y
 MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 9

� �
En 0 0
4
2 @ w0 Hn 0 0 @ 2 h @ 2 /z Ny ðx, 0Þ ¼ Ny ðx, bÞ ¼ Nx ð0, yÞ ¼ Nx ða, yÞ ¼ 0 (26c)
− ðk 1 A − k 2 B Þ þ k1 A k1 A 2 þ
4 @x2 @y2 4 @x @x2
! My ðx, 0Þ ¼ My ðx, bÞ ¼ Mx ð0, yÞ ¼ Mx ða, yÞ ¼ 0 (26d)
Hn @ 2 h @ 2 /z
þ k2 B0 k2 B0 2 þ
4 @y @y2
" !# hðx, 0Þ ¼ hðx, bÞ ¼ hð0, yÞ ¼ hða, yÞ ¼ 0 (26e)
� �
2 2 2 2
Gn 2 @ h @ / z @ h @ / z
− r k1 A0 k1 A 0 2 þ þ k 2 B 0 k2 B 0 2 þ /z ðx, 0Þ ¼ /z ðx, bÞ ¼ /z ð0, yÞ ¼ /z ða, yÞ ¼ 0 (26f)
4 @x @x2 @y @y2
Gn @4h The displacement and rotation fields involved to meet
þ ðk1 A0 − k2 B0 Þð3k2 B0 − 2k1 A0 Þ 2 2 the simply supported boundary conditions are defined in
4 @x @y
! Eq. (27).
3 3 8 9
@ u0 @ v0 8 9
¼ J1 k1 A0 2
þ k2 B0 > u ð x, y, t Þ > >
> U eixt
cos ð ~
mx Þ sin ð ~
n y Þ >
>
@x@t @y@t2 >
>
0 >
> >
>
mn >
>
! > >
< v0 ðx, y, tÞ = X 1 X > ixt
1 < Vmn e sin ðmx ~ Þ cos ðn ~ yÞ > =
@ 4
w0 @ 4
w0 w0 ðx, y, t Þ ¼ Wmn eixt sin ðmx
~ Þ sin ðn ~ yÞ (27)
− J2 k1 A0 2 2 þ k2 B0 2 2 >
>
> hðx, y, t Þ >
> m¼1 n¼1 >
> >
> Hmn eixt sin ðmx
~ Þ sin ðn ~ yÞ >
>
>
@x @t @y @t >
: >
; >
> >
>
! /z ðx, y, tÞ : u e sin ðmx
ixt ~ Þ sin ðn ~ yÞ ;
mn
2 0 @4h 2 0 @4h
− K2 k1 A 2 2 2 þ k2 B 2 2 2 where Umn , Vmn , Wmn , Hmn and umn represent Fourier coeffi­
@x @t @y @t
cients and can be found for each pair of m and n, and x is
(25d) theffiffiffiffiffifrequency ~ ¼ np=b; m
of vibration, n ~ ¼ mp=a; and i ¼
p ffi
! −1: The displacement variables do not depend on the
� �
@ 2
/ z @ 2
h @ 2
/ @ 2
h time variable.
d/z : As55 − k1 A0 2 þ As44 − k2 B0 2 − Ra /z
@x2 @x @y2 @y
� � ! 3.1. Static bending
@u0 @v0 @ 2 w0 @ 2 w0
−L þ þ La þ þ Rðk1 þ k2 Þh
@x @y @x2 @y2 In the case of the static bending behavior, the transverse
! load (q) can be found as
Cn @ 4 w0 @ 4 w0 @ 4 w0 X
1 X
1
− þ þ2 2 2
2 @x4 @y4 @x @y qðx, yÞ ¼ Qmn sin ðmx
~ Þsin ðn
~ xÞ (28)
" ! !# m¼1 n¼1
2 2
Gn 2 0 @ h 0@ h @ 2 /z @ 2 /z where
− r k1 A þ k2 B 2 þ þ
4 @x2 @y @y2 @x2 ða ðb
! 4
2 2 Qmn ¼ qðx, yÞsin ðmx
~ Þsin ðn
~ xÞdxdy (29a)
Hn 0@ h 0@ h @ 2 /z @ 2 /z ab 0 0
þ k1 A 2 þ k2 B 2 þ þ
4 @x @y @y2 @x2
Qmn ¼ q0 , for sinusoidal distributed load, (29b)
@ 2 w0 @ 2 /z
¼ J1s 2
þ K2s
@t @t2 16q0
(25e) Qmn ¼ , for uniform distributed load, (29c)
mnp2
Putting Eqs. (27) and (28) into Eq. (25) results in the
3. Exact solution for simply supported FG plate subsequent system of algebraic equations
2 38 9 8 9
s11 s12 s13 s14 s15 > > Umn >> >
> 0 > >
The following section focuses on analyzing rectangular FG 6 s12 s22 s23 s24 s25 7> > >
> > > >
>
microplate with simply supported boundary conditions and 6 7< Vmn = < 0 =
6 s13 s23 s33 s34 s35 7 Wmn ¼ Qmn (30)
with specific dimensions (length represented as “a” width 6 7
4 s14 s24 s34 s44 s45 5> >
> Hmn >
> >
> > > 0 >
>
>
denoted as “b” and thickness represented as “h”). The >
: >
; : > >
;
s15 s25 s35 s45 s55 umn 0
Navier method will be utilized to find analytical solutions.
This involves expanding the displacement functions where
(u0 , v0 , w0 , h, /z ) and external load q using a double trigono­ An 2 2
metric series. The present model is subject to simply sup­ ~ 2 þ A66 n
s11 ¼ A11 m ~2 þ ~ ðm
n ~ þn ~ 2Þ
4
ported boundary conditions as follows: An
s12 ¼ ðA12 þ A66 Þm ~ − m~
~ n ~2 þn
~ n ðm ~ 2Þ
u0 ðx, 0Þ ¼ u0 ðx, bÞ ¼ v0 ð0, yÞ ¼ v0 ða, yÞ ¼ 0 (26a) 4
Bn
s13 ~ n 2 ðm
¼ ðk2 B0 − k1 A0 Þ m~ ~2 þn ~ 2Þ
4
w0 ðx, 0Þ ¼ w0 ðx, bÞ ¼ w0 ð0, yÞ ¼ w0 ða, yÞ ¼ 0 (26b) ~ n 2 − B11 m
− ðB12 þ 2B66 Þm~ ~3
10 A. TOUNSI ET AL.

�
s14 ¼ Bs11 k1 þ Bs12 k2 m~ − Bs66 ðk1 A0 þ k2 B0 Þm~
~ n2 where mij represent the parts of the generalized mass matrix
Dn and the specific terms are also provided by
− ~ n2
ðk1 A0 − k2 B0 Þm~
4 m11 ¼ I0 , m12 ¼ 0, m13 ¼ −I1 m, ~ m14 ¼ −J1 k1 A0 , m ~ 15 ¼ 0
0
An 2 2 m22 ¼ I0 , m23 ¼ −I1 n~ , m24 ¼ −J1 k2 B n~ , m25 ¼ 0 �
s15 ¼ −L m
~ ~ 2 þ A66 m
s22 ¼ A22 n ~2 þ ~ ðm
m ~2Þ
~ þn m33 ¼ I0 þ I2 ðm ~2 þn ~ 2 Þ, m34 �¼ J2 k1 A0 m
~ 2 þ k 2 B0 n
~ 2 , m35 ¼ J1s
4
m44 ¼ J3 k21 A0 2m~ 2 þ k22 B0 2~
n 2 , m45 ¼ 0, m55 ¼ J3s
~ 3 − ð2B66 þ B12 Þm
s23 ¼ −B22 n ~ 2n
~
(33)
Bn
þ ðk1 A0 − k2 B0 Þm ~ 2n
~ ðm~2 þn ~ 2Þ
4 4. Discussion of numerical results
�
s24 ¼ Bs12 k1 þ Bs22 k2 n~ − Bs66 ðk1 A0 þ k2 B0 Þm
~ 2n
~
This section conclusively verifies the accurateness of the pre­
Dn sented plate theory for the free vibration and static bending
þ ~ 2n
ðk1 A0 − k2 B0 Þm ~
4 responses of simply supported imperfect FG microplates by
s25 ¼ −L n
~ comparing the analytical solution with other available results
~ 4 þ D22 n
s33 ¼ D11 m ~ 4 þ 2ð2D66 þ D12 Þm
~ 2n
~2 in the literature.
The material combination includes aluminum and alu­
2 Fn
~2 þn
þ An ðm ~ 2Þ þ ðk1 A0 − k2 B0 Þ2 m~ 2n
~ 2 ðm
~2 þn ~ 2Þ mina, each with specific material properties.
4
�
s34 ¼ 2Ds66 ðk1 A0 þ k2 B0 Þm
~ 2n ~ 2 − k2 Ds12 m
~ 2 þ Ds22 n ~2 � Ceramic (alumina, Al2O3):qc ¼ 3800kg=m3 ;
�
− k1 Ds11 m~ 2 þ Ds12 n
~2 Ec ¼ 380 GPa, � c ¼ 0:3
� � Metal (aluminum, Al): qm ¼ 2702kg=m3 ;
Cn
þ ~ 2 þ k2 B0 n
k1 A0 m ~ 2 ðm~2 þn ~ 2Þ Em ¼ 70 GPa, � m ¼ 0:3
2
En 2 2 2 In a study by Lam et al. (2003) [70], the material length
þ ðk1 A0 − k2 B0 Þ m ~ n ~
4 scale factor for a homogeneous isotropic epoxy microbeam
Cn 2 2 was determined through an experiment as l ¼ 17:6lm:
s35 ¼ La ðm
~2 þn ~ 2 Þ þ ðm ~ þn ~ 2Þ
2 However, no relevant experimental data for FGMs exists in
s 2 s 2 s s ð 2 2 2
s44 ¼ H11 k1 þ H22 k2 þ 2H12 k1 k2 þ H66 k1 A0 − k2 B0 Þ m~ n
~ the open literature. As a result, the present study adopts the
material length scale parameter as l ¼ 15lm based on the
þ As55 k21 A02 m
~ 2 þ As44 k22 B02 n
~2
� � works of Ke et al. (2012) and Şimşek and Reddy (2013)
2 02 2 02 3 [71,72], and Zhang et al. (2015a, 2015b) [73,74].
þ Gn k1 A þ k2 B − k1 A k2 B m 0 0
~ 2n ~2
2 For convenience, the following non-dimensional forms
1 � Hn 2 02 2 are considered:
þ Gn k21 A0 2m ~ 4 þ k22 B02 n
~4 − kB n ~ � � � �
4 4 2 10Ec h3 a b 10Ec h3 b
s45 ¼ −ðk1 þ k2 ÞR − As55 k1 A0 m ~ 2 − As44 k2 B0 n
~2 w¼ w , ,z , w ¼ w x, , 0 ,
q0 a4 � 2 2 � q0 a4 �2 �
� 3 3
Gn Hn 100c h a b 10Ec h a b
þ k1 A0 m~ 2 þ k2 B0 n
~ 2 ðm ~2 þn ~ 2Þ − ~2
k2 B0 n w^ ¼ 4
w , , 0 , /z ¼ 4
/z , ,
4 4 12q0 a 2 2 q0 a 2 2
Gn 2 2 Hn 2 rxx h sxz h rzz h
s55 ¼ Ra þ As55 m ~ 2 þ As44 n
~ 2 þ ðm ~ þn ~ 2Þ − ~
n r xx ¼ , s xz ¼ , r zz ¼ ,
4 4 q0 a q0 a q0 a
(31) xa2 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi
x¼ qc =Ec , x ^ ¼ xh qc =Ec
h

3.2. Free vibration problem 4.1. Validation

Similarly, the generalized eigenvalue equation can be derived To ensure the model’s accuracy, we compare the predicted
for the free vibration problem as follows: results of FG porous and non-porous results microplates

82 3 2 398 9 8 9
>
> s11 s12 s13 s14 s15 m11 m12 m13 m14 m15 > >>
> Umn >
> >0>
> >
>
> >>
>> >
> >
> > >
<6
6 s12 s22 s23 s24 s25 7
7
6 m12
6 m22 m23 m24 m25 7
7=< Vmn = < 0 =
6 s13 s35 7 26 7
>6 s23 s33 s34 7 − x 6 m13 m23 m33 m34 m35 7
>>
Wmn ¼ 0
> > >
(32)
>
>4 s14 s24 s34 s44 s45 5 4 m14 m24 m34 m44 m45 5>>>
> Hmn >
> >
>0> >
>
: >>
;: >
; >
: > ;
s15 s25 s35 s45 s55 m15 m25 m35 m45 m55 umn 0
 MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 11

Table 1. Comparisons of dimensionless deflections of rectangular non-porous plates under a sinusoidal load.
a/h ¼ 2
b/a ¼ 1 b/a ¼ 2 b/a ¼ 3
K 3D #
Quasi-3D�� HSDT ##
Present 3D #
Quasi-3D�� HSDT ##
Present 3D#
Quasi-3D�� HSDT## Present
0.1 0.5769 0.5786 0.6362 0.5616 1.1944 1.1947 1.2775 1.1721 1.443 1.4427 1.534 1.4158
0.3 0.5247 0.5231 0.5751 0.5222 1.0859 1.0801 1.1553 1.0793 1.3116 1.3043 1.3873 1.3035
0.5 0.4766 0.4724 0.5194 0.47162 0.9864 0.9756 1.0441 0.9749 1.1913 1.1781 1.254 1.1774
0.7 0.4324 0.4262 0.4687 0.4255 0.8952 0.8804 0.9431 0.8798 1.0812 1.0633 1.1329 1.0626
1 0.3727 0.3646 0.4011 0.364 0.7727 0.7535 0.8086 0.753 0.9334 0.9102 0.9719 0.9096
1.5 0.289 0.2797 0.3079 0.2793 0.6017 0.5789 0.6238 0.5785 0.7275 0.6995 0.7506 0.6991
a/h ¼ 4
b/a ¼ 1 b/a ¼ 2 b/a ¼ 3
K 3D# Quasi-3D�� HSDT## Present 3D# Quasi-3D�� HSDT## Present 3D# Quasi-3D�� HSDT## Present
0.1 0.349 0.3486 0.3602 0.3476 0.8153 0.8144 0.8325 0.8131 1.0134 1.0123 1.0325 1.0108
0.3 0.3168 0.3152 0.3259 0.3152 0.7395 0.7364 0.7534 0.7365 0.9190 0.9154 0.9345 0.9155
0.5 0.2875 0.2848 0.2949 0.2848 0.6708 0.6654 0.6819 0.6655 0.8335 0.8271 0.8459 0.8272
0.7 0.2608 0.2571 0.2668 0.2571 0.6085 0.6009 0.6173 0.6009 0.7561 0.7470 0.7659 0.7470
1 0.2253 0.2203 0.2295 0.2203 0.5257 0.515 0.5319 0.5151 0.6533 0.6403 0.6601 0.6404
1.5 0.1805 0.1697 0.1785 0.1697 0.412 0.3973 0.415 0.3973 0.5121 0.4941 0.5154 0.4941
#
Zenkour (2007) [75].
��Belabed et al. (2014) [76].
##
Thai and Kim (2013) [69].

� ð0Þ of isotropic FG porous plates (a/h ¼ 10).

Table 2. Comparisons of dimensionless deflections w
a/h
4 10 100
Quasi-3D [77] Present Quasi-3D [77] Present Quasi-3D [77] Present
K a¼0 a ¼ 0:1 a¼0 a ¼ 0:1 a¼0 a ¼ 0:1 a¼0 a ¼ 0:1 a¼0 a ¼ 0:1 a¼0 a ¼ 0:1
0.1 0.2796 0.3492 0.2798 0.3496 0.7030 0.8779 0.7037 0.8789 0.8867 1.1074 0.8876 1.1086
0.3 0.2528 0.3157 0.2530 0.3160 0.6357 0.7939 0.6364 0.7948 0.8019 1.0014 0.8027 1.0025
0.5 0.2285 0.2854 0.2287 0.2856 0.5746 0.7176 0.5751 0.7183 0.7248 0.9051 0.7255 0.9060
0.7 0.2064 0.2578 0.2066 0.2580 0.5190 0.6482 0.5195 0.6488 0.6547 0.8176 0.6553 0.8184
1 0.1770 0.2211 0.1772 0.2213 0.4452 0.5560 0.4456 0.5566 0.5616 0.7014 0.5622 0.7021
1.5 0.1368 0.1709 0.1369 0.1710 0.3442 0.4298 0.3445 0.4302 0.4342 0.5422 0.4346 0.5427

� ð0Þ of rectangular FG non-porous plates under uniform load (b=a ¼ 3 and a=h ¼ 10).
Table 3. Comparisons of dimensionless deflections w
K
Model 0 0.5 1 2 5
SSDT [78] 1.2583 1.9344 2.5133 3.2267 3.8517
Quasi-3D [79] 1.2544 1.9045 2.4354 3.0816 3.6972
Present 1.2543 1.9044 2.4354 3.0816 3.6971

with only five unknowns, as stated in the previous studies. with the quasi-3D solution from Zenkour (2018) [77]. The
Table 1 displays the dimensionless deflections ðw ð0ÞÞ of FG inclusion of the porosity factor a is implied in Table 2. The
non-porous plate with simply supported boundary conditions present non-porous results (a ¼ 0) are almost more accurate
and under a sinusoidal load and varying power law index val­ than those calculated by the quasi-3D theory of Zenkour
ues. The exponential law EðzÞ ¼ Ec eðz=hþ1=2Þk is utilized to (2018) [77]. The current quasi-3D theory generally provides
determine the effective material properties of the plates comparable results, including the porosity factor (a ¼ 0.1).
decisively [77]. The present non-porous results (a ¼ 0) are In Table 3, the dimensionless transverse displacement w ^ ð0Þ
then compared with the exact 3D solutions by Zenkour (2007) can be found. The non-porous results (a ¼ 0) obtained have
[75], the quasi-3D solutions by Belabed et al. (2014) [76], and been compared with the quasi-3D solutions presented by Lee
the HSDT solutions by Thai and Kim (2013) [69]. et al. (2015) [79] and with those found by Zenkour (2009) [78]
Table 2 contains the dimensionless transverse deflection for rectangular FG macroscopic plates (Al/Al2O3) under a uni­
w ð0Þ of FG porous plates with simply supported boundary form load. In this case, the effectual properties have been
conditions and under a sinusoidal load for several values of determined through power law distribution. The results from
material parameter k, side-to-thickness ratio a/h, and aspect the two tables indicate that the simplified quasi-3D theory with
ratio b/a. The exponential law EðzÞ ¼ Ec eðz=hþ1=2Þk−2a=ð1−aÞ is 5 unknowns accurately predicts the results and aligns well with
utilized to determine the effective material properties of the the predictions made using quasi-3D solutions and 3D solu­
plates decisively [77]. The current calculations are compared tions. It’s important to note that higher-order shear
12 A. TOUNSI ET AL.

Table 4. Comparisons of dimensionless deflections of rectangular FG non-porous microplates under uniform load.
a/h ¼ 5
h/l ¼ 1 h/l ¼ 5 h/l ¼ 1
### ### ### ### ###
K Kirchhoff model Mindlin model Present Kirchhoff model Mindlin model Present Kirchhoff model Mindlin model### Present
0 0.0884 0.123 0.0963 0.3631 0.4479 0.4381 0.4171 0.5147 0.5145
1 0.1686 0.2279 0.182 0.8192 0.9685 0.9505 0.9762 1.1536 1.1532
10 0.4762 0.671 0.5007 1.8409 2.3127 2.3444 2.0905 2.6273 2.8334
a/h ¼ 10
h/l ¼ 1 h/l ¼ 5 h/l ¼ 1
### ### ### ### ###
K Kirchhoff model Mindlin model Present Kirchhoff model Mindlin model Present Kirchhoff model Mindlin model### Present
0 0.0884 0.0972 0.0904 0.3631 0.3844 0.382 0.4171 0.4415 0.4415
1 0.1686 0.1838 0.172 0.8192 0.8567 0.8523 0.9762 1.0205 1.0205
10 0.4762 0.5263 0.4824 1.8409 1.9593 1.9683 2.0905 2.2247 2.2767
###
Reddy and Kim (2012) [80].

2 pffiffiffiffiffiffiffiffiffiffiffiffiffi
Table 5. Dimensionless fundamental frequencies x ¼ x ah qm =Em of square FG porous plates (k ¼ 1).
a=h Model a¼0 a ¼ 0:2 a ¼ 0:5
5 Shahsavari et al. (2018) [49] 8.151 7.641 5.378
Present 8.1512 7.6408 5.3780
10 Shahsavari et al. (2018) [49] 8.818 8.203 5.659
Present 8.8178 8.2027 5.6586
20 Shahsavari et al. (2018) [49] 9.020 8.370 5.738
Present 9.0196 8.3700 5.7382

Table 6. Dimensionless frequencies x

^ comparison for rectangular FG non-porous macroscopic plates.
a/h ¼ 5
Mode (m, n)
1 (1, 1) 2 (1, 2) 3 (2, 2)
Model
k FSDT� Quasi-3D�� Present FSDT� Quasi-3D�� Present FSDT� Quasi-3D�� Present
0 0.2112 0.2121 0.2122 0.4618 0.4659 0.4661 0.6676 0.6757 0.676
0.5 0.1805 0.1819 0.1825 0.3978 0.4041 0.4042 0.5779 0.589 0.5893
1 0.1631 0.164 0.1659 0.3604 0.3676 0.3677 0.5245 0.5362 0.5365
4 0.1397 0.1383 0.1409 0.3049 0.3047 0.3047 0.4405 0.4381 0.4381
10 0.1324 0.1306 0.1318 0.2856 0.2811 0.2812 0.4097 0.4008 0.4009
a /h ¼ 10
Mode (m, n)
1 (1, 1) 2 (1, 2) 3 (2, 2)
Model
k FSDT� Quasi-3D�� Quasi-3D��� Present FSDT� Quasi-3D�� Present FSDT� Quasi-3D�� Present
0 0.0577 0.0578 0.0577 0.0578 0.1376 0.1381 0.1381 0.2112 0.2121 0.2121
0.5 0.049 0.0494 0.0492 0.0494 0.1173 0.1184 0.1184 0.1805 0.1825 0.1825
1 0.0442 0.0449 0.0448 0.0449 0.1059 0.1077 0.1076 0.1631 0.1659 0.1659
4 0.0382 0.0389 0.0389 0.0389 0.0911 0.0923 0.0923 0.1397 0.1409 0.1409
10 0.0366 0.0368 0.0368 0.0368 0.0867 0.0868 0.0868 0.1324 0.1318 0.1318
a /h ¼ 20
Mode (m, n)
1 (1, 1)
Model
K FSDT� Quasi-3D�� Present
0 0.0148 0.0148 0.0148
0.5 0.0125 0.0126 0.0126
1 0.0113 0.0115 0.0115
4 0.0098 0.01 0.01
10 0.0094 0.0095 0.0095
�Hosseini-Hashemi et al. (2011) [83].
��Belabed et al. (2014) [76].
���(Jian Lei et al. 2015).
 MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 13

Table 7. Comparisons of dimensionless static deflections (�

w ð0Þ) of square FG porous microplates under sinusoidal load with simply supported conditions.
k
a /h h /l Method 0 0.5 1 5 10
5 1 2D��� 0.0569 0.0801 0.0989 0.1720 0.2077
Quasi-3D��� 0.0601 0.0838 0.1031 0.1819 0.2215
Present a¼0 0.0601 0.0838 0.1031 0.1819 0.2215
a ¼ 0:1 0.0639 0.0914 0.1151 0.2237 0.2856
a ¼ 0:2 0.0681 0.1006 0.1302 0.2934 0.4066
2 2D��� 0.1432 0.2068 0.2591 0.4359 0.5113
Quasi-3D��� 0.1564 0.2233 0.2781 0.4737 0.5623
Present a¼0 0.1564 0.2233 0.2781 0.4737 0.5623
a ¼ 0:1 0.1662 0.2444 0.3131 0.5919 0.7284
a ¼ 0:2 0.1774 0.2701 0.3591 0.8093 1.0616
4 2D��� 0.2316 0.3424 0.4363 0.7213 0.8275
Quasi-3D��� 0.2609 0.3823 0.4827 0.7977 0.9245
Present a¼0 0.2609 0.3823 0.4827 0.7977 0.9245
a ¼ 0:1 0.2774 0.4202 0.5495 1.0167 1.2106
a ¼ 0:2 0.2960 0.4670 0.6406 1.4622 1.8305
8 2D��� 0.2740 0.4096 0.5265 0.8678 0.9865
Quasi-3D��� 0.3133 0.4651 0.5916 0.9652 1.1064
Present a¼0 0.3133 0.4651 0.5916 0.9652 1.1064
a ¼ 0:1 0.3330 0.5123 0.6773 1.2442 1.4608
a ¼ 0:2 0.3554 0.5712 0.7969 1.8421 2.2668
10 1 2D��� 0.0530 0.0746 0.0926 0.1633 0.1968
Quasi-3D��� 0.0556 0.0777 0.0960 0.1703 0.2066
Present a¼0 0.0556 0.0777 0.0960 0.1703 0.2066
a ¼ 0:1 0.0590 0.0848 0.1073 0.2106 0.2677
a ¼ 0:2 0.0630 0.0933 0.1217 0.2794 0.3852
2 2D��� 0.1283 0.1868 0.2355 0.3938 0.4577
Quasi-3D��� 0.1419 0.2041 0.2551 0.4289 0.5050
Present a¼0 0.1419 0.2041 0.2551 0.4289 0.5050
a ¼ 0:1 0.1508 0.2236 0.2880 0.5392 0.6554
a ¼ 0:2 0.1609 0.2475 0.3316 0.7485 0.9653
4 2D��� 0.1994 0.2992 0.3837 0.6126 0.6911
Quasi-3D��� 0.2320 0.3439 0.4354 0.6934 0.7933
Present a¼0 0.2320 0.3439 0.4354 0.6934 0.7933
a ¼ 0:1 0.2466 0.3786 0.4973 0.8868 1.0335
a ¼ 0:2 0.2631 0.4217 0.5828 1.2949 1.5628
8 2D��� 0.2314 0.3523 0.4554 0.7132 0.7946
Quasi-3D��� 0.2757 0.4150 0.5288 0.8206 0.9265
Present a¼0 0.2757 0.4150 0.5288 0.8206 0.9265
a ¼ 0:1 0.2931 0.4579 0.6078 1.0586 1.2104
a ¼ 0:2 0.3128 0.5117 0.7190 1.5868 1.8575
���(Jian Lei et al. 2015).

deformation theories, such as those proposed by Thai and Kim The dimensionless fundamental frequency of both perfect
(2013) [69] and Zenkour (2009) [78], tend to overvalue the and imperfect FG plates is calculated and presented in Table
displacement of very thick FG plates. This overestimation is 5. Three thickness ratios (a/h) are considered: 5, 10 and 20.
likely due to the influence of normal deformation in these very The results are compared with those of Shahsavari et al.
thick E-FGM plates, a factor that is not accounted for in the (2018) [49], who used a quasi-3D approach. As can be
HSDT [69] and SSDT [78] models. observed in Table 5, there is a high degree of agreement
Table 4 presents an observation of the results achieved between the present results and those previously published
using size-dependent FG Kirchhoff and Mindlin plate mod­ (Table 6).
els [81], which depend on the improved couple stress the­ Table 7 presents the dimensionless frequencies, x of a
ory. The effectual properties are determined by employing rectangular FG microplate with effective material properties.
the power law distribution. The table shows significant dif­ These properties have been estimated based on the distribu­
ferences between the three size dependent plate models. tion of power law. The present results demonstrate excellent
This is because the Kirchhoff model ignores the deformation agreement with those obtained analytically, including quasi-
impacts of both normal and shear, the Mindlin model only 3D solutions [76] and FSDT solutions [78].
studies the former impact, and the developed quasi-3D
model takes into account both impacts.
The material property factors considered in this study are 4.2. Parametric analysis
based on Reddy (2011) [82] and Thai and Choi (2013) [81]:
In this section, we are conducting parametric studies to ana­
3 3 lyze the impact of the material length scale parameter,
Ec ¼ 14:14 GPa, qc ¼ 12:2 � 10 kg=m , � c ¼ 0:38,
power law index, porosity factor, plate thickness, and nor­
Em ¼ 1:44 GPa, qm ¼ 1:22 � 103 kg=m3 , � c ¼ 0:38: mal and shear deformation impacts on the free vibration
14 A. TOUNSI ET AL.

Table 8. Comparisons of dimensionless stress r

� x ða=2, b=2, h=2Þ of square FG porous microplates under sinusoidal load with simply supported conditions.
K
a /h h /l Method 0 0.5 1 5 10
5 1 2D��� 0.2321 0.2796 0.3162 0.5086 0.6655
Quasi-3D��� 0.1878 0.2286 0.2603 0.4059 0.5232
Present a¼0 0.1878 0.2286 0.2603 0.4059 0.5232
a ¼ 0:1 0.1878 0.2321 0.2676 0.4350 0.5843
a ¼ 0:2 0.1878 0.2362 0.2766 0.4688 0.6610
2 2D��� 0.5711 0.7132 0.8185 1.2314 1.5491
Quasi-3D��� 0.4888 0.6096 0.7022 1.0465 1.3087
Present a¼0 0.4888 0.6095 0.7022 1.0465 1.3087
a ¼ 0:1 0.4888 0.6210 0.7285 1.1379 1.4625
a ¼ 0:2 0.4888 0.6351 0.7634 1.2788 1.6861
4 2D��� 0.9086 1.1715 1.3660 1.9544 2.3797
Quasi-3D��� 0.8156 1.0445 1.2198 1.7444 2.1200
Present a¼0 0.8156 1.0445 1.2198 1.7444 2.1200
a ¼ 0:1 0.8156 1.0686 1.2794 1.9313 2.3819
a ¼ 0:2 0.8156 1.0990 1.3630 2.2809 2.8228
8 2D��� 1.0685 1.3974 1.6427 2.3083 2.7721
Quasi-3D��� 0.9793 1.2713 1.4953 2.1005 2.5197
Present a¼0 0.9793 1.2713 1.4953 2.1005 2.5197
a ¼ 0:1 0.9793 1.3034 1.5775 2.3495 2.8451
a ¼ 0:2 0.9793 1.3446 1.6960 2.8538 3.4376
10 1 2D��� 0.4836 0.5801 0.6541 1.0493 1.3774
Quasi-3D��� 0.3787 0.4549 0.5184 0.8310 1.0785
Present a¼0 0.3787 0.4549 0.5184 0.8310 1.0785
a ¼ 0:1 0.3787 0.4611 0.5320 0.8909 1.2080
a ¼ 0:2 0.3787 0.4686 0.5489 0.9585 1.3688
2 2D��� 1.1622 1.4453 1.6561 2.4956 3.1516
Quasi-3D��� 0.9672 1.1951 1.3775 2.0857 2.6225
Present a¼0 0.9672 1.1951 1.3775 2.0857 2.6225
a ¼ 0:1 0.9672 1.2166 1.4282 2.2714 2.9394
a ¼ 0:2 0.9672 1.2432 1.4958 2.5582 3.4039
4 2D��� 1.7954 2.3085 2.6893 3.8314 4.6805
Quasi-3D��� 1.5815 2.0146 2.3517 3.3596 4.0981
Present a¼0 1.5815 2.0146 2.3517 3.3596 4.0981
a ¼ 0:1 1.5815 2.0606 2.4671 3.7201 4.6021
a ¼ 0:2 1.5815 2.1188 2.6301 4.4062 5.4550
8 2D��� 2.0799 2.7149 3.1877 4.4327 5.3392
Quasi-3D��� 1.8799 2.4314 2.8567 3.9688 4.7746
Present a¼0 1.8799 2.4314 2.8567 3.9688 4.7746
a ¼ 0:1 1.8799 2.4928 3.0154 4.4314 5.3702
a ¼ 0:2 1.8799 2.5717 3.2453 5.3865 6.4451
���(Jian Lei et al. 2015).

and static bending characteristics of square FG porous with the increased porosity factor. Upon comparison with
microplates with simply supported boundary conditions and the 2D results, it is incontrovertibly evident that accounting
under a sinusoidal distributed load. The plates’ actual mater­ for the normal deformation effect leads to a clear decrease
ial properties are determined using a power law distribution. in x ð1, 1Þ and an undeniable increase in wð0Þ; revealing the
Tables 7–10 provide information on the dimensionless clear impact of the normal deformation influence and poros­
deflection wð0Þ; normal stresses r xx ; transverse shear stress ity factor on the microplate’s stiffness.
s xz , and frequencies x depend on the current quasi-3D Figure 2 presents the comparison the variation of the
model with only five unknowns and the degraded 2D model deflection versus the normalized microplate length of the
ðuz ¼ 0Þ with varying values of dimensionless thickness simply supported microplate predicted by the new model
(h=l), the width-to-thickness ratio (a=h), porosity factor ðaÞ and by the classical plate theory. Three different values of
and power law index (k). the volume fraction of porosity (i.e. a ¼ 0, 0:1, 0:2) are con­
The data from Tables 7 and 10 undeniably demonstrate sidered. It is observed from Figure 2 that the deflection val­
that reducing the plate thickness results in increased stiff­ ues of the microplate with a ¼ 0 are larger than those with
ness, x ð1, 1Þ and a decrease in wð0Þ: This unequivocally a ¼ 0:1 and 0:2, respectively, with the deflection decreasing
proves that the plate’s size effect is intrinsically linked to its with the increase of volume fraction of porosity. Figure 2
thickness, with the reduction in thickness clearly magnifying shows that a reduction in plate thickness leads to an
the size effect. Furthermore, an increase in the power law increase in stiffness and a reduction in dimensionless deflec­
index unequivocally leads to a decrease in the plate’s stiff­ tion. This indicates that the plate’s size directly impacts its
ness and x ð1, 1Þ; alongside an indisputable increase in w ð0Þ performance, especially when the plate is thinner.
due to the higher aluminum fraction. The present quasi-3D The through-the-thickness distributions of deflections w
theory generally provides comparable results by including are demonstrated in Figure 3a and b for thick (a ¼ 5h) FG
the porosity factor. This factor has a considerable effect on square microplates with various values of k and a. Figure 3a
deflection. The dimensionless deflections w ð0Þ are increasing shows that as the value of k increases, the deflection
 MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 15

Table 9. Comparisons of dimensionless stress �s xz ð0, b=2, 0Þ of square FG porous microplates under sinusoidal load with simply supported conditions.
K
a /h h /l Method 0 0.5 1 5 10
5 1 2D��� 0.0363 0.0386 0.0353 0.0186 0.0203
Quasi-3D��� 0.0427 0.0447 0.0409 0.0231 0.0256
Present (MCST) a¼0 0.0427 0.0447 0.0409 0.0231 0.0256
a ¼ 0:1 0.0427 0.0448 0.0404 0.0191 0.0212
a ¼ 0:2 0.0427 0.0449 0.0398 0.0131 0.0138
2 2D��� 0.1038 0.1073 0.1005 0.0649 0.0717
Quasi-3D��� 0.1110 0.1142 0.1072 0.0707 0.0782
Present (MCST) a¼0 0.1110 0.1142 0.1072 0.0707 0.0782
a ¼ 0:1 0.1110 0.1143 0.1062 0.0607 0.0488
a ¼ 0:2 0.1110 0.1143 0.1048 0.0447 0.0679
4 2D��� 0.1818 0.1859 0.1787 0.1333 0.1466
Quasi-3D��� 0.1850 0.1891 0.1820 0.1363 0.1501
Present (MCST) a¼0 0.1850 0.1891 0.1820 0.1363 0.1501
a ¼ 0:1 0.1850 0.1893 0.1811 0.1227 0.1376
a ¼ 0:2 0.1850 0.1896 0.1798 0.0984 0.1112
8 2D��� 0.2211 0.2261 0.2199 0.1737 0.1905
Quasi-3D��� 0.2221 0.2271 0.2210 0.1747 0.1917
Present (MCST) a¼0 0.2221 0.2275 0.2206 0.1610 0.1808
a ¼ 0:1 0.2221 0.2280 0.2201 0.1357 0.1563
a ¼ 0:2 0.2221 0.2257 0.2217 0.1996 0.2087
10 1 2D��� 0.0371 0.0395 0.0360 0.0190 0.0208
Quasi-3D��� 0.0451 0.0473 0.0432 0.0241 0.0267
Present (MCST) a¼0 0.0451 0.0473 0.0432 0.0241 0.0267
a ¼ 0:1 0.0451 0.0474 0.0427 0.0198 0.0219
a ¼ 0:2 0.0451 0.0475 0.0420 0.0135 0.0142
2 2D��� 0.1065 0.1098 0.1029 0.0668 0.0738
Quasi-3D��� 0.1150 0.1183 0.1111 0.0733 0.0812
Present (MCST) a¼0 0.1150 0.1183 0.1111 0.0733 0.0812
a ¼ 0:1 0.1150 0.1184 0.1100 0.0630 0.0705
a ¼ 0:2 0.1150 0.1185 0.1085 0.0462 0.0506
4 2D��� 0.1843 0.1884 0.1812 0.1358 0.1495
Quasi-3D��� 0.1881 0.1921 0.1849 0.1392 0.1533
Present (MCST) a¼0 0.1881 0.1921 0.1849 0.1392 0.1533
a ¼ 0:1 0.1881 0.1923 0.1840 0.1253 0.1408
a ¼ 0:2 0.1881 0.1925 0.1827 0.1007 0.1141
8 2D��� 0.2225 0.2274 0.2213 0.1752 0.1922
Quasi-3D��� 0.2236 0.2285 0.2223 0.1763 0.1934
Present (MCST) a¼0 0.2236 0.2285 0.2224 0.1763 0.1934
a ¼ 0:1 0.2236 0.2288 0.2220 0.1626 0.1826
a ¼ 0:2 0.2236 0.2293 0.2215 0.1372 0.1582
���(Jian Lei et al. 2015).

increases for a fixed value of a ¼ 0.1 when h/l ¼ 1. thickness (h/l) is small, but it becomes less important as the
Conversely, Figure 3b illustrates that as the value of a plate thickness increases. This suggests that the size effect is
increases, the deflection increases for a fixed value of k ¼ 1 meaningful when the plate thickness is at the micron scale,
when h/l ¼ 4. aligning with experimental observations. Also, the non-
Figure 4 illustrates the variation of the dimensionless dimensional deflection increases as the porosity factor
transverse deflection (w) of FG microplates with the mater­ increases.
ial property gradient index as predicted by MSGT. The com­ The variation of the dimensionless axial normal stress r xx
parison is made for two different values of the of the simply supported square FG porous microplates sub­
dimensionless length scale parameter against the deflection jected to a sinusoidal load is presented in Figure 6 and
curve predicted by the classical 3D and 2D plate theory. It is Table 8. The axial normal stresses are compressive at the
evident that increasing the value of h/l results in a clear upper side and tensile at the lower side of the microplate.
increase in the maximum deflection of FG microplates, It’s important to note that the r xx increases as the plate’s
ultimately converging toward the value predicted by classical thickness rises. Additionally, a rise in the power low index
theory. Furthermore, it is undeniable that the dimensionless leads to an increase in the r xx And when the porosity of the
length scale parameter plays a paramount role in the bend­ plate increases, the axial normal stresses of the FG micro­
ing behavior of FG microplates with higher values of mater­ plates decrease. When considering the normal deformation
ial property gradient index. effect, the present results show a decrease in the r xx com­
The effect of the length scale parameter l on dimension­ pared to the degradation 2D results.
less deflection w of the square FG microplate is illustrated The data presented in Table 9 and Figure 7 indicate that
in Figure 5. The data are generated by the current Quasi-3D the s xz , increases with the thickness of the plate. A compari­
model and the proposed 2D degradation model for two val­ son of these results with those derived from degradation in
ues of the porosity factor (a ¼ 0 and 0.2). The influence of 2D models reveals that the dimensionless shear stress exhib­
the length scale parameter is noticeable only when the plate its an upward trend when the effect of normal deformation
16 A. TOUNSI ET AL.

Table 10. Comparisons of dimensionless natural frequencies x

� (1, 1) of square FG porous microplates with simply supported conditions.
K
a /h h /l Method 0 0.5 1 5 10
5 1 2D��� 12.9565 11.4813 10.5983 8.4762 7.8468
Quasi-3D��� 12.4747 11.1097 10.2849 8.2027 7.5480
Present (MCST) a¼0 12.4756 11.1097 10.2861 8.2014 7.5476
a ¼ 0:1 12.6540 11.1795 10.2628 7.8328 7.0565
a ¼ 0:2 12.8659 11.2641 10.2290 7.2795 6.3088
2 2D��� 8.1759 7.1553 6.5542 5.3491 5.0233
Quasi-3D��� 7.7588 6.8328 6.2857 5.1083 4.7628
Present (MCST) a¼0 7.7614 6.8313 6.2862 5.1068 4.7618
a ¼ 0:1 7.8724 6.8614 6.2429 4.8404 4.4438
a ¼ 0:2 8.0042 6.8966 6.1802 4.4085 3.9307
4 2D��� 6.4393 5.5604 5.0553 4.1718 3.9619
Quasi-3D��� 6.0161 5.2220 4.7737 3.9453 3.7260
Present (MCST) a¼0 6.0146 5.2247 4.7748 3.9458 3.7249
a ¼ 0:1 6.1007 5.2364 4.7159 3.7047 3.4596
a ¼ 0:2 6.2028 5.2482 4.6300 3.2916 3.0090
8 2D��� 5.9205 5.0878 4.6015 3.8080 3.6331
Quasi-3D��� 5.4951 4.7406 4.3143 3.5943 3.4113
Present (MCST) a¼0 5.4905 4.7375 4.3139 3.5912 3.4090
a ¼ 0:1 5.5691 4.7427 4.2482 3.3534 3.1545
a ¼ 0:2 5.6623 4.7462 4.1518 2.9376 2.7109
10 1 2D��� 13.6329 12.0824 11.1415 8.9022 8.2367
Quasi-3D��� 13.2922 11.8250 10.9199 8.7034 8.0198
Present (MCST) a¼0 13.2901 11.8215 10.9270 8.7040 8.0195
a ¼ 0:1 13.4801 11.8929 10.8951 8.3027 7.4906
a ¼ 0:2 13.7059 11.9792 10.8494 7.7048 6.6889
2 2D��� 8.7651 7.6413 6.9954 5.7361 5.3998
Quasi-3D��� 8.3187 7.3006 6.7130 5.4839 5.1335
Present (MCST) a¼0 8.3192 7.2973 6.7063 5.4863 5.1321
a ¼ 0:1 8.4381 7.3258 6.6526 5.1918 4.7892
a ¼ 0:2 8.5795 7.3585 6.5754 4.7098 4.2280
4 2D��� 7.0338 6.0428 5.4762 4.5989 4.3928
Quasi-3D��� 6.5011 5.6087 5.1293 4.3069 4.0903
Present (MCST) a¼0 6.5064 5.6219 5.1335 4.3159 4.0960
a ¼ 0:1 6.5995 5.6304 5.0626 4.0494 3.8153
a ¼ 0:2 6.7100 5.6379 4.9598 3.5819 3.3247
8 2D��� 6.5258 5.5750 5.0315 4.2613 4.0974
Quasi-3D��� 5.9678 5.1146 4.6442 3.9723 3.7809
Present (MCST) a¼0 5.9677 5.1179 4.6579 3.9678 3.7905
a ¼ 0:1 6.0530 5.1194 4.5795 3.7068 3.5261
a ¼ 0:2 6.1544 5.1179 4.4654 3.2362 3.0504
���(Jian Lei et al. 2015).

� vs. Dimensional length x=a of square FG microplates with simply supported boundary conditions and
Figure 2. Variation of dimensionless center deflection w
with a=h ¼ 5, k ¼ 1:
 MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 17

� across the thickness of square FG microplates with simply supported boundary conditions.
Figure 3. Variation of dimensionless center deflection w

� of an FG square microplate.
Figure 4. Effect of the power-law index k on the dimensionless deflection w

� of an FG square microplate (a ¼ 10h, k ¼ 2).

Figure 5. Effect of the material length scale parameter l/h on the dimensionless deflection w
18 A. TOUNSI ET AL.

Figure 6. Variation of dimensionless in-plane normal stress r

� xx throughout thickness of square FG microplates with simply supported boundary conditions.

is described for. Additionally, the neutral plane of FG deformation effects when analyzing FG microplates under
microplates shifts upward, which is in contrast to the behav­ sinusoidal loading conditions.
ior observed in homogeneous plates. This suggests that while The chart in Figure 9 demonstrates how the material
the size effect of the plate is minimal, the influence of nor­ property gradient index affects the natural frequency (x) of
mal deformation on s xz is nearly negligible. FG porous microplates with two different values of the
The graph in Figure 8 illustrates the dimensionless nor­ dimensionless Length scale parameter. The chart compares
mal stress, r zz of square FG microplates with simply sup­ these values with the deflection curve predicted by the clas­
ported boundary conditions and under a sinusoidal load. sical 3D and 2D plate theory. It can be observed that as the
This in-depth analysis is based on both the current and deg­ value of h/l increases, the natural frequency decreases and
radation 2D plate models, and it encompasses various approaches the frequency predicted by the classical plate
width-to-thickness ratios (a/h ¼ 5, 8, 12) and dimensionless model. Additionally, the impact of size effect on the free
thickness values (h/l ¼ 1, 2, 4). The results clearly demon­ vibration response of FG porous microplates is more signifi­
strate that the dimensionless normal stress r zz , from the cant when the material property gradient index is lower.
current model differs significantly from that of the degrad­ Figure 10 illustrates how the length scale parameter
ation 2D model. The normal deformation impact causes the affects the dimensionless frequency (x) of a square FG
microplates to exhibit a flabby behavior in the thickness dir­ plate. The current model (l 6¼ 0) consistently predicts higher
ection, resulting in a decrease in the dimensionless normal frequency values compared to the classical model ðl ¼ 0Þ:
stress and a reduced range of variation. These findings high­ This difference is more pronounced for smaller plate thick­
light the critical importance of accounting for normal nesses but becomes less significant as the thickness
 MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 19

Figure 7. Variation of dimensionless transverse shear stress �s xz throughout thickness of square FG microplates with simply supported boundary conditions.

Figure 8. Difference of dimensionless transverse shear stress r

� zz throughout thickness of simply supported square FG microplates.
20 A. TOUNSI ET AL.

Figure 9. Effect of power-law index on the dimensionless natural frequency x

� of square FG porous microplates.

Figure 10. Effect of material length scale parameter h/l on the dimensionless frequency x
� of a square FG porous microplate (a ¼ 10h, k ¼ 2).

increases. This indicates that the size effect is important and engineering analysis for plate structures solved by analytical
only needs to be taken into account when the plate thick­ methods. It satisfies traction-free boundary conditions at
ness is at the micron scale. both the top and bottom surfaces of the plate without the
need for shear correction factors. The displacement field is
defined as a function of the non-linear variation of in-plane
5. Conclusions displacements through thickness. Additionally, the model
incorporates a hyperbolic distribution of transverse shear
The comparative investigation clearly demonstrates that the stress and accounts for the effects of a variable length scale
current theory is accurate and exhibits greater efficiency, as parameter and a porosity factor (a). The motion equations
it involves only five unknowns. This study introduces a size- are derived by applying Hamilton’s principle. Analytical sol­
dependent model for analyzing the vibration and bending of utions for free vibration and bending problems are obtained
FG porous microplates, which relies on the modified couple for plates with simply supported boundary conditions using
stress theory and quasi-3D Reddy plate theory. In contrast the Navier procedure.
to other high-order theories that require six or more The key findings of this research are outlined as follows:
unknown functions, the proposed model utilizes only five.
The advantages of this model include a reduction in numer­ 1. The proposed model uses five unknown functions, in
ical computation time and a simplification of the contrast to other theories that require six or more.
 MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 21

2. The numerical results demonstrate that incorporating 0[8] A.Z. Hajjaj, N. Jaber, S. Ilyas, F.K. Alfosail, and M.I. Younis,
small-scale effects enhances plate stiffness, which in Linear and nonlinear dynamics of micro and nano-resonators:
review of recent advances, Int. J. Non. Linear Mech., vol. 119,
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The authors extend their appreciation to Taif University, Saudi Arabia, [16] A.C. Eringen, and J.L. Wegner, Nonlocal continuum field theo­
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Disclosure statement gradient continuum theories, Eur. J. Mech. A Solids., vol. 22, no.
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No potential conflict of interest was reported by the author(s).
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@u0 @v0 @ 2 w0 @2h
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24 A. TOUNSI ET AL.

!
@u0 @v0 @ 2 w0 @ 2 w0 � 1 @ 2 v0 @ 2 u0
My ¼ B12 þ B22 − D12 2
− D22 − Ds12 k1 þ Ds22 k2 h Yxz ¼ Bn −
@x @y @x @y2 2 @x2 @x@y
þ L a /z " � �#
3
1 0 0 Þ @ w0 0 @h @/z
(A5) ð
þ Fn k1 A − k2 B þ En k2 B þ (A18)
2 @x2 @y @y @y
� �
@u0 @v0 @ 2 w0 @2h !
Mxy ¼ B66 þ − 2D66 − Ds66 ðk1 A0 þ k2 B0 Þ (A6)
@y @x @x@y @x@y 1 @ 2 v0 @ 2 u0
Yyz ¼ Bn −
2 @x@y @y2
@u0 @v0 @ 2 w0 @ 2 w0 � � �!
Sx ¼ Bs11 þ Bs12 − Ds11 − Ds12 s
− H11 s
k1 þ H12 k2 h 1 @ 3 w0 0 @h @/z
@x @y @x 2 @y2 þ Fn ðk1 A0 − k2 B0 Þ − En k 1 A þ (A19)
2 @x@y2 @x @x
þ R/z
(A7) !
@ 2 w0 2
0 @ h @ 2 /z
Zx ¼ 2Cn þ Gn k2 B þ (A20)
@x@y @x@y @x@y
@u0 @v0 @ 2 w0 @ 2 w0 �
Sy ¼ Bs12 þ Bs22 − Ds12 − Ds22 s
− H12 s
k1 þ H22 k2 h
@x @y @x2 @y2 !
þ R/z @ 2 w0 2
0 @ h @ 2 /z
Zy ¼ −2Cn − Gn k1 A þ (A21)
@x@y @x@y @x@y
(A8)

� �
@u0 @v0 @ 2 w0 @2h @2 h
Sxy ¼ Bs66 þ − 2Ds66 s ð
− H66 k1 A0 þ k2 B0 Þ (A9) Zz ¼ Gn ðk1 A0 − k2 B0 Þ (A22)
@y @x @x@y @x@y @x@y

� � !
@/z 0 @h @ 2 w0 @ 2 w0
Qxz ¼ As55 − k1 A (A10) Zxy ¼ Cn −
@x @x @y2 @x2
!
� � 1 @2 h @ 2 h @ 2 /z @ 2 /z
@/z @h þ Gn k2 B0 2 − k1 A0 þ − (A23)
Qyz ¼ As44 − k2 B0 (A11) 2 @y @x2 @y2 @x2
@y @y

! !
� �
a @u0 @v0 2
a @ w0 @ 2 w0 1 @ 2 v0 @ 2 u0
N z ¼ R /z þ L þ −L þ − Rðk1 þ k2 Þh Wxz ¼ Dn −
@x @y @x2 @y2 2 @x2 @x@y
" � �#
(A12) 1 @ 3 w0 @h @/z
0 0 0
þ En ðk1 A − k2 B Þ 2 þ Hn k2 B þ (A24)
! 2 @x @y @y @y
@ 2 w0 @2h @ 2 /z
Xx ¼ 2An þ Cn k2 B0 þ (A13) !
@x@y @x@y @x@y
1 @ 2 v0 @ 2 u0
Wyz ¼ Dn − 2
! 2 @x@y @y
@ 2 w0 2
@ 2 /z " � �#
0 @ h
Xy ¼ −2An − Cn k1 A þ ¼ −Xx (A14) 1 0 0 @ 3 w0 0 @h @/z
@x@y @x@y @x@y þ En ðk1 A − k2 B Þ − Hn k1 A þ (A25)
2 @x@y2 @x @x
!
@ 2 w0 @ 2 w0 where
Xxy ¼ An − 8 9
@y2 @x2 8 9 >
>
>
1−� > >
>
< A11 B11 D11 Bs11 Ds11 s
H11 = ð h2 <
! � � � =
A12 B12 D12 Bs12 Ds12 s
H12 ¼ kðzÞ 1, z, z2 , f , zf , f 2 1 dz
1 @2h @ 2 h @ 2 /z @ 2 /z : s ; >
> 1 − 2� >
>
þ Cn k2 B0 2 − k1 A0 þ − (A15) A66 B66 D66 Bs66 Ds66 H66 −h2 >
: >
;
2 @y @x2 @y2 @x2 2�
(A26)
!
1 @ 2 v0 @ 2 u0 8 9
Xxz ¼ An − 8 9 > 1 >
2 @x2 @x@y > L > >
> >
>
> >
< a = ð h2 >
<z >
=
" � �# L
1 @ 3 w0 @h @/z ¼ kðzÞ f g 0 dz (A27)
þ Bn ðk1 A0 − k2 B0 Þ 2 þ Dn k2 B0 þ (A16) >R >
> > −h2 >
> 1 − � >
>
2 @x @y @y @y : a; >
> >
>
R : g0 ;
�
!
1 @ 2 v0 @ 2 u0 ðh � �
2
Xyz ¼ An − ðAn , Bn , Cn , Dn , En , Fn , Gn , Hn Þ ¼ 1, f , g, g 0 , fg 0 , f 2 , g 2 , g 0 2 lðzÞl2 dz
2 @x@y @y2
−h2
� �!
1 @ 3 w0 0 @h @/z (A28)
þ Bn ðk1 A0 − k2 B0 Þ − Dn k1 A þ (A17)
2 @x@y2 @x @x