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\DOI{10.5802/crmeca.353}
\datereceived{2026-01-13}
\daterevised{2026-02-26}
\dateaccepted{2026-02-26}
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\dateposted{2026-04-15}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{An analytical method for nonlinear thermo-mechanical buckling
evaluation of functionally graded circular microplates}

\alttitle{M\'{e}thode analytique pour l'\'{e}valuation du flambement
thermo-m\'{e}canique non lin\'{e}aire des microplaques circulaires
\`{a} gradient de propri\'{e}t\'{e}s}

\author{\firstname{Duc} \lastname{Nguyen Minh}\CDRorcid{0009-0005-7056-528X}}
\address{Department of Solid Mechanics, Faculty of Mechanical
Engineering, Le Quy Don Technical University, 236 Hoang Quoc Viet
Street, Nghia Do Ward, Hanoi, 100000, Vietnam}

\author{\firstname{Thom} \lastname{Do Van}\CDRorcid{0000-0002-8570-0625}}
\addressSameAs{1}{Department of Solid Mechanics, Faculty of Mechanical
Engineering, Le Quy Don Technical University, 236 Hoang Quoc Viet
Street, Nghia Do Ward, Hanoi, 100000, Vietnam}

\author{\firstname{Minh} \lastname{Phung Van}\CDRorcid{0000-0002-4457-2479}\IsCorresp}
\addressSameAs{1}{Department of Solid Mechanics, Faculty of Mechanical
Engineering, Le Quy Don Technical University, 236 Hoang Quoc Viet
Street, Nghia Do Ward, Hanoi, 100000, Vietnam}
\email{minhpv@lqdtu.edu.vn}
\email[M. Phung Van]{minhpv.mta@gmail.com}

\shortrunauthors

\keywords{\kwd{FGM circular microplate}
\kwd{Nonlinear thermal buckling}
\kwd{Von K\'{a}rm\'{a}n nonlinearity}
\kwd{Modified couple stress theory}
\kwd{Size-dependent effect}
\kwd{Post-buckling behavior}}

\altkeywords{\kwd{Microplaque circulaire FGM}
\kwd{Flambement thermique non lin\'{e}aire}
\kwd{Non-lin\'{e}arit\'{e} de von K\'{a}rm\'{a}n}
\kwd{Th\'{e}orie modifi\'{e}e des contraintes de couple}
\kwd{Effet d'\'{e}chelle}
\kwd{Comportement post-flambement}}

\begin{abstract} 
This paper analytically investigates the nonlinear static buckling
behavior of functionally graded (FG) circular microplates by
integrating Kirchhoff plate theory, von K\'{a}rm\'{a}n geometric
nonlinearity, and the modified couple stress theory to account for
size-dependent effects. The microplate is concurrently
exposed to a uniform pressure and a uniformly increasing
through-thickness thermal load. A method based on displacement is used,
where the expected movement and bending of the plate are described
using polynomial functions that fit the fixed edges of the plate. This
choice reduces computational cost while maintaining sufficient accuracy
in predicting the nonlinear structural response. By using the Ritz
energy method, we can derive straightforward formulas for the critical
thermal buckling load and how the load relates to deflection after
buckling for the FG circular microplate. The numerical results show
that changes in the volume fraction index, material length-scale
parameter, and shape features are very important in determining how
strong the FGM structure is against buckling and how it behaves after
buckling. The results give important insights and serve as a helpful
guide for designing microscale functionally graded structures that
experience both heat and mechanical stress.
\end{abstract} 

\begin{altabstract} 
Cet article \'{e}tudie analytiquement le comportement de flambement
statique non lin\'{e}aire des microplaques circulaires \`{a} gradient
de propri\'{e}t\'{e}s (FG) en int\'{e}grant la th\'{e}orie des plaques
de Kirchhoff, la non-lin\'{e}arit\'{e} g\'{e}om\'{e}trique de von
K\'{a}rm\'{a}n et la th\'{e}orie modifi\'{e}e des couples de
contraintes afin de prendre en compte les effets d\'{e}pendants de la
taille. La microplaque est simultan\'{e}ment soumise \`{a} une pression
uniforme et \`{a} une charge thermique uniform\'{e}ment croissante
\`{a} travers son \'{e}paisseur. Une m\'{e}thode bas\'{e}e sur les
d\'{e}placements est utilis\'{e}e, dans laquelle les d\'{e}formations
et la flexion attendues de la plaque sont d\'{e}crites \`{a} l'aide
de fonctions polynomiales satisfaisant les conditions d'encastrement
des bords. Ce choix permet de r\'{e}duire le co\^{u}t de calcul tout en
conservant une pr\'{e}cision suffisante pour pr\'{e}voir la r\'{e}ponse
structurelle non lin\'{e}aire. En utilisant la m\'{e}thode
\'{e}nerg\'{e}tique de Ritz, il est possible de d\'{e}river des
expressions simples de la charge critique de flambement thermique ainsi
que la relation entre la charge et la d\'{e}flexion apr\`{e}s
flambement pour la microplaque circulaire FG. Les r\'{e}sultats
num\'{e}riques montrent que les variations de l'indice de fraction
volumique, du param\`{e}tre de longueur caract\'{e}ristique du
mat\'{e}riau et des param\`{e}tres g\'{e}om\'{e}triques influencent
fortement la r\'{e}sistance au flambement de la structure FGM ainsi que
son comportement post-flambement. Ces r\'{e}sultats fournissent des
informations importantes et constituent un guide utile pour la
conception de structures microscopiques \`{a} gradient de
propri\'{e}t\'{e}s soumises \`{a} des sollicitations
thermo-m\'{e}caniques.
\end{altabstract} 

%\input{CR-pagedemetas}

\thanks{Le Quy Don Technical University Research Fund under code
25.01.14.}

\maketitle

\end{noXML}

\section{Introduction}\label{sec1}

In recent years, functionally graded materials (FGMs) have received
considerable attention because they enable smooth variations in
material properties, leading to improved thermo-mechanical performance
compared with conventional laminated composites. FGMs can effectively
alleviate problems like stress concentration, delamination, and thermal
mismatch by customizing the spatial distribution of constituent
{phases, usually} ceramic and {metal, rendering} them especially
appropriate for advanced engineering applications in aerospace,
microelectronics, energy systems, and high-temperature
settings~\cite{1}. As structural components diminish to the
micro-scale, the integration of material length-scale effects is
crucial for precise predictions of mechanical reactions. The modified
couple stress theory (MCST) introduces a single intrinsic length-scale
parameter and has been widely employed to describe size-dependent
behavior in micro- and nano-scale structures with reasonable
mathematical simplicity~\cite{2}.

Circular plates are a key category of structural elements extensively
used in the design of micro-devices, sensors, actuators, and thermal
protection components. The stability characteristics of plates made
from functionally graded materials (FGMs) may exhibit significant
nonlinearity when operating in situations with concurrent mechanical
and thermal loads, particularly under considerable deflections.
Traditional plate theories, while sufficient for macro-scale
assessments, often neglect the synergistic effects of geometric
nonlinearity, material gradation, and size dependency that govern the
behavior of micro-scale structures. Therefore, a comprehensive
theoretical framework that incorporates von K\'{a}rm\'{a}n geometric
nonlinearity, through-thickness material gradation, and microscale
effects is required to accurately capture buckling and post-buckling
responses. Using the Moore--Gibson--Thompson thermal conduction model,
Tiwari \etal~\cite{3} analyzed the visco-thermoelastic vibration
behavior of circular microplate resonators and showed that accounting
for finite thermal wave speeds and relaxation effects leads to
vibration and damping characteristics that differ markedly from those
predicted by classical Fourier-based models. Lin \etal~\cite{4}
investigated vibration-based energy harvesting in functionally graded
circular microplates reinforced with carbon nanotubes and integrated
piezoelectric layers, and demonstrated that both material gradation and
nanotube reinforcement markedly enhance electromechanical conversion
efficiency, which is critical for advanced energy harvesting
applications. Thermo-viscoelastic phenomena in microplates have been
thoroughly investigated. Abouelregal and colleagues~\cite{5} examined
the thermoviscoelastic responses of Kirchhoff circular microplates
using the MGT model enhanced by modified couple stress theory,
demonstrating that size effects and hyperbolic heat conduction
substantially affect transient thermal stresses. Do, Ong, and
Lee~\cite{6} analyzed the nonlinear thermal buckling behavior of porous
microplates reinforced with graphene platelets and showed that both
graphene content and porosity distribution play a significant role in
governing post-buckling stability, especially under high-temperature
conditions. Flexoelectric and spinning microplate devices have garnered
renewed interest. Hosseini \etal~\cite{7} performed a vibration study
of rotating annular flexoelectric microplates, demonstrating that
rotation and flexoelectric coupling together result in significant
stiffening and frequency alterations. Li and Qing~\cite{8} investigated
the size-dependent axisymmetric buckling and free vibration behavior of
functionally graded piezomagnetic microplates based on a nonlocal
integral polar model, and demonstrated that the integral formulation
yields more physically consistent predictions than corresponding
differential models. Recent developments in constitutive modeling have
contributed significantly to the improvement of microplate theories. In
this context, Ahmadi, Fathalilou, and Rezazadeh~\cite{9} employed a
Neo-Hookean hyperelastic formulation to analyze the coupled nonlinear
response of circular microplates supported by micro-pillars, thereby
offering deeper physical insight into pronounced geometric and material
nonlinearities. In addition to mechanics, Yuan and colleagues~\cite{10}
investigated asymmetric optical waveguides using organic microplates,
underscoring their significance in micro-optical and photonic device
engineering. Advanced computational methods and high-order theories
have likewise enhanced the comprehension of microplate dynamics. Hung
\etal~\cite{11} used a modified strain gradient theory with
isogeometric analysis to investigate FG triply periodic minimum surface
(TPMS) microplates, demonstrating that microstructural periodicity and
strain gradient effects considerably influence frequency spectra.
Harazono and colleagues~\cite{12} assessed the accuracy of microplate
handling in robotic laboratory automation systems, providing practical
insights into microplate dynamics in actual operating situations.
Thermoelastic damping, an essential element for micro- and
nano-resonators, has been thoroughly examined in recent scholarly
works. Xiao \etal~\cite{13} proposed a generalized thermoelasticity
formulation to analyze \mbox{thermoelastic} damping in circular nanoplates and
showed that the damping behavior is highly sensitive to relaxation
times and nonlocal heat conduction effects. Peng and
colleagues~\cite{14} examined the memory-dependent and small-scale
influences on the thermoelastic damping of composite microplates
enhanced with graphene nanoplatelets, establishing that visco-thermal
memory and nano-reinforcement enhance damping properties at operating
frequencies. Peng \etal~\cite{15} analyzed size-dependent thermoelastic
damping in functionally graded graphene-reinforced microplates based on
the Moore--Gibson--Thompson model and demonstrated that material
gradation and higher-order heat transport mechanisms have a pronounced
influence on resonant quality factors. Liu and colleagues~\cite{16}
investigated the symmetric and asymmetric vibrational properties of
bi-directionally graded annular microplates, revealing that directional
gradation significantly influences modal interaction and frequency
{separation, crucial} aspects in micro-resonator design. {Ansari and
colleagues}~\cite{17} {developed an analytical solution for the
nonlinear buckling and postbuckling behavior of cylindrical nanoshells
using surface elasticity theory, demonstrating that surface energy
effects significantly modify the critical load and postbuckling
trajectory at the nanoscale. Ansari and colleagues}~\cite{18} {expanded
size-dependent modeling to microplates, examining the thermal buckling
of Mindlin rectangular FGM microplates through the lens of strain
gradient theory and demonstrating the significant impact of material
gradation and intrinsic length scale parameters on thermal stability.
Gholami and colleagues}~\cite{19} {studied how the size affects the
free vibration and buckling of magneto-electro-thermo-elastic
nanoplates using a special plate model, showing how both nonlocal
effects and field interactions together impact the structure's
behavior. 

{Although numerous studies have examined} the mechanical and thermal
buckling behavior of functionally graded plates at the macroscale,
relatively few investigations have focused on circular microplates that
explicitly incorporate size-dependent effects. Current literature often
depends on higher-order shear deformation theories, numerical
discretization methods, or simple linearized models, which might be
computationally intensive or inadequate for accurately representing
nonlinear behavior in the post-buckling phase. In addition, the
availability of analytical solutions for functionally graded circular
microplates under combined thermo-mechanical loading remains scarce
because of the highly nonlinear nature of the governing equations. Jain
\etal~\cite{20} developed a meshfree formulation for analyzing
size-dependent thermal buckling and post-buckling behavior of porous
microplates resting on elastic foundations, and demonstrated that
strong coupling exists among porosity, temperature gradients, and
foundation stiffness under localized heating. Likewise, Chen and his
study team~\cite{21} examined the thermal buckling of graphene
platelet-reinforced composite micro smart plates on elastic
foundations, demonstrating that GPL reinforcement and thermal loading
substantially affect critical buckling temperatures. Tran
\etal~\cite{22} proposed an advanced iBCMO-DNN algorithm for the
stochastic thermal buckling analysis of functionally graded porous
microplates based on modified strain gradient theory, and showed that
the approach achieves high accuracy in capturing variability in
material properties, geometric parameters, and temperature fields.
Advanced modeling of thermoelastic and thermo-viscoelastic behavior
continues to enhance the understanding of microplate systems. Liu and
other researchers~\cite{23} used artificial neural networks to study
the heat buckling of microplates with graphene platelets and integrated
piezoelectric layers. This gave them accurate and computationally
efficient predictions. Zhang \etal~\cite{24} analyzed the thermal
buckling, free vibration, and transient behavior of rotating porous
microbeams reinforced with graphene nanoplatelets and showed that
temperature effects, porosity, and rotational motion interact strongly
to influence the structural response. Joueid \etal~\cite{25} further
investigated on thermoelastic buckling in plates and shells made of
functionally graded materials that rely on temperature and porosity.
They found that critical buckling conditions are very sensitive to
changes in temperature and porosity. Recent research also examines
smart sandwich \mbox{architectures} and microplates including sophisticated
multiphysics couplings. Qin \etal~\cite{26} conducted numerical
analyses to evaluate the critical buckling temperatures of sandwich
microplates with cellular cores and CNT-reinforced piezoelectric
patches, and showed that core topology, electromechanical
reinforcement, and length-scale parameters significantly affect
buckling behavior based on modified strain gradient theory. Tang
\etal~\cite{27} investigated the size-dependent vibration and buckling
behavior of porous functionally graded microplates in thermal
environments based on modified couple stress theory. The dual power-law
distribution of scale parameters helped them understand how
microstructural and thermal effects interact. Salari and
Vanini~\cite{28} analyzed the nonlocal nonlinear static and dynamic
snap-through buckling behavior of thermally post-buckled imperfect
functionally graded circular nanoplates and demonstrated that
temperature effects, initial imperfections, and nonlocal elasticity
strongly govern the nonlinear response. Several recent studies have
looked at how stress, displacement, and instability patterns change
with size in different microplate layouts. Liu and colleagues~\cite{29}
evaluated the impact of thermal loads and micro-scale factors on
stress/strain/displacement distributions, validating notable
discrepancies in microscale responses relative to conventional
expectations. Wen and other researchers~\cite{30} created models of
double-layered microplate systems that took into account couple stress
effects and interlayer interactions. These models showed the
circumstances under which buckling mode changes happen. Tran
\etal~\cite{31} proposed advanced computational methods for the
analysis of vibration, buckling, and transient responses of porous
metal-foam microplates. They showed that porosity and heat fields work
together to control dynamic properties. Additional understanding of
thermal buckling in multi-directional and curved microscale shells has
been attained. {Daikh and colleagues}~\cite{32} {examined porous
functionally graded material (FGM) configurations in sandwich
nanoplates subjected to heat conduction through nonlocal strain
gradient theory, revealing that porosity distribution and small-scale
factors significantly affect thermal buckling resistance. Farajpour and
colleagues}~\cite{33} {introduced a higher-order plate model that
incorporates nonlocal effects and strain gradients for orthotropic
nanoplates exposed to thermal conditions, highlighting the interplay
between nonlocal softening and strain gradient stiffening mechanisms in
influencing the critical buckling load. Bouazza}~\cite{34} created a
detailed model to study how functionally graded rectangular plates bend
under heat at a larger scale, using a method that improves accuracy in
showing how shear forces work without needing extra correction factors.
Khuat Duc \etal~\cite{35} developed an isogeometric modeling
framework with variable length-scale parameters to analyze the dynamic
buckling behavior of thermally loaded doubly curved, bidirectional
functionally graded porous shallow microshells, and showed that
curvature, material gradation, and porosity strongly influence the
buckling response. Li \etal~\cite{36} investigated the vibro-acoustic
response of functionally graded sandwich microplates under coupled
thermal-electric loading and demonstrated that strong interactions
exist between vibro-acoustic behavior, material gradation, and
thermo-electrical fields. {Marin \etal}~\cite{37} {investigated the
impact of internal rotations and alterations in material structure on
the initial boundary value issue of thermoelastic bodies, demonstrating
that the incorporation of microstructural elements significantly
modifies the propagation and behavior of thermoelastic fields. Their
research has shown that employing sophisticated continuum theories is
essential for precisely modeling materials when conventional
thermoelastic concepts are insufficient. From a computational mechanics
perspective, Vlase, Negrean, and their colleagues}~\cite{38} {proposed
an energy-based method that employs the concept of acceleration energy
to formulate motion equations within a three-dimensional finite element
framework. This approach established a robust energy foundation for
formulating dynamic equations and enhanced the stability and
reliability of finite element modeling. Recently, Abouelregal
\etal}~\cite{39} {examined the impact of a specialized heat transfer
model, which accounts for historical influences, on thermoelastic
materials, demonstrating that these factors substantially alter the
behavior of heat waves and stress.}

In this study, the governing equations of a functionally graded
circular microplate are formulated based on Kirchhoff plate theory
combined with modified couple stress theory and von K\'{a}rm\'{a}n-type
nonlinear strain--displacement relations. A displacement-based
methodology using polynomial shape functions that adhere to clamped
boundary constraints is used to guarantee analytical tractability and
enhanced precision. By applying the Ritz energy method, accurate
expressions for the critical thermal buckling load and the nonlinear
post-buckling load--deflection relationships are derived. A parametric
study is performed to investigate how the FGM volume fraction index,
intrinsic length-scale parameter, plate geometry, and thermal loading
influence the stability behavior of the microplate. The findings
provide enhanced insight into the nonlinear thermo-mechanical behavior
of functionally graded material microplates and present practical
recommendations for the optimum design of micro-scale structural
components subjected to coupled loading conditions.

The remainder of this paper is organized as follows. Section~\ref{sec2}
presents the governing equations of the functionally graded circular
microplate, in which the MCST, the formulation of the FGM circular
microplate, and the corresponding mechanical relations are described in
detail.  Section~\ref{sec3} provides a comparative study to validate
the proposed model.  Section~\ref{sec4} presents numerical
investigations along with detailed discussions of the results.
Section~\ref{sec5} summarizes the main conclusions and significant
findings of this study.

\section{Governing equations of a functionally graded circular
microplate}\label{sec2}

The functionally graded circular microplate considered in this study is
treated as a thin elastic structure with deformable behavior, as
illustrated in Figure~\ref{fig1}. The plate is characterized by a
radius ${a}$ and a uniform thickness ${h}$. A cylindrical coordinate
system ($r,\theta, z$)} is employed, with the origin located at the
center of the mid-surface and lying on the $r \theta$-plane. The
microplate is subjected to a uniformly distributed transverse load
$q(r)$. Its kinematic behavior is governed by the von K\'{a}rm\'{a}n
nonlinear plate theory, whereas size-dependent material behavior is
captured using the modified couple stress theory.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}Structural model of the FGM circular microplate.}
\end{figure}

In this work, the earlier-mentioned assumptions are used throughout the
model development, including the derivation of the strain--displacement
relations, the MCST constitutive equations, the energy functional, and
the application of the Ritz method. The model's applicability is
restricted to small strains, moderate rotations, and linear elastic
material behavior to ensure mechanical consistency.

\subsection{Modified couple stress theory}\label{sec21}

According to the modified couple stress theory~\cite{40}, the strain
energy  $U_{\mathrm{int}}$ of a linear elastic isotropic material
occupying the domain $V$ is expressed as follows:
{\begin{equation}\label{eq1}
U_{\mathrm{int}}=\frac{1}{2}\int _{V}\left(\sigma \colon \varepsilon
+m\colon \chi \right) \mathrm{d}V,
\end{equation}}\unskip
Here, $\varepsilon$, ${\chi}$, $m$ (conjugate to ${\chi}$) represent
the strain tensor, symmetric curvature tensor, and the deviatoric
component of the couple-stress tensor, respectively, which are defined
by the following relations~\cite{40}:
{\begin{eqnarray}
&\displaystyle \varepsilon =\tfrac{1}{2}\left[\nabla u+\left(\nabla u\right)^{T}\right],\label{eq2}\vspace*{5pt}\Seqnsplit
&\displaystyle\chi =\tfrac{1}{2}\left[\nabla \theta +\left(\nabla \theta \right)^{T}\right],\label{eq3}\vspace*{5pt}\Seqnsplit
&\displaystyle\sigma =\lambda \,\mathrm{tr}\left(\varepsilon \right)I+2\mu \varepsilon,\vspace*{5pt}\label{eq4}\Seqnsplit
&\displaystyle m=2l^{2}\mu \chi. \label{eq5}
\end{eqnarray}}\unskip
Here, ${\lambda}$ and ${\mu}$ denote the Lam\'{e} constants, while
${l}$ represents the intrinsic material length-scale parameter.
This parameter is mathematically defined as the square root of the
ratio between the curvature modulus and the shear modulus and
characterizes the magnitude of couple-stress effects~\cite{41}. The
symbol ${I}$ denotes the identity tensor. The displacement vector ${u}$
is related to the rotation vector ${\theta}$ by:
{\begin{equation}\label{eq6}
\theta =\textstyle\frac{1}{2}\mathrm{curl}\,u.
\end{equation}}\unskip

Compared with the classical theory, the modified couple stress theory
is distinguished by the use of a symmetric couple-stress tensor and the
introduction of only one intrinsic material length-scale parameter
together with the conventional Lam\'{e} constants~\cite{40,41}.

The stress tensor in Equation~(\ref{eq4}) and the couple-stress tensor
in Equation~(\ref{eq5}) are formulated under the plane-stress
assumption. By expressing the Lam\'{e} constants in terms of the
Young's modulus $E(z)$ and Poisson's ratio ${v}$, the constitutive
relations can be rewritten as follows~\cite{42}:
{\begin{equation}\label{eq7}
\sigma _{\alpha \beta
}=\frac{E(z)}{1-v^{2}}\left[v\varepsilon _{kk}\delta
_{\alpha \beta }+(1-v)\varepsilon _{\alpha \beta }\right],\quad 
m_{\alpha \beta }=2Gl^{2}\chi _{\alpha \beta }
\end{equation}}\unskip
where the Lam\'{e} constants ${\lambda}$ and ${\mu}$ are replaced by
the elastic modulus $E(z)$ and Poisson's ratio $v$. The equations
$\lambda = E(z){\cdot}{\nu}/[(1+\nu)(1 -2\nu)]$ and  ${\mu} = {G} =
E(z)/[2(1 +\nu)]$ show how they are related. The letter ${G}$ stands
for the shear modulus, while the letter ${\delta}_{{ij}}$
signifies the Kronecker delta.

The MCST is an improved model of classical elasticity that adds an
extra part to the strain energy based on curvature and a specific
material length-scale parameter. This analysis assumes the material
behaves in a straightforward elastic way, with any nonlinearity coming
only from how strain relates to displacement based on the von
K\'{a}rm\'{a}n formulation. Since the strains are small and the
stress--strain relationships in MCST stay linear, the idea of total
potential energy still holds true. As a result, combining MCST with
geometric nonlinearity makes sense for studying buckling and what
happens after buckling.

\subsection{FGM circular microplate}\label{sec22}

Functionally graded materials are commonly fabricated by combining
ceramic and metallic constituents or by integrating multiple metallic
phases. The ceramic phase typically exhibits low thermal conductivity
and high temperature resistance. The ductile metallic phase, on the
other hand, helps prevent brittle fracture under steep temperature
gradients~\cite{1}.

A power-law function is adopted to describe the through-thickness
variation of the metal and ceramic volume fractions, denoted by $V_{\mathrm{m}}$ 
and $V_{\mathrm{c}}$, respectively~\cite{43}:
{\begin{equation}\label{eq8}
V_{\mathrm{c}}=\left(\frac{2z+h}{2h}\right)^{k},\quad V_{\mathrm{m}}=1-V_{\mathrm{c}},\quad  
k\geq 0,\quad k=\mathrm{infinity}.
\end{equation}}\unskip

In this formulation, ${z}$ denotes the thickness coordinate within the
range ${-}h/2 \leq z \leq h/2$, ${h}$ represents the plate thickness,
and ${k}$ is the volume fraction exponent with non-negative values. A
linear variation of ceramic and metal constituents through the
thickness is obtained when ${k} = 1$, whereas the plate becomes fully
ceramic for ${k} = 0$. The corresponding thickness-wise distributions
of $V_{\mathrm{c}}$ and $V_{\mathrm{m}}$ governed by the power-law
function are illustrated in Figure~\ref{fig2}.

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}Power-law distribution of  $V_{\mathrm{c}}$ and
$V_{\mathrm{m}}$ through microplate thickness.}
\end{figure}

The behavior of functionally graded materials is strongly influenced by
the volume fraction distribution of their constituents. In this study,
the elastic modulus $E(z)$ and thermal expansion coefficient $\alpha$
are modeled as thickness-dependent properties governed by the Voigt
mixture rule~\cite{44}, while Poisson's ratio $v$ is assumed to remain
unchanged through the thickness~\cite{45}, as presented below:
{\begin{eqnarray}
& \displaystyle E(z)=E_{\mathrm{c}}V_{\mathrm{c}}(z)+E_{\mathrm{m}}V_{\mathrm{m}}(z), \label{eq9}\Seqnsplit
& \displaystyle\alpha (z)=\frac{\alpha _{\mathrm{c}}E_{\mathrm{c}}
V_{\mathrm{c}}(z)+\alpha _{\mathrm{m}}E_{\mathrm{m}}V_{\mathrm{m}}(z)}{E(z)}. \label{eq10}
\end{eqnarray}}\unskip
In many FGM plate models, Poisson's ratio ${v}$ is treated as
thickness-independent. This simplification is motivated by the
relatively minor variation of ${v}$ compared to the elastic
modulus and thermal expansion coefficient, as well as its limited
influence on the overall bending stiffness and buckling response. Such
an assumption is consistent with earlier investigations on FGM
structures~\cite{45}.

The thickness-dependent profiles of the elastic modulus and thermal
expansion coefficient are depicted in Figure~\ref{fig3}.

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Elastic modulus and thermal expansion coefficient
distributions.}
\end{figure}

\subsection{Strain--displacement relationship}\label{sec23}

In the axisymmetric analysis of the functionally graded circular
microplate, the Kirchhoff plate theory is employed to express the
displacement components (${u}_r,u_\theta, u_z$) in terms of the
\mbox{displacements} of a point on the plate's neutral surface~\cite{46}:
{\begin{equation}\label{eq11}
u_{r}(r,z)=u(r)-zw_{,r}(r),\quad u_{\theta }(r,z)=0,\quad u_{z}(r,z)=w(r).
\end{equation}}\unskip
here, $u(r)$ and $w(r)$ represent the radial and transverse
displacements of a point located on the neutral surface of the plate,
respectively, where the radial coordinate ${r}$ varies from 0 to ${a}$.
Partial derivatives with respect to ${r}$ are indicated by a comma
followed by the corresponding subscript.

When a circular plate undergoes large axisymmetric deformation, the
nonlinear strain components that are nonzero according to the von
K\'{a}rm\'{a}n equation are written as~\cite{47}:
{\begin{equation}\label{eq12}
\varepsilon _{rr}(r,z)=u_{,r}+\frac{1}{2}{w_{,r}}^{2}-zw_{,rr},\quad  
\varepsilon _{\theta \theta }(r,z)=\frac{u}{r}-z\frac{1}{r}w_{,r}.
\end{equation}}\unskip

Based on the displacement field defined in Equation~(\ref{eq11}), the
only non-vanishing component of the rotation vector and the
corresponding symmetric curvature tensor are obtained from
Equations~(\ref{eq3}) and~(\ref{eq6}) and can be written in polar
coordinates as:
{\begin{equation}\label{eq13}
\theta _{\theta }=-w_{,r},\quad  \chi _{r\theta }=
\frac{1}{2}\left(\frac{1}{r}w_{,r}-w_{,rr}\right).
\end{equation}}\unskip

\subsection{Stress--strain relationship}\label{sec24}

Based on Hooke's law, the stress--strain relations of the functionally
graded circular microplate under uniformly distributed external
pressure and thermal {loading are given by}~\cite{48}:
{\begin{equation}\label{eq14}
\left\{\begin{array}{l}
\sigma _{r}\\
\sigma _{\theta }
\end{array}\right\}=\left[\begin{array}{ll}
\begin{array}{l}
Q_{11}\\
Q_{21}
\end{array} & \begin{array}{l}
Q_{12}\\
Q_{22}
\end{array}
\end{array}\right]\left\{\left\{\begin{array}{l}
\varepsilon _{r}\\
\varepsilon _{\theta }
\end{array}\right\}-\left\{\begin{array}{l}
\alpha (z)\Delta T\\
\alpha (z)\Delta T
\end{array}\right\}\right\},
\end{equation}}\unskip
where $\Delta T$ is the temperature rise above room temperature without
thermal stress of the structure and $Q_{ij}$ is the reduced stiffness
of the structure determined as follows:
{\begin{equation}\label{eq15}
Q_{11}=Q_{22}=\frac{E(z)}{1-v^{2}},\quad  Q_{12}=Q_{21}=\frac{E(z){\cdot}v}{1-v^{2}}.
\end{equation}}\unskip

\subsection{Formulation of the energy equations}\label{sec25}

The thermoelastic strain potential energy of the axially deformed FGM
circular microplate is {computed as follows}~\cite{49}:
{\advance\jot by 3pt
{\begin{eqnarray}
U_{\mathrm{int}}&=&\frac{1}{2}\int _{V}\left[\sigma
_{r}\left(\varepsilon _{r}-\alpha \Delta T\right)+\sigma _{\theta
}\left(\varepsilon _{\theta }-\alpha \Delta T\right)+2m_{r\theta }\chi
_{r\theta }\right]\mathrm{d}V 
\nonumber\\&
=
&
\frac{1}{2}\int _{V}\left(\sigma _{r}\varepsilon _{r}+\sigma
_{\theta }\varepsilon _{\theta }\right)\mathrm{d}V+\frac{1}{2}\int
_{V}2m_{r\theta }\chi _{r\theta }\mathrm{d}V-\frac{1}{2}\int _{V}\left(\sigma
_{r}\alpha \Delta T+\sigma _{\theta }\alpha \Delta T\right)\mathrm{d}V 
\nonumber\\&
=
&
\uppi {\int }_{-h/2}^{h/2}{\int }_{0}^{a}\left(\sigma _{r}
\varepsilon _{r}+\sigma _{\theta }\varepsilon _{\theta }\right)
r\mathrm{d}r\mathrm{d}z+2\uppi {\int }_{-h/2}^{h/2}{\int }_{0}^{a}m_{r\theta }
\chi _{r\theta }r\mathrm{d}r\mathrm{d}z
\nonumber\\&&
-
\,
\uppi \Delta T{\int }_{-h/2}^{h/2}{\int }_{0}^{a}
\alpha \left(\sigma _{r}+\sigma _{\theta }\right)r\mathrm{d}r\mathrm{d}z.
\label{eq16}
\end{eqnarray}}\unskip

Substituting Equations~(\ref{eq12}) and~(\ref{eq14}) into
Equation~(\ref{eq16}), we obtain:
{\begin{eqnarray}
U_{\mathrm{int}}&=&\uppi {\int }_{-h/2}^{h/2}{\int }_{0}^{a}
\left(Q_{11}u_{,r}+\frac{1}{2}Q_{11}{w_{,r}}^{2}-Q_{11}zw_{,rr}+
Q_{12}\frac{u}{r}-Q_{12}z\frac{w_{,r}}{r}\right)\left(u_{,r}+
\frac{1}{2}{w_{,r}}^{2}-zw_{,rr}\right)r\mathrm{d}r\mathrm{d}z 
\nonumber\\&&
+
\,
\uppi {\int }_{-h/2}^{h/2}{\int }_{0}^{a}\left(Q_{21}u_{,r}+
\frac{1}{2}Q_{21}{w_{,r}}^{2}-Q_{21}zw_{,rr}+Q_{22}\frac{u}{r}
-Q_{22}z\frac{w_{,r}}{r}\right)\left(\frac{u}{r}-\frac{z}{r}w_{,r}\right)
r\mathrm{d}r\mathrm{d}z 
\nonumber\\&&\qquad
+
\,
2\uppi {\int }_{-h/2}^{h/2}{\int }_{0}^{a}\frac{1}{2}Gl^{2}
\left(\frac{1}{r}w_{,r}-w_{,rr}\right)^{2}r\mathrm{d}r\mathrm{d}z 
\nonumber\\&&
-
\,
\uppi \Delta T{\int }_{-h/2}^{h/2}{\int }_{0}^{a}\left[\alpha 
\left(Q_{11}+Q_{12}\right)\left(u_{,r}+\frac{1}{2}{w_{,r}}^{2}+
\frac{u}{r}\right)-\alpha \left(Q_{11}+Q_{12}\right)z
\left(w_{,rr}+\frac{w_{,r}}{r}\right)\right]r\mathrm{d}r\mathrm{d}z.
\label{eq17}
\end{eqnarray}}\unskip
}

The work performed by the external forces is given by~\cite{50}:
{\begin{equation}\label{eq18}
U_{\mathrm{ext}}=2\uppi {\int }_{0}^{a}qwr\mathrm{d}r.
\end{equation}}\unskip
The total energy is obtained by combining Equations~(\ref{eq17})
and~(\ref{eq18}) as follows:
{\begin{equation}\label{eq19}
U_{\mathrm{total}}=U_{\mathrm{int}}-U_{\mathrm{ext}}.
\end{equation}}\unskip

\subsection{Boundary conditions, solution form, and the Ritz energy
method}\label{sec26}

The axisymmetric FGM circular microplate with clamped boundary
conditions along its perimeter is expressed as follows:
{\begin{equation}\label{eq20}
\begin{array}{c}
\mbox{At }r = 0: u=0,\quad  w=\mathrm{finite},\quad  w_{,r}=0. \\
\mbox{At }r = a: u=0,\quad  w=0,\quad  w_{,r}=0.
\end{array}
\end{equation}}\unskip

To enforce the boundary conditions given in Equation~(\ref{eq20}), the
displacement and rotation fields are approximated by the following
expressions:
{\begin{equation}\label{eq21}
u=U\frac{r(a-r)}{a^{2}},\quad  w=W\frac{\left(a^{2}-r^{2}\right)^{2}}{a^{4}}
\end{equation}}\unskip
where the amplitudes of the in-plane displacement ${u}$  and the
transverse deflection ${w}$ are denoted by $U$ and $W$, respectively.

The application of the Ritz energy approach to the total energy
expression in Equation~(\ref{eq19}), after inserting the displacement
fields given in Equation~(\ref{eq21}), leads to:
{\begin{equation}\label{eq22}
\frac{\partial U_{\mathrm{total}}}{\partial U}=\frac{\partial U_{\mathrm{total}}}{\partial W}=0.
\end{equation}}\unskip
The relations in Equation~(\ref{eq22}), when treated using the Ritz
energy approach, lead to a coupled system of two algebraic equations
for the unknown coefficients $U$ and $W$:
{\begin{eqnarray}
&\displaystyle 2a_{11}U+a_{13}W+a_{15}W^{2}=0, \label{eq23}\Seqnsplit
&\displaystyle 2\left(a_{12}+a_{21}+a_{31}\Delta T\right)W+a_{13}
U+3a_{14}W^{2}+2a_{15}UW+4a_{16}W^{3}-Aq=0 \label{eq24}
\end{eqnarray}}\unskip
where the coefficients are determined as follows:
{\begin{eqnarray*}
\begin{array}{ll}
\displaystyle a_{11}=\frac{\uppi }{4}{\int }_{-h/2}^{h/2}Q_{11}\mathrm{d}z,&{\quad}
\displaystyle a_{12}=\frac{32\uppi }{3a^{2}}{\int }_{-h/2}^{h/2}Q_{11}z^{2}\mathrm{d}z ,\\[10pt]
\displaystyle a_{13}=\frac{16\uppi }{5a}{\int }_{-h/2}^{h/2}Q_{11}z\mathrm{d}z,&\quad 
\displaystyle a_{14}=\frac{8\uppi }{5a^{2}}{\int }_{-h/2}^{h/2}Q_{12}z\mathrm{d}z,\\[10pt]
\displaystyle a_{15}=\left(\frac{82\uppi }{315a}{\int }_{-h/2}^{h/2}Q_{12}\mathrm{d}z
-\frac{46\uppi }{315a}{\int }_{-h/2}^{h/2}Q_{11}\mathrm{d}z\right),&\quad
\displaystyle a_{16}=\frac{32\uppi }{105a^{2}}{\int }_{-h/2}^{h/2}Q_{11}\mathrm{d}z,
\end{array} 
\\
\begin{array}{l@{\qquad}l@{\qquad}l}
\displaystyle a_{21}=\frac{32\uppi }{3a^{2}}{\int }_{-h/2}^{h/2}Gl^{2}\mathrm{d}z,&\quad
\displaystyle a_{31}=\frac{-\rmpi }{3}{\int }_{-h/2}^{h/2}\alpha \left(Q_{11}+Q_{12}\right)\mathrm{d}z,
&\quad \displaystyle A=\frac{\uppi a^{2}}{3}.
\end{array}
\end{eqnarray*}}\unskip

Solving Equation~(\ref{eq23}), the expression for the amplitude ${U}$
is obtained as follows:
{\begin{equation}\label{eq25}
U=-\frac{a_{13}W+a_{15}W^{2}}{2a_{11}}.
\end{equation}}\unskip

Substituting the expression of ${U}$ from Equation~(\ref{eq25}) into
Equation~(\ref{eq24}), the resulting equation is:
{\begin{equation}\label{eq26}
\left(4a_{16}-\frac{{a_{15}}^{2}}{a_{11}}\right)W^{3}+\left(3a_{14}-\frac{3a_{13}a_{15}}{2a_{11}}\right)W^{2}+2\left(a_{12}+a_{21}-\frac{{a_{13}}^{2}}{2a_{11}}\right)W+2a_{31}\Delta TW-Aq=0.
\end{equation}}\unskip

Equation~(\ref{eq26}) represents the relationship between the external
pressure load  ${q}$, the thermal load $\Delta T$, and the deflection
amplitude ${W}$. From this equation, the expressions for the external
pressure ${q}$ and the uniformly increased temperature
$\Delta T$ are obtained as follows:
{\begin{eqnarray}
&\displaystyle q=\frac{1}{A}\left(4a_{16}-\frac{{a_{15}}^{2}}{a_{11}}\right)W^{3}+\frac{1}{A}\left(3a_{14}-\frac{3a_{13}a_{15}}{2a_{11}}\right)W^{2}+\frac{2}{A}\left(a_{12}+a_{21}-\frac{{a_{13}}^{2}}{2a_{11}}\right)W+\frac{2a_{31}}{A}\Delta TW, \label{eq27}\Seqnsplit
&\displaystyle\Delta T=-\frac{1}{2a_{31}}\left(4a_{16}-\frac{{a_{15}}^{2}}{a_{11}}\right)W^{2}-\frac{1}{2a_{31}}\left(3a_{14}-\frac{3a_{13}a_{15}}{2a_{11}}\right)W-\frac{1}{a_{31}}\left(a_{12}+a_{21}-\frac{{a_{13}}^{2}}{2a_{11}}\right)+\frac{Aq}{2a_{31}W}. \qquad \label{eq28}
\end{eqnarray}}\unskip

Equations~(\ref{eq27}) and (\ref{eq28}) serve as the basis for
analyzing the nonlinear thermo-mechanical post-buckling response of the
FGM circular microplate.

The critical thermal buckling load according to the bifurcation
criterion is derived by letting ${W}\rightarrow 0$ in
Equation~(\ref{eq28}), yielding:
{\begin{equation}\label{eq29}
\Delta T_{\mathrm{cr}}=-\frac{1}{a_{31}}\left(a_{12}+a_{21}-\frac{{a_{13}}^{2}}{2a_{11}}\right).
\end{equation}}\unskip

For computational convenience, the following nondimensional
transformations are introduced~\cite{51}:
{\begin{equation}\label{eq30}
q=\frac{QEh^{4}}{a^{4}},\quad  w_{0}=\frac{w}{h},\quad  W_{0}=\frac{W}{h}.
\end{equation}}\unskip

\section{Comparative study}\label{sec3}

To verify the reliability of the methodology used in this study, a
comparison between the nondimensional deflection amplitude of the FGM
circular microplate and the results reported by Wang \etal~\cite{51} is
presented in Table~\ref{tab1}. Wang's findings were
achieved by using the principle of minimum total energy to formulate
nonlinear differential equations, converting these equations into
nonlinear algebraic equations by the orthogonal collocation method, and
resolving them using the Newton-Raphson iterative technique. A
nondimensional transformation was implemented for the evenly
distributed load $q=QEh^4/a^4$.  The comparison demonstrates a near
alignment, with a maximum divergence of 4.19\%. Consequently, this
comparison substantiates the validity of the methodology used in the
current study.


Next, we compare the load--deflection curves from Wang's study~\cite{51}
with those from this study in Figure~\ref{fig4}. The results are very
similar, which shows that the two methods agree very well and confirms
that the method used in this study is accurate and reliable.

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig4}Comparison of the load-nondimensional deflection
amplitude curves.}
\end{figure}

\begin{table}%t1
\caption{\label{tab1}Comparison of the nondimensional deflection
amplitude  ${W}_0=w/h$ of an isotropic circular microplate subjected to
a uniformly distributed external pressure ($v=0.3$, $Q=25$)}
\begin{tabular}{cccccccc}
\thead
$h/l$ & {15} & {10} & {5} & {4} & {3} & {2} & {1} \\
\endthead
Wang \etal~\cite{51} & 1.64643 & 1.64033 & 1.60713 & 1.58201 & 1.52748 & 1.37355 & 0.77189 \\
This paper & 1.57875 & 1.57269 & 1.54007 & 1.51578 & 1.46389 & 1.32114 & 0.76269 \\
Difference  (\%) & 4.11\0\0\0 & 4.12\0\0\0 & 4.17\0\0\0 & 4.19\0\0\0 & 4.16\0\0\0 & 3.82\0\0\0 & 1.19\0\0\0
\botline
\end{tabular}
\end{table}

\section{Results and discussion}\label{sec4}
\vspace*{-3pt}

The numerical values presented in the Tables are obtained by directly
substituting the specified material properties, geometric parameters,
and intrinsic length-scale values into the derived closed-form
analytical expressions. After performing the nondimensional
transformations, the resulting algebraic equations are evaluated
numerically using standard computational procedures. No additional
numerical approximation schemes are employed beyond the direct
evaluation of the analytical formulas.

Table~\ref{tab2} shows that the critical thermal buckling load
increases markedly with increasing $l/h$ and decreasing $a/h$.
Moreover, increasing the volume fraction index ${k}$ reduces the
critical buckling load due to reduced stiffness and a higher thermal
expansion coefficient.

\begin{table}%t2
\caption{\label{tab2}Influence of the ratios  ${l/h}$,
${a}/{h}$, and the volume fraction index k on the critical thermal
buckling load of the FGM circular microplate ($E_{\mathrm{c}} =200$~GPa;
$E_{\mathrm{m}}=70$~GPa; ${\alpha}_{\mathrm{c}}=10\times10^{-6}$ 1/K; ${\alpha}_{\mathrm{m}}=23\times10^{-6}$ 1/K;
${v}=0.3$; ${a}=10^{-6}$~m) \vspace*{2pt}}
\begin{tabular}{cccccccc}
\thead
\xmorerows{1}{$l/h$}  & \xmorerows{1}{$a/h$}  & \multicolumn{6}{c}{${k}$} \\\cline{3-8}
 &  & {0.1} & {1} & {3} & {5} & {7} & {10} \\
\endthead
\morerows{2}{0}  & {20} & \0480.1651 & \0326.6435 & \0282.7099 & \0281.9525 & \0280.5594 & \0276.2116 \\
 & {30} & \0213.4067 & \0145.1749 & \0125.6488 & \0125.3122 & \0124.6930 & \0122.7607 \\
 & {40} & \0120.0412 & \0\081.6608 & \0\070.6774 & \0\070.4881 & \0\070.1398 & \0\069.0529\vspace*{6pt} \\
\morerows{2}{0.2}  & {20} & \0562.6910 & \0391.0799 & \0334.4274 & \0329.1014 & \0325.3568 & \0319.0505 \\
 & {30} & \0250.0848 & \0173.8133 & \0148.6344 & \0146.2673 & \0144.6030 & \0141.8002 \\
 & {40} & \0140.6727 & \0\097.7699 & \0\083.6068 & \0\082.2753 & \0\081.3392 & \0\081.3392 \vspace*{6pt}\\
\morerows{2}{0.5}  & {20} & \0995.9519 & \0729.3710 & \0605.9445 & \0576.6330 & \0560.5432 & \0543.9549 \\
 & {30} & \0442.6453 & \0324.1649 & \0269.3086 & \0256.2813 & \0249.1303 & \0241.7577 \\
 & {40} & \0248.9879 & \0182.3427 & \0151.4861 & \0144.1582 & \0140.1358 & \0135.9887 \vspace*{6pt}\\
\morerows{2}{{1}}  & {20} & 2543.3125 & 1937.5534 & 1575.6483 & 1460.6743 & 1400.4945 & 1347.1848 \\
 & {30} & 1130.3611 & \0861.1348 & \0700.2881 & \0649.1886 & \0622.4420 & \0598.7488 \\
 & {40} & \0635.8281 & \0484.3883 & \0393.9120 & \0365.1685 & \0350.1236 & \0336.7962 
\botline
\end{tabular}
\vspace*{2pt}
\end{table}

It should be noted that the present formulation is based on the
classical Kirchhoff plate theory, which neglects transverse shear
deformation. This assumption is appropriate for thin microplates with
sufficiently large slenderness ratios. In the present study, the
investigated range of thickness ratios is  ${a/h}= 20 \div 40$,
corresponding to thin and very thin plates, for which shear effects are
generally negligible. However, for moderately thick microplates with
$a/h< 20$, transverse shear deformation may become significant, and
higher-order shear deformation theories (e.g., Mindlin or HSDT models)
would be more appropriate. Such extensions may be considered in future
work.

The graphical results presented in this study are generated directly
from the closed-form analytical expressions derived in the previous
sections. For each case, the specified material properties, geometric
parameters, and intrinsic length-scale values are substituted into the
nondimensional governing equations to compute the corresponding
numerical values. The resulting data are then processed and plotted
using MATLAB. No experimental data or independent numerical simulations
are employed in constructing the figures.

Figures~\ref{fig5}a and~\ref{fig5}b illustrate the inverse
relationship between the critical thermal buckling temperature
$\Delta T_{\mathrm{cr}}$ and the volume fraction
index ${k}$. It is evident that for any values of the ratios
${l/h}$ and ${a/h}$,
$\Delta T_{\mathrm{cr}}$ decreases rapidly as
${k}$ increases from 0 to approximately 2--4, after which the rate of
decrease gradually slows and approaches an asymptotic trend for larger
values of ${k}$. This indicates that increasing the metallic volume
fraction (i.e., increasing ${k}$) significantly reduces the thermal
stability of the structure.

\begin{figure}
\includegraphics{fig05}
\vspace*{-6pt}
\caption{\label{fig5}Relationship of the ratios  $l/h$, ${a/h}$, and
the volume fraction index ${k}$ to the critical thermal buckling load
of the FGM circular microplate.}
\vspace*{-6pt}
\end{figure}


Figures~\ref{fig5}c and~\ref{fig5}d confirm the trends identified
in  Table~\ref{tab2}. Figure~\ref{fig5}c shows that the critical
thermal buckling temperature increases strongly and nonlinearly with
increasing ${l/h}$, with the size effect becoming particularly
pronounced for ${l/h} > 0.2$. Figure~\ref{fig5}d illustrates the
inverse relationship between $\Delta T_{\mathrm{cr}}$ and ${a/h}$,
indicating that plates with larger surface areas (i.e., increasing
${a}$) or thinner thicknesses (i.e., decreasing ${h}$) exhibit lower
thermal stability.

From the results in Table~\ref{tab2} and the plots in
Figure~\ref{fig5}, it is observed that the critical thermal buckling
temperature $\Delta T_{\mathrm{cr}}$ of the FGM circular microplate is
generally lower than that of a homogeneous circular microplate. That
is, the thermal stability of the FGM circular microplate is inferior to
that of a fully ceramic microplate (i.e., when $k=0$).

The influence of the FGM volume fraction index ${k}$ on the
post-buckling thermo-mechanical behavior is presented in
Figures~\ref{fig6}a and~\ref{fig6}b. The post-buckling mechanical
and thermal load-carrying capacities of the FGM circular microplate
increase as the volume fraction index ${k}$ decreases.
Figure~\ref{fig6}a illustrates the thermal response; the
temperature--deflection curves indicate that the post-buckling thermal
resistance decreases sharply as ${k}$ increases. Plates with smaller
${k}$ values (i.e., ceramic-rich compositions) retain stiffness and
stability much better under thermal loading compared to plates with
larger ${k}$ values (i.e., metal-rich compositions).
Figure~\ref{fig6}b shows the mechanical response; the load--deflection
curves reveal a reduction in post-buckling mechanical resistance as
${k}$ increases. At an ambient temperature increase of $\Delta
T=400$~K, an initial deflection occurs prior to the application of
external pressure for $k = 1, 3, 5$.\looseness=-1

\begin{figure}
\includegraphics{fig06}
\vspace*{-2pt}
\caption{\label{fig6}Influence of the volume fraction index  ${k}$ on
the post-buckling mechanical and thermal behavior of the FGM circular
microplate.}
\end{figure}

In both cases examined in Figure~\ref{fig7}, it is evident that as the
ratio ${a/h}$ increases (i.e., the plate becomes wider and thinner),
the post-buckling mechanical and thermal load-carrying capacities
decrease. The curves corresponding to ${a/h} = 40$ lie significantly
lower than those for ${a/h} = 20$ and ${a/h} = 30$, indicating that the
slenderness of the plate has a strong influence on its post-buckling
resistance.

\begin{figure}
\includegraphics{fig07}
\vspace*{-2pt}
\caption{\label{fig7}Influence of the ratio  ${a/h}$ on the
post-buckling mechanical and thermal behavior of the FGM circular
microplate.}
\vspace*{-2pt}
\end{figure}

Figure~\ref{fig8} illustrates the influence of the ratio ${l/h}$ on the
post-buckling thermo-mechanical behavior. The post-buckling mechanical
and thermal load-carrying capacities of the FGM circular microplate
decrease as the ratio ${l/h}$ decreases. The findings indicate that the
material characteristic length parameter enhances the stiffness,
thereby reducing the deflection of the FGM circular microplate.
However, the size effect becomes significant only when ${l/h}$ is large
and gradually diminishes as this ratio approaches zero (i.e., the
classical model). Moreover, at an ambient temperature increase of
${\Delta}T=400$~K, an initial deflection appears before the application
of external pressure for the cases ${l/h} = 0$ and ${l/h} = 0.2$.

\begin{figure}
\includegraphics{fig08}
\caption{\label{fig8}Influence of the ratio  ${l/h}$ on the
post-buckling mechanical and thermal behavior of the FGM circular
microplate.}
\end{figure}

In both cases examined in Figure~\ref{fig9}, it is observed that the
deflection increases nonlinearly with the radial coordinate ${r/a}$,
and the load-carrying capacity of the structure increases as the ratio
${l/h}$ increases or ${a/h}$ decreases. Figure~\ref{fig9}a shows that
the influence of ${l/h}$ on the deflection is significant when ${l/h}$
is large, but it decreases markedly and may be neglected when ${l/h}$
is small. Therefore, plates modeled using the modified couple stress
theory exhibit greater stiffness compared to those modeled using the
classical plate theory (i.e., $l = 0$). Figure~\ref{fig9}b presents
the influence of ${a/h}$; for the same load level, plates with larger
${a/h}$ values exhibit higher deflections, which is consistent with the
observation that more slender structures deform more easily.

\begin{figure}
\includegraphics{fig09}
\caption{\label{fig9}Nondimensional deflection curves  ${w}_0$ along
the radial coordinate ${r/a}$.}
\end{figure}

Analysis reveals that the material length-scale parameter ${l}$ in the
modified couple stress theory plays a crucial role in enhancing
stiffness and buckling stability. Additionally, it should be
acknowledged that at the micro-scale, the intrinsic length scale may
interact with thermal processes, thereby influencing heat conduction
paths and the distribution of localized thermal
deformation~\cite{15,26}. This work is limited to the aspects of
nonlinear mechanics and pure size effects. For further expansion,
future studies could incorporate the temperature dependence of
parameter $l$ or apply advanced heat conduction models to more
fully elucidate this complex multiphysical relationship.

Figure~\ref{fig10}a illustrates the effect of mechanical loading on
the temperature--deflection curve. As the applied mechanical load
increases, the temperature--deflection curve shifts upward, \mbox{indicating}
that under combined mechanical and thermal loading, the structure
becomes unstable at a lower temperature compared to the case of pure
thermal loading. Figure~\ref{fig10}b depicts the influence of thermal
loading on the load--deflection curve. The load--deflection curves shift
downward as $\Delta T$ decreases, demonstrating that the mechanical
response of the structure is significantly reduced under the influence
of temperature.

\begin{figure}
\includegraphics{fig10}
\caption{\label{fig10}Influence of mechanical loading on the thermal
response and influence of thermal loading on the mechanical
post-buckling response. (a) Influence of mechanical loading on the
post-buckling thermal response.  (b) Influence of thermal loading on
the post-buckling  mechanical response.}
\end{figure}

Figure~\ref{fig12} presents a comparative analysis of the influence of
different FGM volume fraction distribution laws on the post-buckling
thermo-mechanical behavior. We further consider the Sigmoid
distribution pattern as expressed in Equations~(\ref{eq31})
and~(\ref{eq32}).
{\begin{eqnarray}
&\displaystyle V_{\mathrm{c}}=1-\frac{1}{2}\left(\frac{h/2-z}{h/2}\right)^{k}\quad\mbox{where } 0\leq z\leq h/2 \label{eq31}\Seqnsplit
&\displaystyle  V_{\mathrm{c}}=\frac{1}{2}\left(\frac{h/2+z}{h/2}\right)^{k}\quad\mbox{where } -h/2\leq z\leq 0.  \label{eq32}
\end{eqnarray}}\unskip

Variation of $V_{\mathrm{c}}$ through the thickness based on the Sigmoid
distribution is presented in Figure~\ref{fig11}.

\begin{figure}
\includegraphics{fig11}
\caption{\label{fig11}Variation of  $V_{\mathrm{c}}$ through the thickness based
on the Sigmoid distribution.}
\vspace*{5pt}
\end{figure}

\begin{figure}
\includegraphics{fig12}
\caption{\label{fig12}Combined influence of mechanical and thermal
loading on the post-buckling behavior for two different volume fraction
distribution laws. (a) Influence of mechanical loading on the
post-buckling thermal response for two different volume fraction
distribution laws.  (b) Influence of thermal loading on the
post-buckling mechanical response for two different volume fraction
distribution laws.}
\end{figure}

Figure~\ref{fig12}a illustrates the effect of mechanical loading on
the post-buckling thermal response under two different distribution
laws. {The temperature--deflection curves for the two distribution laws
differ significantly. This indicates that the thermal resistance of the
structure depends not only on the volume fraction index} ${k}$ {but
also on the specific distribution pattern of the constituent materials
along the thickness direction}. Similarly, Figure~\ref{fig12}b shows
the influence of thermal loading on the post-buckling mechanical
behavior. The differences between the load--deflection curves for the
two distribution laws demonstrate that the distribution pattern
significantly affects the structural stiffness and mechanical
load-carrying capacity. Therefore, in addition to selecting the volume
fraction index ${k}$, choosing an appropriate gradient distribution law
for the FGM is an important factor in enhancing the load-bearing
performance of FGM circular microplates under combined mechanical and
thermal loading conditions.

The current model employs the Voigt rule of mixtures for the sake of
simplicity. Future research could incorporate more advanced
micromechanical homogenization methods, such as the Mori--Tanaka scheme,
to obtain more accurate effective material properties for FGMs,
particularly when the constituent phases possess significantly
different elastic moduli.

\section{Conclusion}\label{sec5}

In this study, the nonlinear static buckling behavior of functionally
graded circular microplates is analyzed by combining Kirchhoff plate
theory, von K\'{a}rm\'{a}n geometric nonlinearity, and the modified
couple stress theory. The structure is concurrently subjected to a
uniformly distributed external pressure and a uniformly rising
through-thickness thermal load. Using a displacement-based approach,
the assumed forms of the displacement and deflection components were
expressed as polynomial functions that satisfy the clamped boundary
conditions along the plate's perimeter. {This choice of displacement
fields reduces computational complexity while maintaining sufficient
accuracy in evaluating the structural response under loading. By
applying the Ritz energy method, explicit expressions for the critical
thermal buckling load of the FGM \mbox{circular} microplate and the
load--deflection relationships in the nonlinear post-buckling regime
were derived}. The results of the numerical analysis reveal that the
FGM volume fraction index $k$, the material length scale
parameter ${l}$, and geometric factors all play significant roles
in determining the load-carrying capacity and post-buckling behavior of
the structure. Adjusting these parameters can optimize the structural
design to enhance both mechanical and thermal post-buckling
performance. These outcomes provide a scientific basis for the design
and optimization of structures operating under real-world
thermo-mechanical loading conditions.

\section*{Data availability}

The data supporting the conclusions of this investigation are included
in the text.

\section*{Acknowledgment}

This paper is funded by the Le Quy Don Technical University Research
Fund under code 25.01.14.

\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

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