Modeling and Analysis of Thermoelastic Damping in a Piezoelectro-Magneto-Thermoelastic Imperfect Flexible Beam

Abstract

This research addresses the phenomena of thermoelastic damping (TED) and frequency shift (FS) of a thin flexible piezoelectro-magneto-thermoelastic (PEMT) composite beam. Its motion is constrained by two linear flexible springs attached to both ends. The novelty behind the proposed study is to mimic the uncertainties during the fabrication of the beam. Therefore, the equation of motion was derived utilizing the linear Euler–Bernoulli theory accounting for the flexible boundary conditions. The beam’s eigenvalues, mode shapes, and the effects of the thermal relaxation time (t1), the dimensions of the beam, the linear spring coefficients (KL0 and KLL), and the critical thickness (CT) on both TED and FS of the PEMT beam were investigated numerically employing the Newton–Raphson method. The results show that the peak value of thermoelastic damping (Qpeak−1) and the frequency shift (Ω) of the beam increase as t1 escalates. Another observation was made for the primary fundamental mode, where an increase in the spring coefficient KLL leads to a further increase in Ω. On the other hand, the opposite trend is noted for the higher modes. Indeed, the results show the possibility of using the proposed design in a variety of applications that involve damping dissipation.

Full text

Academic Editor: Academic Editor:
Carlos Llopis-Albert
Received: 24 November 2024
Revised: 15 December 2024
Accepted: 19 December 2024
Published: 20 December 2024
Citation: Alneamy, A.M.; Guha, S.;
Tharwan, M.A. Modeling and
Analysis of Thermoelastic Damping in
a Piezoelectro-Magneto-Thermoelastic
Imperfect Flexible Beam. Mathematics
2024,12, 4011. https://doi.org/
10.3390/math12244011
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license
(https://creativecommons.org/
licenses/by/4.0/).
Article
Modeling and Analysis of Thermoelastic Damping in a
Piezoelectro-Magneto-Thermoelastic Imperfect Flexible Beam
Ayman M. Alneamy 1,* , Sayantan Guha 2,* and Mohammed Y. Tharwan 1
1Department of Mechanical Engineering, College of Engineering and Computer Science, Jazan University,
Jazan 45142, Saudi Arabia; mtharwan@jazanu.edu.sa
2
Centre for Data Science, Department of Computer Science and Engineering, Institute of Technical Education
and Research, Siksha ‘O’ Anusandhan (Deemed to be University), Bhubaneswar 751030, Odisha, India
*Correspondence: alneamy@jazanu.edu.sa (A.M.A.); sayantanguha.maths@gmail.com (S.G.)
Abstract: This research addresses the phenomena of thermoelastic damping (TED) and
frequency shift (FS) of a thin flexible piezoelectro-magneto-thermoelastic (PEMT) composite
beam. Its motion is constrained by two linear flexible springs attached to both ends. The
novelty behind the proposed study is to mimic the uncertainties during the fabrication of
the beam. Therefore, the equation of motion was derived utilizing the linear Euler–Bernoulli
theory accounting for the flexible boundary conditions. The beam’s eigenvalues, mode
shapes, and the effects of the thermal relaxation time (
t1
), the dimensions of the beam, the
linear spring coefficients (
KL0
and
KLL
), and the critical thickness (
CT
) on both TED and
FS of the PEMT beam were investigated numerically employing the Newton–Raphson
method. The results show that the peak value of thermoelastic damping (
Q−1
peak
) and the
frequency shift (
Ω
) of the beam increase as
t1
escalates. Another observation was made
for the primary fundamental mode, where an increase in the spring coefficient
KLL
leads
to a further increase in
Ω
. On the other hand, the opposite trend is noted for the higher
modes. Indeed, the results show the possibility of using the proposed design in a variety of
applications that involve damping dissipation.
Keywords: thermoelastic damping; flexible beam; thermal relaxation time; composite beam
MSC: 74K10
1. Introduction
The existence of thermoelastic damping is observed in practically all materials because
it is a primary source of intrinsic material damping. Therefore, several studies have ex-
amined a number of dissipation mechanisms, including surface, thermoelastic damping
(TED), and support looseness [
1
–
3
]. The thermoelastic damping is one of the primary dissi-
pation techniques that creates an upper limit on the achievable Q-factor of the mechanical
structure. Therefore, it is mandatory to investigate and to predict how the thermoelastic
damping affect the overall performance of the system.
Nádai
[4]
analytically investigated the impact of thermoelastic coupling on structure
dynamics. Other studies proposed and thoroughly investigated an approximate one-
dimensional theory of TED [
5
–
7
]. The effect of thermoelastic damping on the different
kinds of structures, including multi-layer-based structures, beams, resonators, plates, rings,
and cylinders, has been demonstrated [8–12].
Applications such as Structural Health Monitoring (SHM), equipment status moni-
toring, seismic sensing, and triplet thermo-mechanical-pressure multi-material systems
Mathematics 2024,12, 4011 https://doi.org/10.3390/math12244011

Mathematics 2024,12, 4011 2 of 17
are among famous candidates [
13
–
15
]. Thus, as the technology further developed, it is
important to take into account the effect of the thermoelastic damping on composite and
micro-scale-based structures [16–18].
Nowadays, composite structures are rapidly changing due to their uses in several
scientific and technical domains. The numerous benefits of fiber-reinforced composites,
such as their strength, low heat conductivity, electro-magnetic characteristics, and light
weight make them extremely relevant in many applications. Due to these limitations, the
fiber-reinforced composites perform better than their monolithic component components.
An example of these fibers is the integrate piezoelectric and piezomagnetic, which is among
the contemporary complicated composites [19].
To better understand the fundamental behavior of these structures, numerous theoreti-
cal and experimental studies have been conducted. These studies have utilized different
approaches such as shell theory, Green’s functions, perturbation theory, micro-topological
textures, micro-economics approach, interfacial fracture model, and minimum energy prin-
ciple models [
20
–
23
]. In addition, Biot
[24]
introduced the classical coupled thermoelasticity
theory, which asserts that any thermal disturbance that occurs at one location inside the
material would be immediately transferred at all other locations. As a result, it depicts an
implausible scenario in which heat spreads at an unbounded rate. To overcome this issue,
Lord and Shulman
[25]
investigated the addition of a thermal relaxation parameter, which
results in a hyperbolic heat conduction equation.
Another study offered a significant thermoelasticity theory that included temperature
rate dependence and two thermal relaxation parameters [
26
], while others are taking into
account various kinds of thermal theories [
27
–
29
]. Comprehensive investigations into the
influence of thermoelastic coupling on the natural frequency of macro-/micro-scale res-
onators are essential for the advancement of frequency-sensitive systems. Duwel et al.
[30]
used an experimental technique to demonstrate the importance of thermoelastic damping
(TED) in MEMS gyroscopes. The study found that variations in TED can significantly
impact the resonator’s quality factor by two to three orders of magnitude.
Schiwietz et al.
[31]
studied the significance of the thermoelastic damping contributions
for high-frequency modes in MEMS gyroscopes. Their findings indicate that thermolastic
damping is critical for the gyroscope’s reaction to high-frequency vibrations, since it restricts
the quality factor of its high-frequency modes. The effect of geometrical non-linearity of a
cantilever based MEMS device on thermoelastic damping also has been investigated [
32
]. They
found that as the beam undergoes large deflection, the effective stiffness of the system impacts
the eigenfrequencies, and therefore, changes the dynamic characteristics of the structure.
In this work, an investigation of the free vibration and dynamic motion of the ther-
moelastic damping (TED) and the frequency shift (FS) of a thin piezoelectro-magneto-
thermoelastic (PEMT) composite beam was performed. It is modeled based on the Euler–
Bernoulli theory and its motion is restrained by linear springs at both ends. To perform
the analysis, different types of boundary conditions were employed, and the eigenvalues
of the first five modes were numerically calculated using the Newton–Raphson technique
for precision. Here, flexible boundaries have been chosen to mimic the reality due to
the imperfections and uncertainties arise during fabrication. Indeed, these factors could
precipitate significant alterations in the overall functionality of the proposed design. In
addition, the influences of various parameters, such as the thermal relaxation time
t1
for
the Green–Lindsay theory, beam dimensions, the linear spring constants
KL0
and
KLL
,
the first five modes, and the critical thickness of the beam
CT
on the TED and FS, were
meticulously examined and discussed.
The rest of the paper is outlined as follows. Section 2presents the proposed design and
the classical theory of thermoelastic damping that has been used extensively for computing

Mathematics 2024,12, 4011 3 of 17
and analyzing the behavior of the thin piezoelectro-magneto-thermoelastic (PEMT), and
Section 3presents the free vibration problem of the beam. Then, in Section 4, a gener-
alized solution is employed to resolve the system equations of motion. Thereafter, an
extensive analysis and discussion were conducted to demonstrate the viability of the inves-
tigation, culminating in a summary of contributions and recommendations for prospective
research endeavors.
2. Design and Mathematical Formulation
2.1. Structural Design and Modeling
The design consists of a thin piezoelectro-magneto-thermoelastic (PEMT) composite
beam composed of a BaTiO3–CoFe2O4combination. The composite layers are oriented along
the (
x−y
) plane and are layered along the
z
direction, respectively. The material properties
of the beam are obtained from closed-form expressions derived employing the Mori–Tanaka
mean field approach along with the magneto-electroelastic Eshelby tensor [33–35].
Thus, the mechanical, electrical, magnetic, and thermal properties of the PEMT com-
posite are dependent entirely on the properties of piezoelectric-thermoelastic BaTiO
3
,
piezomagnetic-thermoelastic CoFe
2
O
4
, volume fraction (V
f
) of BaTiO
3
, and volume frac-
tion (V
m
) of CoFe
2
O
4
. The volume fractions are related to each other using the relation
Vf+ Vm= 1. It is worth noting that a value of V f= 0.55 has been selected for this study.
The beam has a uniform temperature of
T0
, an electric potential of
ϕ0
, and a magnetic
potential of
ψ0
. Using the Cartesian coordinate system, the beam is assumed to have a length
L
varies in a range of (0
≤
x
≤
L), a width
a
varies in a range of (0
≤
y
≤
a), and a thickness
H
varies in a range of (
−
H/2
≤
z
≤
H/2), respectively, as illustrated in Figure 1. It is also
subjected to bending vibrations with small amplitudes around the
x
-axis. The nature of the
deflection of the beam follows the linear Euler–Bernoulli beam theory, which asserts that when
bending occurs, any plane cross-section perpendicular to the beam axis remains plain and
perpendicular to the neutral axis, and the bent beam slopes are also not too steep. Following
these assumptions, we can express the mechanical displacements as [36,37]:
u=−z w′
x,v=0 , w=w(x,t)(1)
where (w′
x) denotes the spatial derivative with respect to x.
x
z
y
O
aL
H
(a)
L
z
x
H
KL0KLL
(b)
Figure 1. Schematics showing (a) the proposed PTFRC composite beam and (b) the beam connected
to a flexible supports at both ends and its parameters.

Mathematics 2024,12, 4011 4 of 17
Taking into account that the piezoelectro-magneto-thermoelastic (PEMT) composite
beam is thin, the rotating inertia and transverse shear deformations are negligible [
38
,
39
].
Then, an electric field is applied across the thickness while keeping the fibers aligned
horizontally in order to keep its electric field in unidirectionally aligned fiber-reinforced
structure [
40
]. As a result, the electric fields acting in the in-plane are disregarded, which
results in
E1=E2=
0 and
E3
is the only non-zero component. Similar to this, the magnetic
fields also follow the relation
H1=H2=
0 with
H3
being the only non-zero component.
Using Equation
(1)
and the additional assumptions in the constitutive equations of a
transversely isotropic piezoelectro-magneto-thermoelastic medium given by [
41
], we obtain
the equations of stresses, electric displacements, and magnetic inductions as follows:
σ11 =−c11z w′′
x+e31 ϕ′
z+q31 ψ′
z−β11T+t1˙
T
σ22 =−c12z w′′
x+e31 ϕ′
z+q31 ψ′
z−β11T+t1˙
T
σ33 =−c13z w′′
x+e33 ϕ′
z+q33 ψ′
z−β33T+t1˙
T
σ23 =σ13 =σ12 =0
D1=D2=0
D3=−e31 z w′′
x−∈33 ϕ′
z−α33 ψ′
z+p3T+t1˙
T
B1=B2=0
B3=−q31 z w′′
x−α33 ϕ′
z−µ33 ψ′
z+λ3T+t1˙
T
(2)
In Equation
(2)
, the mechanical stress components are denoted as
σij =σji
, while the
stiffness constants are
cij =cji
, respectively. The piezoelectric and piezomagnetic constants
are given by
eij
and
qij
. The thermal stress modulii are set to
βii
and the temperature
change is
T
. The electric and magnetic field intensities are expressed as gradients of the
scalar electric and magnetic potentials
ϕ
and
ψ
, respectively, as
Ei=−ϕ,i
and
Hi=−ψ,i
.
The electric displacements and magnetic induction are denoted by
Di
and
Bi
, respectively,
while the dielectric permittivity and magnetic permeability are
ϵij
and
µij
. In addition,
αij
denotes the magnetoelectric constants and the pyroelectric and pyromagnetic constants
are presented as
pi
and
λi
, respectively. Then, we substitute Equation
(2)
in Maxwell’s
equations to obtain:
e31 w′′
x+∈33 ϕ′′
z+α33 ψ′′
z=p3T+t1˙
T′
z(3)
q31 w′′
x+α33 ϕ′′
z+µ33 ψ′′
z=λ3T+t1˙
T′
z(4)
Integrating Equations (3) and (4) with respect to zresults in:
e31 z w′′
x+∈33 ϕ′
z+α33 ψ′
z=p3(T+t1˙
T) + f1(Φ0,Ψ0)(5)
q31 z w′′
x+α33 ϕ′
z+µ33 ψ′
z=λ3(T+t1˙
T) + f2(Φ0,Ψ0)(6)
Recalling that the beam is thin, we consider: σ33 =0 to get
β33T+t1˙
T=−c31z w′′
x+e33 ϕ′
z+q33 ψ′
z(7)
substituting Equation (7) in Equations (5) and (6) yields:
m1ϕ′
z+m2ψ′
z=m3(8)
n1ϕ′
z+n2ψ′
z=n3(9)
where
m1=∈33 −e33p3/β33
,
m2=α33 −q33 p3/β33
,
m3=−z w′′
x(e31 +c13p3/β33 ) +
f1(Φ0
,
Ψ0)
,
n1=α33 −e33λ3/β33
,
n2=µ33 −q33λ3/β33
, and
n3=−z w′′
x(q31 +c13λ3/β33) +

Mathematics 2024,12, 4011 5 of 17
f2(Φ0
,
Ψ0)
. Then, solving Equations
(8)
and
(9)
by Cramer’s rule,
we obtain:
ϕ′
z=−z w′′
xY1+n2f1(Φ0,Ψ0)−m2f2(Φ0,Ψ0)/G(10)
ψ′
z=−z w′′
xY2+m1f2(Φ0,Ψ0)−n1f1(Φ0,Ψ0)/G(11)
where
G=m1n2−m2n1
,
Y1=n2(e31 +c13 p3/β33)−m2(q31 +c13 λ3/β33 )
, and
Y2=m1(q31 +c13λ3/β33 )−n1(e31 +c13 p3/β33 )
. Knowing that the top and bottom sur-
faces of the beam are electrically shorted, meaning that
ϕ=
0 at
z=±H/
2, we integrate
Equation (10) with respect to zto obtain:
ϕ(x,z,t) = h−Y1w′′
xz2/2 −H2/8+zn2f1(Φ0,Ψ0)−zm2f2(Φ0,Ψ0)i/G(12)
Using the constraints that the top and bottom surfaces of the beam are magnetically
shorted, ψ=0 at z=±H/2, then integrating Equation (11) with respect to zto obtain:
ψ(x,z,t) = h−Y2w′′
xz2/2 −H2/8−zn1f1(Φ0,Ψ0) + zm1f2(Φ0,Ψ0)i/G(13)
Then, we drive the expression for the flexural moment of the beam’s cross-section as:
M(x,t) = −ZH/2
−H/2 aσ11zdz =QI w′′
x+β11 MT(14)
where
Q= [c11 G+e31Y1+q31Y2]/G
and
I=aH3/
12 denotes the moment of inertia of the cross-section, while
MT
denotes the
moment owing to thermal impacts and is expressed as:
MT=aZH/2
−H/2(T+t1˙
T)zdz
Therefore, we write the beam’s transverse equation of motion as:
ρA¨
w+M′′
x=0 (15)
where the cross-sectional area sets to
A=aH
and
ρ
is the density. Then, substituting
Equation (14) into Equation (15) yields:
ρA¨
w+QI wiv
x+β11 MT′′
x=0 (16)
The heat conduction equation is also given by:
Kij T,ij −ρCh(˙
T+t0¨
T) = T0[βij(˙
ui,j+δt0¨
ui,j) + pi(˙
Ei+δt0¨
Ei) + λi(˙
Hi+δt0¨
Hi)] (17)
where
i
,
j=
1, 2,3. The thermal conductivity coefficients are indicated by
Kij
, while the
specific heat at constant strain is designated as
Ch
. The thermal relaxation parameters are
denoted as t1and t0, and the tuning parameter sets to δ.
In this study, we have explicitly examined the effects of the Green–Lindsay thermal
theory [
26
] by substituting
δ=
0 and
t1≥t0≥
0 into Equation
(17)
, which can be written
now as:
K11 T′′
x+K33 T′′
z−ρCh(˙
T+t0¨
T) + β11 T0z(˙
w′′
x+δt0¨
w′′
x) = 0 (18)

Mathematics 2024,12, 4011 6 of 17
where
β11 =β11 −p3Y1/G−λ3Y2/G
. Thus, Equations
(12)
,
(13)
,
(16)
, and
(18)
are govern-
ing the transverse vibrations of PEMT composite beams.
2.2. Solution of the Problem
To solve Equations
(12)
,
(13)
,
(16)
and
(18)
, the solution of the time harmonic vibrations
of the considered beam is taken as:
[w(x,t),ϕ(x,z,t),ψ(x,z,t),T(x,z,t)] = [W(x),Φ(x,z),Ψ(x,z),Θ(x,z)]exp[iωt](19)
Then, substituting Equation (19) in Equations (12), (13), (16), and (18) we obtain:
Φ(x,z) = hY1z2/2 −H2/8W′′
x+n2z f1(Φ0,Ψ0)−m2z f2(Φ0,Ψ0)i/G(20)
Ψ(x,z) = h−Y2z2/2 −H2/8W′′
x+m1z f2(Φ0,Ψ0)−n1z f1(Φ0,Ψ0)i/G(21)
QI W iv
x+β11 MT
0′′
x−ρAω2W=0 (22)
K11 Θ′′
x+K33 Θ′′
z−iωσ0ρChΘ+iω σ′
0β11T0z W′′
x=0 (23)
where
MT
0=aσ1ZH/2
−H/2
Θzdz,σ◦=1+iωt0,σ1=1+iωt1,σ′
◦=1+iωδt0. (24)
Taking into account that adiabatic conditions prevail at the top and bottom surfaces
of the beam, we can write
Θ′
z=
0 at
z=±H/
2. This particular scenario is feasible since
micro-/nano-beams commonly function within high-vacuum environments in order to
attain elevated Q-factors. A trial solution that satisfies
Θ′
z=
0 while taking into account
the stability of temperature fluctuations in the plane that is orthogonal to the thickness
direction is presented as follows:
Θ(x,z) = β11T0σ′
0
ρChσ0 z−sin(pz)
pcos(pH
2)!W′′
x(25)
After that, we differentiate
Θ
twice with respect to
z
and substitute the resulting
expression in Equation (23) to obtain:
Θ′′
x=−σ′
0β11T0Kp′
ρChσ0
sin(pz)
pcos(pH
2)W′′
x(26)
where
p′=p2+ (iωρChσ0)/K33 and K=K33/K11
The expression of
MT
0
given in
Equation (24)
, after utilizing the term given in
Equation (25), results in:
MT
0=IT0β11σ1σ′
0
ρChσ0
(1+f(p))W′′
x(27)
where f(p) = 24(Hp/2 −tan(Hp/2))/H3p3. Henceforth, it yields:
MT
0′′
x=IT0β11σ1σ′
0
ρChσ0
(1+f(p))Wiv
x(28)
After that, differentiating
MT
0
again twice with respect to
x
and using the expression
given in Equation (26) to get:

Mathematics 2024,12, 4011 7 of 17
MT
0′′
x=IT0σ1σ′
0Kβ11
ρChσ0
p′f(p)W′′
x(29)
Comparing Equations (28) and (29) results in:
Kp′f(p)∼
=d2
dx2(1+f(p)) (30)
Using Equations (29) and (30) in Equation (22) yields:
Wiv
x−Aρω2W/Dω=0 (31)
where
Dω=c11 I"1+e31Y1+q31Y2
c11G+β2
11T0σ1σ′
0
ρChσ0c11 (1+f(p))1−p3Y1+λ3Y2
β11G#
In contrast to the thermal gradients working in the perpendicular direction, those
acting in the plane of cross-section are significantly greater along the thickness of the beam.
Hence, we have
∂
2Θ/
∂
x2=0, which results in:
p2=−iωσ0ρCh/K33 (32)
This equation shows a modified expression of
p
. Thus, the trial solution of
Equation (25)
now stands for the solution of Equation
(23)
. Hence, Equations
(20)
,
(21)
,
(25)
and (31) represent the complete set of governing equations of the PEMT composite beam.
3. Free Vibration Problem—TED and FS of the PEMT Beam
Furthermore, the eigenvalue of the both-side-restrained beam was obtained following
the free vibration analysis approach. In this case, both edges of the beam are restrained by
linear springs. Thus, the boundary conditions can be written as:
W=0, W,x−L KL0 W′′
x=0 at x=0
W=0, W,x−L KLL W′′
x=0 at x=L(33)
The terms
KL0
and
KLL =IE/LK
denote the dimensionless spring stiffness at both
ends,
x=
0 and
x=L
, respectively.
E
is the Young’s modulus and
K
denotes the spring’s
stiffness constant. Then, as
K
reaches infinity, the beam cannot deflect at both ends meaning
that
KL0 =KLL =
0 and the slope
dW/dx=
0, at both ends. This indicates that the rate
of change of
W
with respect to
x
is equal to zero at
x=
0. It also demonstrates that these
particular conditions, along with
W=
0 at both ends of the beam, collectively constitute
the complete set of boundary conditions. Therefore, to obtain the eigenvalues, we recall
Equation (31) which can be rewritten as:
"d4
dx4−γ4#W=0, γ4=Aρω2
Dω(34)
The solution of the above equation can be given as:
W(x) = N1sin(γx) + N2cos(γx) + N3sinh(γx) + N4cosh(γx)(35)
where
Ni
represents the constants. Substituting the boundary conditions given by Equa-
tion
(33)
into Equation
(35)
, a set of four equations with four unknowns was obtained.

Mathematics 2024,12, 4011 8 of 17
Equating the determinant of the coefficients of
Ni
, where
i=
1, 2, 3, 4, to 0, and then solving
it to obtain the characteristic equation that governs the free vibration of the beam. The
eigenvalues for the first five modes of the characteristic equation are numerically obtained
using the Newton–Raphson method.
This technique is implemented on Mathematica 7.0. Thus, for fixed values of the
undertaken parameters, initial trial guesses have been taken for the actual roots. Then,
numerical simulations were performed for almost 20 iterations to obtain the actual roots
accurately. Although the eigenvalues can be obtained by other numerical techniques, the
Newton–Raphson method has several advantages such as flexibility and ease of implemen-
tation and quadratic rate of convergence, ensuring that the error decreases exponentially
with each iteration.
Considering constant values for the spring coefficient attached to the left end as
KL
0
=
0.5, a beam length of
L=
1000 µm, and varying the spring coefficient attached
to the right end
KLL
, the obtained eigenvalues, expressed in terms of
γiL
, are listed in
Table 1. It should be noted that dividing these values by the beam length
L
yields the exact
eigenvalues of the piezoelectro-magneto-thermoelastic (PEMT) composite beam. The table
elucidates that for each corresponding value of
KLL
, the eigenvalues exhibit an increase in
its magnitude as well as a further increase in the vibration mode.
Table 1. Eigenvalues of the doubly flexible beam.
KL L→0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Modes↓Eigenvalues (Expressed as γiL)↓
1 2.876408 3.184658 3.254607 3.285685 3.303258 3.314558 3.322437 3.328243 3.332700 3.336229
2 6.231937 6.333290 6.362999 6.377175 6.385474 6.390923 6.394775 6.397642 6.399859 6.401624
3 9.402189 9.464342 9.483472 9.492759 9.498244 9.501866 9.504436 9.506353 9.507840 9.509025
4 12.55444 12.59888 12.61289 12.61975 12.62382 12.62652 12.62843 12.62987 12.63098 12.63186
5 15.71685 15.73928 15.74983 15.75504 15.75815 15.76021 15.76167 15.76277 15.76362 15.764300
Now, the expressions representing the thermoelastic damping (TED) and the frequency
shift (FS) can be derived utilizing Equation
(34)
which gives
ω2=γ4Dω/ρA
. Carrying out
some analysis, the corresponding frequency of the PEMT beam is expressed as:
ωk=ω0q1+ϵT(1+f(p)) (36)
where
ω0=γ2
kH
2sc11(1+Λeme )
3ρ,ϵT=ϵT1−p3Y1+λ3Y2
β11G
1+Λeme (37)
Since the order of magnitude of
ϵT
ranges between
−
4 to
−
5,
f(p)
is replaced by
f(ω0)
and Equation (36) is rewritten considering only up to the first order term as:
ωk=ω0[1+ϵT(1+f(ω0))/2](38)
Resolving
ωk
, presented in Equation
(38)
, into its real and imaginary components
yields:
Re[ωk] = ω0[1+ϵT(1+Λ1)/2](39)
Im[ωk] = ω0[ϵTΛ2/2](40)

Mathematics 2024,12, 4011 9 of 17
In the above equations, the expressions Λ1and Λ2are written as:
Λ1=6
ς2cos θ−6√2
ς3cos3θ
2

sin ς′+tan3θ
2sinh(ς′υ)
cos ς′+cosh(ς′υ)
(41)
Λ2=−6
ς2sin θ−6√2
ς3cos3θ
2
−tan3θ
2sin ς′+sinh(ς′υ)
cosh(ς′υ) + cos ς′
(42)
where
ς=Hp′,ς′=1.414 ςcos(θ/2),υ=tan(θ/2)(43)
Then, recalling from Equation
(32)
, which reveals that
p
is a complex term, using
Euler’s formula and replacing ωby ω0, a new expression for pwas obtained as:
p=1.414 exp[iθ/2]p′(44)
where
θ=tan−1−1
t0ω0
and
p′=rω0ρCh
2Kzz qω2
0t2
0+1
. Thereafter, we obtain the expres-
sions of the inverse quality-factor (Q−1) of the TED and frequency shift (Ω) as
Q−1=2
Im[ωk]
Re[ωk]
=|ϵTΛ2|(45)
Ω=
Re[ωk]−ω0
ω0
=|ϵT(1+Λ1)/2|(46)
It is worth noting that at low frequencies, isothermal conditions are prevalent, and
vibrations in that range lose very little energy. Conversely, for the high frequency range,
adiabatic conditions occur. However, the energy dissipated during vibrations in this range
is likewise rather minimal. In addition, the stress and strain are out of phase and the
internal friction reaches its highest value at the Debye peak [8].
On the other hand, for GL theory, the thermal relaxation times are
t0
and
t1
, where
t1
is a multiple of
t0
. In this scenario, the tuning parameter
δ
is taken as 0. For these
considerations,
p
given by Equation
(44)
can be expressed as
p′=qω0ρCh
√2Kzz
and
θ=−π
4
.
Thereafter,
Q−1
and
Ω
for GL theory are obtained from Equations
(45)
and
(46)
using the
modified expressions.
It is worth noting that considering the boundary conditions of the doubly-clamped
(CC) beam yields different characteristics of TED and FS. A comparison to those obtained
for the CC beam to validate these results was utilized [
42
]. The comparison shows good
agreement between the two approaches. Moreover, to validate this work, the influences
of piezoelectro-magnetic quantities are disregarded, and the Classical dynamical coupled
theory is employed in an isotropic medium. As a result, the obtained expressions for TED
and FS match with those observed in [8].
4. Results and Discussion
This section presents numerical simulations and analyses for the proposed design.
To investigate the influence of the geometrical and material properties on the transverse
thermoelastic damping (TED) (
Q−1
) and the frequency shift (FS) (
Ω
), selected particular
values are for thermal relaxation time, beam length, and spring stiffness constants (
KLL
,
KL0) were chosen for the numerical simulations.

Mathematics 2024,12, 4011 10 of 17
The time relaxation
t1
is designated as n
×
t
0
, where
n=
4, 8, 12, the beam length is
set to
L=
1000 µm, the aspect ratio is
AR
=100, the spring stiffness constant at the left end
sets to
KL0 =
0.5, and the right support stiffness
KLL
varies in a range of [0.5–5] with a
step size of 0.5. As a first step, the TED and FS variations are estimated while varying the
beam thickness values. The analysis indicates that once the thickness reaches a designated
Critical Thickness
(CT)
, the damping
Q−1
achieves a maximum peak value called (
Q−1
peak
)
ascribed to the geometrical and material features. Table 2presents the impacts of
KLL
,
the time relaxation
t1
, and the vibrational mode shapes on the Critical Thickness
(CT)
and the peak damping (
Q−1
peak
) values of the piezoelectro-magneto-thermoelastic (PEMT)
composite beam.
The table shows that for a given beam length
L
, spring constant
KLL
, and modes,
the values of
Q−1
peak
are found to be 4.8792, 9.5408, and 14.2500 at
t1=nt0
, where
n=
4, 8,12. Then, as the time relaxation
t1
further increases, the beam’s quality fac-
tors keeps decreasing. Furthermore, it has been established that the values of
CT
stay
constant when each respective mode and value of
KLL
are considered, despite the previ-
ously indicated changes in
t1
. Alternatively, a reduction in the critical thickness has been
observed as the mode and the control parameter further increase. It is worth noting that
the variation in the CT between each pair of
KLL
is more noticeable considering the first
mode. This, in fact, starts decreasing as higher modes approach.
Table 2. The Critical Thickness
CT
and scaled quality factor
Q−1
peak
at different values of
KLL
and
t1(=n ×t0), considering L=1000 µm and K L0 =0.5.
KL L→0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Modes n CT↓Q−1
peak ×(10−2)↓
4o4.8792
1 8 7.5975
7.0990 6.9969 6.9527 6.9280 6.9123 6.9014 6.8933 6.8872 6.8823
9.5408
12
14.2500
4o4.8792
2 8 4.5376
4.4890 4.4750 4.4684 4.4645 4.4620 4.4602 4.4589 4.4578 4.4570
9.5408
12
14.2500
4o4.8792
3 8 3.4494
3.4343 3.4297 3.4275 3.4262 3.4253 3.4247 3.4242 3.4238 3.4236
9.5408
12
14.2500
4o4.8792
4 8 2.8447
2.8380 2.8359 2.8349 2.8343 2.8339 2.8336 2.8334 2.8332 2.8331
9.5408
12
14.2500
4o4.8792
5 8 2.4428
2.4467 2.4456 2.4450 2.4447 2.4445 2.4444 2.4442 2.4442 2.4441
9.5408
12
14.2500
4.1. Impact of K LL on TED
The variation of the TED (
Q−1
) impacted by spring constant
KLL
and mode shapes
has been investigated. Figure 2shows the variation of the
Q−1
influenced by spring
constant
KLL
of the beam taking into account a length of
L=
1000 µm and time relaxation
of
t1=
8
t0
. The results indicate that the TED is negligible as
H≈
0. The variation of
Q−1
keeps increasing as
H
increases until it reaches a peak value referred to as
Q−1
peak
,
as illustrated in Figure 2a–e for all mode shapes, respectively. Another observation was
found where the value of the critical thickness
CT
decreases as the number of mode shape
increases as shown in Figure 2. The results indicate that increasing the values of spring
constant KLL initially increases the thermoelastic damping Q−1.

Mathematics 2024,12, 4011 11 of 17
Furthermore, as
KLL
keeps getting higher, the spring becomes more flexible. This
ensures that during vibrations, the beam undergoes large deformation, which creates
significant strain gradients. This results in more pronounced thermal gradients across the
beam’s thickness, thus amplifying the TED. This pattern prevails until
Q−1
reaches its peak
value as the thickness reaches the critical value. Beyond this point, the influence of
KLL
on
Q−1reverses, and now, increasing values of K LL decrease Q−1.
Additionally, the figure shows that the responses tend to get closer to each other
considering higher values of
KLL
. This, in fact, suggests that the influence of increasing
KLL
on
Q−1
decreases as illustrated in Figure 2a–e. The operating conditions belonging
to this analysis are listed in Table 2. The obtained results may be utilized to fine-tune
KLL
and optimize damping performance to enhance the working efficiency of several
applications including but not limited to energy harvesters, precision instruments, and
micro-electromechanical systems (MEMS) resonators.
2414
810 12
0
0
Q-1 
*10-2
H (

m)
6
2
4
6
8
10
4.9
4.5
4.9
4.5
5.3 5.5
8.8 9.2
KLL:
0.5:5
KLL:
0.5:5
(a) Mode 1
2414
810 12
0
0
Q-1 
*10-2
H (

m)
6
2
4
6
8
10
7.2
7.0
5.1
5.0
3.84 3.91
5.66 5.74
KLL:
0.5:5
0.5:5
KLL:
(b) Mode 2
2414
810 12
0
0
Q-1 
*10-2
H (

m)
6
2
4
6
8
10
7.02
7.00
5.03
5.00
2.94 2.96
4.34 4.38
KLL:
0.5:5
0.5:5
KLL:
(c) Mode 3
2414
810 12
0
0
Q-1 
*10-2
H (

m)
6
2
4
6
8
10
6.7
6.6
5.05
5.00
2.40 2.41
3.59 3.61
KLL:
0.5:5
0.5:5
KLL:
(d) Mode 4
2414
810 12
0
0
Q-1 
*10-2
H (

m)
6
2
4
6
8
10
4.93
4.90
4.93
4.90
1.91 1.92
3.11 3.12
KLL:
0.5:5
0.5:5
KLL:
(e) Mode 5
Figure 2. Impact of
KLL
on the scaled
Q−1
varying against
H
for fixed
L=
1000 µm,
KL0 =
0.5, and
t1=8t0for (a) mode 1, (b) mode 2, (c) mode 3, (d) mode 4, and (e) mode 5.
4.2. Impact of t1on TED
In this study, an investigating of the influence of the time relaxation
t1
on the thermoe-
lastic damping was performed. As can be seen from Table 2, the peak value of
Q−1
peak
remains
constant at
t1=
4
t0
with a value of 4.8792
×
10
−2
,
t1=
8
t0
with a value of 9.5408
×
10
−2
,
and
t1=
12
t0
as it reaches 14.2500
×
10
−2
considering the five mode shapes and a varying

Mathematics 2024,12, 4011 12 of 17
KLL
. Figure 3shows that the variation of
t1
affects the behavior of the TED of a both-side
restrained PEMT beam having a length of
L=
1000 µm and
KL0 =
0.5. In this study, the
spring constant
KLL
sets to 2. The results indicate a linear response for all modes. However,
as mentioned earlier, the peak value
Q−1
peak
is increasing as the time relaxation increases,
which occurs at similar values of CT.
On the other hand, a larger value of the thermal relaxation time
t1
indicates a slower
flow of heat through the beam material during oscillations. This means that the thermal
gradients persist for a longer time during each oscillation cycle. In fact, the TED arises
from the cyclic flow of heat between the beam’s compressed and stretched regions during
vibration. Then, as
t1
increases, the slower heat flow allows thermal gradients to develop
more significantly leading to a greater energy dissipation, and therefore, increasing values
of
Q−1
peak
with increasing
t1
. It is also seen that critical thickness
CT
decreases as the modes
increase from 1 to 5.
It is observed that elevated oscillations reduce the effective strain and thermal gra-
dients, resulting in a reduction of energy dissipation. Consequently, the
CT
exhibits a
low value with the augmentation of mode numbers, signifying that higher modes are less
proficient in inducing thermal gradients throughout the thickness of the beam. Given that
Q−1
for the beam with
t1=
4
t0
is minimal, it can be inferred that the Q-factor for this beam
is maximized.
0
Q-1 
*10-2
2
4
6
10
H (

m)
0
2
4
6
8
Mode 3
Mode 4
t1=4t0
Mode 1
Mode 2
12
14
8
1212 14
Mode 5
(a)t1=4t0
0
Q-1 
*10-2
2
4
6
10
H (

m)
0
2
4
6
8
12
14
8
1212 14
Mode 3
Mode 4
t1=8t0
Mode 1
Mode 2
Mode 5
(b)t1=8t0
0
Q-1 
*10-2
2
4
6
10
H (

m)
0
2
4
6
8
t1=12t0
12
14
8
1212 14
Mode 3
Mode 4
Mode 1
Mode 2
Mode 5
(c)t1=12t0
Figure 3. Scaled
Q−1
varying against
H
for fixed beam length
L=
1000 µm, left linear spring
coefficient
KL0 =
0.5, and right linear spring coefficient
KLL =
2 for different values of the time
relaxation (a)t1=4t0, (b)t1=8t0, and (c)t1=12t0.

Mathematics 2024,12, 4011 13 of 17
4.3. Impact of K LL on FS
In this section, we examine the variation in the frequency shift as a function of the
spring constant
KLL
. Figure 4illustrates the effects of
KLL
on the frequency shift (
Ω
) of a
bilaterally constrained PEMT beam characterized by a fixed aspect ratio of
AR =
100, time
relaxation of
t1=
8
t0
, and
KL0 =
0.5. The findings indicate an increase in the magnitude
accompanied by a peak value, implying the existence of an optimal value of
H
at which
Ω
attains its maximum value. Beyond the peak value,
Ω
exhibits wave-like behavior with
many peaks and troughs, eventually stabilizing as
H
values further increase. The observed
peaks and troughs correspond to nodal positions within the vibration mode of the beam.
The results show that the amplitudes of oscillations decrease with increasing
H
,
representing the phenomenon of energy dispersion. It is worth noting that an increase in
the values of
KLL
leads to an elevation of
Ω
taking into account the fundamental mode.
Furthermore, the peaks of
Ω
shift away from the central value as
KLL
increases, showing
that for increased values of
KLL
, the largest frequency shifts occur at shorter interval.
Figure 4illustrates that the effect of higher
KLL
leads to higher stiffness, and therefore, the
beam becomes stiffer.
H (

m)
0
5010 20 30 40
0
0.01
0.02
0.03
0.05
0.04
KLL=1, 2, 3, 4, 5
(a) Mode 1
H (

m)
0
5010 20 30 40
0
0.01
0.02
0.03
0.05
0.04
KLL=1, 2, 3,
4, 5
(b) Mode 2
H (

m)
0
5010 20 30 40
0
0.01
0.02
0.03
0.05
0.04
KLL=1, 2, 3, 4, 5
(c) Mode 3
H (

m)
0
5010 20 30 40
0
0.01
0.02
0.03
0.05
0.04
KLL=1, 2, 3,
4, 5
(d) Mode 4
H (

m)
0
5010 20 30 40
0
0.01
0.02
0.03
0.05
0.04
KLL=1, 2, 3, 4, 5
(e) Mode 5
Figure 4. The impact of
KLL
on the frequency
Ω
while varying the beam thickness
H
and taking into
account a fixed
AR =
100,
t1=
8
t0
, and
KL0 =
0.5 for a variational mode of (a) mode 1, (b) mode 2,
(c) mode 3, (d) mode 4, and (e) mode 5.
In contrast, a distinct observation is made regarding the alternate modes. In these cases,
increasing
KLL
results in a reduction in the frequency
Ω
. The curves for each individual
mode converge, showing that the influence of different
KLL
values consistently decreases
for ascending modes, as shown in Figure 4. It also indicates that for each specific value of
KLL, the size of Ωgradually decreases as the mode further increases.

Mathematics 2024,12, 4011 14 of 17
4.4. Impact of t1on FS
Another study was conducted to investigate the effect of the time relaxation
t1
on
the frequency shift. Figure 5demonstrates that the variation of
t1
directly impacts the
frequency employing operational parameters to those used in Section 4.3. A similar trend
to that obtained in Figure 4was observed with increasing magnitudes until they reach peak
value. Beyond this point,
Ω
is off and stabilizes, showing that after the optimal value of the
thickness H, it no longer increases considerably.
Furthermore, higher values of t1result in greater frequency shifts. The steeper initial
slope in
Ω
for
t1=
12
t0
compared to the other values of
t1
indicates a faster increase at
higher thermal relaxation times. Figure 5a shows that the first mode is more stable since the
entire beam moves uniformly and smoothly, resulting in a more even energy distribution.
Higher modes, on the other hand, are more susceptible to minor variations in material
qualities, boundary conditions, and external effects.
Indeed, Figure 5shows that for each value of
t1
, the frequency shift
Ω
keeps decreasing
minutely as modes increase. Potential applications include the tuning of thermal relaxation
times to control the frequency response in S/BAW devices used for environmental and
health monitoring. It is worth noting that for sake of completeness, Figure 5shows the
effect of the five modes despite the fact that their impact remains minimal.
H (

m)
0
t1=4t0
50
t1=8t0
t1=12t0
10 20 30 40
0
0.02
0.04
0.06
0.10
0.08
(a) Mode 1
H (

m)
0
t1=4t0
50
t1=8t0
t1=12t0
10 20 30 40
0
0.02
0.04
0.06
0.10
0.08
(b) Mode 2
H (

m)
0
t1=4t0
50
t1=8t0
t1=12t0
10 20 30 40
0
0.02
0.04
0.06
0.10
0.08
(c) Mode 3
H (

m)
0
t1=4t0
50
t1=8t0
t1=12t0
10 20 30 40
0
0.02
0.04
0.06
0.10
0.08
(d) Mode 4
H (

m)
0
t1=4t0
50
t1=8t0
t1=12t0
10 20 30 40
0
0.02
0.04
0.06
0.10
0.08
(e) Mode 5
Figure 5. The influence of the time relaxiation
t1
on the frequency
Ω
while varying the beam
thickness
H
and taking into account a fixed
AR =
100 and
KL0 =KLL =
0.5 for a variational mode
of (a) mode 1, (b) mode 2, (c) mode 3, (d) mode 4, and (e) mode 5.
5. Conclusions
In the present study, a comprehensive examination of the free vibrational characteris-
tics and dynamic behavior of thermoelastic damping (TED) and the frequency shift (FS) for
a thin piezoelectro-magneto-thermoelastic (PEMT) composite beam was investigated. The
beam’s model is formulated in accordance with the principles of the Euler–Bernoulli beam

Mathematics 2024,12, 4011 15 of 17
theory, accounting for two linear spring elements attached to both ends. To conduct the
analytical simulation, flexible boundary conditions were implemented, and the eigenvalues
corresponding to the initial five vibrational modes were computed numerically employing
the Newton–Raphson method to ensure a high degree of accuracy. These boundaries under-
taken were chosen due to the imperfections and uncertainties that commonly arise during
the fabrication process. The findings reveal that the impacts of the thermal relaxation time
t1
, the beam geometries, the linear spring coefficients (
KL0
and
KLL
), the critical thickness
(
CT
), and the mode shapes on the thermoelastic damping (TED) and the frequency shift
(FS) are significant and cannot be neglected.
The results indicate that the frequency shifts are high considering stiffer linear springs
and higher values of
t1
. Another observation was made where the amplitudes of oscillations
decrease with increasing the beam’s thickness
H
resembling a phenomenon called energy
dissipation. The influence of the time relaxation
t1
on the thermoelastic damping was also
studied. The obtained results show a linear response considering all vibrational modes,
where a peak value
Q−1
peak
begins elevating as the time relaxation increases. Finally, the
impact of the spring coefficients on TED was examined, as it leads to a dramatic increase
in TED before the peak value was reached. This effect, in fact, reverses as the TED moves
away from the vicinity of the peak value. Indeed, such factors have the potential to induce
substantial modifications in the overall operational efficacy of the proposed designs that
involve damping dissipation.
Author Contributions: Conceptualization, methodology, visualization, validation and draft writing,
A.M.A., S.G. and M.Y.T.; manuscript revision and supervision, A.M.A. and S.G. All authors have read
and agreed to the published version of the manuscript.
Funding: The authors gratefully acknowledge the funding of the Deanship of Graduate Studies and
Scientific Research, Jazan University, Saudi Arabia, through project number: RG24-S0147.
Data Availability Statement: The original contributions presented in this study are included in the
article. Further inquiries can be directed to the corresponding authors.
Acknowledgments: The first and third authors acknowledge the support of Jazan University and the
second author acknowledges the support of Siksha ‘O’ Anusandhan (Deemed to be University) for
providing the products and services that facilitated this research.
Conflicts of Interest: The authors declare no conflicts of interest.
References
1.
Hao, Z.; Erbil, A.; Ayazi, F. An analytical model for support loss in micromachined beam resonators with in-plane flexural
vibrations. Sens. Actuator A Phys. 2003,109, 156–164. https://doi.org/10.1016/j.sna.2003.09.037.
2.
Yasumura, K.Y.; Stowe, T.D.; Chow, E.M.; Pfafman, T.; Kenny, T.W.; Stipe, B.C.; Rugar, D. Quality factors in micron-and
submicron-thick cantilevers. J. Microelectromech. Syst. 2000,9, 117–125. https://doi.org/10.1109/84.825786.
3.
Yang, J.; Ono, T.; Esashi, M. Energy dissipation in submicrometer thick single-crystal silicon cantilevers. J. Microelectromech. Syst.
2002,11, 775–783. https://doi.org/10.1109/jmems.2002.805208.
4. Nádai, A. Die Elastischen Platten; Springer: Berlin/Heidelberg, Germany, 1925. https://doi.org/10.1007/978-3-662-11487-2.
5.
Zener, C. Internal friction in solids. I. Theory of internal friction in reeds. Phys. Rev. 1937,52, 230. https://doi.org/10.1103/
physrev.52.230.
6.
Zener, C. Internal friction in solids II. General theory of thermoelastic internal friction. Phys. Rev. 1938,53, 90.
https://doi.org/10.1103/physrev.53.90.
7.
Zener, C.; Otis, W.; Nuckolls, R. Internal friction in solids III. Experimental demonstration of thermoelastic internal friction. Phys.
Rev. 1938,53, 100. https://doi.org/10.1103/physrev.53.100.
8.
Lifshitz, R.; Roukes, M.L. Thermoelastic damping in micro-and nanomechanical systems. Phys. Rev. B. 2000,61, 5600.
https://doi.org/10.1103/physrevb.61.5600.
9.
Zhou, H.; Li, P. Nonlocal dual-phase-lagging thermoelastic damping in rectangular and circular micro/nanoplate resonators.
Appl. Math. Model. 2021,95, 667–687.

Mathematics 2024,12, 4011 16 of 17
10.
Li, S.R.; Xu, X.; Chen, S. Analysis of thermoelastic damping of functionally graded material beam resonators. Compos. Struct.
2017,182, 728–736. https://doi.org/10.1016/j.compstruct.2017.09.056.
11.
Qin, Z.; Ding, X.; Ge, X.; Ruan, Z.; Li, H. A mode order optimized disk resonator gyroscope considering thermoelastic damping.
Int. J. Mech. Sci. 2022,236, 107737. https://doi.org/10.1016/j.ijmecsci.2022.107737.
12.
Sobhy, M.; Zenkour, A.M. Magnetic field effect on thermomechanical buckling and vibration of viscoelastic sand-
wich nanobeams with CNT reinforced face sheets on a viscoelastic substrate. Compos. B Eng. 2018,154, 492–506.
https://doi.org/10.1016/j.compositesb.2018.09.011.
13.
Hassani, S.; Mousavi, M.; Gandomi, A.H. Structural health monitoring in composite structures: A comprehensive review. Sensors
2021,22, 153. https://doi.org/10.3390/s22010153.
14.
Banh, T.T.; Lee, D. Comprehensive polygonal topology optimization for triplet thermo-mechanical-pressure multi-material
systems. Eng. Comput. 2024,40, 3295–3317.
15.
Habashneh, M.; Rad, M.M. Reliability based topology optimization of thermoelastic structures using bi-directional evolutionary
structural optimization method. Int. J. Mech. Mater. Des. 2023,19, 605–620.
16.
Phan, M.T.; Trinh, X.T.; Le, Q.C.; Ngo, V.K.T.; Nguyen, C.C. Effect of environmental conditions on quality factors of MEMS
cantilever beam resonator in gas rarefaction. Sens. Imaging 2021,22, 6. https://doi.org/10.1007/s11220-020-00329-9.
17.
Le, Q.C.; Phan, M.T.; Trinh, X.T.; Truong, H.L.; Ngo, V.K.T.; Nguyen, C.C. The combined effects of temperature and relative
humidity on resonant frequency and quality factor of MEMS beam resonators in atmospheric pressure and gas rarefaction.
Microsyst. Technol. 2023,29, 1357–1373. https://doi.org/10.1007/s00542-023-05514-w.
18.
Nguyen, C.C.; Ngo, V.K.T.; Le, H.Q.; Li, W.L. Influences of relative humidity on the quality factors of MEMS cantilever resonators
in gas rarefaction. Microsyst. Technol. 2019,25, 2767–2782. https://doi.org/10.1007/s00542-018-4239-x.
19. Van Suchtelen, J. Product properties: a new application of composite materials. Philips Res. Rep. 1972,27, 28–37.
20.
Nan, C.W. Magnetoelectric effect in composites of piezoelectric and piezomagnetic phases. Phys. Rev. B 1994,50, 6082.
https://doi.org/10.1103/physrevb.50.6082.
21.
Lee, J.; Boyd IV, J.G.; Lagoudas, D.C. Effective properties of three-phase electro-magneto-elastic composites. Int. J. Eng. Sci. 2005,
43, 790–825. https://doi.org/10.1016/j.ijengsci.2005.01.004.
22.
Abdehvand, M.Z.; Roknizadeh, S.A.S.; Sedighi, H.M. Modeling and analysis of a coupled novel nonlinear magneto-electro-
aeroelastic lumped model for a flutter based energy harvesting system. Phys. Scr. 2020,96, 025213. https://doi.org/10.1088/1402-
4896/abcdc7.
23.
Gong, Z.; Zhang, Y.; Pan, E.; Zhang, C. Three-dimensional general magneto-electro-elastic finite element model for multiphysics
nonlinear analysis of layered composites. App. Math. Mech. 2023,44, 53–72. https://doi.org/10.1007/s10483-023-2943-8.
24.
Biot, M.A. Thermoelasticity and irreversible thermodynamics. J. Appl. Phys. 1956,27, 240–253. https://doi.org/10.1063/1.1722351.
25.
Lord, H.W.; Shulman, Y. A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 1967,15, 299–309.
https://doi.org/10.1016/0022-5096(67)90024-5.
26. Green, A.E.; Lindsay, K.A. Thermoelasticity. J. Elast. 1972,2, 1–7. https://doi.org/10.1007/BF00045689.
27.
Abouelregal, A.E.; Soleiman, A.; Sedighi, H.M.; Khalil, K.; Nasr, M. Advanced thermoelastic heat conduction model with two
fractional parameters and phase-lags. Phys. Scr. 2021,96, 124048. https://doi.org/10.1088/1402-4896/ac2f80.
28.
Guha, S.; Singh, A.K. Effects of Initial Stresses on Reflection phenomenon of plane waves at the free surface
of a Rotating Piezothermoelastic Fiber-Reinforced Composite half-space. Int. J. Mech. Sci. 2020,181, 105766.
https://doi.org/10.1016/j.ijmecsci.2020.105766.
29.
Mirparizi, M.; Zhang, C.; Amiri, M.J. One-dimensional electro-magneto-poro-thermoelastic wave propagation in a functionally
graded medium with energy dissipation. Phys. Scr. 2022,97, 045203. https://doi.org/10.1088/1402-4896/ac576f.
30.
Duwel, A.; Gorman, J.; Weinstein, M.; Borenstein, J.; Ward, P. Experimental study of thermoelastic damping in MEMS gyros. Sens.
Actuator A Phys. 2003,103, 70–75. https://doi.org/10.1016/S0924-4247(02)00318-7.
31.
Schiwietz, D.; Weig, E.M.; Degenfeld-Schonburg, P. Thermoelastic damping in MEMS gyroscopes at high frequencies. Microsystems
Nanoeng. 2023,9, 11.
32.
Méndez, C.; Paquay, S.; Klapka, I.; Raskin, J.P. Effect of geometrical nonlinearity on MEMS thermoelastic damping. Nonlinear
Anal. Real World Appl. 2009,10, 1579–1588.
33.
Mori, T.; Tanaka, K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metal. 1973,
21, 571–574. https://doi.org/10.1016/0001-6160(73)90064-3.
34.
Li, J.Y.; Dunn, M.L. Micromechanics of magnetoelectroelastic composite materials: average fields and effective behavior. J. Intell.
Mater. Syst. Struct. 1998,9, 404–416. https://doi.org/10.1177/1045389X9800900602.
35.
Li, J.Y.; Dunn, M.L. Anisotropic coupled-field inclusion and inhomogeneity problems. Philos. Mag. A 1998,77, 1341–1350.
https://doi.org/10.1080/01418619808214256.
36.
Pisano, A.; Fuschi, P.; Polizzotto, C. Euler–Bernoulli elastic beam models of Eringen’s differential nonlocal type revisited within a
C 0-continuous displacement framework. Meccanica 2021,56, 2323–2337.

Mathematics 2024,12, 4011 17 of 17
37.
Iandiorio, C.; Salvini, P. Elastic-plastic analysis with pre-integrated beam finite element based on state diagrams: Elastic-perfectly
plastic flow. Eur. J.-Mech.-A/Solids 2023,97, 104837.
38.
Tzou, H. Piezoelectric Shells: Distributed Sensing and Control of Continua; Kluwer Academic Publishers: Dordrecht, The Netherlands,
1993. https://doi.org/10.1007/978-94-011-1783-8.
39. Ugural, A.C. Stresses in Plates and Shells; Southeast Book Company: Taiwan, China, 1989.
40.
Kumar, A.; Chakraborty, D. Effective properties of thermo-electro-mechanically coupled piezoelectric fiber reinforced composites.
Mater. Des. 2009,30, 1216–1222. https://doi.org/10.1080/01932690802701887.
41.
Hou, P.F.; Teng, G.H.; Chen, H.R. Three-dimensional Green’s function for a point heat source in two-phase
transversely isotropic magneto-electro-thermo-elastic material. Mech. Compos. Mater. 2009,41, 329–338.
https://doi.org/10.1016/j.mechmat.2008.12.001.
42.
Guha, S.; Singh, A.K.; Singh, S. Thermoelastic damping and frequency shift of different micro-scale piezoelectro-magneto-
thermoelastic beams. Phys. Scr. 2023,99, 015203. https://doi.org/10.1088/1402-4896/ad0bbd.
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual
author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to
people or property resulting from any ideas, methods, instructions or products referred to in the content.