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Material gradation effects on twisting statics
of bi-directional functionally graded micro-tubes
Cite as: AIP Advances 14, 025228 (2024); doi: 10.1063/5.0194270
Submitted: 26 December 2023 •Accepted: 17 January 2024 •
Published Online: 14 February 2024
Reza Aghazadeh,1,a) Mohammad Rafighi,2, b) Raman Kumar,3,c) and Mohammed Al Awadh4, d)
AFFILIATIONS
1Department of Aeronautical Engineering, University of Turkish Aeronautical Association, Ankara 06790, Türkiye
2Department of Mechanical Engineering, Ba¸skent University, Ankara 06790, Türkiye
3Department of Mechanical Engineering, Guru Nanak Dev Engineering College, Ludhiana 141006, India
4Department of Industrial Engineering, King Khalid University, Abha 61421, Saudi Arabia
a)Electronic mail: raghazadeh@thk.edu.tr
b)Author to whom correspondence should be addressed: mohammad.rafighi@gmail.com
c)Electronic mail: sehgal91@gndec.ac.in
d)Electronic mail: mohalawadh@kku.edu.sa
ABSTRACT
This study aims to characterize the twisting behavior of bi-directional functionally graded (FG) micro-tubes under torsional loads within the
modified couple stress theory framework. The two material properties involved in the torsional static model of FG small-scale tubes, i.e., shear
modulus and material length scale parameter, are assumed to possess smooth spatial variations in both radial and axial directions. Through the
utilization of Hamilton’s principle, the governing equations and boundary conditions are derived, and then, the system of partial differential
equations is numerically solved by using the differential quadrature method. A verification study is conducted by comparing limiting cases
with the analytical results available in the literature to check the validity of the developed procedures. A detailed study is carried out on
the influences of the phase distribution profile and geometric parameters upon twist angles and shear stresses developed in FG micro-tubes
undergoing external distributed torques.
©2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license
(http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0194270
I. INTRODUCTION
In recent years, because of the superior properties of function-
ally graded materials (FGMs), such as their excellent performance in
harsh environments and minimization of stress concentration that
rises due to discontinuities in properties of constituent phases, they
have become ideal candidates to be used in various technological
applications. This new class of inhomogeneous composite materials
possesses predetermined smooth spatial variations in volume frac-
tions of components in order to combine the best properties of
two distinct constituents, and hence, they can be optimized for a
specific purpose. Although FGMs were first used in the aerospace
industry as thermal barriers,1,2 at present, their applications have
spread to different industries, such as biomedicine3,4 and electron-
ics.5Recently, along with developments in FGM manufacturing
techniques,6–8 these advanced composites have been widely used in
micro- and nano-electromechanical systems (MEMS and NEMS)
in the form of small-scale beams, plates, shells, and tubes. Conse-
quently, mechanical analyses of functionally graded (FG) small-scale
structures have broadly attracted researchers’ attention.
It is experimentally evidenced by several studies9–12 that clas-
sical elasticity approaches are insufficient to capture scaling effects,
i.e., when the size gets smaller, conventional continuum theories fail
to predict the mechanical behavior correctly; therefore, higher order
continuum models have been developed, which utilize length scale
parameters to address the side effects.13–18 One of these nonclassical
theories commonly used in mechanical analyses of small-scale struc-
tures is the modified couple stress theory (MCST) proposed by Yang
et al.18 In this theory, the effects of curvature tensor as a strain gradi-
ent measure are also considered in evaluating strain energy besides
AIP Advances 14, 025228 (2024); doi: 10.1063/5.0194270 14, 025228-1
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classical deformation gradient measures. MCST employs a single-
length scale parameter to construct the higher-order constitutive
relation.
Based on nonclassical theories, numerous studies in the tech-
nical literature focus on different small-scale structural problems,
such as extensional, flexural, and torsional statics and dynamics of
micro- and nano-components. The characterization of the mechan-
ical behavior of FG micro- and nano-tubes/micro- and nano-rods
under torsional loading is one of these fundamental problems that,
because of their growing applications in MEMS and NEMS, have
gained high interest recently.19–21 Some studies have been devoted
to investigating the torsional statics and dynamics of small-scale
tubes made of homogeneous materials using strain gradient the-
ory,22 MCST,23–25 and nonlocal elasticity theory.26–28 Fang et al.29
studied the temperature and size effects on the torsional charac-
teristics of a graphene nanoribbon encapsulated in a single-walled
carbon nano-tube by performing molecular dynamics simulations.
In most studies on the torsion of FG small-scale tubes,
material gradation has been considered to occur only through the
radius. Models for predicting twisting and vibrations of radially
FG nano-tubes based on nonlocal elasticity are presented by some
authors.30–32 Setoodeh et al.33 utilized MCST to put forward an
analytical approach for linear and nonlinear torsional free vibra-
tions of FG micro-/nano-tubes. In another work, MCST is used by
Rahaeifard34 to assess the statics and dynamics of through-radius FG
micro-bars.
In recent years, along with advances in FGM processing tech-
niques, the smooth gradation of constitutional phases in two or three
dimensions has become feasible, yielding structures with enhanced
properties and high applicability. As a result, structural problems
involving two- or three-dimensional FG micro-tubes have gained
significant interest recently. Various studies exist on the mechan-
ics of two-dimensionally FG large-scale beams, plates, and cylinders
based on classical elasticity theory.35–39 However, a limited num-
ber of studies examine the torsional behavior of bi-directional FG
micro-tubes. In a study by Barretta et al.,40 based on Eringen’s
nonlocal elasticity theory, the torsion of viscoelastic circular nano-
beams possessing radially quadratic and arbitrary axial distributions
of material properties is investigated, and in another study based on
a nonlocal elasticity approach, Li and Hu41 examined the torsional
free vibrations of nano-tubes with continuous material gradation in
axial and radial directions. In another study by Li and Hu,42 within
the framework of MCST, a procedure to evaluate the angle of rota-
tion and shear stress distribution in bi-directional FG micro-tubes is
proposed.
All studies mentioned above disregarded the variation of length
scale parameters through the FG medium, which is a simplifying
assumption. Experimental studies conducted by some researchers
revealed that length scale parameters are material properties.43–46
A few studies demonstrate the required devices and setups for the
experimental determination of length scale parameter values.47 For
instance, the length scale parameter employed in the modified cou-
ple stress theory is defined as the square root of the ratio of the
modulus of curvature to the shear modulus.45,46 Both the modulus
of curvature and shear modulus are material constants, suggest-
ing that the length scale parameter itself is a material property.
Another example is provided by Nikolov et al.,43 who illustrated
the dependency of the material length scale parameter in polymers
on cross-link density, chain interactions, and chain stiffness, all
of which are microstructural properties of the constituent. Con-
sequently, FGM gradation rules must also be applied for these
parameters. Although there exist a limited number of studies that
treat small-scale parameters as material properties for analyzing
beams48,49 and plates,50–53 to our best knowledge, no endeavor is
reported on torsion of FG micro-tubes with variable length scale
parameters in the literature. Accordingly, the current study seems
to be the first effort to regard the length scale parameter utilized
in MCST as a material constant in modeling torsional statics of FG
small-scale tubes.
The current study aims to present an MCST-based model
for predicting twisting characteristics of bi-directional micro-tubes
under torsional loads. All material properties, including shear
modulus and length scale parameters, obey power-law and expo-
nential distribution schemes in radial and longitudinal directions.
Hamilton’s principle is employed to derive a system of equa-
tions governing torsional statics of two-dimensionally FG small-
scale tubes. The system is numerically solved using the differential
quadrature method (DQM). Parametric studies are carried out to
illustrate the influences of different geometric and material fea-
tures on the angle of rotation and shear stress distribution of FG
micro-tubes under externally applied distributed torques.
II. THEORETICAL PRELIMINARIES
According to MCST,18 the total strain energy U for a lin-
early elastic, deformable, isotropic material occupying volume Ωis
expressed as follows:
U=1
2∫Ω(σijεij +mij χij)dV, (1)
where εij,χij,σij, and mij represent the components of strain tensor,
symmetric curvature tensor, Cauchy stress tensor, and the deviatoric
part of couple stress tensor, respectively. εij and χij are the first- and
second-order deformation gradient measures, respectively, which
are expressed in terms of displacement field through the following
relations:
εij =1
2(ui,j+uj,i), (2a)
χij =1
2(eipqεqj,p+ejpq εqi,p). (2b)
Here, eijk represents the components of alternating tensor and the
comma stands for differentiation. ui(i =1, 2, 3) designate the dis-
placement field of the bi-directional FG micro-tube at time t, and
for tubes undergoing torsion, they can be written in the following
form:
u1=0, (3a)
u2(x1,x3,t)=−x3ψ(x1,t), (3b)
u3(x1,x2,t)=x2ψ(x1,t). (3c)
Note that displacements of any point along x1,x2, and x3directions
are, respectively, denoted by u1,u2, and u3.ψ(x1,t) designates the
AIP Advances 14, 025228 (2024); doi: 10.1063/5.0194270 14, 025228-2
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FIG. 1. Configuration of radially and axially FG tubes.
rotation angle about the center of twist, located on the x1-axis for
the axisymmetric configuration considered in the current study. The
coordinate system and spatial variations in material properties for a
typical two-dimensional FG tube with length L, inner radius ri, and
outer radius roare shown in Fig. 1.
Incorporation of Eqs. (3a)–(3c) into Eqs. (2a) and (2b) leads to
the following non-zero components of εij and χij:
ε12 =ε21 =−1
2x3
∂ψ
∂x1,ε13 =ε31 =1
2x2
∂ψ
∂x1, (4a)
χ11 =∂ψ
∂x1,χ22 =−1
2
∂ψ
∂x1,χ33 =−1
2
∂ψ
∂x1,
χ12 =χ21 =−1
4x2
∂2ψ
∂x2
1,χ13 =χ31 =−1
4x3
∂2ψ
∂x2
1.
(4b)
Classical and nonclassical stresses are expressed through the use of
associated constitutive relations given by
σ12 =σ21 =−μx3
∂ψ
∂x1,σ13 =σ31 =μx2
∂ψ
∂x1, (5a)
mij =2μl2χij, (5b)
where μis the shear modulus and lis the material length scale
parameter.
III. DERIVATION OF THE GOVERNING EQUATIONS
The system of partial differential equations and boundary
conditions governing the size-dependent torsional statics of bi-
directional FG micro-tubes are derived by employing Hamilton’s
principle, which postulates that
δ∫t2
t1(W−U)dt =0. (6)
Here, Wdenotes the work done by external forces. In the present
study, a distributed torque Teis considered as the external force
whose work is given by
W=∫L
oTe(x1)ψdx1. (7)
The continuous variations in constituent phases of the bi-directional
FG micro-tube considered in the current study are captured by
utilizing two FGM indices, αand β, for axial, x1, and radial, r, direc-
tions, respectively. The material gradation is an exponential function
of x1and obeys a power-law function along r. Consequently, a
typical material property represented by Z, including μand l, is
expressed in the following form:41
Z(r,x1)=Zr(r)Zx1(x1),
Zx1(x1)=expαx1
L,Zr(r)=(Zli −Zlo)ro−r
ro−riβ+Zlo,
Zri =exp(α)Zli,Zro =exp (α)Zlo.
(8)
Note that subscripts “li,” “lo,” “ri,” and “ro” in Eq. (8) are used to
identify left inner (x1=0, r=ri), left outer (x1=0, r=ro), right inner
(x1=L,r=ri), and right outer (x1=L,r=ro) edges, respectively.
Substituting the strain energy Uand work Winto Eq. (6) yields
the following governing equation:
J55
∂
∂x1μx1
∂ψ
∂x1+3A552
∂
∂x1μl2x1
∂ψ
∂x1
−1
4J552
∂2
∂x2
1μl2x1
∂2ψ
∂x2
1+Te=0. (9)
The boundary conditions are obtained as
ψ=0 or J55μx1
∂ψ
∂x1+3A552μl2x1
∂ψ
∂x1
−1
4J552
∂
∂x1μl2x1
∂2ψ
∂x2
1=0, (10a)
∂ψ
∂x1=0 or 1
4J552μl2x1
∂2ψ
∂x2
1=0. (10b)
Note that in this study, the length scale parameter l is also assumed
to be a material property; hence, μl2used in the higher order con-
stitutive relation given by Eq. (5b) is considered a distinct material
property. As a result, Eq. (8) must also be applied for μl2, which
yields
μl2=μr(r)lr(r)2μ(x1)l(x1)2=μl2rμl2x1. (11)
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TABLE I. Comparison of the maximum angle of rotation ψmax (×10−4rad) and the
maximum shear stress τmax (MPa) for radially FG micro-tubes, with llo =lli,α=0.0,
and T0=0.5 N m/m.
β
ψmax (×10−4rad) τmax (MPa)
Present Analytical42 Present Analytical42
0.0 7.1574 7.1574 44.9713 44.9713
0.5 4.2003 4.2003 70.9268 70.9268
1.0 3.5491 3.5491 59.9304 59.9304
2.0 3.1221 3.1221 52.7192 52.7192
10.0 2.7481 2.7481 46.4040 46.4040
The stiffness and inertia terms appearing in Eqs. (9) and (10) can be
evaluated by
J55 =∫Aμrx2
2+x2
3dA =∫Aμrr2dA =2π∫ro
riμrr3dr, (12a)
{A552,J552}=∫Aμl2r1,x2
2+x2
3dA
=∫Aμl2r1,r2dA =2π∫ro
riμl2rr,r3dr, (12b)
where Ais the cross-sectional area.
IV. METHOD OF SOLUTION
The system of partial differential equations can be solved using
the differential quadrature method (DQM). According to DQM, the
mth derivative of the angle of rotation ψat any grid point along the
x1direction, x1i, can be approximated by a weighted sum of ψvalues
at all grid points,
∂mψ(x1)
∂xm
1x1=x1i=N
∑
j=1c(m)
i j ψ(x1j)fori=1 ,2, ... ,N, (13)
where c(m)
i j are the DQM weighting coefficients for the mth
derivative and Nis the number of grid points. Chebyshev nodes
FIG. 2. Angle of rotation of the bi-directional FG micro-tube when (a) llo =lli =0, (b) llo/lli =0.2, (c) llo /lli =1.0, and (d) llo/lli =2.0.
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are used to discretize the tube in a longitudinal direction as
follows:
x1j=L
21−cosπ(j−1)
N−1 for j=1,2 ,... ,N. (14)
By utilizing DQM formulated in Eq. (13), the governing equations
and boundary conditions are recast into the following system of
linear algebraic equations:
DΨ+Q=0, (15)
where Dis known as the coefficient matrix, Qis the forc-
ing vector resulting from the distributed torque, and Ψis an
unknown angular displacement vector containing ψvalues at all grid
points,
Ψ={ψp},p=1,2, ...,N. (16)
V. NUMERICAL RESULTS
Here, the numerical results regarding the twisting statics of
two-dimensionally FG micro-tubes with both ends fixed are gener-
ated and analyzed. The boundary conditions for a fixed-fixed tube
can be written as
ψ=0 at x1=0, L, (17a)
∂2ψ
∂x2
1=0 atx1=0, L. (17b)
The bi-directional FG tube, which is considered for parametric
investigations in the current study, is made of a metal phase [alu-
minum (Al) with μli =48 GPa] at the left inner edge and a ceramic
phase [silicon carbide (SiC) with μlo =129 GPa] at the left outer
edge. The distribution of length scale parameters is achieved through
the ratio llolli. Different distribution patterns for l can be generated
by changing this ratio, which can reveal the influence of variable
length scale parameters upon numerical results. Using llolli ≠1
FIG. 3. Maximum shear stress of the bi-directional FG micro-tube when (a) llo =lli =0, (b) llo/lli =0.2, (c) llo /lli =1.0, and (d) llo/lli =2.0.
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along with non-zero values for αand β, the variations of the length
scale parameter in radial and axial directions can be captured. llolli
=1 and α=0 indicate a constant length scale parameter for the
whole bi-directional FG tube.
The geometric dimensions of the micro-tube are illustrated in
Fig. 1. Unless otherwise mentioned, the inner and outer radii and
the length are taken as ro=50 μm, ri/ro=1/2, and L=120 μm.
In order to keep the length scale parameter around geometric
dimensions, its value is related to the outer radius as ro/lli =2. A
uniformly distributed torque Te=0.1 N m/m is used as external
loading.
Table I shows the maximum angle of rotation ψmax and
maximum shear stress τmax for radially FG micro-tubes com-
pared to the results given by Li and Hu.42 In their study, Li and
Hu42 provided analytical expressions for the angle of rotation and
shear stresses of the radially FG micro-tube (α=0) with constant
length scale parameters undergoing a sinusoidally distributed torque
Te(x1)=T0sin(πx1L). An excellent correspondence is observed,
indicating the accuracy of the procedures developed in the current
study. Note that the magnitude of shear stress τat any point can be
evaluated as
τ=σ2
12 +σ2
13 =
−μx3
∂ψ
∂x12+μx2
∂ψ
∂x12
=
μ∂ψ
∂x12x2
3+x2
2=μr∂ψ
∂x1. (18)
From Eq. (18), it is evident that, at any axial distance, τmax occurs on
the outer surface, r=ro.
Figure 2 illustrates the through-the-length variation of the rota-
tion angle for different values of FGM indices αand β. Each plot
in Fig. 2 is generated using different ceramic-to-metal length scale
parameter ratio values llolli while keeping lli constant. The provided
results indicate that larger values of llolli, which indicate a larger
equivalent length scale parameter, lead to stiffer tubes with small
values of twist angle. It can also be observed that an increase in
the values of α, or β, or both results in a lower angle of rotation.
FIG. 4. Shear stress distribution through the radius of the bi-directional FG micro-tube at x1=Lwhen (a) llo =lli =0, (b) llo/lli =0.2, (c) llo /lli =1.0, and (d) llo/lli =2.0.
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This fact can be justified by knowing that as αand βget larger, the
tube becomes ceramic-rich with a higher equivalent value of shear
modulus. For homogeneous or radially FG tubes, due to the length-
independent material distribution, the maximum value of ψis seen
to be computed at the midspan, x1=L/2. However, for axially FG
tubes whose αis non-zero, the location of ψmax tends to the left side,
which possesses poor material properties with respect to the ones on
the right-hand side. By letting β=0, through-the-radius variations in
volume fractions are neglected, yielding the same distribution profile
for lregardless of the llolli value. Thus, the results of zero-βvalues
in Figs. 2(b)–2(d) are equal to each other.
Through-the-length distribution of maximum shear stresses
τmax, which occurs at r=ro, is depicted in Fig. 3. Although for
α=0, the τmax profile is symmetric about mid-section with a min-
imum value at x1=L/2, for non-zero values of α, maximum values
of τmax are observed at the right end, and the location of minimum
shear stress is shifted to the left. Similar to the conclusion made for
Fig. 2, the cases with β=0 in Figs. 3(b)–3(d) are unaffected by the
change in llolli. Further inspection of Fig. 3 reveals that a rise in β
while keeping αas constant results in a drop in the magnitude of
τmax. However, by increasing αat a constant β-value, the maximum
shear stress on the left- and right-hand sides becomes smaller and
larger, respectively.
The radial distribution of shear stress τat x1=L, where its max-
imum value occurs, for different FGM indices αand βis provided in
Fig. 4. For only-axially FG micro-tubes, i.e., when β=0, the radial
distribution is linear, whereas a non-zero βintroduces nonlinearity
in the distribution profile of shear stress along the radial direction. It
can also be seen that, at a constant β-value, higher α-values result in
a corresponding increase in τat the right end.
Figure 5 illustrates the variation of the maximum rotation angle
concerning ro/lli for classical, i.e., llo =lli =0, and nonclassical, i.e.,
llolli =2, bi-directional FG tubes. The axial FGM index is kept con-
stant as α=0.5, and three different radial FGM index βvalues are
used to produce the results. Note that, in Fig. 5, to change the value
of ro/lli, the value of lli is altered and rois taken as constant. The
sensitivity of ψmax to the changes in the values of length scale para-
meters of phases is observed to be dramatically pronounced as lli gets
FIG. 5. Maximum angle of rotation of the bi-directional FG micro-tube with α=0.5.
larger than or close to ro. However, increasing ro/lli leads to a corre-
sponding rise in the maximum angle of rotation converging to those
predicted by the classical continuum theory.
VI. CONCLUDING REMARKS
The current study presents a model and analysis procedures for
the torsional statics of bi-directional FG micro-tubes based on the
modified couple stress theory. The proposed methods allow consid-
eration of variations of the material properties, including the length
scale parameter, through both radial and axial directions. In order
to solve a system of governing equations and boundary conditions,
a differential quadrature technique is employed. The accuracy of the
developed procedures is verified by making comparisons of certain
limiting cases to the results available in the literature. The numerical
results include through-the-length and through-the-radius distribu-
tions of shear stresses and angles of rotation for different values of
length scale parameters and FGM indices.
The length scale parameter ratio llo/lli determines the length
scale parameter distribution pattern in a bi-directional FG micro-
tube. By observing the sensitivity of the results to this ratio, it can
be concluded that considering spatial variations of the length scale
parameter in a two-dimensional FG micro-tube is indispensable for
the sufficiently accurate prediction of the twisting behavior.
The results also show that FG indices αand β, which character-
ize the phase distribution profile, significantly influence the torsional
statics of the FG tubes. The proposed model and results capture
these effects, which can be utilized as a powerful means for design
purposes.
The influence of changing the effective value of the length scale
parameter to the values with orders close to or beyond the geomet-
rical dimensions of the micro-tube is examined by altering the ratio
ro/lli. From the results provided, it can be found that the larger the
effective length scale parameter is, the smaller the rotation angle
is. The influence of lon the rotation angle vanishes when it gets
sufficiently smaller than geometrical dimensions.
ACKNOWLEDGMENTS
The authors extend their appreciation to the Deanship of
Scientific Research at King Khalid University for funding this
work through a large group research project under Grant No.
RGP2/306/44.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Reza Aghazadeh: Conceptualization (equal); Investigation (equal);
Methodology (equal); Supervision (equal); Validation (equal); Visu-
alization (equal); Writing – review & editing (equal). Mohammad
Rafighi: Formal analysis (equal); Investigation (equal); Methodol-
ogy (equal); Validation (equal); Writing – review & editing (equal).
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Raman Kumar: Data curation (equal); Project administration
(equal); Writing – review & editing (equal). Mohammed Al Awadh:
Data curation (equal); Resources (equal); Writing – review &
editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available
within the article.
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