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Composites Part B 118 (2017) 41e53

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Composites Part B

journal homepage: www.elsevier .com/locate/compositesb

Thermal response of ceramic matrix nanocomposite cylindrical shellsusing Eshelby-Mori-Tanaka homogenization scheme

B. Sobhaniaragh a, *, R.C. Batra b, W.J. Mansur c, d, F.C. Peters c, d

a Young Researchers and Elite Club, Arak Branch, Islamic Azad University, Arak, Iranb Department of Biomedical Engineering and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USAc Modelling Methods in Engineering and Geophysics Laboratory (LAMEMO), COPPE, Federal University of Rio de Janeiro, 21941-596, RJ, Brazild Department of Civil Engineering, COPPE, Federal University of Rio de Janeiro, RJ, Brazil

a r t i c l e i n f o

Article history:Received 10 December 2016Received in revised form5 February 2017Accepted 19 February 2017Available online 8 March 2017

Keywords:Ceramic-matrix compositeCarbon nanotubesTwo-parameter Eshelby-Mori-TanakaschemeTemperature-dependent propertiesSigmoidal power-law distributionThird-order shear deformation theory

* Corresponding author. Young Researchers and EliAzad University, Arak, Iran

E-mail addresses: [email protected],(B. Sobhaniaragh).

http://dx.doi.org/10.1016/j.compositesb.2017.02.0321359-8368/© 2017 Elsevier Ltd. All rights reserved.

a b s t r a c t

We find thermal stresses developed in Ceramic Matrix Composite (CMC) cylindrical shells reinforcedwith aggregated Carbon Nanotubes (CNTs) with heat flux prescribed on the inner surface and temper-ature on the outer surface. Null surface tractions are prescribed on these two surfaces and the cylinderedges are clamped. The material properties are homogenized by using a two-parameter Eshelby-Mori-Tanaka (EMT) approach. Material properties of the ceramic are assumed to depend upon the tempera-ture, and the smooth variation of the CNT volume fraction through the shell thickness is assumed to bedescribed either by a sigmoidal function or profile-O or profile-X often used in the literature. The one-way coupled thermo-mechanical problem is analyzed by first numerically solving the nonlinear heatequation with the Generalized Differential Quadrature Method (GDQM), and then the linear mechanicalproblem by using Reddy's Third-order Shear Deformation Theory (TSDT) and the GDQM. For the samethermal boundary conditions and the volume fraction of CNTs, the maximum hoop, the in-plane shearand the transverse normal stresses developed in the cylinder are highest for the profile-X of CNTs. Theaggregation factor noticeably influences the maximum transverse normal and the maximum hoopstresses developed in the cylinder.

© 2017 Elsevier Ltd. All rights reserved.

1. Introduction

The outstanding mechanical properties of Carbon Nano-Tubes(CNTs) make them unique candidates for reinforcing polymer,ceramic andmetal matrix composites [1]. Whereas there has been aflurry of activity in studying mechanical properties of CNT-reinforced polymer composites not much work has been done instudying CNTs-reinforced ceramic composites. The combination ofextraordinary characteristics of CNTs with intrinsic advantages ofceramic materials such as thermal stability, high corrosion resis-tance, lowmass density and electrical insulation can generate CNT-Ceramic Matrix Composites (CNT-CMCs) with desired functionaland structural properties [2,3]. The potential of developing high-performance CNT-CMCs that can withstand high temperature,

te Club, Arak Branch, Islamic

[email protected]

severe chemical environment, and wear is very appealing withapplications in such diverse areas as gas turbines, aerospace com-ponents and automobiles. Recent investigations on CNT-CMCs forpotential structural applications are described in Refs. [4e15].

Zhan et al. [11,12] fabricated fully dense nanocompositescomprised of CNTs and nanocrystalline alumina matrix by usingspark-plasma sintering, and demonstrated that their electricalconductivity improved over that of pristine alumina. Furthermore,CNT-alumina matrix nanocomposites with enhanced hardness andfracture toughness were successfully fabricated by Mo et al. [13] bya sol-gel process followed by spark plasma sintering. Zhu et al. [14]prepared bulk alumina ceramic composites by adding 2 wt.% CNTsunder an alternating current electric field. By using catalytic py-rolysis of acetylene gas with iron nitrate impregnated alumina, Anet al. [15] prepared CNT-alumina composites by hot-pressing themand investigated their mechanical and tribological properties.

During the solution of thermo-mechanical problems for CNT-ceramic composites, one usually deduces their effective moduliby employing homogenization techniques such as the ExtendedRule of Mixtures (ERM) [16e18] or the Eshelby-Mori-Tanaka (EMT)

B. Sobhaniaragh et al. / Composites Part B 118 (2017) 41e5342

[19e21] method. Shen [16] modified the conventional rule ofmixtures by introducing CNT efficiency parameters that weredetermined by matching the elastic moduli of polymer-basednanocomposites from molecular dynamics simulation results andthe rule of mixtures. The ERM's limitations include disregardingintrinsic features, such as aggregation and waviness, of CNTsdispersed in the matrix. The EMT is based on the equivalent elasticinclusion concept of Eshelby [22] and the technique for estimatingthe average stress in thematrix phase proposed byMori and Tanaka[23]. The EMT has been applied to CNT-Reinforced Composites(CNTRCs) by Odegard et al. [19] and has been improved upon byusing test data [24e27] to account for the degree of the CNT ag-gregation, orientations, curviness and length. The Scanning Elec-tronMicroscopy (SEM) images [28,29] show that CNTs aggregate ina nanocomposite and form local regions of CNTs-concentrationhigher than their average volume fraction in the nanocomposite.Considering the CNT aggregation effects in the EMT model, Sob-haniaragh et al. [21,30] investigated natural frequencies andbending deformations of the CNTRC shells; e.g., see Sobhaniaragh'sPhD thesis [31]. Furthermore, this methodology has been followedby Fantuzzi et al. [32], Kamarian et al. [33], Tornabene et al. [34],and Heshmati [35] for studying effects of agglomeration and dis-tributions of CNTs on free vibration characteristics of CNT-nanocomposites.

Meguid and Sun [36] have experimentally shown that me-chanical properties of CNTRCs deteriorate if the volume fraction ofCNTs scattered within the matrix phase exceeds a limiting value.Thus, a designer should consider this limiting value of the CNTvolume fraction for fulfilling multi-functional requirements. Apossibility is to smoothly vary the volume fractions of CNTs andthus fabricate Functionally Graded Materials (FGMs) to economi-cally achieve the desired multifunctionality. In recent years, severalresearchers have implemented the idea of smooth gradation ofspatial CNT volume fraction in modeling nanocomposite structuresto achieve desired structural response. For example, a large numberof papers [21,32e35,37e46] have been devoted to studying vibra-tional characteristics of FG CNTRCs. The bending, buckling andpostbuckling deformations of such structures have been studied in

Fig. 1. Geometry of a CMC cylindrical sh

Refs. [30,46e52], and the thermal response of FG CNTRCs has beenstudied in Ref. [53]. Using the piezo-electricity theory and the ERM,Alibeigloo [53] studied thermo-elastic deformations of simplysupported CNTRC cylindrical panel integrated with piezoelectriclayers. With the temperature prescribed on the inner and the outersurfaces of the panel, the author concluded that the maximumradial stress as well as the axial and the circumferential displace-ments could be reduced by having a higher concentration of CNTsnear the outer surface.

Motivated by the lack of research activity on thermal analysis ofceramic matrix nanocomposites reinforced by aggregated CNTs, westudy here thermo-elastic response of CNT-CMC cylindrical shellssubjected only to heat flux and temperature loads. A two-parameter EMT homogenization technique is employed that ac-counts for aggregation of CNTs in the ceramic matrix phase, andmaterial properties of the ceramic matrix are assumed to betemperature-dependent. Consequently, the effective mechanicalproperties of CNTRCs are a function of both temperature and po-sition, leading to nonlinear differential equations that cannot beanalytically solved. The analysis of the mechanical problem issimplified by using Reddy's Third-order Shear Deformation Theory(TSDT) [54] with five unknowns. The nonlinear heat equation andthe TSDT equations are numerically solved by employing the two-dimensional Generalized Differential Quadrature Method (GDQM)proposed by Shu [55] that has been shown to provide reasonablyaccurate solutions [56]. The smooth through-the-thickness spatialvariation of CNTs volume fraction is characterized by a sigmoidalpower-law distribution and two other distributions used in theliterature [37,53].

2. Problem description

Consider a CNT-CMC cylindrical shell of finite length L, meanradius a3m, and thickness h. As depicted in Fig. 1, we use globalcylindrical coordinate axes ða1;a2;a3Þ with the origin at the mid-surface of the left edge to label a material point of the cylinder inthe unstressed reference configuration. The a1-, the a2- and thea3-axes are, respectively, along the cylinder length, the

ell reinforced by aggregated CNTs.

B. Sobhaniaragh et al. / Composites Part B 118 (2017) 41e53 43

circumferential direction, and the cylinder thickness.

2.1. Two-parameter Eshelby-Mori-Tanaka (EMT) model

We briefly review a two-parameter EMT model for findingeffective mechanical properties of CNT-CMCs. This homogenizationscheme is based on Eshelby's theory [22] of elastic inclusions.Eshelby's results for a single inclusion in a semi-infinite elastic,isotropic and homogeneous medium are extended by the Mori-Tanaka method [23] to capture effects of numerous in-homogeneities embedded in a finite domain. Due to low bendingstiffness and weak van der Waals forces [29], CNTs dispersed inCNT-CMCs aggregate in some regions resulting in a higher accu-mulation of CNTs there as compared to their average volume frac-tion in the entire CNT-CMCs. The regions with accumulated CNTsare assumed to be spherical and envisioned as inclusions that havematerial properties different from those of the material surround-ing them. Hence, CNTs dispersed in the matrix are sorted accordingto their positions, i.e., embedded either inside or outside the in-clusions, as depicted in Fig. 1. According to the two-parameter EMTmodel, the total volume Vr of the reinforcing phase is divided intotwo different parts [24]:

Vr ¼ Vincr þ Vm

r (1)

with Vincr and Vm

r denoting, respectively, the volume of CNTsembedded in the inclusions (aggregated CNTs) and that dispersedin the matrix. The aggregation of CNTs in the ceramic matrix de-grades its mechanical properties as compared to that of the virginmatrix. This aspect of the CNT-CMCs is described by introducing theaggregation parameters, z and x, defined as

z ¼ Vinc

V; x ¼ Vinc

rVr

(2)

where Vinc is the effective volume of the inclusions, and theparameter z equals the volume fraction of inclusions relative to thevolume V of the representative volume element. For z ¼ 1, there isno aggregation of CNTs in the matrix phase, and for 0 < z < 1 thedegree of aggregation of CNTs decreases with an increase in z. Theparameter x equals the ratio of the CNTs volume embedded in theinclusions to the total volume of the CNTs. Thus for all CNTs accu-mulated in the inclusions, x ¼ 1. For the CNTs to aggregate

x> z (3)

In order to predict the effective elastic moduli of the inclusionsand the matrix phase, it is assumed that the volume fraction ofCNTs varies only in the thickness direction. Furthermore, the CNTsare assumed to be transversely isotropic and randomly oriented inthe spherical inclusions. Whereas some authors have assumed theaxis of transverse isotropy to be along the CNTcentroidal axis, Batraand Sears [57] have postulated it to be a radial line. The effectivebulkmodulus Kinða3Þ and the effective shearmodulus Ginða3Þ of thespherical inclusions are given by [24,31,40]

Kinða3Þ ¼ Km þ frða3Þxðdr � 3KmarÞ3ðz� frða3Þxþ frða3ÞxarÞ

(4)

Ginða3Þ ¼ Gm þ frða3Þxðhr � 2GmbrÞ2ðz� frða3Þxþ frða3ÞxarÞ

(5)

Similarly, the effective bulk modulus Koutða3Þ and the effectiveshear modulus Goutða3Þ of the material outside the inclusions aregiven by

Koutða3Þ ¼ Km þ frða3Þð1� xÞðdr � 3KmarÞ3ð1� z� frða3Þð1� xÞ þ frða3Þð1� xÞarÞ (6)

Goutða3Þ ¼ Gm þ frða3Þð1� xÞðhr � 2GmbrÞ2ð1� z� frða3Þð1� xÞ þ frða3Þð1� xÞbrÞ

(7)

In Eqs. (4)e(7) Km and Gm, respectively, denote the bulk and theshear moduli of the matrix;frða3Þ and fmða3Þ withfrða3Þ þ fmða3Þ ¼ 1, respectively, denote volume fractions of thereinforcing and the matrix phases. Parameters ar ,br ,dr , and hr aredefined by

ar ¼ 3ðKm þ GmÞ þ kr þ lr3ðGm þ krÞ (8)

br ¼15

�4Gm þ 2kr þ lr3ðGm þ krÞ þ 4Gm

Gm þ pr

þ 2½Gmð3Km þ GmÞ þ Gmð3Km þ 7GmÞ�Gmð3Km þ GmÞ þmrð3Km þ 7GmÞ

�(9)

dr ¼ 13

�nr þ 2lr þ ð2kr þ lrÞð3Km þ 2Gm � lrÞ

Gm þ kr

�(10)

hr ¼15

�23ðnr � lrÞ þ 8Gmpr

Gm þ prþ ð2kr � lrÞð2Gm þ lrÞ

3ðGm þ krÞ

þ 8mrGmð3Km þ 4GmÞ3Kmðmr þ GmÞ þ Gmð7mr þ GmÞ

�(11)

in which kr;mr ;nr and lr are Hill's elastic moduli for the CNTs[58,59].

The effective bulk modulus Kða3Þ and the effective shearmodulus Gða3Þ of the CNT-CMCs are given by

Kða3Þ ¼ Koutða3Þ241þ

z

�Kinða3ÞKoutða3Þ � 1

�

1þ d0ða3Þð1� zÞ�

Kinða3ÞKoutða3Þ � 1

�35 (12)

Gða3Þ ¼ Goutða3Þ241þ

z

�Ginða3ÞGoutða3Þ � 1

�

1þ bða3Þð1� zÞ�

Ginða3ÞGoutða3Þ � 1

�35 (13)

where

d0ða3Þ ¼1þ voutða3Þ

3ð1� voutða3ÞÞ(14)

bða3Þ ¼2ð4� 5voutða3ÞÞ3ð1� voutða3ÞÞ

(15)

The Poisson's ratio, voutða3Þ, of the matrix phase is given by

voutða3Þ ¼3Koutða3Þ � 2Goutða3Þ2½3Koutða3Þ þ Goutða3Þ�

(16)

We assume the CNT-CMC to be isotropic [24] and find itseffective Young's modulus Eða3Þ and Poisson's ratio vða3Þ from thefollowing relations


Eða3Þ ¼9Kða3ÞGða3Þ

3Kða3Þ þ Gða3Þ(17)

vða3Þ ¼3Kða3Þ � 2Gða3Þ6Kða3Þ þ 2Gða3Þ

(18)

Furthermore, the thermal expansion coefficient, a, is calculatedby Ref. [47].

aða3Þ ¼frða3ÞErar þ fmða3ÞEmam

frða3ÞEr þ fmða3ÞEm(19)

where ar and am are the thermal expansion coefficients of the CNTsand the matrix, respectively. The thermal conductivity, kða3Þ, of theCNC-CMT is expressed as [60,61].

kða3Þ ¼�1þ qfrða3Þ

3kr

kmð2ak=dþ qþ kr=kmÞ�km (20)

where ak is the Kapitza radius, km and kr are thermal conductivitiesof the matrix and the reinforcing phases, respectively, d and l(q ¼ l=d) are the diameter and the length of the CNTs, respectively.

We note that frða3Þ ¼ V*CNTVCNT ða3Þ where VCNT ða3Þ equals the

CNT distribution through the shell's thickness, and V*CNT is the

volume fraction of CNTs [62] given by

V*CNT ¼

�rCNT

wCNTrm� rCNT

rmþ 1��1

(21)

where rCNT and rm equal, respectively, the mass densities of theCNTs and thematrix, andwCNT equals themass fraction of the CNTs.

Here we have assumed that the CNT volume fraction is given bythe following sigmoidal power-law distribution

VCNT ða3Þ ¼

es~r � 1�es=2 � 1

��esð~r�0:5Þ þ 1

�!g

(22)

in which ~r ¼ ða3 þ h=2Þ=h, the sigmoid exponent s and theparameter g control the CNT-variation profile in the radial (or thethickness) direction as depicted in Fig. 2. The inner surface of the

Fig. 2. Through-the-thickness distribution of the CNT volume fraction for variousvalues of the sigmoid exponent, s, when g ¼ 1.

shell with the heat flux prescribed has lower CNT volume fractionthat should alleviate thermal stresses and provide desired struc-tural response. With an increase in the value of s, the continuouslygraded shell approaches a laminated shell with two lamina havingCNT volume fractions of 0 and V*

CNT on the inner and the outersurfaces, respectively. Here, we compute results for CNT profilesoften used by others [21,37,53] as well as for profile-X and profile-O.It was reported in Ref. [21] that the CNT volume fractions sym-metrically distributed about the shell midsurface are more effectivein reducing (increasing) the natural frequency than the uniform(asymmetric) distributions.

2.2. Temperature-dependent material properties

It is anticipated that continuously graded CNT-CMCs studied inthis paper will be used in high temperature environments andmaterial properties of the ceramic phase are expected to be tem-perature dependent. Accordingly, we consider the temperaturedependence of mechanical properties of the CNT-CMC, and assumethat Young's modulus E, the thermal conductivity k, and the ther-mal expansion coefficient a (denoted by P in Eq. (23)) are given by

PðTÞ ¼ P0�P�1T

�1 þ 1þ P1T þ P2T2 þ P3T

3

(23)

where P0, P�1, P1, P2 and P3 are constants whose values for Aluminaas the ceramic matrix are taken from Refs. [13,63].

2.3. Third-Order Shell Theory (TSDT)

Based on Reddy's [54] TSDT inwhich straight lines normal to theshell mid-surface before deformation do not necessarily remainstraight during the deformation, the following displacement field isassumed:

U1 ¼ u1ða1;a2Þ þ a3f1ða1;a2Þ � Ca33�f1 þ

vu3va1

�

U2 ¼ u2ða1;a2Þ þ a3f2ða1;a2Þ þ Ca33�

u2a3m

� f2 �1

a3m

vu3va2

�

U3 ¼ u3ða1;a2Þ(24)

where C ¼ 4=3h2, and U1, U2 and U3 are displacement components,respectively, along the a1-, the a2- and the a3- axes. Substitutingfrom Eq. (25) into the strain-displacement relations [64] gives

ε11 ¼ ε011 þ a3ε

011 þ a3

2ε

0011 þ a3

3ε

00011

ε22 ¼ ε022 þ a3ε

022 þ a3

2ε

0022 þ a3

3ε

00022

ε12 ¼ ε012 þ a3ε

012 þ a3

2ε

0012 þ a3

3ε

00012

ε13 ¼ ε013 þ a3ε

013 þ a3

2ε

0013 þ a3

3ε

00013

ε23 ¼ ε023 þ a3ε

023 þ a3

2ε

0023 þ a3

3ε

00023

(25)

where

ε011 ¼u1;1; ε

011 ¼f1;1; ε

0011 ¼0; ε

00011 ¼�C

�f1;1þu3;11

�ε022 ¼ð1=a3mÞu2;2þu3=a3m; ε

022 ¼ð1=a3mÞf2;2; ε

0022 ¼0

ε

0022 ¼C

�ð1=a3mÞu2;2�ð1=a3mÞf2;2�ð1=a3mÞu3;22�

ε012 ¼ð1=a3mÞu1;2þu2;1; ε

012 ¼ð1=a3mÞf1;2þf2;1 ε

0012 ¼0

ε

00012 ¼�ðC=a3mÞf1;2�ðC=a3mÞu3;12þðC=a3mÞu2;1�Cf2;1

ε013 ¼f1þu3;1; ε

013 ¼ 0; ε

0013 ¼�3C

�f1þw;1

�ε

00013 ¼0; ε

023 ¼f2þð1=a3mÞu3;1; ε

023 ¼0

ε

0023 ¼3C

��ð1=a3mÞu3;2�f2þðu2=a3mÞ�; ε

00023 ¼0

(26)


Setting s33 ¼0 in Hooke's law for infinitesimal thermo-elasticdeformations, solving it for ε33, and substituting for ε33 inHooke's law for 3-dimensional deformations, the reduced consti-tutive relation for the cylindrical shell is given by

266664s11s22s12s13s23

377775 ¼

266664Q11 Q12 0 0 0Q12 Q22 0 0 00 0 Q66 0 00 0 0 Q44 00 0 0 0 Q55

377775

266664ε11ε22ε12ε13ε23

377775�

266664b11b22000

377775T

(27)

where ½Q � is the matrix of material elasticities, bi the stress modulifor the thermal expansion coefficients [64], and T the change intemperature from that in the stress-free reference configuration.Expressions for the forces and the moments for the shell are givenby

8<:N11N22N12

9=;¼

Zh=2�h=2

8<:s11s22s12

9=;da3;

8<:M11M22M12

9=;¼

Zh=2�h=2

8<:s11s22s12

9=;a3da3;

(28)

8<:

P11P22P12

9=; ¼

Zh=2�h=2

8<:

s11s22s12

9=;a3

3da3;P13P23

�

¼Zh=2

�h=2

s13s23

�a3

3da3

A10 ¼Zh=2

�h=2

Eða3; TÞ1� v2

da3; A11 ¼Zh=2

�h=2

Eða3; TÞa31� v2

da3; A

A13 ¼Zh=2

�h=2

Eða3; TÞa331� v2

da3; T10 ¼ �Zh=2

�h=2

Eða3; TÞb11T1� v2

da3; T

T13 ¼ �Zh=2

�h=2

Eða3; TÞa33b11T1� v2

da3; B10 ¼Zh=2

�h=2

Eða3; TÞ2ð1þ vÞda3; B

B12 ¼Zh=2

�h=2

Eða3; TÞa322ð1þ vÞ da3; B13 ¼

Zh=2�h=2

Eða3; TÞa332ð1þ vÞ da3; B

B15 ¼Zh=2

�h=2

Eða3; TÞa352ð1þ vÞ da3; B16 ¼

Zh=2�h=2

Eða3; TÞa362ð1þ vÞ da3; T

Q13Q23

�¼

Zh=2�h=2

s13s23

�da3;

R13R23

�¼

Zh=2�h=2

s13s23

�a3

2da3

(29)

Substituting from Eqs. (26) and (27) into Eqs. (28) and (29) gives

N11 ¼ A10ε011 þ A11ε

011 þ A12ε

0011 þ A13ε

00011 þ T10

N22 ¼ A10ε022 þ A11ε

022 þ A12ε

0022 þ A13ε

00022 þ T20

N12 ¼ B10ε012 þ B11ε

012 þ B12ε

0012 þ B13ε

00012

M11 ¼ A11ε011 þ A12ε

011 þ A13ε

0011 þ A14ε

00011 þ T11

M22 ¼ A11ε022 þ A12ε

022 þ A13ε

0022 þ A14ε

00022 þ T11

M12 ¼ B11ε012 þ B12ε

012 þ B13ε

0012 þ B14ε

00012

P11 ¼ A13ε011 þ A14ε

011 þ A15ε

0011 þ A16ε

00011 þ T13

P22 ¼ A13ε022 þ A14ε

022 þ A15ε

0022 þ A16ε

00022 þ T23

P12 ¼ B13ε012 þ B14ε

012 þ B15ε

0012 þ B16ε

00012

P13 ¼ B13ε013 þ B14ε

013 þ B15ε

0013 þ B16ε

00013

P23 ¼ B13ε023 þ B14ε

023 þ B15ε

0023 þ B16ε

00023

Q13 ¼ B10ε013 þ B11ε

013 þ B12ε

0013 þ B13ε

00013

Q23 ¼ B10ε023 þ B11ε

023 þ B12ε

0023 þ B13ε

00023

R13 ¼ B12ε013 þ B13ε

013 þ B14ε

0013 þ B15ε

00013

R23 ¼ B12ε023 þ B13ε

023 þ B14ε

0023 þ B15ε

00023 (30)

in which

12 ¼Zh=2

�h=2

Eða3; TÞa321� v2

da3;

11 ¼ �Zh=2

�h=2

Eða3; TÞa3b11T1� v2

da3;

11 ¼Zh=2

�h=2

Eða3; TÞa32ð1þ vÞ da3;

14 ¼Zh=2

�h=2

Eða3; TÞa342ð1þ vÞ da3;

20 ¼ �Zh=2

�h=2


da3;

3


T11 ¼ �Zh=2

�h=2


da3; T23 ¼ �Zh=2

�h=2

Eða3; TÞb22a33T1� v2

da

(31)

Equations governing infinitesimal deformations of the thermo-elastic shell can be written as follows:

a3mvN11

va1� vN12

va2¼ 0

vN22

va2þ a3mvN12

va1þ Q23 þ

Ca3m

vP22va2

þ CvP12va1

� 3CR23 �C

a3mP23 ¼ 0

�Ca3mv2P11va1

2 þ N22 �C

a3m

v2P22va2

2 � Cv2P12va1va2

� a3mvQ23

va1þ 3Ca3m

vR13va1

� vQ23

va2þ 3C

vR23va2

� Ca3m

vP23va2

¼ 0

a3mvM11

va1� Ca3m

vP11vx

þ vM12

va2� C

vP12va2

þ 3CR13 � a3mQ13 ¼ 0

vM22

va2þ C

vP22va2

� va1vM12

va1þ Ca3m

vP12va1

þ a3m

�Q23 � 3CQ23

� 2C1

a3mP23

�¼ 0

(32)

Here we consider shells with edges a1 ¼ 0, L clamped, and theinner and the outer surfaces a3¼ h/2, -h/2 traction free. In order tofurther simplify the problem, we assume the displacement field tohave the following series expansion:

8>>>><>>>>:

u1u2u3f1f2

9>>>>=>>>>;

¼X∞n¼1

8>>>><>>>>:

u1nða1Þcosðna2Þu2nða1Þsinðna2Þu3nða1Þcosðna2Þf1nða1Þcosðna2Þf2nða1Þsinðna2Þ

9>>>>=>>>>;

(33)

While solving the mechanical problem, the temperature field isassumed to be known.

The 3-dimensional steady-state heat conduction equation in theabsence of heat sources can be stated as [65]

vkva3

vTva3

þ kv2Tva32

þ�

a3ma3m þ a3

�2kvT2

va22þ vkva1

vTva1

þ kv2Tva12

þ 1a3m þ a3

kvTva3

¼ 0

(34)

Because of the dependence of the thermal conductivity kða3; TÞupon the temperature, Eq. (34) is non-linear inT. Thermal boundaryconditions considered in this work are:

Heat convection at a1 ¼ 0 : kða3; TÞvTva1

þ haðT � T∞Þ ¼ 0

(35)

Heat convection at a1 ¼ L : kða3; TÞvTva1

þ haðT � T∞Þ ¼ 0

(36)

Temperature prescribed at a3 ¼ h=2 : T ¼ T0 (37)

Heat flux prescribed at a3 ¼ �h=2 : q ¼ qflux (38)

In Eqs. (35) and (36), T∞ ¼ 300K equals the temperature of theambient air, and ha the convective heat transfer coefficient, and weset T0 ¼ 523K.

3. Numerical solution

The one-way coupled thermo-mechanical problem is analyzedby first solving nonlinear Eq. (34) for the temperature field and thenanalyzing the linear mechanical problem. Equation (34) is numer-ically solved by the GDQM-based iterative procedure.

3.1. Thermal problem

The temperature field is assumed to be given by

Tða1;a2;a3Þ ¼X∞n¼1

Tnða1;a3Þsinðna2Þ (39)

Substitution from Eq. (39) into Eq. (34) results in an equationthat involves partial derivatives of Tn with respect to a1 and a3.These are approximated in terms of the values of Tn at discretepoints by the GDQM [31,55] as follows

vrf ð2Þv2r

��ð2¼2iÞ

¼XN2

k¼1

cðrÞik f ð2kÞ; i ¼ 1;2;…;N2 (40)

Here N2 denotes the number of grid points and cðrÞij the corre-sponding weights given by the following recursive relations for thefirst-order derivative, i.e., r ¼ 1.

cð1Þij ¼ Lð1Þð2iÞ�2i � 2j

�Lð1Þ�2j�; i; j ¼ 1;2;…;N2; isj (41)

where

Lð1Þð2iÞ ¼YN2

j¼1;isj

�2i � 2j

�; (42)

The weighting coefficients for the higher-order derivatives arederived by the following iterative relations.


cðrÞij ¼ r

0@cðr�1Þ

ii cð1Þij �cðr�1Þij�

2i � 2j�1A; i; j ¼ 1;2;…;N2; isj r

¼ 2;3; ;…N2 � 1

(43)

cðrÞii ¼ �XN2

j¼1;isj

cðrÞij ; i ¼ 1;2;…;N2; r ¼ 1;2;…N2 � 1

(44)

We employ the Chebyshev-Gauss-Lobatto (CGL) quadraturepoints given by Ref. [40]

2i ¼12

�1� cos

�i� 1N2 � 1

p

��; i ¼ 1;2;…;N2 (45)

as the grid points. As reported in a number of papers including[55,56] the use of these non-uniformly distributed sampling pointsyields results of good accuracy. We note that Eq. (40) is similar tothat obtained by using the smooth symmetric hydrodynamics basisfunctions, e.g., see Refs. [66,67].

The discrete form of Eqs. (34)e(38) can be written in thefollowing compact form

7

Fig. 3. Flow chart for iterative solution procedure of non-linear heat transfer equation.

� ½Abb� ½Abd�½Adb� ½Add�

�TbTd

�¼ ffCg

f0g�

(46)

where subscripts ‘d’ and ‘b’ denote the domain and the boundarypoints, respectively, and the vector ffCg includes terms due to heatconvection conditions prescribed at a1 ¼ 0 and a1 ¼ L. Bycondensation of the boundary degrees of freedom, the tempera-tures at the domain grid points are given by

fTdg ¼ ½B��1½Adb�½Abb��1ffCg (47)

where

½B� ¼ ½Adb�½Abb��1½Abd� � ½Add� (48)

Noted that Eq. (47) is non-linear in temperature due to thetemperature-dependent thermal conductivity, kða3; TÞ. It is itera-tively solved with the procedure described in the flow chart ofFig. 3.

3.2. Mechanical problem

With the temperature distribution known, equations for themechanical problem are discretized by using the GDQM. Denotingdegrees of freedom of the sampling points within and on theboundary of the domain by subscripts d and b, respectively, wearrive at the following set of linear algebraic equations.� ½Abb� ½Abd�½Adb� ½Add�

�dbdd

�¼ fFTbgfFTdg

�(49)

Here fFTbg and fFTdg are load vectors due to the temperature field.Similar to the solution technique for Eq. (46), we get

fddg ¼ ½L��1hfFTdg � ½Adb�½Abb��1fFTbg

i(50)

where

Fig. 4. Variation of the transverse shear stress through the thickness of CNT-reinforcedcylindrical panel subjected to a thermal load (red and blue lines, respectively, corre-spond to V*

CNT ¼ 0:17 and 0.14). (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this article.)

Fig. 5. Convergence of the through-the-thickness non-dimensional thermal stresses and the radial displacement in a linearly graded CNT-CMC cylindrical shella3m=h ¼ 20; g ¼ 1; s ¼ 1; x ¼ 0:7; z ¼ 0:5.

Fig. 6. Variation of the temperature field and the transverse shear stress in the CNT-CMC cylindrical shell for several values of the CNT volume fraction indexða3m=h ¼ 10; a1 ¼ L=2; s ¼ 1; x ¼ 1Þ.



½L� ¼ ½Add� � ½Adb�½Abb��1½Abd� (51)

4. Results and discussion

In order to verify our computational algorithm, we comparecomputed results for the simply-supported cylindrical panelproblem with those of Ref. [53] for the (10, 10) SWCNTs-polymercomposite. In Ref. [53] the ERM homogenization technique isemployed and following values are assigned to different variables:

a3mh

¼ 10; h ¼ 10mm; Q ¼ p

3;

La3m

¼ 3 (52)

Thermal and mechanical loads:

at a3 ¼ h=2 :; T ¼ T0 sin�pa2

Q

; Q ¼ Q0 sin

�pa2Q

;

at a3 ¼ �h=2 : T ¼ T1 sin�pa2

Q

; Q ¼ 0 (53)

Here Q denotes the panel angle, T0 ¼ 300K and T1 ¼ 273K, and theQ0 is peak value of the distributed surface traction applied on theouter surface of the shell. Following [53], we set Q0 ¼ 2.1 GPa. It is

Fig. 7. Through-the-thickness distributions of the temperature, the circumferential and the tvarious CNT volume fraction profiles ða3m=h ¼ 5; a1 ¼ L=2; g ¼ 1; s ¼ 1; x ¼ 1Þ.

clear from results plotted in Fig. 4 that the presently computedtransverse shear stress with z ¼ 1 (i.e., no aggression of CNTs in thematrix phase) and the EMT homogenization scheme agrees wellwith that reported in Fig. 3c of [53]; the nomenclature and the non-dimensionalization used are the same as in Ref. [53]. The magni-tude of the transverse shear stress based on the EMT model isslightly higher than that based on the ERM approach. An increase inV*CNT from 0.14 to 0.17 increases the maximum transverse normal

stress by a factor of about 3.We report below through-the-thickness variations of the ther-

mal stresses, displacements, and the temperature field for contin-uously graded and temperature-dependent CMC cylindrical shellreinforced with different volume fractions of aggregated CNTs andwith clamped edges. Results are presented in terms of non-dimensional displacements and stresses denoted by an * anddefined as (see [68])

�u*1;u

*2;u

*3� ¼ kmðu1;u2;u3Þ

amL2qo; s*i ¼

kmsiamEmLqo

(55)

Here am, km, and Em are properties of the matrix phase of thenanocomposite and a3m denotes the radius of the shell mid-surface.The geometrical parameters of the shell are L ¼ 0:7m; a3m ¼ 0:3m,the matrix material is Alumina with temperature-dependentproperties borrowed from Refs. [13,63] and the reinforcements

ransverse shear thermal stresses in a continuously graded CNT-CMC cylindrical shell for


are armchair (10, 10) SWCNTs [17] with V*CNT ¼ 0:17 unless other-

wise mentioned. In the parametric study provided in this section,the temperature is in Kelvin and the heat flux applied to the shellinner surface qo ¼ 1E6 W=m2K [68]. The heat transfer coefficient ofthe air is assumed to be ha ¼ 100 W=m2K. The shell thickness co-ordinate normalized by h/2 is denoted by h.

A convergence study of the through-the-thickness thermalstresses and the radial displacement is presented in Fig. 5 for theCNTs volume fraction corresponding to s ¼ 1;g ¼ 1 in Eq. (22).These results evince that the solution obtained with the GDQMrapidly converges with an increase in the number of samplingpoints in the a1 and the a2 directions. The convergence of thetransverse shear and the transverse normal stresses is faster thanthat of the in-plane shear stress and the radial displacement. Themaximum values of the three stress components plotted in Fig. 5are of the same order of magnitude.

In Fig. 6 we have exhibited the through-the-thickness variationof the temperature and the transverse shear stress for differentvalues of g in Eq. (22) for the CNTs volume fraction and x ¼ 1, i.e., allCNTs are aggregated in the inclusions. The temperature gradient atthe outer surface, h ¼ 0.5, and the transverse shear stress at themid-surface, h ¼ 0, increase as the value of g is increased from 1/10

Fig. 8. Effect of values of the sigmoidal exponent on the through-the-thickness variatiða3m=h ¼ 50; a1 ¼ L=2; g ¼ 1; x ¼ 1Þ.

to 10. The value of g thatminimizes s23 at the shell mid-surface alsogives the smallest temperature at the shell inner surface.

The effects of several CNT volume fraction profiles, namely X, O,sigmoidal and uniform [21,37,53], on the through-the-thicknessdistributions of the temperature, the circumferential or the hoop,and the transverse shear stresses are depicted in Fig. 7. The O andthe sigmoidal profiles have no CNTs on the shell inner surface. Thetemperature on the shell inner surface is the lowest for the O andthe X profiles and the highest for the uniform profile. Themaximumvalue of the hoop stress is highest for the X profile and itoccurs at the shell outer surface. For the O profile, the hoop stresshas the maximum value at the shell mid-surface. For the O and theX profiles, the slope of the in-plane shear stress is discontinuous atthe shell mid-surface. The four CNT profiles considered give qual-itatively similar through-the-thickness distributions of the trans-verse shear stress with the O (X) profile giving the smallest (largest)value at the shell mid-surface.

The through-the-thickness variations of the temperature, thecircumferential stress, and the out-of-plane transverse shearstresses for different values of s in Eq. (22) are shown in Fig. 8. Withan increase in the value of s, the temperature gradient in going fromthe outer to the inner shell surface increases especially from the

ons of the temperature and the circumferential and the transverse shear stresses

Fig. 10. Temperature field in a continuously graded CNT-CMC cylindrical shell for twovolume fractions of CNTs ða3m=h ¼ 5; a1 ¼ L=2; g ¼ 1; s ¼ 10; x ¼ 1Þ. Here l is non-dimensional coordinate along the a1-direction.

Fig. 9. Impact of aggregation parameter z on the transverse shear thermal stresses in a continuously graded CNT-CMC cylindrical shellða3m=h ¼ 20; a1 ¼ L=2; g ¼ 1; s ¼ 1; x ¼ 0:9Þ.


mid-surface to the inner surface, the maximum value of thetransverse shear stress increases with an increase in the value of sand their qualitative distributions are nearly the same for s¼ 1, 5,10and 50. We note that an increase in the value of s results in lowervolume fraction of CNTs at the shell inner surface. As shown inFig. 2, for s¼ 50 we have a laminated shell with negligible (highest)volume fraction of CNTs on the inner (outer) one-half of the shell.This affects more through-the-thickness distribution of the trans-verse shear stress around the mid-surface than the temperatureand the hoop stress distributions.

In Fig. 9 we have displayed through-the-thickness variations ofthe hoop and the transverse shear stress for the aggregationparameter z ¼ 0.1, 0.3, 0.4, 0.5 and 0.7. The hoop stress at the outersurface of the shell does not vary monotonically with the change inz. For volume fractions of the CNTs equal to 0.11 and 0.17, distri-butions of the temperature on the a1a3- plane are displayed inFig. 10. It should be noted that the temperature is prescribed on theshell outer surface. These plots evince that increasing the CNTvolume fraction from 0.11 to 0.17 decreases the maximum tem-perature induced from 1100 to 1020 K. The temperature in the axialdirection, except at points near the end faces, is uniform becausethe material properties are independent of the a1� coordinate.

5. Remarks

We note that the exact solution for thermoelastic deformationsof rectangular plates is provided in [69] and a higher-order shearand normal deformable plate theory is used in [70] to analyzetransient thermoelastic deformations. Finite deformations ofcurved beams considering both geometric and material non-linearities are studied in [71,72] using a third-order shear andnormal deformable theory.

6. Conclusions

Thermal stresses induced in a CNT-reinforced ceramic matrixcomposite cylindrical shell with temperature-dependent materialproperties have been analyzed by first solving the nonlinear heatconduction equation and then the linear thermoelasticity equationsby using Reddy's TSDT and the GDQM. The effective moduli havebeen deduced by using the Eshelby-Mori-Tanaka scheme consid-ering the agglomeration of CNTs. The cylinder is clamped at theedges and its inner and outer surfaces are traction free. The tem-perature is prescribed on the outer surface and the heat flux on theinner surface. It is found that the thermal design of continuouslygraded CNTRCs using the widely assumed through-the-thicknessCNT variations known as, profile-O and profile-X, give discontinu-ities at the shell mid-surface in slopes of the hoop and the in-planeshear stresses. Furthermore, for the profile-X the peak magnitudeof the stresses is higher than that for other variations of the CNTs.The consideration of the aggregation of CNTs noticeably increasesthe maximum magnitudes of the in-plane thermal stresses.

Acknowledgments

The last two authors would like to acknowledge the CNPq(Brazilian Council for Scientific and Technological Development)with grant number of 306933/2014-4, and of the FAPERJ (ResearchFoundation of the State of Rio de Janeiro) with grant number of203.234/2016.

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