-
Contents lists available at ScienceDirect
Composite Structures
journal homepage: www.elsevier.com/locate/compstruct
Torsion of bi-directional functionally graded truncated conical
cylinders
G.J. Niea,b, Anup Pydahb, R.C. Batrab,⁎
a School of Aerospace Engineering and Applied Mechanics, Tongji
University, Shanghai 200092, ChinabDepartment of Biomedical
Engineering and Mechanics, Virginia Polytechnic Institute and State
University, M/C 0219, Blacksburg, VA 24061, USA
A R T I C L E I N F O
Keywords:Functionally graded materialsAnalytical solutionsRadial
and axial inhomogeneityTruncated conical cylinderTorsion
A B S T R A C T
Analytical solutions are presented for the torsion of
bi-directional functionally graded (FG) linearly elastictruncated
conical cylinders for six functional forms of the shear modulus
varying in both the radial and the axialdirections. Furthermore,
for an arbitrary variation of the shear modulus along the two
directions, we employ aweighted residual approach (WRA) to obtain
accurate numerical solutions for the linear elastic boundary
valueproblem. The influence of the variation of the shear modulus
and of the cone angle on the stress distribution inthe cylinder is
discussed through numerical examples. The analytical solutions
presented herein can serve asbenchmarks for ascertaining the
accuracy of approximate or numerical solutions. The WRA combined
with anoptimization algorithm can be used to find the shear modulus
variation to maximize the torsional stiffness.
1. Introduction
Functionally graded materials (FGMs) have continuously
varyingspatial material properties [1] and are designed to meet
specific en-gineering requirements like a high stiffness-to-weight
ratio [2] anddesired functionalities in specific directions [3].
There is a large body ofresearch on the statics, dynamics and
stability of FG structures [4–13]that indicates the use of these
materials in high-performance en-gineering industries.
FGM prismatic and conical cylinders [14,15] are used as
fuselages ofaircrafts and rockets, pressure vessels, storage tanks,
and pipelines. Areview of the literature on the torsion of these
structures, briefly sum-marized below, reveals that
elasticity-based analytical and approximatenumerical solutions have
mostly been developed when material prop-erties vary either in the
radial direction or along the cylinder axis.
Chen [16] in 1964 analytically studied the torsion of
in-homogeneous prismatic circular cylinders using an Airy stress
function.The shear modulus of the material was assumed to vary only
along theradius according to a power law. With the same kinematic
assumptionsas those of Ref. [16], i.e., a plane section remains
plane and undergoes arotation with respect to an adjacent
cross-section, Lekhnitskii [17]presented analytical solutions for
the torsion of bodies of revolutionwith cylindrical anisotropy.
Rooney and Ferrari [18] derived analyticalsolutions for the torsion
and the flexure of FG circular cylindrical barsby assuming that the
elastic moduli are smooth functions of coordinatesof a point in a
cross-section. Horgan and Chan [19] extended works ofRefs. [17,18]
and provided new insights into effects of the material
inhomogeneity on the torsional response of linearly elastic
isotropicbars. Horgan [20] also studied the torsion of linearly
elastic anisotropicFGM bars by assuming that the elastic
coefficients depend on bothcross-sectional coordinates.
Ecsedi [21,22] presented analytical solutions for
Saint-Venant’storsional problem of inhomogeneous isotropic and
anisotropic cylind-rical bars with their elastic moduli assumed to
be functions of thePrandtl stress function of the corresponding
homogeneous cylindricalbar. For the torsion of FGM shafts of
arbitrary cross-sections, Arghavanand Hematiyan [23], Chen [24],
and Diaco [25] presented analyticalformulations by assuming that
the shear modulus changes continuouslywith one or both
cross-sectional coordinates. Chen et al. [26–29] alsostudied
Saint-Venant’s torsion of composite material bars. Recently, Jogand
Mokashi [30] as well as Katsikadelis and Tsiatas [31],
respectively,employed the finite element and the boundary element
methods toanalyze the torsion of inhomogeneous and anisotropic
non-circularprismatic bars. Some recent studies on the vibration
and buckling ofFGM conical shells [32–38] have obtained numerical
solutions usingthe Galerkin and the differential quadrature
approaches. Shen et al.[39] investigated small-scale effects on the
static and the dynamictorsional behaviors of FGM shafts by using a
strain gradient theory andassuming a power-law variation of the
elastic moduli with respect to theradial coordinate.
For axial variations of the shear modulus, Batra [40]
analyticallystudied the torsion of a circular cylindrical bar made
of an isotropiceither compressible or incompressible linearly
elastic material. Analy-tical results were also obtained for a
transversely isotropic FGM circular
https://doi.org/10.1016/j.compstruct.2018.11.081Received 5
August 2018; Received in revised form 19 November 2018; Accepted 29
November 2018
⁎ Corresponding author.E-mail addresses: [email protected],
[email protected] (G.J. Nie), [email protected] (A. Pydah),
[email protected] (R.C. Batra).
Composite Structures 210 (2019) 831–839
Available online 04 December 20180263-8223/ © 2018 Elsevier Ltd.
All rights reserved.
T
-
cylindrical bar and it was shown that the axial variation of the
shearmodulus can be adjusted to control the angle of twist of a
cross-section.Batra [41] has shown that the same problem for a
nonlinear elasticMooney-Rivlin material has analytical solutions
only for specific axialvariations of the two material moduli.
Barretta et al. [42] extendedBatra’s work to the non-uniform
torsion of linearly elastic isotropic andcomposite beams with
simply and multiply connected cross-sectionsand presented
closed-form solutions for exponential gradations ofYoung’s and the
shear moduli. Dryden and Batra [43] have analyticallyanalyzed
torsional deformations of a FGM Mooney-Rivlin hollow cy-linder with
loads applied on the inner and the outer surfaces of
thecylinder.
Investigations mentioned above have analyzed the torsion of
pris-matic FGM cylinders with material properties graded either
along anaxial direction or within a cross-section. Here, we present
analyticalsolutions for the torsion of bi-directionally graded
isotropic FGMtruncated conical cylinders with the shear modulus
varying both ra-dially and axially. We analytically solve the
problem and find closed-form solutions for six functional
variations of the shear modulus thatcan serve as benchmarks for
comparing various approximate and nu-merical solutions.
For arbitrary spatial variations of the shear modulus in the two
di-rections, we use the weighted residual approach (WRA) to
numericallysolve the linear elastic boundary value problem. We also
investigate thevariation of the stress field with the change in the
gradation parametersand the cone angle and show that the maximum
value of one of the twonon-zero shear stresses on a cross-section
can be reduced by appro-priately tuning these parameters.
The rest of the paper is organized as follows. The governing
equa-tions and boundary conditions for the torsion problem and the
Airystress function approach for solving the linear elastic
boundary valueproblem are described in Section 2. Analytical
solutions are developedfor FGM circular cylinders of radius R0 in
Section 3. Analytical closed-form solutions for FGM conical
cylinders for six functional variations ofthe shear modulus are
presented in Section 4. We describe the WRA fornumerically solving
the problem in Section 5, and analyze five exampleproblems in
Section 6. Conclusions of the work are summarized inSection 7.
2. Formulation of the problem
2.1. Governing equations
Fig. 1 depicts a truncated conical cylinder of length L with a
trac-tion-free mantle and loaded by equal and opposite couples, Mt
, aboutthe longitudinal −z axis, applied at the circular end faces
of radii Ri andRo, respectively. We use cylindrical coordinates r
and θ to define theposition of a material point in a cross-section.
The radius of a cross-section is = + −R z R z z α( ) ( )tanz i 0 ,
where α2 is the angle of the coneand =z z0 is the location of the
inner end face. The conical cylinder ismade of a linearly elastic
and isotropic FGM with the shear modulusG r z( , ) varying in the
radial and the axial directions.
To analyze the problem, we use a semi-inverse method and
assumethat the radial, ur , and the axial, uz, displacements of
points in the cross-section =z constant are zero. That is, all
points only move in the tan-gential direction, and the tangential
displacement, u r z( , )θ , does notdepend on the angular
coordinate θ. The non-zero infinitesimal strainsare
= ∂∂
− = ∂∂
γ ur
ur
γ uz
,rθθ θ
θzθ
(1)
where γrθ and γθz are shear strains on a plane θ =constant.For a
Hookean material, the shear stresses τ r z( , )rθ and τ r z( , )θz
are
given by
= =τ G r z γ τ G r z γ( , ) , ( , )rθ rθ θz θz (2)
In the absence of body forces, the only non-trivial equation
ofequilibrium is
∂∂
+∂∂
+ = ∂∂
+∂
∂=τ
rτz
τr
r τr
r τz
2 ( ) ( ) 0rθ θz rθ rθ θz2 2
(3)
The boundary condition on the traction-free mantle, =r Rz,
is
+ =τ n τ n 0rθ r θz z (4)
where → = → + →n n e n er r z z is a unit vector normal to the
lateral surface ofthe conical cylinder, and →er and →ez are
orthonormal basis vectors in ther - and the z-directions,
respectively (see Fig. 2). We note that = ∂∂nr
zs
and = − ∂∂nzrs where ds is an element on the lateral
boundary.
Boundary conditions on the inner =z z( )0 and the outer =z L( )0
endfaces are
∫ ∫ ==τ rdrdθ| 0π R θz z z02
0
i0 (5)
and
∫ ∫= =M τ r drdθ|t π R θz z z02
02i
0 (6)
For equilibrium, Eqs. (5) and (6) also hold on a circular
cross-sec-tion z =constant. We also require that the shear stresses
and the tan-gential displacement are bounded at r = 0.
We note that Eq. (5) is identically satisfied when τθz is an
evenfunction of the angle θ. For our kinematic assumptions, it is
triviallysatisfied since τθz is independent of θ.
2.2. Airy stress function
Eq. (3) is identically satisfied by using a stress function, ϕ r
z( , ), suchthat
= −∂∂
=∂∂
r τϕz
r τϕr
,rθ θz2 2 (7)
The compatibility condition derived from Eqs. (1), (2) and (7)
is
∂∂
⎛⎝
∂∂
⎞⎠
− ⎛⎝
∂∂
⎞⎠
+ ∂∂
⎛⎝
∂∂
⎞⎠
=r G
ϕr r G
ϕr z G
ϕz
1 3 1 1 0(8)
Using Eq. (7) in Eqs. (4) and (6), the boundary conditions
become
∂∂
∂∂
+∂∂
∂∂
= =ϕz
zs
ϕr
rs
dϕds
0(9)
∫= ∂∂ ==M πϕr
dr π ϕ r z2 | 2 [ ( , )]tR
z zR
0 0 0i i
0 (10)
Hence, ϕ is constant along the lateral boundary of the
cylinder.Eq. (8) along with boundary conditions (9) and (10)
determine the
stress function ϕ r z( , ). From Eqs. (1) and (7), and noting
that= −∂∂
∂∂( )r r ur ur urθ θ θ , the displacement uθ can be found from
the stress
function using the following equations
∂∂
⎛⎝
⎞⎠
= −∂∂r
ur Gr
ϕz
1θ3
∂∂
⎛⎝
⎞⎠
=∂∂z
ur Gr
ϕr
1θ3 (11)
The traction boundary-value problem defined by Eqs. (8)–(10)
canhave solutions that differ by a rigid body motion. We eliminate
it bysetting =u r z( , ) 0θ 0 .
3. Solutions for FGM circular cylinders of radius R0
3.1. Radial gradation, =G r z G r( , ) n0
We note that n = 0 for a homogeneous cylinder. Choosing=ϕ r z C
r( , ) p0 satisfies the compatibility condition (8) if and only
if
G.J. Nie et al. Composite Structures 210 (2019) 831–839
832
-
= +p n4 . The constant C0 is obtained from Eq. (6) as
= +CM
πR2tn00
4
The shear stresses derived from Eq. (7) are
= = = +− ++τ τ C pr M
πRn r0 and
2( 4)rθ θz p tn
n0
3
04
1
(12)
As pointed out by Chen [16], =τθz constant for = −n 1.The
displacement u r z( , )θ is determined from Eq. (11) as
= + − = + −+uCG
n r z z MπG R
n r z z( 4) ( )2
( 4) ( )θ t n0
00
0 04 0 (13)
The angle of twist per unit length, θ̄, is given by
=−
= ++θu
r z zM
πG Rn¯
( ) 2( 4)θ t n
0 0 04 (14)
The results for the shear stress, the displacement and the
twist/length, Eqs. ()(12)–(14), agree with those of Chen [16].
We note that for = − = =n u θ4, ¯ 0θ . Thus, having an
in-finitesimally small rigid cylinder of radius ∊ ≪ 1 surrounded by
a cy-linder of deformable material with =G G r/0 4 will behave as a
rigidcylinder in torsional deformations. Alternatively, a hollow
cylinderwith a circular hole and =G G r/0 4 will undergo null
torsional de-formations. We note that the shear stresses (12)
satisfy the null tractionboundary conditions on the inner surface
of the hollow cylinder.
3.2. Axial gradation, = + −G r z a b z z( , ) ( ( ))n0
Choosing =ϕ r z C r( , ) p0 satisfies the compatibility
condition (8) ifand only if =p 4. The constant C0, the shear
stresses and the dis-placement u r z( , )θ obtained by following
the procedure of sub-Section3.1 are listed below.
=C MπR2
t0
04
= = =τ τ C r MπR
r0 and 4 2rθ θz t004 (15)
=⎧
⎨⎩
+ − − ≠
=
−− −
+ −( )u
a b z z a n
n
(( ( )) ) 1
ln 1θ
C rb n
n n
C rb
a b z za
4(1 ) 0
1 1
4 ( )
0
0 0(16)
The rotation of a cross section, ̂θ , with respect to the
cross-section at=z z0, is given by
̂ = =⎧
⎨⎪
⎩⎪
+ − − ≠
=
−− −
+ −( )θ u
r
a b z z a n
n
(( ( )) ) 1
ln 1θ
MπR b n
n n
MπR b
a b z za
2(1 ) 0
1 1
2 ( )
t
t
04
04
0
(17)
These results agree with those of Batra [40] for the torsion of
axiallygraded circular cylinders. Furthermore, as shown in Ref.
[40], the
Fig. 1. FGM truncated conical cylinder.
Fig. 2. Axial section of the conical cylinder.
G.J. Nie et al. Composite Structures 210 (2019) 831–839
833
-
product̂=∂∂G z C( ) 4
θz 0 is constant. Hence, the shear modulus can be
found to control the angle of twist of a cross-section. The
shear stress τθzdoes not depend upon the gradation parameters but
the tangentialdisplacement does.
3.3. Bi-directional gradation, =G r z G f z r( , ) ( ) n0
Note that the boundary of a circular cylinder is described by =r
R0and independent of z. Choosing =ϕ r z C r( , ) p0 satisfies the
compat-ibility condition (8) for any choice of a continuously
differentiable axialgradation function f z( ) if and only if = +p
n4 and ≠f z( ) 0 every-where in the cylinder. Following the
procedure of sub-Section 3.1, weget
= +CM
πR2tn00
4
= = = +− ++τ τ C pr M
πRn r0 and
2( 4)rθ θz p tn
n0
3
04
1
(18)
=u r z r F z( , ) ( )θ (19)
where F z( ) is determined from
= +dF z
dzM
πG R f z( )
2 ( )t
n0 0
4 (20)
with the boundary condition =F z( ) 00 .The displacement uθ and
the rotation of a cross-section with respect
to the cross-section at ̂= = =z z θ u r F z, / ( )θ0 , can be
tuned by sui-tably choosing f z( ). For example, for
=−
−f z C
G k z z( ) e
2 ( )
k z z0
0
( )
0
02
̂θ is determined by solving Eq. (20) aŝ = −− − − +θ (e e )ek z
z kz k z z z( ) (( ) )0 2 02 0 2 02
For different values of k, Fig. 3 shows the variation of ̂θ
along thelength of the cylinder. By varying k, the rotation can be
made to rise toa constant value along the length of the cylinder.
The shear stress τθzdoes not depend upon f z( ).
4. Analytical solutions for FGM conical cylinders
Noting that ≡ =η αtanrz on the boundary of a cross-section, any
ϕwhich is a continuously differentiable function of η will
satisfyboundary condition (9).
For a general function G r z( , ), we are unable to find ϕ r z(
, ) thatsatisfies Eqs. (8) and (9). Instead, we assume ϕ to be a
function of η andsubstitute it in compatibility condition (8) to
get a partial differential
equation for G r z( , ). We then intuitively find G that
satisfies thisequation.
For the particular case when G is a function of η, the
requiredvariation of G is obtained as
=+ ′G η G
ηη
ϕ( )(1 )
0
2 523 (21)
where the constant G0 is determined by specifying the value of G
at apoint =η η0, and ′ =ϕ dϕ dη/ . The procedure for deriving
stresses anddisplacements is the same as that in sub-Section 3.1.
Here we list resultsfor six expressions for G r z( , ).
4.1. =G η( ) Gη
03
For = ++
ϕ η C( ) η ηη0
(3 2 )3(1 )
2
2 3/2 , the compatibility condition, Eq. (8), is sa-tisfied. The
constant C0, the shear stresses, and the tangential dis-placement
are determined as
= ++
C M απ α α3 (1 tan )
2 tan (3 2tan )t
02 3/2
2 (22)
=+
=−
τ Cη
z ητ ητ
(1 )andθz rθ θz0
2 5/2
3 2 (23)
⎜ ⎟= − ⎛⎝
+−
+⎞⎠
−u
C ηG
ηz z η z z3
(1 ) 1( ( / ) )θ
0
0
2 3/2
2 2 20
2 3/2 (24)
Eqs. (22)–(24) describe the dependence of the circumferential
dis-placement and the shear stresses upon the cone angle α.
4.2. = +G η G pη( ) (1 )n0
Here p and n are gradation parameters. We choose
=
⎧
⎨
⎪⎪
⎩
⎪⎪
+ ⎡⎣+ + −
− + + − ⎤⎦
≠ −
− = −
− −+ +
+ ++
+
( )
{}
( )( )ϕ η
C F η
F η
n
C n
( )
, 1 ;2 ;
, 1 ;2 ;
2
2
ηη
pηn
n n
n n
p ηη
02 3
3(1 ) 2 2 132 2 2
2
2 152 2 2
2
02 3
3(1 )
n2
2 3/2
2
2
2 3/2
(25)
where = ∑ =∞F a b c z a b c z k( , ; ; ) ( ) ( ) /( ) / !k k k
k
k2 1 0 is the hypergeometric
function and = +a a k a( ) Γ( )/Γ( )k is the Pochhammer
parameter. Theconstant C0, determined from Eq. (6), has a long
expression and isomitted. The shear stresses and the tangential
displacement are de-termined as
=⎧
⎨⎪
⎩⎪
≠ −
= −=
++
++
τC n
C nτ ητ
2
2,θz
η pηr η
η p ηr η
rθ θz
0(1 )
(1 )
0( )
(1 )
n4
3 2 5/2
2 2
3 2 5/2 (26)
⎜ ⎟= − ⎛⎝ +
−+
⎞⎠
uC ηG z η η z z3
1(1 )
1( ( / ) )θ
0
02 2 3/2 2
02 3/2 (27)
For the general case when G is not a function of η, we could
findclosed-form solutions when G r z( , ) has the following four
expressions.This list is by no means exhaustive.
4.3. = =+ +G r z G G( , ) e ek r z k z η0 ( ) 0 (1 )02 2
02 2
Here k0 and G0 are constant gradation parameters. Choosing
⎜ ⎟= ⎛
⎝⎜ −
⎛⎝
⎞⎠
⎞
⎠⎟+ +
ϕ r z C( , )η η
01
( 1)
13
1
( 1)
3
2 12 212
, we obtain
=− −( )
C M
π α α2 cos cost
0 13
3 23 (28)
Fig. 3. Variation of ̂θ along the length of the circular
cylinder with the tuningparameter k.
G.J. Nie et al. Composite Structures 210 (2019) 831–839
834
-
= −+
=τC η
z ητ ητ
( 1)andθz rθ θz
0
3 2 52 (29)
⎜ ⎟= −
⎛
⎝
⎜⎜⎜
⎛
⎝⎜ + −
⎛
⎝⎜
⎛⎝
+ ⎛⎝
⎞⎠
⎞⎠
⎞
⎠⎟
⎞
⎠⎟
++ −
+
−⎛⎝
+ − ⎞⎠
+
⎞
⎠
⎟⎟⎟
⎜ ⎟
− +
− ⎛⎝
+ ⎞⎠ ( )( )
( )( )
( )
u C rG
π k z k η z k η zz
k z ηz η
k z η
z η
32 erf ( ( 1) ) erf
e (2 ( 1) 1)( 1)
e 2 1
θ
k z η
k z η zz
zz
0
003/2
02
02 0
2
( 1)0
2 2
3 2
02 2 2
3 2 2
zz
02 2
32
02 2 0 2
0
032
(30)
Here ∫≡ = ∑− =∞ −
+
+x dterf( ) eπ
x tπ n
xn n
20
20
( 1)! (2 1)
n n2 2 1is the error function.
We note that =u r z( , ) 0θ 0 . Whereas stresses do not depend
upon G0and k0, the displacement field, uθ, does. All three solution
variablesdepend upon the cone angle α.
4.4. = + ⩾−+G r z G r z p( , ) ( ) , 4
rz0
2 2 pp
41
Choosing =ϕ r z C η( , ) p0 , where p is an integer ⩾ 4, we
obtain
=C Mπ α2 tan
tp0 (31)
= =τ Cpr
η τ ητandθz p rθ θz0 3 (32)
=⎛
⎝
⎜⎜
+
+
⎞
⎠
⎟⎟( )
uC pr
Gη
η2ln
1θ z
z
0
0
2
2 20(33)
For this choice of G r z( , ), both the stresses and the
displacementfield, uθ, depend upon the gradation parameter p which
is ⩾4 to ensurethat G r z( , ) is defined at =r 0.
4.5. = + ⩾−G r z G r z z r p( , ) ( ) , 4p0 2 2 4
The choice ⎜ ⎟= ⎛⎝
⎞⎠+
ϕ r z C( , ) ηη
p
0( 1)2
12
, where p is an integer ⩾ 4, gives
=C Mπ α2 sin
tp0 (34)
=+
=+
τ Cpr
ηη
τ ητ(1 )
andθzp
rθ θz0 3 2 1p2 (35)
= −+
⎛
⎝
⎜⎜⎜ +
−+
⎞
⎠
⎟⎟⎟
+ + +( )( )u
C p rG p z η η
(2 )1
( 1)1
θ pzz
0
02 2 1
2 21p p2 0 2
(36)
4.6. = + ⩾+ + ++ − −G r z G n p( , ) , 4r zz
pr n p zr0
( ) ( ( ) )p n p
2 21
2 2
4
The stress function ⎜ ⎟⎜ ⎟= ⎛
⎝+ ⎛
⎝⎞⎠
⎞
⎠+ϕ r z C η( , ) p η
η
n
0( 1)2
12
, where n and p are
integers satisfying + ⩾n p 4 and ≠ −n 2, results in
=−
C Mπ α α2 (1 tan sin )
tp n0 (37)
=+ ++
=++
τ Cr
ηpη n p
ητ ητ
( ( ))( 1)
andθz p n rθ θz032
2 1n2 (38)
= −+
⎛
⎝
⎜⎜⎜ +
−+
⎞
⎠
⎟⎟⎟
+ + +( )( )u
C ηn G z η η
(2 )1
( 1)1
θ nzz
0
01 2 1
2 21n n2 0 2
(39)
4.7. Remarks
Recalling that the solution of the linear elasticity equations
is un-ique provided that the stored energy function is
positive-definite for allnon-rigid deformations of the cylinder, we
conclude that solutions ofthe above six example problems are unique
provided that >G r z( , ) 0everywhere in the cylinder. In Eqs.
(5) and (6), the traction boundaryconditions are not satisfied
pointwise. In view of the Saint-Venantprinciple (e.g., see Refs.
[44,45]), the solution away from the loadedends depends only upon
the resultant forces and moments rather thanon the precise
distributions of surface tractions on the end faces.
Thus,displacements and shear stresses found from expressions given
in Sec-tion 4 will agree with their experimental values at points
away from theend faces.
5. Numerical solutions using the Weighted Residual Approach
In order to numerically solve the problem with an arbitrary
varia-tion of the shear modulus G r z( , ), we employ the Weighted
ResidualApproach (WRA).
As mentioned in Section 4, for ϕ a function of η, the
boundarycondition (9) is trivially satisfied. Hence, in the WRA, we
assume that
∑==
ϕ η b η( )i
N
ii
1 (40)
where = …b i N( 1, 2, , )i are unknown coefficients, and N is
the totalnumber of terms in the series expression for ϕ η( ).
Instead of poly-nomials, one could take other functions of η on the
right-hand side ofEq. (40); e.g., iηsin( ), iηexp( ), etc.
Substituting Eq. (40) into Eq. (7), we have
∑ ∑= == =
τ r zr
i b η τ r zr z
i b η( , ) 1 , ( , ) 1θzi
N
ii
rθi
N
ii
31
21 (41)
We require that τθz and τrθ stay bounded as →r 0. Thus,
= = =b b b 01 2 3 (42)
Substituting from Eq. (41) into boundary condition (6) gives
∑ ==
b α Mπ
tan2i
N
ii t
4 (43)
We need −N( 3) linearly independent equations to uniquely
de-termine values of unknown constants = …b i N( 4, 5, , )i . We
insert in Eq.(8) for ϕ from Eq. (40) and the given expression for G
r z( , ), multiplyboth sides by η j, and integrate the resulting
equations over the conicalcylinder domain to get
∑ = = … −=
X b j N0, 0, 1, , 5i
N
ji i4 (44)
where
∫ ∫= +X ψ η rdr dzji zL R
ii j
0
z
0
0
(45)
= ⎡⎣
− + + ⎤⎦
+ ∂∂
⎛⎝
⎞⎠
− ∂∂
⎛⎝
⎞⎠
ψ G r z i ir
i iz
G r zz
iz
G r zr
ir
( , ) ( 4) ( 1) ( , ) ( , )i 2 2
(46)
The stress function and stresses in the FGM conical cylinder can
thenbe determined using Eqs. (40) and (41).
G.J. Nie et al. Composite Structures 210 (2019) 831–839
835
-
6. Example problems
In the following six example problems, we set =z 0.50 m, =L 0.5
m,=α π/4, and =M π2t MNm.
6.1. Accuracy and convergence
We take = +G r z G( , ) ek r z0 ( )02 2
, which corresponds to Case 4.3 inSection 4.
We investigate the convergence of the WRA solution by
takingdifferent values of N and present results for the shear
stress τθz at the
point =r 0.5 m, =z 0.8 m, in Table 1. It is clear that =N 11
provides anaccurate value of τ (0.5, 0.8)θz , and only 8 basis
functions are needed.
For =N 11, the WRA results for the shear stress τθr at =z 0.5 m
andτθz at =r 0.5 m, are compared with the analytical solution, Eq.
(29), inTables 2 and 3, respectively. Clearly, the two sets of
results agree verywell with each other.
6.2. Four typical variations of G r z( , )
The shear modulus is assumed to vary as
=G r z GExample 2: ( , ) ek η0m
1 1 (47)
= +G r z G k ηExample 3: ( , ) (1 )m0 2 2 (48)
= +G r z G k ηExample 4: ( , ) (1 )m0 3 3 (49)
= + +G r z G r zExample 5: ( , ) (1 ) (1 )k m0 4 4 (50)
Here G0 is a reference value of the shear modulus, and ki and
mi=i( 1, 2, 3, 4) are constants such that >G 0. Example 3
corresponds to
Case 4.2 in Section 4.The variations of stress components in the
cylinder for different
values of ki and mi =i( 1, 2, 3, 4) are shown in Figs. 4–7. It
is observedthat ki affects the variation of shear stresses more
than mi does. Becauseof this, the maximum shear stress can be
reduced as desired.
6.3. Effect of different cone angles
For = −k 11 and =m 11 in Eq. (47), and =α π π/6, /5, and π/4,
re-spectively, we have plotted in Fig. 8 variations of the shear
stresses withr on the cross-section =z 1 m. It is observed that for
a given externaltorque increasing the cone angle decreases peak
values of the shearstresses on the cross-section, =z 1 m. The
maximum value of τθz occursat an interior point. However, τrθ has
the maximum value on the outersurface for =α π/4 but at interior
points for =α π/5 and π/6.
Batra and Nie [46] have extended these results to hollow
concicalcylinders with mantles represented by a general
parabola.
Table 1Convergence of τ (0.5, 0.8)θz (in MPa) with increasing
number of trial functions.
N 7 9 11 13
WRA 6.9544 6.9171 6.9147 6.9148Analytical+ 6.9148 6.9148 6.9148
6.9148Error∗ (%) 0.57 0.03 0.00 0.00
+ Present solution (Eq. (29)).∗ = −Error % 100((WRA
Exact)/Exact).
Table 2Comparison of τ r( , 0.5)θr between WRA and the
analytical solution (Eq. (29))(Unit: MPa).
r (m) 0.1 0.2 0.3 0.4 0.5
WRA 18.7389 28.5234 28.7467 24.0035 18.2689Analytical 18.7385
28.5235 28.7469 24.0031 18.2689
Table 3Comparison of τ z(0.5, )θz between WRA and the analytical
solution (Eq. (29))(Unit: MPa).
z (m) 0.6 0.7 0.8 0.9 1.0
WRA 13.3352 9.5981 6.9147 5.0251 3.6974Analytical 13.3350 9.5981
6.9148 5.0251 3.6974
Fig. 4. Variations of shear stresses for Example 2 with k1 and
m1.
G.J. Nie et al. Composite Structures 210 (2019) 831–839
836
-
7. Conclusions
Analytical solutions for the torsion of bi-directional FGM
truncatedconical cylinders with the shear modulus varying both
radially andaxially have been developed. For six bi-directional
variations of theshear modulus, analytical closed form solutions
for stresses and thetangential displacement are presented that will
serve as benchmarks forcomparing approximate and/or numerical
solutions of the problem.
For an arbitrary variation of the shear modulus with the r - and
thez-coordinates, we employed a weighted residual approach (WRA)
to
numerically solve the linear boundary value problem. The
accuracy andconvergence of the WRA are verified by comparing
results with thoseobtained from the analytical solutions. The
influence of different var-iations of the shear modulus and the
cone angle on the shear stresseshas been analyzed in five examples.
The numerical results show that themaximum shear stress on a
cross-section can be reduced by appro-priately tuning the shear
modulus.
Fig. 5. Variations of shear stresses for Example 3 with k2 and
m2.
Fig. 6. Variations of shear stresses for Example 4 with k3 and
m3.
G.J. Nie et al. Composite Structures 210 (2019) 831–839
837
-
Acknowledgments
GJN’s work was supported by the National Natural
ScienceFoundation of China (Nos. 11772232, 11372225 and 11072177).
RCB’swork was partially supported by the US Office of Naval
Research grantN000141812548 with Dr. Y. D. S. Rajapakse as the
Program Manager.
Appendix A. Supplementary data
Supplementary data associated with this article can be found, in
theonline version,
athttps://doi.org/10.1016/j.compstruct.2018.11.081.
References
[1] Suresh S, Mortensen A. Fundamentals of functionally graded
materials. Inst Mater1998.
[2] Birman V, Byrd LW. Modeling and analysis of functionally
graded materials andstructures. Appl Mech Rev
2007;60(5):195–216.
[3] Wetherhold RC, Seelman S, Wang J. The use of functionally
graded materials toeliminate or control thermal deformation. Compos
Sci Technol1996;56(9):1099–104.
[4] Jha D, Kant T, Singh R. A critical review of recent research
on functionally gradedplates. Compos Struct 2013;96:833–49.
[5] Thai H-T, Kim S-E. A review of theories for the modeling and
analysis of func-tionally graded plates and shells. Compos Struct
2015;128:70–86.
[6] Swaminathan K, Naveenkumar D, Zenkour A, Carrera E. Stress,
vibration and
buckling analyses of FGM plates – a state-of-the-art review.
Compos Struct2015;120:10–31.
[7] Dai H-L, Rao Y-N, Dai T. A review of recent researches on
FGM cylindrical structuresunder coupled physical interactions,
2000–2015. Compos Struct2016;152:199–225.
[8] Wu C-P, Liu Y-C. A review of semi-analytical numerical
methods for laminatedcomposite and multilayered functionally graded
elastic/piezoelectric plates andshells. Compos Struct
2016;147:1–15.
[9] Pydah A, Sabale A. Static analysis of bi-directional
functionally graded curvedbeams. Compos Struct 2017;160:867–76.
[10] Pydah A, Batra R. Shear deformation theory using
logarithmic function for thickcircular beams and analytical
solution for bi-directional functionally graded cir-cular beams.
Compos Struct 2017;172:45–60.
[11] Swaminathan K, Sangeetha D. Thermal analysis of FGM plates
– a critical review ofvarious modeling techniques and solution
methods. Compos Struct2017;160:43–60.
[12] Pydah A, Sabale A. Closed-form exact solutions for thick
bi-directional functionallygraded circular beams. Multidiscip Model
Mater Struct.https://doi.org/10.1108/MMMS-12-2017-0156.
[13] Nie G, Zhong Z, Batra R. Material tailoring for reducing
stress concentration factorat a circular hole in a functionally
graded material (FGM) panel. Compos Struct2018;205:49–57.
[14] Torabi J, Kiani Y, Eslami M. Linear thermal buckling
analysis of truncated hybridFGM conical shells. Compos Part B: Eng
2013;50:265–72.
[15] Akbari M, Kiani Y, Eslami M. Thermal buckling of
temperature-dependent FGMconical shells with arbitrary edge
supports. Acta Mech 2015;226(3):897–915.
[16] Chen Y. Torsion of nonhomogeneous bars. J Franklin Inst
1964;277(1):50–4.[17] Lekhnitskii S. Theory of elasticity of an
anisotropic body. Moscow: Mir Pub; 1981.[18] Rooney FJ, Ferrari M.
Torsion and flexure of inhomogeneous elements. Compos Eng
1995;5(7):901–11.
Fig. 7. Variations of shear stresses for Example 5 with k4 and
m4.
Fig. 8. Variations of shear stresses along the radial direction
for three cone angles. The shear modulus is assumed to vary
according to Eq. (47).
G.J. Nie et al. Composite Structures 210 (2019) 831–839
838
-
[19] Horgan C, Chan A. Torsion of functionally graded isotropic
linearly elastic bars. JElast 1998;52(2):181–99.
[20] Horgan CO. On the torsion of functionally graded
anisotropic linearly elastic bars.IMA J Appl Math
2007;72(5):556–62.
[21] Ecsedi I. Some analytical solutions for Saint-Venant
torsion of non-homogeneouscylindrical bars. Eur J Mech-A/Solids
2009;28(5):985–90.
[22] Ecsedi I. Some analytical solutions for Saint-Venant
torsion of non-homogeneousanisotropic cylindrical bars. Mech Res
Commun 2013;52:95–100.
[23] Arghavan S, Hematiyan M. Torsion of functionally graded
hollow tubes. Eur JMech-A/Solids 2009;28(3):551–9.
[24] Chen T. A novel class of graded cylinders neutral to host
shafts of arbitrary cross-sections under torsion. Mech Res Commun
2011;38(1):68–71.
[25] Diaco M. On torsion of functionally graded elastic beams.
Modell Simul Eng 2016:(8464205).
[26] Chen T, Benveniste Y, Chuang P. Exact solutions in torsion
of composite bars:thickly coated neutral inhomogeneities and
composite cylinder assemblages. Proc RSoc Lond A
2002;458(2023):1719–59.
[27] Chen T. An exactly solvable microgeometry in torsion:
assemblage of multicoatedcylinders. Proc R Soc Lond A
2004;460(2047):1981–93.
[28] Chen T, Lipton R. Bounds for the torsional rigidity of
shafts with arbitrary cross-sections containing cylindrically
orthotropic fibres or coated fibres. Proc R Soc LondA
2007;463(2088):3291–309.
[29] Chen T. A universal expression for bounds on the torsional
rigidity of cylindricalshafts containing fibers with arbitrary
transverse cross-sections. J Elast2010;101(1):1–27.
[30] Jog C, Mokashi IS. A finite element method for the
Saint-Venant torsion andbending problems for prismatic beams.
Comput Struct 2014;135:62–72.
[31] Katsikadelis JT, Tsiatas GC. Saint-Venant torsion of
non-homogeneous anisotropicbars. J Appl Comput Mech
2016;2(1):42–53.
[32] Sofiyev A. The vibration and stability behavior of freely
supported FGM conicalshells subjected to external pressure. Compos
Struct 2009;89(3):356–66.
[33] Sofiyev A. Non-linear buckling behavior of FGM truncated
conical shells subjectedto axial load. Int J Non-Linear Mech
2011;46(5):711–9.
[34] Sofiyev A, Schnack E. The vibration analysis of FGM
truncated conical shells restingon two-parameter elastic
foundations. Mech Adv Mater Struct 2012;19(4):241–9.
[35] Ansari R, Torabi J. Numerical study on the buckling and
vibration of functionallygraded carbon nanotube-reinforced
composite conical shells under axial loading.Compos Part B: Eng
2016;95:196–208.
[36] Sofiyev A. Thermoelastic stability of freely supported
functionally graded conicalshells within the shear deformation
theory. Compos Struct 2016;152:74–84.
[37] Sofiyev A, Zerin Z, Kuruoglu N. Thermoelastic buckling of
FGM conical shells undernon-linear temperature rise in the
framework of the shear deformation theory.Compos Part B: Eng
2017;108:279–90.
[38] Van Dung D, Chan DQ. Analytical investigation on mechanical
buckling of FGMtruncated conical shells reinforced by orthogonal
stiffeners based on FSDT. ComposStruct 2017;159:827–41.
[39] Shen Y, Chen Y, Li L. Torsion of a functionally graded
material. Int J Eng Sci2016;109:14–28.
[40] Batra R. Torsion of a functionally graded cylinder. AIAA J
2006;44(6):1363–5.[41] Batra R. Material tailoring in finite
torsional deformations of axially graded
Mooney-Rivlin circular cylinder. Math Mech Solids
2015;20(2):183–9.[42] Barretta R, Feo L, Luciano R. Some
closed-form solutions of functionally graded
beams undergoing nonuniform torsion. Compos Struct
2015;123:132–6.[43] Dryden J, Batra R. Material tailoring and
moduli homogenization for finite twisting
deformations of functionally graded Mooney-Rivlin hollow
cylinders. Acta Mech2013;224(4):811–8.
[44] Toupin RA. Saint-Venant’s principle. Arch Ration Mech Anal
1965;18(2):83–96.[45] Batra R. Saint-Venant’s principle for a
helical spring. J Appl Mech
1978;45(2):297–301.[46] Batra RC, Nie G. Torsional deformations
and material tailoring of orthotropic bi-
directional FGM hollow truncated conical cylinders with curved
lateral surfaces. IntJ Eng Sci 2018;133:336–51.
G.J. Nie et al. Composite Structures 210 (2019) 831–839
839
