Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 1, pp. 343-351, Warsaw 2017
DOI: 10.15632/jtam-pl.55.1.343

A NOVEL APPROACH TO THERMAL AND MECHANICAL STRESSES IN

A FGM CYLINDER WITH EXPONENTIALLY-VARYING PROPERTIES

Kerimcan Celebi

Adana Science and Technology University, Department of Mechanical Engineering, Adana, Turkey

Durmus Yarımpabuc

Osmaniye Korkut Ata University, Department of Mathematics, Osmaniye, Turkey

e-mail: durmusyarimpabuc@osmaniye.edu.tr

Ibrahim Keles

Department of Mechanical Engineering, Ondokuz Mayis University, Samsun, Turkey

Anovel approach is employed to a general solution for one-dimensional steady-state thermal
and mechanical stresses in a hollow thick cylinder made of a functionally graded material
(FGM). The temperature distribution is assumed to be a function of radius, with general
thermal andmechanical boundary conditions on the inside and outside surfaces of the cylin-
der.Thematerial properties, exceptPoisson’s ratio, areassumed tobe exponentially-varying
through the thickness.Forcing functions applied to the innerboundary are internal pressures
which may be in form of steps. These conditions result in governing differential equations
with variable coefficients. Analytical solutions to such equations cannot be obtained except
for certain simple grading functions and pressures. Numerical approachesmust be adopted
to solve the problem in hand. The novelty of the present study lies in the fact that the
ComplementaryFunctionsMethod (CFM) is employed in the analysis. TheComplementary
Functions method (CFM) will be infused into the analysis to convert the problem into an
initial-value problem which can be solved accurately. Benchmark solutions available in the
literature are used to validate the results and to observe the convergence of the numerical
solutions. The solution procedure is well-structured, simple and efficient and it can be re-
adily applied to cylinders. It is also well suited for problems in whichmechanical properties
are graded.

Keywords: thermal stresses, functionally-graded materials, thick cylinder, Complementary
Functions Method

1. Introduction

Pressure vessel structural members such as cylinders, disks and spheres find broad application
fields in the industry, and their vibration analyses are deemed necessary for safe design and ope-
ration.Hollow cylinders and thick-walled cylindrical shells are common components in structural
applications and device systems involving aerospace and submarine structures, civil engineering
structures, machines, pipes, sensors and actuators, etc. These structures are often exposed to
temperature environment and thermal stresses are then induced. In many cases, thermal stres-
ses will significantly depress strength and also affect functionality of structures. Thus, the exact
analysis of thermal stresses is really important (Ying andWang, 2010). There have beenmany
studies, such as Timoshenko andWoinowsky-Krieger (1959), Boley andWeiner (1960), Das and
Navaratna (1962), Das andRath (1972), Stavsky (1963) andThangaratnam et al. (1988), which
focused on thermal stresses in isotropic homogeneous rectangular plates. Yee andMoon (2002)
have been obtained a closed-form analytical solution for the plane thermal stress analysis of
a homogeneously orthotropic hollow cylinder subjected to an arbitrary, transient, asymmetric



344 K. Celebi et al.

temperature distribution. They used a stress function approach for obtaining hoop, radial, and
shear stresses in a hollow cylinder. Shao (2005) presented, by using a multi-layered approach
based on the theory of laminated composites, the solutions of temperature, displacements, and
thermal-mechanical stresses in a functionally graded circular hollow cylinder. Shao et al. (2008)
used complex Fourier series and Laplace transform techniques to investigate transient heat con-
duction and thermo-mechanical stresses in an FGMhollow cylinder. Jabbari et al. (2002, 2003,
2009) derived the exact solution for one-dimensional and two dimensional steady-state thermo-
elastic problems of functionally graded hollow cylinders where material properties varied with
the power product form of the radial coordinate variable. Recently Ruhi et al. (2005) studied
thermoelastic analysis of thickwalled finite length cylinders of functionally gradedmaterials and
achieved results for stress, strain and displacement components through the thickness and along
the length. The results were presented for uniform internal pressure and thermal loading. Ootao
and Tanigawa (2006) analyzed exactly a one-dimensional transient thermoelastic problem of a
functionally graded hollow cylinder whose thermal and thermoelastic constants were assumed
to vary with the power product form of the radial coordinate variable. The resulting governing
differential equation then possessed variable coefficients. General closed-form solutions to such
equations are not available. Noda et al. (2012) studied the transient thermoelastic analysis for an
FGM solid circular disk whosematerial properties were expressed by a piecewise power law. As
it was done in the works cited above, in such situations the solution methods included integral
transformations, development of finite element models, and, in some special cases, series solu-
tionswere attempted. Assuming that thememberwas composed ofmany homogeneous layers of
different properties emulating theFGMbehavior, therewas anotherway of tackling the problem
on hand. All of these approaches required heavy mathematical manipulations and, in the case
of having to discretize the domain into many elements, a high amount of computational time.

In the present paper, the governing differential equation is non-homogeneous with variable
coefficients which include material properties. A novel approach is attempted to obtain displa-
cements, strains and stresses in a simple and efficient manner. The complementary functions
method (CFM), theoretically explained in the literature by Aktaş (1972), Agarwal (1982) and
Roberts and Shipman (1979) is infused into analysis to convert the problem to an initial-value
problem which can be then easily solved by, for example, the fifth-order Runge-Kutta method
(RK5) with great accuracy (Chapra and Canale, 1998). Shell theories or dividing the material
into homogeneous subelements of different properties emulating the graded behavior contains
the customary approach of modeling FGM structural elements. Finite element analysis, series
expansionmethods anddirectmethods are primary solutionmethods used in the literature. The
present paper uses a novel and efficientmethodwhich employes CFM.A thick hollow cylinder of
FGM under one-dimensional steady-state temperature distribution with general types of ther-
mal andmechanical boundary conditions is analysed. Twomaterial models are used: (a) with a
simple power lawwith constant Poisson’s ratio (Jabbari et al., 2002) for which analytical bench-
mark solutions are available, (b) with exponentially-varying properties. It should be emphasized
once again that the solution procedure is not confined to any particular choice of the material
model; it is equally suitable for arbitrary functions defining the gradient variation of material
properties.

2. Solutions by the Complementary Functions Method

TheCFM transforms two-point boundary-value problems to a system of initial-value problems.
It reduces to a particularly simple solution scheme when applied to a given class of problems,
e.g. for an annular disk of inner radius ri and outer radius ro. As it is shown in the proceeding
Sections, under axisymmetric conditions, the governing differential equation of the dependent
variable u(r) in its most general form is



Novel approach to thermal and mechanical stresses... 345

u′′+P(r)u′+Q(r)u=R(r) (2.1)

subject to boundary conditions on the inner (r = ri) and outer (r = ro) surfaces. Here (·)′
denotes the derivative with respect to r. A general closed-form solution of the above equation
cannot be obtained. The complete solution to Eq. (2.1) is

u= bjuj +up j=1,2 (2.2)

where uj and up are, respectively, homogenous and particular solutions. The coefficients bj are

determined via the boundary conditions. CFM begins by assuming ui = Y
(i)
1 and u

′

i = Y
(i)
2 ,

which means

(Y
(i)
1 )
′ =Y

(i)
2 (2.3)

Here, the index i=1,2 refers to homogeneous solutions and i= pmeans the particular solution.
To determine the homogeneous solutions, the right-hand side of Eq. (2.1) is set equal to zero,
and the following is obtained

(Y
(i)
2 )
′ =−P(r)Y (i)2 −Q(r)Y

(i)
1 (2.4)

The system of Eqs. (2.3) and (2.4) can be solved numerically for each homogeneous solution.
TheKronecker delta initial conditions given below are used to assure linear independence of the
solutions (Roberts and Shipman, 1979)

Y
(i)
j = δji j,i=1,2 (2.5)

To obtain the particular solution, Eq. (2.4) is modified as

(Y
(p)
2 )
′ =−P(r)Y (p)2 −Q(r)Y

(p)
1 +R(r) (2.6)

A particular solution needs only to satisfy the differential equation and homogeneous initial
conditions

Y
(p)
j =0 j=1,2 (2.7)

be imposed. Equations (2.3), (2.6), (2.7) constitute a system of equations for the particular
solution alongwith the initial conditions. Thefifth-orderRunge-Kuttamethod (RK5) is used for
all cases considered.Note that by this procedure not only the solutionu(r) itself but also its first
derivative are readily calculated. Applying the boundary conditions prescribed for the particular
probleminhandresults in the following systemofalgebraic equations for thecoefficients b1 and b2

[

A11 A12
A21 A22

][

b1
b2

]

=

[

RHS1
RHS2

]

(2.8)

Here,Aij includes the values of the homogeneous solutions at the boundary points.RHS1 and
RHS2 contain values of the particular solutions. If the cylinder is subjected to internal and
external pressures, they will also be included in the right hand-side terms. On the other hand,
implementing CFM in the heat conduction problem yields RHS1 and RHS2 as prescribed
temperatures along the boundaries. These points will be illustrated in the following Sections.



346 K. Celebi et al.

3. Heat conduction in the radial direction

The heat conduction equation in the steady-state condition for a one-dimensional problem in
polar coordinates and thermal boundary conditions for a FGM hollow cylinder are given, re-
spectively, as

1

r

(

rk(r)T ′(r)
)′

=0 ri ¬ r¬ ro
C11T(ri)+C12T

′(ri)= f1

C11T(ro)+C12T
′(ro)= f2

(3.1)

where k = k(r) is the thermal conduction coefficient, ri and ro are the inner and outer radii
of the hollow cylinder. Cij are the constant thermal parameters related to the conduction and
convection coefficients. The constants f1 and f2 are known constants on the inside and outside
radii.

It is assumed that the nonhomogeneous thermal conduction coefficient k(r) is an exponential
function of r as

k(r)= koe
βr (3.2)

where ko is amaterial constant and β is the inhomogeneity parameter. Using Eq. (3.2), the heat
conduction equation becomes

1

r

(

reβrT ′(r)
)′

=0 (3.3)

Steady-state axisymmetric heat conduction without heat generation is considered. The heat
balance equation in the radial direction for a nonuniform disk yields

T ′′+B(r)T ′ =0 (3.4)

whereB(r)= (1/r)+β and it is varying as a function of the radial coordinate r. The boundary
conductions are temperatures prescribed on the inner and outer surfaces as

T(ri)=Ti and T(ro)=To (3.5)

The complete solution is the homogeneous solution

T = bjTj j=1,2 (3.6)

with

T ′ = bjT
′

j j=1,2 (3.7)

Following the steps outlined in Section 2, the temperature distribution is obtained at the collo-
cation points. The constants bj can now be found by imposing the boundary conditions. This
process results in a system given by Eq. (2.8) where

A11 =T1(ri) A12 =T2(ri)

A21 =T1(ro) A22 =T2(ro)

RHS1=Ti RHS2=To

(3.8)



Novel approach to thermal and mechanical stresses... 347

4. Governing equation

Consider a thick walled cylinder of the inside radius ri and the outside radius ro made of FGM.
The material is graded through the r−direction. Let u be the displacement component in the
radial direction. Then the strain-displacement relations are

εrr =
du

dr
εθθ =

u

r
(4.1)

The stress-strain relations are

σrr =(λ+2µ)εrr +λεθθ− (3λ+2µ)αT(r)
σθθ =(λ+2µ)εθθ +λεrr− (3λ+2µ)αT(r)

(4.2)

where σij and εij (i,j = r,θ) are stress and strain tensors, T(r) is temperature distribution
determined from the heat conduction equation, α is the coefficient of thermal expansion, and λ
and µ are the Lame coefficients related to the modulus of elasticity E and Poisson’s ratio ν as

λ=
νE(r)

(1+ν)(1−2ν)
µ=

E(r)

2(1+ν)
(4.3)

The equilibrium equation in the radial direction, disregarding the body force and inertia terms,
is

∂σrr
∂r
+
σrr−σθθ
r

=0 (4.4)

To obtain the equilibrium equation in terms of the displacement component for the FGM cylin-
der, the functional relationship of thematerial propertiesmust be known.To ascertain the effect
of the inhomogeneity, the properties are considered to vary exponentially across the thickness

E(r)=Eoe
βr α=αoe

βr (4.5)

where Eo and αo are the material constants and β is the inhomogeneity parameter. Poisson’s
ratio varies very little through the thickness in FGM materials. Furthermore, its effects on
thermal and mechanical stresses are insignificant. For simplicity, Poisson’s ratio is assumed to
be constant (Akbari Alashti et al., 2013; Jabbari et al., 2015).
Using relations (4.1)-(4.5), the Navier equation in term of the displacement is

u′′+P(r)u′+Q(r)u=R(r) (4.6)

where

P(r)= (βr+1)
1

r
Q(r)=

( νβr

1−ν
−1
) 1

r2

R(r)=
eβrαo(1+ν)

(1−ν)
(2βT +T ′)

(4.7)

Following the stepsoutlined inSection2, the completedisplacement is obtainedat thecollocation
points as

u= b1u1+ b2u2+up (4.8)

with

u′ = b1u
′

1+ b2u
′

2+u
′

p (4.9)

The coefficients b1, b2 will bedeterminedusing the stress free conditions on inner (σrr(ri)=−Pi)
and outer (σrr(ro)=−Po) boundaries. This step is particularly simple since during the solution
process the first derivative of the radial displacement has already been calculated.



348 K. Celebi et al.

5. Results and discussions

As an example, consider a thick hollow cylinder of the inner radius a ri = 1m and the outer
radius ro = 1.2m. Poisson’s ratio is taken to be 0.3, and the modulus of elasticity and the
thermal coefficient of expansion at the inner radius are Eo = 200GPa and αo = 1.2 ·10−6/◦C,
respectively. The properties are considered to vary exponentially across the thickness. The bo-
undary conditions for temperature are taken as T(ri) = 10

◦C and T(ro) = 0
◦C. The hollow

cylinder has pressure on its inner surface, so the boundary conditions for stresses are assumed
as σrr(ri)=−50MPa and σrr(ro)= 0MPa.
Thenumerical solution in thepresent study is checkedwith the results obtainedbyJabbari et

al. (2002) for the validation purpose. Comparison is illustrated inTables 1-3. It can be observed
that the results are in good agreement with the same results by Jabbari et al. (2002). The
numerical results have been obtained to six-digit accuracy by picking only 11 collocation points.

Table 1.Comparison of CFMwith Jabbari et al. (2002) for a homogenous cylinder (m=0 and
β=0)

r

ri

T/T(ri) u/ri σrr/Pi σθθ/Pi
CFM Jabbari CFM Jabbari CFM Jabbari CFM Jabbari

1 1 1 0.00136642 0.00136642 −1 −1 5.50909 5.50909
1.04 0.784882 0.784882 0.00133799 0.00133799 −0.754183 −0.754183 5.27802 5.27802
1.08 0.577883 0.577883 0.00131239 0.00131239 −0.534644 −0.534644 5.07268 5.07268
1.12 0.378413 0.378413 0.00128932 0.00128932 −0.337716 −0.337716 4.88943 4.88943
1.16 0.185944 0.185944 0.0012685 0.0012685 −0.160351 −0.160351 4.72526 4.72526
1.2 0 0 0.0012497 0.0012497 0 0 4.57766 4.57766

Table 2. Comparison of CFM with Jabbari et al. (2002) for FGM cylinders with constant
Poisson’s ratio and the elastic modulus obeying a simple power law (m=−2 and β=−2)

r

ri

T/T(ri) u/ri σrr/Pi σθθ/Pi
CFM Jabbari CFM Jabbari CFM Jabbari CFM Jabbari

1 1 1 0.00161944 0.00161944 −1 −1 6.62127 6.62127
1.04 0.814545 0.814545 0.00158667 0.00158667 −0.722291 −0.722291 5.84288 5.84288
1.08 0.621818 0.621818 0.00155679 0.00155679 −0.491571 −0.491571 5.19022 5.19022
1.12 0.421818 0.421818 0.00152961 0.00152961 −0.29876 −0.29876 4.63928 4.63928
1.16 0.214545 0.214545 0.00150494 0.00150494 −0.136767 −0.136767 4.1713 4.1713
1.2 0 0 0.00148263 0.00148263 0 0 3.77145 3.77145

Table 3. Comparison of CFM with Jabbari et al. (2002) for FGM cylinders with constant
Poisson’s ratio and the elastic modulus obeying a simple power law (m=2 and β=2)

r

ri

T/T(ri) u/ri σrr/Pi σθθ/Pi
CFM Jabbari CFM Jabbari CFM Jabbari CFM Jabbari

1 1 1 0.00114082 0.00114082 −1 −1 4.51743 4.51743
1.04 0.753093 0.753093 0.00111628 0.00111628 −0.784165 −0.784165 4.70652 4.70652
1.08 0.533109 0.533109 0.00109454 0.00109454 −0.577254 −0.577254 4.89893 4.89893
1.12 0.336271 0.336271 0.00107516 0.00107516 −0.378189 −0.378189 5.09474 5.09474
1.16 0.159442 0.159442 0.00105779 0.00105779 −0.186042 −0.186042 5.29405 5.29405
1.2 0 0 0.00104213 0.00104213 0 0 5.49697 5.49697



Novel approach to thermal and mechanical stresses... 349

Figure 1a showsvariations of temperature along the radial direction for different values of the
inhomogeneity parameter (β).Thefigure shows that as the inhomogeneity parameterβ increases,
the temperature decreases. Figure 1b shows the plot of the radial displacement along the radius.
The magnitude of the radial displacement is decreased as the inhomogeneity parameter β is
increased. The radial and circumferential stresses are plotted along the radial direction and are
shown in Figs. 1c and 1d. The magnitude of the radial stress is increased as β is increased. It
is seen that for β < 1 the hoop stress decreases along the radial direction. For β > 1, the hoop
stress increases as the radius increases, since themodulus of elasticity is an increasing function of
the radius, seeEq. (2.6). Physically, thismeans that the outer layers of the cylinder are biased to
maintain the stress due to their higher stiffness. There is a limiting value for β, where the hoop
stress remains almost constant along the radius. The curve associated with β = 1 shows that
the variation of hoop stress along the radial direction isminor, and is almost uniform across the
radius. To investigate the pattern of stress distribution along the cylinder radius, the effective
stress σ∗ =

√
2|σr−σθ| is plotted along the radial direction for different values of ro/ri and the

inhomogeneity parameter β. Figure 2 is plotted for ro/ri = 1.2. It is interesting to note from
Fig. 2 that for β=1 the effective stress is almost uniform along the radius of the cylinder.

Fig. 1. Radial distribution of: (a) temperature, (b) radial displacement, (c) radial stress and (d) hoop
stress for cylinder

It should be pointed out once again that the purpose of the present work is the introduction
ofCFMto the solution procedure of the class of problems in hand.Converting the two-point bo-
undary value problem to a system of an initial-value problem gives away to the implementation
of well-established numerical schemes. TheRunge-Kuttamethod of fifth-order (RK5) is used to



350 K. Celebi et al.

Fig. 2. Effective stress distribution for ro/ri =1.2

solve the system of equations. The procedure is simple and efficiently implemented. The nume-
rical results have been obtained exact up to six-digit accuracy by picking only 11 collocation
points in RK5.

6. Conclusion

Thispaperpresents a numerical solution for calculation of axisymmetric thermal andmechanical
stresses in a thick hollow cylinder made of FGM. The material properties through the graded
direction are assumed to be nonlinear with a power law distribution and exponentially-varying
properties. Themechanical and thermal stresses are obtained through the CFM of the solution
of the Navier equation. The comparisons of temperature distributions and stress distributions
are presented in form of tables. The numerical results for all cases are shown to exactly match
those reported by Jabbari et al. (2002). Finally, we can conclude that:

• With the unified approach presented in the present study, one would not have to com-
promise on the functional continuity of the material properties. Analysis of any material
model in form of an arbitrary function subject to internal pressure has been analyzed
efficiently and accurately by employing CFM.

• The unifiedmethod used is accurate andmore efficient than the conventional methods.
• Themethod employed in this study allows one to find solutions of continuous functions.
• The CFM of solving the differential equation provides a complete solution, yielding both
thermal stresses and temperature distributions.

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Manuscript received May 20, 2015; accepted for print September 13, 2016