A pressurized functionally-graded cylinder is considered made of the material whose elastic moduli vary with the radial distance according to the power-law relation. Some peculiar features of the mechanical response are noted for an incompressible functionally-graded material with the power of radial inhomogeneity equal to two. In particular, it is shown that the maximum shear stress is constant throughout the cylinder, while the displacement changes proportional to 1/r along the radial distance. No displacement takes place at all under equal pressures applied at both boundaries.A pressurized functionally-graded cylinder is considered made of the material whose elastic moduli vary with the radial distance according to the power-law relation. Some peculiar features of the mechanical response are noted for an incompressible functionally-graded material with the power of radial inhomogeneity equal to two. In particular, it is shown that the maximum shear stress is constant throughout the cylinder, while the displacement changes proportional to 1/r along the radial distance. No displacement takes place at all under equal pressures applied at both boundaries

Lubarda, Vlado

National and University Library of the Republic of Srpska: DOI Srspka / DOI Српска

Vlado Lubarda, On the mechanical response of a pressurized functionally-graded cylinder 
Contemporary Materials, V−1 (2014)                                                                                                                 Page 26 of 29 
 
Original scientific papers                                                                                                                      UDK 621.762:669.15  
doi: 10.7251/COMEN1401026L 
 
 
ON THE MECHANICAL RESPONSE OF A PRESSURIZED  
FUNCTIONALLY-GRADED CYLINDER 
 
Vlado Lubarda* 
Department of NanoEgineering and Mechanical and Aerospace Engineering, 
University of California, San Diego; La Jolla, CA 92093-0448, USA 
Montenegrin Academy of Sciences and Arts, Rista Stijovica 5, 81000 Podgorica, Montenegro 
 
Abstract: A pressurized functionally-graded cylinder is considered made of the ma-
terial whose elastic moduli vary with the radial distance according to the power-law relation. 
Some peculiar features of the mechanical response are noted for an incompressible func-
tionally-graded material with the power of radial inhomogeneity equal to two. In particular, it 
is shown that the maximum shear stress is constant throughout the cylinder, while the dis-
placement changes proportional to 1/r along the radial distance. No displacement takes place 
at all under equal pressures applied at both boundaries. 
Keywords: Elasticity, functionally-graded material, incompressibility, pressurized 
cylinder, radial inhomogeneity, stress amplification.  
 
 
 
 
1. INTRODUCTION 
 
The effects of radial inhomogeneity of ma-
terial on the mechanical response in thick-walled 
cylinders under different loading conditions have 
been studied by many researchers. The rep-
resentative references include [1−9]. Unusual cha-
racteristics of the response associated with even a 
small degree of radial nonuniformity have been 
discussed, such as the stress amplification due to 
enhanced material stiffness, or the stress shielding 
due to softening of material near a hole. These stu-
dies are of importance for the mechanics of compo-
site materials, processing of the functionally-graded 
materials, wood industry, etc. 
In this paper we consider a pressurized func-
tionally-graded cylinder made of the material whose 
elastic moduli vary with the radial distance according 
to the power-law relation. The mechanical response 
is evaluated for compressible and incompressible 
functionally-graded materials. Some distinct features 
of stress response are noted in the case of the material 
with the quadratic-type radial inhomogeneity. It is 
shown that in this case the maximum shear stress is 
constant throughout an incompressible cylinder, 
while the displacement changes proportional to 1/r 
along the radial distance. The displacement in the 
cylinder vanishes identically under equal pressures 
applied at both boundaries. The presented analysis 
can be extended to other types of boundary condi
tions, which correspond to prescribed displacements 
at both boundaries, or mixed boundary conditions, 
along the lines presented in [10].  
 
 
2. FUNCTIONALLY-GRADED CYLINDER 
 
We consider a long hollow cylinder under 
plane strain conditions, made of isotropic material 
which is functionally-graded (nonuniform) in the 
radial direction so that its elastic shear modulus va-
ries according to the power-law relation 
  .mb
b
rr 

                                                    (1) 
The exponent m  is a positive or a negative real 
number, reflecting the degree of nonuniformity of the 
material, and μb  is  the shear modulus at the outer 
boundary r  = b.  If m >  0 ,  the elastic stiffness 
increases outward from the inner (r =  a)  to the outer 
(r = b)  radius of the cylinder, while it decreases for 
m <  0 .  The shear moduli at the two boundaries are 
related by μa  =cmμb ,  where c  = a/b  is the aspect 
ratio of the hollow cylinder. Generally speaking, for 
m >  0  there is a tendency for stress shielding, and for 
m <  0  for stress amplification. Other types of non-
uniformity can be considered, such as the exponential 
variation [5]. The stress-strain relations for infinite-
simally small elastic deformations are 
* Corresponding author: vlubarda@ucsd.edu 
Vlado Lubarda, On the mechanical response of a pressurized functionally-graded cylinder 
Contemporary Materials, V−1 (2014)                                                                                                                 Page 27 of 29 
 
  ,1
2
1
  rr  
  .1
2
1
r                                               (2) 
The coefficient of lateral contraction v  is as-
sumed to be independent of r ,  which simplifies the 
mathematical aspects of the analysis [7−9]. As a 
consequence, the Young's modulus of elasticity also 
varies according to the power-law relation [1,3,9] 
,)(
m
b
b
rErE 

   .12 bbE                            (3) 
 
 
3. GOVERNING DIFFERENTIAL  
 EQUATIONS 
 
If uniform pressure is applied at two bounda-
ries, the circumferential component of displacement 
is zero, and the radial component u  depends on the 
radial distance r  only. Correspondingly, the stress 
components σ r  and σθ  are also r-dependent only. In 
the absence of body force, the equilibrium equation is 
[11] 
.0
rdr
d rr                                                   (4)  
The strain-displacement relations are εr = du/dr, and 
εθ = u/r, with the corresponding Saint-Venant com-
patibility condition  
.0
rdr
d r                                                              (5) 
By the standard procedure it follows that the radial 
stress is governed by the second-order differential 
equation 
  .0
1
2132
2
2 
 rrr mdr
drm
dr
dr 

        (6)  
The general solution of this equation is  
.21 nnr BrAr
                                                          (7)  
the integration constants are A and B, and the expo-
nents n1 and n2 are  
 ,2
2
1
2,1 smn   .1
44
2/1
2 

  m
ms 

     (8)  
The circumferential stress readily follows from (4) 
and is given by  
    .11 21 21 nn BrnArn                              (9)  
The displacement can be conveniently deduced 
from the circumferential strain εθ as u = rεθ. By 
substituting (7) and (9) into the second of (2), the 
circumferential strain is found to be  
 ,
2
1
21
21
nn
m
b BrArr
b  

                       (10) 
with the parameters  
  ,121 11 n     .121 22 n                                             (11)  
Therefore, the radial displacement is  
 ,
2
21
21
mm
b
m
BrArbu                                      (12)  
where  
,111  nmm  .122  nmm                         (13)  
For  a uniform  material  (m = 0),  the parameters are  
n1 = 0, n2 = 2, m1 = −1, and m2 = 1.  
 
 
4. BOUNDARY CONDITIONS  
 
If the uniform pressures p and q are applied at 
the inner and outer boundary, the boundary condi-
tions are σr(a) = −p and σr(b) = −q. The integration 
constants in (7) are then  
,
1
1
2
n
s
n
b
c
qpcA 
 ,
1
2
1
n
s
n
b
c
qpcB 
                       (14)  
so that the radial and hoop stresses can be expressed 
as  
,
11
)(
2112
n
s
nn
s
n
r r
b
c
qpc
r
b
c
qpcr 







           (15) 
    .
1
1
1
1)(
2112
21
n
s
nn
s
n
r
b
c
qpcn
r
b
c
qpcnr 







   
(16) 
The corresponding radial displacement is  
.
112
)(
2112
21 


 







 
m
s
nm
s
n
b r
b
c
qpc
r
b
c
qpcbru    
 (17) 
 
 
5. INCOMPRESSIBLE FUNCTIONAL-
LY-GRADED CYLINDER 
 
If the considered functionally-graded material 
is also elastically incompressible (υ = 1/2), then  
s = |2 — m|. For example, if m < 2 there follows:  
n1 = 0, n2 = 2 — m, m1 = m — 1, m2 = 1, n1 = 0, and  
n2 = (m — 2)/2. Consequently, in this case, 
Vlado Lubarda, On the mechanical response of a pressurized functionally-graded cylinder 
Contemporary Materials, V−1 (2014)                                                                                                                 Page 28 of 29 
 
,
11
)(
2
22
2 m
mm
m
r r
b
c
qp
c
qpcr








                  (18)  
  ,
1
1
1
)(
2
22
2 m
mm
m
r
b
c
qpm
c
qpcr








   (19) 
and  
  .
1
2
4
)( 2 r
b
c
qpmbru mb 
                                  (20) 
If m = 1, the hoop stress σθ is uniform throughout the 
cylinder.  
 
 
5.1. Incompressible material with m  = 2 
 
For this particular type of the material (υ = 1/2 
and m = 2), the governing differential equation for the 
radial stress is, from (6), 
,02
2
2 
dr
dr
dr
dr rr                                                   (21)  
which has the solution  
  .In
In 
q
b
r
c
pqrr 

                                            (22)  
The corresponding hoop stress is  
  ,In
In 
q
b
er
c
pqr 

                                        (23)  
where e is the Naperian base of the natural logarithm.  
The magnitude of the maximum shear stress is 
constant throughout the cylinder and equals to  
.
In 22max c
qpr                                       (24) 
The hoop strain is  
  ,
In 4
2



r
b
c
pqr b                                                 (25) 
so that the displacement varies with the radial dis-
tance according to  
    .
In 4
2
r
b
c
pqrrru b
                                     (26)  
In particular, if equal pressures are applied at both 
boundaries (p = q), no displacement takes place in the 
cylinder made of this particular functionally-graded 
incompressible material. For a thin-walled cylinder, 
with the thickness δ = b — a « b, the approximation  
In c ≈ − δ/b can be used. For a very thick cylinder 
 (b » a), In c →∞ which gives rise to stress amplifi-
cation.  
6. CONCLUSION 
 
We have presented an analysis of elastic re-
sponse of a pressurized hollow cylinder whose stiff-
ness is nonuniform in the radial direction according 
to the power law relationship μ(r) = μb(r/b)m, where b 
is the outer radius of the cylinder. The stress com-
ponents are expressed as linear combinations of two 
power functions of the radial coordinate r, whose 
exponents n1 and n2 depend on the Poisson ratio υ and 
the nonuniformity parameter m. The corresponding 
exponents in the displacement expression m1 and m2 
also depend on υ and m only. Some distinct features 
of mechanical response are noted for incompressible 
functionally-graded material with the quadratic de-
pendence of the elastic stiffness on the radial dis-
tance. It is shown that the maximum shear stress is 
constant throughout the cylinder, while the dis-
placement decreases as b2/r. No displacement takes 
place under equal pressures applied at both bounda-
ries. The obtained results may be of interest for the 
optimization of material properties of functionally- 
graded materials. 
 
 
7. ACKNOWLEDGMENTS 
 
 This research was supported by the Monte-
negrin Academy of Sciences and Arts and the MAE 
endowment fund from UCSD. Valuable discussions 
with Professor Frank E. Talke are kindly acknowl-
edged. 
 
 
8. REFERENCES 
 
[1] S. G. Lekhnitskii, Anisotropic Plates, 2nd 
ed., Gordon and Breach, New York, 1987. 
[2] R. M. Christensen, Properties of carbon 
fibers, J. Mech. Phys. Solids, Vol. 42 (1994) 
681−695. 
[3] C. O. Horgan, A. M. Chan, The pressurized 
hollow cylinder or disk problem for functionally 
graded isotropic linearly elastic materials, J. Elas-
ticity, Vol. 55 (1999) 43−59. 
[4] R. L. Fosdick, G. Royer-Carfagni, The 
constraint of local injectivity in linear elasticity 
theory, Proc. R. Soc. Lond. A, Vol. 457 (2001) 
2167−2187. 
[5] N. Tutanku, Stresses in thick-walled FGM 
cylinders with exponentially-varying properties, Eng. 
Structures, Vol. 29 (2007) 2032−2035. 
[6] R. C. Batra, G. L. Iaccarino, Exact solu-
tions for radial deformations of a functionally graded 
isotropic and incompressible second-order elastic 
Vlado Lubarda, On the mechanical response of a pressurized functionally-graded cylinder 
Contemporary Materials, V−1 (2014)                                                                                                                 Page 29 of 29 
 
cylinder, Int. J. Nonlin. Mechanics, Vol. 43 (2008) 
383−398. 
[7] E. Erdogan, Fracture mechanics of func-
tionally graded materials, Composites Engineering, 
Vol. 5 (1995) 753−770. 
[8] Z. H. Jin, R. C. Batra, Some basic fracture 
mechanics concepts in functionally graded materials, 
J. Mech. Phys. Solids, Vol. 44 (1996) 1221−1235. 
[9] V. A. Lubarda, On pressurized curvili-
nearly orthotropic circular disk, cylinder and sphere 
made of radially nonuniform material, J. Elasticity, 
Vol. 109 (2012) 103−113. 
[10] V. A. Lubarda, Elastic response of ra-
dially nonuniform, curvilinearly anisotropic solid 
and hollow disks, cylinders and spheres, Proc. Mon-
ten. Acad. Sci. Arts, Vol. 20 (2013). 
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of Elasticity, 3rd ed., McGraw-Hill, New York, 1970. 
 
 
 
 
О МЕХАНИЧКОМ ОДГОВОРУ ФУНКЦИОНАЛНО ГРАДИЈЕНТНОГ  
ЦИЛИНДРА ПОД ПРИТИСКОМ  
 
Сажетак: Разматра се функционално градијентни цилиндар под притиском 
чији модули еластичности варирају са радијалном удаљености по степеној функцији. 
Уочене су неке специфичне карактеристике механичког понашања некомпресибилног 
функционално градијентног материјала чији је степен радијалне нехомогености једнак 
два. Показано је да је максимални напон смицања константан у цијелом цилиндру, док 
је помјерање пропорционално са 1/r уздуж радијалног правца. Помјерање је једнако 
нули у цијелом цилиндру када се на његову унутрашњу и спољашњу површину 
примијени једнаки притисак. 
Кључне ријечи: еластичност, функционално градијентни материјал, 
нестишљивост, цилиндар под притиском, радијална нехомогеност, амплификација 
напона. 
 
 


ON THE MECHANICAL RESPONSE OF A PRESSURIZED  FUNCTIONALLY-GRADED CYLINDER

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