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IJE Transactions A: Basics Vol. 22, No. 4, November 2009 - 405

EXACT ELASTICITY SOLUTIONS FOR THICK-WALLED FGSPHERICAL PRESSURE VESSELS WITH LINEARLY AND

EXPONENTIALLY VARYING PROPERTIES

A.R. Saidi*, S.R. Atashipour and E. J omehzadeh

Department of Mechanical Engineering, Shahid Bahonar University of Kerman P.O. Box 76175-133, Kerman, Iran

[email protected] - [email protected] - [email protected]

*Corresponding Author

(Received: October 7, 2008 – Accepted in Revised Form: February 19, 2009)

Abstract In this paper, exact closed-form solutions for displacement and stress components of thick-walled functionally graded (FG) spherical pressure vessels are presented. To this aim, linearvariation of properties, as an important case of the known power-law function model is used todescribe the FG material distribution in thickness direction. Unlike the pervious studies, the vesselscan have arbitrary inner to outer stiffness ratio without changing the function variation of FGM. Afterthat, a closed-form solution is presented for displacement and stress components based on exponentialmodel for variation of properties in radial direction. The accuracy of the present analyses is verifiedwith known results. Finally, the effects of non-homogeneity and different values of inner to outerstiffness ratios on the displacement and stress distribution are discussed in detail. It can be seen thatfor FG vessels subjected to internal pressure, the variation of radial stress in radial direction becomeslinear as the inner stiffness becomes five times higher than outer one. When the inner stiffness is half of the outer one, the distribution of the circumferential stress becomes uniform. For the case in whichthe external pressure is applied, as the inner to outer shear modulus becomes lower than 1/5, the valueof the maximum radial stress is higher than external pressure.

Keywords Thick-Walled Pressure Vessels, Functionally Graded Materials, Linearly-Varying

Properties, Exponentially-Varying Properties

د پ چكيده د ده ي د يد

ده . يي د

هی ده .د

يي د د د

د پ . پ يي د

پ ديد پ .د گ دهي .

FG ه گ

د ی گ .

ك د د.

د / گ هدد د ي .

1. INTRODUCTION

Functionally graded materials (FGMs) are non-homogenous composites with continuous variationof the constituents from one surface of the materialto the other. Such a material was first introduced in

Japan [1]. FGMs were first used as a thermal shield

in industries. New applications have become possibleusing FGM such as energy conversion [2], dental andorthopaedic implants [3,4], thermogenerators andsensors [5] and joining dissimilar materials [6].Another important usage of FGM is wear resistantcoatings and the covering of mechanical parts suchas gears, cams, roller bearing and machine tools

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406- Vol. 22, No. 4, November 2009 IJE Transactions A: Basics

[7,8]. Functionally graded materials are madeby combining different materials using powdermetallurgy methods.

There are many researches in literature thatconsider static analysis of functionally gradedcylindrical and spherical vessels based on numericalapproaches such as finite element method [9-11].However, some semi-analytical solutions are foundfor thick-walled FG pressure vessels. You et al[12] presented a static analysis for thick-walled FGspherical pressure vessels. Stress analysis of thick-walled FG cylinders was carried out by Tutuncu[13]. These analyses are based on exponentialmodel for property variation in thickness directionby using the power series solution. It is noticeablethat in most of analyses carried out in the field of

FGM, the variation of the material properties hasbeen modeled as exponential [14-19] or power-lawfunction [20-25]. Based on power-law model, forall values of FGM power, mechanical propertiesratio of the structure surfaces can have anyarbitrary value. There are a few analytical solutionsfor FG hollow cylinders [26,27]. An analyticalsolution for static analysis of thick-walledcylindrical and spherical FG pressure vessels basedon power-law model has been investigated by

Tutuncu, et al [28]. Eslami, et al [29] studiedthermal and mechanical stresses in FG thick

spheres. However, in their analyses an incompletepower-law model is used for variation of Young’s

modulus as β= rEE 0 . The limitation of this model

is that for specific value of FGM power β , the

inner to outer Young’s modulus ratio can not havearbitrary value. For example, if the propertiesfunction is supposed to vary linearly (e.g. 1=β ), it

causes to have inner to outer Young’s modulusratio equal to radius ratio (e.g. i0io RREE = ).

However, no closed-form solutions were found forthick-walled FG spherical vessels based on

exponential FG model and also based on power-law function without the mentioned limitation.

In this article, in order to vanish the limitation of incomplete power-law model, an exact analyticalsolution is presented for thick-walled functionallygraded spherical pressure vessels with linearly-varying properties in radial direction. In this closed-form solution, the inner and outer surface propertiescan have arbitrary value. Besides, a closed-formsolution has been obtained for thick-walled FG

spherical pressure vessels based on exponentialmodel for material distribution in radial direction.

The inner to outer material property ratio can havearbitrary value. Therefore, the effects of applyingdifferent materials in designing of pressure vesselscan be considered on the displacement and stresscomponents. Finally, the results are verified byother known results. Meanwhile, the effects of different parameters on the solution are discussedand the solutions based on different FGM modelsare compared.

2. PROBLEM ANALYSIS

In order to elastostatics analysis of spherically

symmetric thick-walled functionally graded sphericalpressure vessels, only one equilibrium equationexists as follow

ϕθθ σ=σ=

σ−σ+

σ;0

r2

dr

d rr (1)

Supposing infinitesimal deformation, the strain-displacement relations in spherical coordinate systemare

rr

u;

dr

du

r =ϕε=θε=ε (2)

For linear behavior of material, the stress-strainrelations in spherically symmetric state for a non-homogenous isotropic material are written as

⎥⎦⎤

⎢⎣⎡ νε+θε

ν−μ

=θσ

⎥⎦⎤

⎢⎣⎡

θνε+εν−ν−

μ=σ

r)21(

)r(2

2r

)1()21(

)r(2r

(3)

Where )r(μ is the radius dependant shear modulusand ν is the Poisson’s ratio. With regard to thesmall variation of Poisson’s ratio, it is usuallyconsider constant in analysis of functionally gradedmaterials (e.g. [22-26]).

Case 1.

Linearly-varying properties In this case, it isassumed that the variation of mechanical properties

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is linear through the thickness. Therefore, variationof shear modulus of a functionally graded materialis modeled as

BrA)r( +=μ (4)

Supposing that the shear modulus of inner andouter surfaces are iμ and oμ , respectively, the

constant coefficients A and B are calculated as

iR

oR

ioB;

iR

oR

iR

ooRiA

−

μ−μ=

−

μ−μ= (5)

Using above equation, the Equation 4 can berewritten as

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−μμ−−=μμ

oRiR1oRr1

oi11o)r( (6)

Substituting Equations 2 and 3 into Equation 1yields an ordinary differential equation as

0rr

u

1

2

drr

du

)r(

)r(ru

2r

2

drr

du

r

2

2dr

ru2d

=⎟⎟ ⎠

⎞⎜⎜⎝

⎛

ν−ν

+μμ′

+−+

(7)

Substituting Equation 4 into above equation, thefollowing equation yields

0rur

2

BrA

B

1

2

r

1

drr

du

r

2

BrA

B

2dr

ru2d

=⎟ ⎠

⎞⎜⎝

⎛ −+ν−

ν

+⎟ ⎠

⎞⎜⎝

⎛ ++

+(8)

The differential Equation 8 can be solved in term

of Hypergeometric function as

[ ] [ ]

[ ] [ ] ⎟ ⎠

⎞⎜⎝

⎛ − Ν− Ν− Ν+− Ν−−

+⎟ ⎠

⎞⎜⎝

⎛ − Ν+ Ν+ Ν−− Ν+−

=

r

1

B

A,21,2),1(H

)1(r.L2C

r

1

B

A,21,2),1(H

)1(r.L1C

ru

(9)

Where H is the Hypergeometric function in which

is defined by power series as [30]

1ix

0i )ic)...(1c(c!i

)ib)(ia)...(1b)(1a(ab1

)x],c[],b,a([H

+∑∞

= ++

+++++

=

(10)

and the constant coefficient Ν is

ν−

ν−= Ν

1

53(11)

Substituting Equation 9 into strain-displacementrelations (2) and using stress-strain relations (3),the radial and circumferential stresses are obtainedas

⎭⎬⎫

⎟ ⎠ ⎞⎜

⎝ ⎛ ν− Ν−ν−+ν− Ν+

⎩⎨⎧

⎟ ⎠ ⎞⎜

⎝ ⎛ ν− Ν+ν−+ν− Ν−

ν−

+−=σ

)r(4

f )]1(31[)r(2

f )1(rL2

C

)r(3

f )]1(31[)r(1

f )1(rL1

C

2r)21(

)BrA(2r

(12)

⎭⎬⎫

⎟ ⎠ ⎞⎜

⎝ ⎛ Νν+ν−−ν Ν+

⎩⎨⎧

⎟ ⎠ ⎞

⎜⎝ ⎛

Νν−ν−−ν Ν−

ν−

+

−

=ϕσ=θσ

)r(3f )1()r(2f rL2C

)r(3f )1()r(1f rL1C2r)21(

)BrA(2

(13)

Where )r(f 1 through )r(f 4 functions are defined as

following

⎟ ⎠

⎞⎜⎝

⎛ − Ν± Ν±− Ν±=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟ ⎠ ⎞⎜

⎝ ⎛ − Ν± Ν± Ν±

Ν± Ν Ν±

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

r

1

B

A],21[],1,2[H

)r(4f

)r(3

f

r

1

B

A],22[],3,[H

r

1

)21(

)1)(2(

B

A

)r(2f

)r(1f

m(14)

In Equations 9, 12 and 13, the unknown coefficients

L1C and L2C can be determined by applying the

boundary conditions of inner and outer surfaces.For a spherical vessel subjected to both internal

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hydrostatic pressure iP and external hydrostatic

pressure oP , the boundary conditions can be written

as

oP)oRr(r

iP)iRr(r==σ−==σ (15)

Satisfying the above boundary conditions, yields

}oP)iR(3f )]1(31[)iR(1f )1(iR2oR

)iBRA(iP)oR(3f )]1(31[)oR(1f )1(

oR2iR)oBRA{(

)oBRA)(iBRA(2

1)21(L2C

}oP)iR(4f )]1(31[)iR(2f )1(iR2oR

)iBRA(iP)oR(4f )]1(31[)oR(2f )1(

oR2iR)oBRA{(

)oBRA)(iBRA(2

1)21(L1C

⎟ ⎠ ⎞⎜

⎝ ⎛ ν− Ν+ν−+ν− Ν−

+−⎟ ⎠ ⎞⎜

⎝ ⎛ ν− Ν+ν−+ν−

Ν−+++

−Δν−−=

⎟ ⎠ ⎞⎜

⎝ ⎛ ν− Ν−ν−+ν− Ν

+−⎟ ⎠ ⎞⎜

⎝ ⎛ ν− Ν−ν−+ν−

Ν+++

−Δν−=

(16)

Where the parameter Δ is defined as

⎟ ⎠ ⎞⎜

⎝ ⎛ ν− Ν+ν−+ν−

×⎟ ⎠ ⎞⎜

⎝ ⎛ ν− Ν−ν−+ν− Ν− Ν

−⎟ ⎠ ⎞⎜

⎝ ⎛ ν− Ν−ν−+ν−

×⎟ ⎠ ⎞⎜

⎝ ⎛ ν− Ν+ν−+ν− Ν Ν−

=Δ

)oR(3f )]1(31[)oR(1f )1(

)iR(4f )]1(31[)iR(2f )1(oRiR

)oR(4f )]1(31[)oR(2f )1(

)iR(3f )]1(31[)iR(1f )1(oRiR

(17)

Case 2.

Exponentially-varying properties In order toconsider the effect of different FGM models ondisplacement and stress distribution of a sphericalFG vessel, it is assumed that the variation of functionally graded shear modulus obeys theexponential model as

)rexp(c)r( Γ=μ (18)

Where c and Γ are two coefficients. Supposing

that the shear modulus of inner and outer surfacesare iμ and oμ , respectively, the constants c and Γ

are obtained as

iRoR

)ioln(;iRoR

oR

o

ioc

−

μμ=Γ

−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

μ

μμ= (19)

Substituting the above relations into Equation 18yields the following function for variation of shearmodulus in radial direction of vessel

)oR/iR(1

)o

R/r(1

o

io)r(

−

−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

μ

μ=μμ (20)

Replacing Equation 20 into differential Equation 7,the following differential equation with variablecoefficients can be obtained

0ru1r12r

2

drrdu

)2r(r

1

2dr

ru2d=⎟

⎠

⎞⎜⎝

⎛ −Γν−

ν++Γ+ (21)

The solution of this equation is

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟ ⎠

⎞⎜⎝

⎛ Γ

ν−ν−

−

+⎟ ⎠

⎞⎜⎝

⎛ Γ

ν−ν−

−Γ−

=

r,2

3,

1

31W

WL2

C

r,2

3,

1

31MWL1C

r

)2rexp(ru (22)

In which MW and WW are Whittaker functions

[31]. Substituting relation (22) into Equations 2and 3 yields

{ })r(3

g)1(E2

C)r(1

g)1(E1

C2r

)2rexp(21c2

r

ν−−ν+Γν−

=σ

(23)

[ ]{

⎭⎬⎫

⎥⎦⎤

⎢⎣⎡ ν−ν++ν−ν−

+ν−+νν+

Γν−ν−

=ϕσ=θσ

)r(4

g)21)(1()r(3

g)1(E2

C

)r(2

g)21()r(1

g)1(E1

C

2r

)2r(exp

)21()1(

c2

(24)

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Where

⎥⎦

⎤⎢⎣

⎡Γ⎟

⎠

⎞⎜⎝

⎛ ν−ν−

−=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡Γ⎟

⎠

⎞⎜⎝

⎛ ν−ν−

−=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

r,2

3,1

1

51

2

1W

W)r(4g

)r(3g

r,2

3,1

1

51

2

1M

W)r(2g

)r(1

g

m

m

(25)

It is assumed that the vessel is subjected to internalhydrostatic pressure iP and external hydrostatic

pressure oP , simultaneously. Thus, the boundary

conditions will be presented by relation (15).Satisfying the mentioned boundary conditions resultin the following relation for unknown coefficients as

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

Γ−Γ

⎟ ⎠

⎞⎜⎝

⎛ +Γ

ν−

ν−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

Γ−Γ

⎟ ⎠ ⎞⎜

⎝ ⎛ +Γν+

ν−

=

)iR(1g)oR(3g)oR(1g)iR(3goP)iR(1g)2iRexp(2

oRiP)oR(1g)2oRexp(2iR

)oRiR(2

expc)1(2

)21(E2C

)iR(1g)oR(3g)oR(1g)iR(3goP)iR(3g)2iRexp(2

oRiP)oR(3g)2oRexp(2iR

)oRiR(2

expc)1(2

)21(

E1C

(26)

Substituting E1C and E2C into relations (22-24) will

result in relations for displacement and radial andcircumferential stress distribution.

3. VERIFY ING THE SOLUTION

In order to verify the accuracy of the presentsolutions, results of two specific problems are

compared with known results in literature. The firstproblem is stress analysis of a homogeneous thick-walled pressure vessel (e.g. 1oi →μμ ). Distribution

of radial and circumferential stresses for ahomogeneous thick-walled spherical pressure vesselsare as follows [32]

iP]13)iRoR[(2

23)r/o

R(;iP

13)iR/oR(

13)r/o

R(

r−

+=ϕσ=θσ

−

−−=σ

(27)

In Figure 1, a comparison between stress distributionof a spherical homogeneous vessel based onreference [32] and both linear and exponentialmodel of present analysis is shown.

In second problem, a comparison has beencarried out with the results of reference [28]. Basedon the relation (4), the ratio of the inner to outerelasticity modulus are as follow

oBRA

iBRA

)o

R(E

)iR(E

+

+= (28)

However the variation of the mechanical propertiesin reference [28] was assumed as

β

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ =o

RiR

)o

R(E)iR(E (29)

It can be seen that as the constants A of relation(29) and β of relation (30) are equal to zero and

unity respectively, the present model and themodel of reference (28) becomes the same. InFigure 2 the comparison is presented for thisspecial case. The superscript H denotes thehomogeneous state.

As it is shown in Figures 1 and 2, the present

results have a good agreement with those inliterature.

4. PARAMETRIC STUDY AND DISCUSSION

To present numerical results, some examples havebeen considered. In all of them, the outer to innerradius ratio ( io RR ) of 2 is used. All figures are

depicted for five different inner to outer shearmodulus (

oi

μμ ) as 1/5, 1/2, 1, 2 and 5. In these

figures, the solid line corresponds to the isotropichomogeneous material.

As it is mentioned, in the first case, it isassumed that the variation of properties forfunctionally graded material is linear and in thesecond case, it is supposed that variation of properties is as an exponential function.

In Figures 3a,b the variation of nondimensionalshear modulus across the radial direction based onlinear function and exponential models are depicted,

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respectively. It can be seen that for different innerand outer shear modulus ratio, the variation of shear modulus stay linear.

4.1. Example 1.

4.1.1. A FG spherical Vessel subjected to internalpressure Pi In Figures 4a,b the variation of

nondimensional radial stress through the radialdirection of a thick-walled spherical vessel basedon linear function and exponential model areshown, respectively. Focusing on these figures, itcan be seen that by increasing the inner to outershear modulus ratio, the radial stress distributiontends to be more nonlinear. However, decreasingthis ratio from unity, the state of nonlinearity

r / Ro

σ

/ P

i

0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1

Present

Reference [32]

σr

σθ

Figure 1. A comparison of nondimensional normal stressesfor a homogeneous thick-walled spherical vessel subjected to

internal pressure iP .

r

σ r

/ σ r H

0.6 0.7 0.8 0.9

1

1.1

1.2

1.3

1.4

Present

Reference [28]

Figure 2. A comparison of nondimensional radial stress for aspecific case of FG thick-walled spherical vessel subjected tointernal pressure.

r / Ro

μ

/ μ o

0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

μi

/μo

=5

μi

/μo

=2

μi

/μo

=1

μi

/μo

=1/2

μi

/μo

=1/5

(a)

r / Ro

μ

/ μ o

0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

μi

/μo

=5

μi

/μo

=2

μi

/μo

=1

μi

/μo

=1/2

μi

/μo

=1/5

(b)

Figure 3. (a) Nondimensional shear modulus across the radialdirection for linear variation model and (b) Nondimensionalshear modulus across the radial direction for exponentialmodel.

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decreases with respect to homogeneous one. Inother words, when the stiffness of the outer surfacegets five times higher than inner one, the radialstress distribution nearly varies linear.

In Figures 5a,b the variation of nondimensional

circumferential stress through the radial directionbased on linear and exponential function modelsare shown, respectively. As it can be found inFigures 5a,b by increasing the inner to outerstiffness ratio, the value of circumferential stress at

r / Ro

σ r

/ P

i

0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

-0.4

-0.2

0

μi

/μo

=5

μi

/μo

=2

μi

/μo

=1

μi

/μo

=1/2

μi

/μo

=1/5

(a)

r / Ro

σ r

/ P

i

0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

-0.4

-0.2

0

μi

/μo

=5

μi /μo=2μ

i/μ

o=1

μi

/μo

=1/2

μi

/μo

=1/5

(b)

Figure 4. (a) Nondimensional radial stress across the radialdirection for a thick-walled spherical vessel subjected to

internal pressure iP based on linear variation model and (b)

Nondimensional radial stress across the radial direction for a

thick-walled spherical vessel subjected to internal pressure iPbased on exponential model.

r / Ro

σ θ

/ P

i

0.5 0.6 0.7 0.8 0.9 10

0.3

0.6

0.9

1.2

1.5

μi

/μo=5

μi

/μo=2

μi

/μo=1

μi

/μo=1/2

μi /μo=1/5

(a)

r / Ro

σ θ

/ P

i

0.5 0.6 0.7 0.8 0.9 10

0.3

0.6

0.9

1.2

1.5

μi

/μo=5

μi

/μo=2

μi

/μo=1

μi

/μo=1/2

μi

/μo=1/5

(b)

Figure 5. (a) Nondimensional circumferential stress across theradial direction for a thick-walled spherical vessel subjected to


Nondimensional circumferential stress across the radial

direction for a thick-walled spherical vessel subjected tointernal pressure iP based on exponential model.

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inner surface increases. In this case, the value of circumferential stress decreases as the points closeto outer surface. Another important point that canbe found from these figures is that the differencebetween circumferential stress of surfacesdecreases with decreasing of stiffness of inner toouter surface. For inner to outer shear modulusratio of 1/2, the circumferential stress distributionacross the thickness of the vessel is approximatelyuniform. Whenever the stiffness of inner to outersurface is lower than this value, the value of circumferential stress in the inner surface is lowerthan the value at outer surface.

The variation of nondimensional displacementthrough the radial direction based on linear andexponential function models are shown in Figures

6a,b respectively. As it can be seen in thesefigures, by increasing the stiffness from inner toouter surfaces, the value of radial displacementdecreases and its nonlinearity increases in radialdirection.

4.2. Example 2.

4.2.1. A FG spherical vessel subjected to bothinternal pressure Pi and external pressure Po In order to consider the effects of applying bothinternal and external pressures on the surfaces of thick walled spherical vessel, in this example theresults of a FG spherical vessel subjected to bothinternal and external pressures are presented. It isassumed that the intensity of the internal andexternal pressure is the same. In Figures 7a,b thevariation of the radial stress in radial direction areshown for the linear and exponential FGM modelrespectively. It can be seen that for this loadingconditions, the radial stress in radial direction isuniform and equal to the external pressure whenthe vessel is made of homogeneous material. For

FG vessels, although internal and external pressureis the same, the variation of the radial stress inradial direction is not uniform. Also, it can be saidthat as the inner shear modulus is higher than outerone, the compression stress is more significant thanthe surface pressure. On the other hand, themaximum of compression stress located in a pointother than the surfaces. However, as the outershear modulus is higher than inner one, thecompression stress in the wall of the vessel islower than the surface pressure. In this case, there

is a point in the vessel that the stress is minimum.In Figures 8a,b the variation of the circumferential

stress of spherical vessel in radial direction aredepicted. It can be seen that only for the homogenousmaterial, the distribution of the circumferential

r / Ro

( u r μ o

) / ( R

o P

i )

0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

μi

/μo=5

μi

/μo=2

μi

/μo=1

μi

/μo=1/2

μi /μo=1/5

(a)

r / Ro

( u r μ o

) / ( R

o

P i )

0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

μi

/μo=5

μi

/μo=2

μi

/μo=1

μi

/μo=1/2

μi

/μo=1/5

(b)

Figure 6. (a) Nondimensional radial displacement across theradial direction for a thick-walled spherical vessel subjected to


Nondimensional radial displacement across the radial

direction for a thick-walled spherical vessel subjected tointernal pressure iP based on exponential model.

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stress is uniform. Also, for FG spherical vesselswith this kind of surface conditions, the variationof the circumferential stress in radial direction isnearly linear. This state of variation is less in

exponential variation of the FGM. It can be seenthat for different values of shear ratio, thecircumferential stress in the middle of the wall isnearly equal to the surface pressure.

r / Ro

σ r

/ P

i

0.5 0.6 0.7 0.8 0.9 1-1.2

-1.1

-1

-0.9

-0.8

-0.7

μi

/μo=5

μi

/μo=2

μi

/μo=1

μi

/μo=1/2

μi

/μo=1/5

(a)

r / Ri

σ r

/ P

i

0.5 0.6 0.7 0.8 0.9 1-1.2

-1.1

-1

-0.9

-0.8

-0.7

μi

/μo=5

μi

/μo=2

μi

/μo=1

μi

/μo=1/2

μi

/μo=1/5

(b)


internal and external pressure iP based on linear variation

model and (b) Nondimensional radial stress across the radial

direction for a thick-walled spherical vessel subjected tointernal and external pressure iP based on exponential model.

r / Ro

σ θ

/ P

i

0.5 0.6 0.7 0.8 0.9 1-1.8

-1.5

-1.2

-0.9

-0.6

-0.3

μi

/μo=5

μi

/μo=2

μi

/μo=1

μi

/μo=1/2

μi

/μo=1/5

(a)

r / Ro

σ θ

/ P

i

0.5 0.6 0.7 0.8 0.9 1-1.8

-1.5

-1.2

-0.9

-0.6

-0.3

μi

/μo=5

μi /μo=2μ

i/μ

o=1

μi

/μo=1/2

μi

/μo=1/5

(b)

Figure 8. (a) Nondimensional circumferential stress across theradial direction for a thick-walled spherical vessel subjected to

internal and external pressure iP based on linear variation

model and (b) Nondimensional circumferential stress across the

radial direction for a thick-walled spherical vessel subjected tointernal and external pressure iP based on exponential model.

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4.3. Example 3.

4.3.1. A FG spherical vessel subjected to externalpressure Po In this example, the results of a FGspherical vessel subjected to external pressure on

the outer surface is presented.In Figures 9a,b the variation of the radial stress

in radial direction are depicted for various shearmodulus ratios. It can be seen that as the inner toouter shear modulus ratio approaches 5, thevariation of the radial stress becomes linear and forlower shear ratios, the variation is nonlinear.Another important issue that can be concluded iswhen the inner to outer shear modulus ratio is lowerthan 1/5, the value of the radial stress in some pointsof the vessel is higher than external pressure. This

value is the maximum of the radial stress.In Figures 10a,b the variation of the circumferentialstress is depicted for both linear and exponentialmodels, respectively. In both models, the variationof the circumferential stress is uniform for inner toouter shear ratio close to 2. For higher shear ratiosthe value of the circumferential stress in innersurface is lower than that of the outer surface andvice versa.

5. CONCL USIONIn a functionally graded material, the property of material varies continuously from point to point.By this characteristic, appropriate mechanicalproperties are achievable. Employing such amaterial is commonly used in building the thick-walled pressure vessels.

In this paper exact elasticity solutions have beenpresented to study the static analysis of thick-walled spherical vessels subjected to internal andexternal hydrostatic pressure. The closed-form

solutions have been obtained for displacement andstress fields based on two common FGM models.With regard to use of different compound of materials in FG vessels, at first, the variationfunction of FGM has been assumed to be linearwith respect to thickness direction. By thisproposed function, inner to outer stiffness ratiobecomes independent of the variation function of properties. After that, it has been supposed thatvariation function of FGM obeys exponentialmodel. Some important results of this analysis

have been obtained as follow:

5.1. Only Internal Pressure Applies

• When the stiffness of the outer surface gets

r / Ro

σ r

/ P

o

0.5 0.6 0.7 0.8 0.9 1

-1

-0.8

-0.6

-0.4

-0.2

0

μi

/μo

=5

μi

/μo

=2

μi

/μo

=1

μi

/μo

=1/2

μi

/μo

=1/5

(a)

r / Ro

σ r

/ P

o

0.5 0.6 0.7 0.8 0.9 1

-1

-0.8

-0.6

-0.4

-0.2

0

μi

/μo

=5

μi

/μo

=2

μi

/μo

=1

μi

/μo

=1/2

μi

/μo

=1/5

(b)


external pressure oP based on linear variation model and (b)

Nondimensional radial stress across the radial direction for a

thick-walled spherical vessel subjected to external pressureoP based on exponential model.

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7/28/2019 856200904 a 05



five times higher than inner one, the radialstress distribution nearly varies linear and byincreasing the inner to outer shear modulusratio, the radial stress distribution tends to bemore nonlinear.

• For inner to outer shear modulus ratio of 1/2,the circumferential stress distribution acrossthe thickness of the vessel is approximatelyuniform and by increasing the inner to outer

stiffness ratio, the value of circumferentialstress at inner surface increases.

5.2. The Same Internal and External PressureApplies Simultaneously

• As the shear modulus of the inner surface ishigher than that of the outer surface, thevalue of the compression stress in thevessel is higher than the surface pressure.Conversely, the value of the compressionstress in the vessel is lower than the surface

pressure when the inner shear modulus islower than the outer one.

• The variation of the circumferential stress inradial direction is nearly linear and forarbitrary shear modulus ratios, the value of this stress at the middle of the wall of thevessel becomes equal to the surface pressure.

5.3. Only External Pressure Applies

• For the inner to outer shear ratios of 5, thevariation of the radial stress is linear and

with decreasing this ratio the distributionbecomes nonlinear.

• When the inner to outer shear modulus isless than 1/5, in some point of the wall of thevessel, the value of the radial stress is higherthan external pressure.

• For inner to outer shear ratios close to 2, thevariation of the circumferential stress isuniform.

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r / Ro

σ θ

/ P

o

0.5 0.6 0.7 0.8 0.9 1-3.2

-2.8

-2.4

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-1.6

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