Creep Stress Redistribution Analysis of Thick-Walled FGM Spheres

© 2010 IAU, Arak Branch. All rights reserved.

Journal of Solid Mechanics Vol. 2, No. 2 (2010) pp. 115-128

Creep Stress Redistribution Analysis of Thick-Walled FGM Spheres

S.M.A. Aleayoub1, A. Loghman2,* 1Department of Mechanical Engineering, Arak Branch, Islamic Azad University, Arak, Iran 2Department of Mechanical Engineering, Faculty of Engineering, University of Kashan, Kashan, Iran

Received 17 May 2010; accepted 18 July 2010

ABSTRACT Time-dependent creep stress redistribution analysis of thick-walled FGM spheres subjected to an internal pressure and a uniform temperature field is investigated. The material creep and mechanical properties through the radial graded direction are assumed to obey the simple power-law variation throughout the thickness. Total strains are assumed to be the sum of elastic, thermal and creep strains. Creep strains are time temperature and stress dependent. Using equations of equilibrium, compatibility and stress-strain relations a differential equation, containing creep strains, for radial stress is obtained. Ignoring creep strains in this differential equation, a closed form solution for initial thermo-elastic stresses at zero time is presented. Initial thermo-elastic stresses are illustrated for different material properties. Using Prandtl-Reuss relation in conjunction with the above differential equation and the Norton’s law for the material uni-axial creep constitutive model, radial and tangential creep stress rates are obtained. These creep stress rates are containing integrals of effective stress and are evaluated numerically. Creep stress rates are plotted against dimensionless radius for different material properties. Using creep stress rates, stress redistributions are calculated iteratively using thermo-elastic stresses as initial values for stress redistributions. It has been found that radial stress redistributions are not significant for different material properties. However, major redistributions occur for tangential and effective stresses.

© 2010 IAU, Arak Branch. All rights reserved.

Keywords: FGM Spheres; Thermo-elastic Creep; Stress Redistribution

1 INTRODUCTION

UNCTIONALLY graded materials are used in modern technologies for structural components such as those used in nuclear, aircraft, space engineering and pressure vessels. Elastic, plastic and creep stress analysis of

these components have attracted wide attention due to combining different material properties and loading conditions. Time-dependent creep stress and damage analysis of thick-walled spherical pressure vessels of constant material properties have been investigated by Loghman and Shokouhi [1]. They studied the creep stress and damage histories of thick-walled spheres using the material constant creep and creep rupture properties defined by the theta projection concept [2]. Loading conditions included an internal pressure and a thermal gradient. Although intensive investigation considering creep of thick-walled spheres and cylinders with constant material properties can be found in the existing references- Loghman and Wahab [3], Sim and Penny [4]- little publication can be found dealing with time-dependent creep of FGM spheres and cylinders. Yang [5] presented a solution for time-dependent creep behavior of FGM cylinders. Following Norton’s law for material creep behavior and using equations of equilibrium, strain displacement and stress strain relations and considering Prandtl-Reuss relations for creep strain rate-stress equation, he obtained a differential equation for the displacement rate. There was no exact solution of the equation.

‒‒‒‒‒‒ * Corresponding author. Tel.: +98 361 591 2425; fax: +98 361 555 9930. E-mail address: [email protected] (A. Loghman).

F

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© 2010 IAU, Arak Branch

With some simplifications and using Taylor expansion series, however, he obtained the displacement rate and then the stress rates were calculated. When the stress rates were known, the stresses at any time were calculated iteratively. Creep deformation and stresses in thick-walled cylindrical vessels of FGM subjected to internal pressure were presented by You et.al. [6]. They obtained a closed form solution for steady state creep stresses in FGM cylinders. Thermal stresses were not considered and stress redistributions were not presented. Time-dependent deformation and fracture of multi-material systems at high temperature were presented by Xuan et. al. [7]. They considered a thick-walled sphere of FGM material subjected to an internal pressure. Using Norton’s law for material creep behavior and using equations of equilibrium, compatibility, stress-strain relations and considering the Prandtl-Reuss relations, obtained a differential equation for the radial stress rate. Radial and circumferential stress distributions with different material creep properties after two hundred hours were illustrated. In this work, however, thermal stresses are not included and stress redistributions are not considered.

The main objective of this paper is to present a time-dependent creep stress redistribution analysis of FGM spheres.

2 THEORETICAL ANALYSIS 2.1 Basic formulation for thermo elastic creep stress analysis of FGM spheres

Consider a sphere with an inner radius ir and outer radius 0r subjected to an internal pressure p and a uniform

temperature field T. The total strains are the sum of elastic, thermal and creep strains

11 122 cr r r rc c T= + + + (1a)

12 11 12( ) crc c c T= + + + + (1b)

where 11 121 / , /c E c E= =- and , r are elastic and thermal expansion radial dependent coefficients

respectively. In this study, is considered to be a constant, , rE and are assumed to obey the power-law variation as

0E E r= and

0 .r r= =

The equilibrium and compatibility equations in spherical symmetry are written as

2( )d0

drr

r r

-+ = (2a)

d0

dr

r r

-+ = (2b)

Substituting the radial and circumferential strains from Eqs. (1a) and (1b) into the compatibility Eq. (2b) and

using equilibrium Eq. (2a) the following differential equation for the radial stress is obtained

( ) 122 0 0 0 0

2

2 2 d 2d d 2 (1 2 )(4 ) ( )

d (1 ) 1 1 d 1d

cc cr r

r r

E r E r E rr r T

r rr

+ +-+ - - =- - - -

- - - - (3)

where creep strains, c

r and c on the right hand side of the above differential equation are time, temperature and

stress dependent.

2.2 Thermoelastic stress analysis of FGM spheres

Ignoring the creep strain terms on the right hand side of the above differential equation gives the following differential equation for thermoelastic stress analysis

22

2

d d(4 ) 2

ddr r

rr r Jr Trr

+¢+ - - = (4)

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Creep Stress Redistribution Analysis of Thick-Walled FGM Spheres 117


where

1 2

1

-¢ =-

,

0 02

1

EJ =-

- (5)

It is obvious that the homogeneous solution to Eq. (4) can be obtained by assuming

r Cr= (6)

Substituting Eq. (6) into Eq. (4) one can obtain the following characteristic equation

2 (3 ) 2 0¢+ - - = (7)

The roots of Eq. (7) are

2

1

3 1(3 ) 8

2 2

- ¢= + - + (8a)

22

3 1(3 ) 8

2 2

- ¢= - - + (8b)

Since 0.5 0> > the discriminant of Eq. (7) is always greater than zero. Therefore, 1 and 2 are real and

distinct. The homogeneous solution of Eq. (4) is therefore

1 21 2rh C r C r= + (9)

The particular solution of the differential Eq. (4) may be obtained as

1 2

1 2rp u r u r= + (10)

where

2

1 21

( ) d

( , )

r R ru r

W r r

-= ò (11)

1

1 21

( ) d

( , )

r R ru r

W r r= ò (12)

in which ( )R r Jr T+= is the expression on the right hand side of Eq.(4) and W is defined as

1 2

1 2 2 1

1

12 11 2 1

1 2

( , ) ( )r r

W r r rr r

+ -- -

= = - (13)

Therefore, 1u and 2u may be obtained by the following integration

2

1

2 1

0 0

20 01 1

2 1 12 1

221 d

(1 )( )( 2)( )

E Tr r E T

u r rr

+

+ - ++ -

-= =- - + - +-ò (14)

1

2

2 1

0 0

20 02 1

2 1 22 1

221 d

(1 )( )( 2)( )

E Tr r E T

u r rr

+

+ - ++ -

-=- =-- - + - +-ò (15)

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Substituting Eqs. (14) and (15) into Eq. (10) one can obtain the particular solution

20 0

1 2

2

(1 )( 2)( 2)rp

E Tr + +=-

- + - + + - + (16)

The complete solution of Eq. (4) can be written as

1 2 2

1 2r C r C r Gr + += + - (17)

where

0 0

1 2

2

(1 )( 2)( 2)

E TG =

- + - + + - + (18)

Using Eq. (2a) the expression for the tangential stress may be written as

1 2 21 21 2

2(1 ) (1 ) (1 )

2 2 2C r C r Gr + ++ +

= + + + - + (19)

The constants 1C and 2C are determined from the boundary conditions

at i rr r p= =-

at 0 0rr r = = (20)

Substituting the above boundary condition into Eq. (17) the constants 1C and 2C are obtained

2 2 2

2 1 1 2

2 20 0 0

1

0 0

( )i i

i i

G r r r r prC

r r r r

+ + + +- +=

- (21a)

1 1 1

2 1 1 2

2 20 0 0

2

0 0

( )i i

i i

G r r r r prC

r r r r

+ + + +- -=

- (21b)

Substituting the constants 1C and 2C from Eqs. (21a) and (21b) into Eqs. (17) and (19) and using Eq. (18) the

expressions for radial and circumferential thermoelastic stresses are obtained

2 2

1

2 1 1 2

1 1 2 1 1 2

2

2 1 1 2 2 1 1 2

2 20 0 0 0

1 2 0 0

2 220 0 0 0

0 0 0 0

2 ( ){[ ]

(1 )( 2)( 2)

( ) ( )[ ] }

i ir

i i

i i

i i i i

E T r r r rr

r r r r

r r r r p r r r rr r

r r r r r r r r

+ + + +

+ + + ++ +

-=

- + - + + - + -

- -+ - +

- -

(22a) 2 2

1

2 1 1 2

1 1

2

2 1 1 2

2 1 1 2

2 20 0 0 01

1 2 0 0

2 220 02

0 0

1 20 0

2 ( ){(1 )[ ]

(1 )( 2)( 2) 2

( ) 2(1 )[ ] (1 ) }

2 2

[(1 ) (1 ) )]2 2

i i

i i

i i

i i

i

E T r r r rr

r r r r

r r r rr r

r r r r

p r r r r

r

+ + + +

+ + + ++ +

-= +

- + - + + - + -

- + ++ + - +

-

+ - ++

2 1 1 20 0ir r r -

(22b)

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The above thermoelastic stresses are the initial elastic stress distribution which will be changed with time as the creep process progresses. These thermoelastic stresses and the effective stresses are shown in Figs. 1-3 for different material properties.

2.3 Time-dependent creep stress analysis of FGM spheres

Considering the temperature field to be steady, the differential equation (3) containing creep strains may be rewritten in terms of creep strain rates as follows

122 0 0

2

2 d 2d d 2 (1 2 )(4 ) ( )

d (1 ) 1 d 1d

cc cr r

r r

E r E rr r

r rr

+-+ - - =- - -

- - -

(23)

The creep strain rates are related to the current stresses and the material uniaxial creep constitutive model by the

well-known Prandtl-Reuss equation

[ ]c er r

e

= -

(24a)

[ ]2

c er

e

= -

(24b)

in which the material uniaxial creep constitutive model is defined by the Norton’s equation

( )( ) n re eB r = (25)

where 1

0( ) bB r b r= and ( )n r is considered to be a constant 0( )n r n= in this study. Substituting from Eq. (25) into

Eqs. (24a) and (24b) one can obtain

1( ) [ ]c nr e rB r -= - (26a)

1( )[ ]

2

nc e

r

B r

-

= - (26b)

The Von-Mises and Tresca’s equivalent stress are the same in spherical symmetry

e r = - (27)

In the absence of temperature field, tangential stress is always greater than radial stress. For many applicable

loading combinations, however, tangential stress is greater than radial stress. Therefore, Eqs. (26a) and (26b) may be written as follows

( )c nr eB r =- (28a)

( )

2

nc eB r

= (28b)

Substituting from Eqs. (28a) and (28b) into differential Eq. (23) the following differential equation is obtained

12

2 02

d d 2 (1 2 ) d 3(4 ) {[ ]( ( ) )}

d (1 ) 1 ddnr r

r e

E rr r B r

r r rr

+-+ - - =- +

- -

(29)

The homogeneous solution of this differential equation is therefore

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1 21 2rh D r D r = + (30)

where 1 and 2 are defined in Eqs. (8a) and (8b). The particular solution can be obtained similar to thermo-elastic

solution as

2

2 11 1

2 1

3{[ ]( ( ) )}

1 d( )

ne

rE dr B r

dr ru rr

+ -

+-=

-ò (31)

1

2 12 1

2 1

3{[ ]( ( ) )}

1 d( )

ne

rE dr B r

dr ru rr

+ -

+-=-

-ò (32)

Eqs. (31) and (32) may be written

2 22 101

2 1

[ ( ) d 3 ( ) d ](1 )( )

n ne e

E d du r B r r B r

dr dr

+ - + -= +

- - ò ò (33)

2 22 102

2 1

[ ( ) d 3 ( ) d ](1 )( )

n ne e

E d du r B r r B r

dr dr

+ - + -=- +

- - ò ò (34)

Selecting 22u r + -= and d

d ( ) dd

nev B r

r= and using ( d . d )u v u v v u= -ò ò

integration by part one can

obtain

2 22 101 1

2 1

[ ( ) (1 ) ( ) d ](1 )( )

n ne e

Eu r B r B r

+ - + -= + - +

- - ò (35)

2 22 102 2

2 1

[ ( ) (1 ) ( ) d ](1 )( )

n ne e

Eu r B r B r

+ - + -=- + - +

- - ò (36)

Then, the particular solution may be written as

1 2

1 2rp u r u r = + (37)

The general solution for the radial stress rate may be written

1 2 1 2

1 2 1 2r D r D r u r u r = + + + (38)

Substituting from Eqs. (35) and (36) into Eq. (38) one can get the radial stress rate and then the tangential stress

rate as

1 2 1 1

2 2

101 2 1

2 1

12

[(1 ) ( ) d(1 )( )

(1 ) ( ) d ]

nr e

ne

ED r D r r r B r

r r B r

+ -

+ -

= + + - +- -

- - +

ò

ò

(39)

1 1

2 2

1011 1

2 1

1 20 022 2

2 1

( 1) [ (1 ) ( ) d ]2 (1 )( )

( 1) [ [(1 ) ( ) d ]2 (1 )( ) 2(1 )

ne

n ne e

Er D r B r

E Er D r B r r B

+ -

+ - +

= + + - +- -

+ + - - + -- - -

ò

ò

(40)

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Since inside and outside pressures do not change with time, the boundary conditions for stress rates at the inner and outer surfaces may be written

at 0i rr r = = (41a) at 0 0rr r = = (41b) Using the above boundary conditions, the constants 1D and 2D are obtained

2 1 1

2 1

1 2 1 2 2

0 0

10 01 1

02 1

1 1 12 0

{(1 )[( ) ( ( ) d )(1 )( )(1 ( ) )

( ) d ) ] (1 ) [( ( ) d ) ( ) d ) ]}

i

i

ne r r

i

i

n n ne r r e r r e r r

E rD r B r

r rr

r B r r r B r r B r

- + -=

-

+ - - + - + -= = =

= - +- - -

- - - + -

ò

ò ò ò

(42)

1 2 1

2 1

1 2 2 1 2

0 0

102 1

02 1

1 1 102

{(1 ) [( ( ) d )(1 )( )(1 ( ) )

( ) d ) ] (1 )[( ( ) d ) ( ) ( ( ) d ) ]}

i

i

ni e r r

i

n n ne r r e r r e r r

i

ED r r B r

r

r

rr B r r B r r B r

r

- + -=

-

+ - + - - + -= = =

=- - +- - -

- - - + -

ò

ò ò ò

(43)

Substituting from Eqs. (42) and (43) into Eqs. (39) and (40) the stress rates can be written as follows

2 1 1

2 1

1 2 1 2

0

2 1

0

2 1

10 01

02 1

1 12 0

1

0

02 1

{(1 )[( ) ( ( ) d )(1 )( )(1 ( ) )

( ) d ) ] (1 ) [( ( ) d )

( ) d ) ]}

{(1 )( )(1 ( ) )

i

i

nr e r r

i

i

n ne r r e r r

ne r r

i

E rr B r

r rr

r B r r r B r

r B r r

Er

r

- + -=

-

+ - - + -= =

+ -=

-

= - +- - -

- - - +

-

-- - -

ò

ò òò

1 2 1 1

0

2 2 1 2 2

0

1 1 2 2

1 11

1 102

1 101 2

2 1

(1 ) [( ( ) d ) ( ) d )

(1 )[( ( ) d ) ( ) ( ( ) d ) ]}

[(1 ) ( ) d (1 ) ((1 )( )

i

i

n ni e r r e r r

n ne r r e r r

i

ne

r r B r r B r

rr B r r B r r

r

Er r B r r r

- + - + -= =

+ - - + -= =

+ - + -

- + -

- - + -

+ - + - - +- -

ò ò

ò ò

ò ) d ]neB rò

(44)

1 2 1 1

2 1

1 2 1 2

0

2

0

10 011

02 1

1 12 0

1 101

2 1

( 1) [ {(1 )[( ) ( ( ) )2 (1 )( )(1 ( ) )

( ) ) ] (1 ) [( ( ) )

( ) ) ]} (1 )(1 )( )

i

i

ne r r

i

i

n ne r r e r r

ne r r

E rr r B dr

r rr

r B dr r r B dr

Er B dr r

- + -=

-

+ - - + -= =

+ - +=

= + - +- - -

- - - +

- + - +- -

ò

ò ò

ò

1

2 1 2 1

2 1

1 2

0

1021

02 1

1 12

( ) ]

( 1) [ {(1 ) [( ( ) )2 (1 )( )(1 ( ) )

( ) ) ] (1 )[( ( ) )

i

i

ne

ni e r r

i

n ne r r e r r

B dr

Er r r B dr

r

r

r B dr r B dr

-

- + -=

-

+ - + -= =

+ + - - +- - -

- - - +

ò

ò

ò ò

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2 1 2 2

0

1 10 02

2 1

20

( ) ( ( ) ) ]} [(1 ) ( ) ](1 )( )

2(1 )

n ne r r e

i

ne

r Er B dr r B dr

r

Er B

- + - + -=

+

- - - +- -

--

ò ò

(45)

Radial and tangential stress rates are obtained from Eqs. (43) and (44) and plotted against dimensionless radius for different material properties in Figs. 4 and 5. When the stress rates are known, time-dependent creep stress redistributions at any time it

are obtained iteratively

( ) ( 1) ( ) ( )

1( , ) ( , ) ( , ) di i i ir i r i r ir t r t r t t -

-= + (46a) ( ) ( 1) ( ) ( )

1( , ) ( , ) ( , ) di i i ii i ir t r t r t t -

-= + (46b) where

( )

0

di

ki

k

t t=

=å

(47)

To calculate ( ) ( , )i

r ir t and ( ) ( , )iir t ,

the stresses at the time 1it - are used. The solution at zero time 0it =

corresponds to thermo-elastic stresses.

3 RESULTS AND DISCUSSION

The results presented here in this study are based on the following data for geometry, material properties, and loading conditions in a thick-walled FGM sphere

o

30 6 o1 0/ 1.2, 22000 MPa, 1.1 10 , 5, 8, 0.3, 1.2 10 1 / C,

2,1, 1, 2, 200 C, 0, 100 MPa

o i o o o

o i

r r E b b n

T P P

- -= = = ´ =- = = = ´

= = - - = = =

Initial thermoelastic radial stresses at zero time are shown in Fig. 1. The boundary conditions for radial stresses

at the inner and outer surfaces of the sphere are satisfied for all material properties. There are not significant differences among radial stresses for all material properties. Tangential thermo-elastic stresses at zero time are shown in Fig. 2 for different material properties. They are tensile throughout thickness. For 2 = =- and

1, = =- however, they are highly tensile at the inner surface of the vessel where the radial stresses are highly

compressive. It means the maximum shear stress which is max ( ) / 2r = - will be very high at the inner surface

of the vessel and can cause yielding to occur. Also highly tensile tangential stresses at the inner surface of the sphere will also decrease the fatigue life of the vessel. Therefore in selection of material for such a vessel, the material identified by 2 = = is the best because of the best shear stress distribution throughout thickness. This can be

well understood from Fig. 3 which shows the effective stress distribution throughout thickness for all four cases of material properties. Effective stresses are, indeed, twice as the maximum shear stresses max( 2 )e = . Therefore, it

is clear from Fig. 3, that the material with 2 = = has a uniform and best shear stress distribution throughout

thickness. In Fig. 3, a reference stress has also been identified in which the effective stresses are identical for all material properties.

Radial and tangential stress rates are plotted against dimensionless radius for different material properties in Figs. 4 and 5. Fig. 4 shows that the radial stress rates are zero at the inner and outer surfaces of the sphere which satisfy the boundary conditions. The maximum rate of change of radial stresses belong to 2 = =- and then

1, = =- 1 = = and 2, = = respectively. Therefore, minimum changes in radial stresses with time

will take place for the material identified by 2 = = as will be explained next in radial stress redistributions.

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Creep Stress Redistribution A

Analysis of Thick

Fig. 1 Initial thermosphere for and = =



Fig. 4 Radial stress the FGM sph

1 = = an

k-Walled FGM S

oelastic radial st2, = =- =

2= at zero time.

oelastic tangenti2, = =- =

2= at zero time.

oelastic effective2, = =- =

2= at zero time.

rates versus dimhere for = =nd 2. = =

Spheres

tresses in the FG1,= =- =

ial stresses in the1,= =- =

e stresses in the 1,= =- =

mensionless radiu2,=- = =

123

GM 1 =

e FGM 1 =

FGM 1 =

us in 1,-

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124

Fig. 5 sinside of ththe tangenremaining will take p

discussed identified b

stress redi

discussion to its final surfaces of

S.M.A. Aleay

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Fig. 7 Time-dependredistributionelastic at zerofor the case

©

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2. = =

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Creep Stress Redistribution A

Analysis of Thick



1. = =



k-Walled FGM S

dent effective crn in the FGM spho time to third se

2. = =

dent radial creepsphere from initicted time interval

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1. = =

dent effective cren in the FGM spho time to third se

1. = =

Spheres

reep stress here from initialelected time inte



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126

S.M.A. Aleayyoub and A. Logghman


1 = =-




2 = =-

©

dent radial creepsphere from initicted time interval.

dent tangential crn in the FGM spho time to third se

1. = =-

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1. = =-

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© 2010 IAU, Arak



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Arak Branch

ntial stress redi

urface of the v shear stress at ng with time. tress is twice ribution becomstributions for

as for the case ions for the ma

stributions sho changes will o, 17 for both mtress is not cha

CLUSIONS

endent creep sroperties. Radion occurs for

hear stress distrith time during of material pro

istribution for

vessel are decre the inner surfa This can alsoas the maximu

mes more unifo the case =

2 = = but

aterials =ow little changoccur for tangematerial properanging with tim

stress redistribdial stress redi tangential and

ribution will o creep process

operties in whi

Creep Stress

this case =easing with timace of the vesso be explainedum shear stresorm through t

1 = are sho

t with a higher

1 =- and ges with time ential and effecrties. A referen

me during creep

bution analysisistribution is nd effective stre

ccur throughou belong to this ch there are no

Redistribution A

2 = is show

me while at thesel is decreasind by effectivess at each pointhickness of thwn in Figs. 9-

rate of change

2= =- are

during creep ctive stresses wnce effective sp process.

s of thick-walnot significantesses. The best

ut the thicknes material. A re

o changes in eff

Analysis of Thick



wn in Fig. 7. It

e outer surface ng with time ane stress redistrnt throughout the vessel durin-11. Stress redi

e with time. Ra

shown in Figs

process for bwith time durinstress can be id

lled FGM spht almost for at material is id

ss of the FGM eference effectiffective stress w

k-Walled FGM S

dent tangential cn in the FGM spho time to third se

2. = =-

dent effective crn in the FGM spho time to third se

2. = =-

t is clear that t

of the sphere and at the outer ributions showthickness, it isng creep proceistributions tak

adial, tangentia

s. 12-17. In Fi

oth material png creep procedentified for al

heres is investall material prodentified by sphere. Also mive stress has bwith time at thi

Spheres

creep stress here from initialelected time inte


tangential stres

are increasing. surface of the

wn in Fig. 8. As clear that theess. Time-depeke place in the

al and effective

gs. 12 and 15,

properties. Howess as shown inll cases in whi

tigated for difoperties. But,

2= = in w

minimum chanbeen identified is point.

127

l erval

l erval

sses at

. Thus, vessel As the e shear endent e same

e stress

radial

wever, n Figs. ich the

fferent major

which a

nges in for all

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REFERENCES

[1] loghman A., shokouhi N., 2009, Creep damage evaluation of thick-walled spheres using a long-term creep constitutive model, Journal of Mechanical Science and Technology 23: 2577-2582.

[2] Evans R.W., Parker J.D., Wilsher B., 1992, The theta projection concept a model based approach to design and life extention of engineering plant, International Journal of Pressure Vessels and Piping 50: 60-147.

[3] Loghman A., Wahab M.A., 1996, Creep damage simulation of thick-walled tubes using the theta projection concept, International Journal of Pressure Vessels and Piping 67: 105-111.

[4] Sim R.G., Penny R.K., 1971, Plane strain creep behaviour of thick-walled cylinders, International Journal of Mechanical Sciences 13: 987-1009.

[5] Yang Y.Y., 2000, Time-dependent stress analysis in functionally graded material, International Journal of Solids and Structures 37: 7593-7608.

[6] You L.H., Ou H., Zheng Z.Y., 2007, Creep deformation and stresses in thick-walled cylindrical vessels of FGM subjected to internal pressure, Composite Structures 78: 285-291.

[7] Xuan F.-Zh., Chen J.-J., Wang Zh., Tu Sh.-T., 2009, Time-dependent deformation and fracture of multi-material systems at high temperature, International Journal of Pressure Vessels and Piping 86: 604-615.

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Creep Stress Redistribution Analysis of Thick-Walled FGM Spheres