Optimum material distributions for prescribed apparent fracture toughness in thick-walled FGM circular pipes

International Journal of Pressure Vessels and Piping 78 (2001) 471±484

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Optimum material distributions for prescribed apparent fracture toughness in thick-walled FGM circular pipes A.M. Afsar, H. Sekine* Department of Aeronautics and Space Engineering, Tohoku University, Aoba-yama 01, Aoba-ku, Sendai 980-8579, Japan Received 29 November 2000; revised 13 June 2001; accepted 15 June 2001

Abstract This study treats the inverse problem of evaluating optimum material distributions intending to realize prescribed apparent fracture toughness in thick-walled functionally graded material (FGM) circular pipes. The incompatible eigenstrain induced in the pipes after cooling from the sintering temperature due to the nonhomogeneous coef®cient of thermal expansion is taken into consideration. An approximation method of ®nding stress intensity factors for a crack in the FGM pipes is introduced in which the nonhomogeneous material properties are simulated by a distribution of equivalent eigenstrain. A radial edge crack emanating from the inner surface of the homogenized pipes is considered for the case of a uniform internal pressure applied to the surfaces of the pipes and the crack. The stress intensity factors determined for the crack in the homogenized pipes represent the approximate values of the stress intensity factors for the same crack in the FGM pipes, and are used in the inverse problem of evaluating optimum material distributions intending to realize prescribed apparent fracture toughness in the FGM pipes. Numerical results obtained for a thick-walled TiC/Al2O3 FGM pipe reveal that the apparent fracture toughness signi®cantly depends on the material distributions, and can be controlled within possible limits by choosing an optimum material distribution pro®le. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Composite material; Inverse problem; Fracture toughness; Functionally graded material; Optimum material distribution; Eigenstrain

1. Introduction Functionally graded materials (FGMs) have outstanding advantages over homogeneous materials and conventional composite materials. Therefore, in recent years, much attention has been paid for analyzing the various aspects of FGMs to get an in-depth knowledge for potential applications of these materials as structural and functional elements in aerospace industries, chemical industries, and nuclear power plants. The key distinguishing features of these materials are their continuously varying material distributions and microstructures with space variables, which characterize the FGMs and can be chosen based on a design criterion to satisfy the needs of an application. The design of FGMs invokes the inverse problems by which an optimum material distribution pro®le can be evaluated to realize a prescribed characteristic in the FGMs for high performance and ef®ciency in practical working conditions. A general inverse design procedure for FGMs was addressed by Hirano and Wakashima [1] to determine both the basic material combination and the optimum * Corresponding author. Tel.: 181-22-217-6982; fax: 181-22-217-6983. E-mail address: [email protected] (H. Sekine).

material distribution pro®le with respect to the objective structural shape and the thermomechanical boundary conditions. Markworth and Saunders [2] considered the inverse problem of optimizing an assumed functional form for the spatially dependent material distributions subject to certain constraints such as maximizing or minimizing the heat ¯ux across the material. Further references of the inverse problems to design FGMs with various geometries subject to various constraints can be found in Refs. [3±8]. In all the references mentioned above, the inverse problems were considered in order to design FGMs optimally from the viewpoint of thermal characteristics. The analytical solution to the inverse problems of designing FGMs from the viewpoint of fracture characteristics turns out to be very complicated due to their nonhomogeneous material properties. Furthermore, the incompatible eigenstrain induced in FGMs after cooling from the sintering temperature due to the nonhomogeneous coef®cient of thermal expansion is to be taken into consideration as it has a great in¯uence on their fracture characteristics. Recently, Sekine and Afsar [9] treated the inverse problem to calculate material distribution pro®les from prescribed apparent fracture toughness [10] in semi-in®nite FGMs with an incompatible eigenstrain and arbitrary variation of

0308-0161/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0308-016 1(01)00061-8

472

A.M. Afsar, H. Sekine / International Journal of Pressure Vessels and Piping 78 (2001) 471±484

y Ro

r Ri

θ

x

p

the pipe are denoted by Ri and Ro, respectively. The FGM pipe consists of two constituent materials A and B, and their volume fractions VA and VB are assumed to vary only in the radial direction from the inner surface to the outer surface of the pipe. Hence, all the material properties are functions of r only. When such a pipe is fabricated and cooled from the sintering temperature, an incompatible eigenstrain, which is also a function of r only, is induced in the pipe due to the nonhomogeneous coef®cient of thermal expansion. Since the material of the FGM pipe is isotropic, all the shear components of the incompatible eigenstrain vanish and the normal components become equal, which can be de®ned by

ep …r† ˆ 2a…r†DT A+B

Fig. 1. Analytical model of a thick-walled FGM pipe.

material properties for the case of an edge crack subject to a far-®eld uniform applied load. Later another model [11] of semi-in®nite FGMs was considered to investigate the effects of periodic edge cracks on their brittle fracture characteristics. More recently, the authors considered the inverse problem of designing FGM coatings around a circular hole in an in®nite elastic medium in which the material distribution pro®les were calculated from the prescribed apparent fracture toughness in the coatings [12]. In the present paper, we introduce an approximation method of calculating stress intensity factors for a crack in thick-walled FGM circular pipes with an incompatible eigenstrain and arbitrary variation of material properties. In the course of the approximation, the FGM pipes are homogenized by simulating their inhomogeneous material properties by a distribution of equivalent eigenstrain. Then a radial edge crack emanating from the inner surface of the homogenized pipes is considered for the case of a uniform internal pressure applied to the surfaces of the pipes and the crack. The stress intensity factors obtained for the crack in the homogenized pipes represent the approximate values of the stress intensity factors for the same crack in the FGM pipes. The approximate stress intensity factors are used in the inverse problem of evaluating optimum material distributions intending to realize prescribed apparent fracture toughness in the FGM pipes. 2. Modeling of thick-walled FGM circular pipes A thick-walled FGM circular pipe referred to the Cartesian coordinate system (x, y) and the polar coordinate system (r, u ) having the same origin located at the center of the pipe is modeled as shown in Fig. 1. The inner and the outer radii of

…1†

where a is the coef®cient of thermal expansion of the FGM pipe and DT is the difference between the sintering and room temperatures. For this model of the FGM pipe, we carry out the inverse calculation for optimum material distributions intending to realize prescribed apparent fracture toughness in the FGM pipe for the case of a radial edge crack emanating from the inner surface of the pipe subject to a uniform pressure p applied to the crack surface and the inner surface of the pipe. 3. Approximation method of ®nding stress intensity factors for a crack in thick-walled FGM pipes As mentioned in Section 1, the nonhomogeneous material properties of FGMs complicate the analytical solution to the inverse problem of evaluating material distributions intending to realize prescribed apparent fracture toughness in the FGMs. Therefore, to overcome the complexity and obtain a tractable problem, in this study we adopt a technique to homogenize the FGM pipes simulating their nonhomogeneous material properties by a distribution of equivalent eigenstrain. Then we formulate an approximation method of stress intensity factors for a crack in the FGM pipes, which is used in the subsequent inverse problem. The concept of the approximation method is explained below. First, the FGM pipes are homogenized by simulating the material nonhomogeneities by a distribution of equivalent eigenstrain. The distribution of the equivalent eigenstrain to be determined is such that the elastic ®elds are identical in both the FGM and the homogenized pipes under the same loading conditions. After determining the distribution of the equivalent eigenstrain, a method is formulated to calculate stress intensity factors for a crack in the homogenized pipes. It is noteworthy that the elastic ®elds determined for the uncracked FGM and homogenized pipes are redistributed in the presence of a crack. Since the equivalent eigenstrain is determined from the condition of identical elastic ®elds in the uncracked FGM and homogenized

A.M. Afsar, H. Sekine / International Journal of Pressure Vessels and Piping 78 (2001) 471±484

Pnf Pnf−1 Pi

f

Pi −f1

rn rn-1 ri ri-1

i r0

P0f

473

conditions can be readily derived as " # " # 2 c2i Pfi21 ri2 Pfi i 2 ri s r;f ˆ 12 2 2 1 2 ci 2 ; 1 2 c2i r 1 2 c2i r " # " # 2 c2i Pfi21 ri2 Pfi i 2 ri s u;f ˆ 11 2 2 1 1 ci 2 ; 1 2 c2i r 1 2 c2i r i s z;f ˆ

2ni …c2i Pfi21 2 Pfi † 2 Ei epi 1 2 c2i

(2)

where r ci ˆ i21 ; ri Pfi ˆ pfi 1 pp;f i

Fig. 2. Layering of the FGM pipe.

pipes, the redistributed elastic ®eld in the cracked homogenized pipes cannot exactly represent the redistributed elastic ®eld in the cracked FGM pipes. Therefore, the stress intensity factors calculated for a crack in the homogenized pipes with the equivalent eigenstrain represent approximate values of the stress intensity factors for the same crack in the corresponding FGM pipes and hence the term approximation method has been used.

The ®rst term on the right hand side of the second of Eq. (3) arises from the applied internal pressure p and the second term arises from the incompatible eigenstrain e p. The strain and the displacement components in the ith layer are derived as

eir;f ˆ "

eiu;f ˆ " £

…1 1 ni †…1 2 2ni † Ei …1 2 c2i †

…c2i Pfi21

£

3.1. Equivalent eigenstrain simulating nonhomogeneous material properties of thick-walled FGM pipes The equivalent eigenstrain to simulate the nonhomogeneities of the FGM pipe shown in Fig. 1 is determined from the condition that the elastic ®eld in the FGM pipe due to the uniform pressure p and the incompatible eigenstrain e p to be equal to the elastic ®eld in a homogeneous pipe of the same geometry due to the uniform pressure p, the incompatible eigenstrain e p, together with the equivalent eigenstrain. First, we determine the elastic ®eld in the FGM pipe. For this purpose, we adopt a technique in which the FGM pipe is divided into n number of layers of in®nitesimal thickness as shown in Fig. 2. Each layer is assumed to have constant volume fractions and material properties but different from those in the other layers. The inner and the outer radii of the ith layer are denoted by ri21 and ri, respectively, where r0 ˆ Ri and rn ˆ Ro : The pressures at the inner and the outer surfaces of the ith layer are, respectively, denoted by Pfi21 and Pfi ; which are the resultant of the pressures due to the applied internal pressure p and the incompatible eigenstrain e p. For this system, the stress ®eld in the ith layer for plane strain and axisymmetric

…3†

2

Pfi †

# c2i ri2 f f 2 …P 2 Pi † 1 …1 1 ni †epi ; 1 2 2ni r 2 i21

…1 1 ni †…1 2 2ni † Ei …1 2 c2i †

…c2i Pfi21

2

Pfi †

# c2i ri2 f f 1 …P 2 Pi † 1 …1 1 ni †epi 1 2 2ni r 2 i21 …4†

ufi

" ) ( …1 1 ni †…1 2 2ni †ri 2 f r 1 ri ˆ 1 ci Pi21 ri 1 2 2ni r Ei …1 2 c2i † )# ( r c2i ri 2 Pfi 1 …1 1 ni †epi r 1 ri 1 2 2ni r

…5†

where n i is Poisson's ratio and Ei is Young's modulus of the ith layer of the FGM pipe. The unknown pressures pfi and pp;f are determined by solving the following systems of i simultaneous linear equations, which are obtained from the condition that …ufi 2 ufi11 † vanishes at r ˆ ri :

dfi;i21 ri21 pfi21 1 dfi;i ri pfi 1 dfi;i11 ri11 pfi11 ˆ 0; i ˆ 1; 2; ¼; n 2 1; f p;f f p;f dfi;i21 ri21 pp;f i21 1 di;i ri pi 1 di;i11 ri11 pi11

ˆ ri ‰…1 1 ni11 †epi11 2 …1 1 ni †epi Š; i ˆ 1; 2; ¼; n 2 1

(6)

474

A.M. Afsar, H. Sekine / International Journal of Pressure Vessels and Piping 78 (2001) 471±484

where n 0 is Poisson's ratio and E0 is Young's modulus of the material B. The unknown pressures phi and pp;h are deteri mined by solving the following systems of linear algebraic ho equations obtained from the condition of …uh0 i 2 ui11 † ˆ 0 at r ˆ ri :

where

dfi;i21 ˆ

2ci …1 2 n2i † ; Ei …1 2 c2i †

dfi;i ˆ 2

1 1 ni ‰1 1 c2i 2 2ni Š Ei …1 2 c2i †

2

dfi;i11

dhi;i21 ri21 phi21 1 dhi;i ri phi 1 dhi;i11 ri11 phi11 ˆ 0;

1 1 ni11 ‰1 1 c2i11 2 2ni11 c2i11 Š; Ei11 …1 2 c2i11 †

2c …1 2 n2i11 † ˆ i11 Ei11 …1 2 c2i11 †

i ˆ 1; 2; ¼; n 2 1; (7)

pf0 ˆ p;

p;f pfn ˆ pp;f 0 ˆ pn ˆ 0

…8†

Now we consider a homogeneous pipe of material B and determine the elastic ®eld due to the applied pressure p and the incompatible eigenstrain e p following the same procedure as the FGM pipe. In this case, the stress ®eld in the ith layer is derived as " # " # 2 c2i Phi21 ri2 Phi i 2 ri s r;h0 ˆ 12 2 2 1 2 ci 2 ; 1 2 c2i r 1 2 c2i r " # " # 2 c2i Phi21 ri2 Phi i 2 ri s u;h0 ˆ 11 2 2 1 1 ci 2 ; 1 2 c2i r 1 2 c2i r i s z;h0

2n0 ˆ …c2i Phi21 2 Phi † 2 E0 epi 1 2 c2i

ˆ

phi

1

"

pp;h i

£

eiu;h0 ˆ " £

2

Phi †

…10†

# c2i ri2 h h 2 …P 2 Pi † 1 …1 1 n0 †epi ; 1 2 2n0 r2 i21

…1 1 n0 †…1 2 2n0 † E0 …1 2 c2i †

…c2i Phi21

2

Phi †

1

c2i

2ci11 …1 2 n20 † E0 …1 2 c2i11 †

(14)

p;h phn ˆ pp;h 0 ˆ pn ˆ 0

…15†

Up to this point, the elastic ®eld obtained for the homogeneous pipe is not equivalent to the elastic ®eld obtained for the FGM pipe. To obtain the equivalence, we consider a distribution of equivalent eigenstrain eij;e in the ith layer of the homogeneous pipe, where j ˆ r; u and z. Now for equivalence of the stress ®elds in the FGM and the homogeneous pipes, we can write i i i s r;f ˆ s r;h0 1 s r;e ;

s ui ;f ˆ s ui ;h0 1 s ui ;e ; ri2 2

1 2 2n0 r

i i i s z;f ˆ s z;h0 1 s z;e0

# …Phi21

2

Phi †

1 …1 1

n0 †epi …11†

uh0 i

dhi;i11 ˆ

ph0 ˆ p;

…1 1 n0 †…1 2 2n0 † E0 …1 2 c2i †

…c2i Phi21

2ci …1 2 n20 † ; E0 …1 2 c2i † " 1 1 n0 1 h di;i ˆ 2 …1 1 c2i 2 2n0 † E0 1 2 c2i # 1 2 2 2 …1 1 ci11 2 2n0 ci11 † ; 1 2 c2i11

dhi;i21 ˆ

and

The strain and the displacement components are obtained

eir;h0 ˆ

(13)

where

(9)

where

as

ˆ …1 1 n0 †…epi11 2 epi †ri ; i ˆ 1; 2; ¼; n 2 1

and

Phi

h p;h h p;h dhi;i21 ri21 pp;h i21 1 di;i ri pi 1 di;i11 ri11 pi11

) ( " …1 1 n0 †…1 2 2n0 †ri 2 h r 1 ri ˆ 1 ci Pi21 ri 1 2 2 n0 r E0 …1 2 c2i † )# ( r c2i ri 2 Phi 1 …1 1 n0 †epi r 1 (12) ri 1 2 2n0 r

(16)

i where s j;e is the stress component in the ith layer of the homogeneous pipe due to the equivalent eigenstrain eij;e : The equivalence of the total strains gives

eir;f ˆ eir;h0 1 eir;e 1 eir;e ; eiu;f ˆ eiu;h0 1 eiu;e 1 eiu;e ; eiz;f ˆ eiz;h0 1 eiz;e 1 eiz;e

(17)

A.M. Afsar, H. Sekine / International Journal of Pressure Vessels and Piping 78 (2001) 471±484

where eij;e is the elastic strain component in the ith layer of the homogeneous pipe associated with the equivalent eigeni strain eij;e . The elastic strain eij;e is related to the stress s j;e by Hooke's law as follows: eir;e ˆ eiu;e

eiu;e ˆ

1 i i ˆ ‰s i 2 n0 s r;e 2 n0 s z;e Š; E0 u;e

eiz;e ˆ

1 i ‰s i 2 n0 s r;e 2 n0 s ui ;e Š E0 z;e

…c2i Pfi21



2

Pfi †

2

…c2i Phi21



Phi †

2

2 12

2 11

 # c2i ri2 f f 2 …P 2 Pi † 1 2 2ni r 2 i21

c2i ri2 h 2 …P 2 Phi † 1 2 2n0 r 2 i21

"  c2i 1 1

2 1 1 c2i

#

! …Phi

r

ri2 2

2

Pfi †

2

n0 E0 …1 2 c2i †

…Phi21 2 Pfi21 †

! 2

#

ri …Phi 2 Pfi † r2

2

2n0 ‰n0 …c2i Phi21 2 Phi † 2 ni …c2i Pfi21 2 Pfi †Š E0 …1 2 c2i †

1

epi …E n 2 Ei n0 †; E0 0 i

c2i

2

Phi †

c2i ri2 h 1 …P 2 Phi † 1 2 2n0 r 2 i21

#

eiz;e ˆ

r2 c2i i2

#

! …Phi

r

2

Pfi †

2

n0 E0 …1 2 c2i †

! ri2 1 2 2 …Phi21 2 Pfi21 † r

2 12

r2 c2i i2 r

#

! …Phi

2

Pfi †

2

2n0 ‰n0 …c2i Phi21 2 Phi † 2 ni …c2i Pfi21 2 Pfi †Š E0 …1 2 c2i †

1

epi …E n 2 Ei n0 †; E0 0 i

#

!

r

…c2i Phi21

"

" ! 1 ri2 2 1 ci 1 2 2 …Phi21 2 Pfi21 † E0 …1 2 c2i † r r2 c2i i2

…1 1 n0 †…1 2 2n0 † E0 …1 2 c2i †

" ! 1 ri2 2 ci 1 1 2 …Phi21 2 Pfi21 † 1 E0 …1 2 c2i † r

…1 1 n0 †…1 2 2n0 † E0 …1 2 c2i †

"

2



(18)

…1 1 ni †…1 2 2ni † Ei …1 2 c2i † "



2

Pfi †

# c2i ri2 f f 1 …P 2 Pi † 1 2 2ni r 2 i21

…c2i Pfi21

"

By setting the strain components eiz;f and eiz;h0 to zero for plane strain and making use of Eqs. (2), (4), (9), (11) and (16)±(18) the expressions for the components of the equivalent eigenstrain in the ith layer of the homogeneous pipe can be derived as

eir;e ˆ

…1 1 ni †…1 2 2ni † Ei …1 2 c2i † "

1 i ‰s i 2 n0 s ui ;e 2 n0 s z;e Š; E0 r;e

475

2 ‰n0 …c2i Phi21 2 Phi † 2 ni …c2i Pfi21 2 Pfi †Š E0 …1 2 c2i † 2

2n0 ‰c2i …Phi21 2 Pfi21 † 2 …Phi 2 Pfi †Š E0 …1 2 c2i †

1

epi …E 2 E0 † E0 i

(19)

Although the equivalent eigenstrain derived above and the incompatible eigenstrain epi are piecewise continuous, discontinuities in their values occur at the interfaces between the layers which are physically inadmissible for a FGM having a continuous variation of material properties. For physically admissible results, we obtain the continuous distributions of the equivalent and the incompatible eigenstrains for the nonlayered homogeneous pipe as shown in Fig. 3 by spline interpolation of the piecewise continuous eigenstrains. Including these continuous distributions of the eigenstrains, the resultant stress ®eld in the nonlayered

476

A.M. Afsar, H. Sekine / International Journal of Pressure Vessels and Piping 78 (2001) 471±484

( ) ZRo 2n0 R2i p E0 n 0 2 p p ˆ 2 1 2e 1 2 r e dr 1 2 n0 Ro 2 R2i Ro 2 R2i Ri   Zr 1 n 0 E0 e e e e …er 2 eu † dr 1 C 2…eu 1 n0 ez † 1 1 …1 2 n20 † Ri r

y Ro

ε (r ) + ε (r ) *

s zh

e j

2 E0 …ep 1 eez †

r

Ri

(20)

where

θ

x

p

1 Cˆ 2 Ro 2 R2i 2

B

Fig. 3. Homogenized pipe with a distribution of incompatible and equivalent eigenstrains.

homogeneous pipe is derived as " # R2i p R2o h 12 2 sr ˆ 2 Ro 2 R2i r " E0 1 Zr p 1 2 2 r e dr 1 2 n0 r Ri # ! ZRo R2i 1 p 1 12 2 r e dr R2o 2 R2i Ri r " E0 1 Zr 1 r…eer 1 eeu 1 2n0 eez † dr 2 r 2 Ri 2…1 2 n20 † )# ( Zr 1 R2i e e …er 2 eu † dr 1 C 1 2 2 1 ; r Ri r " # R2i p R2o h 11 2 su ˆ 2 Ro 2 R2i r " E0 1 Zr p 1 2 ep 1 2 r e dr 1 2 n0 r Ri # ! ZRo R2i 1 p 1 11 2 r e dr R2o 2 R2i Ri r " E0 1 2 2…eeu 1 n0 eez † 2…1 2 n20 † 1 Zr r…eer 1 eeu 1 2n0 eez † dr 1 2 r Ri )# ( Zr 1 R2i e e …er 2 eu † dr 1 C 1 1 2 ; 1 r Ri r

R2o

(

ZRo Ri

r…eer 1 eeu 1 2n0 eez † dr

Z Ro 1 …eer 2 eeu † dr Ri r

) …21†

and e p and eej represent the incompatible and the equivalent eigenstrains which are continuous in the entire range of interest. 3.2. Stress intensity factors The resultant stress ®eld determined for an uncracked homogeneous body is redistributed by the presence of a crack. The redistribution of the stress ®eld can be determined by superposing the disturbed stress ®eld due to the presence of the crack on that obtained for the uncracked body, which, to satisfy the boundary condition along the crack surface, yields

s sh 1 s sd ˆ Ts

…22†

where s sh is the stress component along the prospective crack line in the uncracked homogeneous body, s sd is the component of the disturbed stress ®eld due to the presence of the crack, and Ts is the traction applied to the crack surface. The disturbed stress ®eld can be determined by representing the crack by a continuous distribution of edge dislocations. The resultant stress ®eld in the uncracked homogenized pipe shown in Fig. 3 has been determined in the preceding article. Now we consider a radial edge crack of length l emanating from the inner surface of the homogenized pipe in the case of a uniform pressure p applied to the inner surface of the pipe and the crack surface as shown in Fig. 4. The boundary condition along the crack surface given by Eq. (22) reduces to

s ud ˆ 2s uh 2 p;

Ri # r # Ri 1 l; u ˆ 0

…23†

The disturbed stress component s ud is determined representing the crack by a continuous distribution of edge dislocations and using the method of complex potential functions. When a discrete edge dislocation with Burgers vector b runs through a point at a distance h from the center of the pipe as shown in Fig. 5, the complex potential

A.M. Afsar, H. Sekine / International Journal of Pressure Vessels and Piping 78 (2001) 471±484

y

y Ro

ε * (r ) + ε ej (r )

Ro

r

Ri

477

r

Ri l

p

l x

θ

s

p

h

B

B

Fig. 4. A radial edge crack emanating from the inner surface of the homogenized pipe with a distribution of incompatible and equivalent eigenstrains.

h ˆ Ri 1 s; a0 ˆ

a1 ˆ

"

1 2R2o R2i 1 h 1 2 z2h R2o 1 R2i z3 …z 2 h†2 # 1 X 1 0 0 k 0 2k 2 1 a0 1 …a k z 1 a 2k z † (24) z kˆ1

m0b C…z† ˆ p…k0 1 1†

where m0 is the shear modulus of elasticity and k 0 is Kolosov's constant which is equal to 3±4n 0 for plane strain condition. Now for a continuous distribution of edge dislocations over the crack length l, the complex potential functions can be written as " Zl m0 1 2z 1 1 2 F…z† ˆ 2 1 a0 2 z p…k0 1 1† 0 z 2 h Ro 1 Ri # 1 X 1 …ak zk 1 a2k z2k † b…s† ds; kˆ1

Zl m0 C…z† ˆ p…k0 1 1† 0 2

Fig. 5. A discrete edge dislocation at a distance z ˆ h measured from the center of the pipe.

where

functions are given by " m0b 1 2z 1 1 2 F…z† ˆ 2 1 a0 2 z p…k0 1 1† z 2 h Ro 1 R i # 1 X 1 …ak zk 1 a2k z2k † ; kˆ1

"

1 X

1 2R2o R2i 1 h 1 2 z2h R2o 1 R2i z3 …z 2 h†2 #

1 1 a 00 1 …a 0k zk 1 a 02k z2k † b…s† ds z kˆ1

x

θ

(25)

B0 ; 2 R2i †

2…R2o

B 21 2A 01 Ri 2 ; …R4o 2 R4i † …k0 1 1†…R2o 1 R2i †

a21 ˆ ak ˆ

Rf ˆ Ri =Ro ;

A 01 Ri ; …k0 1 1†

DK1 ; DK

a2k ˆ

DK2 ; DK

k $ 2; k $ 2;

DK1 ˆ R2o …1 1 k†…1 2 R2f †Rf2…k21† Bk 1 {1 2 Rf2…k21† }

DK2 ˆ R2o …1 2 k†…1 2 R2f †Rf2…k21† B2k  1 R2…k11† {R2…k21† 2 R24 f } Bk ; i f DK ˆ R4o ‰…1 2 k2 †…1 2 R2f †2 R2…k21† 2 Rf2…k21† f 1 Rf2…k11† 1 R4k f 1 1Š; Bk ˆ A 00k R2k12 2 A 0k R2k12 ; o i

B 2k

Ro2…k21†

;

478

A.M. Afsar, H. Sekine / International Journal of Pressure Vessels and Piping 78 (2001) 471±484

2 A 02k Rk12 ; B 2k ˆ A 002k Rk12 o i a 021 ˆ 2

Table 1 Coef®cients A 0k and A 00k

k0 A 01 Ri ; …k0 1 1†

k

A 0k

A 00k

k$2

……k 2 1†Rk22 …h2 2 R2i †= i k11

a 022

h

…A 000 2 A 00 †R2o R2i ˆ ; …R2o 2 R2i †

a 023 ˆ

R2o R3i R2o 1 R2i

"

2=h

k$4

k21

h

…2h=R2o †

=Rki

2 2 …2…k 2 1†Rk22 o …h 2 Ro †= k11 k22 k21 h † 1 …Ro =h †

The coef®cients A 0k and A 00k are given in Table 1. The stresses can be expressed in terms of the complex potentials F (z), C (z) and their complex conjugates as below [13]

s ud 1 is rdu ˆ F…z† 1 F…z† 1 ‰zF 0 …z† 1 C…z†Š e 2iu ; 0

ˆ F…z† 1 F…z† 2 ‰zF …z† 1 C…z†Š e

2iu

(27)

Ri # r # Ri 1 l Eq. (30) is the singular integral equation for the unknown density function b(s), which is normalized over the interval [21, 1 1] by using the following substitutions: r ˆ Ri 1 t; Wˆ

where

B…S† ˆ b

3r 1 R2o R2i 1 2 2 ; r R2o 1 R2i R2o 1 R2i r 3 " # h2 2 R2i 3 r R2i 2 2 g2 …r; h† ˆ 3h 2 2Ro 2 2 1 2 R4o 2 R4i h…R2o 2 R2i † h R2i …h2 2 R2o † r 2 h…R2o 2 R2i †

" # 1 R3i R4o Ri 2 2 1 R …3h 2 2Ro † 2 2 2 r 3 …R4o 2 R4i † i h

2t 2 1; l

Sˆ

2s 2 1; l (31)

Finally, we obtain

where

g1 …r† ˆ

Tˆ

2Ro l

By using Eqs. (25) and (27) we can obtain the circumferential component of the disturbed stress ®eld along the crack line (u ˆ 0; z ˆ r) as follows:  Zl  1 2m0 d 1 g1 …r† 1 g2 …r; h† b…s† ds su ˆ p…k0 1 1† 0 r 2 h …28†

    1  X k a2k k k 1 ak r 1 1 1 k 12 2 2 r kˆ2 " # 1 1 X 1 X a 02k 0 k 1 a r 1 k 2 kˆ0 k kˆ4 r

…30†

ˆ 2…s uh 1 p†; (26)

1

0

Substitution of Eq. (28) into Eq. (23) yields  Zl  1 2m0 1 g1 …r† 1 g2 …r; h† b…s† ds p…k0 1 1† 0 r 2 h

a 02k ˆ 2…1 2 k†R2i a2…k22† 1 Ri2…k21† a k22 2 Rki A 02…k22† ;

2

kˆ0

†2

2A 01 R …A 00 R 2 A 021 Ro † 1 o 21 2 i ; k0 1 1 Ro 2 R2i

k $ 0;

is rdu

0

k # 21

#

00 a 0k ˆ 2…1 1 k†R2o ak12 1 a2…k12† Ro22…k11† 2 R2k o A k12 ;

s rd

kˆ1

2…hk21 =Rko †

…Rk22 =hk21 † i

Z1 2m 0 p…k0 1 1† 2 1

"

B…S† dS 1 G1 …T†B…S† dS 1 G2 …T; S†B…S† dS T 2S

ˆ 2‰s uh …T† 1 p†Š; 21 # T # 1    l l S ; G1 …T† ˆ g1 T ; 2 2   l l G2 …T; S† ˆ g2 T; S 2 2

#

(32)



(33)

The density function B(S) can be expressed as the product of a fundamental function w(S) which characterizes the bounded-singular behavior of B(S) and a bounded continuous function w (S) in the closed interval 21 # S # 11. Thus we can formulate B…S† ˆ w…S†w…S†

…34†

The fundamental function w(S) can be expressed in the present case as [14] r 11S …35† w…S† ˆ 12S (29)

To numerically solve Eq. (32), we convert this singular integral equation into a system of algebraic equations by

A.M. Afsar, H. Sekine / International Journal of Pressure Vessels and Piping 78 (2001) 471±484

ˆ2

2N 1 1 h ‰s u …Th † 1 pŠ 2

(36)

Ro2 FI

2.5

where w…11† is obtained from Krenk's interpolation formula [16] given by   2j 2 1 N sin p N X 2  2N 1 1  w…S j † w…11† ˆ …39† 2j 2 1 p 2N 1 1 jˆ1 tan 2N 1 1 2 For known values of s uh and p, the solution of Eq. (36) gives the unknowns w (Sj ), which are used in Eq. (39) to obtain the value of w (11). The stress intensity factor is determined from Eq. (38) that represents the approximate value of the stress intensity factor for the same radial edge crack emanating from the inner surface of the FGM pipe shown in Fig. 1. The validity of the approximation method of stress intensity factors has been explained in our previous works [9,12]. It is observed that the error in the stress intensity factors depends on the gradient of the pro®le of Young's modulus of FGMs. For a small crack length, the error remains within acceptable limits even for a large gradient at the crack tip. However, for a large crack length the gradient should be small at the crack tip for the error to be small. 4. Apparent fracture toughness The stress intensity factor at the tip of a crack in homogeneous isotropic materials without any internal stress is expressed in terms of external applied stresses and geometric factors. On the other hand, the stress intensity factor at the tip of a crack in FGMs is expressed in terms of not only external applied stresses and geometric factors but also internal stresses due to the incompatible eigenstrain and material distributions. For brittle materials, fracture

1.0 0.5

The integration and the collocation points in Eq. (36) are calculated from the following expressions [14]   2j 2 1 Sj ˆ cos p ; j ˆ 1; 2; ¼; N; 2N 1 1   2ph T h ˆ cos h ˆ 1; 2; ¼; N (37) ; 2N 1 1 The stress intensity factors can be computed from 2m0 p KI ˆ 2plw…11† …38† k0 1 1

Ro/Ri=1.50 1.75 2.00 2.25 2.50

2.0 1.5

Ro2 − Ri2

using the Gauss±Jacobi integral formula corresponding to the weight function in Eq. (35) in the manner developed by Erdogan et al. [15] to determine the unknowns w (Sj ) as follows: " # N 2m 0 X 1 …1 1 S j † 1 G1 …Th † 1 G2 …Th ; Sj † w…Sj † k0 1 1 jˆ1 Th 2 Sj

479

0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 l /( Ro − Ri ) Fig. 6. The normalized collection factor.

occurs from the crack tip when the stress intensity factor attains the critical value, i.e. the intrinsic fracture toughness. By ignoring the internal stresses and the material distributions in a FGM, let us imagine the FGM with the same geometric con®guration under the external applied stress that corresponds to its fracture stress. Then, we can evaluate the critical value of stress intensity factor through the formula of stress intensity factor for a crack in homogeneous isotropic materials, which is called the apparent fracture toughness of FGMs [10]. Now we consider a radial edge crack of length l emanating from the inner surface of the FGM pipe as shown in Fig. 1. When the uniform applied pressure at which fracture occurs from the crack tip is pc, the apparent fracture toughness of the FGM pipe can be given by the formula of the stress intensity factor for a radial edge crack of length l in a pipe of homogeneous isotropic material as p …40† K Ica ˆ FI pc pl In Eq. (40), FI is a function of geometric parameters, whose values are available in the literature [17] and shown in Fig. 6 for convenience. The critical pressure pc is given by p that satis®es the condition KI ˆ Kc

…41†

where Kc is the local intrinsic fracture toughness of the FGM pipe, which is given by [18] Kc ˆ

E 0 K E0 c

…42†

In Eq. (42), Kc0 is the intrinsic fracture toughness of the homogeneous material B. 5. Inverse problems The inverse problem of evaluating optimum material

480

A.M. Afsar, H. Sekine / International Journal of Pressure Vessels and Piping 78 (2001) 471±484

Table 2 Material properties of TiC and Al2O3 Material

Young's modulus (GPa)

Shear modulus (GPa)

Poisson's ratio

CTE (8C 21)

KIc (MPa m 1/2)

TiC Al2O3

462 380

194.12 150.79

0.19 0.26

7:4 £ 1026 8:0 £ 1026

4.1 3.5

VA

distributions intending to realize prescribed apparent fracture toughness in the FGM pipe is solved by using the formulations developed for the approximation method of ®nding stress intensity factors. Suppose that a pro®le of apparent fracture toughness KIca is prescribed over a region of radial length L measured from the inner surface to the middle point of the nLth layer of the FGM pipe which is divided into n number of layers of in®nitesimal thickness as shown in Fig. 2. Note that the in®nitesimal thickness of the layers in Fig. 2 can be given by Dl ˆ …Ro 2 Ri †=n: Now assume a radial edge crack emanating from the inner surface of the pipe. The crack length li is varied by taking the crack length as li ˆ Dl=2 1 …i 2 1†Dl; where i ˆ 1; 2; ¼; nL ; to ensure that the tip of the crack is located at the middle point of a layer. Considering the volume fractions VAi …i ˆ 1; 2; ¼; n† of the constituent A in each layer of the in®nitesimal thickness in Fig. 2 as design variables, the optimum material distribution i.e. the optimum values of VAi …i ˆ 1; 2; ¼; n† can be evaluated by solving the optimization problem set up as Minimize : Fobj …VA1 ; VA2 ; ¼; VAn † ˆ Subject to : 0 #

VAi

# 1;

nL X iˆ1

(43)

where KIi is the stress intensity factor at the tip of a crack of length li and Kci is the intrinsic fracture toughness of the ith layer of the FGM pipe. In determining KIi by using Eqs. (36)±(39), p is replaced by pc obtained from Eq. (40). The optimization problem in Eq. (43) is solved by using a numerical optimization program ADS [19] in which the BFGS method is used for the unconstrained minimization subproblem, and the one-dimensional search is used for minimizing the unconstrained function by ®rst ®nding bounds and then using polynomial interpolation. The minimum value of the objective function Fobj …VA1 ; VA2 ; ¼; VAn † obtained by the ADS program is compared with a small positive quantity e to satisfy the condition Fobj …VA1 ; VA2 ; ¼; VAn † # e

KA 2 K K 2K 1 VB B ˆ 0; 3KA 1 4m 3KB 1 4m

mA 2 m m 2m 1 VB B ˆ 0; mA 1 Y mB 1 Y
m 9K 1 8m
Yˆ 6K 1 12m VA

(45)

where V is the volume fraction, and the subscripts A and B are used to denote the respective properties of the constituent materials while the nonsubscripted variables are used to denote the effective properties of the FGM. Then Young's modulus E and Poisson's ratio n are calculated by using the expressions Eˆ

9K m ; …3K 1 m†

nˆ

E 21 2m

…46†

The coef®cient of thermal expansion a is determined by using the relation

a ˆ VA aA

KA …3K 1 4m† K …3K 1 4m† 1 VB aB B K…3KA 1 4m† K…3KB 1 4m†

…47†

6. Numerical results and discussion

…KIi 2 Kci †2 ;

i ˆ 1; 2; ¼; n

is necessary to determine the material properties of the FGM pipe. The material properties of the FGM pipe are determined by using the mixture rule [20] according to which the shear modulus of elasticity m and the bulk modulus K are ®rst determined from the relations

…44†

and the corresponding set of the design variables VAi …i ˆ 1; 2; ¼; n† is taken as the solution of the optimization problem. Then the continuous pro®le of the material distribution is obtained by spline interpolation of the design variables VAi …i ˆ 1; 2; ¼; n†: In order to solve the optimization problem in Eq. (43), it

Numerical results are obtained for TiC/Al2O3 FGM pipes in which TiC and Al2O3 represent the materials A and B, respectively. The mechanical and the thermal properties of TiC and Al2O3 are shown in Table 2. The difference between the sintering and room temperatures is taken as 10008C. In numerical calculations, the number of layers of in®nitesimal thickness shown in Fig. 2 is taken as 50, and the value of e in Eq. (44) is taken as 0.1. 6.1. Evaluation of apparent fracture toughness of FGM pipes with prescribed material distributions In order to demonstrate that the approximation method of stress intensity factors can also be applied to a direct approach, the apparent fracture toughness is evaluated for the FGM pipes with prescribed material distributions along the radial direction of the pipes. For this purpose, we consider the FGM pipes with four different pro®les of material distribution as shown in Fig. 7. Shown in this ®gure is the volume fraction VA of TiC against the radial distance normalized by the wall thickness of the pipes. With these prescribed distribution pro®les, the apparent fracture toughness of the pipes is evaluated taking the ratio of the outer radius to the inner radius of the pipes Ro/Ri as a parameter. The apparent fracture toughness KIca is normalized by the

A.M. Afsar, H. Sekine / International Journal of Pressure Vessels and Piping 78 (2001) 471±484

481

3.0

1.2 1.0

2.5

2.0

0.8

2.0

1.5

K Ica 0 Kc

VA

Ro/Ri=2.5

0.6

1.5

0.4

1.0

0.2

0.5 0

0 0

0.2

0.4

r − Ri Ro − Ri

0.6

0.8

1.0

0

0.2

0.4

0.6 l Ro − Ri

0.8

1.0

Fig. 7. Prescribed material distribution pro®les of TiC in a thick-walled TiC/Al2O3 FGM pipe.

Fig. 9. Effects of pipe wall thickness on the apparent fracture toughness obtained for the prescribed linear material distribution pro®le in Fig. 7.

intrinsic fracture toughness Kc0 of Al2O3. The normalized apparent fracture toughness KIca =Kc0 is plotted in Fig. 8 for Ro =Ri ˆ 2:5 with the variation of the normalized crack length l/(Ro 2 Ri). From Figs. 7 and 8 it is revealed that the apparent fracture toughness of a thick-walled FGM pipe depends signi®cantly on the material distribution pro®les. The compressive internal stress due to the incompatible eigenstrain has the effects of reducing the stress intensity factors at the crack tip that eventually increases the apparent fracture toughness. On the other hand, the tensile internal stress has the reverse effects on the apparent fracture toughness. The distribution pro®les shown in Fig. 7 induce a compressive internal stress over a certain portion of the wall thickness at the inner side of the pipes and a balancing tensile internal stress over the remaining

portion of the wall thickness at the outer side. As a result, the apparent fracture toughness up to a certain thickness at the inner side is higher than the intrinsic fracture toughness of the constituents and lower over the remaining portion as can be seen from Fig. 8. Near the inner surface, the apparent fracture toughness shown by the dotted line in Fig. 8 corresponding to the distribution pro®le shown by the dotted line in Fig. 7 is higher than those obtained for other pro®les of the material distribution. This is due to the fact that the distribution pro®le shown by the dotted line is steeper near the inner surface that induces a compressive internal stress whose absolute magnitude is higher than those obtained for other pro®les. The composition pro®les whose steepness is less induce internal stresses with smaller absolute magnitude. For instance, a uniform pro®le having zero steepness induces no internal stress because there is no incompatibility in the eigenstrain. With the linear material distribution pro®le shown by the solid line in Fig. 7, the normalized apparent fracture toughness is evaluated for three different values of the parameter Ro/Ri and the results are depicted in Fig. 9 in order to investigate the effect of the wall thickness on the apparent fracture toughness. It is seen that the apparent fracture toughness increases as the wall thickness increases for the same type of the material distribution pro®le.

5.0

Ro/Ri=2.5

4.0

K Ica Kc0

3.0 2.0 1.0

6.2. Evaluation of optimum material distributions intending to realize prescribed apparent fracture toughness in FGM pipes

0 0

0.2

0.4

0.6 l Ro − Ri

0.8

1.0

Fig. 8. Apparent fracture toughness of a thick-walled TiC/Al2O3 FGM pipe.

The inverse problem of evaluating optimum material distributions is solved to realize prescribed apparent fracture toughness in the FGM pipes, which is higher than the intrinsic fracture toughness of the constituent materials

482

A.M. Afsar, H. Sekine / International Journal of Pressure Vessels and Piping 78 (2001) 471±484

6.0

0.5

5.0

Ro/Ri=2.5

0.4

I

4.0

0.3

3.0

II

VA

K Ica K c0

I II

0.2

2.0

0.1

1.0 0 0

0.2

0.4

0.6

0.8

0

1.0

0

l Ro − Ri

0.2

0.4

r − Ri Ro − Ri

0.6

0.8

1.0

Fig. 10. Prescribed apparent fracture toughness of a thick-walled TiC/Al2O3 FGM pipe.

Fig. 11. Optimum material distribution pro®les of TiC in a thick-walled TiC/Al2O3 FGM pipe.

shown in Table 2. Although it is possible to control the higher apparent fracture toughness of various pro®les in order to meet the requirement of an application, in this study we consider only two examples as shown by the solid portions of the curves I and II in Fig. 10. The broken line in this ®gure represents a rough estimation of the upper limit of the apparent fracture toughness which is drawn through the points of peak values of the apparent fracture toughness as shown by the dash-dot line in Fig. 8. The normalized apparent fracture toughness in example I is controlled such that it increases linearly from 2.0 to 2.85 over the normalized crack length 0.75. In example II, the normalized value of the apparent fracture toughness is 2.0 that does not vary over the same crack length. For the prescribed apparent fracture toughness of Fig. 10, the corresponding optimum material distribution pro®les evaluated by the inverse calculations are shown in Fig. 11. It exhibits the volume fraction VA of TiC versus the normalized radial distance for the parameter Ro =Ri ˆ 2:5: In order to know the characteristics of the FGM pipes after the controlled region, the apparent fracture toughness in that region is calculated for the material distribution pro®les of Fig. 11 and shown by the dotted portions of the curves I and II in Fig. 10. It is noted that the apparent fracture toughness in both cases reduces drastically after the controlled region. The fabrication of the FGM pipes with the material distributions shown in Fig. 11 may be dif®cult by using a single manufacturing process like chemical vapour deposition, plasma spray, powder metallurgy and physical vapour deposition. However, by a suitable combination of these processes, it may be possible to fabricate these FGM pipes according to the material distributions of Fig. 11. The prescribed apparent fracture toughness shown in Fig. 10 is realized by designing the FGM pipes having the material distribution pro®les as shown in Fig. 11. Thus it

can be concluded that the apparent fracture toughness of a thick-walled FGM pipe can be controlled within possible limits by choosing an optimum material distribution pro®le in the FGM pipe. Fig. 12 shows the effects of the wall thickness of the FGM pipes on the material distributions. These results are obtained for the example I of the prescribed apparent fracture toughness shown in Fig. 10. It is noted that up to a certain thickness at the inner side of the pipes the value of VA increases as the parameter Ro/Ri decreases for the same type of prescribed apparent fracture toughness. The variation of the normalized Young's moduli corresponding to the material distribution pro®les of Fig. 11 is shown in Fig. 13. It is noted that the gradients of the pro®les 0.5 Ro/Ri=2.5 Ro/Ri=2.0 Ro/Ri=1.5

0.4

VA

0.3 0.2 0.1 0 0

0.2

0.4

r − Ri Ro − Ri

0.6

0.8

1.0

Fig. 12. Effects of pipe wall thickness on material distribution pro®les obtained for the prescribed apparent fracture toughness of example I in Fig. 10.

A.M. Afsar, H. Sekine / International Journal of Pressure Vessels and Piping 78 (2001) 471±484

results obtained for TiC/Al2O3 FGM pipes reveal that the apparent fracture toughness in the FGM pipes depends signi®cantly on the material distributions. It is also revealed that the apparent fracture toughness in the FGM pipes can be controlled within possible limits by choosing an optimum material distribution pro®le in the FGM pipes.

2.0

E E0

1.6

I II

1.2

483

Acknowledgements

0.8 0.4 0 0

0.2

0.4

0.6 r − Ri Ro − Ri

0.8

1.0

Fig. 13. Variation of normalized Young's modulus in a thick-walled TiC/ Al2O3 FGM pipe.

The authors would like to acknowledge the partial support of the Grant-in-Aid for Scienti®c Research No. 08455051 of the Ministry of Education, Science, Sports and Culture of Japan to HS. One of the authors (AMA) also thankfully acknowledges the ®nancial assistance through the Japanese Government Scholarship (No. 942062) from the Ministry of Education, Science, Sports and Culture of Japan. References

of these normalized Young's moduli are large only near the inner surface of the FGM pipes. However, the crack tip lying near the inner surface indicates a small crack length for which the errors remain within acceptable limits even for a large gradient. 7. Conclusions A solution method of the inverse problem of evaluating optimum material distributions has been developed to realize prescribed apparent fracture toughness in thickwalled FGM circular pipes. The incompatible eigenstrain induced in the FGM pipes after cooling from the sintering temperature due to the nonhomogeneous coef®cient of thermal expansion has been taken into consideration. An approximation method of stress intensity factors is introduced for a crack in the FGM pipes with any arbitrary variation of material properties. In the course of the approximation method, the FGM pipes are homogenized simulating their nonhomogeneous material properties by a distribution of equivalent eigenstrain, which is determined from the condition of identical elastic ®elds in the uncracked FGM and homogenized pipes. A radial edge crack emanating from the inner surface of the homogenized pipes is considered for the case of a uniform internal pressure applied to the surfaces of the pipe and the crack, and the formulations of the stress intensity factors are obtained by using the well-established distributed dislocation technique. The stress intensity factors obtained for the crack in the homogenized pipes represent the approximate values of the stress intensity factors for the same crack in the FGM pipes, and are used in the inverse problem of evaluating optimum material distributions to realize prescribed apparent fracture toughness in the FGM pipes. Numerical

[1] Hirano T, Wakashima K. Mathematical modeling and design. Mater Res Soc 1995;20(1):40±2. [2] Makworth AJ, Saunders JH. A model of structure optimization for a functionally graded material. Mater Lett 1995;22(1±2):103±7. [3] Nadeau JC, Ferrari M. Microstructural optimization of a functionally graded transversely isotropic layer. Mech Mater 1999;31(10): 637±51. [4] Nadeau JC, Meng XN. On the response sensitivity of an optimally designed functionally graded layer. Composites B 2000;31(4):285± 97. [5] Nakamura T, Wang T, Sampath S. Determination of properties of graded materials by inverse analysis and instrumented indentation. Acta Mater 2000;48(17):4293±306. [6] Ootao Y, Kawamura R, Tanigawa Y, Imamura R. Optimization of material composition of nonhomogeneous hollow sphere for thermal stress relaxation making use of neural network. Comput Meth Appl Mech Engng 1999;180(1-2):185±201. [7] Ootao Y, Tanigawa Y, Nakamura T. Optimization of material composition of FGM hollow circular cylinder under thermal loading: a neural network approach. Composites B 1999;30(4):415±22. [8] Ootao Y, Kawamura R, Tanigawa Y, Imamura R. Optimization of material composition of nonhomogeneous hollow circular cylinder for thermal stress relaxation making use of neural network. J Thermal Stresses 1999;22(1):1±22. [9] Sekine H, Afsar AM. Composition pro®le for improving the brittle fracture characteristics in semi-in®nite functionally graded materials. JSME Int J, Ser A 1999;42(4):592±600. [10] Lakshminarayanan R, Shetty DK, Cutler RA. Toughening of layered ceramic composites with residual surface compression. J Am Ceram Soc 1996;79(1):79±87. [11] Afsar AM, Sekine H. Crack spacing effect on the brittle fracture characteristics of semi-in®nite functionally graded materials with periodic edge cracks. Int J Fracture 2000;102(3):L61±6. [12] Afsar AM, Sekine H. Inverse problems of material distributions for prescribed apparent fracture toughness in FGM coatings around a circular hole in in®nite elastic media. Compos Sci Technol, accepted for publication. [13] Muskhelishvili NI. Some basic problems of the mathematical theory of elasticity. The Netherlands: Noordhoff, 1975, 2nd ed., translated from Russian by JRM Radok. [14] Hills DA, Kelly PA, Dai DN, Korsunsky AM. Solution of crack

484

A.M. Afsar, H. Sekine / International Journal of Pressure Vessels and Piping 78 (2001) 471±484

problems. The distributed dislocation technique. Dordrecht: Kluwer Academic Publishers, 1996. p. 41. [15] Erdogan F, Gupta GD, Cook TS. Numerical solution of singular integral equations. In: Sih GC, editor. Mechanics of fracture: methods of analysis and solutions of crack problems, vol. 1. Leyden: Noordhoff, 1973. p. 368±425. [16] Krenk S. On the use of the interpolation polynomial for solution of singular integral equation. Quart Appl Math 1975;32(4):479±84. [17] Bowie OL, Freese CE. Elastic analysis for a radial crack in a circular ring. Engng Fracture Mech 1972;4(2):315±21.

[18] Nair SV. Crack-wake debonding and toughness in ®ber- or whiskerreinforced brittle-matrix composites. J Am Ceram Soc 1990;73(10): 2839±47. [19] Vanderplaats GN, Sugimoto H. A general-purpose optimization program for engineering design. Comput Struct 1986;24(1):13±21. [20] Nan C-W, Yuan R-Z, Zhang L-M. The physics of metal/ceramic functionally gradient materials. In: Holt JB, Koizumi M, Hirai T, Munir ZA, editors. Ceramic transaction: functionally gradient materials, vol. 34. Westerville, Ohio: The American Ceramic Society, 1993. p. 75±82.