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Inverse problems of material distributions for prescribed apparent fracture toughness in FGM coatings around a circular hole in infinite elastic media
Composites Science and Technology 62 (2002) 1063–1077 www.elsevier.com/locate/compscitech
Inverse problems of material distributions for prescribed apparent fracture toughness in FGM coatings around a circular hole in infinite elastic media A.M. Afsara,*, H. Sekineb a
Department of Mechanical Engineering, Bangladesh University of Engineering and Technology (BUET), Dhaka 1000, Bangladesh b Department of Aeronautics and Space Engineering, Tohoku University, Aoba-yama 01, Aoba-ku, Sendai 980-8579, Japan Received 24 October 2000; received in revised form 15 February 2002; accepted 20 February 2002
Abstract This study is concerned with the inverse problem of calculating material distributions intending to realize prescribed apparent fracture toughness in functionally graded material (FGM) coatings around a circular hole in infinite elastic media. The incompatible eigenstrain induced in the FGM coatings after cooling from the sintering temperature, due to mismatch in the coefficients of thermal expansion, is taken into consideration. An approximation method of determining stress intensity factors is introduced for a crack in the FGM coatings in which the FGM coatings are homogenized simulating the nonhomogeneous material properties by a distribution of equivalent eigenstrain. A radial edge crack emanating from the circular hole in the homogenized coatings is considered for the case of a uniform pressure applied to the surfaces of the hole and the crack. The stress intensity factors determined for the crack in the homogenized coatings represent the approximate values of the stress intensity factors for the same crack in the FGM coatings, and are used in the inverse problem of calculating material distributions in the FGM coatings intending to realize prescribed apparent fracture toughness in the coatings. Numerical results are obtained for a TiC/Al2O3 FGM coating, which reveal that the apparent fracture toughness in FGM coatings around a circular hole in infinite elastic media can be controlled within possible limits by choosing an appropriate material distribution profile in the coatings. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: B. Fracture toughness; C. Crack; Functionally graded material
1. Introduction Functionally graded materials (FGMs) are nonhomogeneous solids, which consist of two or more distinct material phases, such as different ceramics or ceramics and metals, and are the mixture of them such that the composition of each changes continuously with space variables. The change in composition induces material and microstructural gradients, and makes the functionally graded materials different in behavior from homogeneous materials and conventional composite materials [1,2]. These materials are tailorable in their properties via the design of the gradients, which, in turn, depend on material distributions. From a mechanics viewpoint the main advantages of material property * Corresponding author. Tel.: +880-2-966-5636; fax: +880-2-8613046. E-mail address: [email protected] (A.M Afsar).
grading appear to be improved bonding strength, toughness, wear and corrosion resistance, and reduced residual and thermal stresses. Some typical applications include thermal barrier coatings of high temperature components in gas turbines, surface hardening for tribological protection and graded interlayers used in multilayered microelectronic and optoelectronic components [1–3]. Although FGMs have outstanding advantages, the applications of these materials in various branches of engineering and technologies necessitate identifying the probable failure modes and designing them against those failures. In designing with FGMs, an important aspect of the problem is the mechanical failure, specifically the fracture failure. Although the absence of discontinuous interfaces in FGMs does largely reduce material property mismatch, cracks may occur when they are subjected to external loadings [1,2]. Very often the process begins with the formation of microcracks at
0266-3538/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(02)00049-0
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locations of corrosion pits, surface flaws, or severe stress concentrations. Generally a number of microcracks coalesce and form a local dominant crack, which would then propagate subcritically under cyclic or sustained loading. The loads or stresses acting on the medium may be mechanically or thermally induced. There are also uncertainties arising from voids and defects that are introduced in FGMs during manufacturing. Even a small quantity of mechanical imperfections can cause a marked influence on their fracture strength. Therefore, the study of the fracture mechanics of these materials appears to be an utmost necessary to understand, quantify and improve their toughness. Over the past few years, there have been a number of works, both analytical and experimental, to study the responses of FGMs to mechanical and thermal loads for various geometries in various fracture mechanisms. In these works, it is reported that the nonhomogeneous material properties of FGMs complicate the analytical studies of crack problems of these materials. Therefore, to overcome the difficulties and obtain a tractable problem, it is often conventional to regard the material properties to be some certain assumed functions of space variables. The most common functions assumed to model the material properties are exponential and power functions. By assuming exponential functions for material properties, crack problems of nonhomogeneous materials have been considered in Refs. [4–10] for mechanical loadings and in Refs. [11–15] for thermal loadings. Power function variations of material properties are considered in Refs. [16–22] to study the crack problems of nonhomogeneous materials. In these studies, various aspects of crack problems of FGMs have been analyzed for various geometries and loading conditions. However, these studies are concerned only with the direct problems in which the fracture characteristics of FGMs can be analyzed only for certain assumed functions for the material properties, e.g. exponential and power functions. An important aspect of FGMs still remaining to be dealt with is the inverse problem in which desired characteristics of FGMs under mechanical and thermal loadings can be prescribed and the corresponding material distributions via the material properties can be obtained by inverse calculations. This approach appears to be quite effective and efficient tool in designing FGMs for obtaining a desired behavior to suit an application. Obviously, for an inverse problem, some specific functions for the material property distributions cannot be assumed as Zuiker [23] pointed out that these assumed property distributions are not sometimes physically realizable for some material distribution profiles predicted by the inverse problem to achieve desired characteristics in FGMs. Therefore, to solve the inverse problem it is necessary to develop a method that can treat any arbitrary variation of material properties.
Furthermore, incompatible eigenstrains [24] induced in FGMs due to mismatch in the coefficients of thermal expansion when they are cooled from the sintering temperature should also be considered. Recently, Sekine and Afsar [25] introduced an approximation method of determining stress intensity factors for a particular case of an edge crack in semi-infinite FGMs with incompatible eigenstrains and arbitrary variation of material properties. The method was applied to the inverse problem of calculating material distributions to attain prescribed apparent fracture toughness corresponding to improved brittle fracture characteristics. Later another model [26] of the semi-infinite FGMs was considered to investigate the effects of periodic edge cracks on the brittle fracture characteristics. In the present study, the approximation method of determining the stress intensity factors is introduced for a crack in FGM coatings around a circular hole in infinite elastic media in which the FGM coatings are homogenized simulating their nonhomogeneous material properties by a distribution of equivalent eigenstrain. A radial edge crack emanating from the circular hole in the homogenized coatings is considered for the case that the surfaces of the hole and the crack are subjected to a uniform pressure. The stress intensity factors determined for the crack in the homogenized coatings represent the approximate values of the stress factors for the same crack in the FGM coatings, and are used in the inverse calculation of material distributions in the FGM coatings around the circular hole intending to realize prescribed apparent fracture toughness in the FGM coatings.
2. Modeling of the problem Consider an FGM coating around a circular hole of radius R in an infinite elastic medium as shown in Fig. 1, which is referred to the Cartesian coordinate system
Fig. 1. Analytical model of an FGM coating around a circular hole in an infinite elastic medium.
A.M. Afsar, H. Sekine / Composites Science and Technology 62 (2002) 1063–1077
(x, y) and the polar coordinate system (r, ) having the same origin located at the center of the circular hole. The constituent materials of the FGM coating are denoted by A and B, and their volume fractions VA and VB are assumed to vary in the radial direction only. Thus, the region R 4 r 4 Rf has a gradation of the properties while the infinite region r > Rf is homogeneous consisting of the material B only. The continuously and arbitrarily varying Young’s modulus, Poisson’s ratio and the coefficient of thermal expansion of the FGM coating are represented by E, and , respectively while the corresponding properties of the homogeneous region are, respectively, denoted by E0 , 0 and 0 . When such a system is cooled from the sintering temperature, an incompatible eigenstrain [24], which is a nonelastic misfit thermal strain in the present case, is induced in the FGM coating due to mismatch in the coefficients of thermal expansion. It is assumed that the material is isotropic and the microstructure is neglected for which all the shear components of the incompatible eigenstrain vanish and the normal components become equal. The normal component of the incompatible eigenstrain " ðrÞ, which is a function of r only, can be defined by " ðrÞ ¼ ð0 ÞT;
ð1Þ
where T is the difference between the sintering and room temperatures. Eq. (1) is written based on the basic assumption that the material is freely contracted without any constraint due to the change in temperature. For this model of the problem, we carry out inverse calculations of material distributions in the FGM coating for plane stress condition intending to realize prescribed apparent fracture toughness considering a radial edge crack emanating from the circular hole and considering that the surfaces of the hole and the crack are subjected to a uniform pressure p.
3. Approximation method of stress intensity factors for a crack in FGM coatings Functionally graded materials are nonhomogeneous solids and, therefore, their nonhomogeneities have to be considered in studying the fracture behavior of these materials. The consideration of these nonhomogeneities complicates the analytical studies due to mathematical difficulties. Thus it is often conventional to regard the material properties to be some certain assumed functions of space variables, for instance, exponential and power functions, in order to simplify the problems. However, in designing with FGMs, i.e. in the inverse problems, in which material distribution profiles have to be determined to achieve desired fracture characteristics, special functional forms of the properties cannot
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be assumed. Since these assumed functional forms of the properties may not be physically realizable for some material distribution profiles obtained by inverse calculations. Therefore, as an alternate approach, an approximation method is introduced in this study to calculate the stress intensity factors for a crack in FGM coatings, which is not restricted to any specific property distributions, but can treat any arbitrary distributions of the properties. The concept of the approximation method is explained below. First, the FGM coatings 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 fields are identical in both the FGM and the homogenized coatings under the same loading conditions. After determining the distribution of the equivalent eigenstrain, a method is formulated to calculate the stress intensity factors for a crack in the homogenized coatings. Since the equivalent eigenstrain is determined from the condition of identical elastic fields in the uncracked FGM and homogenized coatings, the redistributed elastic field in the cracked homogenized coatings cannot exactly represent the redistributed elastic field in the cracked FGM coatings. Therefore, the stress intensity factors calculated for a crack in the homogenized coatings with the equivalent eigenstrain represent the approximate values of the stress intensity factors for the same crack in the corresponding FGM coatings and hence the term approximation method has been used. 3.1. Equivalent eigenstrain The FGM coating around the circular hole shown in Fig. 1 is homogenized simulating the nonhomogeneous material properties by a distribution of equivalent eigenstrain. As stated before, the equivalent eigenstrain is determined from the condition of identical elastic fields in the uncracked FGM and homogenized coatings subjected to the same loading condition. We first consider the FGM coating subjected to a uniform applied pressure p around the circular hole. To determine the elastic field in the FGM coating, a special technique is adopted in which the FGM coating is radially divided into layers of infinitesimal thickness as shown in Fig. 2, which exhibits one half of the elastic medium. Each layer is assumed to have constant material properties but different from those in the other layers. The inner and the outer radii of the ith layer are, respectively, denoted by ri1 and ri, where r0=R and rn=Rf. The region r > Rf is homogeneous. The pressures at the inner and the outer surfaces of the ith layer are, respectively, P fi1 and P fi which are the resultant of the pressures due to the uniform applied pressure p and the incompatible eigenstrain " . For this layered FGM
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1 i h 2 f ðc P Pif Þ Ei ð1 c2i Þ i i1 i 1 þ i r2i 2 f ci ðPi1 Pif Þ þ "i ; þ 2 1 i r
"i; f ¼
h i 2i f 2 f P P c i þ "i ; Ei ð1 c2i Þ i i1
ð4cÞ
ð1 i Þri 2 f r 1 þ i ri ¼ þ cP Ei ð1 c2i Þ i i1 ri 1 i r r 1 þ i ri 2 c þ þ r"i : Pif ri 1 i r i
ð5Þ
"iz; f ¼
u fi
ð4bÞ
In deriving the above equations, it is important to remind that all the normal components of the incompatible eigenstrain are equal, i.e. "i ¼ "r;i ¼ ";i ¼ "z;i , and the shear components are zero. The unknown pressures pfi and p;f are determined i from the condition that the displacements at r ¼ ri are identical for the ith and the (i+1)th layers, which gives Fig. 2. Layering of the FGM coating around a circular hole in an infinite elastic medium.
fi;i1 ri1 pfi1 þ fi;i ri pfi þ fi;iþ1 riþ1 pfiþ1 ¼ 0; ð6aÞ i ¼ 1; 2; . . . ; n 1;
coating, it can be easily shown that the resultant stress components in the ith layer, in axisymmetric case and plane stress condition, are r;i f ¼
f 2 c2i Pi1 r2i Pif 2 ri 1 1 c i 2 ; r2 r 1 c2i 1 c2i
ð2aÞ
i ; f ¼
f 2 c2i Pi1 r2i Pif 2 ri 1 þ 1 þ c i 2 ; r2 r 1 c2i 1 c2i
ð2bÞ fi;i1 ¼
ri1 ; ri
ð3aÞ
Pif ¼ pfi þ p;f i :
ð3bÞ
At the right hand side of Eq. (3b), the first term appears due to the uniform applied pressure p while the second term appears due to the incompatible eigenstrain " . The strain and the displacement components in the ith layer are derived as
1 i h 2 f ðc P Pif Þ Ei ð1 c2i Þ i i1 i 1 þ i r2i 2 f ci ðPi1 Pif Þ þ "i ; 2 1 i r
¼ ri ð"iþ1 "i Þ; i ¼ 1; 2; . . . ; n 1;
ð6bÞ
where
where ci ¼
f ;f f ;f fi;i1 ri1 p;f i1 þ i;i ri pi þ i;iþ1 riþ1 piþ1
2ci ; Ei ð1 c2i Þ
" # 2 2 1 þ c 1 1 þ c 1 iþ1 i i þ iþ1 ; fi;i ¼ Ei 1 c2i Eiþ1 1 c2iþ1 fi;iþ1 ¼
2ciþ1 ; Eiþ1 ð1 c2iþ1 Þ
ð7aÞ
ð7bÞ
ð7cÞ
and pf0 ¼ p;
ð8aÞ
p;f 0 ¼ 0:
ð8bÞ
The stress and displacement components in the region r > Rf are derived as
"ir; f ¼
ð4aÞ r;f ¼
Rf2 Pnf r2
; r > Rf ;
ð9aÞ
A.M. Afsar, H. Sekine / Composites Science and Technology 62 (2002) 1063–1077
;f ¼ f
u ¼
Rf2 Pnf r2
Rf2 Pnf 2 0 r
; r > Rf ;
; r > Rf :
pfn ¼
2 h
20 ci Pi1 Pih þ "i ; 2 E0 ð1 ci Þ
ð13cÞ
ð1 0 Þri 2 h r 1 þ 0 ri P þ c E0 ð1 c2i Þ i i1 ri 1 0 r r 1 þ 0 ri 2 ci þ þ r"i ; Pih ri 1 0 r
ð14Þ
ð9bÞ
"iz;h0 ¼
ð10Þ
uhi 0 ¼
To solve Eq. (6), it is necessary to know the pressures pfn and p;f n at the outer surface of the nth layer. These are determined from the condition that the displacements ufn of the nth layer and uf of the homogeneous region are equal at r ¼ Rf . From this condition, we obtain 2E0 c2n pfn1 ; ð1 þ 0 ÞEn ð1 c2n Þ þ E0 fð1 n Þ þ ð1 þ n Þc2n g
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where Pih ¼ phi þ p;h i :
ð15Þ
The unknown pressures phi and p;h are, respectively, i determined from the following equations:
ð11aÞ p;f n
E0 En "n ð1 c2n Þ þ 2E0 c2n p;f n1 : ¼ ð1 þ 0 ÞEn ð1 c2n Þ þ E0 fð1 n Þ þ ð1 þ n Þc2n g
hi;i1 ri1 phi1 þ hi;i ri phi þ hi;iþ1 riþ1 phiþ1 ¼ 0; ð16aÞ i ¼ 1; 2; . . . ; n 1;
ð11bÞ Now we consider the homogenized infinite medium with the same geometry as Fig. 1 and determine the elastic field in the corresponding coating region following the same procedure as the FGM coating. Note that the coating region has the same distribution of the incompatible eigenstrain defined by Eq. (1). In this case, the pressure at the outer surface of the nth layer are taken same as that of the FGM coating so as to achieve the same elastic field in the region r > Rf . The equivalency in the elastic fields of the region R 4 r 4 Rf is achieved by assuming a distribution of equivalent eigenstrain. First, we determine the elastic field in the ith layer of the coating region due to the uniform applied pressure p and the incompatible eigenstrain " as
i r;h ¼ 0
i ¼ ;h 0
"ir;h0
ð12aÞ
h 2 c2i Pi1 r2i Pih 2 ri 1 þ 1 þ c i 2 ; r2 r 1 c2i 1 c2i
ð12bÞ
1 0 ¼ E0 ð1 c2i Þ
"i;h0
h 2 c2i Pi1 r2i Pih 2 ri 1 1 c i 2 ; r2 r 1 c2i 1 c2i
1þ 1
h Pih Þ ðc2i Pi1
0 r2i 2 h c ðP 0 r2 i i1
ð13aÞ
Pih Þ þ "i ;
1 0 ¼ ðc2 P h Pih Þ E0 ð1 c2i Þ i i1 1 þ 0 r2i 2 h h c ðP Pi Þ þ "i ; þ 1 0 r2 i i1
;h ;h h h hi;i1 ri1 p;h i1 þ i;i ri pi þ i;iþ1 riþ1 piþ1
¼ ri ð"iþ1 "i Þ; i ¼ 1; 2; . . . ; n 1;
ð16bÞ
where hi;i1 ¼
hi;i
2ci ; E0 ð1 c2i Þ
" # 1 1 þ c2i 1 þ c2iþ1 ¼ þ ; E0 1 c2i 1 c2iþ1
hi;iþ1 ¼
2ciþ1 ; E0 ð1 c2iþ1 Þ
ð17aÞ
ð17bÞ
ð17cÞ
and ph0 ¼ p;
ð18aÞ
phn ¼ pfn ;
ð18bÞ
p;h 0 ¼ 0;
ð18cÞ
;f p;h n ¼ pn :
ð18dÞ
Now we assume a distribution of equivalent eigenstrain "ij;e in the ith layer of the coating region, where j=r, and z. From the equivalency of the stress fields in the coating regions, we can write ð13bÞ i i i r;f ¼ r;h þ r;e ; 0
ð19aÞ
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i i i ;f ¼ ;h þ ;e ; 0
ð19bÞ
i i where r;e and ;e are, respectively, the radial and the circumferential stress components in the ith layer of the coating region of the homogenized medium due to the equivalent eigenstrain "ij;e . From the equivalency of the total strains, we can write
"ir;f ¼ "ir;h0 þ eir;e þ "ir;e ;
ð20aÞ
"i;f ¼ "i;h0 þ ei;e þ "i;e ;
ð20bÞ
"iz;f ¼ "iz;h0 þ eiz;e þ "iz;e ;
ð20cÞ
where eij;e is the elastic strain associated with the equivalent eigenstrain "ij;e in the ith layer of the coating region of the homogenized medium. i The elastic strain eij;e is related to the stress j;e by Hooke’s law as follows eir;e ¼ ei;e
1 i i r;e 0 ;e ; E0
ð21aÞ
1 i i ¼ ;e 0 r;e ; E0
ð21bÞ
0 i i r;e þ ;e : E0
ð21cÞ
eiz;e ¼
Combining Eqs. (2), (4), (12), (13) and (19)–(21), the expressions for the components of the equivalent eigenstrain in the ith layer can be derived as 1 þ r2 1 i i i 2 f f f 2 f c P P c P P i i i i1 1 i r2 i i1 Ei 1 c2i 2 2 h 1 0 1 þ r 0 i 2 ci Pi1 Pih c P h Pih 1 0 r2 i i1 E0 1 c2i 2 1 r2i h f f 2 2 ri h c P P 1 P P 1 c þ i i i1 i i i1 r2 r2 E0 1 c2i 2 2 0 r r f f 2 h 2 i h i P ðP c 1 þ P P Þ ; 1 þ c i i i1 i i i1 r2 r2 E0 1 c2i "ir;e ¼
20 2i f h c2i Pi1 c2i Pi1 Pih Pif 2 2 E0 1 ci Ei 1 ci h i 20 f h c2i Pi1 Pi1 Pih Pif ; 2 E0 1 ci
"iz;e ¼
(22c) ð22cÞ Although the equivalent eigenstrain derived above and the incompatible eigenstrain "i are stepwise continuous, discontinuities in their values occur at the interfaces between the layers, which are physically inadmissible for an FGM with continuously varying material properties. Therefore, for physically admissible results, we obtain continuous distributions of the equivalent and the incompatible eigenstrains for the non-layered coating region of the homogenized medium, as shown in Fig. 3, by spline interpolation of the stepwise continuous eigenstrains. Including this equivalent eigenstrain, the resultant stress components in the non-layered homogenized medium as shown in Fig. 3 are derived as " # Rf2 Rf2 pfn þ p;f pR 2 R2 n rh ¼ 2 1 1 r2 r2 Rf2 R 2 R R2 " f ð # ð Rf 1 r R2 1 r" dr þE0 2 r" dr þ 1 2 r R r Rf2 R 2 R ðr ð E0 1 r 1 e "r "e dr 2 r "er þ "e dr þ þ r R 2 Rr 2 R þC 1 2 ; R 4 r 4 Rf ; r
" # 2 2 f ;f 2 R R p þ p pR R2 n n f f h ¼ 2 1þ 2 1þ 2 r r Rf2 R 2 R R2 " f # ðr ð R f 1 R2 1 þE0 " þ 2 r" dr þ 1 þ 2 r" dr r R r Rf2 R 2 R ð ðr E0 1 r e 1 e e e "r "e dr þ 2" þ 2 r "r þ " dr þ r r 2 R R R2 þC 1 þ 2 ; R 4 r 4 Rf ; r ð23bÞ rh ¼
pfn Rf2
ð22aÞ (22a) 1 þ r2 1 i i i 2 f f f 2 f c ¼ P P c P P þ i i i i i1 i1 1 i r2 Ei 1 c2i 2 1 0 1 þ r 0 i 2 2 h h h h ðc P P Þ þ c P P i i1 i i i1 i 1 0 r2 E0 1 c2i 2 2 1 r f f 2 h 2 ri h i P P c 1 þ P P 1 þ c þ i i i1 i i i1 r2 r2 E0 1 c2i 2 2 0 r r f f 2 h 2 i h i ci 1 2 Pi1 Pi1 1 ci 2 Pi Pi ; r r E0 1 c2i
h ¼
r2
pfn Rf2 r2
"i;e
ð22bÞ (22b)
ð23aÞ (23a)
þ
2 p;f n Rf
r2
2 p;f n Rf
r2
; r 5 Rf ;
; r 5 Rf ;
ð23cÞ
ð23dÞ
where C¼
1 Rf2 R 2 ð R f ð Rf e 1 e e 2 e "r " dr ; r "r þ " dr Rf R R r
ð24Þ
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Fig. 3. Homogenized infinite medium containing a circular hole with a distribution of incompatible and equivalent eigenstrains in the coating region.
Fig. 4. A radial edge crack emanating from the circular hole in the homogenized infinite medium with a distribution of incompatible and equivalent eigenstrains in the coating region.
and " and "ej represent, respectively, the incompatible and the equivalent eigenstrain distributions which are continuous over the entire range of interest.
The disturbed stress field is determined by representing the crack by a continuous distribution of edge dislocations. The method of complex potential functions is used to calculate the stress field due to the edge dislocations. The complex potential functions for an edge dislocation at a point z=h in an infinite medium with a circular hole of radius R as shown in Fig. 5, are given by
3.2. Stress intensity factors The resultant stress field in the cracked homogeneous bodies can be determined by the principle of superposition. First, the stress field in the uncracked bodies is determined due to external loadings and eigenstrains. This stress field is disturbed by the presence of a crack. Second, the disturbed stress field is determined. Finally, the resultant stress field in the cracked bodies is obtained by superposing the disturbed stress field on that obtained for the uncracked bodies. Then the boundary condition along the crack surfaces can be written as
ðzÞ ¼
ðzÞ ¼ sd
þ
sh
¼ Ts ;
ð25Þ
where sh is the stress component along the prospective crack line in the uncracked homogeneous bodies, sd is the stress component of the disturbed field due to the presence of a crack, and Ts is the traction applied to the crack surface. The disturbed stress field can be determined by representing the crack by a continuous distribution of edge dislocations. The stress field in the uncracked homogenized infinite medium shown in Fig. 3 has been determined in the preceding article. Now let us consider a radial edge crack of length l emanating from the circular hole in this homogenized infinite medium as shown in Fig. 4. The surfaces of the hole and the crack are subjected to the uniform applied pressure p. In this case, the boundary condition along the crack surfaces given by Eq. (25) reduces to d ¼ h p; R 4 r 4 ðR þ lÞ; ¼ 0:
ð26Þ
0 b ð 0 þ 1Þ " # 1 1 1 R4 1 X þ ak zk ; z h z h3 z2 k¼3
ð27aÞ
0 b ð 0 þ 1 Þ " # 1 1 1 h 2R 2 1 X 0 k þ þ a z ; z h z ðz hÞ2 h z2 k¼3 k ð27bÞ
where h=R+s, b is the Burgers vector, and 0 is Kolosov’s constant which is equal to (30 )/(1+0 ) for plane stress condition. The coefficients ak and a0k are given by ak ¼ ðk 2Þ
R 2ðk1Þ R 2k ðk 1Þ kþ1 ; k 5 3; k1 h h
a0k ¼ ðk 1Þðk 4Þ
ð28aÞ
R 2ðk2Þ R 2ðk1Þ ðk 2Þ2 k1 ; k 5 3: k3 h h ð28bÞ
Now, for a continuous distribution of edge dislocations over the crack length l, the complex potentials can be written as
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A0k ¼ ðk 1Þðk 4Þ ð k 2Þ 2
R 2ðk2Þ ðR þ sÞk3
R 2ðk1Þ ðR þ sÞk1
; k 5 3:
ð32bÞ
Substitution of Eq. (31) into Eq. (26) yields ðl 2 0 1 1 R2 1 ð 0 þ 1Þ 0 t s R þ t R þ s ðR þ tÞ2 1 1 1X 1X þ ð2 kÞAk ðR þ tÞk þ A0 ðR þ tÞk bðsÞds 2 k¼3 2 k¼3 k ¼ h p; 0 4 t 4 l: Fig. 5. A discrete edge dislocation at a distance z=h measured from the center of a circular hole in an infinite medium.
ðzÞ ¼
0 ð 0 þ 1Þ
ðl " 0
# 1 1 1 R 1 X þ ak zk bðsÞds; z h z h3 z2 k¼3 4
ð33Þ Eq. (33) is the singular integral equation for the unknown density function bðsÞ, which is normalized over the interval (1,+1) by using the substitutions
ð29aÞ ðl
T¼
2t 1; l
ð34aÞ
S¼
2s 1; l
ð34bÞ
D¼
2R : l
ð34cÞ
2
0 1 1 h 2R 1 þ ð 0 þ1Þ 0 z h z ðz hÞ2 h z2 1 X þ a0k zk bðsÞds:
ðzÞ ¼
ð29bÞ
k¼3
The stresses can be expressed in terms of the complex potentials ðzÞ, ðzÞ and their complex conjugates as below [27]: d d þ ir ¼ ðzÞ þ ðzÞ þ ½z0 ðzÞ þ ðzÞ e2i ;
d ¼ ðzÞ þ ðzÞ ½z0 ðzÞ þ ðzÞ e2i : rd ir
ð30aÞ
ð30bÞ
By using Eqs. (29) and (30), we can easily obtain the circumferential stress component along the crack line (=0, z=R+t) as follows ðl
2
2 0 1 1 R 1 ð 0 þ 1 Þ 0 t s R þ t R þ s ð R þ t Þ 2 1 1 1X 1X þ ð2 kÞAk ðR þ tÞk þ A0k ðR þ tÞk bðsÞds; 2 k¼3 2 k¼3
d ¼
Finally, we obtain
2 0 ð 0 þ 1Þ ¼
ð1 1
h ðTÞ
ð35Þ
l BðSÞ ¼ b S ; 2
G1 ðT; SÞ ¼
where G2 ðT; SÞ ¼ R 2ðk1Þ ð R þ sÞ
k1
ð k 1Þ
R 2k ðR þ sÞkþ1
; k 5 3;
p; 1 4 T 4 1;
where
ð31Þ
Ak ¼ ðk 2Þ
ð36aÞ
1 DþTþ1 D2 ; ðD þ S þ 1ÞðD þ T þ 1Þ2
ð36bÞ
1 1X ð2 kÞBk ðD þ T þ 1Þk 2 k¼3
þ ð32aÞ
1 þ G1 ðT; SÞ þ G2 ðT; SÞ BðSÞdS TS
1 1X B0 ðD þ T þ 1Þk ; 2 k¼3 k
ð36cÞ
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Bk ¼ ðk 2Þ 1Þ
D 2ðk1Þ ðD þ S þ 1Þ D 2k
k1
ðD þ S þ 1Þkþ1
B0k ¼ ðk 1Þðk 4Þ
ð36dÞ
2 1 N N sin X 2 2N þ 1 ’ S : ’ðþ1Þ ¼ 2 1 2N þ 1 ¼1 tan 2N þ 1 2
ð36eÞ
The solution of Eq. (39) provides the unknowns ’ðS Þ which are used in Eq. (42) to determine the value of ’ðþ1Þ and then the stress intensity factor can be computed by using Eq. (41).
ðk
;
D 2ðk2Þ
ðD þ S þ 1Þ 2ðk1Þ D : 2Þ2 ðD þ S þ 1Þk1
k3
ðk
ð42Þ
3.3. Verification of the approximation method of SIFs 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 ’ðSÞ in the closed interval –14S 4+1. Thus we can formulate BðSÞ ¼ wðSÞ’ðSÞ:
ð37Þ
In the present case, the fundamental function can be given by [28] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ SÞ wðSÞ ¼ : ð38Þ ð1 SÞ Using the Gauss–Jacobi integral formula corresponding to the weight function in Eq. (38) in the manner developed by Erdogan et al. [29], Eq. (35) can be converted to a system of algebraic equations to determine the unknowns ’ðS Þ as follows N 2 0 X 1 ð1 þ S Þ þ G1 ðT ; S Þ þ G2 ðT ; S Þ T S ð 0 þ 1Þ ¼1 ’ðS Þ ¼
2N þ 1 h ðT Þ þ p ; ¼ 1; 2; 3; . . . ; N: 2 ð39Þ
In the preceding article, the approximation method of determining stress intensity factors for a radial edge crack in an FGM coating around a circular hole in an infinite elastic medium has been introduced. The underlying concept of the approximation method is that an FGM is first homogenized by a distribution of equivalent eigenstrain. Then the stress intensity factors are determined for a crack in the homogenized material, which represent the stress intensity factors for the same crack in the FGM. Thus, to verify the validity of the approximation method, we can reasonably choose any relatively simple case for which numerical results are available in literatures. In this study, we have considered the case of a crack of length 2l in a nonhomogeneous infinite medium. The exponential function of Young’s modulus (E/E0=exp(y)) was assumed to vary in the direction parallel to the crack line. The stress intensity factors are calculated by taking l as a variable quantity, and the results are shown in Fig. 6 and Table 1. The solid lines in Fig. 6 represent the results obtained by Delale and Erdogan [5], and the dotted lines represent the results obtained by the present model. It is noted that the results obtained by the present approximation method coincide with the exact results when l=0 and
The integration and the collocation points are given by [28] 2 1 ; ¼ 1; 2; 3; . . . ; N; ð40aÞ S ¼ cos 2N þ 1
2 T ¼ cos ; ¼ 1; 2; 3; . . . ; N: 2N þ 1
ð40bÞ
It can be readily shown that the expression for the stress intensity factor can be derived as KI ¼
pffiffiffiffiffiffiffi 2 l
2 0 ’ðþ1Þ; ð 0 þ 1 Þ
ð41Þ
where ’ðþ1Þ is computed by Krenk’s interpolation formula [30] given by
Fig. 6. Normalized stress intensity factors for a crack in a nonhomogeneous infinite medium having an exponentially varying Young’s modulus E=E0exp(y).
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Table 1 Comparison of normalized stress intensity factors for a crack in a nonhomogeneous infinite medium with an exponentially varying Young’s modulus E=E0exp(y) l
0 0.1 0.2 0.3 0.4 0.5
Exact results by Delale et al.
Approximate results by present method
KI ðp lÞffiffiffi "0 E0 l
KI ðl Þ pffiffiffi "0 E0 l
KI ðp lÞffiffiffi "0 E0 l
KI ðp -lÞffiffiffi "0 E0 l
KI ðp lÞffiffiffi "0 E0 l
KI ð-lÞ pffiffiffiffiffi "0 E0 l
1.000 1.078 1.158 1.245 1.337 1.435
1.000 0.925 0.858 0.793 0.732 0.665
1.000 1.053 1.111 1.174 1.244 1.321
1.000 0.952 0.909 0.871 0.836 0.806
0 0.025 0.047 0.071 0.093 0.114
0 0.027 0.051 0.078 0.104 0.141
deviate from them as the variable l increases. Thus the value of l characterizing the gradient exp(l) of the Young’s modulus E/E0 at the crack tip (y=l) should be small for the errors to remain within acceptable limits. The two points regarding the gradient should be noted in connection with the small value of l: (i) the product of a finite value of and a small value of the crack length l can give a small value of l. However, the gradient exp(l) of the Young’s modulus may be large in this case depending on the value of . This implies that the error is acceptable small for a small crack length even if the gradient at the crack tip is large. (ii) The product of a small value of and a finite value of the crack length l can give a small value of l. In this case, the gradient exp(l) is small. This implies that the error is acceptable small for a large crack length if the gradient is small at the crack tip. From this discussion, we can claim that our proposed approximation method can be used for FGMs having the gradient of Young’s modulus discussed above. Therefore, it is necessary to check the gradient of the Young’s modulus in order to justify the validity of the design by this method.
Difference
the formula of the stress intensity factor for a crack in homogeneous isotropic materials, which is called the apparent fracture toughness of FGMs [31]. Now we consider a radial edge crack of length l in the FGM coating around the circular hole in the infinite elastic medium as shown in Fig. 1. When the uniform applied pressure at which the fracture occurs from the crack tip is pc , the apparent fracture toughness of the FGM coating can be given by the formula of the stress intensity factor for a radial edge crack of length l emanating from a circular hole in a homogeneous isotropic infinite material as pffiffiffiffiffi KIca ¼ FI pc l: ð43Þ In Eq. (43), FI is a function of the geometric parameter l/R, whose values are available in the literature [32] and shown in Fig. 7 for convenience. The critical pressure pc is given by p that satisfies the condition KI ¼ Kc ;
ð44Þ
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 an FGM 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 the brittle materials, fracture 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 an FGM, let us imagine the FGM with the same geometric configuration under the external applied stress which corresponds to its fracture stress. Then, we can evaluate the critical value of the stress intensity factor through
Fig. 7. Non-dimensional stress intensity factors FI for a radial edge crack at a circular hole in an infinite medium. The hole and the crack surfaces are under a uniform pressure.
A.M. Afsar, H. Sekine / Composites Science and Technology 62 (2002) 1063–1077
where Kc is the local intrinsic fracture toughness of the FGM coating, which is given by [33] Kc ¼
E 0 K : E0 c
ð45Þ
interpolation. The minimum value of the objective function Fobj ðV1A ; V2A ; . . . ; VnA Þ obtained by the ADS program is compared with a small positive quantity " to satisfy the condition Fobj ðV1A ; V2A ; . . . ; VnA Þ 4 ";
Kc0
In Eq. (45), is the intrinsic fracture toughness of the homogeneous material B. Eq. (45) slightly overestimates the intrinsic fracture toughness of the FGM when the volume content of the constituent A is near to unity. The maximum error for VA=1 is 3.8% and it decreases as the value of VA decreases. Thus the use of Eq. (45) in determining the intrinsic fracture toughness of the FGM is quite justifiable.
5. Inverse problems The inverse problem of calculating material distributions intending to realize prescribed apparent fracture toughness in the FGM coating is solved by using the formulations developed for the approximation method of stress intensity factors. Suppose that a profile of apparent fracture toughness KIca is prescribed over a region of radial length L measured from the hole surface to the middle point of the nLth layer of the FGM coating which is divided into n number of layers of infinitesimal thickness as shown in Fig. 2. Note that the infinitesimal thickness of the layers in Fig. 2 can be given by l=(Rf – R)/n. Now assume a radial edge crack in the FGM coating around the hole. The crack length li is varied by taking the crack length as li=l/2 + (i–1)l, where i=1,2,. . .,nL, to ensure that the tip of the crack is located at the middle point of a layer. Considering the volume fractions ViA (i=1,2,. . .,n) of the constituent A in each layer of the infinitesimal thickness in Fig. 2 as design variables, the material distribution, i.e. the values of ViA (i=1,2,. . .,n) can be evaluated by solving the optimization problem set up as Minimize
n
:Fobj V1A ; V2A ; . . . ; VA
Subject to
:0 4 ViA
¼
nL P i¼1
KIi
i 2
Kc ;
4 1; i ¼ 1; 2; . . . ; n; ð46Þ
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 coating. In determining KIi by using Eqs. (39)–(42), p is replaced by pc obtained from Eq. (43). The optimization problem in Eq. (46) is solved by using a numerical optimization program ADS [34] 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 first finding bounds and then using polynomial
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ð47Þ
and the corresponding set of the design variables ViA ði ¼ 1; 2; . . . ; nÞ is taken as the solution of the optimization problem. Then the continuous profile of the material distribution is obtained by spline interpolation of the design variables ViA ði ¼ 1; 2; . . . ; nÞ. In order to solve the optimization problem in Eq. (46), it is necessary to determine the material properties of the FGM coating. The material properties of the FGM coating are determined by using the mixture rule [35] according to which the shear modulus of elasticity
and the bulk modulus K are first determined from the relations VA
KA K KB K þ VB ¼ 0; 3KA þ 4
3KB þ 4
ð48aÞ
VA
A
B
þ VB ¼ 0;
A þ Y
B þ Y
ð48bÞ
Y¼
ð9K þ 8 Þ ; ð6K þ 12 Þ
ð48cÞ
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 non-subscripted variables are used to denote the effective properties of the FGM. Then the Young’s modulus E and the Poisson’s ratio are calculated by using the expressions E¼
9K
; ð3K þ Þ
ð49aÞ
¼
E 1: 2
ð49bÞ
The coefficient of thermal expansion is determined by using the relation ¼ V A A
KA ð3K þ 4 Þ KB ð3K þ 4 Þ þ V B B : Kð3KA þ 4 Þ Kð3KB þ 4 Þ
ð50Þ
6. Numerical results and discussion The solution method of the inverse problem of calculating material distributions to realize prescribed apparent fracture toughness has been developed for the general case of constituent materials A and B which can
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be any real materials in practice. In particular, these can be two different ceramics or a ceramic and a metal depending on an application. In this study, some numerical results are provided for an example of a TiC/ Al2O3 FGM coating in which the constituents TiC and Al2O3 correspond to the materials A and B, respectively. The mechanical and the thermal properties of TiC and Al2O3 are shown in Table 2. The characteristic dimensions R and Rf are taken as 10 and 11 mm, respectively, while the difference between the sintering and room temperatures T is taken as 1000 C. In numerical calculations, the number of layers of infinitesimal thickness shown in Fig. 2 is chosen as 50, and the value of " in condition (47) is taken as 0.1. 6.1. Apparent fracture toughness from prescribed material distributions The apparent fracture toughness in the TiC/Al2O3 FGM coating around a circular hole in an infinite elastic medium are predicted for three different prescribed profiles of the material distributions as shown in Fig. 8. This figure shows the distribution of the volume fraction VA of TiC along the radial distance r. The curves I and III represent two different parabolic distributions while the curve II shows a linear distribution. The corresponding fracture characteristics are plotted in Fig. 9, which displays the apparent fracture toughness KIca Table 2 Material properties of TiC and Al2O3 Material
Young’s modulus (GPa)
Shear modulus (GPa)
Poisson’s ratio
CTE (/ C)
KIc (MPa m1/2)
TiC Al2O3
462 380
194.12 150.79
0.19 0.26
7.4 10–6 8.0 10–6
4.1 3.5
Fig. 8. Prescribed material distribution profiles in a TiC/Al2O3 FGM coating around a circular hole in an infinite elastic medium.
versus radial distance r. Fig. 9 reveals that the apparent fracture toughness depends significantly on the material distribution profiles, particularly the higher volume fraction VA provides the higher apparent fracture toughness. It is also noted that all the distribution profiles shown in Fig. 8 offer much higher apparent fracture toughness KIca compared to the intrinsic fracture toughness of the constituent materials. Specifically, the parabolic distribution shown by the curve I in Fig. 8 gives a maximum apparent fracture toughness of about 12 MPa m1/2 at r=10.64 mm. The intrinsic fracture toughness of the single phase Al2O3 is 3.5 MPa m1/2. However, it is observed from Fig. 9 that the apparent fracture toughness of the homogeneous region (r > 11 mm) of Al2O3 is considerably higher than its intrinsic fracture toughness. This is because the homogeneous region of Al2O3 adjacent to the FGM coating is still under the influence of the incompatible eigenstrain in the FGM coating. Thus, it is realized that the improved fracture characteristics can be maintained up to the distance much greater than the FGM coating thickness. However, at a further distance not shown in the figure, it can be predicted that the apparent fracture toughness will be identical to the intrinsic fracture toughness of Al2O3. It is noted from Fig. 9 that there is a large difference (up to 35%) between the apparent fracture toughness of examples I and III even if the difference between the material properties of the constituents shown in Table 2 is small. This is because the apparent fracture toughness is directly proportional to the critical pressure pc at which a crack starts to extend. This critical pressure pc depends not only on the material properties but also on the internal stress induced in the material due to the incompatible eigenstrain. Again, the internal stress depends on the material distribution profile as well as the difference between the sintering and room temperatures.
Fig. 9. Apparent fracture toughness in a TiC/Al2O3 FGM coating around a circular hole in an infinite elastic medium.
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The inverse calculation of material distributions in the TiC/Al2O3 FGM coating around a circular hole in an infinite elastic medium is carried out to realize prescribed apparent fracture toughness in the coating, which is higher than the intrinsic fracture toughness of the constituents shown in Table 2. The higher apparent fracture toughness can be controlled in a manner that can meet a requirement of an application. Among different possible manners of controlling the higher apparent fracture toughness, two examples are considered in this study as shown by the solid portions of the curves I and II in Fig. 10. This figure shows the prescribed apparent fracture toughness KIca as a function of the radial distance r measured from the center of the hole. The broken line represents the apparent fracture toughness obtained for 100% volume fraction VA of TiC, which represents a rough estimation of the upper limit of the apparent fracture toughness. In the example I, the apparent fracture toughness is controlled such that it increases linearly from 8 MPa m1/2 at the surface of the hole to 10 MPa m1/2 at r=10.75 mm. In the
example II, it remains a constant value of 7 MPa m1/2 over the same range as example I. For these prescribed apparent fracture toughness (solid portions of the curve I and II), the material distributions in the FGM coating (10–11 mm) are computed and plotted in Fig. 11. This figure shows the variation of the volume fraction VA of TiC along the radial distance r. It is seen from Fig. 11 that the values of VA are higher over the range from 10.0 to 10.77 mm in the case of the example I. The fabrication of the FGMs with the material distributions shown in Fig. 11 may be difficult by using a single manufacturing process like chemical vapor deposition, plasma spray, powder metallurgy and physical vapor deposition. However, by a suitable combination of these processes, it may be possible to fabricate these FGMs according to the material distributions of Fig. 11. It is also important to know the characteristics of the material outside the controlled region corresponding to the computed material distributions shown in Fig. 11. The dotted portions of the curves I and II as shown in Fig. 10 represent the characteristics of the material, outside the controlled region, that must be appeared if the apparent fracture toughness is controlled in the manner shown by the solid portions. In the case of the example I, it is seen from Fig. 10 that the apparent fracture toughness KIca decreases sharply after controlling and this decreasing trend is retained over rest of the FGM coating (10.75–11 mm) while, in the case of the example II, it remains almost constant over the same region. However, in both the cases, another sharp decrease in the values of the apparent fracture toughness KIca occurs as the homogeneous region (r=11 mm) is approached and the decreasing trend continues for both the cases as r increases. The prescribed apparent fracture toughness, considered in both the examples I and II as shown in Fig. 10
Fig. 10. Prescribed apparent fracture toughness in a TiC/Al2O3 FGM coating around a circular hole in an infinite elastic medium.
Fig. 11. Material distribution profiles in a TiC/Al2O3 FGM coating around a circular hole in an infinite elastic medium.
For the constituent materials shown in Table 2 and T=1000 C, the rough estimation reveals that the internal stress may vary from 0–277 MPa depending on the material distribution profile of the constituents. This large range of variation (0–277 MPa) occurs for the small range of variation in the material properties of the constituents. From this fact, it is realized that a small change in the material properties causes a large change in the internal stress that affects the critical pressure pc which ultimately causes large change in the apparent fracture toughness. 6.2. Calculation of material distribution profiles intending to realize prescribed apparent fracture toughness
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Fig. 12. Variation of normalized Young’s modulus in a TiC/Al2O3 FGM coating around a circular hole in an infinite elastic medium.
is realized by designing the FGM coating having the material distributions shown in Fig. 11. From this fact it can be concluded that an apparent fracture toughness in FGM coatings around a circular hole in an infinite elastic medium can be controlled within possible limits by choosing an appropriate material distribution in the coatings. Fig. 12 displays the variation of normalized Young’s modulus E/E0 corresponding to the material distribution profiles of Fig. 11. The gradients of the profiles of these normalized Young’s moduli are large near the surface of the hole and small over rest of the region (except at the distance of about 10.75 mm in the case of example I). In Section 3.3, it has been discussed that the errors in the stress intensity factors are acceptable small even if the gradient is large at the distance corresponding to a small crack length (near the surface of the hole). The profile shown by the curve I has also a large gradient at the distance of about 10.75 mm. Near this point, the prescribed apparent fracture toughness shown in Fig. 10 corresponding to the material distribution of Fig. 11 may have a large error. As a remedy for such a case, it is recommended that the apparent fracture toughness should be prescribed over the region greater than the actual region in which the apparent fracture toughness is intended to control.
7. Conclusions A solution method of the inverse problem has been developed for calculating material distributions to realize prescribed apparent fracture toughness in an FGM coating around a circular hole in an infinite elastic medium. The incompatible eigenstrain induced in the FGM coating due to mismatch in the coefficients of thermal expansion has been taken into consideration.
An approximation method of determining stress intensity factors is introduced for a crack in the FGM coating with any arbitrary variation of material properties. The FGM coating is homogenized simulating its nonhomogeneous material properties by a distribution of equivalent eigenstrain, which is determined from the condition of identical elastic fields in the uncracked FGM and homogenized coatings. A radial edge crack emanating from the circular hole in the homogenized coating is considered for the case of a uniform pressure applied to the surfaces of the hole 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 determined for the crack in the homogenized coating represent the approximate values of the stress intensity factors for the same crack in the FGM coating, and are used in the inverse problem of calculating material distributions in the FGM coating to realize prescribed apparent fracture toughness in the coating. Numerical results obtained for a TiC/Al2O3 FGM coating reveal that the apparent fracture toughness in the FGM coating significantly depends on the material distributions. It is also revealed that the apparent fracture toughness in the FGM coating can be controlled within possible limits by choosing an appropriate material distribution profile in the coating.
Acknowledgements The authors would like to acknowledge the partial support of the Grant-in-Aid for Scientific 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 financial assistance through the Japanese Government Scholarship (No. 942062) from the Ministry of Education, Science, Sports and Culture of Japan. References [1] Yamanouchi M, Koizumi M, Hirai T, Shiota I. Proceedings of the First International Symposium on Functionally Gradient Materials, Sendai, Japan, 1990. [2] Holt JB, Koizumi M, Hirai T, Munir ZA. Ceramic transaction: functionally gradient materials, vol. 34. Westerville, (OH): The American Ceramic Society; 1993. [3] Ilschner B, Cherradi N. Proceedings of the Third International Symposium on Structural and Functional Gradient Materials. Lausanne, Switzerland: Presses Polytechniques et Universitaires Romands; 1994. [4] Atkinson C, List RD. Steady state crack propagation into media with spatially varying elastic properties. International Journal of Engineering Science 1978;16:717–30. [5] Delale F, Erdogan F, The. crack problem for a nonhomogeneous plane. ASME Transaction. Journal of Applied Mechanics 1983; 50:609–14.
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Inverse problems of material distributions for prescribed apparent fracture toughness in FGM coatings around a circular hole in infinite elastic media A.M. Afsara,*, H. Sekineb a
Department of Mechanical Engineering, Bangladesh University of Engineering and Technology (BUET), Dhaka 1000, Bangladesh b Department of Aeronautics and Space Engineering, Tohoku University, Aoba-yama 01, Aoba-ku, Sendai 980-8579, Japan Received 24 October 2000; received in revised form 15 February 2002; accepted 20 February 2002
Abstract This study is concerned with the inverse problem of calculating material distributions intending to realize prescribed apparent fracture toughness in functionally graded material (FGM) coatings around a circular hole in infinite elastic media. The incompatible eigenstrain induced in the FGM coatings after cooling from the sintering temperature, due to mismatch in the coefficients of thermal expansion, is taken into consideration. An approximation method of determining stress intensity factors is introduced for a crack in the FGM coatings in which the FGM coatings are homogenized simulating the nonhomogeneous material properties by a distribution of equivalent eigenstrain. A radial edge crack emanating from the circular hole in the homogenized coatings is considered for the case of a uniform pressure applied to the surfaces of the hole and the crack. The stress intensity factors determined for the crack in the homogenized coatings represent the approximate values of the stress intensity factors for the same crack in the FGM coatings, and are used in the inverse problem of calculating material distributions in the FGM coatings intending to realize prescribed apparent fracture toughness in the coatings. Numerical results are obtained for a TiC/Al2O3 FGM coating, which reveal that the apparent fracture toughness in FGM coatings around a circular hole in infinite elastic media can be controlled within possible limits by choosing an appropriate material distribution profile in the coatings. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: B. Fracture toughness; C. Crack; Functionally graded material
1. Introduction Functionally graded materials (FGMs) are nonhomogeneous solids, which consist of two or more distinct material phases, such as different ceramics or ceramics and metals, and are the mixture of them such that the composition of each changes continuously with space variables. The change in composition induces material and microstructural gradients, and makes the functionally graded materials different in behavior from homogeneous materials and conventional composite materials [1,2]. These materials are tailorable in their properties via the design of the gradients, which, in turn, depend on material distributions. From a mechanics viewpoint the main advantages of material property * Corresponding author. Tel.: +880-2-966-5636; fax: +880-2-8613046. E-mail address: [email protected] (A.M Afsar).
grading appear to be improved bonding strength, toughness, wear and corrosion resistance, and reduced residual and thermal stresses. Some typical applications include thermal barrier coatings of high temperature components in gas turbines, surface hardening for tribological protection and graded interlayers used in multilayered microelectronic and optoelectronic components [1–3]. Although FGMs have outstanding advantages, the applications of these materials in various branches of engineering and technologies necessitate identifying the probable failure modes and designing them against those failures. In designing with FGMs, an important aspect of the problem is the mechanical failure, specifically the fracture failure. Although the absence of discontinuous interfaces in FGMs does largely reduce material property mismatch, cracks may occur when they are subjected to external loadings [1,2]. Very often the process begins with the formation of microcracks at
0266-3538/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(02)00049-0
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locations of corrosion pits, surface flaws, or severe stress concentrations. Generally a number of microcracks coalesce and form a local dominant crack, which would then propagate subcritically under cyclic or sustained loading. The loads or stresses acting on the medium may be mechanically or thermally induced. There are also uncertainties arising from voids and defects that are introduced in FGMs during manufacturing. Even a small quantity of mechanical imperfections can cause a marked influence on their fracture strength. Therefore, the study of the fracture mechanics of these materials appears to be an utmost necessary to understand, quantify and improve their toughness. Over the past few years, there have been a number of works, both analytical and experimental, to study the responses of FGMs to mechanical and thermal loads for various geometries in various fracture mechanisms. In these works, it is reported that the nonhomogeneous material properties of FGMs complicate the analytical studies of crack problems of these materials. Therefore, to overcome the difficulties and obtain a tractable problem, it is often conventional to regard the material properties to be some certain assumed functions of space variables. The most common functions assumed to model the material properties are exponential and power functions. By assuming exponential functions for material properties, crack problems of nonhomogeneous materials have been considered in Refs. [4–10] for mechanical loadings and in Refs. [11–15] for thermal loadings. Power function variations of material properties are considered in Refs. [16–22] to study the crack problems of nonhomogeneous materials. In these studies, various aspects of crack problems of FGMs have been analyzed for various geometries and loading conditions. However, these studies are concerned only with the direct problems in which the fracture characteristics of FGMs can be analyzed only for certain assumed functions for the material properties, e.g. exponential and power functions. An important aspect of FGMs still remaining to be dealt with is the inverse problem in which desired characteristics of FGMs under mechanical and thermal loadings can be prescribed and the corresponding material distributions via the material properties can be obtained by inverse calculations. This approach appears to be quite effective and efficient tool in designing FGMs for obtaining a desired behavior to suit an application. Obviously, for an inverse problem, some specific functions for the material property distributions cannot be assumed as Zuiker [23] pointed out that these assumed property distributions are not sometimes physically realizable for some material distribution profiles predicted by the inverse problem to achieve desired characteristics in FGMs. Therefore, to solve the inverse problem it is necessary to develop a method that can treat any arbitrary variation of material properties.
Furthermore, incompatible eigenstrains [24] induced in FGMs due to mismatch in the coefficients of thermal expansion when they are cooled from the sintering temperature should also be considered. Recently, Sekine and Afsar [25] introduced an approximation method of determining stress intensity factors for a particular case of an edge crack in semi-infinite FGMs with incompatible eigenstrains and arbitrary variation of material properties. The method was applied to the inverse problem of calculating material distributions to attain prescribed apparent fracture toughness corresponding to improved brittle fracture characteristics. Later another model [26] of the semi-infinite FGMs was considered to investigate the effects of periodic edge cracks on the brittle fracture characteristics. In the present study, the approximation method of determining the stress intensity factors is introduced for a crack in FGM coatings around a circular hole in infinite elastic media in which the FGM coatings are homogenized simulating their nonhomogeneous material properties by a distribution of equivalent eigenstrain. A radial edge crack emanating from the circular hole in the homogenized coatings is considered for the case that the surfaces of the hole and the crack are subjected to a uniform pressure. The stress intensity factors determined for the crack in the homogenized coatings represent the approximate values of the stress factors for the same crack in the FGM coatings, and are used in the inverse calculation of material distributions in the FGM coatings around the circular hole intending to realize prescribed apparent fracture toughness in the FGM coatings.
2. Modeling of the problem Consider an FGM coating around a circular hole of radius R in an infinite elastic medium as shown in Fig. 1, which is referred to the Cartesian coordinate system
Fig. 1. Analytical model of an FGM coating around a circular hole in an infinite elastic medium.
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(x, y) and the polar coordinate system (r, ) having the same origin located at the center of the circular hole. The constituent materials of the FGM coating are denoted by A and B, and their volume fractions VA and VB are assumed to vary in the radial direction only. Thus, the region R 4 r 4 Rf has a gradation of the properties while the infinite region r > Rf is homogeneous consisting of the material B only. The continuously and arbitrarily varying Young’s modulus, Poisson’s ratio and the coefficient of thermal expansion of the FGM coating are represented by E, and , respectively while the corresponding properties of the homogeneous region are, respectively, denoted by E0 , 0 and 0 . When such a system is cooled from the sintering temperature, an incompatible eigenstrain [24], which is a nonelastic misfit thermal strain in the present case, is induced in the FGM coating due to mismatch in the coefficients of thermal expansion. It is assumed that the material is isotropic and the microstructure is neglected for which all the shear components of the incompatible eigenstrain vanish and the normal components become equal. The normal component of the incompatible eigenstrain " ðrÞ, which is a function of r only, can be defined by " ðrÞ ¼ ð0 ÞT;
ð1Þ
where T is the difference between the sintering and room temperatures. Eq. (1) is written based on the basic assumption that the material is freely contracted without any constraint due to the change in temperature. For this model of the problem, we carry out inverse calculations of material distributions in the FGM coating for plane stress condition intending to realize prescribed apparent fracture toughness considering a radial edge crack emanating from the circular hole and considering that the surfaces of the hole and the crack are subjected to a uniform pressure p.
3. Approximation method of stress intensity factors for a crack in FGM coatings Functionally graded materials are nonhomogeneous solids and, therefore, their nonhomogeneities have to be considered in studying the fracture behavior of these materials. The consideration of these nonhomogeneities complicates the analytical studies due to mathematical difficulties. Thus it is often conventional to regard the material properties to be some certain assumed functions of space variables, for instance, exponential and power functions, in order to simplify the problems. However, in designing with FGMs, i.e. in the inverse problems, in which material distribution profiles have to be determined to achieve desired fracture characteristics, special functional forms of the properties cannot
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be assumed. Since these assumed functional forms of the properties may not be physically realizable for some material distribution profiles obtained by inverse calculations. Therefore, as an alternate approach, an approximation method is introduced in this study to calculate the stress intensity factors for a crack in FGM coatings, which is not restricted to any specific property distributions, but can treat any arbitrary distributions of the properties. The concept of the approximation method is explained below. First, the FGM coatings 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 fields are identical in both the FGM and the homogenized coatings under the same loading conditions. After determining the distribution of the equivalent eigenstrain, a method is formulated to calculate the stress intensity factors for a crack in the homogenized coatings. Since the equivalent eigenstrain is determined from the condition of identical elastic fields in the uncracked FGM and homogenized coatings, the redistributed elastic field in the cracked homogenized coatings cannot exactly represent the redistributed elastic field in the cracked FGM coatings. Therefore, the stress intensity factors calculated for a crack in the homogenized coatings with the equivalent eigenstrain represent the approximate values of the stress intensity factors for the same crack in the corresponding FGM coatings and hence the term approximation method has been used. 3.1. Equivalent eigenstrain The FGM coating around the circular hole shown in Fig. 1 is homogenized simulating the nonhomogeneous material properties by a distribution of equivalent eigenstrain. As stated before, the equivalent eigenstrain is determined from the condition of identical elastic fields in the uncracked FGM and homogenized coatings subjected to the same loading condition. We first consider the FGM coating subjected to a uniform applied pressure p around the circular hole. To determine the elastic field in the FGM coating, a special technique is adopted in which the FGM coating is radially divided into layers of infinitesimal thickness as shown in Fig. 2, which exhibits one half of the elastic medium. Each layer is assumed to have constant material properties but different from those in the other layers. The inner and the outer radii of the ith layer are, respectively, denoted by ri1 and ri, where r0=R and rn=Rf. The region r > Rf is homogeneous. The pressures at the inner and the outer surfaces of the ith layer are, respectively, P fi1 and P fi which are the resultant of the pressures due to the uniform applied pressure p and the incompatible eigenstrain " . For this layered FGM
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1 i h 2 f ðc P Pif Þ Ei ð1 c2i Þ i i1 i 1 þ i r2i 2 f ci ðPi1 Pif Þ þ "i ; þ 2 1 i r
"i; f ¼
h i 2i f 2 f P P c i þ "i ; Ei ð1 c2i Þ i i1
ð4cÞ
ð1 i Þri 2 f r 1 þ i ri ¼ þ cP Ei ð1 c2i Þ i i1 ri 1 i r r 1 þ i ri 2 c þ þ r"i : Pif ri 1 i r i
ð5Þ
"iz; f ¼
u fi
ð4bÞ
In deriving the above equations, it is important to remind that all the normal components of the incompatible eigenstrain are equal, i.e. "i ¼ "r;i ¼ ";i ¼ "z;i , and the shear components are zero. The unknown pressures pfi and p;f are determined i from the condition that the displacements at r ¼ ri are identical for the ith and the (i+1)th layers, which gives Fig. 2. Layering of the FGM coating around a circular hole in an infinite elastic medium.
fi;i1 ri1 pfi1 þ fi;i ri pfi þ fi;iþ1 riþ1 pfiþ1 ¼ 0; ð6aÞ i ¼ 1; 2; . . . ; n 1;
coating, it can be easily shown that the resultant stress components in the ith layer, in axisymmetric case and plane stress condition, are r;i f ¼
f 2 c2i Pi1 r2i Pif 2 ri 1 1 c i 2 ; r2 r 1 c2i 1 c2i
ð2aÞ
i ; f ¼
f 2 c2i Pi1 r2i Pif 2 ri 1 þ 1 þ c i 2 ; r2 r 1 c2i 1 c2i
ð2bÞ fi;i1 ¼
ri1 ; ri
ð3aÞ
Pif ¼ pfi þ p;f i :
ð3bÞ
At the right hand side of Eq. (3b), the first term appears due to the uniform applied pressure p while the second term appears due to the incompatible eigenstrain " . The strain and the displacement components in the ith layer are derived as
1 i h 2 f ðc P Pif Þ Ei ð1 c2i Þ i i1 i 1 þ i r2i 2 f ci ðPi1 Pif Þ þ "i ; 2 1 i r
¼ ri ð"iþ1 "i Þ; i ¼ 1; 2; . . . ; n 1;
ð6bÞ
where
where ci ¼
f ;f f ;f fi;i1 ri1 p;f i1 þ i;i ri pi þ i;iþ1 riþ1 piþ1
2ci ; Ei ð1 c2i Þ
" # 2 2 1 þ c 1 1 þ c 1 iþ1 i i þ iþ1 ; fi;i ¼ Ei 1 c2i Eiþ1 1 c2iþ1 fi;iþ1 ¼
2ciþ1 ; Eiþ1 ð1 c2iþ1 Þ
ð7aÞ
ð7bÞ
ð7cÞ
and pf0 ¼ p;
ð8aÞ
p;f 0 ¼ 0:
ð8bÞ
The stress and displacement components in the region r > Rf are derived as
"ir; f ¼
ð4aÞ r;f ¼
Rf2 Pnf r2
; r > Rf ;
ð9aÞ
A.M. Afsar, H. Sekine / Composites Science and Technology 62 (2002) 1063–1077
;f ¼ f
u ¼
Rf2 Pnf r2
Rf2 Pnf 2 0 r
; r > Rf ;
; r > Rf :
pfn ¼
2 h
20 ci Pi1 Pih þ "i ; 2 E0 ð1 ci Þ
ð13cÞ
ð1 0 Þri 2 h r 1 þ 0 ri P þ c E0 ð1 c2i Þ i i1 ri 1 0 r r 1 þ 0 ri 2 ci þ þ r"i ; Pih ri 1 0 r
ð14Þ
ð9bÞ
"iz;h0 ¼
ð10Þ
uhi 0 ¼
To solve Eq. (6), it is necessary to know the pressures pfn and p;f n at the outer surface of the nth layer. These are determined from the condition that the displacements ufn of the nth layer and uf of the homogeneous region are equal at r ¼ Rf . From this condition, we obtain 2E0 c2n pfn1 ; ð1 þ 0 ÞEn ð1 c2n Þ þ E0 fð1 n Þ þ ð1 þ n Þc2n g
1067
where Pih ¼ phi þ p;h i :
ð15Þ
The unknown pressures phi and p;h are, respectively, i determined from the following equations:
ð11aÞ p;f n
E0 En "n ð1 c2n Þ þ 2E0 c2n p;f n1 : ¼ ð1 þ 0 ÞEn ð1 c2n Þ þ E0 fð1 n Þ þ ð1 þ n Þc2n g
hi;i1 ri1 phi1 þ hi;i ri phi þ hi;iþ1 riþ1 phiþ1 ¼ 0; ð16aÞ i ¼ 1; 2; . . . ; n 1;
ð11bÞ Now we consider the homogenized infinite medium with the same geometry as Fig. 1 and determine the elastic field in the corresponding coating region following the same procedure as the FGM coating. Note that the coating region has the same distribution of the incompatible eigenstrain defined by Eq. (1). In this case, the pressure at the outer surface of the nth layer are taken same as that of the FGM coating so as to achieve the same elastic field in the region r > Rf . The equivalency in the elastic fields of the region R 4 r 4 Rf is achieved by assuming a distribution of equivalent eigenstrain. First, we determine the elastic field in the ith layer of the coating region due to the uniform applied pressure p and the incompatible eigenstrain " as
i r;h ¼ 0
i ¼ ;h 0
"ir;h0
ð12aÞ
h 2 c2i Pi1 r2i Pih 2 ri 1 þ 1 þ c i 2 ; r2 r 1 c2i 1 c2i
ð12bÞ
1 0 ¼ E0 ð1 c2i Þ
"i;h0
h 2 c2i Pi1 r2i Pih 2 ri 1 1 c i 2 ; r2 r 1 c2i 1 c2i
1þ 1
h Pih Þ ðc2i Pi1
0 r2i 2 h c ðP 0 r2 i i1
ð13aÞ
Pih Þ þ "i ;
1 0 ¼ ðc2 P h Pih Þ E0 ð1 c2i Þ i i1 1 þ 0 r2i 2 h h c ðP Pi Þ þ "i ; þ 1 0 r2 i i1
;h ;h h h hi;i1 ri1 p;h i1 þ i;i ri pi þ i;iþ1 riþ1 piþ1
¼ ri ð"iþ1 "i Þ; i ¼ 1; 2; . . . ; n 1;
ð16bÞ
where hi;i1 ¼
hi;i
2ci ; E0 ð1 c2i Þ
" # 1 1 þ c2i 1 þ c2iþ1 ¼ þ ; E0 1 c2i 1 c2iþ1
hi;iþ1 ¼
2ciþ1 ; E0 ð1 c2iþ1 Þ
ð17aÞ
ð17bÞ
ð17cÞ
and ph0 ¼ p;
ð18aÞ
phn ¼ pfn ;
ð18bÞ
p;h 0 ¼ 0;
ð18cÞ
;f p;h n ¼ pn :
ð18dÞ
Now we assume a distribution of equivalent eigenstrain "ij;e in the ith layer of the coating region, where j=r, and z. From the equivalency of the stress fields in the coating regions, we can write ð13bÞ i i i r;f ¼ r;h þ r;e ; 0
ð19aÞ
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i i i ;f ¼ ;h þ ;e ; 0
ð19bÞ
i i where r;e and ;e are, respectively, the radial and the circumferential stress components in the ith layer of the coating region of the homogenized medium due to the equivalent eigenstrain "ij;e . From the equivalency of the total strains, we can write
"ir;f ¼ "ir;h0 þ eir;e þ "ir;e ;
ð20aÞ
"i;f ¼ "i;h0 þ ei;e þ "i;e ;
ð20bÞ
"iz;f ¼ "iz;h0 þ eiz;e þ "iz;e ;
ð20cÞ
where eij;e is the elastic strain associated with the equivalent eigenstrain "ij;e in the ith layer of the coating region of the homogenized medium. i The elastic strain eij;e is related to the stress j;e by Hooke’s law as follows eir;e ¼ ei;e
1 i i r;e 0 ;e ; E0
ð21aÞ
1 i i ¼ ;e 0 r;e ; E0
ð21bÞ
0 i i r;e þ ;e : E0
ð21cÞ
eiz;e ¼
Combining Eqs. (2), (4), (12), (13) and (19)–(21), the expressions for the components of the equivalent eigenstrain in the ith layer can be derived as 1 þ r2 1 i i i 2 f f f 2 f c P P c P P i i i i1 1 i r2 i i1 Ei 1 c2i 2 2 h 1 0 1 þ r 0 i 2 ci Pi1 Pih c P h Pih 1 0 r2 i i1 E0 1 c2i 2 1 r2i h f f 2 2 ri h c P P 1 P P 1 c þ i i i1 i i i1 r2 r2 E0 1 c2i 2 2 0 r r f f 2 h 2 i h i P ðP c 1 þ P P Þ ; 1 þ c i i i1 i i i1 r2 r2 E0 1 c2i "ir;e ¼
20 2i f h c2i Pi1 c2i Pi1 Pih Pif 2 2 E0 1 ci Ei 1 ci h i 20 f h c2i Pi1 Pi1 Pih Pif ; 2 E0 1 ci
"iz;e ¼
(22c) ð22cÞ Although the equivalent eigenstrain derived above and the incompatible eigenstrain "i are stepwise continuous, discontinuities in their values occur at the interfaces between the layers, which are physically inadmissible for an FGM with continuously varying material properties. Therefore, for physically admissible results, we obtain continuous distributions of the equivalent and the incompatible eigenstrains for the non-layered coating region of the homogenized medium, as shown in Fig. 3, by spline interpolation of the stepwise continuous eigenstrains. Including this equivalent eigenstrain, the resultant stress components in the non-layered homogenized medium as shown in Fig. 3 are derived as " # Rf2 Rf2 pfn þ p;f pR 2 R2 n rh ¼ 2 1 1 r2 r2 Rf2 R 2 R R2 " f ð # ð Rf 1 r R2 1 r" dr þE0 2 r" dr þ 1 2 r R r Rf2 R 2 R ðr ð E0 1 r 1 e "r "e dr 2 r "er þ "e dr þ þ r R 2 Rr 2 R þC 1 2 ; R 4 r 4 Rf ; r
" # 2 2 f ;f 2 R R p þ p pR R2 n n f f h ¼ 2 1þ 2 1þ 2 r r Rf2 R 2 R R2 " f # ðr ð R f 1 R2 1 þE0 " þ 2 r" dr þ 1 þ 2 r" dr r R r Rf2 R 2 R ð ðr E0 1 r e 1 e e e "r "e dr þ 2" þ 2 r "r þ " dr þ r r 2 R R R2 þC 1 þ 2 ; R 4 r 4 Rf ; r ð23bÞ rh ¼
pfn Rf2
ð22aÞ (22a) 1 þ r2 1 i i i 2 f f f 2 f c ¼ P P c P P þ i i i i i1 i1 1 i r2 Ei 1 c2i 2 1 0 1 þ r 0 i 2 2 h h h h ðc P P Þ þ c P P i i1 i i i1 i 1 0 r2 E0 1 c2i 2 2 1 r f f 2 h 2 ri h i P P c 1 þ P P 1 þ c þ i i i1 i i i1 r2 r2 E0 1 c2i 2 2 0 r r f f 2 h 2 i h i ci 1 2 Pi1 Pi1 1 ci 2 Pi Pi ; r r E0 1 c2i
h ¼
r2
pfn Rf2 r2
"i;e
ð22bÞ (22b)
ð23aÞ (23a)
þ
2 p;f n Rf
r2
2 p;f n Rf
r2
; r 5 Rf ;
; r 5 Rf ;
ð23cÞ
ð23dÞ
where C¼
1 Rf2 R 2 ð R f ð Rf e 1 e e 2 e "r " dr ; r "r þ " dr Rf R R r
ð24Þ
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Fig. 3. Homogenized infinite medium containing a circular hole with a distribution of incompatible and equivalent eigenstrains in the coating region.
Fig. 4. A radial edge crack emanating from the circular hole in the homogenized infinite medium with a distribution of incompatible and equivalent eigenstrains in the coating region.
and " and "ej represent, respectively, the incompatible and the equivalent eigenstrain distributions which are continuous over the entire range of interest.
The disturbed stress field is determined by representing the crack by a continuous distribution of edge dislocations. The method of complex potential functions is used to calculate the stress field due to the edge dislocations. The complex potential functions for an edge dislocation at a point z=h in an infinite medium with a circular hole of radius R as shown in Fig. 5, are given by
3.2. Stress intensity factors The resultant stress field in the cracked homogeneous bodies can be determined by the principle of superposition. First, the stress field in the uncracked bodies is determined due to external loadings and eigenstrains. This stress field is disturbed by the presence of a crack. Second, the disturbed stress field is determined. Finally, the resultant stress field in the cracked bodies is obtained by superposing the disturbed stress field on that obtained for the uncracked bodies. Then the boundary condition along the crack surfaces can be written as
ðzÞ ¼
ðzÞ ¼ sd
þ
sh
¼ Ts ;
ð25Þ
where sh is the stress component along the prospective crack line in the uncracked homogeneous bodies, sd is the stress component of the disturbed field due to the presence of a crack, and Ts is the traction applied to the crack surface. The disturbed stress field can be determined by representing the crack by a continuous distribution of edge dislocations. The stress field in the uncracked homogenized infinite medium shown in Fig. 3 has been determined in the preceding article. Now let us consider a radial edge crack of length l emanating from the circular hole in this homogenized infinite medium as shown in Fig. 4. The surfaces of the hole and the crack are subjected to the uniform applied pressure p. In this case, the boundary condition along the crack surfaces given by Eq. (25) reduces to d ¼ h p; R 4 r 4 ðR þ lÞ; ¼ 0:
ð26Þ
0 b ð 0 þ 1Þ " # 1 1 1 R4 1 X þ ak zk ; z h z h3 z2 k¼3
ð27aÞ
0 b ð 0 þ 1 Þ " # 1 1 1 h 2R 2 1 X 0 k þ þ a z ; z h z ðz hÞ2 h z2 k¼3 k ð27bÞ
where h=R+s, b is the Burgers vector, and 0 is Kolosov’s constant which is equal to (30 )/(1+0 ) for plane stress condition. The coefficients ak and a0k are given by ak ¼ ðk 2Þ
R 2ðk1Þ R 2k ðk 1Þ kþ1 ; k 5 3; k1 h h
a0k ¼ ðk 1Þðk 4Þ
ð28aÞ
R 2ðk2Þ R 2ðk1Þ ðk 2Þ2 k1 ; k 5 3: k3 h h ð28bÞ
Now, for a continuous distribution of edge dislocations over the crack length l, the complex potentials can be written as
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A0k ¼ ðk 1Þðk 4Þ ð k 2Þ 2
R 2ðk2Þ ðR þ sÞk3
R 2ðk1Þ ðR þ sÞk1
; k 5 3:
ð32bÞ
Substitution of Eq. (31) into Eq. (26) yields ðl 2 0 1 1 R2 1 ð 0 þ 1Þ 0 t s R þ t R þ s ðR þ tÞ2 1 1 1X 1X þ ð2 kÞAk ðR þ tÞk þ A0 ðR þ tÞk bðsÞds 2 k¼3 2 k¼3 k ¼ h p; 0 4 t 4 l: Fig. 5. A discrete edge dislocation at a distance z=h measured from the center of a circular hole in an infinite medium.
ðzÞ ¼
0 ð 0 þ 1Þ
ðl " 0
# 1 1 1 R 1 X þ ak zk bðsÞds; z h z h3 z2 k¼3 4
ð33Þ Eq. (33) is the singular integral equation for the unknown density function bðsÞ, which is normalized over the interval (1,+1) by using the substitutions
ð29aÞ ðl
T¼
2t 1; l
ð34aÞ
S¼
2s 1; l
ð34bÞ
D¼
2R : l
ð34cÞ
2
0 1 1 h 2R 1 þ ð 0 þ1Þ 0 z h z ðz hÞ2 h z2 1 X þ a0k zk bðsÞds:
ðzÞ ¼
ð29bÞ
k¼3
The stresses can be expressed in terms of the complex potentials ðzÞ, ðzÞ and their complex conjugates as below [27]: d d þ ir ¼ ðzÞ þ ðzÞ þ ½z0 ðzÞ þ ðzÞ e2i ;
d ¼ ðzÞ þ ðzÞ ½z0 ðzÞ þ ðzÞ e2i : rd ir
ð30aÞ
ð30bÞ
By using Eqs. (29) and (30), we can easily obtain the circumferential stress component along the crack line (=0, z=R+t) as follows ðl
2
2 0 1 1 R 1 ð 0 þ 1 Þ 0 t s R þ t R þ s ð R þ t Þ 2 1 1 1X 1X þ ð2 kÞAk ðR þ tÞk þ A0k ðR þ tÞk bðsÞds; 2 k¼3 2 k¼3
d ¼
Finally, we obtain
2 0 ð 0 þ 1Þ ¼
ð1 1
h ðTÞ
ð35Þ
l BðSÞ ¼ b S ; 2
G1 ðT; SÞ ¼
where G2 ðT; SÞ ¼ R 2ðk1Þ ð R þ sÞ
k1
ð k 1Þ
R 2k ðR þ sÞkþ1
; k 5 3;
p; 1 4 T 4 1;
where
ð31Þ
Ak ¼ ðk 2Þ
ð36aÞ
1 DþTþ1 D2 ; ðD þ S þ 1ÞðD þ T þ 1Þ2
ð36bÞ
1 1X ð2 kÞBk ðD þ T þ 1Þk 2 k¼3
þ ð32aÞ
1 þ G1 ðT; SÞ þ G2 ðT; SÞ BðSÞdS TS
1 1X B0 ðD þ T þ 1Þk ; 2 k¼3 k
ð36cÞ
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Bk ¼ ðk 2Þ 1Þ
D 2ðk1Þ ðD þ S þ 1Þ D 2k
k1
ðD þ S þ 1Þkþ1
B0k ¼ ðk 1Þðk 4Þ
ð36dÞ
2 1 N N sin X 2 2N þ 1 ’ S : ’ðþ1Þ ¼ 2 1 2N þ 1 ¼1 tan 2N þ 1 2
ð36eÞ
The solution of Eq. (39) provides the unknowns ’ðS Þ which are used in Eq. (42) to determine the value of ’ðþ1Þ and then the stress intensity factor can be computed by using Eq. (41).
ðk
;
D 2ðk2Þ
ðD þ S þ 1Þ 2ðk1Þ D : 2Þ2 ðD þ S þ 1Þk1
k3
ðk
ð42Þ
3.3. Verification of the approximation method of SIFs 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 ’ðSÞ in the closed interval –14S 4+1. Thus we can formulate BðSÞ ¼ wðSÞ’ðSÞ:
ð37Þ
In the present case, the fundamental function can be given by [28] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ SÞ wðSÞ ¼ : ð38Þ ð1 SÞ Using the Gauss–Jacobi integral formula corresponding to the weight function in Eq. (38) in the manner developed by Erdogan et al. [29], Eq. (35) can be converted to a system of algebraic equations to determine the unknowns ’ðS Þ as follows N 2 0 X 1 ð1 þ S Þ þ G1 ðT ; S Þ þ G2 ðT ; S Þ T S ð 0 þ 1Þ ¼1 ’ðS Þ ¼
2N þ 1 h ðT Þ þ p ; ¼ 1; 2; 3; . . . ; N: 2 ð39Þ
In the preceding article, the approximation method of determining stress intensity factors for a radial edge crack in an FGM coating around a circular hole in an infinite elastic medium has been introduced. The underlying concept of the approximation method is that an FGM is first homogenized by a distribution of equivalent eigenstrain. Then the stress intensity factors are determined for a crack in the homogenized material, which represent the stress intensity factors for the same crack in the FGM. Thus, to verify the validity of the approximation method, we can reasonably choose any relatively simple case for which numerical results are available in literatures. In this study, we have considered the case of a crack of length 2l in a nonhomogeneous infinite medium. The exponential function of Young’s modulus (E/E0=exp(y)) was assumed to vary in the direction parallel to the crack line. The stress intensity factors are calculated by taking l as a variable quantity, and the results are shown in Fig. 6 and Table 1. The solid lines in Fig. 6 represent the results obtained by Delale and Erdogan [5], and the dotted lines represent the results obtained by the present model. It is noted that the results obtained by the present approximation method coincide with the exact results when l=0 and
The integration and the collocation points are given by [28] 2 1 ; ¼ 1; 2; 3; . . . ; N; ð40aÞ S ¼ cos 2N þ 1
2 T ¼ cos ; ¼ 1; 2; 3; . . . ; N: 2N þ 1
ð40bÞ
It can be readily shown that the expression for the stress intensity factor can be derived as KI ¼
pffiffiffiffiffiffiffi 2 l
2 0 ’ðþ1Þ; ð 0 þ 1 Þ
ð41Þ
where ’ðþ1Þ is computed by Krenk’s interpolation formula [30] given by
Fig. 6. Normalized stress intensity factors for a crack in a nonhomogeneous infinite medium having an exponentially varying Young’s modulus E=E0exp(y).
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Table 1 Comparison of normalized stress intensity factors for a crack in a nonhomogeneous infinite medium with an exponentially varying Young’s modulus E=E0exp(y) l
0 0.1 0.2 0.3 0.4 0.5
Exact results by Delale et al.
Approximate results by present method
KI ðp lÞffiffiffi "0 E0 l
KI ðl Þ pffiffiffi "0 E0 l
KI ðp lÞffiffiffi "0 E0 l
KI ðp -lÞffiffiffi "0 E0 l
KI ðp lÞffiffiffi "0 E0 l
KI ð-lÞ pffiffiffiffiffi "0 E0 l
1.000 1.078 1.158 1.245 1.337 1.435
1.000 0.925 0.858 0.793 0.732 0.665
1.000 1.053 1.111 1.174 1.244 1.321
1.000 0.952 0.909 0.871 0.836 0.806
0 0.025 0.047 0.071 0.093 0.114
0 0.027 0.051 0.078 0.104 0.141
deviate from them as the variable l increases. Thus the value of l characterizing the gradient exp(l) of the Young’s modulus E/E0 at the crack tip (y=l) should be small for the errors to remain within acceptable limits. The two points regarding the gradient should be noted in connection with the small value of l: (i) the product of a finite value of and a small value of the crack length l can give a small value of l. However, the gradient exp(l) of the Young’s modulus may be large in this case depending on the value of . This implies that the error is acceptable small for a small crack length even if the gradient at the crack tip is large. (ii) The product of a small value of and a finite value of the crack length l can give a small value of l. In this case, the gradient exp(l) is small. This implies that the error is acceptable small for a large crack length if the gradient is small at the crack tip. From this discussion, we can claim that our proposed approximation method can be used for FGMs having the gradient of Young’s modulus discussed above. Therefore, it is necessary to check the gradient of the Young’s modulus in order to justify the validity of the design by this method.
Difference
the formula of the stress intensity factor for a crack in homogeneous isotropic materials, which is called the apparent fracture toughness of FGMs [31]. Now we consider a radial edge crack of length l in the FGM coating around the circular hole in the infinite elastic medium as shown in Fig. 1. When the uniform applied pressure at which the fracture occurs from the crack tip is pc , the apparent fracture toughness of the FGM coating can be given by the formula of the stress intensity factor for a radial edge crack of length l emanating from a circular hole in a homogeneous isotropic infinite material as pffiffiffiffiffi KIca ¼ FI pc l: ð43Þ In Eq. (43), FI is a function of the geometric parameter l/R, whose values are available in the literature [32] and shown in Fig. 7 for convenience. The critical pressure pc is given by p that satisfies the condition KI ¼ Kc ;
ð44Þ
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 an FGM 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 the brittle materials, fracture 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 an FGM, let us imagine the FGM with the same geometric configuration under the external applied stress which corresponds to its fracture stress. Then, we can evaluate the critical value of the stress intensity factor through
Fig. 7. Non-dimensional stress intensity factors FI for a radial edge crack at a circular hole in an infinite medium. The hole and the crack surfaces are under a uniform pressure.
A.M. Afsar, H. Sekine / Composites Science and Technology 62 (2002) 1063–1077
where Kc is the local intrinsic fracture toughness of the FGM coating, which is given by [33] Kc ¼
E 0 K : E0 c
ð45Þ
interpolation. The minimum value of the objective function Fobj ðV1A ; V2A ; . . . ; VnA Þ obtained by the ADS program is compared with a small positive quantity " to satisfy the condition Fobj ðV1A ; V2A ; . . . ; VnA Þ 4 ";
Kc0
In Eq. (45), is the intrinsic fracture toughness of the homogeneous material B. Eq. (45) slightly overestimates the intrinsic fracture toughness of the FGM when the volume content of the constituent A is near to unity. The maximum error for VA=1 is 3.8% and it decreases as the value of VA decreases. Thus the use of Eq. (45) in determining the intrinsic fracture toughness of the FGM is quite justifiable.
5. Inverse problems The inverse problem of calculating material distributions intending to realize prescribed apparent fracture toughness in the FGM coating is solved by using the formulations developed for the approximation method of stress intensity factors. Suppose that a profile of apparent fracture toughness KIca is prescribed over a region of radial length L measured from the hole surface to the middle point of the nLth layer of the FGM coating which is divided into n number of layers of infinitesimal thickness as shown in Fig. 2. Note that the infinitesimal thickness of the layers in Fig. 2 can be given by l=(Rf – R)/n. Now assume a radial edge crack in the FGM coating around the hole. The crack length li is varied by taking the crack length as li=l/2 + (i–1)l, where i=1,2,. . .,nL, to ensure that the tip of the crack is located at the middle point of a layer. Considering the volume fractions ViA (i=1,2,. . .,n) of the constituent A in each layer of the infinitesimal thickness in Fig. 2 as design variables, the material distribution, i.e. the values of ViA (i=1,2,. . .,n) can be evaluated by solving the optimization problem set up as Minimize
n
:Fobj V1A ; V2A ; . . . ; VA
Subject to
:0 4 ViA
¼
nL P i¼1
KIi
i 2
Kc ;
4 1; i ¼ 1; 2; . . . ; n; ð46Þ
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 coating. In determining KIi by using Eqs. (39)–(42), p is replaced by pc obtained from Eq. (43). The optimization problem in Eq. (46) is solved by using a numerical optimization program ADS [34] 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 first finding bounds and then using polynomial
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ð47Þ
and the corresponding set of the design variables ViA ði ¼ 1; 2; . . . ; nÞ is taken as the solution of the optimization problem. Then the continuous profile of the material distribution is obtained by spline interpolation of the design variables ViA ði ¼ 1; 2; . . . ; nÞ. In order to solve the optimization problem in Eq. (46), it is necessary to determine the material properties of the FGM coating. The material properties of the FGM coating are determined by using the mixture rule [35] according to which the shear modulus of elasticity
and the bulk modulus K are first determined from the relations VA
KA K KB K þ VB ¼ 0; 3KA þ 4
3KB þ 4
ð48aÞ
VA
A
B
þ VB ¼ 0;
A þ Y
B þ Y
ð48bÞ
Y¼
ð9K þ 8 Þ ; ð6K þ 12 Þ
ð48cÞ
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 non-subscripted variables are used to denote the effective properties of the FGM. Then the Young’s modulus E and the Poisson’s ratio are calculated by using the expressions E¼
9K
; ð3K þ Þ
ð49aÞ
¼
E 1: 2
ð49bÞ
The coefficient of thermal expansion is determined by using the relation ¼ V A A
KA ð3K þ 4 Þ KB ð3K þ 4 Þ þ V B B : Kð3KA þ 4 Þ Kð3KB þ 4 Þ
ð50Þ
6. Numerical results and discussion The solution method of the inverse problem of calculating material distributions to realize prescribed apparent fracture toughness has been developed for the general case of constituent materials A and B which can
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be any real materials in practice. In particular, these can be two different ceramics or a ceramic and a metal depending on an application. In this study, some numerical results are provided for an example of a TiC/ Al2O3 FGM coating in which the constituents TiC and Al2O3 correspond to the materials A and B, respectively. The mechanical and the thermal properties of TiC and Al2O3 are shown in Table 2. The characteristic dimensions R and Rf are taken as 10 and 11 mm, respectively, while the difference between the sintering and room temperatures T is taken as 1000 C. In numerical calculations, the number of layers of infinitesimal thickness shown in Fig. 2 is chosen as 50, and the value of " in condition (47) is taken as 0.1. 6.1. Apparent fracture toughness from prescribed material distributions The apparent fracture toughness in the TiC/Al2O3 FGM coating around a circular hole in an infinite elastic medium are predicted for three different prescribed profiles of the material distributions as shown in Fig. 8. This figure shows the distribution of the volume fraction VA of TiC along the radial distance r. The curves I and III represent two different parabolic distributions while the curve II shows a linear distribution. The corresponding fracture characteristics are plotted in Fig. 9, which displays the apparent fracture toughness KIca Table 2 Material properties of TiC and Al2O3 Material
Young’s modulus (GPa)
Shear modulus (GPa)
Poisson’s ratio
CTE (/ C)
KIc (MPa m1/2)
TiC Al2O3
462 380
194.12 150.79
0.19 0.26
7.4 10–6 8.0 10–6
4.1 3.5
Fig. 8. Prescribed material distribution profiles in a TiC/Al2O3 FGM coating around a circular hole in an infinite elastic medium.
versus radial distance r. Fig. 9 reveals that the apparent fracture toughness depends significantly on the material distribution profiles, particularly the higher volume fraction VA provides the higher apparent fracture toughness. It is also noted that all the distribution profiles shown in Fig. 8 offer much higher apparent fracture toughness KIca compared to the intrinsic fracture toughness of the constituent materials. Specifically, the parabolic distribution shown by the curve I in Fig. 8 gives a maximum apparent fracture toughness of about 12 MPa m1/2 at r=10.64 mm. The intrinsic fracture toughness of the single phase Al2O3 is 3.5 MPa m1/2. However, it is observed from Fig. 9 that the apparent fracture toughness of the homogeneous region (r > 11 mm) of Al2O3 is considerably higher than its intrinsic fracture toughness. This is because the homogeneous region of Al2O3 adjacent to the FGM coating is still under the influence of the incompatible eigenstrain in the FGM coating. Thus, it is realized that the improved fracture characteristics can be maintained up to the distance much greater than the FGM coating thickness. However, at a further distance not shown in the figure, it can be predicted that the apparent fracture toughness will be identical to the intrinsic fracture toughness of Al2O3. It is noted from Fig. 9 that there is a large difference (up to 35%) between the apparent fracture toughness of examples I and III even if the difference between the material properties of the constituents shown in Table 2 is small. This is because the apparent fracture toughness is directly proportional to the critical pressure pc at which a crack starts to extend. This critical pressure pc depends not only on the material properties but also on the internal stress induced in the material due to the incompatible eigenstrain. Again, the internal stress depends on the material distribution profile as well as the difference between the sintering and room temperatures.
Fig. 9. Apparent fracture toughness in a TiC/Al2O3 FGM coating around a circular hole in an infinite elastic medium.
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The inverse calculation of material distributions in the TiC/Al2O3 FGM coating around a circular hole in an infinite elastic medium is carried out to realize prescribed apparent fracture toughness in the coating, which is higher than the intrinsic fracture toughness of the constituents shown in Table 2. The higher apparent fracture toughness can be controlled in a manner that can meet a requirement of an application. Among different possible manners of controlling the higher apparent fracture toughness, two examples are considered in this study as shown by the solid portions of the curves I and II in Fig. 10. This figure shows the prescribed apparent fracture toughness KIca as a function of the radial distance r measured from the center of the hole. The broken line represents the apparent fracture toughness obtained for 100% volume fraction VA of TiC, which represents a rough estimation of the upper limit of the apparent fracture toughness. In the example I, the apparent fracture toughness is controlled such that it increases linearly from 8 MPa m1/2 at the surface of the hole to 10 MPa m1/2 at r=10.75 mm. In the
example II, it remains a constant value of 7 MPa m1/2 over the same range as example I. For these prescribed apparent fracture toughness (solid portions of the curve I and II), the material distributions in the FGM coating (10–11 mm) are computed and plotted in Fig. 11. This figure shows the variation of the volume fraction VA of TiC along the radial distance r. It is seen from Fig. 11 that the values of VA are higher over the range from 10.0 to 10.77 mm in the case of the example I. The fabrication of the FGMs with the material distributions shown in Fig. 11 may be difficult by using a single manufacturing process like chemical vapor deposition, plasma spray, powder metallurgy and physical vapor deposition. However, by a suitable combination of these processes, it may be possible to fabricate these FGMs according to the material distributions of Fig. 11. It is also important to know the characteristics of the material outside the controlled region corresponding to the computed material distributions shown in Fig. 11. The dotted portions of the curves I and II as shown in Fig. 10 represent the characteristics of the material, outside the controlled region, that must be appeared if the apparent fracture toughness is controlled in the manner shown by the solid portions. In the case of the example I, it is seen from Fig. 10 that the apparent fracture toughness KIca decreases sharply after controlling and this decreasing trend is retained over rest of the FGM coating (10.75–11 mm) while, in the case of the example II, it remains almost constant over the same region. However, in both the cases, another sharp decrease in the values of the apparent fracture toughness KIca occurs as the homogeneous region (r=11 mm) is approached and the decreasing trend continues for both the cases as r increases. The prescribed apparent fracture toughness, considered in both the examples I and II as shown in Fig. 10
Fig. 10. Prescribed apparent fracture toughness in a TiC/Al2O3 FGM coating around a circular hole in an infinite elastic medium.
Fig. 11. Material distribution profiles in a TiC/Al2O3 FGM coating around a circular hole in an infinite elastic medium.
For the constituent materials shown in Table 2 and T=1000 C, the rough estimation reveals that the internal stress may vary from 0–277 MPa depending on the material distribution profile of the constituents. This large range of variation (0–277 MPa) occurs for the small range of variation in the material properties of the constituents. From this fact, it is realized that a small change in the material properties causes a large change in the internal stress that affects the critical pressure pc which ultimately causes large change in the apparent fracture toughness. 6.2. Calculation of material distribution profiles intending to realize prescribed apparent fracture toughness
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Fig. 12. Variation of normalized Young’s modulus in a TiC/Al2O3 FGM coating around a circular hole in an infinite elastic medium.
is realized by designing the FGM coating having the material distributions shown in Fig. 11. From this fact it can be concluded that an apparent fracture toughness in FGM coatings around a circular hole in an infinite elastic medium can be controlled within possible limits by choosing an appropriate material distribution in the coatings. Fig. 12 displays the variation of normalized Young’s modulus E/E0 corresponding to the material distribution profiles of Fig. 11. The gradients of the profiles of these normalized Young’s moduli are large near the surface of the hole and small over rest of the region (except at the distance of about 10.75 mm in the case of example I). In Section 3.3, it has been discussed that the errors in the stress intensity factors are acceptable small even if the gradient is large at the distance corresponding to a small crack length (near the surface of the hole). The profile shown by the curve I has also a large gradient at the distance of about 10.75 mm. Near this point, the prescribed apparent fracture toughness shown in Fig. 10 corresponding to the material distribution of Fig. 11 may have a large error. As a remedy for such a case, it is recommended that the apparent fracture toughness should be prescribed over the region greater than the actual region in which the apparent fracture toughness is intended to control.
7. Conclusions A solution method of the inverse problem has been developed for calculating material distributions to realize prescribed apparent fracture toughness in an FGM coating around a circular hole in an infinite elastic medium. The incompatible eigenstrain induced in the FGM coating due to mismatch in the coefficients of thermal expansion has been taken into consideration.
An approximation method of determining stress intensity factors is introduced for a crack in the FGM coating with any arbitrary variation of material properties. The FGM coating is homogenized simulating its nonhomogeneous material properties by a distribution of equivalent eigenstrain, which is determined from the condition of identical elastic fields in the uncracked FGM and homogenized coatings. A radial edge crack emanating from the circular hole in the homogenized coating is considered for the case of a uniform pressure applied to the surfaces of the hole 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 determined for the crack in the homogenized coating represent the approximate values of the stress intensity factors for the same crack in the FGM coating, and are used in the inverse problem of calculating material distributions in the FGM coating to realize prescribed apparent fracture toughness in the coating. Numerical results obtained for a TiC/Al2O3 FGM coating reveal that the apparent fracture toughness in the FGM coating significantly depends on the material distributions. It is also revealed that the apparent fracture toughness in the FGM coating can be controlled within possible limits by choosing an appropriate material distribution profile in the coating.
Acknowledgements The authors would like to acknowledge the partial support of the Grant-in-Aid for Scientific 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 financial assistance through the Japanese Government Scholarship (No. 942062) from the Ministry of Education, Science, Sports and Culture of Japan. References [1] Yamanouchi M, Koizumi M, Hirai T, Shiota I. Proceedings of the First International Symposium on Functionally Gradient Materials, Sendai, Japan, 1990. [2] Holt JB, Koizumi M, Hirai T, Munir ZA. Ceramic transaction: functionally gradient materials, vol. 34. Westerville, (OH): The American Ceramic Society; 1993. [3] Ilschner B, Cherradi N. Proceedings of the Third International Symposium on Structural and Functional Gradient Materials. Lausanne, Switzerland: Presses Polytechniques et Universitaires Romands; 1994. [4] Atkinson C, List RD. Steady state crack propagation into media with spatially varying elastic properties. International Journal of Engineering Science 1978;16:717–30. [5] Delale F, Erdogan F, The. crack problem for a nonhomogeneous plane. ASME Transaction. Journal of Applied Mechanics 1983; 50:609–14.
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