This article was downloaded by: [University of Chicago Library]On: 24 May 2013, At: 00:56Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number:1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK

Journal of Thermal StressesPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/uths20

TRANSIENT THERMALSINGULAR STRESSES OFMULTIPLE CRACKING IN A W-Cu FUNCTIONALLY GRADEDDIVERTOR PLATESei UedaPublished online: 19 Jan 2011.

To cite this article: Sei Ueda (2002): TRANSIENT THERMAL SINGULAR STRESSES OFMULTIPLE CRACKING IN A W-Cu FUNCTIONALLY GRADED DIVERTOR PLATE, Journal ofThermal Stresses, 25:1, 83-95

To link to this article: http://dx.doi.org/10.1080/014957302753305899

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private studypurposes. Any substantial or systematic reproduction, redistribution,reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make anyrepresentation that the contents will be complete or accurate or up todate. The accuracy of any instructions, formulae, and drug doses shouldbe independently verified with primary sources. The publisher shall notbe liable for any loss, actions, claims, proceedings, demand, or costs ordamages whatsoever or howsoever caused arising directly or indirectly inconnection with or arising out of the use of this material.

TRANSIENT THERMAL SINGULAR STRESSESOF MULTIPLE CRACKING IN A W± CU

FUNCTIONALLY GRADED DIVERTOR PLATE

Sei Ueda

Department of Mechanical EngineeringOsaka Institute of Technology

Osaka, Japan

W e consider the transient thermal singular stresses of multiple cracking in a func-tionally graded divertor plate due to a thermal shock. The plate is made of a gradedlayer bonded between a homogeneous substrate and a homogeneous coating, and it issubjected to a cycle of heating and cooling on the coating surface of the plate. Thesurface layer contains a parallel array of embedded or edge cracks perpendicular to theboundaries. The thermal and elastic properties of the material are dependent on thetemperature and the position. Finite element calculations are carried out, and thetransient thermal stress intensity factors are shown graphically.

Functionally graded materials (FGMs) have gained considerable importance in ex-tremely high-temperature environments such as nuclear reactors and chemical plants[1]. For example, the tungsten (W)-copper (Cu) graded composite has been devel-oped for the divertor plate of the International Thermonuclear Experimental Re-actor (ITER) [2]. These functional gradations open a new avenue for optimizingboth material and component structures to achieve high performance and materiale� ciency. At the same time, it posts many challenging mechanics problems, in-cluding the thermal stress distributions and fracture of FGMs. In the quest for thesolution of these problems, extensive studies have been carried out, theoretically, onthermal stress distribution [3± 6], thermal fracture [7± 11], and fracture due to athermal shock [1, 12± 15]. The experimental observation of cracking in FGMs due toa thermal shock has been made by Kawasaki and Watanabe [16], and the experimentshowed that many cracks were initiated at or in the surface layer.

The present study focuses on the transient thermal singular stresses of multiplecracking in a W-Cu functionally graded divertor plate due to a thermal shock withtemperature-dependent properties. The plate is made of a graded layer bondedbetween a homogeneous substrate and a homogeneous coating and is subjected toa cycle of heating and cooling on the coating surface of the plate. The surface layer

Received 31 March 2000; accepted 9 October 2000.Address correspondence to Professor Sei Ueda, Department of Mechanical Engineering, Osaka

Institute of Technology, 5-16-1 Omiya, Asahi-ku, Osaka 535-8585 Japan. E-mail: ueda@med.oit.ac.jp

Journal of Thermal Stresses, 25:83± 95, 2002Copyright # 2002 Taylor & Francis0149-5739 /02 $12.00+ .00

83

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:56

24

May

201

3

contains a parallel array of embedded or edge cracks perpendicular to theboundaries. The thermal and elastic properties of the material are dependent onthe temperature and the position. Using a ® nite element method, numerical resultsfor the transient thermal stress intensity factors are obtained and presented ingraphic form.

STATEMENT OF THE PROBLEM

The schematic divertor plate [2] used for computations is shown in Figure 1. Thedivertor plate of thickness h examined in this study consists of the tungsten coatingof thickness hW, the FGM layer of thickness hFGM, and the copper substrate ofthickness hCu. A rectangular Cartesian coordinate system (x; y; z) is introduced inthe divertor plate in such a way that the x-axis is the thickness direction and theperiodic array of cracks of length 2a0 = b ± a (0 4 a < b < hW) is distributedperpendicular to the y-axis a distance 2c apart. Here, the subscripts x; y; z will beused to refer to the coordinate directions and the subscripts W, FGM, Cu to in-dicate the tungsten coating, the FGM layer, and the copper substrate, respectively.rW and rCu denote the mass densities of the tungsten and the copper. The thermaland mechanical properties of the tungsten and the copper are functions of thetemperature f. The tungsten has the coe� cient of thermal expansion aW(f),thermal conductivity lW(f), speci® c heat cW(f), Young’s modulus EW(f), andPoisson’s ratio nW(f). The thermal and mechanical properties of the copper areaCu(f), lCu(f), cCu(f), ECu(f), nCu(f).

As a ® rst-order approximation, the mass density r(x) and the e� ective thermaland mechanical properties a(x; f), l(x; f), c(x; f), E(x; f), n(x; f) of the divertorplate can be obtained using the rule of mixture

Figure 1. A schematic of multiple cracking in a W-Cu functionally graded divertor plate.

84 S. UEDA

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:56

24

May

201

3

r(x) = V (x)rW ‡ {1 ± V (x)}rCu

a(x; f) = V (x)aW(f) ‡ {1 ± V (x)}aCu(f)

l(x; f) = V (x)lW(f) ‡ {1 ± V (x)}lCu(f)

c(x; f) = V (x)cW(f) ‡ {1 ± V (x)}cCu(f)

E(x; f) = V (x)EW(f) ‡ {1 ± V (x)}ECu(f)

n(x; f) = V (x)nW(f) ‡ {1 ± V (x)}nCu(f)

(1)

where V (x) is the volume fraction of the tungsten phase and is chainging from one tozero. If, in addition to V (hW) = 1, we require that V (hW ‡ hFGM) = 0, that is, thematerial at x = hW is pure tungsten and at x = hW ‡ hFGM is pure copper, then inthis case, V (x) is a power-law function of x

V (x) =

1 0 4 x 4 hW

1 ±x ± hW

hFGM

p

hW 4 x 4 hW ‡ hFGM

0 hW ‡ hFGM 4 x 4 h

8>>>><

>>>>:

(2)

where the constant p, 0 4 p < 1, is essentially the nonhomogeneity parameter ofthe FGM layer. In the limiting cases, E(x; f) of the FGM layer would approachECu(f) for p ! 0 and EW(f) for p ! 1. A value of p = 1:0 corresponds to thelinear variation of E(x; f), for 0 < p < 1 the FGM layer is Cu-rich, and for1 < p < 1 it is W-rich.

We assume that the divertor plate is suddenly heated from the initial tem-perature FA to the boundary temperature FA on the Cu surface (x = h) and tothe boundary temperature FB on the W surface (x = 0). After the steady state isachieved in the material at time t = tA, the W surface of the material is suddenlycooled to the initial temperature FA. The combination of temperatures is one ofthe combinations to which the divertor plate is subjected when the reactor goesfrom the normal operation to the transient operation. It is also assumed that theresulting transient thermal stress problem is quasi-static, that is, the inertia e� ectsare negligible.

HEAT CONDUCTION PROBLEM

The initial and boundary conditions are

f(x; 0) = FA 0 4 x 4 h (3)

f(0; t) = FA ‡ (FB ± FA){H(t) ± H(t ± tA)}

f(h; t) = FA(4)

TRANSIENT THERMAL SINGULAR STRESSES OF MULTIPLE CRACKING 85

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:56

24

May

201

3

where H(t) denotes the Heaviside unit step function in time. The transient tem-perature distributions f(x; t) may be determined from the solutions of the nonlineardi� usion equation:

@

@xl(x; f)

@f(x; t)@x

= r(x)c(x; f)@f(x; t)

@t0 < t (5)

Using the analytical method introduced by the author [6], the solution of Eq. (5) withEqs. (3) and (4) for arbitary functions l(x; f), r(x), c(x; f) can be obtained.

FINITE ELEMENT ANALYSIS OF THE THERMAL STRESSPROBLEM

Let the displacement components in the x- and y-directions be labeled by ux(x; y; t)and uy(x; y; t). The stress components sxx(x; y; t), syy(x; y; t), and sxy(x; y; t) arefound as

sxx(x; y; t)

syy(x; y; t)

sxy(x; y; t)

2

64

3

75 =

m(x; f){1 ‡ k(x; f)}k(x; f) ± 1

m(x; f){3 ± k(x; f)}k(x; f) ± 1

0

m(x; f){3 ± k(x; f)}k(x; f) ± 1

m(x; f){1 ‡ k(x; f)}k(x; f) ± 1

0

0 0 m(x; f)

2

66666664

3

77777775

£

@ux(x; y; t)@x

± eT(x; f)

@uy(x; y; t)@y

± eT(x; f)

@ux(x; y; t)@y

‡ @uy(x; y; t)@x

2

6666666664

3

7777777775

(6)

where m(x; f) = E(x; f)=2{1 ‡ n(x; f)} are the shear moduli of the plate andk(x; f) = 3 ± 4n(x; f). While the stress± free temperature is assumed to be FA, thestresses at high temperatures would be relaxed due to an inelastic deformation after along time passes [17]. Then it is assumed that the stress components at time tA arealso zero and the thermal strain eT(x; f) is given by

eT(x; f) =

Z f(x;t)

FA

a(x; j) dj 0 4 t 4 tA

Z f(x;t)

f(x;tA)a(x; j) dj tA 4 t

8>>><

>>>:

9>>>=

>>>;(7)

86 S. UEDA

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:56

24

May

201

3

The symmetry and boundary conditions can be written as

syy(x; 0; t) = 0 a < x < b

uy(x; 0; t) = 0 0 4 x 4 a; b 4 x 4 h(8)

sxy(x; 0; t) = 0 0 4 x 4 h (9)

uy(x; c; t) = uy0(t) 0 4 x 4 h (10)

sxy(x; c; t) = 0 0 4 x 4 h (11)

sxx(0; y; t) = 0 0 4 y 4 c (12)

sxy(0; y; t) = 0 0 4 y 4 c (13)

sxx(h; y; t) = 0 0 4 y 4 c (14)

sxy(h; y; t) = 0 0 4 y 4 c (15)

The uniform displacement uy0(t) in the y-direction can be found from the condition

Z h

0syy(x; c; t) dx = 0 (16)

The ® nite element formulation of problems in elasticity may be found in manytextbooks and monographs. In the conventional approach of applying the method toelasticity problems involving nonhomogeneous materials, for the sake of simplicity,the mechanical properties of the medium are assumed to be constant within eachelement. Ordinarily the sti� ness matrix [K] relating the nodal point displacements{u} and loads {F} through

[K]{u} = {F} (17)

may be expressed as

[K] =

Z

O[B]T [D][B] dV (18)

where O is the volume, [D] is the material sti� ness or the elasticity matrix, and [B]is the matrix that relates the strains to the displacements {u} and contains thegradients of the interpolating functions. In this study, the elasticity matrix [D] isassumed to be a function of the space coordinate [11] and the temperature. Theintegral in Eq. (18) is evaluated using a Gaussian integration technique, and [D] isspeci® ed at each Gaussian integration point. For the plane strain problem underconsideration the two-dimensional integral resulting from Eq. (18) may then beexpressed as

[K] =Xn

i= 1

Xn

j= 1

[B]Tij [D]ij[B]ijJijwiwj (19)

TRANSIENT THERMAL SINGULAR STRESSES OF MULTIPLE CRACKING 87

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:56

24

May

201

3

where the subscripts i and j designate the coordinates of the Gaussian integrationpoints and refer to the fact that the parent quantities are evaluated at thesepoints; Jij is the determinant of the Jacobian matrix; and wi; wj are the Gaussianweights.

Using the most common de® nition of the stress intensity factor as the strength ofthe stress singularity ahead of the crack, the mode I thermal stress intensity factors atthe crack tips x = a and x = b are

KIa(t) = limx!a

��������������������2p(a ± x)

psyy(x; 0; t)

KIb(t) = limx!b

��������������������2p(x ± b)

psyy(x; 0; t)

(20)

Applying the displacement method to Eq. (20), the thermal stress intensity factorsKIa(t) and KIb(t) can be obtained as

KIa(t) = limx!a

m(x; f)2{1 ± n(x; f)}

�����������2p

x ± a

ruy(x; 0; t)

KIb(t) = limx!b

m(x; f)2{1 ± n(x; f)}

�����������2p

b ± x

ruy(x; 0; t)

(21)

In the problems treated thus far concerning FGMs, even relatively coarse grid 8-node isoparametric elements and n = 9 seem to give quite adequate results. The ® niteelement mesh contains 826 elements and 2649 nodes, and 6-node singular elements[18] are placed around the crack tips at x = a and x = b (y = 0), respectively.

NUMERICAL RESULTS AND DISCUSSION

Numerical calculations have been carried out at the temperatures of FA = 293 [K](room temperature), FB = 1300 [K] and for the geometric parameters ofhW=h = 0:138, hFGM=h = 0:586, hCu=h = 0:276 [2]. The parameter p is varied from0.2 to 5.0. Using the experimental results [19± 21], the mass densities and the thermaland mechanical properties of the tungsten and the copper are approximated by thefollowing functions of the temperature f

rW = 1930 [kg=m3]

aW(f) = 4:6 [10± 6=K]

lW(f) = 162:3 ‡ 0:07f ‡ 2:0 £ 10± 5f2 [W=mK]

cW(f) = 10:011����f

p± 0:1362f [J=kgK]

EW(f) = 411:4 ± 0:044f [GPa]

nW(f) = 0:2876 ‡ 8:0 £ 10± 6f

(22)

88 S. UEDA

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:56

24

May

201

3

rCu = 8930 [kg=m3]

aCu(f) = 15:37 ‡ 7:18 £ 10± 3f ± 4:0 £ 10± 6f2 [10± 6=K]

lCu(f) = 401:4 ± 0:0625f [W=mK]

cCu(f) = 36:675����f

p± 0:8333f [J=kgK]

ECu(f) = 128:0 ± 0:029f [GPa]

nCu(f) = 0:4459 ± 3:09 £ 10± 4f

(23)

The previous study [6] indicated that the e� ects of the temperature-dependenc e ofthe properties on the transient temperature and stress distributions are pronounced.It also noted that the stress near the coating surface is compressive under theheating process and is tensile under the cooling process. Moreover, experimentalobservations of cracking in FGMs due to a thermal shock showed that many surfacecracks were initiated under the cooling process [16]. The present study focuses onthe transient behavior of the thermal stress intensity factors under the coolingprocess.

In the ® rst series of calculations, we consider the e� ect of the temperature- de-pendence of the thermal and elastic properties on the thermal stress intensity factorsfor the parameter p = 1:0 and crack spacing c=hW = 1:1. In all cases a normalizingstress is used to present the results in dimensionless form

s0 =EW(FA)aW(FA)(FB ± FA)

1 ± nW(FA)(24)

Figures 2 and 3 show the normalized stress intensity factors KIa(t)=s0(pa0)1=2,

KIb(t)=s0(pa0)1=2 of the embedded crack for a=hW = 0:2, b=hW = 0:8 (a0=hW = 0:3)

Figure 2. The effect of temperature dependence of the properties on the thermal stress intensity factors ofthe embedded crack for c=hW = 1:1, a=hW = 0:2, b=hW = 0:8, and p = 1:0.

TRANSIENT THERMAL SINGULAR STRESSES OF MULTIPLE CRACKING 89

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:56

24

May

201

3

and KIb(t)=s0(pb)1=2 of the edge crack for a=hW = 0:0; b=hW = 0:4, respectively. Inthe ® gures, F = kCu(FA)(t ± tA)=h2 denotes the Fourier number under the coolingprocess, where the thermal di� usivity kCu(FA) is given by

kCu(FA) =lCu(FA)

rCucCu(FA)(25)

The solid and dotted lines represent results with and without the temperature-de-pendent properties. In temperature-independen t calculations, the values of thethermal and elastic properties at FA are used. It can be seen that the temperature-dependent parameters make a signi® cant impact on the transient behaviors of thethermal stress intensity factors.

In the next set of calculations, we study the e� ect of the crack spacing c=hW

on the transient themal stress intensity factors. Figures 4 to 6 show the nor-malized stress intensity factors KIa(t)=s0(pa0)

1=2 and KIb(t)=s0(pa0)1=2 of the

embedded crack (a=hW = 0:2, b=hW = 0:8) for p = 0:2, 1.0, and 5.0, respectively.The solid lines represent KIa(t) and the dotted lines KIb(t). The magnitude ofKIa(t) is greater than that of KIb(t). In both cases, KIa(t) and KIb(t) initiallyincrease, go through maxima, and then tend to decrease, as the Fourier numberF increases. The values of KIa(t) and KIb(t) increase with increasing crack spa-cing c=hW. The e� ect of c=hW on the normalized stress intensity factorKIb(t)=s0(pb)1=2 of the edge crack (a=hW = 0:0; b=hW = 0:4) for p = 0:2; 1:0; and5.0 are plotted in Figures 7 to 9. The same tendencies as the embedded crackshown in Figures 4 to 6 are observed, and it may be also seen that as c decreasesKIb(t) decreases.

In the ® nal set of calculations, we consider the e� ects of the gradation parameterp on the thermal stress intensity factors. Figures 10 and 11 demonstrate the

Figure 3. The effect of temperature dependence of the properties on the thermal stress intensity factors ofthe edge crack for c=hW = 1:1, a=hW = 0:0, b=hW = 0:4, and p = 1:0.

90 S. UEDA

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:56

24

May

201

3

maximum values of the stress intensity factors KMAXIa =s0(pa0)1=2, KMAX

Ib =s0(pa0)1=2 ofthe embedded crack (a=hW = 0:2, b=hW = 0:8) and KMAX

Ib =s0(pb)1=2 of the edgecrack (a=hW = 0:0, b=hW = 0:4) for p = 0:2, 1.0, and 5.0, respectively. The maximumvalues of the stress intensity factors increase with increasing crack spacing c=hW andgradation parameter p. For the embedded crack, the reduction in the maximumvalues of the stress intensity factors owing to the functional gradation of theinterface layer can be signi® cant.

Figure 4. The effect of crack spacing on the thermal stress intensity factors of the embedded crack fora=hW = 0:2, b=hW = 0:8, and p = 0:2.

Figure 5. The effect of crack spacing on the thermal stress intensity factors of the embedded crack fora=hW = 0:2, b=hW = 0:8, and p = 1:0.

TRANSIENT THERMAL SINGULAR STRESSES OF MULTIPLE CRACKING 91

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:56

24

May

201

3

CONCLUSIONS

The ® nite element analysis of the thermal shock fracture behavior of the functionallygraded W-Cu divortor plate with temperature-dependent properties was shown inthis study. Based on the numerical results, we make the following conclusions.

1. Temperature-dependent changes in the thermal and elastic properties of Cu andW signi® cantly a� ect the thermal shock behavior of the transient stress intensityfactors.

Figure 6. The effect of crack spacing on the thermal stress intensity factors of the embedded crack fora=hW = 0:2, b=hW = 0:8, and p = 5:0.

Figure 7. The effect of crack spacing on the thermal stress intensity factors of the edge crack fora=hW = 0:0, b=hW = 0:4, and p = 0:2.

92 S. UEDA

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:56

24

May

201

3

2. For each crack spacing and gradiation parameter considered, initially thevalues of the transient thermal stress intensity factors of the embedded andedge cracks increase with time. They then reach maximum values and decreasewith time.

3. The peak values of the thermal stress intensity factors increase with the crackspacing, indicating that as crack density becomes large the parallel crack tends toarrest at a relatively small crack length.

4. The dependence of the stress intensity factors on the gradation is seen to bequite strong. In addition, a peak appears on each curve; its value and thecorresponding crack spacing depend on the value of the gradation parameter.

Figure 8. The effect of crack spacing on the thermal stress intensity factors of the edge crack fora=hW = 0:0, b=hW = 0:4, and p = 1:0.

Figure 9. The effect of crack on spacing the thermal stress intensity factors of the edge crack fora=hW = 0:0, b=hW = 0:4, and p = 5:0.

TRANSIENT THERMAL SINGULAR STRESSES OF MULTIPLE CRACKING 93

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:56

24

May

201

3

REFERENCES

1. J. N. Reddy and C. D. Chin, Thermomechanical Analysis of Functionally Graded Cylinders andPlates, J. Thermal Stresses, vol. 21, pp. 593± 626, 1998.

2. Y. Itoh, M. Takahashi, and H. Takano, Development of Tungsten/Copper Graded Composite forHigh-Heat Flux Components, FGM News, vol. 5, pp. 3± 8, 1996. (Japanese)

3. R. L. Williamson, B. H. Rabin, and J. T. Drake, Finite Element Analysis of Thermal ResidualStresses at Graded Ceramic-Metal Interfaces. Part I. Model Description and Geometrical E� ects, J.Appl. Phys., vol. 74, no. 2, pp. 1310± 1320, 1993.

Figure 10. The effect of the gradation parameter on the maximum stress intensity factors of the embeddedcrack for a=hW = 0:2, and b=hW = 0:8.

Figure 11. The effect of the gradation parameter on the maximum stress intensity factors of the edge crackfor a=hW = 0:0 and b=hW = 0:4.

94 S. UEDA

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:56

24

May

201

3

4. J. T. Drake, R. L. Williamson, and B. H. Rabin, Finite Element Analysis of Thermal Residual Stressesat Graded Ceramic-Metal Interfaces. Part II. Interface Optimization for Residual Stress Reduction, J.Appl. Phys., vol. 74, no. 2, pp. 1321± 1326, 1993.

5. S. Ueda and M. M. Gasik, Thermal-Elasto-Plastic Analysis of W-Cu Functionally Graded MaterialsSubjected to a Uniform Heat Flow by Micromechanical Model, J. Thermal Stresses, vol. 23, pp. 395±409, 2000.

6. S. Ueda, Thermoelastic Analysis of W-Cu Functionally Graded Materials Subjected to a ThermalShock by micromechanical model, J. Thermal Stresses, vol. 24, pp. 19± 45, 2001.

7. N. Noda and J. H. Jin, Thermal Stress Intensity Factors for a Crack in a Functionally GradientMaterial, Int. J. Solids Structures, vol. 30, no. 6, pp. 1039± 1056, 1993.

8. M. Nemat-Alla and N. Noda, Study of an Edge Crack Problem in a Semi-In® nite FunctionallyGraded Medium with Two Dimensionally Nonhomogeneous Coe� cients of Thermal Expansionunder Thermal Load, J. Thermal Stresses, vol. 19, pp. 863± 888, 1996.

9. F. Erdogan and B.H. Wu, Crack Problems in FGM Layers under Thermal Stresses, J. ThermalStresses, vol. 19, pp. 237± 265, 1996.

10. G. Bao and L. Wang, Multiple Cracking in Functionally Graded Ceramic/Metal Coatings, Int. J.Solids Structures, vol. 32, no. 19, pp. 2853± 2871, 1995.

11. Y.-D. Lee and F. Erdogan, Residual/Thermal Stresses in FGM and Laminated Thermal BarrierCoatings, Int. J. Fracture, vol. 69, pp. 145± 165, 1994/1995.

12. J. H. Jin and N. Noda, Transient Thermal Stress Intensity Factors for a Crack in a Semi-In® nite Plateof a Functionally Gradient Material, Int. J. Solids Structures, vol. 31, no. 2, pp. 203± 218, 1994.

13. J. H. Jin and R. C. Batra, Stress Intensity Relaxation at the Tip of an Edge Crack in a FunctionallyGraded Material Subjected to a Thermal Shock, J. Thermal Stresses, vol. 19, pp. 317± 339, 1996.

14. N. Noda, Thermal Stress Intensity Factor for Functionally Gradient Plate with an Edge Crack, J.Thermal Stresses, vol. 20, pp. 373± 387, 1997.

15. S. Ueda, Thermal Shock Fracture in a W-Cu Divertor Plate with a Functionally Graded Non-homogeneous Interface, J. Thermal Stresses, in press.

16. A. Kawasaki and R. Watanabe, Thermal Shock Fracture Mechanism of Functionally GradientMaterials as Studied by Burner Heating Test, in B. Ilchner and N. Cherradi (eds.), Proc. 3rd Inter-national Symposium on Structural and Functionally Gradient Materials, pp. 397± 404, Presses Poly-techniques et Universitaires Romandes, Switzerland, 1994.

17. K. Kokini and B.D. Choules, Surface Thermal Fracture of Functionally Graded Ceramic Coatings:E� ect of Architecture and Materials, Composites Eng., vol. 5, pp. 865± 877, 1995.

18. J. R. Yeh, The Mechanics of Multiple Transverse Cracking in Composite Materials, Int. J. SolidsStructures, vol. 25, no. 12, pp. 1445± 1455, 1989.

19. J. Gittus, Creep, Viscoelasticity and Creep Fracture in Solids, p. 606, Applied Science Publishers,London, 1976.

20. R. W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, p. 605, John Wiley& Sons, New York, 1976.

21. W. Johnson and P. B. Mellor, Engineering Plasticity, p. 646, van Nostrand Reinhold Co., London,1973.

TRANSIENT THERMAL SINGULAR STRESSES OF MULTIPLE CRACKING 95

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:56

24

May

201

3