www.elsevier.com/locate/compstruct

Composite Structures 79 (2007) 174–179

Scattering of anti-plane stress waves by a crackin a non-homogeneous orthotropic medium

Li Ma a,b,*, Wu Nie b, Lin-Zhi Wu a, Li-Cheng Guo a

a Center for Composite Materials, Harbin Institute of Technology, Harbin 150001, Chinab School of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China

Available online 28 February 2006

Abstract

Scattering of anti-plane harmonic waves by a finite crack in the functionally graded orthotropic medium is investigated by means ofthe Schmidt method. By using the Fourier transform and defining the jump of displacement components across the crack surface as theunknown function, a pair of dual integral equations are derived. To solve the dual integral equations, the jump of the displacement com-ponents across the crack surface is expanded in a series of Jacobi polynomial. The dynamic stress intensity factor is obtained as functionsof the incident wavelength, gradient parameter of the functionally graded materials. Numerical examples are provided to show the effectsof material properties upon the dynamic fracture behavior of the functionally graded materials.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Functionally graded orthotropic materials; Scattering of stress wave; Dynamic stress intensity factor

1. Introduction

Functionally graded materials (FGMs) to be use as,inter alia, superheat-resistive materials have promisedattractive applications in furnace liners, space structures,and fusion reactors. FGMs consist to two distinct materialphases, such as ceramic and metal alloy phases, and is amixture of them such that the composition of each changescontinuously along one direction. The change in micro-structure induces chemical, material, and microstructuralgradient, and makes functionally graded materials differentin behavior from homogeneous materials and traditionalcomposite materials [1]. These materials are tailorable intheir properties via the design of the gradients in chemistryand microstructure that is possible within them.

Experiments have shown that cracks occur in function-ally graded materials (see above references). The

0263-8223/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruct.2005.12.002

* Corresponding author. Address: Center for Composite Materials,Harbin Institute of Technology, Harbin 150001, China. Tel./fax: +86 45186402323.

E-mail address: mali@hit.edu.cn (L. Ma).

knowledge of crack growth and propagation in function-ally graded materials is important in designing componentsof FGMs and improving its fracture toughness. From thefracture mechanics viewpoint, a crack in FGMs may exhi-bit the complex behavior due to the varying of the mechan-ical properties of the material. The fracture behavior ofFGMs has been studied for both static and dynamic prob-lems. Delale and Erdogan [2] studied analytically thebehavior of a crack in an infinite functionally graded platewith varying elastic material constants in the direction par-allel to the crack. Their study indicates that Young’s mod-ulus has a significant effect on the stress field, but the effectof Poisson’s ratio can be ignored. The generalized mixedmode problem for an arbitrarily orientated crack in FGMswas discussed by Konda and Erdogan [3]. Shbeeb et al.[4,5] investigated the arbitrarily oriented multi-crack prob-lem in the infinite FGMs by using stress function andsingular integral equations. Jin and Batra [6] summarizedthe crack-tip fields in non-homogeneous materials andobtained the stress intensity factor for an edge crack in afunctionally graded strip. Numerical simulation [7,8] andexperimental studies [9–14] have also been conducted toinvestigate the static fracture behavior of FGMs.

L. Ma et al. / Composite Structures 79 (2007) 174–179 175

In all these studies, FGMs are assumed to be isotropic.However, because of the nature of the techniques used inprocessing, the graded materials are seldom isotropic.For example, FGMs processed by using a plasma spraytechnique have generally a lamella structure [15], whereasprocessing by electron beam physical vapor depositionwould lead to a highly columnar structure [16]. Thus, theorthotropic properties should be considered in studyingthe mechanics of FGMs. However, due to the problemcomplexity, up to now, only a few researcher consideredthe crack problem for functionally graded orthotropicmaterials. For example, Ozturk and Erdogan [17] analyzedthe mode I crack problem for an infinite inhomogeneousorthotropic medium. Chen et al. [18] investigated the tran-sient response of the internal crack in a functionally gradedorthotropic strip.

In this paper, we deal with the dynamic orthotropicfunctionally graded material solid containing a Griffithcrack under the action of normal anti-plane harmonicstress wave loading. Fourier transform was used to reducethe two-dimensional wave problem to the solution of dualintegral equation. To solve the dual integral equations, thejump of the displacement components across the crack sur-face is expanded in a series of Jacobi polynomial. Thedynamic stress intensity factor is obtained as functions ofthe incident wavelength, gradient parameter of the func-tionally graded materials. Numerical examples are pro-vided to show the effects of material properties upon thedynamic fracture behavior of the functionally gradedmaterials.

2. Formulation of the problem

Consider a functionally graded orthotropic mediumwith a crack of length 2a along x-axis, as shown inFig. 1. The coordinates x and y are assumed as the princi-pal axes of orthotropy. Assume the material propertiesvary in the y-axis. Under the anti-plane deformation, thefunctionally graded orthotropic material has followingstress–strain relationship:

sxz ¼ lxðyÞexz; syz ¼ lyðyÞeyz ð1Þ

where lx(y) and ly(y) are the shear modulus.

–a a

y

1

2 x

Fig. 1. A finite Griffith crack in the functionally graded orthotropicmedium.

An anti-plane harmonic wave moves in the positive y

direction and impinges normally on the crack surfaces.The scattered wave field generated by the crack can bedescribed by displacement component W in the z direction.In response to the incident harmonic wave, the displace-ment component can be written as

W ¼ w expðixtÞ ð2Þ

where x is the wave frequency and t is the time variable. Ifthe strain–displacement relations are used, the equation ofmotion can be written in terms of displacement componentas follows:

lxðyÞo2Wox2þ lyðyÞ

o2Woy2þ

dlyðyÞdy

oWoy¼ qðyÞ o

2Wot2

ð3Þ

In order to simplify the complexity of mathematics, wewill focus this study on a special class of FGM in whichthe variations of material properties have the same materialgradient parameter. Therefore, we assume

lxðyÞ ¼ lx0 expðbyÞ; lyðyÞ ¼ ly0 expðbyÞ;qðyÞ ¼ q0 expðbyÞ ð4Þ

where b is the gradient of material properties and is a po-sitive or negative constant. lx0, ly0 and q0 are the shearmoduli and mass density at y = 0. This model includesthe effects of varying elastic stiffness constants and massdensity, but ignores the effect of the spatial variation ofwave speed. It should be mentioned that this effect is sec-ond order in nature if the spatial variation of wave speedis not significant in comparison with that of the elastic stiff-ness constants and mass density [19,20].

The scattered wave field is assumed to by symmetricalwith respect to the y-axis. Since the time-factor exp(ixt)is a common factor in all equations, it is dropped out here-after. From Eqs. (3) and (4), we obtain

o2wox2þ a

o2woy2þ ab

owoyþ x2

c2sh

w ¼ 0 ð5Þ

where a ¼ ly0

lx0, c2

sh ¼lxðyÞqðyÞ ¼

lx0

q0.

3. Solution of the problem

For the present problem, it is convenient to dividethe functionally graded orthotropic medium into tworegions, namely material 1 in the upper half plane andmaterial 2 in the lower half plane. The governing Eq. (5)are solved by using the Fourier transform technique. Thegeneral expressions for displacement components can bewritten as

w1ðx; yÞ ¼2

p

Z 1

0

A1ðsÞec1y cosðsxÞds

w2ðx; yÞ ¼2

p

Z 1

0

A2ðsÞec2y cosðsxÞds ð6a; bÞ

176 L. Ma et al. / Composite Structures 79 (2007) 174–179

where

c1 ¼�ab�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðabÞ2 þ 4a s2 � x2

c2sh

� �r

2a;

c2 ¼�abþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðabÞ2 þ 4a s2 � x2

c2sh

� �r

2a

Now consider the boundary and the continuity condi-tions (see Fig. 1). The boundary conditions for the scat-tered wave field at y = 0 can be written as

sð1Þyz ðx; 0þÞ ¼ sð2Þyz ðx; 0

�Þ ¼ �s0; jxj 6 a

sð1Þyz ðx; 0þÞ ¼ sð2Þyz ðx; 0

�Þ; jxj > a

w1ðx; 0þÞ � w2ðx; 0�Þ ¼ 0; jxj > a ð7a–cÞ

In the present paper, the wave is vertically incident andwe only consider that s0 is positive. To solve the presentproblem, the jump of the displacement components acrossthe crack surface can be defined as follows:

f ðxÞ ¼ w1ðx; 0þÞ � w2ðx; 0�Þ; x 6 a

0; x > a

�ð8Þ

Applying the Fourier transform and the boundary con-dition, it can be obtained

c1A1ðsÞ ¼ c2A2ðsÞA1ðsÞ � A2ðsÞ ¼ �f ðsÞ ð9a; bÞ

A superposed bar indicates the Fourier transform in thispaper. It is defined as follows:

�f ðsÞ ¼Z 1

0

f ðxÞ cosðsxÞdx;

f ðxÞ ¼ 2

p

Z 1

0

�f ðsÞ cosðsxÞds ð10Þ

By solving Eq. (9) with two unknown functions, we have

A1ðsÞ ¼c2

c2 � c1

�f ðsÞ; A2ðsÞ ¼c1

c2 � c1

�f ðsÞ ð11Þ

The problem therefore reduces to the determination ofunknown functions A1(s) and A2(s). The boundary condi-tions can be applied to yield a pair of integral equations,

2

p

Z 1

0

�f ðsÞ cosðsxÞds ¼ 0; jxj > a

2

p

Z 1

0

c1c2

c2 � c1

�f 1ðsÞ cosðsxÞds ¼ � s0

ly; jxj 6 a ð12a; bÞ

Due to the assumed symmetry about the y-axis in geom-etry and the applied loading. It is sufficient to consider theproblem for 0 6 x 61 only. From the nature of the dis-placement along the crack line, it can be found that thejump of the displacement across the crack surface arefinite, differentiable and continuous functions. Therefore,

the jump of the displacement across the crack surface canbe expanded as the following series

f ðxÞ ¼X1n¼1

anP ð1=2;1=2Þ2n�2

xa

� �1� x2

a2

� �1=2

; 0 6 x 6 a

f ðxÞ ¼ 0; x > a ð13a; bÞ

where an are unknown coefficients to be determined andP ð1=2;1=2Þ

n ðxÞ is a Jacobi polynomial. The Fourier transfor-mation of Eq. (13) is [21]

�f ðsÞ ¼X1n¼1

anGn1

sJ 2n�1ðsaÞ ð14Þ

With

Gn ¼ 2ffiffiffippð�1Þn�1 C 2n� 1

2

� ð2n� 2Þ! ð15Þ

where C(x) and Jn(x) are Gamma and Bessel functions,respectively.

Substituting Eq. (13) into Eq. (12), it can be shown thatEq. (13a) is automatically satisfied. After integration withrespect to x in [0,x], Eq. (13b) reduces to

X1n¼1

anGn

Z 1

0

1

s2

c1c2

c2 � c1

J 2n�1ðsaÞ sinðsxÞds ¼ � ps0

2lyx ð16Þ

From the relation [22]

Z 1

0

1

sJ nðsnÞ sinðgsÞds ¼

sin½n sin�1ðg=nÞ�n

; n > g

nn sinðnp=2Þ

n½gþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2 � n2

q�n; g > n

8>>>><>>>>:

ð17ÞThe semi-infinite integral in Eq. (16) can be modified asZ 1

0

1

s2uðsÞJ 2n�1ðsaÞ sinðsxÞds

¼ d2n� 1

sin ð2n� 1Þ sin�1 xa

� �h i

þZ 1

0

1

s1

suðsÞ � d

�J 2n�1ðsaÞ sinðsxÞds ð18Þ

where uðsÞ ¼ c1c2

c2�c1, and lims!1

1s uðsÞ ¼ lims!1

1s

c1c2

c2�c1¼ d.

This constant can be obtained by using the Mathematicaprogram and are independent of the gradient parameterb. Also, that constant equal to one of the homogeneousmaterial case. The semi-infinite integral in the right sideof Eq. (18) can be evaluated directly, so coefficients an inEq. (16) can now be solved by the Schmidt method [23–25]. For brevity, Eq. (16) can be rewritten as

X1n¼1

anEnðxÞ ¼ UðxÞ; 0 < x < a ð19Þ

where En(x) and U(x) are known functions and coefficientsan are unknown and will be determined. A set of functionsPn(x) which satisfy the orthogonality condition

1.2

1.3

L. Ma et al. / Composite Structures 79 (2007) 174–179 177

Z a

0

P mðxÞP nðxÞdx ¼ Nndmn; N n ¼Z a

0

P 2nðxÞdx ð20Þ

can be constructed from the function En(x), such that

P nðxÞ ¼Xn

i¼1

Min

MnnEnðxÞ ð21Þ

where Min is the cofactor of the element din of determinantDn, which is defined as

Dn ¼

d11 d12 . . . d1n

d21 d22 . . . d2n

. . . . . . . . . . . .

dn1 dn2 . . . dnn

���������

���������; dij ¼

Z a

0

EiðxÞEjðxÞdx ð22Þ

From Eqs. (19)–(22), we obtain

an ¼X1j¼n

qj

Mnj

Mjjð23Þ

where

qj ¼1

Nj

Z a

0

UðxÞP jðxÞdx ð24Þ

k 3

0.0 0.2 0.4 0.6 0.8 1.0 1.20.8

1.0

1.2

1.4

1.6

1.8

βa=1.5

βa=1.0

βa=0.5

βa=0.0

a /cshω

Fig. 3. Variation of normalized stress intensity factor with normalizedwave number ax/csh at different gradient parameter ba for isotropic FGM(a = 0).

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60.8

0.9

1.0

1.1

k 3

Result of Sih [28]

Present result

a /cshω

Fig. 2. Comparison of normalized dynamic stress intensity factors of theisotropic homogeneous material for Sih’ results and present results.

4. Dynamic stress intensity factors

Since coefficient an is known, the entire perturbationstress field can be obtained. In fracture dynamic mechanics,however, it is important to determine the perturbationstress syz in the vicinity of crack tips. syz along the crackline can be expressed as

sð1Þyz ðx; 0Þ ¼2ly0

p

X1n¼1

anGn

Z 1

0

1

sc1c2

c2 � c1

J 2n�1ðsaÞ cosðsxÞds

ð25ÞObserving the expressions in Eq. (25), the singular por-

tion of the stress field can be obtained, from the relations[22]

Z 1

0

J nðsnÞcosðgsÞds¼

cos½nsin�1ðg=nÞ�ffiffiffiffiffiffiffiffiffiffiffiffiffiffin2�g2

q ; n> g

� nn sinðnp=2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffig2�n2

qgþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffig2�n2

q �n ; g> n

8>>>>>><>>>>>>:

ð26ÞThe singular part of the stress field can be expressed as

follows:

s ¼2ly0d

p

X1n¼1

anGnH nðxÞ ð27Þ

where

H nðxÞ ¼ð�1Þna2n�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 � a2p

xþffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 � a2ph i2n�1

ð28Þ

Thus, we obtain the stress intensity factor KIII as follows:

KIII ¼ limx!aþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pðx� aÞ

ps ¼

4ly0dffiffiffiap

X1n¼1

anC 2n� 1

2

� ð2n� 2Þ! ð29Þ

5. Results and discussion

Numerical calculations are carried out for several repre-sentative problems in this section. As discussed in theworks [26,27], the Schmidt method is satisfactory if the first10 terms of the infinite series in Eq. (19) are retained.

First, we restrict our attention to the dynamic crackproblem in the isotropic homogeneous materials. The nor-malized dynamic stress intensity factors k3 ¼ KIII=s0

ffiffiffiffiffiffipap

given by Sih [28] are compared with the present results inFig. 2. A very good agreement can be observed in the

0.0 0.4 0.8 1.2 1.6

0.5

1.0

1.5

2.0

α=0.5

a /cshω

k 3

α=1.0

α=1.5

α=2.0

Fig. 4. Variation of normalized stress intensity factor with normalizedwave number ax/csh at different gradient parameter a for homogeneousmaterials (b = 0).

178 L. Ma et al. / Composite Structures 79 (2007) 174–179

figure. And from these results, it can be seen that our anal-ysis process is reasonable and effective.

Fig. 3 illustrates the variation of the dynamic normal-ized stress intensity factor k3 ¼ KIII=s0

ffiffiffiffiffiffipap

with the nor-malized wave number ax/csh at different gradientparameter ba for isotropic functionally graded materials(i.e. a = 0). With the increase of ba, the peak values of nor-malized dynamic stress intensity factor at the crack tip willincrease. This phenomenon is caused by the gradient ofFGM.

For different a, the variation of the dynamic normalizedstress intensity factor k3 with the normalized wave numberax/csh are shown in Fig. 4. From this figure, it can be seenthat with the increase of a, the values of normalizeddynamic stress intensity factor at the crack tip willdecrease.

6. Conclusions

Scattering of anti-plane harmonic stress waves by afinite Griffith crack in a functionally graded orthotropicmaterial is determined in this study. The analysis is basedupon an integral transform technique. The Fredholm inte-gral equation is solved by using the Schmidt method. Thepresent method is applied to illustrate the fundamentalbehavior of a crack in FGMs under the dynamic loading.The numerical results show that the gradient and ortho-tropy of the material property has a considerable effecton the fracture behavior of the FGMs.

Acknowledgements

The present work is supported by National Natural Sci-ence Foundation of China under Grant Nos. 10502017 and10432030, National Science Foundation for DistinguishedYoung Scholars under Grant No. 10325208 and the China

Postdoctoral Science Foundation under Grant No.2005037640.

References

[1] Miyamoto Y, Kaysser WA, Rabin BH, editors. Functionally gradedmaterials: design, processing and applications. Boston: Kluwer Aca-demic publishers; 1999.

[2] Delale F, Erdogan F. The crack problem for a nonhomogeneousplane. J Appl Mech 1983;50:609–14.

[3] Konda N, Erdogan F. The mixed mode crack problem in anonhomogeneous elastic plane. Eng Fract Mech 1994;47:533–45.

[4] Shbeeb NI, Binenda WK, Kreider KL. Analysis of the driving forcesfor multiple cracks in an infinite nonhomogeneous plate, Part I:theoretical analysis. J Appl Mech 1999;66:492–500.

[5] Shbeeb NI, Binenda WK, Kreider KL. Analysis of the driving forcesfor multiple cracks in an infinite nonhomogeneous plate, Part II:numerical solution. J Appl Mech 1999;66:501–5.

[6] Jin ZH, Batra RC. Some basic fracture mechanics concepts infunctionally graded materials. J Mech Phys Solids 1996;44:1221–35.

[7] Anlas G, Santare MH, Lambros J. Numerical calculation of stressintensity factors in functionally graded materials. Int J Fract2000;104:131–43.

[8] Marur PR, Tippur HV. Numerical analysis of crack-tip fields infunctionally graded materials with a crack normal to the elasticgradient. Int J Solids Struct 2000;37:5353–70.

[9] Marur PR, Tippur HV. Evaluation of mechanical properties offunctionally graded materials. J Test Eval 1998;26:539–45.

[10] Butcher RJ, Rousseau CE, Tippur HV. A functionally gradedparticulate composite: preparation, measurements and failure anal-ysis. Acta Mater 1999;47:259–68.

[11] Carpenter RD, Liang WW, Paulino GH, Gibeling JC, Munir ZA.Fracture testing and analysis of a layered functionally gradedTi/TiB beam in 3-point bending. Mater Sci Forum 1999;308–311:837–42.

[12] Carpenter RD, Paulino GH, Munir ZA, Gibeling JC. A noveltechnique to generate sharp cracks in metallic/ceramic functionallygraded materials by reverse 4-point bending. Scripta Mater2000;43:547–52.

[13] Rousseau CE, Tippur HV. Compositionally graded materials withcracks normal to the elastic gradient. Acta Mater 2000;48:4021–33.

[14] Li H, Lambros J, Cheeseman BA, Santare MH. Experimentalinvestigation of the quasi-static fracture of functionally gradedmaterials. Int J Solids Struct 2000;37:3715–32.

[15] Sampath S, Herman H, Shimoda N, Saito T. Thermal sprayprocessing of FGMs. MRS Bull 1995;20(1):27–31.

[16] Kaysser WA, Ilschner B. FGM research activities in Europe. MRSBull 1995;20(1):22–6.

[17] Ozturk M, Erdogan F. Mode I crack problem in an inhomogeneousorthotropic medium. Int J Eng Sci 1997;35(9):869–83.

[18] Chen J, Liu Z, Zou Z. Transient internal crack problem for anonhomogeneous orthotropic strip (mode I). Int J Eng Sci2002;40:1761–74.

[19] Meguid SA, Wang XD, Jiang LY. On the dynamic propagation of afinite crack in functionally graded materials. Eng Fract Mech2002;69:1753–68.

[20] Jiang LY, Wang XD. On the dynamic propagation in an interphasewith spatially varying elastic properties under inplane loading. Int JFract 2002;114:225–44.

[21] Erdelyi A, editor. Table of the integral transforms, vol. 1. NewYork: McGraw-Hill; 1954.

[22] Gradshteyn IS, Ryzhik IM. Table of integral, series and prod-ucts. New York: Academic Press; 1980.

[23] Morse PM, Feshbach H. Methods of theoretical physics, vol. 1. NewYork: McGraw-Hill; 1958.

L. Ma et al. / Composite Structures 79 (2007) 174–179 179

[24] Yau WF. Axisymmetric slipless indentation of an infinite elasticcylinder. SIAA J Appl Math 1967;15:219–27.

[25] Itou S. Three-dimensional waves propagation in a cracked elasticsolid. ASME J Appl Mech 1978;45:807–11.

[26] Zhou ZG, Shen YP. Investigation of the scattering of harmonic shearwaves by two collinear cracks using the non-local theory. ACTAMech 1999;135:169–79.

[27] Ma L, Wu L-Z, Zhou Z-G. Dynamic stress intensity factors aroundtwo parallel cracks in a functionally graded layer bonded to dissimilarhalf planes subjected to anti-plane incident time-harmonic stresswave. Int J Eng Sci 2004;42(2):187–202.

[28] Sih GC. Mechanics of Fracture 4: elastodynamic crack prob-lems. Leyden: Noordhoff International Publishing; 1977.