This article was downloaded by: [Brown University]On: 21 May 2013, At: 04:10Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Philosophical Magazine LettersPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tphl20

Dynamic stress around two holesburied in a functionally gradedpiezoelectric material layer underelectro-elastic wavesXue-Qian Fang a , Jin-Xi Liu a , Xiao-Hua Wang b & Le-Le Zhang aa Department of Engineering Mechanics, Shijiazhuang RailwayInstitute, Shijiazhuang 050043, P.R. Chinab School of Computing and Informatics, Shijiazhuang RailwayInstitute, Shijiazhuang 050043, P.R. ChinaPublished online: 22 Mar 2010.

To cite this article: Xue-Qian Fang , Jin-Xi Liu , Xiao-Hua Wang & Le-Le Zhang (2010): Dynamicstress around two holes buried in a functionally graded piezoelectric material layer under electro-elastic waves, Philosophical Magazine Letters, 90:5, 361-380

To link to this article: http://dx.doi.org/10.1080/09500831003680752

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 study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representationthat the contents will be complete or accurate or up to date. The accuracy of anyinstructions, formulae, and drug doses should be independently verified with primarysources. The publisher shall not be liable for any loss, actions, claims, proceedings,demand, or costs or damages whatsoever or howsoever caused arising directly orindirectly in connection with or arising out of the use of this material.

Philosophical Magazine LettersVol. 90, No. 5, May 2010, 361–380

Dynamic stress around two holes buried in a functionally graded

piezoelectric material layer under electro-elastic waves

Xue-Qian Fanga*, Jin-Xi Liua, Xiao-Hua Wangb and Le-Le Zhanga

aDepartment of Engineering Mechanics, Shijiazhuang Railway Institute, Shijiazhuang050043, P.R. China; bSchool of Computing and Informatics, Shijiazhuang Railway

Institute, Shijiazhuang 050043, P.R. China

(Received 21 August 2009; final version received 3 February 2010)

A theoretical method is presented to study the multiple scatteringof electro-elastic waves resulting from two subsurface holes in a function-ally graded piezoelectric material (FGPM) layer bonded to a homogeneouspiezoelectric material, and the dynamic stress around the holes is alsopresented. The analytical solutions of wave fields are expressed byemploying wave function expansion method, and the expanded modecoefficients are determined by satisfying the boundary conditions at thesurface and around the holes. The mechanical and electrical boundaryconditions at the free surface of the structure are satisfied by using theimage method. Analyses show that the piezoelectric property and thedistance between the two holes express great effect on the dynamic stressaround the holes, and the effect increases with the decrease of the thicknessof FGPM layer. If the material properties of the homogeneous piezoelectricmaterial are greater than those at the surface of the structure, the dynamicstress increases dramatically due to the piezoelectric property and thedistance between the two holes. The angular distribution of electricdisplacement under different parameters is also presented. The accuracyand efficiency of the solving method are demonstrated by the comparisonof selected results with the solutions obtained by using finite elementsoftware ANSYS.

Keywords: functionally graded piezoelectric material layer; two circularholes; dynamic stress concentration factor; image method

1. Introduction

Piezoelectric materials have long been attractive materials in the smart systemsof aerospace, automotive, medical, and electronic fields due to the intrinsic couplingcharacteristics between their electric and mechanical fields. However, as piezoelectricmaterials are being extensively used as actuators or transducers in the technologiesof smart and adaptive systems, the mechanical reliability and durability of thesematerials become increasingly important. To meet the demand of advancedpiezoelectric materials in lifetime and reliability and with the help of the development

*Corresponding author. Email: fangxueqian@163.com

ISSN 0950–0839 print/ISSN 1362–3036 online

� 2010 Taylor & Francis

DOI: 10.1080/09500831003680752

http://www.informaworld.com

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3

in modern material processing technology, the concept of functionally gradedmaterials has recently been extended into the piezoelectric materials [1]. Wavetheory can simulate many types of loading (static loading, dynamic loading,coupling loading, and so on). The study of wave propagation in functionallygraded piezoelectric material (FGPMs) layers can provide great help in obtaininghigher strength and toughness of materials. It is also a theoretical backgroundof the non-destructive analysis of FGPM microstructures by using ultrasonictechnique.

Numerical modeling, simulation, and analysis of FGPMs with discontinuities,such as holes, cracks, and inclusions have attracted lots of interests in recent years.Understanding how geometrical discontinuities influence the stress distributionin FGPMs is critical for design applications. A considerable amount of work hasbeen completed with regard to the determination of static stress concentrationfactors for common discontinuities [2–5]. The understanding of dynamic stressin FGPMs subjected to dynamic loading is more important to many areas of designincluding crashworthiness, high-speed impact, and transportation of hazardousmaterials. However, considering the complexity of wave scattering resulting from thenon-homogeneous property of FGPMs and the complexity of multiple scatteringfrom the scatterer and the boundary, relatively little work has been done with regardto the wave propagation in FGPMs. Recently, Chen et al. [6] have considered theelectromechanical impact response of FGPMs with a crack using integral transformtechnique. Ma et al. [7] have investigated the stress and electric displacementintensity factors of two collinear cracks subjected to anti-plane shear waves inFGPMs. Most recently, Fang et al. have studied the dynamic stress from a circularcavity [8] and a circular inclusion [9] buried in a FGPM layer, and both thedisplacement field and the piezoelectric field were considered. In engineering,investigation on two discontinuities exist in FGPMs is more representative.However, the multiple interactions of electro-elastic waves between the two holesin a FGPM layer make this problem more complex.

The main objective of this article is to investigate the multiple scattering ofelectro-elastic waves resulting from two subsurface holes in a FGPM layer bondedto a homogeneous piezoelectric material, and the mutual influence of the two holesare also considered. The incidence of anti-plane shear waves at the surface of thestructure is applied, and both the displacement field and piezoelectric field inFGPMs are presented. The mechanical and electrical boundary conditions at the freesurface are satisfied by using the image method. Wave function expansion method isapplied to express the wave fields and electric potentials [10]. The expanded modecoefficients are determined by satisfying the boundary conditions at the free surfaceand around the two holes. Addition theorem for Bessel functions is used toaccomplish the translation between different coordinate systems of the actual andimage holes. The analytical solutions of the dynamic stress concentration factor(DSCF) and electric displacement around the holes are presented, and the numericalsolutions are graphically illustrated. The ANSYS program (copyright by SAS IP,Inc.) is used to validate the solving method in this article. The effects of thepiezoelectric property, the incident wave number and the relative position of the twosubsurface holes in the FGPM layer on the DSCFs and electric displacement arealso analyzed.

362 X.-Q. Fang et al.

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3

2. Wave motion equations in FGPMs and their solutions

Consider a FGPM layer bonded to a homogeneous piezoelectric material, asdepicted in Figure 1. The mechanical and electrical short conditions at the free surfaceare considered. c144, e

115, "

111, and �

1 are the elastic stiffness, piezoelectric constant,dielectric constant, and density of materials at the surface of FGPM layer, and c244,e215, "

211, and �2 those of the homogeneous piezoelectric material. The material

properties in FGPM layer vary smoothly along the x direction. Let two circular holeslie in the FGPM layer. The radii of the two holes are a and b, respectively. The distancebetween the centers of the two holes is d. The distances between the centers of thetwo holes and the upper edge of the layer are the same, and denoted as h1, and thatbetween the centers of the holes and the lower edge of the FGPM layer is h2.

All materials exhibit transversely isotropic behavior and are polarized in thez-direction. Let an anti-plane shear wave with frequency ! hit the surface of theFGPM layer in the positive x-direction. A state of anti-plane strain is consideredsuch that the piezoelectric solids experience displacement only in the z-direction.The governing equations can be simplified, if we are only interested in theout-of-plane displacement component and in-plane electric components, i.e.

ux ¼ uy ¼ 0, u ¼ uzðx, y, tÞ, ð1Þ

Ex ¼ Exðx, y, tÞ, Ey ¼ Eyðx, y, tÞ, Ez ¼ 0, ð2Þ

in which the electric field intensities Ex and Ev are related to the electric potential �by the following relations:

Ex ¼ �@�

@x, Ey ¼ �

@�

@y: ð3Þ

The mechanically and electrically coupled constitutive equations can be writtenas follows:

�zx ¼ c44ðxÞ@u

@xþ e15ðxÞ

@�

@x, �zy ¼ c44ðxÞ

@u

@yþ e15ðxÞ

@�

@y, ð4Þ

Dx ¼ e15ðxÞ@u

@x� "11ðxÞ

@�

@x, Dy ¼ e15ðxÞ

@u

@y� "11ðxÞ

@�

@y, ð5Þ

ax

ay

bx

by

1h

2h

aobo

Anti-plane shear waves

Distribution of material properties

aθarbθ br

d

Figure 1. Schematic representation of the subsurface holes and the incident elastic waves ina FGPM layer.

Philosophical Magazine Letters 363

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3

where �zi, u, Dj, and � ( j¼ x, y) are the shear stress, anti-plane displacement, in-planeelectric displacement, and electric potential, respectively; c44(x) is the elastic stiffnessof graded materials measured in a constant electric field, "11(x) is the dielectricconstant of graded materials measured in constant strain, and e15(x) is the

piezoelectric constant of graded materials.The anti-plane governing equation and Maxwell’s equation in FGPMs are

described as follows:

@�zx@xþ@�zy@y¼ �ðxÞ

@2u

@t2, ð6Þ

@Dx

@xþ@Dy

@y¼ 0: ð7Þ

Substituting Equations (4) and (5) into (6) and (7), the following equations can beobtained:

@c44ðxÞ

@x

@u

@xþ c44ðxÞ

@2u

@x2þ@e15ðxÞ

@x

@�

@xþ e15ðxÞ

@2�

@x2þ c44ðxÞ

@2u

@y2þ e15ðxÞ

@2�

@y2¼ �ðxÞ

@2u

@t2,

ð8Þ

@e15ðxÞ

@x

@u

@xþ e15ðxÞ

@2u

@x2�@"11ðxÞ

@x

@�

@x� "11ðxÞ

@2�

@x2þ e15ðxÞ

@2u

@y2� "11ðxÞ

@2�

@y2¼ 0: ð9Þ

For mathematical convenience, it is assumed that all material properties varycontinuously, and have the same exponential function distribution along the x

direction in the layer, i.e.

c44ðxÞ ¼ c044e2�x, e15ðxÞ ¼ e015e

2�x, "11ðxÞ ¼ "011e

2�x, �ðxÞ ¼ �0e2�x: ð10Þ

According to the continuous condition of the material properties in the layer andat the position of x¼ h2, the constants �, c044, e015, "

011, and �0can be calculated

as follows:

� ¼1

2ðh1 þ h2Þln

c244c144

� �, ð11Þ

c044 ¼ c144e2�h1 , e015 ¼ e115e

2�h1 , "011 ¼ "111e

2�h1 , �0 ¼ �1e2�h1 : ð12Þ

In the above formulations, it is assumed that the ratio c244=c144 is equal to e215=e

115,

"211="111, and �2/�1. Though the variations are unrealistic, it would allow us to

comprehend the effects of material properties of FGPMs and the distance betweenthe two holes on the dynamic stress around the holes, and can provide references

for the non-destructive detection in FGPMs.Substituting Equation (10) into (8) and (9), the following equations are obtained:

2�c044@u

@xþ c044r

2uþ 2�e015@�

@xþ e015r

2� ¼ 2��0@2u

@t2, ð13Þ

2�e015@u

@xþ e015r

2u ¼ 2�"011@�

@xþ "011r

2�: ð14Þ

364 X.-Q. Fang et al.

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3

Here, r2 ¼ @2=@x2 þ @2=@y2 is the two-dimensional Laplace operator in the variables

x and y.Assume that another electro-elastic field is expressed as follows:

¼ �� �1u, ð15Þwhere �1 ¼ e015="

011.

From Equations (13) and (14), the following equations can be obtained:

r2uþ 2�@u

@x¼

1

c2SH

@2u

@t2, ð16Þ

2�@

@xþ r2 ¼ 0, ð17Þ

where cSH ¼ffiffiffiffiffiffiffiffiffiffiffiffi�e=�0

pwith �e ¼ c044 þ ðe

015Þ

2=ð"011Þ being the wave speed of electro-

elastic waves.The steady solution of this problem is investigated. Assuming that u ¼ u0Ue�i!t,

Equation (16) can be changed into

r2Uþ 2�@U

@xþ k2U ¼ 0, ð18Þ

where ! is the frequency of the incident waves, and k ¼ !=cSH is the wave number

of incident waves.To solve Equation (18), the solution can be proposed as follows:

U ¼ e��xwðx, yÞ, ð19Þ

where w(x, y) is the function introduced for derivation.Substituting Equation (19) into (18), one can see that the function w(x, y) should

satisfy the following equation:

r2wþ �2w ¼ 0, ð20Þ

where � ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk2 � �2Þ

p. It is noted that k4� should be satisfied.

According to Equations (15)–(17), one can see that there exist electro-elastic

waves with the form of u ¼ u0Ue�i!t ¼ u0 expð��xÞeið�x�!tÞ, which denotes the

propagating wave with its amplitude of vibration attenuating in the xdirection.Similarly, the solution of in Equation (17) has the following form:

¼ 0e��xeiði�x�!tÞ: ð21Þ

Note that all field quantities have the same time variation e�i!t that is suppressed

in all subsequent representations for notational convenience.According to Equations (17) and (20), the general solutions of the scattered field

of electro-elastic waves resulting from the embedded holes in FGPMs can be

described, using wave function expansion method [10], as follows:

us ¼ e��r cos X1

n¼�1

anHð1Þn ð�rÞe

in, ð22Þ

s ¼ e��r cos X1

n¼�1

bnHð1Þn ði�rÞe

in, ð23Þ

Philosophical Magazine Letters 365

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3

where (r, ) is the corresponding cylindrical coordinate system shown in Figure 1,Hð1Þn ð�Þ is the nth Hankel function of the first kind, and an and bn determined bysatisfying the boundary conditions are the mode coefficients of the scattered waves.Note that Hankel function Hð1Þn ð�Þ denotes the outgoing wave and satisfies theradiation condition at infinity. The solution of the scattered–reflected waves has thesame form as that of the scattered waves [8].

3. Boundary conditions of this structure

The mechanical and electrically open conditions at the free surface of the FGPM

layer can be given as follows:

�xzð�h1, yÞ ¼ 0, Dxð�h1, yÞ ¼ 0: ð24Þ

Without loss of generality, the case that the two holes are free of traction isinvestigated. For the holes, the boundary conditions around it are that the radialshear stress is equal to zero, and the electric potential and normal electricdisplacement are continuous. They can be expressed as follows:

�rzjr¼p¼@ut

@r

����r¼p

þ �2@�t

@r

����r¼p

¼ 0, ð p ¼ a, bÞ, ð25Þ

Djr¼p¼ e15@ut

@r

����r¼p

�"11@�t

@r

����r¼p

¼ �"0@�I

@r

����r¼p

, ð p ¼ a, bÞ, ð26Þ

�tjr¼p ¼ �Ijr¼p, ð p ¼ a, bÞ, ð27Þ

where �2 ¼ e015=c044 and �3 ¼ "

011="0. Superscripts t and I denote the total wave field

and the electric potential inside the holes. Note that "0 ¼ 8:85� 10�12 F=m is thedielectric constant of vacuum.

4. The multiple scattering of electro-elastic waves around the two holes and

the total wave field

Consider the electro-elastic waves propagating along the positive x direction in the

FGPM structure. In the local coordinate systems (rp, p) of the holes, the incidentwaves can be expanded as follows [10].

uðinÞ1p ¼ u0e

��xpþh1ði���Þei�xp ¼ u0e��rp cos pþh1ði���Þ

X1n¼�1

inJnð�rpÞeinp , ð p ¼ a, bÞ,

ð28Þ

where u0 is the amplitude of the incident waves, � is the wave number of thepropagating waves, and Jnð�Þ is the nth Bessel function of the first kind. Note that a

366 X.-Q. Fang et al.

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3

and b denote, respectively, the two holes in Figure 1, and the superscript (in) denotes

the incident waves.Similarly, the incident field (in) is expressed as follows:

ðinÞ1p ¼ �1u0e��ðxpþ2h1Þe��xp ¼ �1u0e

��ðrp cos pþ2h1ÞX1

n¼�1

inJnði�rpÞeinp , ð p ¼ a, bÞ:

ð29Þ

In the local coordinate systems (rp, p) of the holes, the scattered field can be

described as follows:

uðsÞ1p ¼ e��rp cos p

X1l¼1

X1n¼�1

Aln1H

ð1Þn ð�rpÞe

inp

¼ e��rp cos pX1

n¼�1

�Apn1H

ð1Þn ð�rpÞe

inp , ð p ¼ a, bÞ, ð30Þ

ðsÞ1p ¼ �1e��rp cos p

X1l¼1

X1n¼�1

Bln1H

ð1Þn ði�rpÞe

inp

¼ �1e��rp cos p

X1n¼�1

�Bpn1H

ð1Þn ði�rpÞe

inp , ð p ¼ a, bÞ, ð31Þ

where �An ¼P1

l¼1 Aln and �Bn ¼

P1l¼1 B

ln are the total scattering coefficients of the

two holes. Note that l denotes the lth mode coefficients of scattered waves.When the electro-elastic wave propagates in the FGPM layer, it is scattered by

the two holes at first. Then, the outgoing scattered wave from one hole is not only

reflected on the straight surface ðxa ¼ xb ¼ �h1Þ, but is scattered by the other hole.

In addition, the scattered and reflected waves at the surface are scattered by the two

holes again. This complex phenomenon is shown in Figure 1.To satisfy the mechanical and electrical boundary conditions at the free surface,

the image method is applied. In Figure 2, the reflected waves at the edge of FGPM

layer are described by the scattered waves resulting from the virtual image holes.

The distance between the virtual image holes and the straight boundary is also h1.

The magnitudes of the incident waves and scattered waves of the actual and image

holes are the same, however, the directions of them are opposite. So, the boundary

conditions at the free surface can be satisfied.For the image holes, the waves propagate in the negative x0 direction, and can be

expressed as follows:

uðinÞ2p ¼ u0e

�x0pþh1ði���Þe�i�x0p ¼ u0e

�x0pþh1ði���ÞX1

n¼�1

i�nJnð�r0pÞe

in0p , ð p ¼ a, bÞ, ð32Þ

ðinÞ2p ¼ �1u0e�x0p�2�h1e�iði�Þx

0p ¼ �1u0e

�x0p�2�h1X1

n¼�1

i�nJnði�r0pÞe

in0p , ð p ¼ a, bÞ: ð33Þ

Philosophical Magazine Letters 367

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3

Likewise, in the local coordinate systems ðr0p, 0pÞ, the scattered fields resulting

from the image holes can be described as follows:

uðsÞ2p ¼ e�r

0p cos

0p

X1n¼�1

�Apn2H

ð1Þn ð�r

0pÞe

in0p , ð p ¼ a, bÞ, ð34Þ

ðsÞ2p ¼ �1e�r0p cos

0p

X1n¼�1

�Bpn2H

ð1Þn ði�r

0pÞe

in0p , ð p ¼ a, bÞ: ð35Þ

where �Apn2 and �B

pn2 determined by satisfying the boundary conditions are the

scattering mode coefficients resulting from the image holes.From Equations (28), (29), (32), and (33), the total incident fields of elastic

and electro-elastic waves, the original incident wave plus their image as the reflected

wave from the free surface, are written as, respectively,

uðinÞ ¼ uðinÞ1p þ u

ðinÞ2p , ð p ¼ a, bÞ: ð36Þ

�ðinÞ ¼ �1 uðinÞ1p þ u

ðinÞ2p

� �þ ðinÞ1p þ

ðinÞ2p , ð p ¼ a, bÞ: ð37Þ

Then, the total elastic field u(t) and total electro-elastic field �ðtÞ around the two

holes are produced by the superposition of the incident fields, the scattered fields,

and the reflected fields at the free surface, i.e.

uðtÞ ¼ uðinÞ þ uðsÞ1a þ u

ðsÞ1b þ u

ðsÞ2a þ u

ðsÞ2b , ð38Þ

�ðtÞ ¼ �ðinÞ þ �1 uðsÞ1a þ u

ðsÞ1b þ u

ðsÞ2a þ u

ðsÞ2b

� �þ ðsÞ1a þ

ðsÞ2a þ

ðsÞ1b þ

ðsÞ2b : ð39Þ

ax

ay

bx

by

1h

2h aobo

bx′

bo′

ax′

ay′ao′by′

ar

aθ

br

bθ

br′

bθ ′ ar′aθ′1h

Figure 2. Sketch of the image method at the surface of FGPMs.

368 X.-Q. Fang et al.

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3

Now, from the boundary conditions (24) at the surface of the structure, thefollowing can be obtained:

@uðtÞ

@x¼@�ðtÞ

@x¼ 0, at x ¼ �b: ð40Þ

From Equations (36) and (37), it is clear that the total incident fields havesatisfied these conditions.

Applying the boundary conditions (40) to Equations (38) and (39), and from thefact that ra ¼ r0a, rb ¼ r0b,

0a ¼ � a, and

0b ¼ � b on the plane boundary of the

semi-infinite structure, and the identity Hð1Þn ein ¼ Hð1Þ�n, the following relationsbetween the total mode coefficients of scattered waves are obtained:

�Apn2 ¼ �

�Apn1,

�Bpn2 ¼ �

�Bpn1: ð41Þ

From Equation (41), the relations between the total scattering coefficients of theimage holes and those of the actual holes are obtained.

Then, by using the following translational addition theorems of Besselfunctions [11]

Hð1Þn ðkr2Þein2 ¼

X1m¼�1

eiðn�mÞ1,2Hð1Þm�nðkr1,2ÞJmðkr1Þeim1 , ð42Þ

the scattered fields uðsÞ2p and ðsÞ2p of the image holes can be represented in the local

coordinate systems (rp, p). After some manipulations, the total scattered fieldsaround the actual holes a and b are, respectively, expressed as follows:

uðsÞa ¼ uðsÞ1a þ u

ðsÞ2a þ u

ðsÞ1b þ u

ðsÞ2b

¼ e��ra cos aX1

n¼�1

�Aan1H

ð1Þn ð�raÞe

ina þX1

n¼�1

X1m¼�1

�Abm1hð1ÞmnJnð�raÞe

ina

" #

þ e�ð2h1þra cos aÞX1

n¼�1

X1m¼�1

�Aam1�hð2ÞmnJnðkraÞe

ina þX1

n¼�1

X1m¼�1

�Abm1�hð3ÞmnJnðkraÞe

ina

" #,

ð43Þ ðsÞa ¼

ðsÞ1a þ

ðsÞ2a þ

ðsÞ1b þ

ðsÞ2b

¼ e��ra cos aX1

n¼�1

�Ban1H

ð1Þn ði�raÞe

ina þX1

n¼�1

�Bbn1hð4ÞmnJnði�raÞe

ina

" #

þ e�ð2h1þra cos aÞX1

n¼�1

X1m¼�1

�Bam1�hð5ÞmnJnði�raÞe

ina þX1

n¼�1

X1m¼�1

�Bbm1�hð6ÞmnJnði�raÞe

ina

" #,

ð44ÞuðsÞb ¼ u

ðsÞ1b þ u

ðsÞ2b þ u

ðsÞ1a þ u

ðsÞ2a

¼ e��rb cos bX1

n¼�1

�Abn1H

ð1Þn ð�rbÞe

inb þX1

n¼�1

�Aan1hð7Þmnð�rbÞe

inb

" #

þ e�ð2h1þrb cos bÞX1

n¼�1

X1m¼�1

�Abm1�hð8ÞmnJnð�rbÞe

inb þX1

n¼�1

X1m¼�1

�Aam1�hð9ÞmnJnð�rbÞe

inb

" #,

ð45Þ

Philosophical Magazine Letters 369

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3

ðsÞb ¼ ðsÞ1b þ

ðsÞ2b þ

ðsÞ1a þ

ðsÞ2a

¼ e��rb cos bX1

n¼�1

�Bbn1H

ð1Þn ði�rbÞe

inb þX1

n¼�1

�Ban1hð10Þmn ði�rbÞe

inb

" #

þ e�ð2h1þrb cos bÞX1

n¼�1

X1m¼�1

�Bbm1�hð11Þmn Jnði�rbÞe

inb þX1

n¼�1

X1m¼�1

�Bam1�hð12Þmn Jnði�rbÞe

inb

" #,

ð46Þ

where �hðiÞmn ði ¼ 1, 2, . . . , 12Þ are shown in the Appendix.Inside the two holes, the elastic wave field vanishes, and only the electric field

exists. The electric potential inside the actual holes is standing wave, and is expressed

as follows:

�Ip ¼ �1X1

n¼�1

Cpn1Jnði�rpÞe

inp , ð p ¼ a, bÞ, ð47Þ

where Cpn1 are the mode coefficients of the electric potential inside the actual holes.

5. Determination of scattering mode coefficients and DSCF

By substituting the expressions of wave fields into the boundary conditions of the

two actual holes, one can obtain a set of system of linear equations for the unknown

total scattering coefficients �Aan1,

�Abn1,

�Ban1,

�Bbn1,

�Can1,

�Cbn1

� . Multiplying by e�is at both

sides of these linear equations, and then integrating from � to , the following

simplified formula can be obtained:

ð1þ �1�2Þ e��a cos a �a cos a F1aH

�Aas1 þ F1a

J

X1m¼�1

�hð1Þms�Abm1

!"(

þ G1aH

�Aas1 þ G1a

J

X1m¼�1

�hð1Þms�Abm1

#

þ e�ð2h1þa cos aÞ �a cos aF1aJ

X1m¼�1

�hð2Þms�Aam1 þ

X1m¼�1

�hð3Þms�Abm1

!"

þ G1aJ

X1m¼�1

�hð2Þms�Aam1 þ

X1m¼�1

�hð3Þms�Abm1

!#)

þ �1�2 e��a cos a �a cos a F2aH

�Bas1 þ F2a

J

X1m¼�1

�hð4Þms�Bbm1

!"(

þ G2aH

�Bas1 þ G2a

J

X1m¼�1

�hð4Þms�Bbm1

#

370 X.-Q. Fang et al.

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3

þ e�ð2h1þa cos aÞ �a cos aF2aJ

X1m¼�1

�hð5Þms�Bam1 þ

X1m¼�1

�hð6Þms�Abm1

!"

þ G2aJ

X1m¼�1

�hð5Þms�Bam1 þ

X1m¼�1

�hð6Þms�Abm1

!#)

¼ �ð1þ �1�2Þ ise��a cos aþh1ði���Þ þ i�se�a cos aþh1ð��i�Þ �

F1aJ

� �1�2 ise��ða cos aþ2h1Þ þ i�se�a cos a �

F2aJ ; ð48Þ

ð1þ �1�2Þ e��a cos b �a cos b F1bH

�Abs1 þ F1b

J

X1m¼�1

�hð7Þms�Aam1

!"(

þ G1bH

�Abs1 þ G1b

J

X1m¼�1

�hð7Þms�Aam1

#

þ e�ð2h1þa cos bÞ �a cos bF1bJ

X1m¼�1

�hð8Þms�Abm1 þ

X1m¼�1

�hð9Þms�Aam1

!"

þ G1bJ

X1m¼�1

�hð8Þms�Abm1 þ

X1m¼�1

�hð9Þms�Aam1

!#)

þ �1�2 e��a cos b �a cos b F2bH

�Bbs1 þ F2a

J

X1m¼�1

�hð10Þms�Bam1

!"(

þ G2bH

�Bbs1 þ G2b

J

X1m¼�1

�hð10Þms�Bam1

#

þ e�ð2h1þa cos aÞ �a cos aF2aJ

X1m¼�1

�hð11Þms�Bam1 þ

X1m¼�1

�hð12Þms�Abm1

!"

þ G2bJ

X1m¼�1

�hð11Þms�Bam1 þ

X1m¼�1

�hð12Þms�Abm1

!#)

¼ �ð1þ �1�2Þ ise��b cos bþh1ði���Þ þ i�se�b cos bþh1ð��i�Þ �

F1bJ

� �1�2 ise��ðb cos b�h1Þ þ i�se�ðb cos bþ2h1Þ �

F2bJ ; ð49Þ

�3 e��a cos a �a cos a F2aH

�Bas1 þ F2a

J

X1m¼�1

�hð4Þms�Bbm1

!"(

þG2aH

�Bas1 þ G2a

J

X1m¼�1

�hð4Þms�Bbm1

#

þ e�ð2h1þa cos aÞ �a cos aF2aJ

X1m¼�1

�hð5Þms�Bam1 þ

X1m¼�1

�hð6Þms�Abm1

!"

þG2aJ

X1m¼�1

�hð5Þms�Bam1 þ

X1m¼�1

�hð6Þms�Abm1

!#)

� Cas1G

2aJ ¼ ��3 ise��ða cos a��h1Þ þ i�se�ða cos aþ2h1Þ

�F2aJ

Philosophical Magazine Letters 371

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3

�3 e��b cos b �b cos b F2bH

�Bbs1 þ F2b

J

X1m¼�1

�hð10Þms�Bam1

!"(

þG2bH

�Bbs1 þ G2b

J

X1m¼�1

�hð10Þms�Bam1

#

þ e�ð2h1þb cos bÞ �b cos bF2bJ

X1m¼�1

�hð11Þms�Bbm1 þ

X1m¼�1

�hð12Þms�Aam1

!"

þG2bJ

X1m¼�1

�hð11Þms�Bbm1 þ

X1m¼�1

�hð12Þms�Aam1

!#)

� Cbs1G

2bJ ¼ ��3 ise��ðb cos b��h1Þ þ i�se�ðb cos bþ2h1Þ

� F2bJ ; ð50Þ

e��a cos a �Aas1F

1aH þ F1a

J

X1m¼�1

�hð1Þms�Abm1

!(

þ e�ð2h1þa cos aÞF1aJ

X1m¼�1

�hð2Þms�Aam1 þ

X1m¼�1

�hð3Þms�Abm1

!)

þ e��a cos a F2aH

�Bas1 þ F2a

J

X1m¼�1

�hð4Þms�Bbm1

!

þ e�ð2h1þa cos aÞF2aJ

X1m¼�1

�hð5Þms�Bam1 þ

X1m¼�1

�hð6Þms�Abm1

!� Ca

s1F2aJ

¼ �u0 ise��a cos aþh1ði���Þ þ i�se�a cos aþh1ð��i�Þ �

F1aJ

�þ ise��ða cos a��h1Þ þ i�se�ða cos aþ2h1Þ �

F2aJ

; ð51Þ

e��b cos b �Abs1F

1bH þ F1b

J

X1m¼�1

�hð7Þms�Aam1

!(

þ e�ð2h1þb cos bÞF1bJ

X1m¼�1

�hð8Þms�Abm1 þ

X1m¼�1

�hð9Þms�Aam1

!)

þ e��b cos b F2bH

�Bbs1 þ F2b

J

X1m¼�1

�hð10Þms�Bam1

!

þ e�ð2h1þb cos bÞF2bJ

X1m¼�1

�hð11Þms�Bbm1 þ

X1m¼�1

�hð12Þms�Aam1

!� Cb

s1F2bJ

¼ �u0 ise��b cos bþh1ði���Þ þ i�se�b cos bþh1ð��i�Þ �

F1bJ

�þ ise��ða cos a��h1Þ þ i�se�ða cos aþ2h1Þ �

F2aJ

: ð52Þ

Here, the following notations are used:

F1pH ¼ Hð1Þn ð�pÞ, ð53Þ

372 X.-Q. Fang et al.

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3

F1pJ ¼ Jnð�pÞ, ð54Þ

G1pH ¼ sHð1Þs ð�pÞ � �pH

ð1Þsþ1ð�pÞ

h i, ð55Þ

G1pJ ¼ sJsð�pÞ � �pJsþ1ð�pÞ½ �, ð56Þ

F2pH ¼ Hð1Þn ði�pÞ, ð57Þ

F2pJ ¼ Jnði�pÞ, ð58Þ

G2pH ¼ sHð1Þs ði�pÞ � i�pHð1Þsþ1ði�pÞ

h i, ð59Þ

G2pJ ¼ sJsði�pÞ � i�pJsþ1ði�pÞ½ �, ð p ¼ a, bÞ: ð60Þ

According to the DSCF definition, the DSCF is the ratio of the hoop shear stress

around the holes and the maximum stress resulting from the incident waves [10].

Thus, the DSCF around the circular hole in FGPMs is expressed as follows:

DSCF ¼ ��z ¼�z�0

��������: ð61Þ

Here,

�z ¼1

rc44@ut

@þ e15

@�t

@

� : ð62Þ

It should be noted that �0 ¼ u0�ek denotes the maximum stress resulting from the

incident waves.Thus, the DSCF around the hole a in FGPMs is expressed as follows:ZDSCF ¼

e2�a cos a

ka

@ut

@þe015�e

@ t

@

� �

¼e�a cos a

kaeh1ði���Þ �a sin a

X1n¼�1

inJnð�aÞeina þ

X1n¼�1

inþ1nJnð�aÞeina

" #(

þ e2�a cos a�h1ði���Þ ��a sin aX1

n¼�1

i�nJnð�aÞeina þ

X1n¼�1

i1�nnJnð�aÞeina

" #

þ �a sin aX1

n¼�1

Aan1H

ð1Þn ð�aÞe

ina þX1

n¼�1

X1m¼�1

�Abm1hð1ÞmnJnð�aÞe

ina

!"

þX1

n¼�1

inAan1H

ð1Þn ð�aÞe

ina þX1

n¼�1

X1m¼�1

in �Abm1hð1ÞmnJnð�aÞe

ina

!#

þ e2�ðh1þa cos aÞ ��a sin aX1

n¼�1

X1m¼�1

�Aam1�hð2ÞmnJnð�aÞe

ina

"

þX1

n¼�1

X1m¼�1

�Abm1�hð3ÞmnJnð�aÞe

ina

!

Philosophical Magazine Letters 373

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3

þX1

n¼�1

X1m¼�1

in �Aam1�hð2ÞmnJnð�aÞe

ina þX1

n¼�1

X1m¼�1

in �Abm1�hð3ÞmnJnð�aÞe

ina

" #)

þ�1e

015

�ee�a cos a�2�h1 �a sin a

X1n¼�1

inJnði�aÞeina þ

X1n¼�1

inþ1nJnði�aÞeina

" #(

þ e2�ða cos aþh1Þ ��a sin aX1

n¼�1

i�nJnði�aÞeina þ

X1n¼�1

i1�nnJnði�aÞeina

" #

þ e�a cos a �a sin aX1

n¼�1

�Ban1H

ð1Þn ði�aÞe

ina þX1

n¼�1

�Bbn1hð4ÞmnJnði�aÞe

ina

!" #

þX1

n¼�1

in �Ban1H

ð1Þn ði�aÞe

ina þX1

n¼�1

in �Bbn1hð4ÞmnJnði�aÞe

ina

" #

þ e�ð2h1þ3a cos aÞ ��a sin aX1

n¼�1

X1m¼�1

�Bam1�hð5ÞmnJnði�aÞe

ina

"

þX1

n¼�1

X1m¼�1

�Bbm1�hð6ÞmnJnði�aÞe

ina

!

þX1

n¼�1

X1m¼�1

in �Bam1�hð5ÞmnJnði�aÞe

ina þX1

n¼�1

X1m¼�1

in �Bbm1�hð6ÞmnJnði�aÞe

ina

):

ð63Þ

The electric displacement around the hole a is expressed as follows:

D�a ¼ e�ða cos aþ2h1Þ �a sin aX1

n¼�1

inJnði�aÞeina þ

X1n¼�1

inþ1nJnði�aÞeina

" #

� ��a sin aX1

n¼�1

i�nJnði�aÞeina þ

X1n¼�1

i1�nnJnði�aÞeina

" #

þ e�a cos a �a sin aX1

n¼�1

�Ban1H

ð1Þn ði�aÞe

ina þX1

n¼�1

�Bbn1hð4ÞmnJnði�aÞe

ina

!" #

þX1

n¼�1

in �Ban1H

ð1Þn ði�aÞe

ina þX1

n¼�1

in �Bbn1hð4ÞmnJnði�aÞe

ina

" #

þ e�ð2h1þ3a cos aÞ ��a sin aX1

n¼�1

X1m¼�1

�Bam1�hð5ÞmnJnði�aÞe

ina

"

þX1

n¼�1

X1m¼�1

�Bbm1�hð6ÞmnJnði�aÞe

ina

!

þX1

n¼�1

X1m¼�1

in �Bam1�hð5ÞmnJnði�aÞe

ina þX1

n¼�1

X1m¼�1

in �Bbm1�hð6ÞmnJnði�aÞe

ina : ð64Þ

Similarly, the electric displacement around the hole b can be obtained.

374 X.-Q. Fang et al.

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3

6. Numerical examples and discussion

Fatigue failures often occur in the regions with high stress concentrations, so anunderstanding of the distribution of dynamic stress around the holes is very usefulin the structural design. According to the expression of DSCF, the DSCFs aroundthe circular holes are computed. For convenience, it is assumed that the two holes areof the same radius a.

In the following analysis, it is convenient to make the variables dimensionless.To accomplish this step, we may introduce a characteristic length a, where a is theradius of the holes. The following dimensionless variables and quantities have beenchosen for computation: the incident wave number is k� ¼ ka ¼ 0:1� 3:0, theposition of the hole beneath the surface of FGPM structure is h�1 ¼ h1=a ¼ 1:1� 5:0and h�2 ¼ h2=a ¼ 1:1� 5:0, the distance between the centers of the two holes isd� ¼ d=a ¼ 2:1� 8:0, and the ratio of the material properties of FGPMs isp ¼ c244=c

144 ¼ 0:2� 5:0.

To validate the present dynamical model, comparison with other methods isgiven in Figure 3. In Figure 3a, finite element analyzes are performed using ANSYSversion 9.0. During the modeling, the ratios of h�1 and h�2 are h�1 ¼ 1:2 and h�2 ¼ 2,respectively. The distance ratio d* is d*¼ 7.0. The ratio of the material properties ofFGPMs is p¼ 2.0. The problem is modeled in two-dimensional strain state and thefree-meshing procedure is used for meshing. The elements near the holes are taken assmall as possible in order to simulate the dynamic stress distribution near the holesmore accurately. In the grading direction, the thickness of FGPM layer is dividedinto 10 elements, and constant material properties for each element are used.An anti-plane periodic dynamic load of frequency 1.4 is applied at the surface of thestructure. The stress is normalized using Equation (61). The normalized DSCFsaround the circle hole a are shown in Figure 3a. For comparison, the DSCFsobtained from the method in this article are also plotted in this figure. The excellentagreement can be found in Figure 3a.

Figure 3. Comparison with the results obtained from other methods: (a) comparison withANSYS program and (b) comparison with Ref. [12].

Philosophical Magazine Letters 375

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3

Comparison with the previous literatures is also given. Figure 3b illustrates theangular distribution of the dynamic stress around the hole a with parameters: p¼ 1.0,e15 ¼ "11 ¼ 0, h�1 ¼ 5:0, and d*¼ 8.0. p¼ 1.0 means that the functionally gradedmaterials reduce to the homogeneous materials. e15 ¼ "11 ¼ 0 implies that thepiezoelectric effect is not taken into consideration. When the distance ratio ish�1 ¼ 5:0, the effect of the surface of the structure can be ignored. When the distancebetween the two holes is d*¼ 8.0, the mutual influence of them can be ignored.It can be seen that the angular distribution of DSCFs is symmetric about bothaxes when the dimensionless wave number is small. So, the effect of the surface ondynamic stress disappears. When k*¼ 1.0, the maximum value of DSCFs isabout 2.1, and appears near the positions of ¼ =2, 3=2. Through comparison,it is found that the results coincide with those in an infinite homogeneousmaterial [8,12].

Figure 4 displays the angular distribution of the DSCFs around the circle hole awith different values of d*. It is noted that p¼ 2.0 means that the material propertiesof FGPMs increase along the x direction. p¼ 0.2 means that the material propertiesof FGPMs decrease along the x-direction. In Figure 4a, it can be seen that themaximum dynamic stress has a trend of shifting toward the illuminate side of thehole. This phenomenon is caused by the stronger scattering of electro-elastic wavesat the boundary of the hole. When the distance between the two holes becomessmaller, the maximum dynamic stresses around the hole a increases greatly. At thepositions facing the hole b, the variation of dynamic stresses is greater. Thisphenomenon is caused by the multiple scattering of electro-elastic waves between thetwo holes. At the positions near the hole b, and at the positions far from the hole b,the variation of dynamic stresses is little.

In Figure 4b, it can be seen that the maximum dynamic stress has a trend ofshifting toward the shadow side of the hole. However, due to the influence of thesurface, the dynamic stresses near the surface are still greater. The dynamic stressesnear the positions of ¼ �=2 are the maximum. Comparing with the results inFigure 4a, it is clear that the effect of the distance between the holes on the dynamicstress is greater when the material properties of FGPMs increase along thex-direction

In Figure 4c, the effect of boundary conditions disappears in this case. It can beseen that the maximum dynamic stress occurs at the position of the shadow sides.Comparing with the results in Figure 4b, it is clear that when the holes are far fromthe surface, the effect of the distance between the two holes on the dynamic stressdecreases.

In Figure 4d, the value of h�2 becomes great. Comparing with the results inFigure 4c, it is clear that the dynamic stress around the hole a decreases with theincrease of h�2. The effect of the distance between the two holes also decreases.So, the dynamic stress decreases with the increase of thickness of FGPMs.

Comparing the results in Figure 4e with those in Figure 4a, it can be seen that theboundary effect at the surface disappear, and the maximum dynamic stress occurs atthe positions near ¼ �=2. With the increase of the buried depth of the subsurfaceholes, the effect of the distance between the two holes decreases. Comparing withthe results in Figure 3b, it is clear that the dynamic stress increases due to thepiezoelectric properties of FGPMs.

376 X.-Q. Fang et al.

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3

Figure 4f is the case of higher frequencies. Comparing with the results inFigure 4e, it can be seen that the dynamic stress increases with the increase of wavefrequency. The effect of the distance between the two holes also increases with thewave frequency.

Figure 5 shows the angular distribution of the electric displacements aroundthe circle hole a with different values of d*. In Figure 5a, it can be seen that when the

Figure 4. Angular distribution of DSCF around the hole a in FGPM layer; 1 d*¼ 7.0;2 d*¼ 2.5. (a) k� ¼ 1:0, h�1 ¼ 1:2, h�2 ¼ 1:2, p ¼ 2:0; (b) k� ¼ 1:0, h�1 ¼ 1:2, h�2 ¼ 1:2, p ¼ 0:2;(c) k� ¼ 1:0, h�1 ¼ 5:0, h�2 ¼ 1:2, p ¼ 0:2; (d) k� ¼ 1:0, h�1 ¼ 5:0, h�2 ¼ 5:0, p ¼ 0:2;(e) k� ¼ 1:0, h�1 ¼ 5:0, h�2 ¼ 5:0, p ¼ 2:0; (f) k� ¼ 1:5, h�1 ¼ 5:0, h�2 ¼ 5:0, p ¼ 2:0.

Philosophical Magazine Letters 377

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3

material is homogeneous, the electric displacement is symmetrical about the x- andy-axes. The electric displacements around the hole are much less than the DSCFsaround the hole. The distance between the two holes expresses little effect onthe electric displacements around the hole. In Figure 5b, it can be seen that themaximum electric displacement occurs near the illuminate side of the hole whenp¼ 2.0. Comparing with the results in Figure 4a, it is clear that the effect of thedistance between the two holes on the electric displacement is less than that onthe dynamic stress. When p¼ 0.2, it can be seen that the distribution of electricdisplacement express little variation. It is also found that the effect of the distancebetween the two holes on the electric displacement is very little when p¼ 0.2.

7. Conclusions

The propagation and multiple scattering of electro-elastic waves in a FGPM layerwith two circular holes are investigated theoretically by employing image methodand wave functions expansion method. The analytical and numerical solutions ofthis problem are presented. For the homogeneous materials, our results are in good

Figure 5. Angular distribution of electric displacement around the hole a in FGPM layer;1 d*¼ 7.0; 2 d*¼ 2.5. (a) k� ¼ 1:0, h�1 ¼ 5:0, h�2 ¼ 5:0, p ¼ 1:0; (b) k� ¼ 1:0, h�1 ¼ 1:5,h�2 ¼ 5:0, p ¼ 2:0; (c) k� ¼ 1:0, h�1 ¼ 5:0, h�2 ¼ 5:0, p ¼ 0:2.

378 X.-Q. Fang et al.

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3

agreement with the solutions in previous literatures. Comparing with the solutionin the static case, analysis shows that the piezoelectric property has great effect on thedynamic stress in the region of higher frequencies. In the dynamic case, the obtainedresults are validated by comparison with those from ANSYS program.

In contrast to the case of the single hole in FGPMs, it is found that the distancebetween the two holes expresses great effect on the dynamic stress at the oppositepositions of the holes. When the material properties of FGPMs increase along thex-direction, the effect of the distance between the two holes on the dynamic stressis greater. With the increase of the buried depth of the holes, the effect of the distancebetween the two holes on the dynamic stress decreases. When the thickness of thefunctionally graded layer is small, the dynamic stresses near the surface and theopposite sides of the two holes increases greatly. With the increases of wavefrequency, the effect of the distance between the two holes on the dynamic stressincreases. The effects of the piezoelectric property, the incident wave number, andthe relative position of the two subsurface holes in the FGPM layer on the dynamicstress is greater than that on the electric displacement.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Foundationnos. 10672108, 10972147).

References

[1] C. Li and G.J. Weng, ASME J. Appl. Mech. 69 (2001) p.481.[2] B.L. Wang and N. Noda, Theor. Appl. Fract. Mech. 35 (2001) p.93.[3] B.L. Wang, Mech. Res. Commun. 30 (2003) p.151.

[4] C.-H. Chue and Y.-L. Ou, Int. J. Solids Struct. 42 (2005) p.3321.[5] Z.-G. Zhou and L.-Z. Wu, Int. J. Eng. Sci. 44 (2006) p.1366.[6] J. Chen, Z.X. Liu and Z.Z. Zou, Theor. Appl. Fract. Mech. 39 (2003) p.47.[7] L. Ma, L.-Z. Wu, Z.-G. Zhou, L.-C. Guo and L.-P. Shi, Eur. J. Mech. A. Solids 23 (2004)

p.633.[8] X.Q. Fang, Int. J. Solids Struct. 45 (2008) p.5716.[9] X.-Q. Fang, J.-X. Liu, X.-H. Wang, T. Zhang and S. Zhang, Compos. Sci. Tech. 69

(2009) p.1115.[10] Y.-H. Pao and C.C. Mow, Diffraction of Elastic Waves and Dynamic Stress

Concentrations, Crane, Russak, New York, 1973.

[11] J.A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941.[12] S.K. Datta, K.C. Wong and A.H. Shan, ASME J. Appl. Mech. 51 (1984) p.798.

Appendix

�hðiÞmn ði ¼ 1, 2, . . . , 6Þ are expressed as follows:

�hð1Þmn ¼ e�iðn�mÞabHð1Þn�mð�dabÞ, ðA1Þ

�hð2Þmn ¼ ð�1ÞnþmH

ð1Þnþmð�daa0 Þ, ðA2Þ

Philosophical Magazine Letters 379

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3

�hð3Þmn ¼ e�iðn�mÞab0Hð1Þn�mð�dab0 Þ, ðA3Þ

�hð4Þmn ¼ e�iðn�mÞabHð1Þn�mði�dabÞ, ðA4Þ

�hð5Þmn ¼ ð�1ÞnþmH

ð1Þnþmði�daa0 Þ, ðA5Þ

�hð6Þmn ¼ e�iðn�mÞab0Hð1Þn�mði�dab0 Þ, ðA6Þ

�hð7Þmn ¼ e�iðn�mÞbaHð1Þn�mð�dbaÞ, ðA7Þ

�hð8Þmn ¼ ð�1ÞnþmHð1Þnþmð�dbb0 Þ, ðA8Þ

�hð9Þmn ¼ e�iðn�mÞba0Hð1Þn�mð�dba0 Þ, ðA9Þ

�hð10Þmn ¼ e�iðn�mÞbaHð1Þn�mði�dbaÞ, ðA10Þ

�hð11Þmn ¼ ð�1ÞnþmHð1Þnþmði�dbb0 Þ, ðA11Þ

�hð12Þmn ¼ e�iðn�mÞba0Hð1Þn�mði�dba0 Þ, ðA12Þ

where dij denotes the distance between the holes ða, b, a0, b0Þ.

380 X.-Q. Fang et al.

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

4:10

21

May

201

3