Integral Equations, Boundary Value Problems And Related Problems 9789814452885, 9789814452878

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INTEGRAL EQUATIONS, BOUNDARY VALUE PROBLEMS AND RELATED PROBLEMS

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INTEGRAL EQUATIONS, BOUNDARY VALUE PROBLEMS AND RELATED PROBLEMS Dedicated to Professor Chien-Ke Lu on the occasion of his 90th birthday Yinchuan, Ningxia, China

19- 23 August 2012

Editor

Xing Li Ningxia University, China

'~ World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224

USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

INTEGRAL EQUATIONS, BOUNDARY VALUE PROBLEMS AND RELATED PROBLEMS Copyright© 2013 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereof,' may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission .fi-om the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-4452-87-8

Printed in Singapore.

v

PREFACE This proceedings volume is a collection of 26 selected papers of the 15th conference of integral equations, boundary value problems and related problems which held at Ningxia University during August 19-23, 2012. The conference was co-sponsored by Chinese Society of Mathematics, Association for Science and Technology of Ningxia, and Mathematics Society of Ningxia. The conference organizers were Ningxia University, Wuhan University, Peking University, Fudan University, Sun Yat-sen University, Beijing Normal University, Hebei University, Hebei Normal University and Renmin University of China. The conference brought active researchers together to reassess recent advances on integral equations, boundary value problems and their applications, and to provide guidance for relevant research topics for the future. The theory and application of integral equations and boundary value problems is important subject within applied mathematics. Integral equations and boundary value problems are used as mathematical models for many and varied physical situations, and they also occur as reformulations of other mathematical problems. The conference topics focus on the theory and analysis of integral equations,integral operators and boundary value problems; Applications of integral equations and boundary value problems to mechanics and physics; Approximate solutions of integral equations and boundary value problems; Boundary value problems for partial differential equations and functional equations; And some problems in Clifford analysis. As editors we express our gratitude to the contributors of the volume for submitting their manuscripts. Thanks also to Prof. Huili Han, Dr. Shenghu Ding and Mr. Pengpeng Shi from the School of Mathematics and Computer Science at Ningxia University for having organized the conference and unified the manuscripts for this volume. We also thank Dena of World Scientific Publishing Company for making the publication of this volume possible. Finally,the chief editor would like to thank the financial support

vi

of 211 fund of Ningxia University and the National Natural Science Foundation of China (10962008, 51061015).

Chief Editor: Xing Li, Ningxia University

Editorial Board: Guochun Wen, Peking University Jin Cheng, Fudan University Daiqing Dai, Sun Yat-Sen University Jinyuan Du, Wuhan University Zuoliang Xu, Renmin University of China

vii

ORGANIZING COMMITTEES EDITORIAL BOARD for the 15th conference of integral equations, boundary value problems and related problems Xing Li (Chief Editor) Guochun Wen Jin Cheng Daiqing Dai Jinyuan Du Zuoliang Xu

-Ningxia University, China -Peking University, China - Fudan University, China -Sun Yat-Sen University, China -Wuhan University, China - Renmin University of China, China

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ix

CONTENTS

Preface Organizing Committees Some Properties of a Kind of Singular Integral Operator for K-Monogenic Function in Clifford Analysis L.P. Wang, Z.L. Xu and Y. Y. Qiao

v Vll

1

Some Results Related with Mobius Transformation in Clifford Analysis Z.X. Zhang

11

The Scattering of SH Wave on the Array of Periodic Cracks in a Piezoelectric Substrate Bonded a Half-Plane of Functionally Graded Materials J.Q. Liu, X. Li, S.Z. Dong, X. Y. Yao and C.F. Wang

19

Anti-Plane Problem of Two Collinear Cracks in a Functionally Graded Coating-Substrate Structure S.H. Ding and X. Li

34

A Kind of Riemann Boundary Value Problem for Triharmonic Functions in Clifford Analysis

49

L.F. Gu A New Dynamical Systems Method for Nonlinear Operator Equations 59 X.J. Luo, F.C. Li and S.H. Yang A Class of Integral Inequality and Application W.S. Wang

67

X

An Efficient Spectral Boundary Integral Equation Method for the Simulation of Earthquake Rupture Problems W.S. Wang and B. W. Zhang

75

High-Frequency Asymptotics for the Modified Helmholtz Equation in a Half-Plane H.M. Huang

86

An Inverse Boundary Value Problem Involving Filtration for Elliptic Systems of Equations Z.L. Xu and L. Yan

93

Fixed Point Theorems of Contractive Mapping in Extended Cone Metric Spaces H.P. Huang and X. Li

104

Positive Solutions of Singular Third-Order Three-Point Boundary Value Problems B. Q. Yan and X. Liu

115

Modified Neumann Integral and Asymptotic Behavior in the Half Space Y.H. Zhang, G. T. Deng and Z.Z. Wei

130

Piecewise Tikhonov Regularization Scheme to Reconstruct Discontinuous Density in Computerized Tomography J. Cheng, Y. Jiang, K. Lin and J. W. Yan

140

About the Quaternionic Jacobian Conjecture H. Liu

153

Interaction between Antiplane Circular Inclusion and Circular Hole of Piezoelectric Materials L.H. Chang and X. Li

162

Convergence of Numerical Algorithm for Coupled Heat and Mass Transfer in Textile Materials M.B. Ge, J.X. Cheng and D.H. Xu

170

Haversian Cortical Bone with a Radial Microcrack X. Wang

184

xi

Spectra of Unitary Integral Operators on L 2 (IR.) with Kernels k(xy) D. W. Ma and G. Chen

195

The Numerical Simulation of Long-Period Ground Motion on Basin Effects Y. Q. Li and X. Li

211

Complete Plane Strain Problem of a One-Dimensional Hexagonal Quasicrystals with a Doubly-Periodic Set of Cracks X. Li and P.P. Shi

224

The Problem about an Elliptic Hole with III Asymmetry Cracks in One-Dimensional Hexagonal Piezoelectric Quasi crystals H.S. Huo and X. Li

235

The Second Fundamental Problem of Periodic Plane Elasticity of a One-Dimensional Hexagonal Quasicrystals J. Y. Cui, P.P. Shi and X. Li

246

The Optimal Convex Combination Bounds for the Centroidal Mean H. Liu and X.J. Meng

258

The Method of Fundamental Solution for a Class of Elliptical Partial 263 Differential Equations with Coordinate Transformation and Image Technique L.N. Wu and Q. Jiang Various Wavelet Methods for Solving Fractional Fredholm-Volterra Integral Equations P.P. Shi, X. Li and X. Li

275

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1

SOME PROPERTIES OF A KIND OF SINGULAR INTEGRAL OPERATOR FOR K-MONOGENIC FUNCTION IN CLIFFORD ANALYSIS* LIPING WANGa,b, ZUOLIANG xua, YUYING QIAObt a

school of Information, Renmin University of China, Beijing 100872, China

This paper has two parts. In the first part, we study the boundedness for a kind of singular integral operator, which is related to the Cauchy-type integral of k-monogenic function in Clifford Analysis. In the second part, we show the Holder continuous of the singular integral operator.

Keywords: Clifford Analysis, k-monogenic function, singular integral operator, Holder continuous.

AMS No: 30G30, 30G35.

1. Introduction

Clifford analysis is a kind of function theory which was established in order to study the solution of Dirac equation in spinor fields. And it is as such a direct generation to higher dimensions of the classical function theory of holomorphic function in the complex analysis. Clifford analysis is an important branch of modern analysis that studies functions defined on Rn with their values in Clifford algebra space, which is an associative and incommutable algebra structure. It possesses both theoretical and applicable values to many fields, such as elastic mechanics, fluid mechanics and so on. Since last century, a number of mathematicians have made great efforts in real and complex Clifford analysis. For example Brack, Relanghe, Sommen1 11, Wen1 21, Huang, Qiao1 3 , 4 letc. In addition, the solutions of partial differential equation and their boundary value problems have been applied in many subjects such as *This research is supported by NSFC (No.l0971224, No.l1171349) and Natural Science Foundation of Hebei Province(No.A2010000346). tCorresponding author, E-mail: [email protected].

2

physics, chemistry and engineering technology. And singular integral operators are one of the basic tools to solve partial differential equation and their boundary value problems. Hence for many years studying the properties of singular integral operators have been a hot and significative topics. References [5-9] studied some properties and application of Cauchy type integral operator and singular integral operator in higher dimensional. And when we solve some problem of degenerate equations and degenerate system in higher dimensional space, we need define and study some singular generalized integral operator in ordered to give a solution of degenerate equations and degenerate system. In this paper, we define a kind of singular integral operator. Firstly, we discuss the boundedness for the singular integral operators in an nonempty open bounded connected set in Rn+, which is related to the Cauchy-type integral of k-monogenic function in Clifford Analysis. Secondly, we discuss the Holder continuous of the singular integral operator.

2. Preliminaries Let e 1 , · · · , en be an orthogonal basis of the Euclidean space Rn, and let An(R) be the 2n-dimensional Clifford algebra with basis {eA: eA = ecv 1 ···eah}, where A= {a1,··· ,ah} ~ {1,··· ,n},1:::; a1 < a 2 < · · · < ah :::; n. and e0 = e0 = 1. Hence the real Clifford algebra is composed of elements having the type a= l:xAeA, XA E R. Its module is A

defined as lal = (

1

~ lxAI 2 )

2

The associative and noncommutative multiplication of the basis m

An(R) is governed by the rules e7=-1, i=1,2,··· ,n, { eiej = -ejei, 1:::; i,j:::; n, i eol

e(l'2.

0

•

eo.h

=

e(l'l02···(l'h'

cJ j, 1 :::; a1

< · · · < ah :::; n.

In addition, Suppose D is an nonempty open bounded connected set in Rn+ = {x = (x1,x2,··· ,xn) ERn lxn?: 0} and the boundary 8D is a differentiable, oriented and compact Liapunov surface in Rn+. The function f which is defined in D with values in An(R) can be expressed as f(x) = 2:: fA(x)eA, where the fA(x) is a real-valued function. In the paper, A

let f(x) E C("l(D,An(R))

= {flf: D--+ An(R), f(x) = l:fA(x)eA}, A

where fA(x) has continuous r-times differentials.

3

And the Dirac operator is defined as Df

n

DJ

DJ

n

.

n

DJ

n

DJ

= l:ei Dx = 2:: l:e;.eA DxA, f D = l:axe;. = l:l:eAei DxA. i i A i i A t

t

t

t

f is called a left(right) k-monogenic function if Dk f(x) = O(f(x)Dk O)(r ~ k, k < n, k E Z). f(x) E LP(D), which means DJ f(x) E LP(D),j

=

k-1

0, 1, · · · , k- 1 and Lp[f(x), D] =

2:: Lp[Dj f(x), D].

j=O

In the following, we define a kind of singular integral operator and give some necessary lemma. Then we discuss some properties of the singular integral operator.

Definition 2.1 Let D be as stated above and f(x) E cCrl(D,An(R)), we define a kind of singular integral operator

(T[f])(y)

~( -1)j

=

j=O

{ AJ+l (x- y)Hl dxDj f(x),

Jo

Wn

lx- Yin

where Xn is the n-th component of x, f(x) E V'(D), 0 < a < 1 is a positive constant and Wn is the area of the unit sphere in Rn, Aj is as stated the reference {10}, it is irrelevant to the x,y. Remark 1: When y E n- = Rn- 0, T[f] is a normal generalized integral. When y E 0, T[f] is its Cauchy principal value. Remark 2: When a = 0, n a normal Teodorescu operator.

Lemma 2.1 [lO]

For any x, y 1 , y 2 E Rn, when j ~ 0, we have

(x- Yl)H 1 I lx - Y1ln ::;

[~

= 2, k = 1, the singular integral operator is

(x- Y2)H 1 -

(lxl

lx - Y2ln I

+ IY2I)j

.·

~ lx- Yll 2 1x- Y2ln-t •t=l

Remark: When j

=

+~ ~ •t=l

(lxl + IY2I)j-i ]IYl _ Y2l· lx- Ylln-t

0 , The second part is vanishing.

Lemma 2. 2[ll] Suppose D is a bounded domain in Rn, n ~ 2, and let a and {3 satisfy 0 < a, {3 < n, a + {3 > n, then for all x1, x2 E Rn such that

4

3. The main results

Theorem 3.1 Let DC Rn(n > 2) be as stated above, j(x) E LP(D), p > n+2 - - , Then we have

1-a

IT[f]l:::; M(n,p,D)Lp[j,D], where

p, q, {3 0 ,

are

positive

constants

and

satisfy

1

1

-p +-q

=

1,

f3o=2(1-a)_ 4(q-1). n

nq

Proof By Holder formula, we have

L [1I

k-1

< J 2 L p[f, D ] -

j=O

1

·

rr

1

Xn

.

X

-

Y

1

l(n-j-1)q]

Ci

dx

I~- ;ztlq(n-j-1)(1- a:~] l _,

where d = maxxn,

x = (x1, x2, .. · ,

xESl

f3o=2(1-a)_ 4(q-1). n

S1

ldxl laql

nq

Xn_l),

Y = (y1, Y2, .. · , Yn-1)·

5

n+2 For p >--,we have 1-a

1 0 and takenin case of J(T(x)) < 0. Theorem 5.2. Let f, g E C 1 (B(y0 , R),C(V2,o)) then

J

2

g(y)dCYvf(y) =±(ad-be)

~(~.m

nC(B(y

j g(T(x)) lex+ ex+ d dl

0,

R), C(V2,o)),

ex+ d

3

Mx lex+ dl 3 f(T(x)),

al*

(18) where the right side of {18) is taken+ in case of J(T(x)) in case of J(T(x))

< 0, 0* ~

> 0 and taken-

r- 1 (B(y0 , R))

Acknowledgements

This work is supported by a DAAD-K.C. Wong Education Foundation and NNSF for Young Scholars of China (No. 11001206).

18

References 1. H. Begehr, Iterations of Pompeiu operators. Mem. Diff. Eq. Math. Phys. 12 (1997), 3-21. 2. H. Begehr, Iterated integral operators in Clifford analysis. Journal for Analysis and its Applications 18 (1999), 361-377. 3. H. Begehr, Representation formulas in Clifford analysis. Acoustics, Mechanics,and the Related Topics of Mathematical Analysis. World Scientific Singapore, 2002, 8-13. 4. H. Begehr, Zhang Zhongxiang, Du Jinyuan, On Cauchy-Pompeiu formula for functions with values in a universal Clifford algebra. Acta Mathematica Scientia 23B(1) (2003), 95-103. 5. F. Brack, R. Delanghe and F. Sommen, Clifford Analysis. Research Notes in Mathematics 76. Pitman Books Ltd, London, 1982. 6. R. Delanghe, On regular analytic functions with values in a Clifford algebra. Math. Ann. 185 (1970), 91-111. 7. R. Delanghe, On the singularities of functions with values in a Clifford algebra. Math. Ann. 196 (1972), 293-319. 8. R. Delanghe, Clifford analysis: History and perspective. Computational Methods and Function Theory. 1 2001, 107-153. 9. R. Delanghe, F. Brackx, Hypercomplex function theory and Hilbert modules with reproducing kernel. Proc. London Math. Soc. 37 1978, 545-576. 10. R. Delanghe, F. Sommen, V. Soucek, Clifford algebra and spinor-valued functions. Kluwer, Dordrecht, 1992. 11. S. L. Eriksson, H. Leutwiler, Hypermonogenic functions and Mobius transformations. Advances in Applied Clifford Algebras, ll(s2), 2001, 67-76. 12. K. Giirlebeck, W. Sprossig. Quaternionic analysis and elliptic boundary value problems. Akademie-Verlag, Berlin, 1989. 13. V. Iftimie, Functions hypercomplex. Bull. Math. Soc. Sci. Math. R. S. Romania. 9(57), 1965, 279-332. 14. R. KrauBhar, A characterization of conformal mappings in R 4 by a formal differentiability condition. Bulletin de la Societe Royale des Sciences de Liege. 70, 2001, 35-49. 15. E. Obolashvili, Higher order partial differential equations in Clifford analysis. Birkhauser, Boston, Basel, Berlin, 2002. 16. E. M. Olea, Morera type problems in Clifford analysis. Rev. Mat. Iberoam. 17, 2001, 559-585. 17. J. Peetre, T. Qian, Mobius covariance of iterated Dirac operators. J. Austral. Math. Soc. (Series A) 56, 1994, 403-414. 18. T. Qian, J. Ryan, Conformal transformations and Hardy spaces arising in Clifford analysis. J. Operator Theory. 35, 1996, 349-372. 19. Xu Zhenyuan, Boundary value problems and function theory for spininvariant differential operators. Ph.D. thesis, Ghent State University, 1989. 20. Zhang Zhongxiang, On k-regular functions with values in a universal Clifford algebra. J. Math. Anal. Appl. 315(2), 2006, 491-505. 21. Zhang Zhongxiang, Some properties of operators in Clifford analysis. Complex Var., elliptic Eq. 6(52), 2007, 455-473.

19

THE SCATTERING OF SH WAVE ON THE ARRAY OF PERIODIC CRACKS IN A PIEZOELECTRIC SUBSTRATE BONDED A HALF-PLANE OF FUNCTIONALLY GRADED MATERIALS JUNQIAO LIU and XING LI*

School of civil and water conservancy engineering,Ningxia University, Yinchuan, 750021,P.R.of China *E-mail: [email protected] www.nxu.edu.cn JUNQIAO LIUt, SHUZHUAN DONG, XIYAN YAO and CAIFENG WANG

Faculty of Applied Mathematics, Yuncheng University, Yuncheng,044000, P.R. of China t E-mail: liujunqiao @yeah. net

The interaction of SH wave with the array of periodic cracks in a piezoelectric substrate bonded to a functionally graded materials (FGJVI) coatings is considered. The governing equations along with permeable crack boundary, regularity and continuity conditions across the interface are reduced to a coupled set of Hilbert singular integral equations which are solved approximately by applying Cheyshev polynomials. Numerical results for the normalized dynamic stress intensive factors (NDSIF) and the normalized electric displacement intensive factors (NEDIF) are presented. The effects of geometric parameters and the physical parameters, and the effects of frequency and angles of SH wave on the NDSIF and the NEDIF are discussed.

Keywords: Functionally graded materials;Singular integral equations;The array of periodic cracks;Piezoelectric substrate;Intensity factors.

1 Introduction In recent years, the piezoelectric materials (PZM) have been found to have wide applications in the smart systems of aerospace, automotive, medical and electronic fields due to the intrinsic coupling characteristic between their electric and mechanical fields [1]. Due to its brittleness, PZM have a tendency to develop critical cracks during the manufacturing and the poling process. These defects greatly affected the mechanical integrity and elec-

20

tromechanical performance of PZM [2,3]. Zhou studied the interaction of SH wave with two cracks in the same line of the piezoelectric media in 2001 [4]. Narita and Shindo studied the scattering of SH wave from the crack in composite plane with PZM, where the dual integral equation are obtained by using Fourier transform [5,6]. Wang and Meguid studied the interaction of SH wave with some cracks in PZM by using integral transform and Chebyshev polynomials [7,8]. Gu and Yu discussed the elastic wave scattering by an interface crack between a piezoelectric layer and an elastic substrate in 2002 [9]. Ueda.S studied diffraction of anti-plan shear waves in a piezoelectric laminated with a vertical crack by the integral transform and the singular integral equation method in 2003 [10]. The scattering of SH wave from a crack in the piezoelectric substrate bonded with a half-space of FGM is studied by Li X [11]. The scattering of SH wave by a vertical crack in a FGM coated the piezoelectric strip was considered in 2010 by Liu [12]. Liu studied the scattering of SH wave by the array of periodic cracks in magneto-electro-elastic composites in 2011 [13]. However, in this paper, the scattering of SH wave from the array of periodic cracks in a piezoelectric substrate bonded to a half-plane of functionally graded materials (FGM) is studied. Based on the Fourier integral transform and traditional permeable crack electrical boundary conditions, the Hilbert singular integral equation can be obtained by defining the dislocation density function. The singular integral equation is solved by the approximation of Chebyshev polynomials and the numerical calculations are carried out [14]. The effects of geometric parameters and the physical parameters, the effects of frequency and angles of SH wave on the NDSIF and the NEDIF are discussed.

2 Description of the problem Consider FGM perfectly bonded to a piezoelectric substrate weakened by a periodic array of cracks, as shown in Fig.l. Assume that the crack of length 2c lies in the periodic array of 2L. The thickness between the crack and the bonded line is h. Refer to a cartesian coordinate system (x, y, z) located at the center of the crack in which z- axis is the poling axis of the piezoelectric media. The shear modulus and the mass densities of the FGM are expressed by exponential form as p,(y)

=

C44

exp[J(y- h)], p(y) =Po exp[J(y- h)]

(1)

21

y FGMs I

l

PZM II

f

-2L-c -2L+c -L -c

-3L

c

X

PZM Ill

~wave e

Fig. 1.

2L+c 3L

L 2L-c

Geometry of the crack problem

where C44, p 0 , 1 are the elastic modulus, the mass density and the graded coefficient in the PZE respectively. p,(y) and p(y) are the shear modulus and the mass densities of the FGM. Let time-harmonic SH wave originating at y ---+ oo be incident with an incidence angle e on the cracks. The antiplane shear stress act on the materials, so the displacement components are written as

u(x, y, t) = 0, v(x, y, t) = 0, w(t) (x, y, t) = w(i) (x, y, t) + w(x, y, t)

(2)

where wCt) (x, y, t), wCi) (x, y, t) and w(x, y, t) are the total field, incidence field and scattering field, respectively. In the incidence field, the displacement and the electric potential are expressed as

wCil(x,y,t)

=

A 0 exp{-i[a 2(xcosB+ysinB) +wt]}

q}i) (x, y, t)

= e 15 w(i) (x, y,

cu

t)

(3)

(4)

e,

where A 0 , w, e 15 , c: 11 , t are the amplitude, the incidence angle, the frequency, the piezoelectric coefficient, dielectric coefficient and time, respectively. i = A, a2 = .!;!_ (the wave number), c2 = r;;ii (the wave ~

v~

velocity). We regard the piezoelectric substrate as two layers, the part below the crack as the third layer and the part above the crack as the second layer. The first layer is the FGM. The constitutive relations of the piezoelectric

22

substrate are expressed as T31 (x, y, t)

=

elfitP.:r(x, y, t)

+ C44W,x(x, y, t)

(5) (6) (7) (8)

The constitutive relations of the FGM can be written as O':n =

C44

exp[r(y- h)]w,x(x, y, t); 0'32

= C44

exp[r(y- h)]w.y(x, y, t) (9)

where T3.J, 0' 3.J and D 7(j = 1, 2) represent the stress of PZM, the stress of FGM and the electric displacement of PZM, respectively. Substituting (3) and (4) into (6), the stress in the incidence field will be written as T:g) (x,

where C*

y, t) =

=

-iAoa2C* sine exp { -i[a2(x cos e +

y sin e)+ wt]}

(10)

2 e,s+C44Cll' Ell

Substituting (5)

~

(9) into the following governing equations

aT;n (x, y, t) ax

+

aT:i2(X,

y, t)

ay

a 2w(x, y, t) = Po at2

(11)

aD1(x,y,t) + aD2(x,y,t) = 0 ax ay a(j3l(x, y, t) ax

+

a0'32(x, y, t) ay

= Po

(12)

a 2w(x, y, t) at2

(13)

We can obtain three equations as follows,

e15(tP,xx(x, y, t)+t/>.yy(x, y, t))+c44(W,xx(x, y, t)+w,yy(x, y, t)) =Po a

2

w~~; y, t) (14)

-EutP.xx(x, y, t)

+ e15W,xx(x, y, t)- cut/>,yy(x, y, t) + e15W,yy(x, y, t) = 0 (15)

w,xx(x, y, t)

+ /Wy(x, y, t) + W,yy(x, y, t) =

Po

c44

a2 w(x, y, t) at 2

(16)

23

The electric permeable boundary conditions are appropriate to a silt crack in PZM. Then the boundary conditions can be written as follows,

T;3t(x,O) = T3;(x,O) = -T~2(x,O),Dt(x,O) = n;-(x,O) = n;(x,O), (17) Tiz(x,O)

=

T3;(x,O),Dt(x,O)

=

n;-(x,O),

¢+(x, 0) = ¢-(x, 0), w(x, o+) = w(x, o-)(a < lxl < L)

(18)

a1;(x, h) = T;{;(x, h), D2 (x, h) = 0, w+(x, h) = W-(x, h), (lxl < +oo) (19) w(-L,y)

=

w(L,y); ¢(-L,y)

=

(20)

¢(L,y);

(23)

3. Conversion of the problem

Because of the same time as in the incidence field, the displacement and electric potential in the scattering field are expressed as follows,

w(x, y, t) = w(x, y) exp( -iwt), ¢(x, y, t) = e1.s w(x, y, t)

(24)

En

Substituting eq. (24) into eqs. (14) "" (16) c44 \7

2w(x, y)

+ e15 \7 2¢(x, y) = -p0 w 2 w(x, y)

(25) (26)

'1'7 2

v

Let m2

2

obtain

=

2 w ( x, y )+ 1 8w(x,y) -__ p 0 w w ( x, y ) 8

pocuw2 E11C44+et 5 '

Y

f(x y) = ,!.(x y) '

o/

(27)

C44

'

e,s E11

w(x y) m2 = '

'

1

pow2. C44

We can

24

\7 2 w(x, y)

+1

ow(x, y) oy

+ miw(x, y) = 0

(28)

We introduce the finite Fourier integral transforms as follows

w2(x,y)

=

1 1 ~ mrx L[2a2w0 + L._.,a2wncosy]

(29)

n=l

(30)

(31)

(32)

(33) eqs. (25) "' (27) can be rewritten as

a2w0 = A2w0 cosm2y+B2w0 sin m2y, a2wn = A2wn exp(AoY)+B2wn exp( -AoY)

(34) a:>wO = A:>wO cosm2y+B:>wO sin m2y, a:>wn = A:>wn exp(AoY)+B:>wn exp( -AoY)

(35) a1wo = A1wo exp(A:>Y)+Blwo exp(A4Y), a1wn = A1wn exp(AIY)+Blwn exp( -A2Y)

(36) a210

=

A210Y + B2fO, a2fn

=

A21n exp(AJY) + B21n exp( -AJY)

(37)

V1

(39)

where

A·

_ -{ ±

3.4-

2

2 -

4mi A _ mr ' f- L

25

The stress and the electric displacement can be written 632 = C44 exp(i(y-

h))±[~a~wo +

f

a~wn cos n~x]

(40)

n=l I

2

00

I

CXJ

_ e15 ( a2JO """"' 1 " n1rx) eL5 + C44E11 ( a2wo """"' 1 n1rx) L 2 + ~ a2fn cos L + Len 2 + ~ a2wn cos L

T32 -

n=l

n=l

(41) ( 42)

1

D2

I

= -Ell [2a3JO +

L oo

n7rx

I

a:Jfn cos

(44)

L]

n=l Applying the boundary conditions, we can obtain A1wo = A1wn = A3fO = A3wo = A210 = A2wo = 0

(45)

A2wn exp(.\oh) + B2wn exp( -Aoh) = Blwn exp( -.\2h)

(47)

exp(AJh)A2fn- exp( -AJh)B2.fn = 0

(48)

-EnC44A2 exp( -.\2h)B1wn = Ao(ei 5 +c44En)[A2wn exp(.\oh)-B2wn exp(.\oh)] (49)

el5(A2wn + B2wn) + En(A2fn + B2Jn) = el5A:1wn + cnA:1fn

(51)

We define the following dislocation density function g

( ) = a[w (x, o+)- w (x, o-)l X

OX

We can obtain A:1wn- A2fn- B2fn

1 =~ .f

lc -c

I I , X

::;

a

g(x)sinAJUdu

(52)

(53)

26

Applying the boundary condition (18)

L1

1+c L -,\-g(u) G1(n) sin .A 1ucos .A 1xdu co

-c

=

·

T('l(x)

(54)

f

n=O

Let . G1(n) ei 5 - 2c~ 4 c:I 1 a- lnn - - - n-+DO AJ - e15(ei5 + 2c44Cll)- 2c44CI1

(55)

we obtain integral equation in which the unknown function is g(x).

11+c

4L

(cot

-c

11+c

1r(u-x) 1r(u+x) +cot )g(u)du+2L 2L L

k 1 (x, u)g(u)du

-c

T(il(x)

= --

a (56)

where k 1 (x, u) =

L= (m- a) sin(..\1u) cos(.A1x)

(57)

n=O

4. Solution of the problem To solve (56), the following normalized quantity are

(x,u,L,h) (g( u), k 1(u, x), Tj;l (x))

=

=

(x,u,L,h)/c

(g( uj c), k 1(uj c, xj c), Tj~l (x/ c))

(58) (59)

For simplicity in what follows, the bar appearing with the dimensionless quantities is omitted. The eq.(56) can be written

11+1

(cot 4L _ 1

11+

1 1r(u-x) 1r(u+x) +cot )g(u)du+k 1 (x,u)g(u)du 2L 2L L _1

T(il(x)

= --

a (60)

2

where T(i) (x) = " 15 ~~: 4 c 11 ( -ia2Ao) sine exp( -ia2x cos B). Considering eqs. (19) and (54), we have [

1 1

g (u)du

=

0

(61)

27

The solution to eq. (60) can be expressed (62) where T2j+1(u) is the first kind of Chebyshev polynomials, Bj is constant to be determined. Substituting eq. (62) into eq. (60), we can obtain the equation as follows,

~wz( n(uz-xj) n(uz+xj))"!'() ~4wzk( )"!'( )_4LTo(xj) L....- -:;;: cot £"' 2.5

2 1.5

0.5

0.2

0.4

0.6

1.4

1.6

1.8

2

1.8

2

X

Fig. 4.

y y y y y

5

y

The effects of the NDSF on w, Ao

=0.2 =0.4 =0.6 =0.8 =0.9 =1.0

4

3

2

0.2

0.4

1.2

1.4

1.6

X

Fig. 5.

The effects of the NDSF on w, 1

Fig.3 presents the curve about effects of incident angles of SH wave on the NDSIF for the case 1 = 1, h/c = 1. When e increasing from ; 0 to -if,

31

h =0.8 h=1.0

o. 7

--e---

h =1.2 h =1.6 h=2.0 h=2.5

Fig. 6.

The effects of the NDSF on w, h

the variation of the NDSIF is not obviously. When the incidence angle is ~, the NDSIF are maximum. From Fig.3, the thickness h affects more the maximum of the NDSIF and the NEDIF. With h increasing, the maximum of the NDSIF and the NEDIF are increasing, too. That is the FGM have function to stop the NDSIF and the NEDIF increasing. Seen the Fig.4, when the amplitude increasing, the NDSIF and the NEDIF are also increasing. At the same time the maximum become bigger. Observed Fig.5, when the length of crack increasing, the NDSIF and the NEDIF are also increasing.

6 Conclusions In the present paper, the scattering of SH wave on the array of periodic cracks in a piezoelectric substrate bonded a half-plane of functionally graded materials has been investigated. Using the Fourier transforms, the elasticity equations were converted analytically into a system of singular integral equations with Hilbert kernels. The singular integral equations are solved by the numerically using Chebyshev polynomials. From Figs.2-6, it can be concluded that for a fixed value of the parameter wajc 2 , the nor-

32

malized dynamic stress intensity factor K;II increases for increasing values of the graded parameter/, the angle of incidence e, and the amplitude A 0 , respectively. Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos.10962008 and 51061015) and Research Fund for the Doctoral Program of Higher Education of China (No. 20116401110002). References 1. Rao S. S., Sunar M.(1994). Piezoelectricity and ite use in disturbance sensing and control of flexible structures: a survey. Appl. Mech. Rev., 47: 113~123. 2. Chen Z. T., Meguid S. A.(2000). The transient response of a piezoelectric strip with a vertical crack under electromechnical impact load[J]. International Journal of Solids and Structures, 37: 6051~6062. 3. Meguid S. A., Wang X. D.(1998). Dynamic antiplane behavior of interacting cracks in a piezoelectric medium. International Journal Fracture, 91: 391~403.

4. Zhou Z. G., Li H. C., Wang B.(2001). Investigation of the scattering of antiplane shear waves by two conlinear cracks in a piezoelectric material using a new method[J]. Acta Mechanica, 147(1):87~97. 5. Narita F., Shindo Y.(1999). Scattering of antiplane shear waves by a finite crack in piezoelectric laminates[J]. Acta Mechnica, 134(1):27~43. 6. Narita F., Shindo Y. (1998). Scattering of love waves by a surface-beaking crack in piezoelectric layered media[J]. JSME International Journal series A, 41: 40~48. 7. Wang X. D., Meguid S. A.(2000). Modeling and analysis of the dynamic behavior of piezoelectric materials containing interacting cracks [J]. Mechanics of Materials, 32(7):723~737. 8. Wang X. D. (2001). On the dynamic behavior of interaction interfacial cracks in piezoelectric media[ J]. International Journal of solids and Structures, 38(8): 831~851. 9. Gu B.,Yu S.W., Feng X. Q .. ( 2002). Elastic wave scattering by an interface crack between a piezoelectric layer and elastic substrate[J]. International Journal of Fracture, 116: L29~L34. 10. Sei Ueda. (2003).Diffraction of antiplane shear waves in a piezoelectric laminate with a vertical crack[J]. European Journal of Mechanica A/Solids, 22: 413~422.

11. Li X., Liu JQ.(2009). Scattering of the SH wave from a crack in a piezoelectric subsrate bonded to a half-space of functionally graded materials[J]. Acta Mechnica, 208:299~308. 12. Liu JQ., Duan HQ. Li X. (2010). The scattering of SH wave on a vertical crack in a coated piezoelectric strip[J]. Chinese Journal of Solid Mechanics, 31(4): 385~391.

33

13. Liu JQ., Li X. (2011). The scattering of SH wave on a Periodic Array of cracks in Magneto-electro-elastic Composites[J]. Chinese Journal of Engineering Mathematics, 28( 4): 4 70cv4 74. 14. F. Erdogan and G.D.Gupta.(1972). On the Numerical solution of singular integral equations[J]. Quarterly of Applied Mathematics, 1: 525cv534.

34

ANTI-PLANE PROBLEM OF TWO COLLINEAR CRACKS IN A FUNCTIONALLY GRADED COATING-SUBSTRATE STRUCTURE SHENG-HU DING*, XING LI School of Mathematics and Computer Science, Ningxia University, Yinchuan, 750021, China *Corresponding author (Tel: +86 951 2061405; fax: +86 951 2061405. Email: dshsjtu200g@163. com).

A theoretical treatment of anti plane crack problem of two collinear cracks on the two sides of and perpendicular to the interface between a functionally graded orthotropic strip bonded to an orthotropic homogeneous substrate is put forward. The problem is formulated in terms of a singular integral equation with the crack face displacement as the unknown variable. Numerical calculations are carried out, and the influences of the orthotropy and nonhomogeneous parameters on the mode III stress intensity factors are investigated.

Keywords: Functionally graded orthotropic strip; Collinear cracks; Singular integral equation; Stress intensity factor

1. Introduction

With the application of functionally gradient materials in engineering, most of the current researches on the fracture analysis of the FGMs interface have been devoted to the FGMs interlayer and the interface between the functionally graded material (FGM) coating and the homogeneous substrate [1-5]. Ding et al. [6,7] studied the problem of the model III crack in a functionally graded piezoelectric layer bonded to a piezoelectric half-plane. The dynamic fracture problem of the weak-discontinuous interface between a FGM coating and a FGM substrate have been studied by Li et al.[8,9]. The mode III fracture problem of a cracked functionally graded piezoelectric surface layer bonded to a cracked functionally graded piezoelectric substrate are studied by Chen and Chue [10]. Introducing a hi-parameter exponential function to simulate the continuous variation of material properties, two bonded functionally graded finite strips with two collinear cracks are investigated by Ding and Li [11].

35

Their processing techniques mean that FGMs are seldom isotropic [12,13]. Therefore, the failure behaviors of orthotropic FGMs attracted some researchers' attention. Supposing the material properties to be onedimensionally dependent, Chen and Liu [14] studied the transient response of an embedded crack and edge crack perpendicular to the boundary of an orthotropic functionally graded strip. Wang and Mai [15] studied the static and transient solutions for a periodic array of cracks in a functionally graded coating bonded to a homogeneous half-plane. A theoretical treatment of mode I crack problem is put forward for a functionally graded orthotropic strip. Guo et al.[16,17] considered the internal crack and edge crack perpendicular to the boundaries respectively. Assuming the variation of material properties is a power law function, Feng et al. [18] investigated the dynamic fracture problem of a mode III crack, which is also parallel to the boundary of an orthotropic FGM strip. Though there are lots of papers related to the crack problem of FGMs, very few papers on the antiplane crack problem of two collinear cracks on the two sides of and perpendicular to the interface between a functionally graded orthotropic strip bonded to a homogeneous orthotropic substrate are published. Such a problem is important not only because a practical FGM may subject to an antiplane mechanical deformation, but also because it can provide a useful analogy to the most important in-plane crack problem. The material properties are assumed to vary continuously along the x-direction. The crack problem can be reduced into a system of singular integral equations after applying the integral transform technique. The influences of geometrical and physical parameters and crack interactions on the stress intensity factor were analyzed. 2.

Description of the problem

Consider a graded coating bonded to a homogeneous substrate. There is two collinear cracks vertical to the interface between two mediums, as shown in Fig.l. Each strip contains an internal crack with crack length (bj- aj)(j = 1, 2) perpendicular to the interface, respectively. The principal axes of the medium are along the x and y-axes. By adjusting the thickness, h 1 and h 2 , of the two bonded materials, the model in Fig.1 can be generalized to represent a FGM coating with finite thickness bonded to a homogeneous substrate with finite or infinite thickness. In this paper we assume that the medium has orthotropic properties. To make the analysis tractable, it is further assumed that the shear moduli of the graded region is given by the following two-parameter expression

36

~X(x)=~Xoe~x

FGM Coating

~y(x)=~yoe~x

y

Homogeneous Substrate

h2

j

b2 X

Fig. 1.

The crack geometry in bonded materials.

f.Lx ( X)

=

f.Lxoe

(1x

,

f.Ly ( x ) = f.Lyoe

(J:r

,

(1)

where f.L:rO and J.Lyo are elastic constants on the surface x = 0. Many authors have used the assumptions shown in Eq.(1) for FGM applications [9,14,16,17]. The basic elasticity equations for nonhomogeneous materials undergoing antiplane shear deformations are

(2)

Tyz .

=

aw

J.L .4(x, y)8y ,

(3)

where w is the antiplane displacement, Trz and Tyz are the shear stresses, and f.L:r(x, y) and J.Ly(x, y) are the shear moduli. Substituting Eq.(3) to Eq.(2) becomes

for the substrate.

(5)

37

2 J3h 1=1n0.1 J3h 1=1n1.0

1.8

J3h 1=1n10.0

1.6

"' 0

Klll(-b1 )h:oc~-5

Klll(-a1)h:oc~·5

~

(.)

.!:::.0

1.4

'::>:.-

....

1.2

·• 0.1

Fig. 2.

0.2

0.3

0.4

0.5

Characteristic curves of the normalized SIFs vary with qjh1 at different (3h1

(h2/h1 = 1.0, c2/h2 = 0.0).

1.5 Isotropic

1.45

Orthotropic I

1.4

"' 0

Orthotropic II

1.35 Klll(-b1 )h:oc~· 5

~

(.)

0

..!:::. '::>:.-

Klll(-a1)h:oc~·5

1.3 1.25 1.2

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

C/h 1

Fig. 3. Comparison of SIFs for collinear cracks between different FGM coating-substrate structures (h2/h1 = 5.0, (3h1 =ln0.5, c2/h2 = 0.2).

3. Derivation of the singular integral equations

Employing the Fourier transform on Eqs.(4) and (5), the solutions for w 1 and w 2 become wl(x,y)

=

1 -2 7r

joe-= Cl(a)ern 1 ye-iO!Xda+~ lor= (Al(a)en x +A2(a)en x)sinayda, 1

2

7r

(6)

38

1.4 cjh 2 =0.0 C/h 2=0.1

1.3

C/h 2=0.2

"' =

~

'-'= ..t:

Klll(-b1 )h:oc~·5

1.2

Klll(-a1 )h:oc~· 5

">£.-

1.1

·•·• 0.90

0.05

0.1

0.15

...

•·•·

. ·•

·•

...

0.2

0.25

0.3

0.35

...

........

•

0.4

C/h 1

Fig. 4. Effect of qjh1 on the SIFs for single carck and collinear cracks in an orthotropic I FGM coating-substrate structure (h2/h1 = 1.0, f3h1 =ln0.5).

1.2 1.18

c,th 1=0.0

1.16

c,fh 1=0.1

1.14

c,th 1=0.2

ON "'(.)

= 1.12

Klll(b2)h:oc~ 5

1.1

Klll(a)ltoc~·s

..t:

">£.-

.··

1.08

...

1.06

•

1.04 1.02 10

0.05

0.1

0.15

0.25

0.3

0.35

0.4

Fig. 5. Effect of c2/h2 on the SIFs for single crack and collinear cracks in an orthotropic I FGM coating-substrate structure (h2/h1 = 1.0, f3h2 =ln0.5).

39

1.4,-----,-----,-----,----,,----,-----,-----,-----, J3h 1 =1n0.5

1.35

J3h 1 =1n1.0

1.3 ~ ~ '-' .!::.0

1.25

"::1::.-

1.2

J3h 1 =1n2.0

Klll(-b1)11:oc~·5 Klll(-a1 )h:oc~· 5

1.1

0.05

0.1

0.15

0.25

0.3

0.35

0.4

Fig. 6. The influence of f3h1 on the SIFs for collinear cracks in an orthotropic I FGM coating-substrate structure (h2/h1 = 1.0, c2/h2 =0.2).

1.25 J3h 2 =1n0.5 J3h 2 =1n1.0

1.2

J3h 2 =1n2.0

0

.

Klll(b)l1:oc~·5

"''-' 1.15 N

Klll(a2)h:oc~· 5

.!::.0

•

"::1::.-

1.1

1.0

0.05

0.1

0.15

0.25

0.3

0.35

0.4

Fig. 7. The influence of f3h2 on the SIFs for collinear cracks in an orthotropic I FGM coating-substrate structure (h2/h1 = 1.0, qjh1 =0.2).

where

40

1.4 h/h,=1.0

1.3

h/h,=2.0 h/h,=10.0

"'

au~

1.2

..!:::.0 ":>::.-

1.1

•

0.90

0.05

0.1

0.15

.. ..

...

.....

. ...

0.2

0.25

0.3

0.35

0.4

c,th 1

Fig. 8. The influence of h2/h1 on the SIFs for collinear cracks in an orthotropic I FGM coating-substrate structure (f3h1 =ln0.5, c2/h1 =0.2).

1.18 hz'h,=1.0

1.16

hz'h,=2.0

1.14

hz'h,=10.0

1.12

"' 0

N

(..)

0

Klll(b2)hoc~· 5

1.1

..!:::.

-,:.-

Klll(a2)h:oc~·5 1.08

.

1.06

...

1.04 1.02 10

• 0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

c,th 1

Fig. 9. The influence of h2/h1 on the SIFs for collinear cracks in an orthotropic I FGM coating-substrate structure (f3h1 =ln0.5, c2/h1 =0.2).

41

(8)

Pl

= -p2 = -

I

a

I

/11,

(9) and C1 (a), C2(a), ... , B2(a) are unknown functions to be obtained from the continuity and boundary conditions. The boundary and continuity conditions can be stated as

w1(0,y) = w2(0,y),

Txzl(O,y) = Trz2(0,y),

T.rzl( -h1, y) = 0,

Txz2(h2, y)

=

0,

-CXJ

-CXJ

,

(17)

UX2

where x = x 1 for x < 0, and x = x 2 for x > 0. By applying Fourier transform, the unknown functions C 1 (a) and C 2 ( a) can be expressed in the form of dislocation functions as

(18) After employing the continuity conditions Eqs. (10) and (11) and the Fourier inverse transform, the following can be obtained

42

ql exp (q1h2)B1

+ q2 exp (q2h2)B2 = -1

1= .

2n -=

zp

iap exp ( -iph2)

2 2 Pl + a

C2(p)dp. (22)

By solving Eqs. (19)-(22) for functions A1 , A2 , B 1 and B 2 , we have Du D21 A1(a) = fDlR1- fDlR2

D:n

+ fDlR:>-

D41 fDlR4,

(23)

where

(29)

with D = (dij)(i,j = 1- 4) is a square coefficient matrix of order 4 in Eqs.(23)-(26), IDI is the determinant of D, and Dij are the cofactors of the elements d,1 (i,j = 1- 4).

43

From the load conditions (14) and (15) on the crack surfaces, the following singular integral equations can be obtained

where

(33)

K13(x 1 , t 2 ) = :

1

1= ~ {a (~~~ en

111

-

~~~ en

2

"

1

)

(

-1)1+ 1 R 72 } da, (35) (36)

44

K 23 (x2, t2) = : 1

laoc

t,

{a (

~~~ e'll

12

-

~~~ eq

212

)

(

-1)1+ 1R 72 } da,

(38) where A is an arbitrary positive constant. The solutions of the singular integral equations Eqs. (31) and (32) with the Cauchy type kernel are (39) Consider two collinear internal cracks in a FGM coating bonded to a homogeneous substrate (Fig.1). For this case, note that the singular term 1 - - is associated with two embedded cracks in two materials and leads to

t-x

the standard square-root singularity for the unknown function g1(tl) and g2(t2). It may easily be shown that for -b1 > -h1 and -a1 < 0, and a2 > 0 and b2 < h 2, Kij ( i = 1, 2, j = 1 - 3) remain bounded in the closed interval -b1 ::::; (x1, t1)::::; -a1 and a2::::; (x2, t2) ::::; b2, respectively. In order to change Eqs. (31) and (32) into standard form, we define dimensionless quantities t1 = u1(b1- al)/2- (b1 t2 = u2(b2- a2)/2

+ a1)/2,

x1 = r1(b1- a1)/2- (b1

+ (b2 + a2)/2,

x2 = r2(b2- a2)/2

fJ(Uj)

Fj(Uj)

=

gj(tj),

=

+ al)/2,

+ (b2 + a2)/2,

Gj(tj),

j = 1, 2,

(40)

then Eq.(39) becomes (41)

where Fj(uj)(j

= 1, 2) are bounded functions, which can be expressed as 00

00

n=O

n=O

where Tn is the Chebyshev polynomial of the first kind and An and En are unknown constants. F 1(ul) and F 2(u 2) must fulfill the following singlevaluedness as

45

1 [ 1 F1(ul)j )1- ui du1 = 0,

1 [ 1 F2(u2)/ )1-

u~ du2 = 0.

(43)

The results of stress intensity factors (SIFs) become CXJ

Kni( -b1) = /L1/Lyoe-flb 1 j(b1- ai)/2

2) -1)nAn,

(44)

n=1 co

Kni( -al) = -JL1/Lyoe-f3a 1 jr:-:(b_1___a_1-,---)/-,-2

L

An,

(45)

n=1 co

Knr(a2) = /L1/Lyo}(b2- a2)/2

L( -1t Bn,

(46)

n=1 CXJ

Knr(b2) = -JL1/Lyo}(b2- a2)/2

L

Bn.

(47)

n=1

4. Results and discussion

In the following numerical computations, the shear loads T1 and T2 applied on the crack surfaces are assumed to be equal when cracks in FGM coating and homogeneous substrate, respectively. T1 and T2 are constant tractions, which are also assumed to equal To. Numerical calculations were carried out for three kinds of materials (i.e. isotropic, orthotropic I and II), whose mechanical properties can be found in [14]. The only difference between orthotropic I and II is that their reinforced directions are perpendicular to each other. For internal cracks problem, the stress intensity factors at the crack tips are normalized by Toy'cl or Toy'c2, where c1 = (b 1 - ai)/2, c2 = (b 2 - a 2)/2. First of all, let us verify the validity of the analytical solution procedure. We compare the normalized SIFs of a central crack in a isotropic FGM coating bonded to a homogeneous strip, namely h 2/h 1 = 1.0 and (b 2 2 ) = 0, with the results provided by Yang and Zhou [19]. Fig.2 illustrates the variation of normalized SIFs for bonded isotropic medium with {3h 1 =ln0.1, lnl.O and ln10.0. The numerical result is in good agreement with that presented by Yang and Zhou [19]. Consider a practical case that a FGM coating is bonded to a homogeneous substrate with internal cracks. Fig.3 compares the effect of material properties on the SIFs for internal cracks in different FGM coatingsubstrate structures. The isotropic strip has a very similar curve of SIFs to that of orthotropic I strip and the orthotropic II strip. It can be found

;a

46

that the material properties (i.e., isotropic, orthotropic I and orthotropic II) will influence the SIFs. The value of SIFs of the orthotropic I strip is maximal. Therefore, it is necessary to take the orthotropy into account when analyzing the mechanical behaviors of FGMs. Figs. 4 and 5 show the influence of the crack length cl/h 1 and c2/h 2 on the SIFs for two internal cracks in an orthotropic I FGM coating-substrate structure. Obviously, the crack length will influence the SIFs, and this further shows the necessity of taking the internal cracks into account when analyzing the mechanical behaviors of FGMs. Figs. 6 and 7 illustrate the variation of the normalized SIFs with different nonhomogeneous constant (3h 1 and (3h 2 in an orthotropic I FGM coating-substrate structure. It can be seen that Kn 1 ( -a 1 )/T0 c~· 5 increases and Kni( -bl)/T0 c~· 5 decreases with the increasing of (3h 1 . However, both Kni(a 2 )/T0 cg· 5 and Kni(b 2 )/T0 cg· 5 increase with the increasing of (3h 2 • With the increasing of c2 /h 1 , both Kn 1 (a 2 )/T0 cg· 5 and Kn 1 (b 2 )/T0 cg· 5 generally increase. When (3 is positive, the material properties on the crack tip -a 1 and b2 are stronger than ones on the crack tip -b 1 and a 2 , and Kni( -al)jT0 c~· 5 and Kni(b 2 )/T0 cg· 5 are greater than Kni( -b 1 )/T0 c~· 5 and Kn 1 (a 2 )/Tocg· 5 . These results show where the stronger the material properties are, the larger the normalized stress intensity factors at the crack tip. Figs. 8 and 9 show the influence of different thickness ratio h 2 /h 1 on SIFs for an orthotropic I FGM coating-substrate structure when (3h 1 =ln0.5. It can be found that the SIFs of crack tip decrease with an increasing of h2/h1. For different h2/h1, Kni( -bl)/Toc~"'5 are always greater than Kn 1 (-al)/Toc~·fi. But Kn 1 (b 2 )/T0 cg.s are greater than Kn 1 (a 2 )/T0 cg"'5 when cl/h 1 < 0.2, and Kn 1 (a 2 )/T0 cg.s are greater than Kn 1 (b 2 )/T0 cg"'5 when cl/h 1 > 0.2.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (51061015,11261045) and research fund for the doctoral program of higher education of China (20116401110002).

References 1. Ang, W.T., Clements, D.L., 1987. On some crack problems for inhomogeneous elastic materials. International Journal of Solid and Structures 23, 1089-1104.

47

2. Chen, Y.F., Erdogan, F., 1996. The interface crack problem for a nonhomogeneous coating bonded to a homogeneous substrate. Journal of The Mechanics and Physics of Solids 44, 771-787. 3. Li, C.Y., Weng, G.J., 2001. Dynamic stress intensity factors of a cylindrical interface crack with a functionally graded interlayer. Mechanics of Materials 33, 325-333. 4. Noda, N., Wang B.L., 2002. The collinear cracks in an inhomogeneous medium subjected to transient load. Acta Mechanica 153, 1-13. 5. Huang, G.Y., Wang, Y.S., Yu, S.W., 2004. Fracture analysis of a functionally graded interfacial zone under plane deformation. International Journal of Solid and Structures 41, 731-743. 6. Ding, S.H., Li, X., 2008a. An anti-plane shear crack in bonded functionally graded piezoelectric materials under electromechanical loading. Computational Materials Science 43, 337-344. 7. Ding, S.H., Li, X., 2008b. Periodic cracks in a functionally graded piezoelectric layer bonded to a piezoelectric half-plane. Theoretical and Applied Fracture Mechanics 49, 313-320. 8. Li, Y.D., Jia B., Zhang, N., Tang L.Q., Dai, Y., 2006. Dynamic stress intensity factor of the weak/micro-discontinuous interface crack of a FGM coating. International Journal of Solids and Structures 43, 4795-4809. 9. Li, Y.D., Kang Y.L., Yao D., 2008. Dynamic stress intensity factors of two collinear mode-III cracks perpendicular to and on the two sides of a bi-FGM weak-discontinuous interface. European Journal of Mechanics A/Solids 27 , 808-823. 10. Chen, Y.J., Chue, C.H., 2010. Mode III fracture problem of a cracked FGPM surface layer bonded to a cracked FGPM substrate. Archive of Applied Mechanics 80, 285-305. 11. Ding, S.H., Li, X., 2011. Mode-l crack problem for functionally graded layered structures. Internation Journal of Fracture 168, 209-226. 12. Sampath, S., Herman, H., Shimada, N., Saito, T., 1995. Thermal spray processing of FGMs. Mrs Bulletin 20, 27-31. 13. Kaysser,W.A., Ilschner, B., 1995. FGM research activities in Europe. Mrs Bulletin 20, 22-26. 14. Chen, J., Liu, Z.X., 2005. Transient response of a mode III crack in an orthotropic functionally graded strip. European Journal of Mechanics A/Solids 24, 325-336. 15. Wang B.L., Mai Y.-W., 2006. Periodic antiplane cracks in graded coatings under static or transient loading. Journal of Applied Mechanics 73, 134-142. 16. Guo, L.C.,Wu, L.Z., Zeng, T.,Ma, L., 2004. Mode I crack problem for a functionally graded orthotropic strip. European Journal of Mechanics A-Solids 23, 219-234. 17. Guo, L.C.,Wu, L.Z., Zeng, T., 2005. The dynamic response of an edge crack in a functionally graded orthotropic strip. Mechanics Research Communications 32, 385-400.

48

18. Feng,W.J., Zhang, Z.G., Zou, Z.Z., 2003. Impact failure prediction of mode III crack in orthotropic functionally graded strip. Theoretical and Applied Fracture Mechanics 40, 97-104. 19. Yong, H.D., Zhou, Y.H, 2006. Analysis of a mode III crack problem in a functionally graded coating-substrate system with finite thickness. International Journal of Fracture 141, 459-467

49

A KIND OF RIEMANN BOUNDARY VALUE PROBLEM FOR TRIHARMONIC FUNCTIONS IN CLIFFORD ANALYSIS LONGFEIGU

School of Mathematics and Statistics, Wuhan University, 430072/Wuhan Hubei, P. R. China E-mail: [email protected] www. whu. edu. en

In this article, we mainly deal with the boundary value problem for triharmonic function with value in universal Clifford algebra:

where (j = tor D =

1, ... , 5)

()[2

£: ek 0 ~'"' u(x)

k=l

·

is a Liapunov surface in R/', the Dirac opera=

I.:;eAuA(x) are unknown functions with values A

in an universal Clifford algebra Cl(Vn,n)· Under some hypotheses, it is proved that the boundary value problem has a unique solution.

Keywords: Symplectic schemes; elastodynamic; anti-plane shear; boundary integral method.

1. Introduction and preliminaries

Riemann boundary value problems theory in complex plane has been systematically developed in [7 ,8]. It is an interesting topic to generalize the classical Riemann boundary value problems theory to Clifford analysis. In [4,6,10,12], etc, many interesting results about boundary value problem and Riemann Hilbert problems for monogenic functions in Clifford analysis are presented. In [11], Green's function for the Dirichlet problem for poly harmonic equations were studied. In this article, with the aim to study the Riemann boundary value problem for triharmonic functions. At first, based on the higher order Cauchy integral representation formulas in [5, 9] and the Plemelj formula, we give some properties of triharmonic functions in

50

Clifford analysis, for example, the mean value theorem, the Painleve theorem, etc. Furthermore, on the basis of the above results, we consider the following Riemann boundary value problems:

(1)

and

6 3 [u](x) = 0 x x { u+(x) = u-(x)G(x) + g(x), (DJu)+(x) = (DJu)-(x)Ai + fj(x), x u(oo)=O

Rn\oD E 8D E 8D E

(2)

where (j = 1, ... , 5). In (1) and (2), A and Aj are invertible constants, we denote the inverse elements as A- 1 and Aj 1 . u(x), (DJu)(x), g(x), fj(x) E Hi3(8D,Cl(Vn,n)),j = 1, ... ,5, 0 < f3 s; 1. The explicit solutions for (1) is shown and the boundary value problem (2) is solvable under some hypotheses. Let V,,,, (0 s; s s; n) be an n-dimensional ( n 2': 1) real linear space with basis { e1, e2, ... , en}, Cl(V,,,,) be the universal Clifford algebra over Vn,s· For more information on Cl(V,,,,)(O s; s s; n), we refer to [1, 2, 3]. Throughout this article, suppose D be an open, bounded non-empty subset of Rn with a Liapunov boundary 8D, denote o+ = D, o- = Rn \0. In this article, for simplicity, we shall only consider the case of s = n, the operator D which is written as n

D

0

= "'ek-;:;: C'(D, Cl(Vn n))--+ C'- 1 (D, Cl(V,, n)) L......t ux ' ' k=l

k

Let u be a function with value in Cl(Vn,n) defined in D, the operator D acts on the function u from the left and from the right being governed by the rule

Definition 1.1. A compact surface r is called Liapunov surface with Holder exponent a, if the following conditions are satisfied: • At each point

X

E

r

there is a tangential space.

51

• There exists a number r, such that for any point x E r the set r B, (X) (Liapunov ball) is connected and parallel lines to the outer normal a(x) intersect at not more than one point. • The normal a(x) is Holder continuous on r, i.e. there are constants C > 0 and 0 -Ro(>-)1 -)1

.(tj- 0) and >.(tj + 0) the left limit and right limit of>.(() as (-+ tj(j = 1,2,3,4) on r, and noting that ei'Pj

= >.(tj- 0)' /j =

A(tj + 0

~ln >.(tj- 0) = JrZ A(tj + 0

IPj - Kj. 7r

(3.5)

100

By (2.10), we can get 471 = ~' 472 = -~, 473 = ~' 474 = 3; . In order to seek the bounded solutions , it is required that 0 < rJ < 1, in which Kj(j = 1,2,3,4) are integers, hence K 1 = K 3 = K 4 = O,K2 = -1. This shows that the index

(3.6) By using the results of Chapter 1 in [9], we can obtain the solution of Problem B for analytic functions is unique. It is easy to see that ( u) = 0, thus Z*(() = 0, i.e. Z 1 (() = Z 2 ((). This completes the proof. In the following, we prove the existence of Problem B for (2.7) by using Newton imbedding method. In order to seek the approximate solution of Problem B, we consider the complex equation with a parameter E [0, 1] :

e

Z.(- BF((, Z, Zc) =

B((), B(()

=

(1- B)F((, 0, 0),

(3.7)

and first suppose that F((, Z, Zc) = 0 in the Ern = 1/m (m is a positive integer) neighborhood Urn of points tj (j = 1, 2, 3, 4), it sufficies to multiply F((, Z, Zd by the function

TJm(()

=

{

u]=1 {I(- til
.(()(()] = r(()- Re[>.(()W(()], ( E f* = f\{t1, t2, t:,, t4}. Introduce a function g(() = f1~= 1 -

1

C[J(D() n WPo(D(), herein Po(2

t

( ( - 1 )'1,

(3.10)

it can be derived g(()Z(() E

2 N, where N is a positive integer. This shows that S(Zn Z 1) ---+ 0 as n, l ---+ oo. Following the completeness of the Banach space B = Cp(Dc) n W1; 0 (Dm), there is a function Z*(()g(() E B, such that when n ---+ oo,

S(Z- Z*) = C;3[g(Zn- Z*), D(] + LPo[l(g(Zn- Z*))cl +l(g(Zn- Z*))(l, De]---+ 0, from (3.12) it follows that Z*(() is a solution of Problem B for (3.11), i.e. (3. 7) for E E. It is easy to see that the positive constant 6 is independent of e 0 (0 1. But

lx- YIP

s; lx- ziP+ lz- YIP

is impossible for all x > z > y. Actually, Taking account of the inequality (a+ b)P > aP + bP, for all a, b > 0, we arrive at lx- yiP = lu + viP = (u + v)P > uP+ vP = (x- z)P + (z- y)P = lx- ziP+ lz- YIP, for all x > z > y. Thus, (d3) in Definition 1.1 is not satisfied, i.e. (X, d) is not a cone metric space. Example 1.5. Let X= {1,2,3,4}, E 0, y 2': 0}. Define d: X x X ---+ E by

d(x,y)

=

=

IR. 2 , P

{(x,y) E E

X>

{e(l'x- Yl-1, lx- Yl-1), if x =I= y, , if X= y.

6 5 But it is not a cone metric space since the triangle inequality is not satisfied.

Then (X, d) is an extended cone metric space with the coefficient s

=

-

107

Indeed, d(1, 2)

Example 1.6.

= 2:=

lxniP

> d(1, 4) + d(4, 2), The set X=

[P

d(3, 4)

with 0

> d(3, 1) + d(1, 4).

< p < 1, where

< oo}, together with the function d: X x

[P = {

{xn}

C ~:

X--+~+'

n=l

where X= {xn}, y = {Yn} E [P. Put E = ll, P = {{xn} E E: Xn ~ 0, for all n ~ 1}. Letting the mapping d : X x X --+ E be defined by d(x,y) = { 11 C;;Ylk::: 1 , we conclude that (X,d) is an extended cone metric space with the coefficient s

1

= 2 "P > 1.

Definition 1. 7. Let (X, d) be an extended cone metric space, x E X and {xn} a sequence in X. Then (i) { Xn} converges to x whenever for every c E E with e « c there is a natural number N such that d(xn, x) « c for all n ~ N. We denote this by lim Xn = x or Xn--+ x(n--+ oo). n--+CXJ

(ii) { Xn} is a Cauchy sequence whenever for every c E E with e « c there is a natural number N such that d(xn, Xrn) « c for all n, m ~ N. (iii) (X, d) is a complete extended cone metric space if every Cauchy sequence is convergent. A same method as in [8], the following assertions will be used in the sequel(in particular when dealing with extended cone metric spaces in which the cone need not be normal). (p1) If u 0, which satisfies

It is easy to see that a(t)

y-~ + y 2

:f

y"'(t)-

129 4

r3 --7

1 + r3

1r

> - (for example r = 1) such that 8

r

------------~---------------

(1+;~~D1 1 g(rx(1-x))a(x)dx

Meanwhile,

lim y-++=

1

>-

2

.

j(t, y) = +oo, lim f(t, y) = +oo. Consequently, the y

y-+0

conclution of Theorem 1 and Theorem 2 for BVP (4.1) hold.

1 1

Note that

o yl(1+x)(1-x)

rl

In=

x2

Jo

7r

dx =-.Because 4

(2k- 1)!!

7r

xn (2k)!! . 2' n = 2k, { 2 J(1+x)(1-x)dx= ( k- 2)!! n=2k-l. (2k- 1)!!'

References [1] R.P.Agarwal & D.O'Regan, A survey of recent results for initial and boundary value problems singular in the dependent variable. [2] D.R.Anderson, Multiple positive solutions for a three-point boundary value problem, Math.Comput.Modelling.27 (1998) 49-57. [3] Z.Du,W.Ge,X.Lin, A class of third-order mutiple-point boundary value problems, J .Math.Anal.Appl.294 (2004) 104-112. [4] D.Guo, V.Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press,New York,1988. [5] Y.Liu, Structure of a class of singular boundary value problem with superlinear effect, J.Math.Anal.Appl.284 (2003) 78-89. [6] Y.Sun, Positive solutions of singular third-order three-point boundary value problem, J.Math.Anal.Appl. 306 (2005) 589-603. [7] Q.Yao, Positive solutions of singular third-order three-point boundary value problems, J.Math.Anal.Appl. 354 (2009) 207-212.

130

MODIFIED NEUMANN INTEGRAL AND ASYMPTOTIC BEHAVIOR IN THE HALF SPACE* YANHUI ZHANG*a a

GUAN TIE DENG*b

ZHENZHEN WEIM

Department of Mathematics, Beijing Technology and Business University, Beijing 100048, China b Sch. Math. Sci. & Lab. Math. Com. Sys., Beijing Normal University, Beijing 100875, China

The main objective is to give the asymptotic behavior at infinity for the Neumann integral and the modified Neumann integral in the half space of JR". The asymptotic behavior hold outside an explicitly defined exceptional set as [10].

Keywords: Behavior.

Neumann Integral;

Modified Neumann Integral; Asymptotic

AMS No: 31B05, 35J05.

1. Introduction and main results

Let Rn (n ?: 2) denote the n-dimensional Euclidean space. For x in the upper half-space H = {x = (x', Xn) E lltn, Xn > 0}, where x' E lltn- 1 is the project of x onto the hyperplane aH = lltn- 1 . We recall that the Laplace equation

t:.u=

a2u

a2u

a2u

-+-+···+axi ax~ ax?, =0

admits the fundamental solution1 11 1 -loglxl,n=2, 21f E(x)= { 1 In lxln-2' n > 2.

*Project supported by PHR(IHLB 201008257 and IHLB 201106206) and Scientific Research Common Program of Beijing Municipal Commission of Education(KM 200810011005) and College Students Scientific Research and Undertaking Action Project(PXM 2012-014213-000067)

131

rn =

( 2-

1) , and Wn is the surface area of the unit sphere in IR.n. It satisfies Wn

n

I:!.E(x- y) = 6(x- y), x, y

E

!Rn

when 6 is the Dirac distribution; and the Laplace operator has the property[ 2l (1.1)

for any f is a Schwartz function and }(0 is the Fourier transform. Motivated by (1.1), (-6.)"'l'(O O,m(c)(B(x,t) nJR;.n- 1 ) > s(~)n- 1 }.

(2.4) The measure

(2.5) defined by m(E)(E) = m(E n

{y'

E

JR;.n-l : IY'I ?: Rc} ),

(2.6)

134

and function

(2.7) Write

Then

where denote

N3[f](x) =2rn

r

Xnf~y;,~2 dy'.

}{y'EJRn-LIY'I.), we get

IN~E)[f](x)l

r

=2rn(l + lxln- 2)xn

dm(E)/(~]2

}{y'EJRn-LO::;Ix-y'l9lxl}

< 2"'tn (1 + lxln-2)x n -

1

3lxl d

0

lx- Y I

(c)( )

mx tn-2 t

::; Alxls. Moreover INJcl[f](x)l

::;2/nXn

1+ly',l:~22dm(s)(y')

r }{y'EJRn-Lix-y'I2'3IJ,I}

= arg min fa, f

into a problem solving an Euler equation according to the following theorem:

Theorem 2.2. Let fa is the minimum of the piecewise Tikhonov regularization functional

fey= IIRf- glli2(z) + allfll~l(u}= 1 D,)' then f'y is the unique solution that satisfies the Euler equation {

R*Rf

+ af- a6.f = R*g, R*Rf

=

R*g,

(7)

147

r

Proof. Let minimize ,f,(f). Then the Frechet derivative of .fc,(f) is zero at f", i.e.

We can compute the Frechet derivative of

io: (!" + (3v)

,f, (f) as follows

io: (!")

-

(3 =

~ (11Rr- g + f3Rvlll2(Z) + allr + f3vll~'(u~~ 1 D,) -IIRr- glll2(z)-

=

allrll~'(u~~ 1 D,))

~ ( (Rr- g, (3Rv)L2(Z) + /3 2 IIRvlll2(Z) +a(!", f3v) Hl(u~~ 1 D.;)

=

(R*Rr- R* g, v)£2(!1) +a (fn' v) Hl (u7~1 Di)

+ af3 2 llvll~'(u~~ 1 D,))

+ (3\R*Rv, v)£2(!1)

+ af3 (v' v) Hl (u7~1 Di) + a(fa,VJH1(u}~ 1 D.;)

-+ (R*Rfa- R*g,v)£2(!1)

(as (3-+ 0).

(8)

By the definition of H 1 norm above, we can computer the inner product in (8) as follows:

- L 1 v6.(r . XDJ dx k

D;

i=l

k

{

1

= ~ }Di(r. XDJvdx + u}~lDD; v

- L 1 v6.(r . XDJ dx k

i=l

D;

a(r · XD)

an

i

ds(x)

148

Therefore, we have

(R*Rj"'- R*g, v)L2(n) +a (fa- !J.j"', vh2cuz= 1 D;) = 0,

't:/v E H6(D),

i.e. f"' is the solution to the Euler equations: {

R*Rf

+ af- a!J.f =

R*g,

in D :=

u7=

1 Di,

on 8D := u7~ 1 8Di.

R*Rf = R*g,

D

Once more, by using the formula (2.1), we obtain a differential-integral equation as follows: 2 {_I-1 -IJ(y)dy+af(x)-a!J.f(x)= { g(B,x·B)de, {

Jn

x- y

} 51

2 {_I-1 -IJ(y)dy= { g(B,x·B)de, Jn x- y } 51

in D, on 8D.

Suppose we know the location of discontinuous interface of f in Example.II.1 i.e. the location of the circle, we apply this piecewise Tikhonov regularization method with a regularization parameter a = 0.001 numerically. The regularization solution j, !v14 such that for any x E [O,H],

0 < m1 0,

and let

(11) Similarly, G± = FTg±, H± = FTh±, etc. Then g+(x) = Tg+(x)

1

00

=I:

k+(xy)f+(y) dy (Tk+)(x

G+(t) = u(t)F+( -t)

+ laoc k_(xy)f-(y) dy,

+

y)(Tf+)(y) dy

+I:

(Tk_)(x

+ v(t)F_ (-t).

Similarly, G_(t) = v(t)F+( -t)

+ u(t)F_( -t).

Combining the preceding two equations, we obtain

-t)) .

( G+(t)) = W(t) (F+( G_(t) F_(-t) Replacing t by -t yields ( G+( -t))

G_( -t)

=

W( -t) (F+(t)).

F_(t)

We then combine the last two equations to obtain

G(t)

=

Z(t)F(t),

+

y)(Tf_)(y) dy,

201

where

G(t) = (

~=~~; ),

G+( -t) G_( -t)

F(t) = (

~=~~; ).

F+( -t) F_( -t)

(12)

Note that Ga(t) = G(t)- aF(t). Thus,

G,y(t)

=

(Z(t)- ai)F(t).

(13)

Since the map

r .. f

r+

(F+(t)) F_(t)

is an isometry, we see that K - a! is invertible if and only if the matrix (Z(t)- ai)- 1 is essentially bounded. Therefore, (i) and (ii) are equivalent. (iii) B (iv): By the same reasoning as in the last case, we have G(t) = Z(t)F(t) and H(t) = Z(t)G(t). Hence, H(t) = Z 2 (t)F(t), or

Q(t) 0 ) H(t) = ( O Q( -t) F(t). It follows that

( H+(t))

H_(t)

= Q(t) (F+(t)) F_(t)

'

or

(14) and hence

It is now clear that (iii) and (iv) are equivalent. (ii) B (vi): We have (Z(t)- ai)- 1 = det(Z(t)- ai)- 1 · Ma(t), where Ma(t) is the adjoint matrix of (Z(t) - a!) and hence it is essentially bounded. It follows that (ii) and (vi) are equivalent. (iii) B (vii): The reasoning is the same as in the previous case. (vi) B (vii): It follows from the identity

(Z(t)- ai) (

0) = (a-l(Q(t)a2J) W(t)) o -ai '

I a- 1 w(-t) I

(15)

that det(Z(t)- a!)= det(Q(t)- a 2 I).

(16)

202

Therefore (vi) and (vii) are equivalent. (iv) -R (v): By what has been proved, (i) and (iv) are equivalent. It follows that (iv) and (v) are equivalent. (vii) -R (viii): It is straightforward to verify that

(17) from which the equivalence of (vii) and (viii) follows.

D

We call (w, z) defined by (10) the compound symbol of k and that of Kk. Corollary 2.7. Let K E 2. Then Corollary 2.8. Let K E 2 1Y(K 2 ) = Rw U Rz.

-~Y(K)

= IY(K).

D

with compound symbol (w, z). Then 1Y(K) 2 =

Corollary 2.9. Let K E 2. Then K is real if and only if ~Y(K)

D

= {±1}.

Proof. Let K E 2 be a real operator. Then K 2 =I, hence 1Y(K) 2 = {1} by Proposition 2.3. Take an arbitrary nonzero f E L 2 (ffi.). Then (K +I)(KI)f = 0. If (K -I)f = 0, then 1 E 1Y(K). If (K -I)f -j. 0, then ( -1) E 1Y(K). Either way, IY(K) = {±1}, by the main theorem. Conversely, suppose that IY(K) = {±1}. Then 1Y(K 2 ) = {1} = Rw URz, by Corollary 2.8. It follows that w(t) = z(t) = 1, and hence Q(t) =I. By (14), K 2 =I. Since K K =I, we see that K = K. D

3. Eigenfunctions and Integral Equations

Let K E Band a E 1Y(K). Then K- a! is not invertible, hence it is either non-injective or non-surjective. Suppose that K- a! is non-injective. Then there is a nonzero function f E L 2 (ffi.) such that (K - ai)f = 0. The function f is called an eigenfunction of K associated with the eigenvalue a. The kernel of (K- ai),

N(K- ai) = {f E L 2 (ffi.) : (K- ai)f = 0} is called the eigenspace of K associated with a. The following proposition is likely known in the literature; we include a short proof here for the convenience of the reader. Proposition 3.1. Let K be a unitary operator in !J1J and a E C Then (K- ai) is invertible if and only if it is surjective.

203

Proof. Assume the contrary. Then there is a K- o:I that is surjective and non-injective. Hence there is a nonzero f E L 2 (IR.) with (K- o:I)f = 0 and an hE L 2 (IR.) with (K- o:I)h =f. Then 0 c/= (J, f)= ((K- o:I)h, f)= (h, (K- a!) f)= (h, 0)

=

0.

This is a contradiction.

D

Note that Proposition 3.1 does not hold for a general operator in !!lJ. For instance, the "back-shift" operator L defined by

- { 0,

L(ej ) -

ej-l,

where { e 1 , e 2 , ... injective.

}

if j = 1, . . 1f J > 1,

is an orthonormal basis of L 2 (JR.), is surjective, but non-

Theorem 3.2. Let K E 2 Let

with compound symbol (w, z), and o:

E

a(K).

A= A,= {t E JR.: w(t) = o: 2 }, B = Ba = {t E JR.: z(t) = o: 2 }. Then N(K- o:I) is nontrivial if and only if m(A U B) > 0. Furthermore, the following hold: (i) If N(K- o:I) is nontrivial, then it is infinite dimensional. (ii) N(K- o:I) is nontrivial if and only if N(K + o:I) is nontrivial. (iii) If m(A) > 0 and m(B) = 0, then every function in N(K ± o:I) is even. (iv) If m(A) = 0 and m(B) > 0, then every function in N(K ± o:I) is odd. Proof. Note that the sets A, B are symmetric about the origin, since w, z are even functions. Suppose that m(A U B) = 0. We shall prove that N(K- o:I) = 0. Let f E N(K- o:I). By (13), we have

(Z(t)- o:I)F(t)

=

0,

(18)

where F(t) is defined by (11) and (12). By the assumption m(A U B) = 0 and (16), (17), the matrix (Z(t)- o:I) is nonsingular almost everywhere on JR.. It follows that F = 0, and hence f = 0. Now we assume that m(A) > 0. Let A+ = An {t : t > 0} and A_ = A n { t : t < 0}. Choose a measurable function p( t) on A+ such that

p(t?

= o:- 1 (u(t)

+ v(t)).

(19)

204

FortE A_, let p(t) = 1/p( -t). The function p(t) is now defined on A and, since a 2 = w(t) on A, equality (19) holds for t E A. It follows that -ap(t)

+ (u(t) + v(t))p( -t)

=

0,

tEA.

The last equality is equivalent to

(Z(t))- ai)

p(t)) (~~)

~

= 0, tEA.

(20)

( -t)

Thus, for each function q(t) E L~(A), the set of even, complex-valued, square-integrable, measurable functions on A, we have

(Z(t))- ai)

l ) :~~~:~~~

(-t)q( -t)

=

0,

(21)

tEA.

( -t)q( -t)

By (8), lu(t) + v(t)l = 1, and hence lp(t)l = 1. Recall that defined by (6) and (9). Define TA: L~(A)---+ L 2 (IR.) by

TA(q) =

r- 1(XAPq

r

is the operator

C)).

Then IITA(q)ll = v'2llqll, hence TA is injective. By (21), TA(L~(A)) C N(Kai). Therefore, N(K- ai) is infinite dimensional. Note that each function in TA(L~(A)) is even. Suppose that m(A) > 0 and m(B) = 0. Then rank(Z(t)- ai) = 4 a.e. (almost everywhere) on JR. \A, and rank(Z(t)- a 2 I)< 4 a. e. on A. We now show that rank(Z(t)- a 2 I) = 3 a. e. on A. We can verify that

2Q(t)

= (w(t) + z(t) w(t)- z(t)) w(t)- z(t) w(t) + z(t) ·

Since w(t) = a 2 -=f. z(t), we see that the (1, 2) entry of Q(t) is non-zero a. e. on A. Hence,

Q(t)- a 2 I

"I- 0

a.e. on A.

(22)

It follows from (15) and (22) that

rank(Z(t)- ai) = 3 a. e. on A. Now let

f E N(K- ai) and

(23)

205

Then

F+(t) ) F_(t) ( (Z(t))- od) F+( -t) = 0,

tEA.

(24)

F_(-t) Since Z(t) - o:I is nonsingular a.e. on JR;. \A, equality (24) implies that = 0 a.e. on JR;. \A. By (23), the two vectors in (20) and (24) are proportional. It follows that there exists a q E L~(A) such that

rj(t)

(

F+(t) ) F_(t) F+( -t) F _ ( -t)

~p(t)q(t) p(t)q(t)

)

p( -t)q( -t) (-t)q( -t)

.

Thus, f = TA(q). Therefore TA(L~(A)) = N(K- o:I), and every function in N ( K ± o:I) is even. The case when m(B) > 0 is similar. We define a function r(t) on B such that r(t)r( -t) = 1, r(t) 2 = o:- 1 (u(t)- v(t)). It follows that

-o:r(t) + (u(t)- v(t))r(-t)

=

0,

t E B,

and

(Z(t))- o:I) (

~~;i -r( -t)

)

=

0, t

E B.

Then the map TB : L~(B) -+ L 2 (JR;.) defined by TB(s) =

r- 1 (XBTS ( ~ 1 ))

is injective and satisfies TB(L~(B)) c N(K- o:I). Therefore, N(K- o:I) is infinite dimensional. Note that each function in TB (L~ (B)) is odd. If m(A) = 0 and m(B) > 0, then TB(L~(B)) = N(K- o:I). Hence, every function in N(K- o:I) is odd. D Corollary 3.3. Let K E 2. If O"(K) is finite, then every o: E O"(K) is an eigenvalue with an infinite dimensional eigenspace. D

206

Corollary 3.4. Let K E 2? with compound symbol (w, z). If Awn Az = 0, then for each eigenvalue of K, the eigenspaces N(K ± ai) consist entirely of either even functions or odd functions. D Proposition 3.5. Let K C 2? be a real operator. Then u(K) = {±1}, L 2 (JR.) = N ( K- I) EB N ( K +I), and N ( K ±I) are both infinite dimensional. Proof. By Corollary 2.9, u(K) = {±1}. It is clear that

L 2(IR.) =(I+ K)(L 2(IR.)) EB (I- K)(L 2(IR.)). It is also clear that (I+ K)(L 2(IR.))

c N(K- I). Iff

E

N(K- I), then

f= f+Kf + f-Kf = f+Kf E (I+K)(L2(IR.)). 2

2

2

Hence (I +K)(L 2(IR.)) = N(K -I). Similarly, (I -K)(L 2(IR.)) = N(K +I). Therefore L 2(IR.) = N(K- I) EB N(K +I). By Theorem 3.2, N(K ±I) are infinite dimensional.

D

An example of a real k EX can be found in [3, Example 3.8]: k(x) = b"(x _ 1) _ xsi~(blog lxl). 7rlxl3/2log lxl By applying the preceding theorems, we obtain the following.

"I-

Theorem 3.6. If a

±1, the Fredholm equation of second kind

_1 f(1/x)-

1

xysin(blog lxyl) I 1">/21 I I f(y) dy = g(x) xy · og xy has a unique solution, which is given by -af(x) + lxl

IR 1r

2 -1 (1- a )f(x) = ag(x) + lxl g(1/x)-

rxysin(blog lxyl)

}IR 1rlxyl 3/ 2 log lxyl g(y) dy.

The solution spaces of the homogeneous Fredholm equations ±f(x) + lxl-1 f(1/x)-

rxysin(blog lxyl) f(y) dy = 0

JIR 7rlxYI 3 / 2 log lxyl

are infinite dimensional. The inhomogeneous Fredholm equations

_1 . ](1/x)-

1

xysin(blog lxyl) . I I"J/21 I I j(y) dy = g(x) xy · og xy are solvable if and only if the right-side function g( x) satisfies the companion homogeneous Fredholm equations ±f(x) + lxl

=t=g(x)+lxl respectively.

_1 g(1/x)-

IR 1r

1

xysin(blog lxyl) I 13/21 I lg(y)dy=O, xy · og xy

IR 1r

D

207

Theorem 3. 7. Let n 2': 2 be an integer and let K E !JlJ be such that Kn Then

Proof. Let Pj =I+ (3-j K

Then L,~=l Pj

=

+ ((3-j K) 2 + · · · + ((3-j K)n-l

for j

=

= I.

1, ... , n.

ni, hence L 2 (IR.)

=

P1(L 2 (IR.))

+ · · · + Pn(L 2 (IR.)).

It suffices to show that N(K- j3.il) = Pj(L 2(IR.)). Since (K- j3JI)Pj = 0, it follows that N(K- j3J I) :=J Pj(L 2(IR.)). Suppose that f E N(K- j3J I). Then Pvf = 0 for v -=f. j, since Pv = II.\,iv(I- (3-.\ K). It follows that nf = 'L, Pvf = P.if and f E Pj(L 2(IR.)). Thus N(K- j3J I) = Pj(L 2(IR.)) and the proof is complete. D

An operator K E 2? is said to be Fourier-like if K 2f(x) = f(-x) for each f E L 2 (IR.). We now determine the spectra and eigenspaces of Fourierlike operators. Some of our work here is inspired by [15], where the spectrum of the Fourier transform is determined, and methods to find eigenfunctions are described. Let K E 2? be a Fourier-like operator with symbol (u, v). Then (see [3, Theorem 5.2])

u(-t) = v(t),

v(-t) = u(t).

(25)

z(t)

(26)

Equalities (8) and (25) imply that

w(t)

=

1,

=

-1.

By Theorem 3.2, Proposition 3.7, and (26), we have the following. Theorem 3.8. Let K E 2? be a Fourier-like operator. Set, for v = 1, 2, 3, 4,

Pv = I+ i-v K

+ i-2v K2 + i-3v K3.

Then the following hold: (i) cr(K) = {±1, ±i}. (ii) N(K- ivl) = Pv(L 2(IR.)), for v = 1,2,3,4. (iii) L 2 (IR.) = EB~=lN(K- ivi). (iv) The eigenspaces N(K- iv I) are infinite dimensional. (v) N(K ±I) consists of even functions. (vi) N(K ± il) consists of odd functions.

D

208

For Fourier-like kernels, an example is given in [3, Example 3.12]): k(x) = 1- i 1 + sgn x ( -iS(x _ 1) + 1- isgn(ln lxl) . min(lxl lxl- 1)) 2

JlXT

2

'

1+i1-sgn x( -u'( x + 1) + 1+isgn(lnlxl) · mm . (I x II x 1-1)) . + 2

JlXT

2

'

There are plenty of other Fourier-like kernels; but for most of them, finding their explicit forms is intractable. For each positive integer n the following is a Fourier-like kernel: k ( )= nX

lal

1=

0}

1- i ,itlogl:rl { cos(tltln- 1 + 7r/4), for X> 27TJ2TXT _ 00 e isin(t1W'- 1 +7T/4),forx

0 . The doubly-periodic fundamental

parallelogram in the aperiodical plane of the quasicrystals body will be denoted by P00 . Its vertices are

±w1 ± w2 • The

boundary of P00 will be denoted by

with the positive direction taken to be anticlockwise, and

r = u;=l r,.

r

It will

be assumed that there exists a system of m cracks distributed in ~HI, denoted ~

by lj

=

ajb/j

= l, ... ,m) , which are smooth and non-intersecting curve

segments. The positive directions of lj are taken to be from ai to bi , and we denote 1

=

2.= ';'=

1

.

The system of cracks inside Pmn (m, n = 0, ±1, ...) is

congruent to 1 in P00 . Without loss of generality, we may assume that the origin is not on the cracks. If there are only Griffith cracks in the quasi crystals body along the x1 -axis,

the direction of the atom periodic arrangement, assume that the quasicrystals is in complete plane strain (CPS) state, which is a specific case of the three-

226

dimensional elasticity state. Exactly, the quasicrystals subjected to combined antiplane shear and inplane loads in the aperiodical plane, ( X 2 ,x3 -plane). Obviously, the elastic equilibrium in the aperiodical plane is reduced twodimensional (plane) elastic problem. Different from classics complete plane strain formula, in our case, considering the contribution of phonon and phanon stress, the generalized complete plane strain (CPS) state is 0"22 = LT22(x2,X:;),LT;;;; = O";;;;(x2,X:;),LT2:l = 0"2:l(x2,X:;) H:;:; = H:;:; (x 2, x:1), H:12 = H:12 (x 2, X;;)

(1.1)

0"12 = LT12(x2,X;;),LTl:l = 0"1:l(x2,X:;)

ell = e1, W:n = e2 where e1 , e 2 are real constants. These equations of classics CPS problem similar to the above equations in the absence of phanon physical quantitys were first introduced by Glingauss in reference in 1975. And a creativeness achievement of solving various doublyperiodic CPS problems of elasticity theory attributed the success to Li Xing in reference [5] in 2001. Similarly, we can resolve the special three-dimensional elastic system into two linearly independent plane elastic systems by the superposition principle of forces; one is the generalized plane strain state

0"22(x~,X;;),LT;;;;

0"22

1H:;:; -

O";;;;(X~:X;;),LT2:l

= 0"2:l(x2,X:;)

(I)

H:12 (x 2,x;1) 0"12 = LTl:l = 0, ell = e1, W:n = 0

l

H;;;;(x 2,x:1),H:12 -

and another is the longitudinal displacement state CT22

0"12

= 0":;:; = CT2:1 = H:;:; = H:12 = 0

LT12(x2,x:;),LT1:J = LT1:;(x2,x:;),H:n

ell - 0, W:n

-

=

H;n(x2,x:J)

(II)

e2

For the elastic system (1), the phonon and phanon stress components can be expressed by analytic functions cp1(zk),.2 II

~

~·i

1

1

~"' +(8),1

81\)w2(t) + (81'\- 82\)w2(t)

-

I

(4.4)

·3

f ,[g7(t)- gj(t)]dt + \ ~ \ irh J,[9t(t)- 9j(t)]d"f; + 8:; + 8 irh J [g+(t)- g-:-(t)]dt. + 'h J [j+(t)- r(t)]dt. \ +\

+A

~A

2

'h

1

~z

2

±

~

·

'1 111

2

·

J

1

~

~·i

.l

J

~/"

J

·

·

.l

J

2A[Re(\7iw1) + iRe(\71w2 )] + 2B[Re(A.2r2w1) + iRe(A._z72w2)]

'hJ

Uhf

=G~ 1 +iG~ 2 +Re{,\1 ..\...z~i ~/[w1 (t)+w2 (t)]d~ +-\\-;- ~1 ,[w1 (t)+w2 (t)]dt.z A.

7]

+~___l

J,[g+(t)- g::-(t)]dt

A..z+A..zm;

1

1

+-\\ i7]2 J,[w1 (t) + w2 (t)]dt.z + ~

1

2}

7]

+iRe{\\~

J,[w1(t) +

m;

w2 (t)]d~

(4.5)

~ 7]2 J, [g+(t)g-:-(t)]dt2 } 1 1 A._z+A..zm

1

Without loss of generality, assume that (~b 2

+

J;8,~)Re[A2 r2 (w 1

+ iw2 )]

+c

(4.6)

0

Then, the determinant of the system of equations (4.4 )-( 4.5) is nonzero. Hence, we can obtain A, B, C uniquely from that. Letting z---+ t0 , so ; ---+ lf,z2 ---+ ~,z:1 ---+~~,and using the Plemelj formulae, taking equations (4.3), either for the positive or negative boundary, we get the same equations

Re{\A.z j[w1(t) + w2 (t)][((t1 - tf)dt1 m 1 +ARe(..)Ar/) + BRe(A.zBr:/)

g7(to) + gj(to) =

2

,\2

- Re{\

+\

1

J

((~ - tg)d~]} (4.7)

+

-

~i ,r"[gi (t)- gi (t)JW2

-

Zz)dt~J + e.zj

231

where w 3 ( t) is an unknown real function, E is an undermined complex comstant. Substituting equation (4.9) into the second formulae of (3.5), we get a system of algebraic equations for the unknown real constant Re E, lrn E

(wk- wk)ReE- i(wk

+ wk)IrnE = lwkl ~ + ~Re{rhf w3 (t)dt}

(4.10) m 1 Because of the determinant of the system of equations (4.1 0) is nonzero. we can obtain E uniquely from that.Substituting (4.9) into the first equation of (4.10), either for the positive or negative boundary, we get the same equations as 2

~ J w 3 (t)dlog[a(t- t 0 )] + dm(Et 0 ) - iC* = 2Kz

't

a(t- to)

0

(4.11)

J

5. Unique Solvability It is now be proved that if ci,c3i'j

=

1,2, ... ,rnwill be suitably chosen (also

uniquely), then equations (4.7) and (4.8) are solvable in ~m. Similarly to the work of [5], for this purpose, the homogeneous equation, obtain from equation (4.7) and (4.8) for'I;}(t) = o,cztf(t) = O,ThJ(t) = O,e1 = 0 will be considered and it

232

will be shown that it has no non-trivial solutions. This mean if wf(t),wg(t) be any of

solution wf(t)

this

equation,

Cy

after

= wg(t) = Oeverywhere on1.

=

cfj,C,1j

be

= c{ij

taken,

the

It obvious that (2.32) is satisfied in this

case. Let ~Pf(z1 ),cpg(z2 ),cp~(z3 ),A0 ,E 0 ,C0 be the corresponding values of cp1 (;),

cp2 (z 2 ),cp3 (z 3 ),A,E,C detennined by equations (3.1), (4.1-4.3), and equations (4.5), (4.6) for w1 (t)

wf(t),w 2 (t)

=

=

wg(t) .It is easy to verify that they satisfy

the corresponding boundary conditions (4.8) and (4.9), which from the first fundamental problem under homogeneous conditions. By the uniqueness theorem [11 ], the elastic region may only have a rigid motion as ~Pf(z1 )

= ikr1z1 + D1 ,cpg(z2 ) = ikr2z2 +

D2 ,cp~(z 3 )

= ikr3 z 3 + D 3

(5.1)

Referring to formulae (4.1 )-(4.3), we have

~Pf (z1 ) =

ikri z1 + D1

~Pg(z 2 ) =

ikr2 z2 + D2 =

~Pg(z;1 )

ikr;3z:l + D:1

=

1

=

~ J [wf (t) +

wg (t) ](( t1 - z1 )dt1 + A0 r1z1

1

27rl

I

-~J Jw~(t) + 21f'l -1

wg(t)]((t2 - z2)dt2 + E 0 r2 z2

(5.2)

(-81\ - 82\ + 83>-1 + 84 \)w~(t) ) + - 82:\2)wg(t) W:l - z;;)dt, + Cz:;

[

= 2Ki J~~~~ +(83\ -

81\)w~(t) (8~:\I

Considering quasi-periodicity of the two sides of equations, we obtain

A 0 = E 0 = C 0 = ik

1

-.

J [w (t) + w (t)]dt 1

27rz r _1_J

2Ki

l

0 1

0 2

(-81\

(5.3)

_ _

1

=

-82\

0

+83 \

0

)

0

+84 \)w1 (t) -

-

0

r 1" +(83\ -81 ~)w2 (t)+(84 \ -82 ~)w2 (t)

_

(5.4)

dt., - 0 ·

Taking equations (5.4) into account, from equations (4.5) and (4.6) we have

A0 = E 0 = C 0 = 0 Hence, from equations (5.3) and (5.5) we have k = 0, Furthermore, equations (5.2) can be reduced by

~J [w 0 (t) + w20 (t)]((t 2Jri -,1 1 . 1

D1

=

D2

= -

Dl

= 2Jri

;i

2

1

rJw2(t)

J,~"

[

(5.5)

z1 )dt1

+ wg(t)]((t2 -

z2)dt2

(-81\ - 82\ + 83,\1 + 84 \)w2(t) +(8:3·\ - 81>.2)wg(t) +

(b~:\1- b~\)wg(t)

I

W:J -

(5.6)

z3)dt3

233

Considering w1 (t) be a real function which describe the contribution of phanon, and w 2 (t) a the complex functions which describe that of phanon on the cracks, and using the Plemelj formulae from (5.6), and we get immediately

wf(t)

=

wg(t)

0, t E

=

r

(5.7)

We now turn to prove the equation (4.11) to be uniquely solvable. To do this, we must prove w:~(t)

0 under homogeneous conditions of the first

=

fundamental problem. In this case, we have (e.g. see [5])

cp 0 (zm)

=

2 ~'i J. W;~(t)((t- zm)dt + E

0

z

=

Ez

(5.8)

Considering quasi-periodicity of (5.9), we obtain

~J w~(t)dt = 27rz

O,E 0

=0

(5.9)

1

By the Plemelj formulae from (5.8), we get immediately w~(t)

=

0, t E

r

(5.10)

Now, we have proved the system of singular integral equations, which is constituted by equations (4.7) and (4.8), and the Fredholm integral equation of the second kind, which is constituted by equation (4.11 ), are uniquely solvable, respectively. Thus, after obtaining the unique solution w1 (t), w 2 ( t) from the above

singular

integral

equation,

we

may

obtain

the

functions

cp; (z1 ), cp;(z2 ), cp:~ (z:1 ) from equations (4.1)-(4.3), furthermore, the stress functions cp1 (z1 ),cp 2 (z 2 ),cp 3 (z 3 ) will be obtained from equations (2.1). And after obtaining the unique solution w:3 (t) from the above Fredholm equation of the second kind, the functions cp(zrrJ may be obtained from equations (4.9).

References 1.

2. 3. 4. 5.

Fan T. Y., et al., The strict theory of complex variable function method of sextuple harmonic equation and applications, J. Math. Phys., 51 (20 10), 053519. Fan T. Y., Mathematical theory of elasticity of quasicrystals and its applications, Science Press, Beijing, 2010. Lekhnitskii S. G., Theory of elasticity of an anisotropic body. Holden Day. San Francisco,1963. Li X., On the mathematical problem of composite materials with a doubly periodic set of cracks, Appl. Math. Mech. (PRC) 14(1993), 1143-1150. Li X., Applications of doubly quasi-periodic boundary value problems in elasticity theory, Shaker Verlag, Aachen, 2001.

234

6.

7.

8.

9.

I 0.

11. 12. 13.

14.

Li X., Complete plane strain problem of a nonhomogeneous elastic body with a doubly-periodic set cracks, Z. angenw. Math. Mech, 6(2001), 377391. Li X., The effect of a homogeneous cylindrical inlay on cracks in the doubly-periodic complete plane strain problem, Int. J. Fract. 4(2001 ), 40341. Li X., General solution for complete plane strain problem of a nonhomogeneous body with a doubly- periodic set of inlays, Z. angenw. Math. Mech, 9(2006), 682-690. Liu G. T., He Q. L. and Guo R. P., The plane strain theory for onedimensional hexagonal quasicrystals in aperiodical plane, Acta Physica Sinica, 6(2009), S 118-S 123(Chinese). Lu C. K., On the complex Airy functions in doubly-periodic plane elasticity, J. Math. (PRC), 6(1986), 319-330 (Chinese). Lu C. K., Complex variables methods in plane elasticity, World Scientific, Singapore, 1995 (Chinese). Muskhelishvili N. 1., Some basic problems of the mathematical theory of elasticity, Noordhoff, Groningen, 1956. Zheng K., On the fundamental problems in an anisotropic plane with a doubly-periodic set cracks, Comm. On Appl. Math. and Comput. 1(1993), 14-20( Chinese). Zheng K., On the doubly-periodic fundamental problems in plane elasticity, Acta Mathematical Scientia 8(1988), 95-104(Chinese).

235

THE PROBLEM ABOUT AN ELLIPTIC HOLE WITH III ASYMMETRY CRACKS IN ONE-DIMENSIONAL HEXAGONAL PIEZOELECTRIC QUASICRYSTALS* HUASONG HUO, XING LI **

(School of Mathematics and Computer Science, Ningxia University, Yinchuan 750021) In this paper, by introducing the appropriate conformal transformation and using the complex variable function method, the problem about an elliptic hole with III asymmetry cracks in one-dimensional hexagonal piezoelectric quasicrystals is sovlved and the SIFs at the crack tip are obtained in the case of impermeable and permeable boundary conditions. In some special onditions, the cracks can be turned into a Griffith crack. By imitation computation, we can get the SIFs at the tip of the Griffith crack. especially, when the quasicrystals become vestigial the crystal, we can also get the SIFs at the tip of the Griffith crack, which is in accordance with the know results.

Key words: one-dimensional hexagonal piezoelectric quasicrystals, elliptic hole with III asymmetry cracks, complex variable method, SIFs

0. Introduction Quasicrystal is found in recent years a new structure. It is its special structure led to its characteristics changed, For example, in force, thermal, electrical, magnetic and related physical and chemical properties 11 -4Jetc. For quasicrystal defects research has made certain achievements 15 -8J, however, the piezoelectric quasicrystal research are rare, Piezoelectric for quasi crystal material functional widely used has certain practical significance. This paper, by using generalized complex variable method, With the help of the appropriate conformal mapping, the problem about along the quasi-periodic direction through two asymmetric elliptical hole edge cracks in one-dimensional hexagonal piezoelectric quasicrystals is researched, we get the SIFs at the tip of cracks in electric impermeability and electric permeability two circumstances. In extreme cases, it can degenerate into existing relevant conclusion. *This work is supported by the National Natural Science Foundation of China (10962008; 51061015; 11261045) and Research Fund for the Doctoral Program of Higher Education of China (20 116401110002). **Corresponding author. Tel.: +86 951 2061405. E-mail address: [email protected] (X. Li).

236

1. Basic equation

x1 -x2 is the coordinate cycle plane in one-dimensional hexagonal piezoelectric quasicrystals, the coordinate axis x:3 is the quasi-periodicity direction, establish rectangular coordinate system, Electric polarization direction for x 3 direction, u1 , u2 , u3 are phonon displacement component, w is phason displacement,¢ is the electric potential; ~11 , 6 2, ~33 , 2~23 , 26 1,26 2, w1, w2, w 3 , are the phonon,phason strains and electric fields; an, a 22 , a:1:1, a 2;1, a;11 , a 12 , H 1, H 2, H;1, D 1, D 2 , D:1 are the phonon, phason stress components and electric displacements, according to references[?], The generalized Hooke's law for one-dimensional hexagonal piezoelectric quasicrystal with point group 6 mm is given. all a22

= =

Cllc:ll + C12E22 + C1:;E:;:; + R1w:1 - e:~1E:1' C12cll + c22c22 + C1:lE:;:; + ~W:;

a33 = C13c:ll + C13E22 + C33c33 +

-

~w3 -

e~1E:l' e~3E3,

a23 = a32 = 2C44c:31 + R3w2 - ei:;E2, a13 = a31 = 2C44c:31 + R3w1 - ei5E1, a12 H:;

= =

a21

=

2Cfific:12' 2

R1 (ell + c22) + ~E:;:; + k1w:; - e:nE:l'

(I)

H 1 = 2R3c:31 + k2w1 - e~0 E1 , H2 = 2R3c:32 + k2w2 - e~5E2, D3 = eL(c:ll + E22) + e~3E33 + e~3w3+ E33 E3,

D1 = 2eioc::n + e~5w1 + Ell E1' D2

=

1

2

2et5E:l2 + e15w2+ Ell E2.

where CZJ , kZ are elastic constants in the phonon and phason fields; R,. are phonon-phason coupling elastic constants; e~i, e7i are piezoelectric coefficients; Ell , E33 are two dielectric coefficients.

Geometric equation:

(2)

237

Regardless of the energy balance equation: alan

8 1a 21

+ 82a12 + 8:lal:l + 8 2a 22 + 8 3a2:3

=

0,

=

0,

+ 8 2a:12 + 8:;0":;:1 = 0, 8 1H 1 + 8 2 H 2 + 8:1H:1 = 0,

(3)

8 1a:n

81D1

+ 82D2 + 83D;3

0.

=

In one-dimensional hexagonal piezoelectric quasicrystals, When the defect through the x:1 direction, material geometric properties will not with quasiperiodicity direction change, in that way

a.,·u.Z d

=

O,a.,w

0,8.1a ZJ

=

d

,

l

=

0,8.,1H Z

0,8,.1D,.

=

0,( ·i,j

=

=

1,2,3)

(4)

Substituting (4) into (1)-(3), we obtain c44 \72U:;

= 0,

+ R:l\72w+ei5 \72¢

2

2

2

2

R:3\7 u:3 + K 2\7 w + e15 \7 ¢ = 0, 1

2,

2

2

e15 \7 u:l + e15 \7 w-

When

R3

E?is

R3

K2

E?i.;

1

2

E?is

(5)

-

\7 ¢ - 0.

1

c4,~

E?is

E44

2

2

-

+c

0, type (5) can be written as a further

E44

(6)

From (6) we know, this problem can be thought of as three harmonic equation of solving problem, using the complex function theory knowledge, using u:1 , w, ¢ express as three analytic function of the real component. (7)

where

Re

express

as

'P;(z)(i

= 1,2,3)

express

z

+ ix2 , i

.W.

= x1

=

analytic as

any

function three

of of

the the

real analytic

component, function,

Now, using the complex variable function method to solve the above elastic balance issue.

238

Using the nature of the analytic function and Cauchy-Riemann theorem, from (I)-( 4) obtain

(8)

where cp; =dcpi / dz, for simplicity, using cpi express cpi (z), (i

=

1, 2, 3).

2. Two asymmetric elliptical hole edge crack problem (Impermeable problem) 2.1 The permeability of stress field

If an one-dimensional hexagonal piezoelectric quasicrystals has two asymmetric elliptical hole edge crack along the quasi-periodic direction penetration

( x3 direction), elliptic semi-major axis for a long, elliptical short half shaft forb long, craze for c1 - a, c2 - a long , elliptical hole and the boundary of the crack for L long, Tis a quasi-cycle the direction of shear force in infinity. The following chart

I

1

I

---- ...

'

---.,. ' )--~2 {' '; 0,

lim g'(t) 1--H=

= -oo,

(10)

( 11) From (11) we clearly see that g'(t) 0 is defined by

dn Dau(t) =-rn-au(t) dt 17 where

aE R,n-l 0 and 1 is the gamma function[l3]. In the paper, we consider Fredholm-Volterra integral equations involves fractional integral term as:

Lm 1Ia

/l,17 17 17

~I

~~

y(t)+ u 1f0 k 1(s,t)y(s)ds+ u 2 f0 k 2 (s,t)y(s)ds = f(t, y(t)),

n=l

where

0'.5.a17 ,0'.5.s,t'.5.l,A17

A,8 E R. 2

277

2.2. Some properties of wavelets Recently, wavelets have proved to be a wonderful mathematical tool, and have been applied extensively to find the approximate solution of problems. They constitute a group of functions constructed from dilation and transformation of a single function called the mother wavelets function. When the dilation parameter and the translation parameter vary continuously, we have various families of wavelets. 2.2.1. Some properties of Haar wavelets The orthogonal set of Rationalized Harr functions is a family of square waves with magnitude of ±1in some intervals and zeros elsewhere. The first wavelets function is h0 (t). The mother wavelets function is h1(t) which constructs a group of functions[ 14]:

h0 (t) =I,O::=;t