Green's Function and Boundary Elements of Multifield Materials

Green's function and boundary elements of multifield materials Qing-Hua Qin Hardcover: 266 pages Publisher: Elsevier Sc...

Author: Qing-Hua Qin

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Green's function and boundary elements of multifield materials Qing-Hua Qin

Hardcover: 266 pages Publisher: Elsevier Science (July 18, 2007) ISBN-10: 0080451349 ISBN-13: 978-0080451343

Contents Preface Acknowledgements Notation Chapter 1 Introduction.....................................................................................................1 1.1 Foundation of Green’s function.....................................................................................1 1.2 Dirac delta function.......................................................................................................5 1.3 Basic equations of piezoelectricity.................................................................................7 1.4 Linear theory of magnetoelectroelastic materials...........................................................12 1.5 Two solution procedures in anisotropic multifield materials............................................17 References..................................................................................................................23

Chapter 2 Green’s function for electroelastic problems...............................................25 2.1 Introduction................................................................................................................25 2.2 Green’s function by Radon transforms.........................................................................26 2.3 Green’s function by the potential function approach......................................................27 2.4 Green’s function by Fourier transforms........................................................................32 2.5 Half-plane problem.....................................................................................................38 2.6 Solution of bimaterial problems....................................................................................41 2.7 Bimaterial solid with an interface crack........................................................................42 2.8 Elliptic hole and inclusion.............................................................................................51 2.9 Arbitrarily shaped hole................................................................................................61 2.10 Semi-infinite cracks.....................................................................................................68 2.11 Anti-plane problems in a cracked solid........................................................................69 2.12 Dynamic Green’s function...........................................................................................73 2.13 Green’s functions for an infinite medium with a crack...................................................77 2.14 Approximate expressions of Green’s function..............................................................80 2.15 Green’s functions for time-dependent thermo-piezoelectricity.......................................82 References.................................................................................................................88

References.................................................................................................................88

Chapter 3 Green’s function for thermoelectroelastic problems..................................92 3.1 Introduction...............................................................................................................92 3.2 Green’s function in free space.....................................................................................92 3.3 Half plane solid..........................................................................................................93 3.4 Bimaterial solid..........................................................................................................95 3.5 Elliptic hole problems.................................................................................................97 3.6 Arbitrarily shaped hole problems................................................................................99 3.7 Elliptic inclusion problems........................................................................................102 References..............................................................................................................117

Chapter 4 Green’s function for magnetoelectroelastic problems.............................118 4.1 Introduction.............................................................................................................118 4.2 3D Green’s functions by Radon transform.................................................................118 4.3 3D Green’s functions by potential function approach..................................................120 4.4 Green’s functions of magnetoelectric problems...........................................................124 4.5 Half plane problems................................................................................................. 127 4.6 Bimaterial problems..................................................................................................130 4.7 Problems with an elliptic hole or a crack....................................................................131 4.8 Green’s functions of thermomagnetoelectroelastic problems.......................................134 4.9 Dynamic Green’s functions of magnetoelectroelastic media.........................................142 References...............................................................................................................149

Chapter 5 Boundary element method for piezoelectric materials.............................151 5.1 Introduction.............................................................................................................151 5.2 Boundary integral equation.......................................................................................151 5.3 BE formulation.........................................................................................................164 5.4 Evaluation of u* and t* 5.5 Evaluation of singular integrals...................................................................................170

5.6 Crack tip singularity by evaluating hypersingular integrals...........................................174 5.7 Multi-domain problems............................................................................................178 5.8 Application of BEM to fracture analysis....................................................................179 5.9 Mixed BEM-homogenization method........................................................................182 5.10 Numerical assessments.............................................................................................186 References...............................................................................................................191

Chapter 6 Boundary element method for thermopiezoelectric materials...................194 6.1 Introduction..............................................................................................................194 6.2 BEM for thermopiezoelectric problems.....................................................................194 6.3 Application of BEM to determine SED intensity factors.............................................205 6.4 Effective properties of cracked piezoelectricity..........................................................206 6.5 Numerical examples.................................................................................................210 References...............................................................................................................213

Chapter 7 Trefftz boundary element method...............................................................215 7.1 Introduction..............................................................................................................215 7.2 Anti-plane piezoelectric problems..............................................................................215 7.3 Plane piezoelectricity.................................................................................................224 References................................................................................................................239

Appendix A Radon Transform.........................................................................................241 Appendix B The constants appeared in Section 4.3................................244 Appendix C Numerical Integration...................................................................................246 Index...............................................................................................................................249

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Preface Green’s functions play an important role in the solution of numerous problems in the mechanics and physics of solids. It is the heart of many analytical and numerical techniques such as singular-integral-equation methods, boundary-element methods, eigenstrain approaches, and dislocation methods. In general, the term “Green’s function” refers to a function, associated with a given boundary value problem, which appears as an integrand for an integral representation of the solution to the problem. In 1828, an English miller from Nottingham published a mathematical essay on Green’s function. Although the essay generated little response at that time, Green’s function, however, had since found wide applications in areas ranging from classical electrostatics to modern multi-field theory. Research to date on Green’s function and the corresponding boundary element formulation of multi-field materials has resulted in the publication of a great deal of new information and has led to improvements in design and fabrication practices. Articles have been published in a wide range of journals attracting the attention of both researchers and practitioners with backgrounds in the mechanics of solids, applied physics, applied mathematics, mechanical engineering and materials science. However, no extensive, detailed treatment of this subject has been available up to the present. It now appears timely to collect significant information and to present a unified treatment of these useful but scattered results. These results should be made available to professional engineers, research scientists, workers and students in applied mechanics and material engineering, e.g. physicists, metallurgists and materials scientists. The objective of this book is to fill this gap, so that readers can obtain a sound knowledge of Green’s function and its boundary element implementation of multi-field materials and structures. This volume details the development of each of the techniques and ideas, beginning with a description of the basic concept of Green’s function from a mathematical point of view. From there we progress to the derivation and construction of Green’s function for multifield problems including piezoelectric, thermopiezoelectric, and magnetoelectroelastic problems and show how they arise naturally in the response of multifield solid to external loads. Green’s function and boundary elements of multifield materials is written for researchers, postgraduate students and professional engineers in the areas of solid mechanics, Physical science and engineering, applied mathematics, mechanical engineering, and materials science. Little mathematical knowledge beyond the usual calculus is required, although convenience matrix presentation is used throughout the book. This book consists of six chapters and three appendices. The first chapter gives a brief description of Green’s function and linear theory of multi-field material in order to establish notation and to provide a common source for reference in later chapters. It describes in detail the method of deriving Green’s function and boundary value equations of multi-field problems. Chapter 2 deals with Green’s function of piezoelectric materials, beginning with a discussion of Green’s function by Radon transforms and ending with a brief description of dynamic Green’s function in three-dimensional piezoelectric solids. Chapter 3 presents Green’s functions in theropiezoelectric problems. Green’s functions of infinite thermopiezoelectric solids with a half-plane boundary, a bimaterial interface, an elliptic, an arbitrarily shaped hole, and elliptic inclusion are presented. Chapter 4 is concerned with applications of Green’s function to magnetoelectroelastic problem. Chapter 5 describes the development of boundary element formulation using the reciprocity theorem and the Green’s function described in the previous chapters. The final chapter talks about an alternative boundary element formulation which is

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Preface

suitable for modelling multi-field problem with discontinuity and demonstrate the power and versatility of the discontinuity boundary element formulation in treating fracture problems of multi-field materials. I am indebted to a number of individuals in academic circles and organizations who have contributed in different, but important, ways to the preparation of this book. In particular, I wish to extend my appreciation to X.Q. Feng, E. Pan, and S.W. Yu. Special thanks go to Jonathan Agbenyega and other staff members of Elsevier for their commitment to excellence in all aspects of the publication of this book. Finally, I wish to acknowledge the individuals and organizations cited in the book for permission to use their material. Thanks are particularly to Elsevier, John Wiley & Sons, and Birkhäuser Verlag at Basel of Switzerland for their permission to use the invaluable materials published by them.

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Acknowledgements Pages 26–27: Dunn ML, Electroelastic Green’s functions for transversely isotropic media and their application to the solution of inclusion and inhomogeneity problems, Int J Eng Sci, 32, 119–131, 1994 Pages 29–31: Dunn ML and Wienecke HA, Green’s functions for transversely isotropic piezoelectric solids, Int J Solids Struct, 33, 4571–4581, 1996 Section 2.8: Lu P and Williams FW, Green’s function of piezoelectric materials, with an elliptic hole or inclusion, Int J Solids Struct, 35, 651–664, 1998 Section 2.10: Chen BJ, Xiao ZM and Liew KM, A line dislocation interacting with a semi-infinite crack in piezoelectric solid, Int J Eng Sci, 42, 1–11, 2004 Section 2.11: Chen BJ, Liew KM and Xiao ZM, Green’s function for anti-plane problems in piezoelectric media with a finite crack, Int J Solids Struct, 41, 5285–5300, 2004 Section 2.12: Wang X and Zhong Z, Two-dimensional time-harmonic dynamic Green’s functions in transversely isotropic piezoelectric solids, Mech Res Commun, 30, 589–593, 2003 Section 3.7: Qin QH, Thermoelectroelastic solution on elliptic inclusions and its application to crackinclusion problems, Appl Math Modelling 25, 1–23, 2000 Section 4.2: Pan E, Three dimensional Green’s functions in anisotropic magnetoelectroelastic bimaterials, Z Angew Math Phys, 53, 815–838, 2002 Section 4.3: Hou PF, Ding HJ and Chen JY, Green’s function for transversely isotropic magnetoelectroelastic media, Int J Eng Sci, 43, 826–858, 2005 Section 4.4: Li JY, Magnetoelectric Green’s functions and their application to the inclusion and inhomogeneity problems, Int J Solids Struct, 39, 4201–4213, 2002 Section 4.7: Liu J, Liu X, Zhao Y. Green’s functions for anisotropic magnetoelectroelastic solids with an elliptical cavity or a crack. Int J Eng Sci, 39, 1405–1418, 2001

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Acknowledgements

Section 5.4: Kogl M and Gaul L, A boundary element method for transient piezoelectric analysis, Eng Analysis Boun Elements, 24, 591–598, 2000 Section 5.5: Davi G and Milazzo A, Multidomain boundary integral formulation for piezoelectric materials fracture mechanics, Int J Solids Srtuc, 38, 7065–7078, 2001

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Notations a A b B bi be bm Bi Cijkl (or cij) Cv Di E emij e~lij Ei EiJMl F fij g G gij hi – hn h* h(z) Hi k kij KMN (n) M(j) ni pi q0 q–s q0 q* s S, H, L T –t i Ta T0 T U ui Ui V

matrix defined in Eq (1.118) matrix defined in Eq (1.130) matrix defined in Eq (1.122) or line dislocation in Eq (2.61) matrix defined in Eq (1.130) body force per unit volume electric charge density body electric current magnetic induction elastic stiffness constant specific heat per unit mass electric displacement Young’s modulus piezoelectric constant piezomagnetic constants electric field generalied material constant defined in Eq (1.115) Lekhnitskii stress function in Eq (1.148) elastic compliance Gibbs energy function shear modulus inverse piezoelectric constants heat flow prescribed surface heat flow line heat source function defined in Eq (2.99) magnetic field 2 1/2 =(k11k22 – k12 ) defined after Eq (3.2) constant of heat conduction =EiMNjninj defined in Eq (2.306) =-iB(j)A(j)-1 defined after Eq (2.85) unit vector outward normal to boundary Γ eigenvalue of the material equation determined from Eq (1.118) = T / 4πi +h* / 4πk defined in Eq (3.6) prescribed surface charge generalized line force = AT q0 + BT b defined in Eq (2.70) entropy density matrices defined in Eq (1.142) small temperature change prescribed surface traction absolute temperature reference temperature line temperature discontinuity internal energy density mechanical displacement displacement and electric potential defined in Eq (1.114) induction function in Eq (1.148)

0080451349-Preface.fm Page xii Saturday, December 23, 2006 2:10 PM

xii

νi z*α αij βij Γ δ δij εij (or εp) κij λ λij μij Πij ρ σij (or σp) υ φ ϕ χi ψ ∇2

Notations

pyromagnetic constants or material eigenvalues in Section 2.3 = z α / pα electromagnetic constants dielectric impermeability boundary of solution domain Ω variational symbol or Dirac delta function Kronecker delta strain dielectric constant Lamé constant thermal-stress coefficient magnetic permeabilities stress and electric displacement defined in Eq (1.114) mass density stress Poisson’s ratio electric potential Generalized stress function defined in Eq (1.121) pyroelectric coefficient magnetic potential 2

2 ¶ ¶ - , two-dimensional Laplacian operator = -------2- + ------2 ¶x ¶y

1

Chapter 1

Introduction

1.1 Foundation of Green’s function Green's function is a basic solution to a linear differential equation, a building block that can be used to construct many useful solutions. In heat conduction, we know that the Green's function represents the temperature at a field point due to a unit heat source applied at the source point. In elastostatics, the Green's function stands for the displacement in the solid due to the application of a unit point force. In general, the exact form of Green's function depends on the differential equation, the body shape, and the types of boundary condition present. Green's functions are named in honor of British mathematician and physicist George Green (1793-1841), who pioneered the concept in the 1830s. To show how Green's function arises, and to initiate further study of the method, a typical one-dimensional boundary value problem is first considered and solved by fairly elementary methods. Consider a taut string of length l whose transverse displacement, u(x), is governed by the following differential equation (see Chapter 1 in [1]) u¢¢( x ) + l 2u( x ) = - f ( x );

0£ x£l

(1.1)

and by the boundary conditions

u (0) = u (l ) = 0

(1.2)

where the prime ¢ stands for differentiation, Ȝ is a known constant, and f(x) is a known function of x. To solve the boundary value problem (1.1) and (1.2) we employ the method of variation of parameters. To this end, we start by considering the characteristic equation of the associated homogeneous equation to Eq (1.1): r 2 + l2 = 0, which factors (r + il )(r - il ) having the roots r1 = -il and r2 = il,

where i = - 1. The general solution of the associated homogeneous equation is then given by u ( x) = c1eilx + c2e -ilx

(1.3)

Noting that eilx = cos lx + i sin lx, Eq (1.3) can be rearranged in the form u ( x) = A cos lx + B sin lx

(1.4)

where A = c1 + c2 and B = i (c1 - c2 ). By the method of variation of parameters, we assume that A and B are unknown functions of x such that

u ( x) = A( x) cos lx + B( x) sin lx

(1.5)

Now we have two unknown functions here and so two equations are required to determine the two unknowns. The first equation can be obtained by plugging the proposed solution (1.5) into Eq (1.1). The second equation can come from a variety of places. We are going to obtain our second equation simply by making an assumption that will make our work easier. This can be done by setting[1]

A¢( x) cos lx + B¢( x) sin lx = 0

(1.6)

Using Eq (1.6) we can then find that Eq (1.5) constitutes a solution provided that

- lA¢( x) sin lx + lB¢( x) cos lx = - f ( x)

(1.7)

Green’s function and boundary elements of multifield materials

2

Solving Eqs (1.6) and (1.7), we obtain

A¢( x) =

f ( x) sin lx

l

;

B¢( x) =

- f ( x) cos lx l

(1.8)

Thus, the functions A and B can be written in the form

A( x) =

1

f ( x) sin lxdx + c; l³

B ( x) = -

1 f ( x) cos lxdx + d l³

(1.9)

where c and d are two constants to be determined by the boundary conditions (1.2). Substituting Eq (1.9) into (1.5) we have u ( x) =

cos lx x sin lx x f ( y ) sin lydy f ( y ) cos lydy ³ c l l ³d

(1.10)

After some mathematical manipulation we arrive at u ( x) = ³

x 0

l f ( y ) sin lx sin l (l - y ) f ( y ) sin ly sin l (l - x) dy dy + ³ x l sin ll l sin ll

(1.11)

x

= ³ f ( y )G ( x, y )dy 0

in which the function G(x,y) is a two-point function of (x,y), known as the Green’s function for the boundary value problem (1.1) and (1.2). It is defined as[1] sin ly sin l (l - x) ; 0 £ y £ x, °° l sin ll G ( x, y ) = ® ° sin lx sin l (l - y ) ; x £ y £ l l sin ll ¯°

(1.12)

where x is usually known as field point(or observation point) and y the source point. It can be seen from Eqs (1.11) and (1.12) that the two-point Green’s function, G(x,y), is independent of the force term f(x), and depends only upon the differential equation considered and the boundary conditions which are imposed. Consequently, the solution to all such problems as that examined above but having different forcing terms is known in the form of Eq (1.11) once the kernel G(x,y) has been determined. The Green’s function G(x,y) associated with the above boundary value problem can also be defined by the following auxiliary boundary value problem: G ¢¢ + l2 G = -d( x - y ) ;

G (0, y ) = G (l , y ) = 0

x¹ y

(1.13)

(1.14)

where y is fixed and 0 < y < l. d( x - y ) is a two point symbolic function known as the Dirac delta function, which is discussed in the next section. Although the problem (1.13) and (1.14) is quite similar to the boundary value problem (1.1) and (1.2), the forcing function in Eq (1.13) is a Dirac delta function rather than an arbitrary function f(x). This means that solving the problem for G will be somewhat simpler than solving the corresponding problem for u, and once G has been found for a particular differential operator, say L, and set of boundary conditions, the function G may be used for solving (1.1) and (1.2) for any number of times where only the function f(x) changes from time to time. It is this feature of Green’s function, coupled with its

Introduction

3

physical interpretation, that makes it most useful in engineering applications. The discussion above is for a simple and quite special operator only. To prepare the way for further study of such functions and their use in solving boundary value problems, let us consider a more general boundary value problem:

Lu ( x) = - f ( x) , u (a ) = a ,

u (b) = b

(1.15) (1.16)

where we introduce as a separate quantity for our consideration the differential operator L [in Eq (1.1), L = d 2 / dx 2 + l2 ], a and b are the boundary points of the solution domain, and Į and ȕ are two prescribed constants. Since this is a linear problem the solution to Eqs (1.15) and (1.16) can be assumed to consist of a particular part u p and a homogeneous part u h as u = u h + u p , where u p satisfies the boundary value problem

Lu p ( x) = - f ( x) ; u p (a) = u p (b) = 0

(1.17)

and u h is the solution of Lu h ( x) = 0; u h (a) = a ; uh (b) = b

(1.18)

For a second-order linear ordinary differential operator, the general solution for u h is of the form u h = c1u1 ( x) + c 2 u 2 ( x)

(1.19)

where u1 and u 2 are linearly independent solutions of the homogeneous differential equation (1.18)1, and the constants c1 and c 2 are determined by imposing the non-homogeneous boundary conditions (1.18)2,3. To determine the particular solution u p , Eq (1.17)1 is rewritten as u p ( x) = - L-1 ( x) f ( x)

(1.20)

where L-1 is the inverse of the differential operator L. Since L is a differential operator, it is reasonable to expect its inverse to be an integral operator. We expect the usual properties of inverse to hold: LL-1 = L-1 L = I

(1.21)

where I is the identity operator. On the other hand, if we consider the problem as an example of heat conduction, f(x) would be the distribution function of heat source density. We first consider the heat source concentrated at a point y, and G ( x, y ) is a function of the temperature representing a unit of heat source at the point y and satisfying the boundary conditions. Because of the continuity of f(x), all other solutions are simply superpositions of this function. Therefore, the particular solution should be b

u p ( x) = ³ G ( x, y ) f ( y )dy a

The consistency of Eqs (1.20) and (1.22) yields

(1.22)


4

b

- L-1 f ( x) = ³ G ( x, y ) f ( y )dy a

(1.23)

If we formally apply the differential operator L to both sides of (1.23), and assume commutativity of L with integration, we find that b

f ( x) = - ³ L[G ( x, y )] f ( y )dy

(1.24)

L[G ( x, y )] = -d ( x - y )

(1.25)

a

This will lead to

Besides, the boundary conditions (1.17)2,3 require that b

b

u p (a ) = ³ G (a, y ) f ( y )dy = 0 ;

u p (b) = ³ G (b, y ) f ( y )dy = 0

a

a

(1.26)

Since f(y) can represent any functions of y, the relations (1.26) are possible only if G (a, y ) = G (b, y ) = 0

(1.27)

Therefore the Green’s function we are seeking is a solution of the boundary value problem L[G ( x, y )] = -d ( x - y ) ® ¯ G (a, y ) = G (b, y ) = 0

(1.28)

As an illustration, consider the two-dimensional Laplace’s operator L = Ñ2 =

¶2 ¶2 + 2 2 ¶x1 ¶x2

(1.29)

The Green’s function for this particular differential operator is known to be

G (x, y ) = -

1 ln r 2p

(1.30)

where x =( x1 , x 2 ) and y = ( y1 , y 2 ) are two position vectors in two-dimensional space and 1/ 2

r = ª¬( x1 - y1 )2 + ( x2 - y2 ) 2 º¼

(1.31)

It can be seen that the Green’s function here represents the potential at the observation point x due to a point charge at the source point y. It is also found that the Green’s function depends on the distance between the source and observation point only. From the discussion above it is as yet unclear how the Green’s functions for a given differential operator can be derived. In the rest of this section we follow the results given in [2] to show the procedure for deriving Green’s functions using Fourier transform. To this end, consider the Helmholtz equation in three dimensions: ª ¶2 º ¶2 ¶2 Lu = « 2 + 2 + 2 + l 2 » u = (Ñ3 + l 2 )u = 0 ¬ dx1 dx2 dx3 ¼

(1.32)

where Ñ3 is the three-dimensional Laplace’s operator. From the discussion above the related free-space (no boundary conditions are specified) Green’s function can be obtained by considering

5

Introduction

L(x)G (x, y ) = -d(x - y ) where x = ( x1 , x 2 , x3 ) and three-dimensional space, and

y = ( y1 , y 2 , y 3 )

are

(1.33) two

position

d(x - y ) = d( x1 - y1 )d( x 2 - y 2 )d( x3 - y 3 )

vectors

in

(1.34)

is the three-dimensional Dirac delta function. The differential equation (1.33) can be solved using a variety of methods. Here we use the Fourier transform method given in [2] to solve this problem. Since we are calculating only the free-space component of the Green’s function, we can use a single variable r=x-y, as the free-space Green’s function will depend only on the relative distance between points x and y and not on their absolute positions. We begin by defining a Fourier transform pair [2] uˆ (ȟ ) =

1 (2p) 3

³

¥ -¥

u (r )e -iȟ×r dr, u (r ) = ³

¥ -¥

uˆ (ȟ )e iȟ×r dȟ

(1.35)

where ȟ = (x1 , x 2 , x 3 ) represents the coordinates in the frequency domain. Applying the forward transformation (1.35)1 to Eq (1.33) for the Green’s function we obtain

Gˆ (ȟ ) =

1 (2p) 3 (x 2 - l2 )

(1.36)

where x 2 = ȟ × ȟ . The Green’s function in physical space is then given through the inverse integral G (r ) =

1 ( 2 p) 3

e iȟ×r ³-¥ (x 2 - l2 )dȟ ¥

¥ ½ p ¥ xe irx xe -irx dȟ dȟ ¾ ® 2 2 3 ³ -¥ 2 2 ³ ¥ ir (2p) ¯ (x - l ) (x - l ) ¿ p = {R1 - R2 } ir (2p) 3

=

(1.37)

where r = r × r = [( x1 - y1 ) 2 + ( x 2 - y 2 ) 2 + ( x3 - y 3 ) 2 ]1 / 2 and the integrals R1 and R2 can be evaluated by considering a contour in the complex x space [2]: xeirx d x = ipeilr -¥ (x - (l + ie))(x + (l + ie))

(1.38)

xe - irx d x = -ipeilr -¥ (x - (l + ie))(x + (l + ie ))

(1.39)

R1 = lim ³ e®0

R2 = lim ³ e®0

¥

¥

The Green’s function is then given by G (r ) =

1 irl p {ipe ilr + ipe ilr } = e 4pr ir (2p) 3

(1.40)

1.2 Dirac delta function Having already used the concept of the Dirac delta function in Section 1.1 above, we


6

now introduce it in a way which demonstrates how this concept fits into the familiar realm of functions and integrals, and we provide some physical interpretations. The Dirac delta function is important in the study of phenomena of a point action or an impulsive force, such as an action which is highly localized in space and/or time, including the point force in solid mechanics; the impulsive force in rigid body dynamics; the point mass in gravitational field theory; and the point heat source in the theory of heat conduction. The point action or the impulse is usually described by the Dirac delta function d(x) [2,3]. The function d(x) is defined to be zero when x ¹ 0 , and infinite at x = 0 in such a way that the area under the function is unity. With this in mind, a concise definition of d(x) can be given in the form d( x) = 0 if x ¹ 0;

³

b -a

d( x)dx = 1

(1.40)

for any a > 0 and b > 0. This is a weak definition of d(x), since the limits of integration are never precisely zero. The definition (1.40) is, however, sufficient for work with Green’s functions [2]. Alternatively, d(x) can also be mathematically defined as 0 ( x ¹ 0) d( x ) = ® ¯¥ ( x = 0)

(1.41)

As mentioned in [2,3], the Dirac delta function has several special properties which are useful for deriving Green’s functions. They are: (i) Physical dimension of d(x) Since the definition of the Dirac delta function requires that the product d(x)dx is dimensionless, the dimension of d(x) is the inverse of its argument x. (ii) Sifting property. If function f(x) is continuous at x = y, we have

³

b a

f ( x) if a < x < b d( x - y ) f ( y )dy =® if x Ï [a,b] ¯ 0

(1.42)

(iii) Unit impulse property. With a any real constant, we have d(ax) =

1 d( x ) a

(1.43)

(iv) Derivative property

³

b -a

f ( x)d¢( x)dx = - f ¢(0)

(1.44)

(v) Integral property.

³

x -¥

d(y)dy = H ( x) ;

dH ( x - y ) = d( x - y ) dx

(1.45)

where H(x) is the Heaviside unit step function defined as 0 if x < 0 H ( x) = ® ¯1 if x > 0

(1.46)

7

Introduction

(vi) If f(x) has real roots x n (i.e. f ( x n ) = 0 ), then d( f ( x)) = ¦ n

d( x - x n ) f ¢( x n )

(1.47)

The sum applies to all real roots. The proof of these properties can be found in [2,4,5]. 1.3 Basic equations of piezoelectricity In this section, we recall briefly the three-dimensional formulation of linear piezoelectricity that appeared in [6-9]. Here, a three-dimensional Cartesian coordinate system is adopted where the position vector is denoted by x (or xi). In this book, both conventional indicial notation xi and traditional Cartesian notation (x, y, z) are utilized. In the case of indicial notation we invoke the summation convention over repeated indices. Moreover, vectors, tensors and their matrix representations are denoted by bold-face letters. The three-dimensional constitutive equations for linear piezoelectricity can be derived by considering the full Gibbs energy function per unit volume g [9]

g = U - E m Dm - T a s

(1.48)

where U, s, Dm and Em are the internal energy density, entropy density, electric displacement and electric field, respectively, T a =T0+T is the absolute temperature, where T0 is the reference temperature, and T a small temperature change: T 0 (m=1,2,3,4) and z m is the complex conjugate of z m . According to the residual theory of complex integrals, Eq (2.66) can be further written as GMR (x) = -

4 1 Im ¦ 2p x m =1

AMR (p + zq) 4

a9 (z m - z m )

Õ

(z m - z k )(z m - z k )

k =1, k ¹ m

in which the roots of D(p + zq) are assumed to be distinct from each other.

(2.69)

38


2.5 Half-plane problem In the previous sections, we derived Green’s function in free space using the Radon transform, the potential function approach, and the Fourier transform. Green’s functions for piezoelectric solids with defects such as a half-plane boundary, bimaterial interface, and an elliptic hole boundary are discussed in the remaining sections of this chapter. In this section, Green’s functions in a half-plane piezoelectric solid subjected to a line force-charge q 0 and a generalized line dislocation b are presented through use of Stroh’s formalism. The boundary condition on the infinite straight boundary of the half-plane is free of surface traction-charge. Owing to the linear property of the problem, the principle of superposition is used and this problem can be divided into two problems (see Fig. 2.4): (i) An infinite piezoelectric plate is subjected to the line force-charge q 0 and the line dislocation b both located at z0(x10, x20) in an infinite domain; and (ii) the infinite straight boundary of the half-plane is subjected to loadings making the surface traction-charge free. We start by considering an infinite domain subjected to a line force-charge q 0 and a line dislocation b, both located at z0(x10, x20). The general solution (1.128) and (1.129) is in this case in the form [2,41,42] U = 2 Re{A ln( za - za 0 ) q f },

j = 2 Re{B ln( za - za 0 ) q f }

(2.70)

where q f is a complex vector to be determined, A and B are now 4´4 matrices (5´5 matrix in Section 1.5) expressed by

A = [a1 a 2 a3 a 4 ],

B = [b1 b 2 b3 b 4 ]

(2.71)

and T

T

U = {u1 u2 u3 f} , j = {j1 j2 j3 j4 } , za = x1 + pa x2 ,

za 0 = x10 + pa x20 , (a = 1 - 4)

(2.72)

39

Green’s function of electroelastic problem

za

x2

·

(q 0 , b)

· z0

x1

j=0

0 (a)

za

x2

·

(q 0 , b )

· z0

za

x2

·

x1

j (q 0 , b)

x1

+

0

*

*

- j (q 0 , b)

0

(c)

(b)

Fig. 2.4 (a) Half-plane subjected to q 0 and b;

(b) problem (i), j * is induced by

q 0 and b at x2=0; (c) problem (ii) Since ln( za - za 0 ) is a multi-valued function we introduce a cut along the line defined by x2=x20 and x1 £ x10 . Using the polar coordinate system (r,q) with its origin at z0(x10, x20) and q=0 being parallel to the x1-axis, the solution (2.70) applies to

-p < q < p,

r>0

(2.73)

Therefore ln( za - za 0 ) = ln r ± ip,

at q = ±p for a = 1-4

(2.74)

Due to this relation, Eq (2.70) must satisfy the conditions

U(p) - U(-p) = b,

j (p) - j (-p) = q 0

4p Re(iAq f ) = b,

4p Re(iBq f ) = q 0

(2.75)

which lead to (2.76) It is easy to prove that Eq (2.76) satisfy following equilibrium conditions of the force and single-valued conditions of the generalized displacement U:

³

C

d j = q0 ,

³

C

dU = b

where C represents an arbitrary closed curve enclosing the point z0. Eq (2.76) can be written as

(2.77)

40


ª A A º iq f ½ 1 b ½ ¾= ® ¾ « »® ¬ B B ¼ ¯-iq f ¿ 2p ¯q 0 ¿

(2.78)

It follows from Eq (1.139) that iq f ½ 1 ª B T A T º b ½ ® ¾= « T T »® ¾ ¯-iq f ¿ 2p ¬B A ¼ ¯q 0 ¿

(2.79)

Hence

qf =

1 ( A T q 0 + BT b ) 2pi

(2.80)

The traction-charge on the half-plane boundary induced by the applied q 0 and b is then given as I I P2IJ ( x1 ) = {s12I s22 s32 D2I } = j IJ ,1 = T

1 ° 1 °½ Im ®B ( A T q 0 + BT b ) ¾ p °¯ x1 - za 0 ¿°

(2.81)

Therefore the second problem is the half-plane solid subjected to loading of Ȇ 2IIJ ( x1 ) = -Ȇ 2I J ( x1 ) at the infinite straight boundary of the half-plane, which is equivalent to the condition that j = j I + j II = 0

at x2 = 0

(2.82)

where the following relations have been used [2]: Pn = j, s

(2.83)

where n is the normal direction of the boundary, s is the arc length measured along the half-plane boundary, Pn represents the surface traction-charge vector. Integrating Eq (2.83) and ignoring the induced integral constant, which represents rigid body motion, results in Eq (2.82). Therefore, to satisfy the boundary condition on the infinite straight boundary (2.82), the solution can be assumed in the form [2,43] U = U I + U II = j = j I + j II =

4 1 1 Im{A ln( za - za 0 ) q*} + ¦ Im{A ln( za - zb 0 ) qb } , p p b=1

(2.84)

4 1 1 Im{B ln( za - za 0 ) q*} + ¦ Im{B ln( za - zb 0 ) qb } p p b=1

(2.85)

where qb are unknown constants to be determined and

q* = A T q 0 + BT b

(2.86)

Substituting Eq (2.85) into Eq (2.82) yields j=

4 1 1 Im{B ln( x1 - za0 ) q*} + ¦ Im{ln( za - zb0 )Bqb } = 0 p p b=1

(2.87)

41

Green’s function of electroelastic problem

Noting that Im( f ) = - Im( f ) , we have Im{B ln( x1 - za 0 ) q*} = - Im{B ln( x1 - za 0 ) q*} ,

(2.88)

and 4

ln( x1 - za 0 ) = ¦ ln( x1 - zb 0 )Ib ,

(2.89)

I i = d ia = diag[d i1 , d i 2 , d i 3 , d i 4 ]

(2.90)

b=1

where

Equation (2.87) now yields qb = B -1BI bq* = B -1BI b ( AT q 0 + BT b)

(2.91)

If the boundary x2 = 0 is a rigid surface, then U = 0,

at x2 = 0

(2.92)

The same procedure shows that the solution is given by Eqs (2.84) and (2.85) with qb = A -1AI b ( AT q 0 + BT b)

(2.93)

2.6 Solution of bimaterial problems We now consider a bimaterial piezoelectric plate subjected to a line force-charge q 0 and a line dislocation b, both located in the upper half-plane at z0(x10, x20), for which the upper half-plane ( x2 > 0) is occupied by material 1, and the lower half-plane ( x2 < 0) is occupied by material 2 (see Fig. 2.5). They are rigidly bonded together so that

U (1) = U ( 2) , j (1) = j ( 2 ) ,

at x2 = 0

(2.94)

where the superscripts (1) and (2) label the quantities relating to materials 1 and 2. The equality of traction-charge continuity comes from the relation ¶j / ¶s = Pn , where Pn is the surface traction-charge vector on a curve boundary (the interface in the bimaterial problem). When points along the interface are considered, integration of Pn(1) = Pn(2) provides j(1) = j (2) , since the integration constants which correspond to rigid body motion can be neglected.

42


za

·

x2

(q 0 , b)

material 1

· z0

x1 0 interface

material 2

Fig. 2.5 Bimaterial plate subjected to q 0 and b To satisfy the continuity condition (2.94) on the interface of the bimaterial plate, the Green’s function solution may be assumed, in a similar way as treated in the half-plane problem, in the form [2,43] U (1) =

4 1 1 Im{A (1) ln( za(1) - za(1)0 ) q (1)*} + ¦ Im{A (1) ln( za(1) - zb(1)0 ) qb(1) } , p b=1 p

(2.95)

j (1) =

4 1 1 Im{B (1) ln( za(1) - za(1)0 ) q (1)*} + ¦ Im{B (1) ln( za(1) - zb(1)0 ) qb(1) } p p b=1

(2.96)

for material 1 in x2>0 and 4 1 U (2) = ¦ Im{A (2) ln( za(2) - zb(1)0 ) qb(2) } , p b=1

(2.97)

4 1 j(2) = ¦ Im{B (2) ln( za(2) - zb(1)0 ) qb(2) } b=1 p

(2.98)

(1) (2) for material 2 in x20), and

f0(2) ( z (2) ) =

1 4 ¦ ln( za(2) - z0(1)k ) B**k q(1)* , 2pi k =1

(2.110)

for material 2 (x20) is occupied by a piezoelectric material (material 1), and the lower half-plane (x20 and U (2) =

3 1 Im ¦ [A (2) ln( z - z0b qb(2) ] p b=1

(2.153)

j (2) =

3 1 Im ¦ [B (2) ln( z - z0b qb(2) ] p b=1

(2.154)

for material 2 in x2 z (t 20 )

(3.149)

j =1

j (2) (2) j (2) - j in which e j = [z t(2) ] / j. Imposition of the continuity conditions 0 + ( a2 t / a1t ) z t 0

(3.95) on the interface leads to ¥

¥

j =1

j =1

f1 (s ) + d ln s + f 1 ( s) + d ln s = ¦ [c j + G *j c j ]s - j + ¦ [c j + G *j c j ]s j ¥

¥

j

j

+ q 0 {ln s - ¦ e j s - j } + q 0 {ln s - ¦ e j s j }

- f1 (s) + d ln s + f1 (s) - d ln s =

-

k (2) k (1)

¥

¦ [c j - G*j c j ]s- j j =1

k (2) k (1)

(3.150) ¥

¦ [c

j

- G*j c j ]s j

j =1

¥ ¥ k (2) q0 k (2) q0 -j {ln s s } + {ln s e ¦j j ¦j e j s j } k (1) k (1)

(3.151)

where s = e iq stands for a point located on the unit circle in the zt-plane, and q is a polar angle. Comparison of the coefficients of the “ln” terms on both sides of Eqs (3.150) and (3.151) yields


110

d=

1 1 k (2) k (2) (1 + (1) )q0 - (1 - (1) )q0 2 2 k k

(3.152)

In order to use the analytical continuation method, Eqs (3.150) and (3.151) are rewritten, by deleting the “ln” terms, as ¥

¥

j =1

j =1

f 1 ( s ) - ¦ [c j + G *j c j - q 0 e j ]s - j = - f 1 ( s ) + ¦ [c j + G j* c j - q 0 e j ]s j

- f1 (s) -

k (2) k (1)

¥

¦ [c j - G*j c j + q0e j ]s- j = - f1 (s) j =1

k (2) k (1)

¥

¦ [c

j

(3.153)

- G*j c j + q0 e j ]s j

j =1

(3.154) Similar to the treatment for the case of loading applied outside the inclusion, application of the method of analytical continuation to Eqs (3.153) and (3.154) yields ¥

f 1 ( z ) = ¦ [c j + G *j c j - q 0 e j ]z - j j =1 ¥

z Î W1

(3.155)

f 1 ( z ) = ¦ [c j + G c j - q 0 e j ]z * j

j

zÎW2

j =1

for Eq (3.153), and f1 (z ) = -

k (2) k (1)

k (2) f1 (z ) = - (1) k

¥

¦ [c

j

- G*j c j + q0 e j ]z - j

z Î W1

j =1

(3.156)

¥

¦ [c

j

* j

- G c j + q0 e j ]z

j

z Î W2

j =1

for Eq (3.154). Solving Eqs (3.155) and (3.156) for cj yields cj =

k (1) - k (2) (G0 - G j G j / G0 ) -1 (q0 e j - G j q0 e j / G0 ), 2k (1)

(j = 1, 2," , ¥)

(3.157)

where G0 = (1 + k (2) / k (1) ) / 2, G j = (1 - k (2) / k (1) )G*j / 2

(3.158)

Having obtained the solution of ck, the functions gi in Eqs (3.1)4 and (3.2)2 can be given by g1 ( zt(1) ) = q0 {a1(1)t z t(1) (ln z t(1) - 1) + a2(1)t (1 + ln z t(1) ) / z t(1) } ¥

+ (c1 + G1*c1 - q0 e1 )a1(1)t ln z t(1) + ¦ G4 j z t(1) - j

(3.159)

j =1

(2) (2) g 2 ( zt(2) ) = q0 {a1(2) - 1) + a2(2)t (1 + ln z t(2) ) / z t(2) } t z t (ln z t ¥

(2) - q0 e1a1(2) + ¦ [G5 j z t(2) j + G6 j z t(2) - j ] t ln z t j =1

where

(3.160)

111

Green’s function for thermoelectroelastic problems

G4 j = -[(c j +1 + G*j +1c j +1 - q0 e j +1 )a1(1)t - a2(1)t (c j -1 + G*j -1c j -1 - q0 e j -1 )t j1 / j (2) G5 j = (a1(2) t c j -1t j1 - a2 t c j +1 ) / j (2) 1t

* j +1 j +1

G6 j = [a (G c

(3.161) (2) 2t

* j -1 j -1

- q0 e j +1 ) - a (G c

- q0 e j -1 )t j1 ] / j

with tij = 1 for i ¹ j, tij = 0 for i = j . 3.7.2.2 Green’s functions for electroelastic fields. From Eqs (3.1)4 and (3.2)2 the particular piezoelectric solution induced by thermal load can be written as U (pi ) = 2 Re[c( i ) gi (z t(i ) )], j (pi ) = 2 Re[d ( i ) gi (z t( i ) )]

(i=1,2)

(3.162)

As stated previously, the particular solutions (3.162) do not generally satisfy the conditions (3.95)3,4 along the interface. We therefore need to find a corrective isothermal solution for a given problem so that, when it is superimposed on the particular thermoelectroelastic solution, the interface conditions (3.95)3,4 will be satisfied. Owing to the fact that f(za) and g(zt) have the same order to affect the SED in Eq (3.2)2, possible function forms may be taken from the partition of g(zt) in order to satisfy the boundary conditions considered. They are f1( j ) ( zk( j ) ) = aF1 ( z (k j ) ) / 2, f 2( j ) ( zk( j ) ) = ipk( j )bF2 ( z (k j ) ) / 2, f 3( j ) ( zk( j ) ) = a1(kj ) ln z (k j ) , ¥

¥

k =1

k =1

f4( j ) ( z ( j ) ) = ¦ z (aj ) k rk( j ) , f5( j ) ( z ( j ) ) = ¦ z (aj )-k s(k j )

(j=1,2)

(j=1,2)

(3.163)

(3.164)

where f k( j ) are 4-component vectors, rk( j ) and s (k j ) are constant vectors with 4-componants to be determined, and F1 (z ) = z -1 (1 + ln z ) - z (1 - ln z ), F2 (z ) = z -1 (1 + ln z ) + z (1 - ln z ),

(3.165)

The Green’s functions for the electroelastic fields can thus be chosen as 3

5

k =1

k =4

3

5

k =1

k =4

U ( j ) = 2 Re{¦ A ( j ) f k( j ) ( z (aj ) ) q (k j ) + ¦ A ( j )fk( j ) ( z ( j ) ) + c ( j ) g j ( z t( j ) )] j ( j ) = 2 Re{¦ B( j ) f k( j ) ( z (aj ) ) q (k j ) + ¦ B( j )fk( j ) ( z ( j ) ) + d ( j ) g j ( z t( j ) )]

(3.166)

(3.167)

Substituting Eqs (3.166) and (3.167) into Eq (3.95) provides q1(1) = q0 X1[ A (2) -1 (c(2) - c(1) ) - B (2) -1 (d (2) - d (1) )],

(3.168)

q1(2) = q0 X 2 [ A (1) -1 (c(1) - c(2) ) - B (1) -1 (d (1) - d (2) )],

(3.169)

(1) -1 q (1) X1[ A (2) -1 ( p1*(2)c(2) - p1*(1)c(1) ) - B (2) -1 ( p1*(2)d (2) - p1*(1)d (1) )], 2 = q0 P

(3.170)


112

(2) -1 q (2) X 2 [ A (1) -1 ( p1*(1)c(1) - p1*(2)c(2) ) - B (1)-1 ( p1*(1)d (1) - p1*(2)d (2) )], 2 = q0 P

q3(1) = a1(1)k q3(2) = a1(2) k

-1

X1[ A (2) -1 (b*(1)c(1) - b*(2)c(2) ) - B (2) -1 (b*(1)d (1) - b*(2)d (2) )],

-1

X1[ A (1) -1 (b*(2)c(2) - b*(1)c(1) ) - B (1) -1 (b*(2)d (2) - b*(1)d (1) )],

(3.171) (3.172) (3.173)

¥

A (1)f5(1) (s) + A (1) f 4(1) (s) - A (2)f5(2) (s) - A (2) f 4(2) (s) + ¦ (c(1)G4 j - c (2)G5 j - c(2)G6 j )s - j j =1

¥

=A (2)f 4(2) (s) + A (2) f5(2) (s) - A (1)f 4(1) (s) - A (1) f5(1) (s) - ¦ (c (1)G4 j - c(2)G5 j - c (2)G6 j )s j j =1

(3.174) ¥

B (1)f5(1) (s) + B (1) f 4(1) (s) - B (2)f5(2) (s) - B (2) f 4(2) (s) + ¦ (d (1)G4 j - d (2)G5 j - d (2)G6 j )s- j j =1

¥

=B (2)f 4(2) (s) + B (2) f5(2) (s) - B (1)f 4(1) (s) - B (1) f5(1) (s) - ¦ (d (1)G4 j - d (2)G5 j - d (2)G6 j )s j j =1

(3.175) where * ( 2) P ( j ) = diag[ p1( j ) p2( j ) p3( j ) p4( j ) ] , b*(1) = a1(1) = - q 0 a1(t2 ) e1 (3.176) t ( c1 + G1 c1 - q 0 e1 ), b*

It is found that Eqs (3.175) and (3.176) can be obtained from Eqs (3.129) and (3.130) by replacing G1j by G4j, G2j by G5j, and G3j by G6j. Therefore, the procedure for determining the four sets of constant vectors, i.e., rk( j ) and s (k j ) ( j = 1, 2) , operates in the same way as that treated in Section 3.7.1.2. The results of rk( j ) and s(k j ) in Eq (3.164) have the same form as those in Eqs (3.136)-(3.139), but G1j , G2j, G3j in those equations are now replaced by G4j, G5j, and G6j. 3.7.3 Green’s functions for thermal loads applied on the interface 3.7.3.1 Green’s functions for thermal fields. When (x10, x20) is on the interface, say i q0 x10 = a cos q0 , x20 = b sin q0 , z t(1)0 = z t(2) , the proper choice of f 0 and f 0* should 0 =e reflect the singularity properties of both inclusion and matrix. By referring to the Eq (20) given in Yen et al [10], they may be assumed in the form f 0 (z t(1) ) = q0 ln(z t(1) - eiq0 )

(3.177)

¥

f1 (z t(1) ) = ¦ a* j z t(1) - j + q1 ln(z t(1)-1 - e - iq0 )

(3.178)

f 0* ( z t( 2 ) ) = q 2 ln[( z t( 2 ) - z t(02 ) ) / a1(t2 ) ]

(3.179)

j =1

With the use of the series representation ¥

ln(1 - x ) = - ¦ x j / j

(3.180)

j =1

and the relation (3.148), Eq (3.179) may be rewritten as ¥

j (2) - j (2) f 0* ( z (2) - eiq0 ) - ¦ [( a2(2)t e - iq0 / a1(2) / j ]} t ) = q2 {ln( z t t ) zt j =1

(3.181)

113


The continuity conditions (3.95)1,2 on the interface give q1 =

k (1) - k (2) 2k (1) q , q2 = (1) q0 (1) (2) 0 k +k k + k (2)

(3.182)

To determine the remaining unknown functions, f1 and f2, the method of analytical continuation presented in the previous sections will be used. As described in Eqs (3.105) and (3.106), the holomorphic properties provide f11 (z ) = f 22 (z ) + f 21 (z ),

z Î W1

f11 (z ) = f 22 (z ) + f 21 (z ),

z Î W2

(2)

f11 (z ) = k [ f 22 (z ) - f 21 (z )] / k (1) ,

z Î W1

f11 (z ) = k (2) [ f 22 (z ) - f 21 (z )] / k (1) ,

z Î W2

(3.183)

where ¥

¥

f 11 (z ) = ¦ a * j z - j , f 21 (z ) = ¦ c j z j , j =1

¥

j =1

* j

f 22 (z ) = ¦ [G c j -e* j ]z

(3.184)

-j

j =1

j where e* j = q2 (a2(2)t e -iq0 / a1(2) t ) / j .

Solving Eq (3.183) yields a* j = c j + G*j c j - e* j ,

cj =

k (1) - k (2) (G0 - G jG j / G0 ) -1 ( e* j - G j e* j / G0 ), 2k (1)

(3.185) j = 1, 2," , ¥

(3.186)

Thus the corresponding functions gi are given as g1 ( zt(1) ) = {a1(1)t [q0 F3 (z t(1) , eiq0 ) + q1 F4 (z t(1) -1 , e -iq0 )] + a2(1)t [q0 F4 (z t(1) , eiq0 ) ¥

+q1 F3 (z t(1) -1 , e - iq0 )] + a*1a1(1)t ln z t(1) + ¦ G7 j z t(1) - j }

(3.187)

j =1

i q0 (2) (2) g 2 ( zt(2) ) = q2 {a1(2) ) + a2(2)t F4 (z t(2) , eiq0 )} - e*1a1(2) t F3 (z t , e t ln z t ¥

+ ¦ [G8 j z t(2) j + G9 j z t(2) - j ]

(3.188)

j =1

where F3 ( x , y ) = ( x - y )[ln( x - y ) - 1]

(3.189)

1 1 1 F4 ( x , y ) = ( - ) ln( x - y ) + ln x x y y

(3.190)


114

G7 j = -[a1(1)t a*( j +1) - t j1a2(1)t a*( j -1) ] / j, (2) G8 j = ( a1(2) t c j -1t j1 - a2 t c j +1 ) / j = G5 j , (2) 1t

* j +1 j +1

G9 j = -[a ( G c

(3.191)

(2) 2t

* j -1 j -1

- e*( j +1) ) - a ( G c

- e*( j -1) )t j1 ]/ j

For the same reason as given in relation to Eqs (3.166) and (3.167), the corresponding electroelastic solutions may be assumed in the form 5

7

U ( j ) = 2 Re{¦ A ( j ) f k( j ) ( z (aj ) ) q (k j ) + ¦ A ( j )fk( j ) ( z ( j ) ) + c ( j ) g j ( z (t j ) )] k =1

k =6

5

7

j ( j ) = 2 Re{¦ B( j ) f k( j ) ( z (aj ) ) q (k j ) + ¦ B( j )fk( j ) ( z ( j ) ) + d ( j ) g j ( z (t j ) )] k =1

(3.192)

(3.193)

k =6

where f1( j ) ( z (aj ) ) = a[ q0 F3 ( z (aj ) , eiq0 ) + q0 F4 ( z (aj ) , eiq0 ) + q1F3 ( z (aj ) -1 , e - iq0 ) + q1F4 ( z (aj )-1 , e - iq0 )] / 2 f 2( j ) ( z (aj ) ) = ipa( j )b[ - q0 F3 ( z (aj ) , eiq0 ) + q0 F4 ( z (aj ) , eiq0 )

(j=1,2)

(3.194)

(j=1,2)

(3.195)

+ q1F3 ( z (aj ) -1 , e - iq0 ) - q1F4 ( z (aj )-1 , e - iq0 )]/ 2 f 3( j ) (z (aj ) ) = a[ q2 F3 ( z (aj ) , eiq0 ) + q2 F4 ( z (aj ) , eiq0 )]/ 2 f 4( j ) (z (aj ) ) = ipa( j )b[ - q2 F3 (z (aj ) , eiq0 ) + q2 F4 ( z (aj ) , eiq0 )]/ 2

f 5( j ) ( z (aj ) ) = a1(aj ) ln z (aj ) ¥

¥

k =1

k =1

f6( j ) ( z ( j ) ) = ¦ z (aj ) k rk( j ) , f7( j ) ( z ( j ) ) = ¦ z (aj )-k s(k j )

(j=1,2)

(3.196)

(j=1,2)

(3.197)

The continuity conditions (3.95)3,4 now provide

q1(1) = X1 (B (2)-1d (1) - A (2)-1c(1) ),

q1(2) = X 2 ( A (1)-1c(1) - B (1)-1d (1) ),

(3.198)

*(1) (1) -1 (1) q (1) q1 , 2 = p1 P

*(1) (2) -1 (2) q (2) q1 , 2 = p1 P

(3.199)

q3(1) = X1 ( A (2)-1c(2) - B (2)-1d (2) ), q3(2) = X 2 (B (1)-1d (2) - A (1)-1c(2) ),

(3.200)

*(2) (1) -1 (1) q (1) P q3 , 4 = p1

(3.201)

q5(1) = a1(1)k q5(2) = a1(2) k

-1

*(2) (2) -1 (2) q (2) P q3 , 4 = p1

X1[ A (2) -1 (b**(1)c(1) - b**(2)c(2) ) - B (2) -1 (b**(1)d (1) - b**(2)d (2) )],

-1

X 2 [ A (1) -1 (b**(2)c(2) - b**(1)c(1) ) - B (1) -1 (b**(2)d (2) - b**(1)d (1) )]

(3.202) (3.203)

while the remaining unknown coefficients have the same form as those in Eqs (3.136)~(3.139) except that G1j, G2j and G3j should be replaced by G7j, G8j and G9j. 3.7.3 Green’s functions for an elliptic hole


115

3.7.3.1 Boundary conditions. When the inclusion becomes an elliptic hole with finite permittivity and thermal conductivity but vanishing elastic stiffness, the Green's function can be derived based on the results in the last section. In this case, the thermal and electric fields are still continuous across the hole surface, but the hole is free of traction. The boundary conditions on the hole can thus be written as T t (1) n = {t1 t2 t3 Dn } = t 0 = I 4 D0

J(1) = J(0) ;

(3.204)

where I4={0 0 0 1}T, ti (i=1,2,3) are the rectangular coordinates of surface force, t0 is surface traction-charge vector in the hole, the subscript "0" stands for the quantity associated with hole material, and Dn and D0 are the normal components of electric displacement in the matrix and the hole, respectively. Noting that tn=j,s, the boundary condition (3.204)2 can be further written as j (1)

s

= I 4 ³ D0ds = I 4 ³ ( D20dx1 - D10dx2 ) s

s

(3.205)

By writing the electric potential f(0) (z), where z0=x1+ix2, inside the hole as f(0) ( z0 ) = 2 Re f 0 ( z0 )

(3.206)

the electric displacement Di0 can be expressed as D10 = -2 k0 Re[df 0 / dz0 ], D20 = 2k0 Im[df 0 / dz0 ]

(3.207)

where k0 is the dielectric constant of the hole material. Using the above relations, Eq (3.205) can be further written as j (1)

s

= 2k 0I 4 Im[ f 0 ( z0 )]

(3.208)

In addition, the continuous condition of electric potential across the hole surface requires f(1) = 2 Re f 0 ( z0 )

(3.209)

3.7.3.2 Green’s function for holes. Since the interface condition for thermal field is the same as that in section 3.7.2, the Green's functions given in Eqs (3.187) and (3.188) are still available, but Eq (3.188) should be rewritten in the following form, using the subscript "0" here instead of the subscript "2": i q0 (0) (0) g 0 ( zt(0) ) = q2 {a1(0) ) + a2(0)t F4 (z t(0) , eiq0 )} - e*1a1(0) t F3 (z t , e t ln z t ¥

+ ¦ [G8 j z t(0) j + G9 j z t(0) - j ]

(3.210)

j =1

and z (t 0) is related to z t( 0 ) (x1+ p1*(0) x2) by (0) (0) (0) -1 zt(0) = a1(0) t z t + a2 t z t

(3.211)

The boundary conditions (3.208) and (3.209) together with Eqs (3.187) and (3.210) suggest that the solutions f(0), f(1) and j(1) be chosen as


116 5

f(0) ( z 0 ) = 2 Re[¦ f k 0 (z 0 )qk 0 + f 60 (z 0 ) + f 70 (z 0 ) + c4(0) g0 ( z (0) t )]

(3.212)

k =1

5 (1) (1) (1) f(1) ( z ) = 2 Re[ A (1) 4 {¦ f k a ( z a ) q k + f6 ( z ) + f7 ( z )} + c4 g1 ( z t )]

(3.213)

k =1

5

(1) (1) j (1) = 2 Re[B1{¦ f k a ( z a ) q (1) k + f6 ( z ) + f7 ( z )} + d g1 ( z t )] k =1 (1) k

where z a = z (1) a , qk0 and q

(3.214)

= {q k 1 q k 2 q k 3 q k 4 }T are constants to be determined,

(1) (1) (1) (1) } , Aij are the components of matrix A, and A (41) = ( A41 A42 A43 A44

f1a ( z a ) = a[ q0 F3 ( z a , eiq0 ) + q0 F4 (z a , eiq0 ) + q1F3 ( z a-1 , e - iq0 ) + q1F4 ( z a-1 , e - iq0 )]/ 2 f 2 a ( z a ) = ipab[ - q0 F3 ( z a , eiq0 ) + q0 F4 ( z a , eiq0 ) + q1F3 ( z a-1 , e - iq0 ) - q1F4 (z a-1 , e - iq0 )] / 2 f 3a ( z a ) = a[ q2 F3 ( z a , eiq0 ) + q2 F4 ( z a , eiq0 )]/ 2 f 4 a ( z a ) = ipab[ - q2 F3 ( z a , eiq0 ) + q2 F4 ( z a , eiq0 )] / 2

f 5a ( z a ) = a1(1)a ln z a ; f 50 ( z 0 ) = ln z 0 ¥

¥

k =1

k =1

(a = 0-4)

(3.215)

(a =0-4)

(3.215)

(a =0-4)

(3.217)

(a =1-4)

f 6 ( z ) = ¦ z km rk(1) , f 7 ( z ) = ¦ z m- k s (k1) ¥

¥

k =1

k =1

(a =1-4)

f 60 ( z 0 ) = ¦ z 0k rk 0 , f 70 ( z 0 ) = ¦ z 0- k s k 0

(3.218) (3.219)

(3.220)

where p m = p m(1) , p0=i, rk(1) = {rk 1 rk 2 rk 3 rk 4 }T and s k(1) = {s k 1 s k 2 s k 3 s k 4 }T are two four-component vectors to be determined. The interface condition (3.209) provides 4

q10 = A (41) q 1(1) + c 4(1) ,

q 20 = -i ¦ A4(1j) p j q 2(1j) - it 1 c 4(1)

(3.221)

j =1 4

q 30 = A (41) q (31) + c 4(1) ,

q 40 = -i ¦ A4(1j) p j q 4(1j) + it 2 c 4(1)

(3.222)

j =1

q50 = A (41) a1(m1) q 5(1) + a *1 a1(1t) c 4(1) + e*1 a1(t0 ) c 4( 0 )

(3.223)

The remaining unknowns rj0, sj0, q (k1) , r j(1) and s (j1) can also be determined through use of the boundary conditions (3.208), (3.209) and Eq (3.94), and we will omit those details, which are tedious and algebraic. 3.7.3.3 Cracks. By letting b ® 0 in Eq (3.55), the problem discussed above becomes an infinite piezoelectric solid containing a slit crack of length 2a. In this case, Eqs

117


(3.187) and (3.188) are reduced to a iq0 iq0 (1) -1 , e - iq0 ) + q0 F4 (z (1) g1 ( zt(1) ) = {q0 F3 ( z (1) t , e ) + q1 F4 ( z t t ,e ) 2 ¥

(3.224)

-1 (1) - j +q1F3 ( z (1) , e - iq0 ) + a*1 ln z (1) } t t } + ¦ G7 j z t j =1

g0 ( zt(0) ) =

aq2 iq0 (0) {F3 (z t(0) , eiq0 ) + F4 ( z (0) t , e )} - e*1 ln z t 2 ¥

(3.225)

+ ¦ [G8 j z t(0) j + G9 j z t(0) - j ] j =1

(0)

Therefore the solutions f , f(1) and j(1) should be chosen as f(0) ( z 0 ) = 2 Re[

¦

f k 0 ( z 0 )qk 0 + f 60 (z 0 ) + f 70 ( z 0 ) + c4(0) g0 ( z (0) t )]

(3.226)

k =1,3,5

f(1) ( z ) = 2 Re[ A (1) 4 {

¦

(1) (1) f k a ( z a ) q (1) k + f6 ( z ) + f7 ( z )} + c4 g1 ( z t )]

(3.227)

k =1,3,5

j (1) = 2 Re[B(1) {

¦

(1) f k a ( z a ) q (1) k + f6 ( z ) + f7 ( z )} + d1 g1 ( z t )]

(3.228)

k =1,3,5

Similarly, the unknown constants in Eqs (3.226)-(3.228) can be determined by using the boundary conditions (3.208) and (3.209).

References [1] Qin QH, Thermoelectroelastic Green’s function for a piezoelectric plate containing an elliptic hole, Mech Mat, 30, 21-29, 1998 [2] Qin QH and Mai YW, Thermoelectroelastic Green’s function and its application for bimaterial of piezoelectric materials, Arch Appl Mech, 68, 433-444, 1998 [3] Qin QH, Thermoelectroelastic analysis of cracks in piezoelectric half-plane by BEM, Compu Mech, 23, 353-360, 1999 [4] Qin QH, Green's function for thermopiezoelectric materials with holes of various shapes, Arch Appl Mech, 69, 406-418, 1999 [5] Qin QH, Thermoelectroelastic Green's function for thermal load inside or on the boundary of an elliptic inclusion, Mech Mat, 31, 611-626, 1999 [6] Qin QH, Thermoelectroelastic solution on elliptic inclusions and its application to crack-inclusion problems, Appl Math Modelling 25, 1-23, 2000 [7] Qin QH, Fracture mechanics of piezoelectric materials, Southampton: WIT Press, 2001 [8] Stagni L, On the elastic field perturbation by inhomogeneous in plane elasticity, ZAMP, 33, 313-325, 1982 [9] Hwu C and Yen WJ, On the anisotropic elastic inclusions in plane elastostatics. J Appl Mech, 60, 626-632, 1993 [10] Yen WJ, Hwu C and LiangYK, Dislocation inside, outside or on the interface of an anisotropic elliptical inclusion, J Appl Mech, 306-311, 1995

118

Chapter 4 Green’s function for magnetoelectroelastic problems 4.1 Introduction This chapter deals with applications of the Green’s function approach to solving coupled magnetoelectroelastic problems. Green’s functions by Radon transform in full space are described first, followed by a discussion of Green’s functions of magnetoelectroelastic problems using the potential function approach. The discussion is then extended to the cases of magnetoelectric plate, magnetoelectroelastic solid with half-plane boundary, bimaterial interface, and elliptic hole. Finally, Green’s functions for thermomagnetoelectroelastic solids with various defects are presented. Application of the Green’s function approach to the analysis of magnetoelectroelastic solids has been considered by many distinguished researchers over the past decades [1-13]. For example, Pan [1] derived three-dimensional Green’s functions in anisotropic magnetoelectroelastic full-space, half-space, bimaterials based on the extended Stroh formalism and two-dimensional Fourier transforms. Soh et al [2] presented explicitly a 3D Green's function for an infinite three-dimensional transversely isotropic magnetoelectroelastic solid based on the potential theory. Huang et al [3] obtained magnetoelectroelastic Eshelby tensors in an inclusion resulting from the constraint of the surrounding matrix of piezoelectric-piezomagnetic composites. Liu et al [4] obtained Green’s functions for an infinite 2D anisotropic magnetoelectroelastic medium containing an elliptic cavity or a crack. Li [5] presented explicit expressions of Green’s function for magnetoelectric problems. Researchers in [6-8] used the potential function approach to derive Green’s functions for various magnetoelectroelastic problems. Alshtis et al [9] obtained the Green’s functions of an angularly inhomogeneous magnetoelectroelastic medium subjected to a line defect at the origin of the coordinate system. Jiang and Pan [10] derived Green’s functions for a 2D polygonal inclusion problem in magnetoelectroelastic full-, half-, and bimaterial-planes. Recently, Qin [11-13] obtained Green’s functions for magnetoelectroelastic solids with various defects. The defects may be an elliptic hole or a Griffith crack, a half-plane boundary, a bimaterial interface, wedge boundary, or a rigid inclusion. Chen et al [14] derived a set of dynamic Green’s functions of magnetoelectroelastic medium based on potential approach. For the present, we restrict our attention to the problems treated in [1, 4, 5, 8, 10-14]. 4.2 3D Green’s functions by Radon transform 4.2.1 Integral expressions for the Green’s function In this section, the results presented in [1] for 3D Green’s functions developed using Radon transform are described. In doing so, we consider an anisotropic magnetoelectroelastic solid subjected to an extended concentrated force f at the origin (0,0,0). The governing equations and constitutive relations of this problem are defined by Eqs (1.112) and (1.113), respectively. By comparing Eq (1.112) with Eq (1.33), the governing equation for determining Green’s function corresponding to Eq (1.112) can be written as

EiJMnGMR ,in (x) + d JRd (x) = 0

(4.1)

119

Green’s function for magnetoelectroelastic problems

which is in the same form as that of Eq (2.4), but has different components, where the components GIJ (x) represent the elastic displacement at x in the xI-direction (for I=1,2,3) or electric potential (for I=4) or magnetic potential (for I=5) due to a unit point force in the xJ-direction (for J=1,2,3) or a unit point charge (for J=4) or a point electric current (for J=5) at the origin, d JR is the fifth-rank Kronecker delta [1]. To make the derivation tractable, the 5´5 matrix, K JM (n) = EiJMl ni nl , defined in -1 Section 2.2 [just after Eq (2.5)] is again used here. By integrating K JM (n)d(n × x) with respect to n, taking its second derivatives with respect to x, and multiplying the results by EiJMl, we obtain the identity [1]

EiJMl

¶2 ¶xi ¶xl

³

n×x = 0

-1 K JM (n)d(n × x)d W(n) = d JR Ñ3 ³

d(n × x) n ×x = 0

n

2

d W(n)

(4.2)

Making use of the plane representation of the Dirac delta function [15] d( x ) =

1 d(n × x) Ñ3 ³ d W(n) , 2 2 n × x = 0 8p n

(4.3)

where n is defined in Appendix A, Eq (4.2) can be rewritten as EiJMl

¶2 ¶xi ¶xl

³

n×x = 0

-1 K JM (n)d(n × x)d W(n) = -8p2 d JR d(x)

(4.4)

The comparison between Eqs (4.1) and (4.4) provides GJM (x) =

1 8p2

³

n×x = 0

-1 K JM (n)d(n × x)d W(n)

(4.5)

4.2.2 Explicit expression for Green’s functions As was pointed out in [15], the integral (4.5) can be transformed into a one-dimensional infinite integral and the results can then be reduced to a summation of five residues. This is achieved by expressing the vector n in terms of a new, orthogonal, and normalized system (d-m-t) as shown in Fig. 2.1. The new base (d-m-t) is chosen as t=

x , r= x r

(4.6)

The remaining two unit vector orthogonal to t can be obtained as d=

t´v , m = t ´ d, t´v

(4.7)

where v is an arbitrary unit vector different from t (i.e, v ¹ t ), and the symbol ‘´’ represents cross product of any two vectors. The vector n can thus be expressed in terms of the new system (d-m-t) as

n = xd + zm + ht

(4.8)

n × x = xd × x + zm × x + ht × x = r h

(4.9)

It is clear that


120

Therefore, in the new system (d-m-t), the integral (4.5) becomes 1 -1 K JM (xd + zm + ht )d(r h)d W(x, z, h) 8p 2 ³ n × x = 0 AJM (xd + zm + ht ) 1 = 2³ d(r h)d W(x, z, h) 8p n×x =0 D(xd + zm + ht )

GJM (x) =

(4.10)

where AJM (n) is the adjoint matrix of K JM (n) and D(n) = det K JM (n) is the determinant of K JM (n) . Carrying out the integration of (4.10)2 with respect to h provides [15] GJM (x) =

¥ A 1 JM (d + z m ) dz 2 ³ -¥ 4p r D(d + zm)

(4.11)

Since the inverse of KJM exists [15], its determinant D(n) does not have real roots. Therefore, the tenth-order polynomial equation of z

D (d + zm) = 0

(4.12)

has ten roots, five of them being the conjugate of the remainder. With these roots, we can write the polynomial (4.12) as 10

5

k =0

k =1

D(d + zm) = ¦ ak +1z k = a11 Õ (z - z k )(z - z k ),

(4.13)

where a11 is the coefficient of z10. In terms of the residues at the poles, the Green’s function (4.11) can finally be expressed explicitly as ª º « » AJM (d + z mm) 1 » GJM (x) = Im ¦ « 5 » 2pr m =1 « « a11 (z k - z k ) Õ (z m - z k )(z m - z k ), » k =1( k ¹ m ) ¬ ¼ 5

(4.14)

4.3 3D Green’s functions by potential function approach In the previous section, we discussed Green’s functions for magnetoelectroelastic solids by the Radon transform approach. This section describes Green’s functions for the same problem but derived through use of the potential function approach. The discussion below follows the results presented in [8]. 4.3.1 General solutions for magnetoelectroelastic solids Consider a transversely isotropic magnetoelectroelastic solid subjected to an extended concentrated force f at the origin (0,0,0). The governing equations and constitutive relations of this problem are defined by Eqs (1.90)-(1.97). The general solution for the solid with distinct eigenvalues can be assumed in the form

121

Green’s function for magnetoelectroelastic problems 4 4 ¶v j § · U = D ¨ iv 0 + ¦ v j ¸ , wm = ¦ s j kmj , ¶z j j =1 j =1 © ¹ 4 4 ¶ 2v § · s1 = 2¦ (c66 - w1 j s 2j ) 2 j , s 2 = 2c66 D 2 ¨ iv 0 + ¦ v j ¸ , ¶z j j =1 j =1 © ¹ 4

s zm = ¦ wmj j =1

¶ 2v j ¶z

2 j

(m=1,2,3)

(4.15)

4 § ¶v j · ¶v t zm = D ¨ s0rmi 0 + ¦ s j wmj ¸, ¨ ¶z0 j =1 ¶z j ¸¹ ©

,

where v 0 and v j are potential functions to be determined, U = u x + iu y = eiq (ur + iuq ), w1 = u z , w2 = f, w3 = y, z j = s j z s1 = s x + s y , s 2 = s x - s y + 2it xy = e 2iq (sr - sq + 2itrq ), t z1 = t xz + it yz = eiq (trz + itqz ), t z 2 = Dx + iDy = eiq ( Dr + iDq ), t z 3 = Bx + iBy = eiq ( Br + iBq ), s z1 = s z , s z 2 = Dz , s z 3 = Bz , r1 = c44 , r2 = e15 , r3 = e15 , kmj = w1 j = c44 (1 + k1 j ) + e15k2 j + e15k3 j ,

bmj a j s 2j

, D=

¶ ¶ +i , ¶x ¶y

w2 j = e15 (1 + k1 j ) - k11k2 j - a11k3 j ,

w3 j = e15 (1 + k1 j ) - a11k2 j - m11k3 j ,

(4.16)

and a j , s j , bmj are constants listed in Appendix B. 4.3.2 Green’s function for 3D magnetoelectroelastic solids For a transversely isotropic magnetoelectroelastic solid subjected to an extended concentrated force f at the origin (0,0,0), the corresponding Green’s functions can be obtained based on the general solution presented above. We consider below the case in which all eigenvalues are distinct. For problems with multiple eigenvalues the details of the solution procedure are similar to the case with distinct eigenvalues, and have been documented in [8]. 4.3.2.1 Solution due to a point force Pz in z-direction, a point charge Q, and a magnetic monopole J. In this case the problem is axially symmetric about the z-axis and the related potential functions can be assumed in the form [8]: v 0 = 0, v j = M j sign( z ) ln R*j

(j=1-4)

(4.17)

where M j are constants to be determined, sign(z) is the signum function defined by Eq (2.225), and R*j is defined in Eq (2.17). Substituting Eq (4.17) into Eq (4.15) we have 4

U = sign( z )¦ M j j =1

4

s zm = -¦ M j wmj j =1

4 x + iy 1 , wm = ¦ M j s j kmj , * Rj Rj R j =1 j

zj R

3 j

4

, t zm = -¦ M j s j wmj j =1

x + iy R 3j


122

4 4 ª 2 z x2 + y2 § 1 1 ·º M j (2c66 - w1 j s 2j ) j3 , s1 = 2c66sign( z )¦ M j « + 2 » ¨ ¸ ¦ * 2 * ¨ * ¸ R j R j © R j R j ¹ »¼ Rj j =1 j =1 «¬ R j R j 4 ª y 2 - x 2 - 2ixy § 1 1 ·º s 2 = 2c66sign( z )¦ M j « ¨¨ + * ¸¸ » 2 * Rj Rj j =1 © R j R j ¹ ¼» ¬«

(4.18)

There are four unknowns M j in Eq (4.18). We need to find four conditions to determine the four unknowns. The displacements ux and uy are bounded on the z-axis, which provides 4

¦M

j

=0

(4.19)

j=1

The remaining three equations can be established by considering the requirements of the force, charge, and electric current balances. These requirements are enforced by integrating the traction, the normal component of the electric displacement, and magnetic induction over the surface of a small spherical cavity centered at the origin, and requiring these to balance the point force Pz, the point charge Q, and a magnetic monopole J. These conditions lead to 4

¦w

1j

Mj =

j =1

4

Pz , 4p

¦w

2j

Mj =

j =1

-Q , 4p

4

¦w

2j

Mj =

j =1

-J 4p

(4.20)

Solving Eqs (4.19) and (4.20), yields

M j = u j Pz + b j Q + g j J ,

(j=1-4)

(4.21)

where 1 u1 ½ ª 1 °u ° « w ° 2 ° « 11 w12 ® ¾= ° u3 ° «w21 w22 °¯u4 °¿ «¬ w31 w32

1 w13 w23 w33

1 º w14 »» w24 » » w34 ¼

-1

0 ½ ° ° °1 ° ® ¾, °0 ° °¯0 °¿

1 b1 ½ ª 1 °b ° « w ° 2 ° « 11 w12 ® ¾= °b3 ° «w21 w22 « ¯°b4 ¿° ¬ w31 w32

1 g1 ½ ª 1 °g ° «w ° 2 ° « 11 w12 ® ¾= ° g 3 ° «w21 w22 « ¯° g 4 ¿° ¬ w31 w32

1 w13 w23 w33

1 º w14 »» w24 » » w34 ¼

-1

0 ½ ° ° °0 ° ® ¾ °0 ° °¯1 °¿

1 w13 w23 w33

1 º w14 »» w24 » » w34 ¼

-1

0 ½ ° ° °0 ° ® ¾, °1 ° °¯0 °¿

(4.22)

4.3.2.2 Solution due to point forces Px in x-direction and Py in y-direction. For a point force Px applied at the origin, the solution can be assumed in the form v0 =

¡jx ¡0 y , v j = * , (j=1-4) * R0 Rj

where ¡ j (j=0-4) are constants to be determined. Substituting Eq (4.23) into Eq (4.15) we have

(4.23)

123


ª 1 x( x + iy ) º ª 1 y ( y - ix) º 4 U = -¡ 0 « * +¦¡j « * », , *2 » R0 R0 ¼ j =1 «¬ R j R j R*2 »¼ j ¬ R0 4 ª 4 1 x2 + y2 § 1 2 · º °½ ° s1 = 2 x ¦ ¡ j ®(2c66 - w1 j s 2j ) 3 - c66 « + ¨ ¸» ¾ , *2 ¨ Rj R 2j R*2 R*j ¸¹ »¼ ° j =1 «¬ R j R j j © Rj ¯° ¿

ª iy x - iy xy ( y - ix) § 1 2 ·º s 2 = 4c66 ¡ 0 « 3 + ¨ + * ¸» *2 2 *2 R0 R0 © R0 R0 ¹ ¼» ¬« 2 R0 R0 R0

t zm

4 ª 2( x + iy ) x( x - iy ) 2 § 1 2 ·º - 2c66 ¦ ¡ j « + ¨¨ + * ¸¸ » , *2 2 *2 R j R j © R j R j ¹ ¼» j =1 ¬« R j R j ª 1 y ( y - ix) § 1 1 ·º = s0rmsign( z ) ¡ 0 « ¨ + * ¸» * 2 * R0 R0 © R0 R0 ¹ »¼ «¬ R0 R0 4 ª 1 x( x + iy ) § 1 1 ·º - sign( z )¦ s j wmj ¡ j « ¨¨ + * ¸¸ » , * 2 * R j R j © R j R j ¹ ¼» j =1 ¬« R j R j 4

wm = -sign( z )¦ ¡ j s j kmj j =1

x , R j R*j

4

s zm = ¦ ¡ j wmj j =1

x , R 3j

(4.24)

There are five unknowns ¡ j (j=0-4) in Eq (4.24), which can be determined by enforcing the requirement that the solution be continuous on z = 0, plus the force balance condition. Consideration of the continuity of wm and tmz on z = 0 yields [8] 4

4

¦s k

j mj

¡ j = 0,

j =1

s0rm ¡ 0 +¦ s j wmj ¡ j = 0, (m=1-3)

(4.25)

j =1

By substituting rm and wmj in Eq (4.16) into Eq (4.25)2 and after some manipulation, Eq (4.25)2 can be simplified to only one independent equation as 4

¦s ¡ j

j

=0

(4.26)

j =1

which together with Eq (4.25)1 consists of four independent equations for determining the unknown constants ¡ j . Consideration of the equilibrium of a layer cut by two planes of z = ±e in the neighborhood of the plane of z = 0 gives

³

¥ -¥

[t zx ( x, y, e) - t zx ( x, y, -e)]dxdy + Px = 0

(4.27)

Substitution of t z1 in Eq (4.24) into Eq (4.27), after some manipulation, yields 4

2pc44 s0 ¡ 0 - 2pc44 ¦ s j ¡ j = - Px

(4.28)

j =1

Solving Eqs (4.25)1, (4.26) and (4.28) for unknown constants ¡ j , we obtain ¡ j = l j Px (j=0-4)

(4.29)


124

where

l0 = -

1 , 4ps0 c44

l1 ½ ° ° 1 °l 2 ° ® ¾= ° l 3 ° 4pc44 °¯l 4 °¿

ª s1k11 «s k « 1 21 « s1k31 « ¬ s1

s2 k12 s2 k22 s2 k32 s2

s3k13 s3 k23 s3 k33 s3

s4 k14 º s4 k24 »» s4 k34 » » s4 ¼

-1

0 ½ °0 ° ° ° ® ¾ °0 ° °¯1 °¿

(4.30)

The solution (4.29) is for the point force Px applied at the origin. When the point forces Px and Py are simultaneously applied at the origin, the corresponding potential functions are v0 = l0

Px y - Py x * 0

R

,

vj = lj

Px x + Py y R*j

(j=1-4)

(4.31)

from which the Green’s functions of the magnetoelectroelastic field can be determined. 4.4 Green’s functions for magnetoelectric problems In this section we consider the coupling between magnetic and electric fields only. The discussion follows the results presented in [5]. In the case of static linear magnetoelectric problems where there is no free electric charge and current, the basic equations governing the magnetic and electric fields are [5]

Di ,i = 0,

Bi ,i = 0

Di = kij E j + aij H j , Bi = aij E j + mij H j Ei = -f,i ,

(4.32)

H i = -y ,i

Using the shorthand notation described in Section 1.5, the constitutive equations, Gaussian equations, and gradient equations can be written as P iJ ,i = EiJMnU J ,in = 0,

P iJ = EiJMnU M , n

(4.33)

where the magnetoelectric fields UJ, generalized stress P iJ , and stiffness constant EiJMn are defined as D , J = 1, -f, J = 1, P iJ = ® i UJ = ® -y = , J 2, ¯ ¯ Bi J = 2,

EiJMn

kin , J = M = 1, °a , J = 1, M = 2, ° = ® in °ain , J = 2, M = 1, °¯ min , J = 2, M = 2

(4.34)

(4.35)

By comparing with Eq (1.33), the Green’s function corresponding to Eq (4.33) is defined by the differential equation EiJMnGMR ,in (x - x) + d JRd (x - x) = 0 (4.36)

The Green’s function GMR (x - x) can be obtained by applying the Radon transform defined in Appendix A to Eq (4.36) above [5]:

125


EiJMl zi zl

¶2 ˆ GMR (n, w - n × x) + d JR d(w - n × x) = 0 2 ¶w

(4.37)

Multiplying the inverse of K JM (n) = EiJMl ni nl to both sides of Eq (4.37) and then taking its inverse transform, we obtain 1 GMR (x - x) = 2 8p

³

-1 (n)d[n × (x - x)]dS (n) K MR

(4.38)

n =1

Denoting t = (x - x) / x - x (see Fig. 2.1), and utilizing the property of the Dirac delta

function given in Eq (1.43), we have GMR (x - x) =

1 8p2 x - x

³

-1 (n)d[n × t ]dS (n) K MR

(4.39)

n =1

t n q m g n*

d

Fig. 4.1 Relationship between t, n, and n* The relationship between vectors n and t is shown in Fig. 4.1, where n* lies in the plane m-d normal to t. Making use of the property of the Dirac delta function again, Eq (4.39) can be further written as GMR (x - x) =

1 8p x - x 2

³

-1 (n* )d g (n* ) K MR

(4.40)

n =1

where n = 1 represents a unit circle, which is the intersection of the unit sphere with plane m-d (see Fig. 4.1). For the Green’s functions defined by the integral (4.40), Li [5] obtained their explicit expressions for transversely isotropic media. The discussion here follows his results. For a transversely isotropic magnetoelectric medium, the matrix KJM is a 2´2 matrix given by ª k (n 2 + n 2 ) + k33 n32 K JM = « 11 12 22 2 ¬a11 (n1 + n2 ) + a33n3

a11 (n12 + n22 ) + a33 n32 º » m11 (n12 + n22 ) + m33 n32 ¼

(4.41)


126

The inverse of KJM can then be obtained as -1 K JM =

ª m11 (n12 + n22 ) + m33n32 1 « det K JM ¬ -a11 (n12 + n22 ) - a33 n32

-a11 (n12 + n22 ) - a33 n32 º » k11 (n12 + n22 ) + k33 n32 ¼

(4.42)

For the sake of simplicity, assume the source point is located at the origin, i.e., x = (0, 0, 0) , and the field point is lying in the x1-x3 plane, with tanq = x1 / x3 . For such a configuration, the intersection between the unit sphere n = 1 and the plane m-d (see Fig. 4.1) is a unit circle represented by h12 + h22 = 1, with h3=0 corresponding to the plane m-d. Without loss of generality, choose h2 to coincide with a unit vector in the x2-direction. Thus the variable n is related to hi by n1 = h1 cos q, n2 = h2 , n3 = -h1 sin q

(4.43)

By introducing a complex variable z = h1 + ih2 = eig , we have z + z -1 z - z -1 dz , h2 = , dg = 2 2i iz

h1 =

(4.44)

Making use of Eq (4.44), the integral (4.40) can be expressed in terms of the complex variable z and then evaluated using Cauchy’s residue theorem. To use the Cauchy’s residue theorem, express g = det K JM in terms of z and q as [5] g=

2 ( k33m33 - a 33 ) sin 4 q (1 - A1 )(1 - A2 )h1 (z )h2 (z ), 16z 4

(4.45)

where hi (z ) = z 4 + 2 Bi z 2 + 1, A1 = A2 =

Bi =

Ai cos 2 q + sin 2 q + Ai , (1 - Ai ) sin 2 q

2 2 -b - b 2 - 4(a11 - k11m11 )(a33 - k33m33 ) 2 - k33m33 ) 2(a33 2 2 -b + b 2 - 4(a11 - k11m11 )(a33 - k33m33 ) 2 - k33m33 ) 2(a 33

,

(4.46)

,

with b = k33m11 + k11m33 - 2a11a33

(4.47)

Noting that if z * is a root of hi (z ) = 0 , so are -z* , 1/ z* , and - 1/ z* [5], so in general, there two sets of roots: (i) ±ci with modulus less than unity; and (ii) ±bi with moduli great than unity. Eq (4.45) can thus be factored as g=

2 ( k33m33 - a 33 ) sin 4 q (1 - A1 )(1 - A2 )(z 2 - c12 )(z 2 - c 22 )(z 2 - b12 )(z 2 - b22 ) 4 16z

(4.48)

Substituting Eq (4.48) into Eq (4.40) yields GMR (x) = C

zLMR (z )d z , (z - c )(z - c 22 )(z 2 - b12 )(z 2 - b22 ) n =1

³

2

2 1

2

(4.49)


127

where (1 + z 2 ) 2 (m11 cos 2 q + m33 sin 2 q) - (1 - z 2 ) 2 m11 , MR = 11 ° 2 2 2 2 2 2 °-(1 + z ) (a11 cos q + a 33 sin q) + (1 - z ) a11 , MR = 12 LMR (z ) = ® 2 2 2 2 2 2 °-(1 + z ) (a11 cos q + a 33 sin q) + (1 - z ) a11 , MR = 21 2 2 2 2 2 2 ° ¯ (1 + z ) ( k11 cos q + k33 sin q) - (1 - z ) k11 , MR = 22

(4.50)

and C=

1 2ip x ( k33m33 - a ) sin 4 q(1 - A1 )(1 - A2 ) 2

2 33

(4.51)

Now define

tMR (z ) =

zLMR (z ) , (z - b12 )(z 2 - b22 ) 2

(4.52)

which is analytical inside z = 1 . Since bi > 1 , Eq (4.49) can be rewritten as GMR (x) = C

tMR (z )d z , (z - c12 )(z 2 - c 22 ) n =1

³

2

(4.53)

According to Cauchy’s residue theorem, Eq (4.53) can be further written as ª t ( c ) - t (c ) º GMR (x) = 8ipC « MR 1 2 MR2 2 » (c1 - c 2 ) ¬ ¼

(4.54)

4.5 Half-plane problems 4.5.1 Green’s function for magnetoelectroelastic plate with horizontal half-plane boundary The Green’s functions for a half-plane magnetoelectroelastic plate subjected to loadings q 0 and b can be obtained in the same way as that treated in Section 2.5 if the half-plane boundary is in horizontal direction. Accordingly, the generalized displacement vector U and stress function j can be assumed in the form: U= j=

5 1 1 Im{A ln( za - za 0 ) q*} + ¦ Im{A ln( za - zb 0 ) qb } , p b=1 p

(4.55)

5 1 1 Im{B ln( za - za 0 ) q*} + ¦ Im{B ln( za - zb 0 ) qb } p b=1 p

(4.56)

where q* and qb are in the same form as those in Eqs (2.86) and (2.91), respectively, but now each of them has five components instead of four. 4.5.2 Green’s function for magnetoelectroelastic plate with vertical half-plane boundary 4.5.2.1 Coordinate transformation. When the half-plane boundary is in the vertical direction (or even arbitrarily oriented) instead of the horizontal direction (Fig. 4.2), the approach described in Section 2.5 is no longer applicable. The reason is that,


128

unlike in the case of horizontal boundary where za = x1 + pa x2 becomes a real number on the boundary x2 = 0, za is, in general, neither a real number nor a pure imaginary number on the vertical boundary x1 = 0, which complicates the related mathematical derivation. To circumvent this problem, a new coordinate variable is introduced [11]: za* = za / pa

(4.57)

In this case za* is a real number on the vertical boundary x1 = 0. This coordinate transformation can be used for both half-plane and bimaterial problems with a vertical boundary.

x2

· z0 (x10 , x20 )

o

x1

Fig. 4.2 Magnetoelectroelastic plate with vertical half-plane boundary 4.5.2.2 Green’s function for magnetoelectroelastic plate with vertical half-plane boundary. Let the material occupy the region x1 > 0 , and a line force-charge q 0 and a line dislocation b be applied at z0 (x10, x20) (see Fig. 4.2). To satisfy the boundary conditions on x1 = 0, the solution can be assumed in the form [11] U= ĳ=

5 1 1 Im{A ln( za* - za* 0 ) q*} + ¦ Im{A ln( za* 0 - zb*0 ) qb } , p b=1 p

(4.58)

5 1 1 Im{B ln( za* - za* 0 ) q*} + ¦ Im{B ln( za* 0 - zb*0 ) qb } p p b=1

(4.59)

where q* is given in Eq (2.86) and q b are unknown constants to be determined. Consider first the case in which the surface x1 = 0 is traction-free, so that

j=0

for x1 = 0

(4.60)

Substituting Eq (4.59) into Eq (4.60) yields ĳ=

5 1 1 Im{B ln( x2 - za* 0 ) q*} + ¦ Im{B ln( x2 - zb*0 ) qb } = 0 p b=1 p

Using the following relations

(4.61)

129


Im{B ln( x2 - za* 0 ) q*} = - Im{B ln( x2 - za*0 ) q} ,

(4.62)

and 5

ln( x 2 - z a* 0 ) = ¦ ln( x 2 - z b*0 )I b ,

(4.63)

I b = d ba = diag[d b1 , d b 2 , d b3 , d b 4 , d b5 ] ,

(4.64)

qb = B -1BIb q* = B -1BIb ( AT q 0 + BT b)

(4.65)

b =1

where

Eq (4.61) yields

If the boundary x1 = 0 is a rigid surface, then U = 0,

at x1 = 0

(4.66)

The same procedure shows that the solution is given by Eqs (4.58) and (4.59) with q b = A -1 A I b ( A T q 0 + B T b )

(4.67)

It is noted that Eqs (4.65) and (4.67) have the same forms as those of Eqs (2.91) and (2.93). The final version of the Green’s function can thus be written in terms of zk as 5 1 1 U = Im{A ln( za - za 0 ) / pa q*} + ¦ Im{A ln( za / pa - zb0 / pb ) qb } (4.68) p p b=1 ĳ=

5 1 1 Im{B ln( za - za 0 ) / pa q*} + ¦ Im{B ln( za / pa - zb 0 / pb ) qb } p b=1 p

h

· z 0 ( x10 , x20 )

(4.69)

z - plane

x2 z 0 (x 0 , h 0 )

·

q0

o

x1

o

x

Fig. 4.3 Magnetoelectroelastic solid with arbitrarily oriented half-plane boundary 4.5.2.2 Green’s function for a solid with an arbitrarily oriented half-plane boundary. If the half-plane boundary is in an angle ș 0 ( ș 0 ¹ 0 ) with positive x1-axis, the corresponding Green’s function can be obtained by introducing a new mapping function [11]


130

z = z q0 / p

( ș0 ¹ 0 )

(4.70)

which maps the boundary q = ș 0 in the z-plane onto the real axis in the z-plane (x+ih) (Fig. 4.3). Following the procedure in section 4.5.2.1 it can be shown that the resulting Green’s functions can be expressed as U=

5 1 1 Im{A ln( zap / q0 - zap 0/ q0 ) q*} + ¦ Im{A ln( zap / q0 - zbp0/ q0 ) qb } p p b=1

(4.71)

ĳ=

5 1 1 Im{B ln( zap / q0 - zap 0/ q0 ) q*} + ¦ Im{B ln( zap / q0 - zbp0/ q0 ) qb } p b=1 p

(4.72)

where q* and qb

have, respectively, the same forms as those given in Eqs (2.86) and

(4.65). 4.6 Bimaterial problems 4.6.1 Green’s function for magnetoelectroelastic bimaterial with horizontal interface Similar to the discussion of half-plane problems in the previous section, the Green’s functions for a magnetoelectroelastic bimaterial solid subjected to loadings q 0 and b can also be obtained in the same way as that treated in Section 2.6 if the interface is in the horizontal direction. Consider a bimaterial magnetoelectroelastic plate subjected to loadings q 0 and b applied in the upper half-plane at z0(x10, x20), in which the upper half-plane ( x2 > 0) is occupied by material 1 and the lower half-plane ( x2 < 0) is occupied by material 2 (see Fig. 2.5). Following the procedure described in Section 2.6, the generalized displacement vector U and stress function j are, in this case, assumed in the form U (1) =

5 1 1 Im{A (1) ln( za(1) - za(1)0 ) q (1)*} + ¦ Im{A (1) ln( za(1) - zb(1)0 ) qb(1) } , p b=1 p

(4.73)

j (1) =

5 1 1 Im{B (1) ln( za(1) - za(1)0 ) q (1)*} + ¦ Im{B (1) ln( za(1) - zb(1)0 ) qb(1) } p p b=1

(4.74)

for material 1 in x2>0 and 5 1 U (2) = ¦ Im{A (2) ln( za(2) - zb(1)0 ) qb(2) } , p b=1

(4.75)

5 1 j(2) = ¦ Im{B (2) ln( za(2) - zb(1)0 ) qb(2) } b=1 p

(4.76)

(1) (2) for material 2 in x2 0, where zb*(1) , z a*(i ) = z a( i ) / p a( i ) (i = 1,2), 0 = zb 0 / pb

q (1)* , qb(1) , and qb(2) are the same as in Eqs (4.73)-(4.76). It is found that Eqs (4.77)-(4.80) can be obtained from Eqs (4.73)-(4.76) by replacing za(1) , za(1)0 , zb(1)0 , za(2) , and zb(1)0 with za*(1) , za*(1) , zb*(1) , za*(2) , and zb*(1) . 0 0 0 x2 · z 0 ( x10 , x 20 )

o

x1

Material 1

Material 2 Interface

Fig. 4.4 Magnetoelectroelastic bimaterial with vertical interface Finally, if the interface is in an angle ș 0 ( ș 0 ¹ 0 ) with positive x1-axis, the corresponding Green’s function can be obtained from Eqs (4.73)-(4.76) by replacing za(1) , za(1)0 , zb(1)0 , za(2) , and zb(1)0 with za(1) p / q0 , za(1)0 p / q0 , zb(1)0 p / q0 , za(2) p / q0 , and zb(1)0 p / q0 [11].

4.7 Problems with an elliptic hole or a crack In the previous two sections, Green’s functions for magnetoelectroelastic solids with half-plane boundary and bimaterial interface were discussed. Extension of the procedure to the case of magnetoelectroelastic solids containing an elliptic hole is described in this section. Green’s functions for an infinite magnetoelectroelastic solid with an elliptic hole induced by electroelastic loads are derived using Stroh formalism. The loads may be a generalized line force q 0 and a generalized line dislocation b. The discussion here is based on the results presented in [4].


132

4.7.1 Basic formulation of the hole problem Consider an infinite magnetoelectroelastic solid with an elliptic hole subjected to a generalized line dislocation b and a generalized line force q 0 at a point z0(x10,x20) outside the ellipse (see Fig. 2.7). The hole is assumed to be free of force, charge and electric current along its surface, but to be filled with a homogeneous gas of dielectric constant kc and permeability mc. As indicated in [4], electric and magnetic fields can permeate in a vacuum and can exist inside the hole of a magnetoelectroelastic material. The electric and magnetic potentials in the hole satisfy the Laplace equation Ñ 2 fc = 0 ,

Ñ 2 y c = 0 in Wc

(4.81)

and the constitutive relations are Dic = kc Eic = -kc

¶fc ¶y c , Bic = mc H ic = -m c , ¶xi ¶xi

i=1,2

(4.82)

where superscript c refers to the variables associated with hole medium. As treated in Section 2.8, the general solution to Eq (4.81) can be assumed in the form fc ( z ) = 2 Re[ f fc ( z )], y c ( z ) = 2 Re[ f yc ( z )],

(4.83)

Making use of Eqs (4.82) and (4.83), the resultants of the normal components of electric displacement and magnetic induction along an arc can be written as S Dc = -2 kc Im[ f fc ( z )] , S Bc = -2m c Im[ f yc ( z )]

(4.84)

Y c = {fc , y c }T = 2 Re{f c ( z )} , S c = {S Dc , S Bc }T = -2P c Im{f c ( z )}

(4.85)

If we define

with ª kc f c ( z ) = { f fc ( z ), f yc ( z )}T , P c = « ¬0

0º », mc ¼

(4.86)

the boundary conditions of the hole problem can be expressed as gT U = Y c , j = -gS c along the hole boundary,

(4.87)

{sij , Di , Bi } ® 0 at infinity,

(4.88)

and the balance equations are defined by Eq (2.77), where ª0 0 0 1 0 º g=« » ¬0 0 0 0 1 ¼

T

(4.89)

4.7.2 Green’s function for the hole problem Using the mapping functions (2.162), (2.182), and superposition principle of linear problems, the general expressions for the fields outside and inside the ellipse can be written in the form U = 2 Re{A[f0 (ȗ) + f1 (ȗ)] , and

j = 2 Re{B[f0 (ȗ) + f1 (ȗ)] in W

(4.90)

133


Y c == 2 Re{f c (z 0 )} , S c = -2P c Im{f c (z 0 )} in Wc

(4.91)

where W and Wc are defined in Fig. 2.7, z a and z 0 are related to za and z by Eqs (2.162) and (2.182), respectively. Following the procedure in Section 2.8, functions f0, f1, and fc are assumed in the form f0 = f 0 (z a ) = ln( za - za 0 q f ,

(4.92)

¥

f1 (ȗ) = f1 (z a ) = ¦ z a- k d k ,

za ³ 1 ,

(4.93)

k =1

¥

f c (z 0 ) = ¦ (z 0k + rin2 k z 0- k )e k , rin £ z 0 £ 1 ,

(4.94)

k =1

where q f is defined by Eq (2.80), dk and ek are constants to be determined, a -b , a and b are defined in Fig. 2.7. a+b In order to obtain d k and e k from the boundary condition (4.87), the function f0

rin =

should also be expressed in terms of zk. As treated in Section 2.8, Eq (4.92) is written in terms of Laurent series expansions in the form ¥

(

)

f0 = ¦ z ka c k + z a- k c - k , k =1

1 £ z a £ za0

(4.95)

where ck = -

1 § 1 · ¨ ¸ k © z a0 ¹

k

qf ,

c-k = -

1 § da · ¨ ¸ k © caz a 0 ¹

k

qf ,

(4.96)

while ca and d a are defined by Eq (2.164), and z a 0 is defined by Eq (2.172). Substituting Eqs (4.93)-(4.95) and noting that z 0 = z a = eiq on the hole boundary, the condition (4.87) yields ¥

¥

k =1

k =1

gT Re ¦{A[c k eik q + (c - k + d k )e - ik q ]} = Re ¦ ( eik q + rin2 k e -ik q )e k , ¥

¥

k =1

k =1

Re ¦{B[c k eik q + (c - k + d k )e - ik q ]} = gP c Im ¦ ( eik q + rin2 k e - ik q )e k

(4.97) (4.98)

Comparing the coefficients of eik q and e - ik q in Eqs (4.97) and (4.98), we have

gT Ad k - ( rin2 k e k + ek ) = -gT ( Ack + Ac - k ), gT Ad k - (e k + rin2 k ek ) = -gT ( Ac- k + Ac k ), Bd k + igP c ( rin2 k e k - ek ) = -( Bck + Bc - k ), Bd k + igP c (e k - rin2 k ek ) = -( Bc- k + Bc k ) Solving Eq (4.99) for d k and e k , we obtain

(4.99)


134

d k = -c - k - B -1Bck + iB -1gP c ( ek - rin2 k e k ), e k = ( rin4 k W - VW -1V ) -1 ( rin2 k Yk - VW -1Yk )

(4.100)

where

W = I + gT MgP c ,

V = I - gT MgP c , Yk = 2igT L-1Bc k

(4.101)

with I being the 2´2 unit matrix, M and L being defined by Eqs (2.217) and (1.142). Substituting Eq (4.100) into Eq (4.93) and using relation (2.193), f1 can be expressed in the form 5 § § d /c · 1 · -1 f1 = - ln ¨1 - a a ¸ q f - ¦ ln ¨1 ¸¸ B BIbq f ¨ z a za 0 ¹ b=1 © © z a zb 0 ¹

¥

+ i¦ z

-k a

-1

c

(4.102)

2k in k

B gȆ ( ek - r e )

k =1

The corresponding Green’s functions can thus be obtained by substituting Eqs (4.92) and (4.102) into Eqs (4.90) and (4.91). When the minor axis 2b of the ellipse approaches zero, the elliptic hole reduces to a slit crack of length 2a. In this case, the Green’s function can be obtained by considering that c0 = d0 = ca = da = a / 2 . When b=0, Eq (4.102) reduces 5 § § d /c · 1 · -1 -1 f1 = - ln ¨ 1 - a a ¸ q f - ¦ ln ¨1 ¸¸ B [(LM ) - I )BIbq f ¨ z a za 0 ¹ b=1 © © z a zb 0 ¹ 5

+¦ b=1

(4.103)

§ 1 · -1 B (LM ) -1 BIbq f ln ¨1 ¨ z z ¸¸ 0 a b © ¹

where az a = za + za2 - a 2 ,

az a 0 = za 0 + za2 0 - a 2

(4.104)

Eq (4.103) indicates that f1 is independent of Pc. This means that the Green’s functions for a magnetoelectroelastic plate with a crack are not related to Pc. 4.8 Green’s functions for thermomagnetoelectroelastic problems In the previous chapter, applications of Green’s function to thermopiezoelectric problems were presented. Extension of the procedure to including a magnetic field is described in this section. The discussion here follows the results presented in [12,13]. 4.8.1 Basic equation for thermomagnetoelectroelastic problems For a thermomagnetoelectroelastic solid where all fields are functions of x1 and x2 only, the complete set of governing equations for coupled thermo-electro-magnetoelastic problems is defined by Eqs (1.98)-(1.101). Using the shorthand notation described in Section 1.5.1 and assuming that no free electric charge, electric current, body force, and heat source exist, Eqs (1.98) and (1.99) can be rewritten as [12,13]: qi ,i = 0, together with

P iJ ,i = 0

(4.105)

135


qi = - kijT, j ,

P iJ = EiJMnU M ,n - ciJ T

(4.106)

in which l ij , J £ 3, ° ciJ = ® ci , J = 4, ° n , J = 5, ¯ i

(4.107)

The general solution to Eq (4.105) is in the same form as that of Eq (3.1), but the vectors here have five components. 4.8.2 Green’s function for half-plane problems 4.8.2.1 Green’s function for thermal field. The half-plane boundary considered here is in the vertical direction (x1=0 on the boundary in our analysis, see Fig. 4.2). In the analysis the boundary faces of the half-plane are assumed to be thermal-insulated, free of traction force, external electric current and charge. The boundary condition along the half-plane boundary can thus be written as

J=ĳ=0

(4.108)

The half-plane solution can be obtained by considering the full-space solution plus some modification term to satisfy the condition on the boundary of the half-plane [Eq (4.107)]. To this end, the general solution for temperature and heat-flow function can be assumed in the form [12] T = 2 Re[ g ¢( z t* )] = 2 Re[ f 0 ( z t* ) + f1 ( z t* )]

(4.109)

J = 2k Im[ g ¢( zt* )] = 2k Im[ f 0 ( zt* ) + f1 ( zt* )]

(4.110)

where zt* = zt / p1* , and f0 is in the form [12] f 0 ( z t* ) = q 0 ln( z t* - z t*0 )

(4.111)

with q0 defined by Eq (3.6). For the half-plane in the zt* = zt / p1* system, the perturbation function can be assumed in the form f1 ( z t* ) = q1 ln( z t* - z t*0 )

(4.112)

Substituting Eqs (4.111) and (4.112) into Eq (4.110), the condition (4.108)1 yields Im[q0 ln( x 2 - z t*0 ) + q1 ln( x 2 - z t*0 )] = 0

(4.113)

Noting that z t* = x 2 on the half-plane boundary and Im( f ) = - Im( f ) , we have Im[q0 ln( x 2 - z t*0 )] = - Im[q 0 ln( x 2 - z t*0 )]

(4.114)

q1 = q 0

(4.115)

Eq (4.113) now yields

The function g in Eq (3.1) can then be obtained by integrating f0 and f1 with respect to zt, which yields g ( zt ) = q0 f * ( zt* , zt*0 ) + q0 f * ( zt* , zt*0 )

(4.116)

136


where the function f * is defined by Eq (3.7). 4.8.2.2 Green’s function for magnetoelectroelastic field. As treated previously, the general solution of the thermomagnetoelectroelastic problem can be written as

U = U p + Uh , ĳ = ĳ p + ĳh

(4.117)

where subscripts ‘p’ and ‘h’ refer, respectively, to the particular and homogeneous solutions. From Eqs (3.1) and (3.2) the particular solution of a magnetoelectroelastic field induced by thermal loading can be written as

U p = 2 Re[cg ( zt )],

ĳ p = 2 Re[dg ( zt )]

(4.118)

To satisfy the boundary condition (4.108), the homogeneous solution Uh and jh can be assumed in the form [12]

U h = 2 Re[cq0 f * ( zk* , zt*0 ) + cq0 [ f * ( zk* , zt*0 )]

(4.119)

ĳ h = 2 Re[dq0 f * ( zk* , zt*0 )] + dq0 f * ( zk* , zt*0 )]

(4.120)

where zk* is defined by Eq (4.57). For simplicity, denote f ( zk* ) = q0 f * ( zk* , zt*0 ) + q0 f * ( zk* , zt*0 )

(4.121)

Substitution of Eqs (4.119)-(4.121) into (3.2)2, and then into (4.108)2, leads to

q = -B -1d

(4.122)

Thus Green’s functions for the magnetoelectroelastic field of the half-plane problem can be written as U = 2 Re[- Af (z * )B -1d + cg ( z t )], ĳ = 2 Re[-Bf (z * )B -1d + dg ( z t )] (4.123) where f (z * ) = diag[ f ( z1* ) f ( z 2* ) f ( z 3* ) f ( z 4* ) f ( z 5* )]. 4.8.3 Green’s function for bimaterial problems We now consider a bimaterial solid whose interface is on x2-axis (x1=0). It is assumed that the left half-plane (x10) is occupied by material 2 (Fig. 4.4). They are rigidly bonded together so that T (1) = T ( 2 ) , J (1) = J ( 2) ,

U (1) = U ( 2 ) ,

ĳ (1) = ĳ ( 2 ) , at x1=0

(4.124)

where the superscripts (1) and (2) label the quantities relating to materials 1 and 2 respectively. 4.8.3.1 Green’s function for thermal field in bimaterial solids. For a bimaterial subjected to a line temperature discontinuity Tˆ and a line heat source h*, both located in the left half-plane at z 0 ( x10 , x 20 ) as shown in Fig. 4.4, the general solution for the bimaterial solid can be assumed in the form [12] T (1) = 2 Re[ f 0 ( zt(1)* ) + f1 ( zt(1)* )], J(1) = 2k (1) Im[ f 0 ( zt(1)* ) + f1 ( zt(1)* )] , x1 < 0 , (4.125) T ( 2) = 2 Re[ f 2 ( z t( 2 )* )],

J ( 2) = 2k ( 2) Im[ f 2 ( z t( 2)* )] ,

x1 > 0

(4.126)

where the function f0 is again given in Eq (4.111). To satisfy the interface condition (4.124)1,2, the functions f1 and f2 are taken as

137


f1 ( z t(1)* ) = q1 ln( z t(1)* - z t(01)* ) ,

(4.127)

f 2 ( z t( 2 )* ) = q 2 ln( z t( 2 )* - z t(01)* )

(4.128)

With the substitution of Eqs (4.111), (4.127), and (4.128) into Eqs (4.125) and (4.126), the continuity condition (4.124)1,2 provides q1 =

k (1) - k ( 2) q0 , k ( 2 ) + k (1)

q2 =

2k (1) q0 k (1) + k ( 2 )

(4.129)

Therefore the function g for the present bimaterial problem can be written in the form * (1)* g1 ( zt(1)* ) = q0 f * ( zt(1)* , zt(1)* , zt(1)* 0 ) + q1 f ( zt 0 )

(4.130)

g 2 ( zt(2)* ) = q2 f * ( zt(2)* , zt(1)* 0 )

(4.131)

4.8.3.2 Green’s function for magnetoelectroelastic field in bimaterial solids. To use the condition (4.124)3,4 we first consider the particular solution due to the thermal field. Using Eqs (4.130) and (4.131), the particular solution for the magnetoelectroelastic field can be written as (1)* (1) * (1)* U (1) ) = 2 Re[q0c(1) f * ( zt(1)* , zt(1)* , zt(1)* p ( zt 0 ) + c q1[ f ( zt 0 )] ,

(4.132)

(1)* (1) * (1)* ĳ (1) ) = 2 Re[q0d (1) f * ( zt(1)* , zt(1)* , zt(1)* p ( zt 0 ) + d q1 f ( zt 0 )]

(4.133)

for x1 < 0 , and (2)* U (2) ) = 2 Re[c(2) q2 f * ( zt(2)* , zt(1)* p ( zt 0 )] ,

(4.134)

(2)* ĳ (2) ) = 2 Re[d (2) q2 f * ( zt(2)* , zt(1)* p ( zt 0 )]

(4.135)

for x1 > 0 . For the same reason as in Section 3.3.2, a corrective solution needs to be constructed in such a way that when it is superimposed on the particular solutions (4.132)-(4.135) the interface condition (4.124) will be satisfied. Owing to the fact that f(zk) and g(zt) have the same rule affecting U and j in Eqs (3.1) and (3.2), possible function forms come from the partition of solution g(zt). This is f1 ( za(1)* ) = f * ( za(1)* , zt(1)* f 2 ( za(1)* ) = f * ( za(1)* , zt(1)* 0 ), 0 )

(4.136)

(2)* * (2)* (1)* f3 ( za(2)* ) = f * ( za(2)* , zt(1)* 0 ) , f 4 ( za ) = f ( za , zt 0 )

(4.137)

Thus the resulting expressions of U(i) and j(i) can be given as

U (1) = 2 Re{A (1) [ f1 ( za(1)* ) q11 + f 2 ( za(1)* ) q12 ] + q0c(1) f * ( zt(1)* , zt(1)* 0 ) + c(1) q1 f * ( zt(1)* , zt(1)* 0 )},

ĳ(1) = 2 Re{B (1) [ f1 ( za(1)* ) q11 + f 2 ( za(1)* ) q12 ] + q0d (1) f * ( zt(1)* , zt(1)* 0 ) + d (1) q1 f * ( zt(1)* , zt(1)* 0 )} for x1 < 0 , and

(4.138)

(4.139)


138

U (2) = 2 Re{A (2) [ f 3 ( za(2)* ) q 21 + f 4 ( za(2)* ) q 22 ] + c(2) q2 f * ( zt(2)* , zt(1)* 0 )},

j

(2)

= 2 Re{B

(2)

f3 ( z

(2)* a

) q 21 + f 4 ( z

(2)* a

(2)

*

(2)* t

) q 22 ] + d q2 f ( z

,z

(1)* t0

)}

(4.140)

(4.141)

for x1 > 0 . The substitution of Eqs (4.138)-(4.141) into Eq (4.124)3,4 yields q11 = M 1 [(B ( 2) -1d ( 2) - A ( 2 ) -1c ( 2 ) )q 2 - (B ( 2 ) -1d (1) - A ( 2 ) -1c (1) )q 0 ]

(4.142)

q 21 = M 2 [(B (1) -1d (1) - A (1) -1c (1) )q 0 - (B (1) -1d ( 2 ) - A (1) -1c ( 2 ) )q 2 ]

(4.143)

q12 = -M 1 (B ( 2) -1d (1) - A ( 2) -1c (1) )q1

(4.144)

q 22 = M 2 (B (1) -1d (1) - A (1) -1c (1) )q1

(4.145)

where M 1 = (B ( 2) -1B (1) - A ( 2) -1 A (1) ) -1 , M 2 = (B (1) -1B ( 2) - A (1) -1 A ( 2) ) -1 . Thus, the explicit expression of the thermomagnetoelectroelastic Green’s functions for the bimaterial solid can be obtained by substituting Eqs (4.142)-(4.145) into Eqs (4.138)-(4.141). 4.8.4 Green’s function for elliptic hole problems The hole problem to be considered here is illustrated in Fig. 2.7, showing an infinite two-dimensional thermomagnetoelectroelastic plate containing an elliptic hole (with the limit b=0, a crack) with semi-major axis a and semi-minor axis b. The plate is subjected to a line heat source h* and a line temperature discontinuity Tˆ , both located at z0(x10,x20) (Fig. 2.7). In this subsection, the hole is assumed to be filled with a homogeneous gas (air or vacuum) of permittivity (kc) and permeability (mc), where superscript ‘c’ refers to the quantities associated with the hole medium [12]. Therefore, induced electric and magnetic fields exist in the hole, denoted by Wc, and can be governed by the equations Ñ 2 fc = 0 ,

Ñ 2yc = 0 ,

in Wc

(4.146)

with the constitutive relations Dic = kc Eic = -kc f,ci ,

Bic = kc H ic = -mc y ,ci

i=1,2 in Wc

(4.147)

The general solutions to Eqs (4.146) and (4.147) are thus given by fc = 2 Re[ f1c ( z )] ,

y c = 2 Re[ f 2c ( z )] ,

z = x1 + ix 2

(4.148)

Defining the potential, electric displacement and magnetic induction function f c ( z)½ fc ½ U c = ® c ¾ = 2 Re ® 1c ¾ , ¯y ¿ ¯ f2 ( z)¿

ªb c ĳ c = 2 Re{« 1 ¬0

0 º f1c ( z ) ½ ¾} »® b2c ¼ ¯ f 2c ( z ) ¿

(4.149)

where b1c = -i kc , b2c = -imc , we have P c1 = {D1c , B1c }T = -ĳ,2c ,

P c2 = {D2c , B2c }T = ĳ,1c

(4.150)

Using the above expression, the magnetoelectroelastic boundary conditions along the surface of the hole can be written as

139


gT U = U c , ĳ = gĳc P ij ® 0

on G

at infinity

(4.151) (4.152)

where g is defined by Eq (4.89). As in Section 2.8, single-valued mapping (2.182) z = c0z 0 + d 0 z 0-1

(4.153)

is used here. The Green’s functions for thermal field here are the same as those in Section 3.5. We will not repeat that here. To satisfy the conditions (4.151), the solutions U and j can be chosen to be Eqs (3.64) and (3.65), i.e. 4

U = 2 Re[¦ A f k (z a ) q k + cg (z t )]

(4.154)

k =1 4

ĳ = 2 Re[¦ B f k (z a ) q k + dg (z t )]

(4.155)

k =1

and Uc and jc are assumed in the form 4

U c = 2 Re ¦ [ f kc (z 0 ) q ck ]

(4.156)

k =1

4

ĳc = 2 Re ¦ [B c f kc (z 0 ) q ck ]

(4.157)

k =1

where fk is defined by Eq (3.63), and ª -i k c Bc = « ¬ 0

0 º », -imc ¼

f kc (z 0 ) = Fk (z 0 )

(4.158)

in which Fk is given in Eq (3.61). The condition (4.151) provides 2 Re[gT ( Aqi + cri )] = 2 Re[q ci ] , 2 Re[Bqi + dri ] = 2 Re[gB c q ci ] , (i=1-4) (4.159) where r1 = ct q0 , r2 = d t q0 , r3 = d t q0 , r4 = ct q0

(4.160)

Solving Eqs (4.159) yields qi = (B - gB c gT A) -1 (gB c gT c - d)ri , q ci = gT ( Aqi + cri ) , (i=1-4)

(4.161)

A crack of length 2a can be formed by letting the minor axis b of the ellipse approach zero. The solutions for a crack in an infinite magnetoelectroelastic plate can then be obtained from the formulation above by setting b=0. In this case, Eqs (2.162), (3.54), and (4.153) are reduced to za =

a a z a + z a-1 ) , ca = da = ( 2 2

(4.162)


140

zt =

z=

a a z t + z t-1 , ct = d t = 2 2

(

)

a z 0 + z 0-1 , 2

(

)

c0 = d 0 =

a 2

(4.163)

(4.164)

4.8.5 Green’s functions for a wedge or semi-infinite crack Consider an infinite magnetoelectroelastic wedge whose symmetric line extends infinitely in the negative direction of the x1-axis (Fig. 4.5). The wedge angle is denoted by 2q0. The solid is subjected to a temperature discontinuity Tˆ and a heat source h*, both at a point z0 ( x10 , x20 ) as shown in Fig. 4.5. The wedge faces are assumed to be thermal-insulated, free of force, external electric current and charge. The boundary condition along the two wedge faces can thus be written as J=ĳ=0 (4.167) 4.8.5.1 General solution for thermal field As in Section 4.8.2, the general solution for temperature and heat-flow function can be assumed in the form T = 2 Re[ g ¢( z t )] = 2 Re[ f 0 (z t ) + f 1 (z t )]

(4.168)

J = 2k Im[ g ¢( z t )] = 2k Im[ f 0 (z t ) + f1 (z t )]

(4.169)

where f0 is chosen to represent the solutions associated with the unperturbed thermal fields and f1 is a function corresponding to the perturbed field due to the wedge boundary. Here z t and z t 0 are related to zt and zt 0 (= x10 + p1* x20 ) by the mapping functions [13] z t = z 1t / l and z t 0 = z 1t 0/ l

(4.170)

where l= p /(2p - 2q 0 ) and z t = x + ih maps the wedge boundary q = ±(p - q 0 ) in the zt-plane into the imaginary axis in the z t -plane (Fig. 4.5). Therefore the solution domain is mapped into the right half plane axis in the z t -plane. For a given loading condition, the function f0 can be obtained easily since it is related to the solution of homogeneous media. When an infinite space is subjected to a line heat source h* and the thermal analog of a line temperature discontinuity T0, both located at (x10,x20), the function f0 can be chosen in the form f 0 (z t ) = q0 ln(z t - z t 0 )

(4.171)

and q0 is defined by Eq (3.6). For the half-plane in the z t = x + ih system, the perturbation function can be assumed in the form [13]: f1 (z t ) = q1 ln(-z t - z t 0 ) (4.172) Substituting Eqs (4.171) and (4.172) into Eq (4.169), the condition (4.167)1 yields

141


Im[q 0 ln(ih - z t 0 ) + q1 ln(-ih - z t 0 )] = 0

(4.173)

Noting that Im( f ) = - Im( f ) , we have (4.174) Im[q0 ln(ih - z t 0 )] = - Im[q0 ln(-ih - z t 0 )] Equation (4.173) now yields q1 = q 0 (4.175) Having obtained the solution of f0 and f1, the function g ¢( z t ) can now be written as g ¢( z t ) = q 0 ln( z tl - z tl0 ) + q0 ln(- z tl - z tl0 ) Substituting Eq (4.176) into Eqs (4.168) and (4.169) yields T = 2 Re[q 0 ln( z tl - z tl0 ) + q 0 ln(- z tl - z tl0 )]

(4.177)

J = 2k Im[q 0 ln( z tl - z tl0 ) + q0 ln(- z tl - z tl0 )]

(4.178)

(4.176)

The function g in Eq (3.1) can thus be obtained by integrating the functions of f0 and f1 with respect to zt, which leads to g ( zt ) = q0 fˆ1 ( zt ) + q0 fˆ2 ( zt )

(4.179)

where fˆ1 ( zt ) = lzt (-1 + 2 F1 (1/ l,1,1 + 1/ l, ztl / ztl0 ) + zt ln( ztl - ztl0 ) fˆ2 ( zt ) = lzt (-1 + 2 F1 (1/ l,1,1 + 1/ l, - ztl / ztl0 ) + zt ln(- ztl - ztl0 )

(4.180)

with 2F1(a,b,c,z) being a hypergeometric function defined in [16] 2

F1 (a, b, c, z ) =

b -1 1t G (c ) (1 - t ) c -b -1 dt ³ G(b)G(c - b) 0 (1 - tz ) a

(4.181)

or series expansion 2

F1 (a, b, c, z ) = 1 +

¥ (a ) n (b) n n ab a (a + 1)b(b + 1) 2 z+ z +" = ¦ z 1!c 2!c(c + 1) n = 0 n!(c ) n

(4.182)

where G(x) is a gamma function. 3.2 Green’s functions for magnetoelectroelastic fields The general solution of the thermomagnetoelectroelastic problem can be written as

U = U p + Uh , ĳ = ĳ p + ĳh

(4.183)

where subscripts ‘p’ and ‘h’ refer, respectively, to the particular and homogeneous solutions. From Eqs (3.1) and (3.2) the particular solution of magnetoelectroelastic field induced by thermal loading can be written as

U p = 2 Re[cg ( z t )], ĳ p = 2 Re[dg ( z t )]

(4.184)

To satisfy the boundary condition (4.167)2 along the wedge boundary, a corrective isothermal solution is required so that, when superimposed on the particular


142

thermomagnetoelectroelastic solution, the surface conditions (4.167)2 will be satisfied. Owing to the fact that f(za) and g(zt) have the same order of effect on stress and electric displacement in Eqs (3.1) and (3.2) [note that the term Bf(z)q in Eq (3.2) is now replaced by B( f1 ( za ) q1 + f 2 ( za ) q 2 ) ], possible function forms come from the partition of g(zt). They are f1 ( za ) = lza (-1 + 2 F1 (1/ l,1,1 + 1/ l, zal / ztl0 ) + za ln( zal - ztl0 ) f 2 ( za ) = lza (-1 + 2 F1 (1/ l,1,1 + 1/ l, - zal / ztl0 ) + za ln(- zal - ztl0 )

(4.185)

The substitution of Eqs (4.184) and (4.185) into (3.2), and then into (4.167), leads to

q1 = -B -1dq 0 ,

q 2 = -B -1dq0

(4.186)

Substituting Eq (4.186) into Eqs (3.1) and (3.2), the Green’s functions can then be written as U = 2 Re[- A( f1 ( za ) q0 + f 2 ( za ) q0 )B -1d + cg ( zt )], ĳ = 2 Re[-B( f1 ( za ) q0 + f 2 ( za ) q0 )B -1d + dg ( zt )]

(4.187)

When q0=0, i.e, l=1/2, Eq (4.187) represents Green’s functions for the case of a semi-infinite crack in an infinite magnetoelectroelastic solid. h

x2

z t - plane

· z t 0 (x 0 , h 0 )

· z0 (x10 , x20 ) q0 q0

o

x1

o

x

Fig. 4.5: Wedge-shaped magnetoelectroelastic plate and its mapping in the z-plane 4.9 Dynamic Green’s functions of magnetoelectroelastic media In this section, the potential approach is used to derive dynamic Green’s functions of magnetoelectroelastic problems. The discussion follows the results presented in [14]. 4.9.1 Dynamic potentials Consider a linear transversely isotropic magnetoelectroelastic medium with the poling direction along the x3-axis being perpendicular to the isotropic plane, whose motion in Euclidean space is described in terms of the independent variables x = {xi} and t. The governing equations of electric and magnetic fields are still given by Eqs (1.98)2,3, while the elastic equilibrium equations (1.98)1 are, in this case, modified as

143


s ij , j + bi = rui

(4.188)

where r is the mass density of the magnetoelectroelastic medium, and the dots over a variable represent temporal derivatives. For simplicity, the discussion below considers an infinite two-dimensional ‘quasi-plane’ magnetoelectroelastic medium with transversely isotropic symmetry only. All field quantities depend only on the plane space vector r =(x,y) in the plane of transverse isotropy. These assumptions mean that

U = {u1 , u 2 , u 3 , f, y} = {u1 , u 2 , u 3 , f, y}( x, y ) ¶ (×) = 0 ¶x3

(4.189)

In view of Eq (4.189), Eqs (4.188), (1.59)2 and (1.82)1 can be reduced to ¶s11 ¶s12 ¶ 2u + + b1 = r 21 ¶x ¶y ¶t ¶s12 ¶s22 ¶ 2u + + b2 = r 22 ¶x ¶y ¶t ¶s13 ¶s 23 ¶ 2u + + b3 = r 23 ¶x ¶y ¶t ¶D1 ¶D2 + + be = 0 ¶x ¶y ¶B1 ¶B2 + + bm = 0 ¶x ¶y

(4.190)

Substituting Eqs (1.90)-(1.92) into Eq (4.190) leads to the following governing equation T (Ñ,

¶ )U + F = 0 ¶t

(4.191)

where ¶ ª º TI (Ñ, ) 0 « » ¶ ¶t T (Ñ, ) = « » ¶ » ¶t « TII (Ñ, ) 0 «¬ ¶t »¼ ª ¶ 2 c11 - c12 ¶ 2 ¶2 -r 2 « c11 2 + 2 ¶x ¶t 2 ¶y ¶ TI (Ñ, ) = « 2 « ¶t c11 + c12 ¶ « 2 ¶x¶y ¬

c11 + c12 ¶ 2 2 ¶x¶y

º » » ¶ 2 c11 - c12 ¶ 2 ¶2 » -r 2 » c11 2 + ¶y ¶t ¼ 2 ¶x 2

(4.192)

(4.193)


144

ª ¶2 « c44Ñ 2 - r 2 ¶t « ¶ TII (Ñ, ) = « e15Ñ 2 ¶t « e15Ñ 2 « ¬« Ñ2 =

e15Ñ 2 -k11Ñ 2 -a11Ñ 2

º e15Ñ 2 » » -a11Ñ 2 » -m11Ñ 2 » » »¼

¶2 ¶2 + ¶x 2 ¶y 2

(4.194)

(4.195)

F = {b1 , b2 , b3 , be , bm }T

(4.196)

Introducing following notations U I = {u1 , u2 }T , U II = {u3 , f, y}T , FI = {b1 , b2 }T ,

FII = {b3 , be , bm }T

(4.197)

Eq (4.191) can be further written as ¶ ª º 0 «TI (Ñ, ¶t ) » U I ½ FI ½ « »® ¾+® ¾ = 0 U F ¶ « TII (Ñ, ) » ¯ II ¿ ¯ II ¿ 0 ¶t ¼» ¬«

(4.198)

where the operator matrix T is divided into two parts in which TI corresponds to a pure elastic part which acts on u1 and u2 in the isotropic plane and TII corresponds to the magnetoelectroelastic coupling of u3, f and y. ˆ (r, t ) of Eq (4.191) are thus determined The magnetoelectroelastic potentials G by the following differential equations[14]: T (Ñ,

¶ ˆ )G (r, t ) + I 5Q s (r )d(t ) = 0 ¶t

(4.199)

where I5 represents a 5´5 unit matrix, Qs(r) denotes the characteristic function of the inclusion S embedded in the medium, characterizing the spatial distribution of generalized unit forces. Based on this explanation, the dynamic magnetoelectroelastic potential can be regarded as generalized displacement due to a source distribution of generalized forces represented by an inclusion. On the other hand, the dynamic Green’s function G (r, t ) for Eq (4.191) can be defined as T (Ñ,

¶ )G (r, t ) + I 5d 2 (r )d(t ) = 0 ¶t

(4.200)

Making use of Eqs (4.199) and (4.200), dynamic potential and Green’s function can be related by ˆ (r, t ) = ¥ G (r - r ¢, t )Q (r ¢)dr ¢ = G (r - r ¢, t )dr ¢ G s ³ ³ -¥

S

(4.201)

where S denotes the surface of the inclusion under consideration. Similar to the character of the operator T in Eq (4.192), the dynamic potential can also be decomposed as

145


ªˆ 0 º ˆ (r, t ) = «G I (r, t ) G » ˆ (r, t ) » G «¬ 0 ¼ II

(4.202)

ˆ (r, t ) and G ˆ (r, t ) satisfy, respectively[14] where G I II ¶ ˆ )G I (r, t ) + I 2 Q s (r )d(t ) = 0, ¶t ¶ ˆ TII (Ñ, )G II (r , t ) + I 3 Q s (r )d(t ) = 0 ¶t TI (Ñ,

(4.203)

ˆ (r, t ) is written in matrix form as As presented in [14,17], G I ª ¶2 « 2 ˆ (r, t ) = 1 « ¶x G I 2 r« ¶ « ¶x¶y ¬

¶2 º » 1 ¶x¶y » ˆ ˆ I 2 gˆ 2 ( h - h2 ) + 2 ¶ » 1 c66 ¶y 2 »¼

(4.204)

where functions hˆ1 , hˆ2 and gˆ 2 are determined by the equations [14] § 1 ¶2 · ¨ Ñ 2 - 2 2 ¸ gˆ i + Q s (r )d(t ) = 0 ci ¶t ¹ ©

(i=1,2)

(4.205)

§ 1 ¶2 · ¶2 ˆ hi + Q s (r )d(t ) = 0 Ñ ¨ 2 ¸ ci2 ¶t 2 ¹ ¶t 2 ©

(i=1,2)

(4.206)

where ¶ 2 hˆi (i=1,2) ¶t 2

(4.207)

c c11 c -c , c1 = 66 , c66 = 11 12 r r 2

(4.208)

gˆ i = c1 =

The solution of the second equation of Eq (4.203) can be obtained by taking inverse of the operator matrix [14] ˆ (r, t ) = T * (Ñ, ¶ ) gˆ (r, t )I G 3 II II II ¶t where TII* (Ñ,

(4.209)

¶ ¶ ) is the adjoint matrix of TII (Ñ, ) and gˆ II (r, t ) satisfies ¶t ¶t QII (Ñ,

in which QII (Ñ,

¶ ) gˆ II (r, t ) + Q s (r )d(t ) = 0 ¶t

(4.210)

¶ ¶ ) is the determinant of TII (Ñ, ) : ¶t ¶t

QII (Ñ,

§ ¶ ¶ º 1 ¶2 · ª 2 ) = det «TII (Ñ, ) » = ( k11m11 - a11 )cˆ44 ¨ Ñ 2 - 2 2 ¸ Ñ 22 ¶t ¶t ¼ c3 ¶t ¹ ¬ ©

(4.211)


146

cˆ44 = c44 +

e152 k11 + e152 m11 - 2e15e15a11 2 k11m11 - a11 c3 =

cˆ44 r

(4.212)

(4.213)

Using two new functions defined by [14] 2 gˆ 3 = ( k11m11 - a11 )cˆ44Ñ 22 gˆ II

§ 1 ¶2 · 2 gˆ s = ( k11m11 - a11 )cˆ44 ¨ Ñ 2 - 2 2 ¸ Ñ 2 gˆ II c3 ¶t ¹ ©

(4.214)

two new equations can be induced from Eq (4.210) as § 1 ¶2 · Ñ ¨ 2 ¸ gˆ 3 + Q s (r )d(t ) = 0 c32 ¶t 2 ¹ © Ñ 2 gˆ s + Q s (r )d(t ) = 0

(4.215)

Considering the character of Eq (4.215)2, it is reasonable to assume that gˆ s = gˆ 4 (r )d(t ) . Thus we have Ñ 2 gˆ 4 + Q s (r ) = 0

(4.216)

¶ ) into Eq (4.209) and making use of Eq ¶t ˆ (r, t ) becomes (4.214), the dynamic potential G II

Substituting the expression of TII* (Ñ,

ª a* b* c* º ˆ (r, t ) = « b 0 0 » gˆ 3 (r, t ) G II « * » a cˆ «¬ c* 0 0 »¼ * 44 ª0 0 0º ½ gˆ (r, t ) 2 1 - ««0 1 0 »» ®m11 gˆ 4 (r )d(t ) + 3 [e15 - m11 (cˆ44 - c44 )]¾ ˆ a* c 44 ¿ «¬0 0 0 »¼ ¯ ª0 1 + «« 0 a* «¬ 0 ª0 1 - ««0 a* «¬0

0 0º ½ gˆ (r, t ) 0 1 »» ®a11 gˆ 4 (r )d(t ) + 3 [e15e15 - a11 (cˆ44 - c44 )]¾ ˆ c 44 ¿ 1 0 »¼ ¯ 0 0º ½ gˆ (r, t ) 2 0 0 »» ® k11 gˆ 4 (r )d(t ) + 3 [e15 - k11 (cˆ44 - c44 )]¾ ˆ c 44 ¿ 0 1 »¼ ¯

(4.217)

where 2 a* = k11m11 - a11 , b* = e15m11 - e15a11 , c* = e15 k11 - e15a11

4.9.2 Green’s functions

(4.218)

147


Based on the discussion above, the dynamic Green’s function G (r, t ) is governed by the functions gi (i=1,2,3,4) and hi (i=1,2) which are similar to the functions gˆ i (i=1,2,3,4) and hˆ (i=1,2). The functions gi (i=1,2,3,4) and hi (i=1,2) are, in this case, i

determined by replacing Qs(r) by d(r): § 1 ¶2 · Ñ ¨ 2 ¸ gi + d(r )d(t ) = 0 ci2 ¶t 2 ¹ © § 1 ¶2 · ¶2 ¨ Ñ 2 - 2 2 ¸ 2 hi + d(r )d(t ) = 0 ci ¶t ¹ ¶t ©

(i=1,2,3)

(4.219)

(i=1,2)

(4.220)

Ñ 2 g 4 + d(r ) = 0

(4.221)

Eq (4.221) is the Laplace equation and its fundamental solution can be found in many text books as g 4 (r ) = -

1 ln r 2p

where r = ( x 2 + y 2 )1/ 2 . The functions gi(r,t) and hi(r,t) have been presented in [18] as 1 H (t - r / ci ) gi (r , t ) = , (i=1,2,3) 2p t 2 + r 2 / ci2

(4.222)

(4.223)

· H (t - r / ci ) ª § ct c 2t 2 r2 º 2 «t ln ¨ + », (i=1,2) (4.224) 1 t ¸ ¸ 2p r2 c2 » «¬ ¨© r ¹ ¼ where H(x) is the Heaviside step function defined by 1 if x > 0 H ( x) = ® (4.225) ¯0 if x < 0 Similar to Eqs (4.204) and (4.217), the corresponding expressions of dynamic Green’s function are given by hi (r , t ) =

ª ¶2 ¶2 º « 2 » ¶x¶y » 1 ¶x 1 (h1 - h2 ) + G I (r, t ) = « 2 I 2 g2 2 » c r« ¶ ¶ 66 « 2 » ¬ ¶x¶y ¶y ¼

(4.226)

148


ª a* b* c* º g (r, t ) G II (r, t ) = «« b* 0 0 »» 3 a cˆ «¬ c* 0 0 »¼ * 44 ª0 0 0 º ½ g (r, t ) 2 1 « [e15 - m11 (cˆ44 - c44 )]¾ - «0 1 0 »» ®m11 g 4 (r )d(t ) + 3 a* cˆ44 ¿ «¬0 0 0 »¼ ¯ (4.227) ª0 0 0º ½ g (r, t ) 1 [e15e15 - a11 (cˆ44 - c44 )]¾ + «« 0 0 1 »» ®a11 g 4 (r )d(t ) + 3 a* cˆ44 ¿ «¬ 0 1 0 »¼ ¯ ª0 0 0 º ½ g (r, t ) 2 1 « - «0 0 0 »» ® k11 g 4 (r )d(t ) + 3 [e15 - k11 (cˆ44 - c44 )]¾ a* cˆ44 ¿ «¬0 0 1 »¼ ¯

By means of the Fourier transform, expressions of gi (i=1,2,3,4) and hi (i=1,2) in the space-frequency domain can also be obtained [14,17], namely w + ix 1 g j (r , w + ix) = iH 0(1) ( r ), 4 cj h j ( r , w + ix) = -

w + ix 1 H 0(1) ( r ), 4(w + ix) 2 cj

( j=1,2,3)

(4.228)

( j=1,2)

(4.229)

where i = -1 and the infinitesimal damping constant x ® 0+ is introduced in order to guarantee the causality of the derived Green’s function [18] and H 0(1) (×) is the Hankel function of the first kind. On the basis of the expressions of g j (r , w + ix) and h j (r , w + ix), the dynamic Green’s functions in the space-frequency domain can be obtained as ª ¶2 ¶2 º « 2 » ¶x¶y » (1) i « ¶x G I (r, t ) = {H 0 (k2 r ) - H 0(1) (k1r )} + k22 I 2 H 0(1) (k2 r ) (4.230) rw2 « ¶ 2 ¶2 » « 2 » ¬ ¶x¶y ¶y ¼


149

ª a* b* c* º 2 (1) ik H (k r ) G II (r, t ) = «« b* 0 0 »» 3 0 23 4a*rw «¬ c* 0 0 »¼ ª0 0 0º m ½ ik 2 H (1) (k r ) 1 « - «0 1 0 »» ® - 11 ln r + 3 0 2 3 [e152 - m11 (cˆ44 - c44 )]¾ a* 2p 4rw ¿ «¬0 0 0 »¼ ¯ (4.231) ª0 0 0º 2 (1) a ½ ik H (k r ) 1 + «« 0 0 1 »» ®- 11 ln r + 3 0 2 3 [e15e15 - a11 (cˆ44 - c44 )]¾ 2p a* 4rw ¿ «¬ 0 1 0 »¼ ¯ ª0 0 0 º k ½ ik 2 H (1) (k r ) 1 « - «0 0 0 »» ® - 11 ln r + 3 0 2 3 [e152 - k11 (cˆ44 - c44 )]¾ a* 2p 4rw ¿ «¬0 0 1 »¼ ¯

where k j = w / c j + ix (j=1,2,3) and the result when x ® 0+ has been used, which is omitted in the derivation process. It is easy to prove that the dynamic functions GI(r,w) and GII(r,w) defined in Eqs (4.230) and (4.231) fulfil TI (Ñ, w)G I (r, w) + I 2 d(r ) = 0, TII (Ñ, w)G II (r, w) + I 3d(r ) = 0

(4.232)

References [1] Pan E, Three dimensional Green’s functions in anisotropic magnetoelectroelastic bimaterials. Z Angew Math Phys, 53, 815-838, 2002 [2] Soh AK, Liu JX, Hoon KH. Three-dimensional Green's functions for transversely isotropic magnetoelectroelastic solids. Int J Non Sci Num Simulation, 4, 139-148, 2003 [3] Huang JH, Chiu YH, Liu HK. Magneto-electro-elastic Eshelby tensors for a piezoelectric-piezomagnetic composite reinforced by ellipsoidal inclusions. J Appl Phys, 83, 5364-5370, 1998 [4] Liu J, Liu X, Zhao Y. Green’s functions for anisotropic magnetoelectroelastic solids with an elliptical cavity or a crack. Int J Eng Sci, 39, 1405-1418, 2001 [5] Li JY, Magnetoelectric Green’s functions and their application to the inclusion and inhomogeneity problems, Int J Solids Struct, 39, 4201-4213, 2002 [6] Wang X and Shen YP, The general solution of three-dimensional problems in magnetoelectroelastic media, Int J Eng Sci, 40, 1069-1080,2002 [7] Ding HJ, Jiang A, Hou PF and Chen WQ, Green’s function for two-phase transversely isotropic magnetoelectroelastic media, Eng Ana Boun Elements, 29, 551-561, 2005 [8] Hou PF, Ding HJ and Chen JY, Green’s function for transversely isotropic magnetoelectroelastic media, Int J Eng Sci, 43, 826-858, 2005 [9] Alshits VI, Kirchner HOK and Ting TCT, Angularly inhomogeneous piezoelectric piezomagnetic magnetoelectric anisotropic media, Philos Mag Lett,

150


71, 285-288, 1995 [10] Jiang X and Pan E, Exact solution for 2D polygonal inclusion problem in anisotropic magnetoelectroelastic full-, half-, and bimaterial-planes, Int J Solids Struct, 41, 4361-4382, 2004 [11] Qin QH, Green’s functions of magnetoelectroelastic solids with a half-plane boundary or bimaterial interface, Philos Mag Lett, 84, 771-779, 2004 [12] Qin QH, 2D Green’s functions of defective magnetoelectroelastic solids under thermal loading, Eng Ana Boun Elements, 29, 577-585, 2005 [13] Qin QH, Green’s functions of magnetoelectroelastic solids and applications to fracture analysis, pp93-106, in Proc. of 9th Int Conf on Inspection, Appraisal, Repairs & Maintenance of Structures, Fuzhou, China on 20-21 October, 2005, Ed. Ren WX, K.C. Gary Ong and John S.Y. Tan, CI-Premier PTE LTD [14] Chen P, Shen YP and Tian XG, Dynamic potentials and green’s functions of a quasi-plane magnetoelectroelastic medium with inclusion, Int J Eng Sci, 44, 540-553, 2006 [15] Pan E and Tonon F, Three-dimensional Green’s functions in anisotropic piezoelectric solids, Int J Solids Struct, 37, 943-958, 2000 [16] Seaborn JB, Hypergeometric functions and their applications, New York, Springer-Verlag, 1991 [17] Michelitsch TM, Levin VM and Gao HJ, Dynamic potentials and Green’s functions of a quasi-plane piezoelectric medium with inclusion, Proc R Soc London A, 458, 2393-2415, 2002 [18] Levin VM, Michelitsch TM and Gao HJ, Propagation of electroacoustic waves in the transversely isotropic piezoelectric medium reinforced by randomly distributed cylindrical inhomogeneities, Int J Solids Struct, 39, 5013-5051

151

Chapter 5 Boundary element method for piezoelectricity 5.1 Introduction In the previous four chapters we described Green’s functions in piezoelectric and piezomagnetic materials. Applications of these Green’s functions to boundary element method (BEM) are described in this and next chapter. It should be noted that application of BEM to piezoelectric materials has been the subject of fruitful scientific attention by many a distinguished researcher (e.g. Lee and Jiang [1], Lee [2], Denda and Mansukh [3], Sanz et al [4], and others). Lee and Jiang [1] derived the boundary integral equation of piezoelectric media by the method of weighted residuals for plane piezoelectricity. Lu and Mahrenholtz [5] presented a variational boundary integral equation for the same problem. Ding et al [6] developed a boundary integral formulation which is efficient for analyzing crack problems in piezoelectric material. Rajapakse [7] discussed three boundary element methods (direct boundary method, indirect boundary element method and fictitious stress-electric charge method) in coupled electroelastic problems. Xu and Rajapakse [8] and Rajapakse and Xu [9] extended the formulations in [7,8] to the case of piezoelectric solids with various defects (cavities, inclusions, cracks, etc.). Liu and Fan [10] established a boundary integral equation in a rigorous way and addressed the question of degeneration for problems of cracks and thin shell-like structures. Pan [11] derived a single domain BE formulation for 2D static crack problems. Denda and Lua [12] developed a BEM formulation using Stroh’s formalism to derive the fundamental solution but did not show any numerical results. Davi and Molazo [13] used the known subdomain method to formulate a multidomain BEM, well suited for crack problems, by modeling crack faces as boundaries of the different subdomains. Groh and Kuma [14] developed a direct collocation boundary element code with a subdomain technique for analyzing crack problems and calculating stress intensity factors. Zhao et al [15] presented a boundary integral-differential model for interfacial cracks in 3D piezoelectric solids. Liew and Liang [16] presented Green’s functions for transversely isotropic piezoelectric bimaterials based on the solution of Ding et al [17] for distinct eigenvalue materials and applied them to the solution of the problem of cavities in an infinite domain. Sanz et al [4] presented a general BEM for 3D fracture problems in piezoelectric materials. Khutoryaansky et al [18] introduced a BE formulation for time-dependent problems of linear piezoelectricity. Time-dependent BEM for piezoelectricity was also discussed in [19,20]. Recently, Ding and Jiang [21] extended the BEM approach to 2D problems in magnetoelectroelastic media. In addition, for the application of BEM in piezoelectric materials, the work presented in [22-29] should also be mentioned. 5.2 Boundary integral equation 5.2.1 Governing equations In this section, the theory of piezoelectricity presented in Chapter 1 is briefly summarized for deriving the corresponding boundary integral equation. Under the condition of a static deformation, the governing equations for a linear and generally anisotropic piezoelectric solid consist of [11] (i) Equilibrium equations

sij , j + bi = 0, (ii) Constitutive relations

Di ,i + be = 0

(5.1)


152

sij = cijkl e kl - emij Em ,

Dk = ekij eij + k mk Em

(5.2)

(iii) Elastic strain-displacement and electric field-potential relations e ij =

1 (u i , j + u j ,i ) , E i = -f ,i 2

(5.3)

(iv) Boundary conditions ui = ui ti = sij n j = ti

on Gu ½ ¾, on Gt ¿

f=f Dn = Di ni = -qs

on G f ½ ¾ on G D ¿

(5.4)

5.2.2 Boundary integral equation for piezoelectricity Several approaches have been used in the literature to establish boundary integral equations of piezoelectric materials, such as the variational approach [5], weighted residual approach [1,2,19,30], and Betti’s reciprocity theorem [11,14,18,20,22,31,32]. A brief discussion of the above three approaches which have been widely used in BE analysis is given here. (1) Variational method. The generalized variational principle below is due to Lu and Mahrenholtz [5] and is applicable to any boundary value problems of piezoelectric materials. It is based on a modified functional with six kinds of independent variable, i.e. displacements and electric potential in the domain, and displacements, tractions, electric potential and surface charge on the boundary. Since all the boundary conditions considered have been introduced into the framework of the modified variational expression, no additional constraints have to be satisfied when assuming the variables on the boundary. The functional used for deriving boundary integral equation is of the form 1 1 P (u, f, u , t, f , qs ) = ³ [ cijkl ui , j uk ,l - kij f,i f, j + eikl f,i uk ,l - bi ui - be f]d W W 2 2 - ³ [ti (ui - ui ) - q s (f - f )]d G - ³ ti ui d G + ³ qs f d G G

Gt

GD

(5.5)

- ³ ti (ui - ui )d G + ³ qsi (f - f)d G, Gu

Gf

in which the variables assumed to be independent are: (a) displacement field, u, with components ui; (b) electric potential in the domain, f; (c) boundary displacement field, u , with components, ui ; (d) boundary traction, t, with components, ti ; (e) boundary electric potential, f ; (f) boundary surface charge, qs . Lu and Mahrenholtz [5] showed that the stationary condition of the functional (5.5) leads to Eqs (5.1), (5.4), and ui = ui , f = f , t j = tj , Dn = - qs on G

(5.6)

The proof can be made by taking variations with respect to the independent variables as

153

Boundary element method for piezoelectricity

dP (u, f, u , t, f , qs ) = - ³ [(sij , j + bi )dui + ( Di ,i + be )df]d W W

+ ³ [(ti - ti )dui + ( Dn + qs )df + (f - f )dq s - (ui - ui )dtj ]d G G

+ ³ (ti - ti )dui d G - ³ Gt

-³

Gu

GD

(qs - qs )df d G

(5.7)

(ui - ui )dti d G + ³ (f - f)dq s d G = 0 Gf

Therefore, the Eular equations for expression (5.7) are Eqs (5.1), (5.4), and (5.6), as the quantities dui , df , dui , df , dti , and dq s may be arbitrary. This indicates that the boundary conditions (5.4) and (5.6) have been included in the framework of the variational formulation (5.5). The independent boundary variables ui , f , ti , and qs in Eq (5.5) are, in this case, not required to satisfy the boundary conditions (5.6). Therefore, the continuity requirements of the boundary variables are relaxed in the variational expression (5.5) which may be convenient for interpolating boundary variables. To convert the variational expression (5.5) into a boundary expression, the first integration in Eq (5.5) is transformed into a boundary integral. This can be done by integrating by parts and making use of relations (5.2) and (5.3). Then Eq (5.5) can be rewritten as 1 1 P (u, f, u , t, f , qs ) = ³ [- sij , j ui - Di ,i f - bi ui - be f]d W W 2 2 1 [(ti - 2ti )ui + ( Dn + 2qs )f]d G - ³ ti ui d G + ³ qs f d G Gt GD 2 ³G qs f d G, ti ui d G - ³ + ³ ti ui d G - ³ q s fd G + ³ Gu

Gf

(5.8)

G-Gf

G-Gu

The functional (5.8) can be taken as the basis for the deriving BE formulation. (2) Weighted residual method. The method of weighted residuals can also be used to establish the boundary integral equations presented above. As indicated in [1,2], the electroelastic weighted residual statement for the boundary value problem (5.1) and (5.4) may be expressed as

³

W

(sij ,i + b j )ui*d W = ³ (ti - ti )ui*d G + ³ (ui - ui )ti*d G Gt

Gu

(5.9)

* * * where ui and ti = s ij n j are the displacement and traction respectively corresponding to the weighted field. By carrying out integration by parts and using the constitutive equation (5.2) and noting that - qs* = Di*ni , Eq (5.9) can be written as [1]

³

W

(s*ij , j ui + Di*,i f + bi ui* + be f* )d W = ³ (ti*ui - ti ui* )d G + ³ (ui ti* - ui*ti )d G Gt

Gu

+ ³ (qs* f - f*qs )d G - ³ Gf

GD

(qs f* - qs*f)d G

(5.10)

Lee and Jiang [1] mentioned that Eq (5.10) can also obtained by choosing Eqs (5.1)2 and (5.4)2 instead of Eqs (5.1)1 and (5.4)1 in the weighted residual statement (5.9). To convert Eq (5.10) into a boundary integral equation, choose the weighted functions such that they satisfy Eq (5.1), i.e., s*ij , j + d( x - x )ei = 0, Di*,i + d( x - x ) = 0 (5.11)


154

where ei is the unit vector in xi-direction. Then, the first two terms of the first integral in Eq (5.10) reduce to * * (5.12) ³ W sij , j ui d W = -ui (x)ei , ³ W D j , j fd W = -f(x) Further, if the point force in each direction and the point charge are taken as independent of each other, the weighting functions can be written in the following form u *j = uij* (x - x)ei , t *j = tij* (x - x)ei , f* = f*i (x - x)ei , qs* = qsi* (x - x)ei (5.13) After some mathematical manipulation [1,2], the boundary integral equation for the three displacement components and for the electric potential can be obtained as follows uI (x) + ³ (tij* (x, x)u j (x)d G(x) - ³ qsi* (x, x)f(x)d G(x) G G = ³ uij* (x, x)b j (x)d W (x) + ³ f*i (x, x)be (x)d W (x) W W + ³ uij* (x, x)t j (x)d G(x) - ³ f*i (x, x)qs (x)d G(x) G

(5.14)

G

where u = {uI } = {u1 , u2 , u3 , - f} . Eq (5.14) is the piezoelectric analogue of Somigliana’s identity and gives values of the displacement and electric potential at any internal point in terms of the boundary field values, the body forces and charge, and the fundamental solution. Note that if x is taken to the boundary, the integrals have a singularity which needs to be taken into account. For example, if x lies on a smooth boundary, uI (x) is replaced by uI (x) / 2. The boundary integral equation (5.14) applies for static elastoelastic problems only. To solve time-dependent thermo-piezoelectric problems, a boundary integral equation developed in [19] is presented below. Note that for time-dependent thermopiezoelectric problems, Eqs (2.375) and (5.1)-(5-4) are both space- and timedependent. To eliminate the time variable in these equations, applying the Laplace transform [33] with respect to time to Eqs (2.375) and (5.1) leads to sij , j + bi = 0 (5.15) Di ,i + be = 0 (5.16) hi ,i + T0b s + bt = 0 (5.17) where the symbol ( ) over a variable indicates that the variable is expressed in the transform domain, b is the Laplace transformed variable, and s(0) is assumed to be zero without loss of generality. In the transformed domain, the electroelastic weighted residual statement for the boundary value problem (5.15) and (5.16) may be expressed as [19] * (5.18) ³ (sij , j + bi )ui d W = 0 W

where ui* is the displacement in the transformed domain corresponding to the weighting field. By using integration by parts, divergence theorem and constitutive equations, the following expression can be obtained


³

W

(s*ij , j ui + Di*,i f + bi ui* + be f* )d W + ³ (ui*ti - ti *ui - f*qs + fqs* )d G G + ³ (c m Em* T - c m EmT * + l ij eij* T - l ij eijT * )d W = 0

155

(5.19)

W

In order to eliminate the last two volume integrals as required in the boundary integral equation, apply the weighting function field and actual field to the linear heat conduction equation such that kijT,ij - T0b s - bt = 0 (5.20) kijT,ij* - T0bs * - bt* = 0 (5.21) Then, making use of Eqs (5.20) and (5.21), and after somewhat lengthy mathematical manipulation, the following expression is obtained [19] * * * * ³ W (cm EmT - cm EmT + lij eijT - lij eijT )d W (5.22) 1 ª = (bt*T - bt T * )d W - ³ (qn*T - qn T * )d G º ³ W G ¬ ¼ bT0

where qn = - k ij T, j ni . Substituting Eq (5.22) into Eq (5.19) and applying the inverse Laplace transform to Eq(5.19), yields [19] * * * * ³ (s ij , j ´ui + D i,i ´ f + bi ´ ui + be ´ f )d W W

+

1 T0

³

W

(bt* ´ T - bt ´ T * - qn* ´ T - qn ´ T * )d W

(5.23)

+ ³ (qs* ´ f + qs ´ f* - ti ´ ui* - ti* ´ ui )d G = 0 G

in which the notation of the convolution integral t

f ´ g = ³ f (t - t) g (t)d (t) 0

and the property of the inverse Laplace transform L-1 (bf ) = f , ( L-1 ( f ) = f )

(5.24)

(5.25)

have been used. To make the first three volume integrals vanish, Jiang [19] considered the weighting field as an infinite thermo-piezoelectric body subjected to unit forces, electric charge, and pulse heat source equal to the following three states separately: (1) bi* = d(x - x) H (t )dij e j , be* = bt* = 0 (5.26) (2) be* = d(x - x) H (t ), bi* = bt* = 0 (3) bt* = d(x - x)d(t ), bi* = be* = 0

(5.27) (5.28)

Jiang [19] also mentioned that the heat source with respect to time t is subjected to the Dirac delta function while the other two loads are subjected to the step function of time. This is purely a mathematical consideration and can be easily explained by noting that bt* in the integral does not involve the time derivative while both


156

s *ij , j and D i*,i do. If no body forces, volume charge, and heat source are considered, the

boundary integral equation (5.23) becomes 1 uI (x) + ³ [t Ij* (x, x) ´ u j (x) - qsI* (x, x) ´ f(x) + qnI* (x, x) ´ T (x)]d G(x) W T0 1 = ³ [uIj* (x, x) ´ t j (x) - f*I (x, x) ´ qs (x) + TI* (x, x) ´ qn (x)]d G(x) G T0

(5.29)

where u = {uI } = {u1 , u2 , u3 , - f, - T / T0 }, uij* represents the displacement in the jth direction at field point x due to a point force in the ith direction applied at the source point x interior to G, u4i* denotes the ith displacement at x due to a point electric charge at x , u5i* stands for the ith displacement at x due to a pulse heat source acting at time zero at x , and so on. (3) Reciprocity theorem method. In addition to the variational approach and weighted residual methods discussed above, the reciprocity theorem can also be used to establish the boundary integral equation. The presentation below is from the development in [12,18,31,32,34]. (a) reciprocity theorem in electroelastic problems [12,18] Consider two electroelastic states, namely State 1: {U I(1) }T = {u1(1) , u2(1) , u3(1) , f(1) }T ; {bI(1) }T = {b1(1) , b2(1) , b3(1) , be(1) }T ;

(5.30)

{t I(1) }T = {t1(1) , t2(1) , t3(1) , -qs(1) }T . State 2: {U I(2) }T = {u1(2) , u2(2) , u3(2) , f(2) }T ; {bI(2) }T = {b1(2) , b2(2) , b3(2) , be(2) }T ;

(5.31)

{t I(2) }T = {t1(2) , t2(2) , t3(2) , -qs(2) }T . The first state represents the solution to piezoelectric problems with finite domains and general loading conditions; the second state is of an artificial nature and represents the fundamental solution to the case of a fictitious infinite body subjected to a point force or a point charge. Further, introduce the compatible field {e ij , ui , Ei , f } , which satisfies Eqs (5.3). The principle of virtual work is given by

³

W

bI(1)U I d W + ³ t I(1)U I d G = ³ (sij(1) e ij - Di(1) E i )d W G

W

(5.32)

On the other hand, for a linear piezoelectric solid, one can show that the following reciprocal property of Betti type holds sij(1) eij(2) - Di(1) Ei(2) = sij(2) eij(1) - Di(2) Ei(1)

(5.33)

By substituting Eq (5.33) into Eq (5.32), the following reciprocal relation can be obtained

³

W

bI(1)U I(2) d W + ³ t I(1)U I(2) d G = ³ bI(2)U I(1) d W + ³ t I(2)U I(1) d G G

W

G

(5.34)

(b) static boundary integral equation To convert Eq (5.34) into a boundary integral equation, assume that state (1) is the actual solution for a body W with the boundary G and state (2) is the solution for a


157

fictitious infinite body subjected to a point force at x in the xm direction with no bulk charge distribution (i.e. be(2) = 0) , namely bi(2) (x) = d(x - x)dim , (m=1, 2, 3) (5.35) The displacement, electric potential, stress, and electric displacement induced by the abovementioned point force were discussed in Chapter 2. Using the solution given in Chapter 2, the variables ui(2) , f(2) , sij(2) , and Di(2) can be written in the form

ui(2) = uij* (x, x)e j , f(2) = f*j (x, x)e j = u4* j (x, x)e j , * * sij(2) = S*ijm (x, x)em = [cijkl ukm ,l ( x, x) - elij u4 m ,l ( x, x)]em , * * * Di(2) = Dim (x, x)em = [eikl ukm ,l ( x, x) + k il u4 m ,l ( x, x)]em Making use of Eq (5.4), the traction and surface charge can be obtained as * ti(2) = tim (x, x)em = S*ijm (x, x)n j em , * -qs(2) = t4*m (x, x)em = Dim (x, x)ni em

(5.36)

(5.37)

Substituting all the above quantities associated with state (2) into Eq (5.34) yields ui (x) = ³ u *Ji (x, x)bJ (x)d W(x) - ³ t Ji* (x, x)u J (x)d G(x) W G (5.38) + ³ u *Ji (x, x)t J (x)d G(x) G

where i=1,2,3, J=1-4, u4 = -f , b4 = be , and t4 = -qs . Next, assume that state (2) of the fictitious infinite body is subjected to a point charge at x with no body force distribution, i.e. be(2) (x) = d(x - x), bi(2) (x) = 0 (m=1, 2, 3) (5.39) The resulting displacement and electric potential induced by this point charge are given by ui(2) = ui*4 (x, x), (5.40) * f(2) = f*4 (x, x) = u44 ( x, x ) Substituting the solution in Eq (5.40) into Eqs (5.2) and (5.4), we have * sij(2) = S*ij 4 (x, x) = cijkl uk* 4,l (x, x) - elij u44, l ( x, x), * Di(2) = Di*4 (x, x) = eikl uk* 4,l (x, x) + kil u44, l ( x, x), ti(2) = ti*4 (x, x) = S*ij 4 (x, x)n j , * (x, x) = Di*4 (x, x)ni - qs(2) = t44 Substituting Eqs (5.39)-(5.41) into Eq (5.34), we obtain f(x) = ³ u*J 4 (x, x)bJ (x)d W(x) - ³ t J* 4 (x, x)u J (x)d G(x) W G + ³ u *J 4 (x, x)t J (x)d G(x) G

Combining Eq (5.38) with Eq (5.42), we have

(5.41)

(5.42)


158

uI (x) = ³ u *JI (x, x)bJ (x)d W(x) - ³ t JI* (x, x)u J (x)d G(x) W G + ³ u *JI (x, x)t J (x)d G(x)

(5.43)

G

where I, J =1-4. Obviously, the outcome of Eq (5.43) is identical to that of Eq (5.14) which was obtained from the weighted residual method. Making use of Eqs (5.2) and (5.43), the corresponding stresses and electric displacements are expressed as * * P iJ (x) = ³ DKiJ (x, x)bK (x)d W(x) - ³ S KiJ (x, x)u K (x)d G(x) W G (5.44) * + ³ DKiJ (x, x)t K (x)d G(x) G

where

¶t * (x, x) ¶u * (x, x) * * S KiJ (x, x) = EiJMn MK (x, x) = EiJMn MK , DKiJ (5.45) ¶xn ¶xn The integral representation formula for the generalized traction components can be obtained from Eqs (5.4) and (5.44) as t J (x) = ³ VIJ* ( x, x)bI ( x)d W( x) - ³ WIJ* (x, x)uI (x) d G(x) W G (5.46) + ³ VIJ* (x, x)t I (x)d G(x) G

where * VIJ* (x, x) = DIkJ (x, x)nk (x), * WIJ* (x, x) = S IkJ (x, x)nk (x)

(5.47)

It can be seen from Eq (5.38), (5.42), (5.47) that to obtain the fields at internal points, the boundary data of traction, displacement, electric potential and the normal component of electric displacement need to be known throughout the boundary G. For this purpose, one can examine the limiting forms of Eqs (5.38), (5.42) and (5.47) as x approaches the boundary. To properly circumvent the singular behaviour when x approaches x , Chen and Lin [22] assumed a singular point x on the boundary surrounded by a small hemispherical surface of radius e, say G e , centred at the point x with e ® 0. Since the asymptotic behaviour of Green’s function in piezoelectric solids at r = x - x ® 0 is mathematically similar to that of uncoupled elasticity, Eqs (5.38), (5.42), and (5.47) can be rewritten as [22] cki uk (x) = ³ u *Ji (x, x)bJ (x)d W(x) - ³ t Ji* (x, x)u J (x)d G(x) W G + ³ u *Ji (x, x)t J (x)d G(x)

(5.48)

G

bf(x) = ³ u *J 4 (x, x)bJ (x)d W(x) - ³ t J* 4 (x, x)u J (x)d G(x) W G + ³ u *J 4 (x, x)t J (x)d G(x)

(5.49)

cki tk (x) = ³ VJi* (x, x)bJ (x)d W(x) - ³ WJi* (x, x)u J (x)d G(x) W G + ³ VJi* (x, x)t J (x)d G(x)

(5.50)

G

G

159


-bqs (x) = ³ VJ*4 (x, x)bJ (x)d W(x) - ³ WJ*4 (x, x)u J (x)d G(x) W G + ³ VJ*4 (x, x)t J (x)d G(x)

(5.51)

G

where x Î G, and the coefficients cki and b are defined as cki (x) = d ki + lim ³ tki* (x, x)d G(x) e® 0

(5.52)

Ge

* b(x) = 1 + lim ³ t44 (x, x)d G(x)

(5.53)

Ge

e® 0

In the field of BEM, the coefficients cki and b are usually known as boundary shape coefficients, cii (x) = b(x) =1 if x ÎW, cii (x) = b(x) =1/2 if x is on the smooth boundary [29]. Using the concept of boundary shape coefficients, Eq (5.43) can be rewritten as cKI uK (x) = ³ u*JI (x, x)bJ (x)d W(x) - ³ t JI* (x, x)u J (x)d G(x) W G (5.54) + ³ u JI* (x, x)t J (x)d G(x) G

where x Î G, and cK 4 = c4 K = bd K 4 . It is more convenient to work with matrices rather than continue with the indicial notation. To this effect the generalized displacement U, traction T, body force b, and boundary shape coefficients C are defined as [6] t1 ½ b1 ½ u1 ½ ª c11 c12 c13 °t ° °b ° «c c c °u ° ° ° ° ° ° ° U = ® 2 ¾ , T = ® 2 ¾ , b = ® 2 ¾ , C = « 21 22 23 « c31 c32 c33 ° u3 ° ° t3 ° °b3 ° « °¯-f°¿ °¯ -qs °¿ °¯be °¿ ¬0 0 0

0º 0 »» 0» » b¼

T

(5.55)

Similarly, the fundamental solution coefficients can be defined in matrix form as * ª u11 « * u U* = « 21 * «u31 « * «¬u41

T

u12* u13* u14* º * * * » u22 u23 u24 » , * * * » u32 u33 u34 * * * » u42 u43 u44 »¼

ª t11* «* t T* = « 21 * «t31 «* «¬t41

t12* t13* t14* º * * * » t22 t23 t24 » * * * » t32 t33 t34 * * * » t42 t43 t44 »¼

T

(5.56)

where uij* and tij* (i=1,2,3) represent the elastic displacement and traction, respectively, in the jth direction at a field point x due to a unit point load acting at the source point x , u4* j and t4* j denote the jth elastic displacement and traction, respectively, at x due to a unit electric charge at x , f*i and -qsi* (i=1,2,3) stand for the electric potential and surface charge, respectively, at x corresponding to a unit force in the ith direction at x , f*4 and -qs*4 represent the electric potential and surface charge, respectively, at x due to a unit electric charge at x . Using the matrix notation defined in Eqs (5.55) and (5.56), the integral equation (5.54) can be written in matrix form as

CU = ³ U*bd W - ³ T*Ud G + ³ U*Td G W

G

G

(5.57)

Using Eq (5.57), the generalized displacement at point, say point xi , can be obtained


160

by enforcing a point load at the same point. In this case Eq (5.57) becomes C( x i )U( x i ) = ³ U* ( x i , x )b( x )d W( x ) - ³ T* ( x i , x )U( x )d G( x ) + ³ U* ( x i , x )T( x )d G( x ) W

G

G

(5.58) (c) boundary integral equation for crack problems [34] Consider a finite three-dimensional piezoelectric solid containing several cracks. Let L be the union of all cracks, L+ and L- represent the union of the positive side and negative side of the cracks. The displacements and electric potential can, in this case, still be evaluated using Eq (5.43) if the boundary G is replaced by G+L, i.e. uI (x) + ³ t JI* (x, x)u J (x)d G(x) + ³ + - t JI* (x, x)u J (x)d G(x) L +L G (5.59) = ³ u*JI (x, x)bJ (x)d W(x) + ³ u JI* (x, x)t J (x)d G(x) W G + ³ + - u*JI (x, x)t J (x)d G(x) L +L

To simplify the integral equation (5.59), denote the elastic displacements and electric potential discontinuities as Du = u + - u - for J = 1, 2,3 (5.60) Du J = ® j +j -j J =4 ¯ Df = f - f for Then, using the relations t IJ* (x, x + ) = -t IJ* (x, x - ) = t IJ*+ (x, x) and uIJ* (x, x + ) = uIJ* (x, x - ) , Eq (5.59) can be simplified as uI (x) + ³ t JI* (x, x)u J (x)d G(x) + ³ + t JI*+ (x, x)Du J (x)d G(x) L G (5.61) = ³ u*JI (x, x)bJ (x)d W(x) + ³ u JI* (x, x)t J (x)d G(x) W

G

As indicated in [34], for a finite piezoelectric solid with several embedded flat cracks, there two parts of the boundary. One is the external boundary G, and the other is the union of cracks L. If the source point x is taken to the boundary G, the boundary integral equation can be derived from Eq (5.61) as cIK uK (x) + ³ t JI* (x, x)u J (x)d G(x) + ³ + t JI*+ (x, x)Du J (x)d G(x) L G xÎG (5.62) = ³ u*JI (x, x)bJ (x)d W(x) + ³ u *JI (x, x)t J (x)d G(x) W

G

When the source point x is taken to the boundary L, using the free boundary conditions of traction and surface charge on crack faces, the hypersingular integral equation can be obtained as *+ * ³>L+ SKiJ (x, x)DuK (x)d G(x) + ³ G S KiJ (x, x)uK (x)d G(x) x Î L+ (5.63) * * = ³ DKiJ (x, x)bK (x)d W(x) + ³ DKiJ (x, x)t K (x)d G(x) W

where

>³

G

means that the integral must be interpreted as a finite-part integral. The

first integral in Eq (5.63) has the singular order r -3 and a hypersingular order. Eqs (5.62) and (5.63) form the basis for developing the BE formulation. (d) dynamic boundary integral equation [31,32]

161


The reciprocity theorem approach described above can also be used to derive the dynamic boundary integral equation. To doing this, consider the following two dynamic electroelastic states State 1: {U I(1) }T = {u1(1) , u2(1) , u3(1) , f(1) }T ; {bI(1) }T = {b1(1) , b2(1) , b3(1) , be(1) }T ;

(5.64)

(1) {t I(1) }T = {t1(1) , t2(1) , t3(1) , -qs(1) }T , sij(1), j + bi(1) = rui(1) , D (1) j , j + be = 0 .

State 2: {U I(2) }T = {u1(2) , u2(2) , u3(2) , f(2) }T ; {bI(2) }T = {b1(2) , b2(2) , b3(2) , be(2) }T ;

(5.65)

(2) {t I(2) }T = {t1(2) , t2(2) , t3(2) , -qs(2) }T , sij(2), j + bi(2) = rui(2) , D (2) =0. j , j + be

State (1) again represents the solution to piezoelectric problems with finite domains and general loading conditions. The second state stands for the fundamental solution to the case of an infinite piezoelectric solid subjected to an impulsive point force and an impulsive point charge. Furthermore, define a function X ij as X ij = ³ bk(i ) ´ uk( j ) d W + ³ tk(i ) ´ uk( j ) d G + ³ be(i ) ´ f( j ) d W - ³ qs( i ) ´ f( j ) d G W

G

W

G

(5.66)

where the first two terms represent mechanical work and the last two terms represent electrical work. It is easy to show that X12 = X 21 . The proof can be accomplished by recognizing that

³

G

(tk(1) ´ uk(2) - qs(1) ´ f(2) )d G = ³ ( - bk(1) ´ uk(2) + ruk(1) ´ uk(2) + sij(1) ´ ui(2) , j )d W W

+ ³ (-be(1) ´ f(2) + Di(1) ´ f,(2) i )d W

(5.67)

W

as a consequence of the divergence theorem and the field equations (5.1)-(5.4). Inserting this relation into Eq (5.66) with i=1 and j=2 leads to (1) (2) X12 = ³ (ruk(1) ´ uk(2) + sij(1) ´ ui(2) , j + Di ´ f,i ) d W W

(5.68)

Furthermore, for linear piezoelectricity the following relation holds true: (1) (2) (2) (1) (2) sij(1) ´ ui(2) ´ f,(1) , j + Di ´ f,i = sij ´ ui , j + Di i

(5.69)

ruk(1) ´ uk(2) = ruk(2) ´ uk(1)

(5.70)

X12 = X 21

(5.71)

and since

it follows that

Eq (5.71) is the starting point for derivation of the dynamic boundary integral equation for piezoelectricity using the reciprocity theorem approach. This derivation is based on two independent loading conditions for the second state, where a unit force and a unit charge are applied at a source point x . We first consider that state (2) is subjected to a point force at x in the xm direction with no bulk charge distribution (i.e. be(2) = 0) , namely bi(2) (x) = d(x - x)d(t )dim , (m=1, 2, 3) (5.72) The displacement, electric potential, stress, and electric displacement induced by the abovementioned point force were discussed in Chapter 2. Using the solution given in


162

Chapter 2, the variables ui(2) , f(2) , sij(2) , and Di(2) can be written in the form

ui(2) = uij* (x, x, t )e j , f(2) = f*j (x, x, t )e j = u4* j (x, x, t )e j , * * sij(2) = S*ijm (x, x, t )em = [cijkl ukm ,l ( x, x, t ) - elij u4 m ,l ( x, x, t )]em , * * * Di(2) = Dim (x, x, t )em = [eikl ukm ,l ( x, x, t ) + k il u4 m ,l ( x, x, t )]em Making use of Eq (5.4), the traction and surface charge can be obtained as * ti(2) = tim (x, x, t )em = S*ijm (x, x, t )n j em , * - qs(2) = t4*m (x, x, t )em = Dim (x, x, t )ni em

(5.73)

(5.74)

Substituting all the above quantities associated with state (2) into Eq (5.71) leads to u j ( x, t ) = ³

G

t

³

0

uIj* (x, x, t - t)t I (x, t)d td G(x) t (x, x, t - t)uI (x, t)d td G(x)

-³

G

³

t * 0 Ij

+³

W

³

0

t

xÎW

(5.75)

[u Ij* (x, x, t - t)bI (x, t)d td W(x)

which is the representation formula for the elastic displacements. Similarly, the representation formula for the electric potential can be obtained by considering following loading condition: be(2) (x) = d(x - x)d(t ), bi(2) = 0 (m=1, 2, 3) (5.76) The resulting displacement and electric potential induced by this point charge are given by ui(2) = ui*4 (x, x, t ), (5.77) * f(2) = f*4 (x, x, t ) = u44 ( x, x, t ) Substituting the solution in Eq (5.77) into Eq (5.4), we have * sij(2) = S*ij 4 (x, x, t ) = cijkl uk* 4,l (x, x, t ) - elij u44, l ( x, x, t ), * Di(2) = Di*4 (x, x, t ) = eikl uk* 4,l (x, x, t ) + kil u44, l ( x, x, t ), ti(2) = ti*4 (x, x, t ) = S*ij 4 (x, x, t )n j , * - qs(2) = t44 (x, x, t ) = Di*4 (x, x, t )ni

(5.78)

Substituting Eqs (5.76)-(5.78) into Eq (5.71), we obtain f(x, t ) = ³

³

0

uI*4 (x, x, t - t)t I (x, t)d td G(x)

-³

G

³

t * 0 I4

+³

W

³

0

G

t

t (x, x, t - t)uI (x, t)d td G(x)

t

[u I*4 (x, x, t - t)bI (x, t)d td W(x)

Combining Eq (5.75) with Eq (5.79), we have

xÎW

(5.79)

163


u J ( x, t ) = ³

u IJ* (x, x, t - t)t I (x, t)d td G(x)

t

G

³

0

-³

G

³

t * 0 IJ

+³

W

³

0


t

xÎW

(5.80)

[u IJ* (x, x, t - t)bI (x, t)d td W(x)

where I, J =1-4. Making use of Eqs (5.2) and (5.80), the corresponding stresses and electric displacements are expressed as t * P iJ (x, t ) = ³ ³ DKiJ (x, x, t - t)bK (x, t)d td W(x) W

0

t

-³

G

³

0

+³

G

³

0

t

* S KiJ (x, x, t - t)uK (x, t)d td G(x)

(5.81)

* DKiJ (x, x, t - t)t K (x, t)d td G(x)

The corresponding integral representation formula for the generalized traction components can be obtained from Eqs (5.4) and (5.81) as [31] W

³

0

VIJ* (x, x, t - t)bI (x, t)d td W(x)

-³

G

³

0

+³

G

³

0

t J ( x, t ) = ³

t

t

t

WIJ* (x, x, t - t)uI (x, t)d td G(x)

(5.82)

VIJ* (x, x, t - t)t I (x, t)d td G(x)

where * VIJ* (x, x, t - t) = DIkJ (x, x, t - t)nk (x), * WIJ* (x, x, t - t) = S IkJ (x, x, t - t)nk (x)

(5.83)

As with the treatment in Eqs (5.48)-(5.51), the dynamic boundary integral equations corresponding to Eqs (5.80) and (5.82) can be obtained by examining the limiting process when x ® G : cJK u K (x, t ) = ³ -³

G

+³ cJK t K (x, t ) = ³ -³ +³

³

G

³

t 0

uIJ* (x, x, t - t)t I (x, t)d td G(x)


t * 0 IJ

W

³

W

³ t

G

³

0

G

³

0

t

t 0 t 0

xÎG

(5.84)

xÎG

(5.85)

[u IJ* (x, x, t - t)bI (x, t)d td W(x) VIJ* (x, x, t - t)bI (x, t)d td W (x)

WIJ* (x, x, t - t)uI (x, t)d td G(x) VIJ* (x, x, t - t)t I (x, t)d td G(x)

5.2.3 Boundary integral equation for magnetoelectroelastic solid The boundary integral equations discussed in Section 5.2.2 above apply for electroelastic problems only. For magnetoelectroelastic materials, the corresponding boundary integral equation can be obtained similarly [21]. To this end, consider a


164

magnetoelectroelastic solid with the domain W and its boundary G. The basic equations governing this problem are Eqs (1.82), (1.83), (1.88), (1.89), (5.1), (5.3), and (5.4). Eq (5.54) still applies for this boundary value problem if we replace U, U*, T, T*, and b by [21] u1 ½ °u ° °° 2 °° U = ® u3 ¾ , ° -f ° ° ° °¯-y °¿ ª u11* « * «u21 * * U = «u31 « * «u41 «u * ¬ 51

where

t1 ½ °t ° °° 2 °° T = ® t3 ¾ , °-q ° ° s° °¯ m °¿

b1 ½ ª c11 c12 c13 °b ° «c c c °° 2 °° « 21 22 23 b = ® b3 ¾ , C = « c31 c32 c33 « °b ° «0 0 0 ° e° «¬ 0 0 0 °¯bm ¿° T

u12* u13* u14* u15* º * * * * » u22 u23 u24 u25 » * * * * » , u32 u33 u34 u35 * * * * » u42 u43 u44 u45 » * * * * » u52 u53 u54 u55 ¼

ª t11* «* «t21 * * T = «t31 «* «t41 «t * ¬ 51

t12* t13* t14* t15* º * * * * » t22 t23 t24 t25 » * * * * » t32 t33 t34 t35 * * * * » t44 t45 t42 t43 » * * * * » t52 t53 t54 t55 ¼

* c(x) = 1 + lim ³ t55 ( x, x ) d G ( x ) e®0

Ge

0 0 0 b 0

0º 0 »» 0» » 0» c »¼

T

(5.86)

T

(5.87)

(5.88)

and where u4* j and t4* j ( j=1-3) are, respectively, displacement components and surface tractions in the xj-direction at x due to a unit electric charge at x , u5* j and t5* j ( j=1-3) are, respectively, displacement components and surface tractions in the xj-direction at x due to a unit current at x , ui*4 , ti*4 , ui*5 , and ti*5 (i=1-3) are, respectively, electric potential, surface charge, magnetic potential, and surface magnetic induction at x due * * * * , u45 , t44 , and t45 are, respectively, electric potential, magnetic to a unit force at x , u44 potential, surface charge, and surface magnetic induction at x due to a unit electric * * * * , u55 , t54 , and t55 are, respectively, electric potential, magnetic charge at x , u54 potential, surface charge, and surface magnetic induction at x due to a unit current at x. 5.3 BE formulation The analytical solution to the boundary integral equations developed in Section 5.2 is not, in general, possible to obtain due to the complexity of the geometry, loading condition and the kernels. Therefore a numerical procedure via discretization of the boundary into a number of elements must be used to solve the equations. In the following, we consider first the BE implementation of Eq (5.8), and then discuss the discretization form of Eqs (5.48)-(5.51). 5.3.1 BE implementation of Eq (5.8) [5] In order to obtain a weak solution of Eq (5.8), the independent functions in Eq (5.8) are approximated as the product of known functions by unknown parameters. To eliminate the first domain integrals in Eq (5.8), the displacement u and electric potential f in the domain are expressed in terms of linear combinations of fundamental solutions:

165


u = U*uT su + U*fT s f , f = ĭ*uT su + ĭ*fT s f

(5.89)

where the notation in Eq (5.89) has the same form and meaning as in [5], i.e. su and s f are unknown parameters, U*u and U*f are the matrices whose coefficients are displacements at x due to a unit force acting in one of the xi directions and a unit electric charge at x on the boundary, respectively, while the coefficients of ĭ*u and ĭ*f are electric potentials at x due to a unit force acting in one of the xi directions and a unit electric charge at x on the boundary, respectively. The fundamental solutions given in Chapter 2 can be used for this purpose. Similarly, the boundary variables in Eq (5.8) may be interpolated as follows u = NTuu du + NTuf d f ,

f = NTf u du + NTff d f ,

t = NTtt pt + NTtqs p qs ,

qs = NTqs u pt + NTqs qs p qs

(5.90)

where Nij are the matrices whose components are nonsingular interpolation functions of the type used in standard BEs, and du , d f , pt , and p qs are unknown vectors. As u and f are defined by Eq (5.89), the first integrals in Eq (5.8) vanish if one excludes from the domain W the singularities occurring at the source points on the boundary. Substituting Eqs (5.89) and (5.90) into Eq (5.8) and using following notation

su ½ °du ½° ° pt °½ s = ® ¾, d = ® ¾, p = ® ¾ °¯d f °¿ ¯s f ¿ ¯°p qs ¿°

(5.91) T

ª U* ĭ*u º ªTu* ȍ*u º ª Fuu Fuf º F=« = ³ « *u dG » *»« * *» G U «¬ f ĭ f »¼ «¬Tf ȍf »¼ ¬ Ffu Fff ¼

(5.92)

T

ª U*u ĭ*u º ª N tt -N qs t º ªG uu G uf º G=« = » dG » ³G « * *»« ¬G fu G ff ¼ ¬« U f ĭf ¼» «¬ N tqs -N qs qs »¼ T

(5.93) T

ª N q t º ª N f u º ª N º ª N uu º ª Luu Luf º = ³ « tt » « » dG dG - ³ « s » « L=« » » G-Gu N G-Gf «¬ N qs qs »¼ «¬ N ff » ¬L fu L ff ¼ ¬« tqs ¼» «¬ N uf »¼ ¼

(5.94)

ª N q t º ª N º ªU º U = « u » = ³ « tt »ud G - ³ « s » fd G G G u f ¬« U f ¼» ¬« N tqs ¼» ¬« N qs qs ¼»

(5.95)

ª N f u º ª N uu º ª Tt º T=« »=³ « td G - ³ « »qs d G » Gt N Gf N «¬ ff » ¬«Tqs ¼» ¬« uf ¼» ¼

(5.96)

ª U*u ĭ*u º b ½ ªBu º dW B=« »=³ « * *»® ¾ W U ¬Bf ¼ ¬« f ĭf »¼ ¯be ¿

(5.97)

1 P = -sT B + sT Fs - sT Gp + pT Ld - dT T + pT U 2

(5.98)

we have


166

where Tu* , Tf* and ȍ*u , ȍ*f are the matrices whose components are the tractions and surface charges obtained from the displacements U*u , U*f and the electric potentials ĭ*u , ĭ*f , respectively.

Performing the vanishing variation of P with respect to s, d, and p, we have ˆ - Gp - B = 0 Fs

(5.99)

-G T s + Ld + U = 0

(5.100)

T

L p-T = 0

(5.101)

1 Fˆ = (FT + F) 2

(5.102)

where

The unknowns s and p can be obtained by solving Eqs (5.99) and (5.101). Then substituting the solution s into Eq (5.100), the final system of equations for the solution of the problem can be written as Kd = Q

(5.103)

ˆ , R = (G -1 )T L, Q = T + RT B - KL-1U K = RT FR

(5.104)

where

5.3.2 BE implementation for Eqs (5.48)-(5.51) In order to obtain a numerical solution of Eqs (5.48)-(5.51), as in the usual BE approach, the boundary G and the domain W are discretized into a series of boundary elements and internal cells, respectively. The boundary vectors U and T are defined at each element “j” in terms of their values at nodal points of the element as U = NU j , T = NTj

(5.105)

where Uj and Tj are the element nodal displacements, electric potential, tractions, and surface charge of dimensions 4´M for a three-dimensional electroelastic solid, M being the number of nodes within the element. The interpolation function matrix N is a 4´4M array of shape functions, i.e. [29] ª N1 0 0 0 «0 N 0 0 1 N=« « 0 0 N1 0 « ¬ 0 0 0 N1

" NM 0 " 0 NM

0 0

"

0

0

NM

"

0

0

0

0 º 0 »» 0 » » NM ¼

(5.106)

in which components Ni represent the shape function which is the same as that in conventional finite element formulation. For example, for an 8-node isoparametric element as shown in Fig. 5.1, the shape functions Ni (i =1-8) may be given in the form

N1 =

1 (1 - s )(1 - t )( - s - t - 1) , 4

N2 =

1 (1 + s )(1 - t )( s - t - 1) , 4

(5.107a)

N3 =

1 (1 + s )(1 + t ) ( s + t - 1) , 4

N4 =

1 (1 - s )(1 + t ) ( - s + t - 1) , 4

(5.107b)

167


N5 =

1 1 - s 2 (1 - t ) , 2

)

N6 =

1 (1 + s ) 1 - t 2 , 2

)

(5.107c)

N7 =

1 1 - s 2 (1 + t ) , 2

N8 =

1 (1 - s ) 1 - t 2 2

)

(5.107d)

(

(

)

(

(

t 2

5 1

1 1

1 6

s

8

1 4

7

3

Fig. 5.1 8-node isoparametric element

Substitution of the approximation (5.105) into Eqs (5.48)-(5.51) yields K Cu ( x i ) U u ( x i ) = ¦ s =1

}

U*u ( x, x i )b(x )d W( x )

Ws

{³ } b(x )f(x ) = ¦ {³ U (x, x )b(x)d W(x)} - ¦ {³ T (x, x ) Nd G( x)}U + ¦ {³ U ( x, x ) Nd G( x)}T C (x )T ( x ) = ¦{³ V ( x, x )b( x )d W( x )} - ¦{³ W ( x, x ) Nd G( x )}U + ¦{³ V ( x, x ) Nd G( x )}T -b( x ) q ( x ) = ¦{³ V ( x, x )b( x )d W( x )} - ¦{³ W ( x, x ) Nd G( x )}U + ¦{³ V ( x, x ) Nd G(x )}T NE

-¦ k =1

{³

{³

}

NE T ( x , x i ) Nd G ( x ) U k + ¦

* u

Gk

k =1

Gk

U ( x, x i )Nd G(x ) Tk

(5.109)

* u

K

i

i

* f

Ws

s =1

i

NE

NE

* f

Gk

k =1

i

k

k =1

* f

Gk

i

(5.110)

k

K

u

i

t

i

s =1

* t

Ws

i

NE

NE

k =1

* t

Gk

i

k

k =1

Gk

* t

i

(5.111)

k

K

i

s

i

Ws

s =1

* qs

i

NE

k =1

NE

Gk

* qs

i

k

k =1

Gk

* qs

i

(5.112)

k

where the summation from 1 to NE indicates summation over the NE boundary elements, G k is the surface of the kth element, and the summation from 1 to K indicates summation over the K internal cells where the body force integrals are to be computed. Eqs (5.109) and (5.110) are usually suitable for problems without cracks or


168

discontinuities, and Eqs (5.111) and (5.112) are effective for solving crack problems. The matrices U u , U *u , Tu* , U *f , Tf* , Tt , Vt* , Wt* , Vq*s , Wq*s are defined by ªu11* u1 ½ ° ° * « * U u = ®u2 ¾ , U u = «u12 °u ° «u13* ¯ 3¿ ¬

* u 21

* u 31

* u 22 * u 23

* u 32 * u 33

* ªt11* º u 41 «* » * * u 42 » , Tu = «t12 * » «t13* u 43 ¬ ¼

* * * U*f = {u14* , u24 , u34 , u44 },

* t 21

* t 31

* t 22 * t 23

* t 32 * t 33

* º t 41 * » t 41 » * » t 43 ¼

* * * Tf* = {t14* , t24 , t34 , t44 }

ªW11* W21* W31* W41* º ªV11* V21* V31* V41* º t1 ½ « » « » ° ° Tt = ®t 2 ¾ , Vt* = «V12* V22* V32* V42* » , Wt* = «W12* W22* W32* W41* » °t ° «W13* W23* W33* W43* » «V13* V23* V33* V43* » ¯ 3¿ ¬ ¼ ¬ ¼ Vq*s = {V14* , V24* , V34* , V44* },

Wq*s = {W14* , W24* , W34* , W44* }

(5.113)

Combining Eq (5.109) with Eq (5.110), and Eq (5.111) with Eq (5.112), we have K C(xi )U(xi ) = ¦ s =1

Ws

}

U* (x, xi )b(x)d W(x)

{³ C(x )T(x ) = ¦ {³ V (x, x )b(x)d W(x)} - ¦ {³ W (x, x )Nd G(x)}U + ¦ {³ NE

-¦ k =1

{³

{³

}

NE T (x, xi )Nd G(x) U k + ¦ *

Gk

Gk

k =1

}

U (x, xi ) Nd G( x) Tk *

(5.114)

K

*

i

i

i

Ws

s =1

NE

NE

*

k =1

i

Gk

k

k =1

Gk

}

V (x, xi )Nd G(x) Tk *

(5.115)

After performing the discretization with boundary elements and internal cells and carrying out the integrals, Eq (5.14) is reduced to a system of algebraic equations: N N K C(xi )U(xi ) + ¦ H ij U j = ¦ Gij T j + ¦ Bis( u )

(5.116)

N N K Cxi )T(xi ) + ¦ Wij U j = ¦ Vij T j + ¦ Bis(t )

(5.117)

j =1

j =1

j =1

s =1

j =1

s =1

where N is the number of nodes, and U j and T j are the generalized displacement and traction at node j. The inference matrices H, G, V, and W are evaluated by H ij = ¦ ³ T* (xi , x)N q (x)d G(x), Gij = ¦ ³ U* (xi , x)N q (x)d G(x) t

Gt

t

Gt

Vij = ¦ ³ V * (xi , x)N q (x)d G(x), Wij = ¦ ³ W* (ȟ i , x)N q (x)d G(x) t

Gt

t

Gt

(5.118)

where the summation extends to all the elements to which node j belongs and q is the number of order of the node j within element t, and ª Vt* º V = « * », «¬ Vqs »¼ *

ª Wt* º W =« *» «¬ Wqs »¼ *

(5.119)

169


The pseudo-loading component Bis( u ) and Bis( t ) is defined as Bis( u ) = ³ U* (ȟ i , x)b(x)d W(x) , Ws

Bis( t ) = ³ V* (ȟ i , x)b(x)d W(x) Ws

(5.120)

If we define H ij = H ij + dij C (xi ) and Vij = Vij + dij C (xi ) , Eqs (5.116) and (5.117) can be further written in matrix form as HU=GT+B(u)

(5.121)

(t)

(5.122)

WU=VT+B

The vectors U and T represent all the values of generalized displacements and tractions before applying boundary conditions. These conditions can be introduced by collecting the unknown terms to the left-hand side and the known terms to the righthand side of Eqs (5.121) and (5.122). This gives the final system of equations, i.e. EX=R

(5.123)

By solving the above system all boundary variables are fully determined. 5.4 Evaluation of uij* and tij* For illustration, we list below the explicit expressions of fundamental solutions presented in [28], although the Green’s functions presented in Chapter 2 can also be used for evaluating integrals (5.118) and (5.120). For a two-dimensional piezoelectric solid, the corresponding fundamental solution is given by [28] 1 3 ½ uIJ* ( z K , z K ) = - Re ®¦ AJR M RI ln( z R - z R ) ¾ p ¯ R =1 ¿

(5.124)

1 3 p n -n ½ t IJ* ( z K , z K ) = Re ®¦ FJR M RI R 1 2 ¾ zR - zR ¿ p ¯ R =1

(5.125)

where zK is defined in Eq (1.126), n = {n1 , n2 } is the outward unit normal at the observation point x, pR and the matrix A is determined by Eqs (1.118) and (1.130), and Fij is given by 3

FIK = ¦ [c2 IR1 + c2 IR 2 pK ] ARK

(5.126)

R =1

and the matrix M is obtained from M = A -1 (B -1 + B -1 ) -1 , with B = iAF -1

(5.127)

* * The terms DKiJ and S KiJ appearing in Eq (5.44) are obtained by differentiation of

uij* and tij* as * * * * DKiJ = EiJMl uMK S KiJ = EiJMl tMK ,l , ,l ,

* * while VKJ* , and WKJ* in Eq (5.46) are related to DKiJ and S KiJ by Eq (5.47).

The derivatives of uij* and tij* are evaluated from

(5.128)


170

d + pR d k 2 ½ 1 3 u IJ* ,k ( zM , zM ) = Re ®¦ AJR M RI k1 ¾ p ¯ R =1 zR - zR ¿

(5.129)

p n -n ½ 1 3 t IJ* ,k ( zM , zM ) = Re ®¦ FJR M RI (d k1 + pR d k 2 ) R 1 22 ¾ p ¯ R =1 ( zR - zR ) ¿

(5.130)

5.5 Evaluation of singular integrals The accuracy of BEM for piezoelectric problems is critically dependent upon proper evaluation of the boundary integrals. The integrals (5.118) and (5.120) present a singular behavior of the order O(1/r) and O(1/r2) for the generalized displacement and traction fundamental solutions, where r is the distance from a source point to the element under evaluation. The discussion which follows illustrates the basic procedure in treating singular integrals, taken from the results in [20]. 5.5.1 Weakly singular integrals Integrals of the kernels U* in Eq (5.118) show a weak singularity of the type O [ln( z K - z K 0 )] when the source point and the field point are coincident or they are a short distance apart in comparison with the size of the element, which can be numerically integrated without difficulty by using a special quadrature including a logarithm (see Appendix C for details of Gauss quadrature formulae). 5.5.2 Non-hypersingular integrals [28] The kernels T* appearing in Eq (5.118) show a strong singularity of O [1/( z K - z K )] as x ® x . Integration of such kernels over the element G j that contains the source point x can be achieved as follows: It is obvious that integrals of the type * (5.131) ³ T ( x j , x) N q ( x) d G ( x) Gj

contain the basic integral as I K = ³ [ pK n1 (x) - n2 (x)] Gj

1 N q (x)d G(x) (K=1-4) zK - zK

(5.132)

where n1, n2 are the components of the external unit normal to the boundary at the observation point x (see Fig. 5.2). Define rK = z K - z K = ( x1 - x1 ) + pK ( x2 - x2 ) (5.133) It follows that drK drK dx1 drK dx2 = + = - n2 + pK n1 d G dx1 d G dx2 d G

(5.134)

Eq (5.134) is the key for all the transformations proposed below, and this illustrates that the Jacobian drK / dG of the coordinate transformation that maps the geometry of the boundary element G j onto the complex plane rK is included in the fundamental solution itself for the piezoelectric case.

171


x2 G

dG x x

·

·

dx2 dx1

x1 ( n1 , n2 ) = ( dx1 / d G, - dx2 / d G)

Fig. 5.2 Outward unit normal at boundary point x Making use of Eq (5.134), Eq (5.132) can be rewritten as IK = ³

Gj

1 N q (x)drK rK

(5.135)

which can be decomposed into the sum of a regular integral plus a singular integral with a known analytical solution IK = ³

Gj

1 1 (N q (x) - 1)drK + ³ drK Gj r rK K

(5.136)

* Integration of the kernels DKiJ and VIJ* can be achieved in a similar way as for T* kernels since they contain singularities of the same type when x ® x . From Eq (5.129), we have the singular integral of the type [28]

1 I K¢ = ³ [ pK n1 (x) - n2 (x)] N q (x)d G(x) (K=1-4) Gj rK

(5.137)

which can be regularized as follows 1 dr 1 N q (x)drK I K¢ = ³ [ pK n1 (x) - n2 (x) - K ] N q (x)d G(x) + ³ Gj G j d G rK rK

(5.138)

The first integral in Eq (5.138) is regular and the second integral can be easily evaluated. 5.5.3 Hypersingular integrals [28] Note that the integration of WIJ* in Eq (5.118) has a hypersingularity of the order O(1/r2) as x ® x . From Eq (5.130) it follows that the hypersingular integral in Eq (5.118) is of the form I K¢¢ = ³ [ pK n1 (x) - n2 (x)] Gj

1 1 N q (x)d G(x) = ³ 2 N q (x)drK Gj r rK2 K

(K=1-4)

(5.139)

As indicated in [28], the integral (5.139) can be again decomposed into the sum of a regular integral plus singular integrals with known analytical solutions by using Eq (5.134) and the first two terms of the series expansion of the shape function Nq at x , considered as a function of the complex space variable rK


172

N q (rK ) = N q

rK = 0

+

dN q drK

rK + 0(rK2 ) » N q 0 + N q¢ 0 rK

(5.140)

rK = 0

Thus, I K¢¢ can be written as I K¢¢ = ³

Gj

1 1 N q (x)drK = ³ 2 [N q (x) - N q 0 - N q¢ 0 rK )drK 2 G j rK rK

+N q 0 ³

Gj

1 1 N q (x)drK + N q¢ 0 ³ N q (x)drK Gj r rK2 K

(5.141)

The first integral in (5.141) is regular and the other two can be easily evaluated analytically.

5.5.4 Self-adaptive subdivision method An alternative method presented in [29] for evaluating singular integrals is introduced in this section. As was pointed out in [29], the advantage of this technique is the possibility of monitoring the convergence of the matrix coefficients. The basic steps of the method are as follows:

A

B

D

C

A

B (b)

(a)

A1

B1

D1

C1

A1

B1

(c)

B2

(d)

A2

D2 C2 A2 B2

(e)

(f)

Fig. 5.3 Self-adaptive subdivision scheme

173


(i) The normalized element is divided into two triangles or four rectangles (see Fig. 5.3a,b) and the matrices H and G are evaluated over one of them, for example subelement A. (ii) The sub-element is divided (see Fig. 5.3c,d) and the the matrices H and G are are obtained as the sum of the integrals evaluated over each of these new sub-elements. (iii) If the difference in the sum of the absolute values of the coefficients of H and G calculated in step 1 and step 2 is greater than a prescribed tolerance, the procedure is repeated over each new sub-element until convergence is achieved (see Fig. 5.3e,f). (iv) The procedure is repeated in the rest of the element, i.e., triangle B and rectangles B, C, and D, respectively.

5.5.5 Singular integral in time-domain boundary integral equation [31] It is noted that Eq (5.84) have a Cauchy-type singularity of the order 1 2 ° ° x-x t IJ* (x, x, t - t) µ ® ° 1 ° ¯ x-x

when x ® x, for 3D, (5.142)

when x ® x, for 2D

The time-domain singular integrals can be evaluated by using a two-state conservation integral presented in [31]. The two-state conservation integral for linear dynamic piezoelectricity is stated as

³ {e G

}

(2) * (1) (2) º e P iJ(2) ´ U J(1),t ns - ¬ª P (1) jI ´ U I , k - rd IM U M ´ U I d jk ¼ n j d G

rki rst

(2) (1) º - ³ ª¬bI(1) ´ U I(2) , k + bI ´ U I , k ¼d W = 0 W

(5.143)

where e rki is the permutation tensor, and d , for I , M = 1, 2,3, d*IM = ® im otherwise ¯ 0,

(5.144)

Proof of the state can be obtained by considering the following conservation integral of linear dynamic piezoelectricity: ª1 º J kD = ³ « (P mN ´ U N ,m + rd*IM UM ´ U I )d jk - P jI ´ U I ,k »n j d G G 2 ¬ ¼

(5.145)

- ³ bI ´ U I ,k d W = 0 W

Wang and Zhang [31] indicated that the conditions for the validity of Eq (5.145) are the zero initial conditions: U I ( x, t ) = U I ( x, t ) = 0

for t = 0

(5.146)

and the absence of singularities with the piezoelectric solid under consideration. The proof of Eq (5.145) is straightforward by applying the divergence theorem and the properties of the Riemann convolution. To this end, let U I , P iJ , and bI be in the form


174

(2) (1) (2) U I = U I(1) + U I(2) , P jI = P (1) jI + P jI , bI = bI + bI

(5.147)

Substituting Eq (5.147) into Eq (5.145), we have J kD (U I ) = J kD (U I(1) ) + J kD (U I(2) )

{

}

(2) * (1) (2) º = ³ e rki e rst P iJ(2) ´ U J(1),t ns - ª¬P (1) jI ´ U I , k - rd IM U M ´ U I d jk ¼ n j d G G

(5.148)

(2) (1) º - ³ ª¬bI(1) ´ U I(2) , k + bI ´ U I , k ¼d W = 0 W

Since the first two terms in Eq (5.148) vanish identically, the two-state conversation integral (5.143) holds true for linear piezoelectric solids. Consider that the abovementioned state (1) represents the solution to piezoelectric problems with finite domains and general loading conditions. The second state stands for the fundamental solution to the case of an infinite piezoelectric solid subjected to an impulsive point force and an impulsive point charge. By using Eq (5.148) and the sifting property of the Dirac delta function, a representation formula for the generalized displacement gradients U L ,k (x, t ) can be obtained, as * * * * º U L ,k (x, t ) = ³ e rki e rst DJiL (5.149) ´ U J ,t ns - ª¬uIL , k ´ P jI - rd IM u IL ´ U M d jk ¼ n j d G G

{

}

where x Ï G. Making use of the constitutive equation (5.2) and the expression of ti and qs in Eq (5.4), the following 3D dynamic boundary integral equation on generalized traction can be obtained as * * t I (x, t ) = E pILk n p (x, t ) ³ e rkm e rst DJmL ´ UM nk º¼ d G (5.150) ´ U J ,t ns - ª¬u*JL ,k ´ t J - rd*IM uIL G

{

}

The corresponding 2D time-domain boundary integral equation can be obtained from Eq (5.150) as * t I (x, t ) = EaILb na (x, t ) ³ DJ*gL ´ (nb¶ g - ng ¶ b )U J - ª¬u*JL ,b ´ t J - rd*IM uIL ´ UM nb º¼ d G G

{

}

(5.151) 5.6 Crack tip singularity by evaluating hypersingular integrals Based on the Green’s functions presented in Section 2.3, Qin and Noda [34] developed an approach for evaluating the hypersingular integrals appearing in Eq (5.63) and provided explicit expressions of stress and electric displacement field near the crack tip. It should be mentioned that the Green’s functions presented in Section 2.3 are for a point force or charge applied at origin only. For the case of a point load acting at x = ( x , y, z ) rather than 0, the solution still applies if x, y, z in Section 2.3 are replaced by x - x , y - y , z - z . Using the solution given in Section 2.3, the hypersingular integral in Eq (5.63) can be rewritten as

>³

L+

1 2 [c44 D0 v0 (2dab - 3r,a r,b ) + k11 (dab - 3r,a r,b )]Dub (x)d G(x) = -ta 0 (x) 3 r

(5.152)

1 [k33 Du3 (x) + k34 Df(x)]d G(x) = -t30 (x) 3 r

(5.153)

1 [k43 Du3 (x) + k44 Df(x)]d G(x) = -qs 0 (x) 3 r

(5.154)

>³ >³

L+

L+

175


where a, b = 1, 2, x Î L+ , ti 0 and qs 0 stand for the mechanical and electric loads on the crack surfaces due to internal or external loads and can be obtained from the solution for the loads of the uncracked solid, and kij is given by 3

w f k11 = ¦ [c44 ( Ai + Di vi ) + e15 Bi ][c44 (vi l uv i + l i ) + e15 l i ]

(5.155)

i =1

3

w f k33 = ¦ (-c13l uv i + c33l i vi + e33 vi l i )(c33 Ai vi + e33 Bi vi - c13 Di )

(5.156)

i =1 3

w f k34 = ¦ (-c13l uv i + c33 l i vi + e33 vi l i )(e33 Ai vi - k 33 Bi vi - e31 Di )

(5.157)

i =1 3

w f k43 = ¦ (-e31l uv i + e33 l i vi - k33vi l i )(c33 Ai vi + e33 Bi vi - c13 Di )

(5.158)

i =1 3

w f k34 = ¦ (-e31l uv i + e33 l i vi - k 33 vi l i )(e33 Ai vi - k33 Bi vi - e31 Di )

(5.157)

i =1

w f with Ai,Bi,Di,vi, l uv i , l i , and l i being defined in Section 2.3. When using Eqs (5.152)-(5.154) for analyzing singularity of generalized stress near a crack tip, it is convenient to use a local coordinate system as shown in Fig. 5.4. That is, the x1-axis is the tangent line of the crack front at point x0, x2-axis is the internal normal line in the crack plane, and x3-axis is normal to the crack. Then the displacement and electric potential discontinuities between crack surfaces near the point x0 can be assumed as

Du i (x) = g k (x 0 ) x 2l k , Df(x) = F (x 0 ) x 2l 4 ,

0 < Re(l k ) < 1

(5.158)

where g k ( x 0 ) and F (x 0 ) are non-zero constants related to point x0, l k is the singular index at the crack front. Consider now the singular point x0 on the boundary surrounded by a small semi-circular surface of radius e, say G e , centred at the point x0 with e ® 0. Using the main-part analytical method [35], the following relations can be obtained Du1 dx1dx2 @ -2pl1 g1 (x0 ) x2l1 -1 cot(l1p) r3

(1.159)

Du1 2 ( x1 - x1 ) 2 dx1dx2 @ - pl1 g1 (x0 ) x2l1 -1 cot(l1p) 5 3 r

(1.160)

Du2 4 ( x2 - x2 ) 2 dx1dx2 @ - pl 2 g 2 (x 0 ) x2l2 -1 cot(l 2 p) 5 r 3

(1.161)

Du1 ( x1 - x1 )( x2 - x2 )dx1dx2 @ 0 5 r

(1.162)

Du3 dx1dx2 @ -2pl 3 g3 (x0 ) x2l3 -1 cot(l 3 p) 3 r

(1.163)

Df dx1dx2 @ -2pl 4 F (x 0 ) x2l4 -1 cot(l 4 p) r3

(1.164)

>³

Ge

>³

Ge

>³

Ge

>³

Ge

>³

Ge

>³

Ge


176

x

x3 x1

Ge

r q

x2 e

x0

Fig. 5.4 A semi-circular domain Ge on the crack surface Making use of the above relations, Eqs (5.152) and (5.154) yield cot(l i p) = 0, (i=1-4)

(5.165)

Then the singular index is obtained as l1 = l 2 = l 3 = l 4 =

1 2

(5.166)

Considering Eq (5.158) and the main-part analytical method, the following relations can be obtained, which are necessary for deriving crack-tip fields: 1 § 3( x1 - x1 ) 2 · (5.167) 1 ¨ ¸ Du1dx1dx2 @ 0 ³Ge R03 © R02 ¹ pg1 (x 0 ) q 1 § 3( x2 - x2 ) 2 · (5.168) 1 cos 0 ¸ Du1dx1dx2 @ ³Ge R03 ¨© 2 R02 r r ¹ 0 pg (x ) q 1 § 3v02 x32 · 1Du1dx1dx2 @ 1 0 cos 0 3 ¨ 2 ¸ 2 R0 © R0 ¹ r0 r

(5.169)

qj pg (x ) 1 § 3( x2 - x2 ) 2 · 1¸¸ Du3 dx1dx2 @ 3 0 cos , (j=1,2,3) 3 ¨ 2 ¨ Rj © Rj 2 rj r ¹

(5.170)

³

Ge

³

Ge

³

Ge

³

Ge

2 2 q pg (x ) 1 § 3v j x3 · Dui dx1dx2 @ i 0 cos j 13 ¨ 2 ¸ ¨ ¸ Rj © Rj ¹ 2 rj r

3v j x3 ( x2 - x2 ) R

5 j

Du3 dx1dx2 @ -

qj pg3 (x0 ) sin , (j=1,2,3) 2 rj r

where r0 = (cos 2 q + v02 sin 2 q)1/ 2 , q0 = tg -1 (v0 tgq), rj e

iq j

(5.171)

(5.172)

= cos q + iv j sin q. Using the

relations (5.167)-(5.172), the singular stresses and electric displacement near the crack front can be expressed as

177


pg1 (x0 )

2 s13 = c44 D0 v02

s 23 = s33 = D3 = -

q0 2

cos

r0 r

pg 2 (x0 ) 3 u u 1 q p Ai g i cos i + ¦ 2 r i =1 ri r

¦ [ g (x ) A

pg 2 (x0 ) 3 u w 1 q p Ai g i sin i ¦ 2 r i =1 ri r

¦ [ g (x ) A

pg 2 (x0 ) r

q 1 p sin i 2 ri r

3

¦ Aiu gif i =1

3

3

w i

0

(5.173)

+ F (x0 ) Aif ]g ui

q 1 sin i (5.174) 2 ri

+ F (x0 ) Aif ]g iw

q 1 cos i (5.175) 2 ri

i =1 3

3

w i

0

i =1

3

¦ [ g (x ) A 3

0

w i

+ F (x0 ) Aif ]g if

i =1

q 1 cos i (5.176) 2 ri

where Aiu = c44 ( Ai + Di vi ) + e15 Bi , Aiw = c33 Ai vi + e33 Bi vi - c13 Di , Aif = e33 Ai vi - k 33 Bi vi - e31 Di , g iu = c44 ( vi luvi + lwi ) + e15lfi g iw = c13luvi - c33vi lwi ) - e33 vi lfi g if = e31l iuv - e33l iwvi + k 33vi l if

(5.177)

Using the traditional definitions of the stress intensity factors and the electric field intensity factor

K I = lim s 33 ( r, q) q= 0 r ®0 ° s 32 ( r, q) q=0 °° K II = lim r ®0 ® s 31 ( r, q) q=0 ° K III = lim r ®0 ° K = lim D ( r, q) °¯ IV r ®0 3 q=0

2r , 2r , (5.178)

2r , 2r ,

The above singular stresses and electric displacement can be rewritten in terms of field intensity factor as s13 = s 23 = s 33 = D3 =

KI 2r KI 2r KI 2r

K III 1 q cos 2 2r r0

f 21 ( q) + f 31 ( q) + f 41 ( q) +

K II 2r K II 2r K II 2r

f 22 ( q) + f 32 ( q) + f 42 ( q) +

(5.179) K IV 2r K IV 2r K IV 2r

f 24 ( q)

(5.180)

f 34 ( q)

(5.181)

f 44 ( q)

(5.182)

where f 21 ( q) = -

1 k 33 k 44 - k 34 k 43 f 22 ( q) =

3

¦ (k

44

Aiw - k 43 Aif ) g iu

i =1

q 1 3 u u 1 Ai g i cos i ¦ k11 i =1 2 ri

1 ri

sin

qi 2

(5.183)

(5.184)


178

f 24 ( q) = -

3 q 1 1 ( k 33 Aiw - k 34 Aif ) g iu sin i ¦ 2 k 33k 44 - k 34 k 43 i =1 ri

(5.185)

f 31 ( q) = -

3 q 1 1 ( k 44 Aiw - k 43 Aif ) g iw cos i ¦ 2 k 33k 44 - k 34 k 43 i =1 ri

(5.186)

f 32 ( q) = f 34 ( q) = f 41 ( q) =

q 1 3 u w 1 sin i Ai g i ¦ 2 k11 i =1 ri

3 q 1 1 ( k 34 Aiw - k 33 Aif ) g iw cos i ¦ 2 k 33k 44 - k 34 k 43 i =1 ri

(5.187)

(5.188)

3 q 1 1 ( k 44 Aiw - k 43 Aif ) g if cos i ¦ 2 k 33k 44 - k 34 k 43 i =1 ri

(5.189)

q 1 3 u f 1 ¦ Ai g i r sin 2i k11 i =1 i

(5.190)

f 42 ( q) = f 41 ( q) = -

3 q 1 1 ( k 34 Aiw - k 33 Aif ) g if cos i ¦ 2 k 33k 44 - k 34 k 43 i =1 ri

(5.191)

5.7 Multi-domain problems Eq (5.18) is suitable for problems with a single solution domain only. If the solution domain is made up piece-wise of different materials the problem can be solved by the multi-domain BEM [13]. The multi-domain approach presented in [13] is based on the division of the origin domain into homogeneous subregions (see Fig. 5.5) so that Eq (5.121) still holds for each single subdomain, and one can write H ( i ) U ( i ) - G ( i ) T( i ) = B ( i ) ,

(i = 1, 2,! , J )

(5.192)

where J is the number of subregions and the superscript (i) indicates quantities associated with the ith subregion. To obtain the solution it is necessary to restore domain unity by enforcing the generalized displacement and traction continuity conditions along the interfaces between contiguous subdomains. Let us introduce a partition of the linear algebraic system given by Eq (5.192) in such a way that the generic vector can be written as [13]

{

y ( i ) = yG( ii)1 " yG(iiJ)

T

}

(5.193)

where the vector yG( iij) collects the components of y(i) associated with the nodes belonging to the interface Gij between the ith and jth subdomain, with the convention that Gii stands for the external boundary of the ith subdomain (see Fig. 5.5). Based on this arrangement, the interface compatibility and equilibrium conditions are given by [13] U (Giij) = U (Gjij) , TG( iij) = -TG( ijj ) , (i = 1," J - 1; j = i + 1," J )

(5.194)

It should be noted that if the ith and jth subdomain have no common boundary, yG( iij) is

179


a zero-order vector and Eq (5.194) is no longer valid. The system of Eq (5.192) and the interface continuity conditions (5.194) provide a set of relationships which, together with the external boundary conditions, allows derivation of the electroelastic solution in terms of generalized displacement and traction on the boundary of each subdomain. It should be mentioned that the multi-domain approach described here is suitable for modeling general fracture problems in piezoelectric media [13].

·

Gii

·

Gij

·

Wi

Gik

G jj

Wj

· ·

Wk

G jk

· G kk

Fig. 5.5 Multi-domain configuration 5.8 Application of BEM to fracture analysis Over the years, several special boundary elements have been presented to capture the crack-tip singularity of piezoelectric materials. For illustration, three typical boundary elements presented in [11,28, 36] are discussed here. 5.8.1 One quarter-point quadratic crack-tip element The boundary element formulation described in Section 5.3 is inefficient for modeling crack-tip elements. To capture the square root characteristics of the generalized relative crack displacement (GRCD) near the crack-tip, a quarter-point quadratic crack-tip element is developed by setting [28] z=2

r -1 L

(5.195)

where z is the boundary element natural coordinate, r is the distance from a field point to the crack-tip, and L the element length (see Fig. 5.6). In the element, the collocation points NC1, NC2, and NC3 for the quarter-point element are located at z1 = -3 / 4, z 2 = 0, and z 3 = 3 / 4 , respectively (see Fig. 5.6). In such a case, the distance r from the collocation nodes of the quarter-point element to the crack-tip follows from Eq (5.195) r1 =

L L 49 L at NC1, r2 = at NC2, r3 = at NC3 64 4 64

(5.196)

If we use the conventional polar coordinate system (r,q), centered at the crack-tip and such that q = ±p are the crack faces, the GRCD DU across the two opposite crack faces may be written in terms of the generalized stress intensity factors (GSIF), as


180

[28] Du1 ½ K II ½ 8r ° ° ° ° DU = ® Du2 ¾ = Re[B] ® K I ¾ p ° Df ° °K ° ¯ ¿ ¯ IV ¿

(5.197)

where KI and KII are the conventional mode I and mode II SIF, respectively, and KIV is the electric displacement intensity factor. Particularizing Eq (5.197) for the collocation node NC1, the following one-point formula for direct evaluation of the GSIF can be obtained Du1NC1 ½ K II ½ 2p ° ° ° ° K = ® KI ¾ = 2 (Re[B]) -1 ®Du2NC1 ¾ L °K ° ° Df NC1 ° ¯ IV ¿ ¯ ¿

(5.198)

L 49L/64 L/4 L/64 Crack-tip

NC1

·

z=-1 z=-3/4

NC2

·

·

NC3

z=3/4

z=0

z=1

Fig. 5.6 Configuration of quarter-point element

5.8.2 An alternative crack-tip element To effectively evaluate crack-tip fields, Pan [11] constructed the following crack-tip element with its tip at z=-1 3

DU = ¦ F k DU k

(5.199)

k =1

where the superscript k (k=1,2,3) denotes the GRCD at nodes z=-2/3,0,2/3, respectively. The shape functions Fk are defined by [11] 3 3 z + 1[-5 + 18(z + 1) - 9(z + 1) 2 ], 8 1 F2 = z + 1[5 - 8(z + 1) + 3(z + 1) 2 ], 4 3 3 F3 = z + 1[1 - 4(z + 1) + 3(z + 1) 2 ], 8 5 F1 =

(5.200)

For the GSIF calculation, Pan [11] employed the extrapolation method of the GRCD, which requires an analytical relation between the generalized displacement and the GSIF. This relation can be expressed as DU(r ) = 2

where K is defined in Eq (5.198).

2r (iAB -1 )K p

(5.201)

181


5.8.3 8-node crack element For a 3D cracked piezoelectric solid, Wippler and Kuna [36] developed a set of 8node quadrilateral elements including regular elements and non-conformal elements. The regular element is used to model a smooth surface, while the non-conformal element is used to model surfaces with edges and corners. Special crack front elements modeling the radial dependence of the field quantities around the crack front are used in the vicinity of cracks. Fig. 5.7 shows the possible element types and their node and side numbering for the geometry discretization.

D D· D·

4

8

1

D· D· D· D· D· 7

·

3

S3

S4

S2

6

S1

5 (a)

2

D· D· D·

Dt ·D D· s ·D D· ·D ·D D· ·

(b)

D ·D ·D ·D ·D

·

(c)

Fig. 5.7 Types of conformal and non-conformal boundary elements In this setting the shape functions for a regular element are defined by Eq (5.107). For non-conformal interpolation the node coordinates depend on the discontinuous side. If, for example, element four (S4) is the discontinuous element, the node coordinates are

sN = {-D, 1, 1, - D, 0.5(-D + 1), 1, 0.5( -D + 1), - D} t N = {-1, - 1, 1, 1, - 1, 0, 1, 0}

(5.202)

which leads the discontinuous shape functions used near edges and corners. D is the discontinuity parameter (D0 in the element located at the right of the node m, s0 and 4 1 U (2) = ¦ Im{A (2) ln( za(2) - zb(1)0 ) qb(2) } , b=1 p

(6.55)

4 1 j (2) = ¦ Im{B (2) ln( za(2) - zb(1)0 ) qb(2) } b=1 p

(6.56)

qb(1) = B (1) -1[I - 2(M (1) -1 + M (2) -1 ) -1 L(1) -1 ]B (1) I b BT b ,

(6.57)

qb(2) = 2B (2) -1 (M (1) -1 + M (2) -1 ) -1 L(1) -1B (1) Ib BT b ,

(6.58)

where

with M ( j ) = -iB ( j ) A ( j ) -1 as the surface impedance matrix. (ii) a plate containing an elliptic hole and multiple cracks 1 1 4 U(b) = Im[ A ln(z a - z a 0 BT ]b + ¦ Im[ A ln(z a-1 - zb 0 B -1BIb BT ]b (6.59) p p b=1 j(b) =

1 1 4 Im[B ln(z a - z a 0 BT ]b + ¦ Im[B ln(z a-1 - zb 0 B -1BIb BT ]b (6.60) p p b=1

(iii) half-plane problems 1 1 4 U(b) = Im[ A ln( za - za 0 BT ]b + ¦ Im[ A ln( za - zb0 B -1BIb BT ]b (6.61) p p b=1 j(b) =

1 1 4 Im[B ln( za - za 0 BT ]b + ¦ Im[B ln( za - zb 0 B -1BIb BT ]b (6.62) p p b=1

(iv) a plate containing a hole of various shapes For the case of a plate containing a hole of various shapes the expressions of j(b) and U(b) have the same form as those of Eqs (6.59) and (6.60), but the relation between z a and za is defined by Eq (2.223) (v) wedge problems

203

Boundary element method for discontinuity problems

U(b) =

1 1 4 Im[ A ln( zkl - zkl0 BT ]b + ¦ Im[ A ln(- zkl - zbl0 B -1BI b BT ]b (6.63) p p b=1

j(b) =

1 1 4 Im[B ln( zkl - zkl0 BT ]b + ¦ Im[B ln(- zkl - zbl0 B -1BI b BT ]b (6.64) p p b=1

As treated earlier, the boundaries L and G are divided into a series of boundary elements, for which the EDEP discontinuity may be approximated through linear interpolation as M

b( s ) = ¦ b m Fm ( s ) .

(6.65)

m =1

With the approximation (6.65), the EDEP and SED functions can be expressed in the form M

U(z ) = ¦ Im[ ADm (z )]b m , m =1

M

j(z ) = ¦ Im[BDm (z )]b m

(6.66)

m =1

where Dm has different forms for different problems. The function Dm is given below for five typical problems. (i) bimaterial problems Dm (z ) =

4 1 ( m -1) ( m -1) * T T ½ lm -1 + s B ln( ) ln( zab z + ds ® ¦ 0 ) B BI b B ¾ aa 0 ³ l p m-1 ¯ b=1 ¿ lm -1

(6.67)

4 ½l -s 1 (m) (m) ds + ³ ® ln( zaa ) BT + ¦ ln( zab ) B*BIb BT ¾ m 0 0 p lm ¯ b=1 ¿ lm

(m) ( m) (m) (m) * (1) -1 where zaa [I - 2(M (1) -1 + M (2) -1 ) -1 L(1) -1 ] . 0 = za - za 0 , zab 0 = za - zb 0 , B = B

(i) a plate containing an elliptic hole D m (z ) =

4 ½l +s 1 ( m -1) T B z z + ds ln( ) ln(z a-1 - zb( m0 -1) ) B -1BIb BT ¾ m -1 ¦ a a0 ³ l p m-1 b=1 ¿ lm -1

{

4 ½l -s 1 ° + ³ ® ln(z a - z (am0) ) BT + ¦ ln(z a-1 - zb( m0 ) ) B -1BI bBT ¾ m ds l m p °¯ b=1 ¿ lm

(6.68)

(ii) half-plane problems Dm (z ) =

4 1 m -1 T m -1 -1 T ½ lm -1 + s B ln( ) ln( zab z + ds ® ¦ 0 ) B BI b B ¾ aa 0 ³ l p m-1 ¯ b=1 ¿ lm -1

4 1 m m -1 T ½ lm - s ds + ³ ® ln( zaa ) BT + ¦ ln( zab 0 0 ) B BI b B ¾ p lm ¯ b=1 ¿ lm

(6.69)

(iv) a plate containing multiple cracks and a hole of various shapes (see Section 4.11). It has the same form as that in Eq (6.68), but z a is related to za by Eq (2.223).


204

(v) wedge problems Dm (z ) =

5 l +s 1 { ln( zal - zal 0( m -1) ) BT + ¦ ln(- zal - zbl0( m -1) ) B -1BI bBT } m -1 ds ³ p lm-1 lm -1 b=1

+

5 l -s 1 { ln( zal - zal 0( m ) ) BT + ¦ ln(- zal - zbl0( m ) ) B -1BIb BT } m ds ³ l p m lm b=1

(6.70) In particular the displacement at node j is given by M

U[ f (daj ) ] = ¦ Im{ADm [ f (d aj ) ]}b m .

(6.71)

m =1

Substituting Eq (6.66) into Eq (6.52), we have M ª º §M · P (b) = ¦ «bTi × ¨ ¦ k ij b j ¸ / 2 - g i » i =1 ¬ « © j =1 ¹ ¼» where k ij =

1 1 Im[DTi (z 0j -1 )BT ]ds l j -1 ³ l j-1 lj

gj = ³

l j -1 + l j

³

lj

Im[DTi (z 0j )BT ]ds

(6.72)

(6.73) (6.74)

G j F j ( s )ds

and G j = -t qn when node j is located at the boundary L, G j = t 0 - t qn for the other nodes. The minimization of Eq (6.72) leads to a set of linear equations: M

¦k

ij

b j = gi .

(6.75)

j =1

Similarly, the final form of the linear equations to be solved is obtained by selecting the appropriate equation from Eqs (6.71) and (6.75). Equation (6.71) will be chosen for those nodes at which the EDEP is prescribed, and Eq (6.75) will be chosen for the remaining nodes. Once the EDEP discontinuity b has been found, the SED at any point can be expressed as M

P 1 = -¦ Im[BPD¢m (z )]b m , m =1

M

P 2 = ¦ Im[BD¢m (z )]b m

(6.76)

m =1

Therefore, the surface traction-charge vector Pn in a coordinate system local to a particular crack line, say the ith crack, can be expressed in the form P n = W(ai ){-P 1 sin a i +P 2 cos a i }T

(6.77)

where W(ai ) is defined by ª cos a sin a « - sin a cos a W(a ) = « « 0 0 « 0 ¬ 0

0 0 1 0

0º 0 »» 0» » 1¼

(6.78)

Using Eq (6.77) we can evaluate the SED intensity factors by the following definition

205


K (c) = {K II K I K III K D }T = lim 2pr P n (r ) r ®0

(6.79)

6.3 Application of BEM to determine SED intensity factors In practical computations the SED intensity factors may be evaluated in several ways, such as extrapolation formulae, traction formulae, J-integral formulae [6], leastsquare method [7,8], and others [6]. In our analysis, the method of least-square is used, since only the EDEP field obtained from the BEM is required in the procedure. Therefore not much computer time is required for the calculation of K-factors. Moreover, it is very easy to implement the method into our BEM computer program. That is why we select the least-square method rather than any other to calculate SED intensity factors. 6.3.1 Relation between SED function and SED intensity factors In order to take into account the crack-tip singularity of the SED field we choose the mapping function [9] zk - zk 0 = w(xk ) = x 2k ,

(k=1, 2, 3, 4, t)

(6.80)

where zk0 is the coordinate of the crack tip under consideration. Recall that the general expressions for the EDEP field and SED function of a linear thermopiezoelectric solid are [2] U = 2 Re[ Af (z ) + cg ( zt )],

j = 2 Re[Bf (z ) + dg ( zt )]

(6.81)

The EDEP and SED fields in x-plane can then be written as U = 2 Re[ Af (x) + cg (xt )],

P 2 = 2 Re[Bf ¢(x) / x + dg ¢(xt ) / xt ]

(6.82)

With the usual definition, the vector of SED intensity factor, K, is evaluated by K = lim x 2pP 2 = 2 2p lim Re[Bf ¢(x) + dg ¢(xt )] x® 0

x®0

(6.83)

The functions f and g near the crack tip can be assumed as simple polynomials of x 2n

2n

j =1

j =1

f (x) » ¦ (s j + is 2 n + j )x j , g (xt ) » ¦ rj xtj

(6.84)

where rj ( j=1, 2, 3, ! , 2n) are known complex constants, and sj are real constant vectors to be determined. On the crack surface which is traction-charge free, i.e., j =0, substitution of Eq (6.84) into Eq (6.81)2 yields 2n

j = 2 Re ¦ [B(s j + is 2 n + j )x j + drj xtj ] = 0

(6.85)

j =1

Noting that xj= xtj =(- x) j/2 along the crack surface, where x is the distance from crack tip to the point concerned, we have s 2 n + j = B -R1[-B I s j + Im{drj }] , s 2 n + j = B -I 1[B R s j + Re{drj }] ,

for j=1, 3, 5, ! , 2n-1,

(6.86)

for j=2, 4, 6, ! , 2n,

(6.87)


206

in which BR=Re(B), BI=Im(B). Substituting Eqs (6.86) and (6.87) into Eq (6.84), and then into Eqs (6.82) and (6.83), yields 2n

U = ¦ [Q j (x)s j + S j (x, xt )],

(6.88)

j =1

-1 R

-1 R

K = 2 2p Re[B(I - iB B I )s1 + iBB Im(dr1 ) + dr1 ]

where Q j (x) = [I - iB -R1B I ]x j , S j (x, xt ) = iB -R1Im[drj ]x j +drj x tj , ( j = 1, 3,! , 2n - 1), (6.89) Q j (x) = [I + iB -I 1B R ]x j , S j (x, xt ) = iB -I 1Re[drj ]x j +drj x tj , ( j = 2, 4,! , 2n)

(6.90)

6.3.2 Simulating K by BEM and least-square method The least-square method may be developed by considering the residual vector for EDEP field at point k (k=1, 2, ! , m) 2n

R k = ¦ [Q j (x k )s j + S j (x k , xtk )] - U k ,

(k=1, 2, ! , m)

(6.91)

j=1

where Uk is the EDEP vector at point k obtained from the BEM given in the previous section. The minimum for the sum of the squares of the residual vector p = {s}T [Q]T [Q]{s} - 2{s}T [Q]T ({U} - {S}) + terms without {s}

(6.92)

[Q]T [Q]{s} = [Q]T ({U} - {S}) ,

(6.93)

provides where {s}={s1 , s 2 ,! , s 2 n }T ,

{U} = {U1 , U 2 ,! , U m }T ,

(6.94)

2n

{S} = {S1* , S*2 ,! , S*m }T , S*k = ¦ S j (x k , xtk ) ,

(6.95)

j=1

ª Q11 «Q 12 [Q] = « «! « ¬Q1m

Q 21 Q 22 ! Q2m

! Q 2 n ,1 º ! Q 2 n ,2 »» , ! ! » » ! Q 2 n ,m ¼

Q jk = Q j (x k )

(6.96)

(6.97)

Once the unknown vector {s} has been obtained from Eq (6.93), the SED intensity factor K can be evaluated from Eq (6.88)2. In the calculation, an appropriate number m can be set to obtain the required accuracy. 6.4 Effective properties of cracked piezoelectricity 6.4.1 Effective properties and concentration matrix P In Section 5.8 we presented formulations for predicting effective properties of


207

piezoelectricity with inclusions or holes. For piezoelectricity weakened by cracks, the effective material properties can be analyzed by the combination of micromechanics models presented in Section 5.8 and the BE Eqs (6.71) and (6.75). To this end, define the overall elastic, piezoelectric, and dielectric constants of the piezoelectric solid by [10] ı = C* İ - (e * ) T E (6.98) D = e*İ + ț * E

Fig. 6.3 A typical RVE with cracks

Using the definition (6.98), the following two types of boundary condition can be used to evaluate overall material properties: a) Uniform traction, ı 0 , and electric displacement, D0, on boundary G of the RVE (see Fig. 6.3): ı = ı 0 , D = D0

(6.99)

b) Uniform strain, e0, and electric field, E0, on boundary G of the RVE: İ = İ0 , E = E0

(6.100)

Since the material behavior is linear, the principle of superposition is used to decompose the load (e0, E0) into two elementary loadings (e0, E0=0) and (e0=0, E0). For the loading case (e0, E0=0), Eq (6.98) becomes ı = C *İ 0 D = e *İ 0

(6.101)

Using relations (5.219) and (6.101), we have C * = C1 + (C 2 - C1 ) A 2 v 2 e * = e1 + (e 2 - e1 ) A 2 v 2

(6.102)

where the concentration tensor A2 is defined by the linear relation İ2 = A 2İ 0

(6.103)

Following the average strain theorem [11], the strain tensor, İ 2 , can be expressed as


208

( İ2 )ij =

1 2W 2

³

¶W 2

(ui n j + u j ni )dS

(6.104)

where ui is the ith component displacement vector, and ni the ith component of unit outward vector normal to the boundary. When inclusions become cracks, Eq (6.104) cannot be directly used to calculate the average strain. This problem can be bypassed by considering cracks to be very flat voids of vanishing height and thus also of vanishing volume. Multiplying both sides of Eq (6.104) by v2 and considering the limit of flattening out into cracks, i.e. v 2 = W 2 / W ® 0 , one obtains lim (İ 2 v 2 ) ij =

v2 ® 0

v2 2W 2

³

L

(Du i n j + Du j ni )dS =

1 (Du i n j + Du j ni )dS 2W ³ L

(6.105)

where Du i is the jump of displacement across the crack faces, L = l1 È l 2 È ! È l N , li is the length of the ith crack, N the number of cracks within the RVE under consideration. For convenience, we define P = lim ( A 2 v 2 ) v2 ®0

(6.106)

where P can be calculated by the relation: (Pİ 0 ) ij = lim (İ 2 v 2 ) ij = v2 ® 0

1 (Du i n j + Du j ni )dS 2W ³ L

(6.107)

Substituting Eq (6.106) into Eq (6.102) and considering C 2 ® 0 and e 2 ® 0 when inclusions become cracks, yields C * = C1 (I - P) e * = e1 (I - P )

(6.108)

On the other hand, for the loading case (e0=0, E0), Eq (6.98) leads to

ı = -(e * ) T E 0 D = ț *E 0

(6.109)

Similar to the treatment in Eq (6.102), we have e * = e1 + B T2 (e 2 - e1 )v 2 ț * = ț 1 + (ț 2 - ț 1 )B 2 v 2

(6.110)

where the tensor B2 is defined by

E2 = B 2 E 0

(6.111)

By comparing Eqs (6.102)2 and (6.110)1, it is evident that (e 2 - e1 ) A 2 = B T2 (e 2 - e1 )

(6.112)


209

Therefore the effective constitutive law (6.98) is completely defined once the concentration factor A2 has been determined. The estimation of integral (6.104) or (6.105) and thus A2 (or P) is the key to predicting the effective electroelastic moduli C* , e * , and ț * . Calculation of integral (6.104) or (6.105) through the use of the BEM is the subject of the following section. Eqs (6.71) and (6.75) are used to evaluate Dui in Eq (6.107). The matrix P can then be predicted through use of Eq (6.107).

6.4.2 Algorithms for self-consistent and Mori-Tanaka approaches The algorithm used for predicting effective properties of cracked piezoelectricity based on the self-consistent or Mori-Tanaka approach is similar to that in Section 5.8 and is described in detail as below.

(a) Algorithms for self-consistent BEM approach (a) Assume initial values of material constants C *( 0 ) , e *( 0 ) , and ț *( 0 ) ( C *( 0 ) =C0, e *( 0 ) =e0 and ț *( 0 ) =k0 are used as initial value in our analysis)

(b) Solve Eqs (6.71) and (6.75) for b m (i ) (= Dum (i ) ) using the values of C *( i -1) , e *(i -1) , and ț *(i -1) , where the subscript (i) stands for the variable associated with the the ith iterative cycle. (c) Calculate P(i) in Eq (6.106)) by way of Eq (6.107) using the current values of

b m (i ) , and then determine C *(i ) , e *(i ) , and ț *(i ) by way of Eqs (6.102) and (6.110). (d) If e ( i ) = F(*i ) - F(*i -1) / F(*0) £ e , where e is a convergent tolerance, terminate the iteration; F may be C, e or k, otherwise take C *(i ) , e *(i ) , and ț *(i ) as the initial values and go to Step (b).

(b) Mori-Tanaka-BEM approach As indicated in Section 5.8.4, the key assumption in the Mori-Tanaka theory is that the concentration matrix P MT is given by the solution for a single crack embedded in an intact solid subject to an applied strain field equal to the as yet

210


unknown average field in the solid, which means that the introduction of cracks in the solid results in a value of İ 2 given by

İ 2 = A 2DIL İ1

(6.113)

where A 2DIL is the concentration matrix related to the dilute model. As such, it is easy to prove that [11]

P MT = P DIL (I + P DIL ) -1

(6.114)

where I is unit matrix. It can be seen from Eq (6.114) that the Mori-Tanaka approach provides explicit expressions for effective constants of defective piezoelectric solids. Therefore, no iteration is required with the mixed Mori-Tanaka-BEM. 6.5 Numerical examples As an illustration, the proposed BE model is applied to the following two numerical examples in which an inclusion and a crack are involved. In all the calculations, the materials for the matrix and the elliptic inclusion are assumed to be BaTiO3 and Cadmium Selenide, respectively. The material constants for the two materials are as follows: (i) Material constants for BaTiO3 c11 = 150 GPa, c12 = 66 GPa, c13 = 66 GPa, c33 = 146 GPa, c44 = 44 GPa, * a11 = 8.53 ´10-6 K -1 , a*33 = 1.99 ´10-6 K -1 , l 3 = 0.133 ´105 N/CK,

e31 = -4.35 C/m 2 , e33 = 17.5 C/m 2 , e15 = 11.4 C/m 2 , k11 = 1115 k0 , k33 = 1260 k0 , k0 = 8.85 ´10-12 C2 / Nm 2 = permittivity of free space (ii) Material constants for Cadmium Selenide c11 = 74.1 GPa, c12 = 45.2 GPa, c13 = 39.3 GPa, c33 = 83.6 GPa, c44 = 13.2 GPa, g11 = 0.621´ 106 NK -1m -2 , g3 = -0.294 ´ 105 CK -1m -2 ,

g 33 = 0.551´ 106 NK -1m -2 ,

e31 = -0.160 Cm -2 ,

e33 = 0.347 Cm -2 ,

e15 = 0.138 Cm -2 , k11 = 82.6 ´ 10-12 C2 N -1m -2 , k33 = 90.3 ´10-12 C2 N -1m -2 * where cij is elastic stiffness, a11 and a*33 are thermal expansion constants, l3 and g3 are pyroelectric constants, eij is piezoelectric constants, and gij is a piezothermal constant. Since the values of the coefficient of heat conduction for BaTiO3 and Cadmium Selenide could not be found in the literature, the values k33/k11=1.5 for BaTiO3 and k33/k11=1.8 for Cadmium Selenide, k13=0 and k11=1 W/mK are assumed. In our analysis, plane strain deformation is assumed and the crack line is assumed to be in the x1-x2 plane, i.e., D3=u3=0. Therefore the stress intensity factor vector K* now has only three components (KI, KII, KD). In the least-square method, the SED intensity factors are affected by the parameters n, dmax and dmin, where n is the number of terms in Eq (6.91), dmax is the

211


maximum distance from crack tip to the n-point at which the residual vectors are calculated, and dmin is the minimum distance. In our analysis dmin is set to be 0.05c.

x2

B a

2b

2.5b

A b a

x1

Fig. 6.4 Geometry of the crack-inclusion system Example 1: Consider a crack of length 2b and an inclusion embedded in an infinite plate as shown in Fig. 6.4. The uniform heat flow hn0 is applied on the crack face only. In our analysis, the crack was modelled by 40 linear elements. Table 1 shows that the numerical results for the coefficients of stress intensity factors bi at point A (see Fig. 6.4) vary with dmax when the crack angle a = 0D and n=5, 10, 15, respectively, where bi are defined by K I = hn 0 c pc g 33b1 / k1 , K II = hn 0 c pc g11b2 / k1 ,

(6.115)

K D = hn 0 c pcc3b D / k1

Table 6.1 The BEM results for coefficients bi vs dmax in Example 1 dmax/c 0.5

1.0

1.5

SIEM

n 5 10 15 5 10 15 5 10 15

b1 1.230 1.224 1.222 1.225 1.221 1.220 1.231 1.227 1.226 1.207

b2 0.328 0.323 0.322 0.321 0.318 0.318 0.328 0.324 0.323 0.311

bD 0.755 0.747 0.745 0.749 0.745 0.743 0.757 0.752 0.750 0.732


212 1.4 1.2 1 0.8 0.6 0.4 0.2

0

20

40

60

80

100

Fig. 6.5 The coefficients bi versus crack angle a For comparison, the singular integral equation method (SIEM) given in [12,13] was used to obtain corresponding results. It can be seen from Table 6.1 that the results obtained using dmax=c are often closer to those obtained by SIEM than by using dmax=0.5c or dmax=1.5c. This is because more data can be included into the leastsquare method for a large dmax, but a too large dmax may not represent the crack-tip properties and can cause errors. Figure 6.5 shows the results of coefficients bi as a function of crack angle a when dmax=c and n=15. It is evident from the figure that all the coefficients bi are not very sensitive to the crack angle, but vary slightly with it. It is also evident that the two numerical models (BEM and SIEM) provide almost the same results. Example 2: Consider a rectangular thermopiezoelectric plate containing a crack and an inclusion as shown in Fig. 6.6. In the calculation, each side of the outer boundary is modelled by 50 linear elements and the crack is divided into 40 linear elements, and dmax=b and n=15 are used. In Fig. 6.6 the coefficients of SED intensity factors bi at point A (see Fig. 6.6) are presented as a function of crack orientation angle a. Numerical results for such a problem are not yet available in the literature. Therefore for comparison, the well-known finite element method [14] is used to obtain corresponding results. In the calculation, an eight-node quadrilateral element model has been used. In addition, the three nodes along one of the sides of each of the quadrilateral elements are collapsed at the crack tip and the two adjoining mid-points are moved to the quarter distances [15], in order to produce 1/r1/2 type of singularity. It can be seen from Fig. 6.7 that the values of bi are more sensitive to crack orientation than those in Example 1. They reach their peak values at a = 37D for b1 , a = 42D for b2 , and a = 50D for b D , respectively. It is also evident from Fig. 6.7 that the maximum discrepancy between the numerical results obtained from the two models is less than 5%.

213


h2 = 0 hn = h0

T=0

A

B

2b

a 1.5b b

T=0

3.5a

a

hn = 0

b

h2 = 0

4a

Fig. 6.6 Configuration of the crack-inclusion system in Example 2 (a=2b)

Present method

2

Finite element method

b1 1.5

b2

1

bD 0.5

0

0

20

40

60

80

100

a(deg.)

Fig. 6.7 SED intensity factors versus crack angle a

References [1] Qin QH, Thermoelectroelastic analysis of cracks in piezoelectric half-plane by BEM, Computa Mech, 23, 353-360, 1999 [2] Qin QH and Lu M, BEM for crack-inclusion problems of plane thermopiezoelectric solids, Int J Numer Meth Eng, 48, 1071-1088, 2000 [3] Qin QH and Mai YW, BEM for crack-hole problems in thermopiezoelectric materials, Eng Frac Mech, 69, 577-588, 2002 [4] Qin QH, Green’s functions of magnetoelectroelastic solids and applications to fracture analysis, pp93-106, in Proc. of 9th Int Conf on Inspection, Appraisal, Repairs & Maintenance of Structures, Fuzhou, China, 20-21 Oct, 2005 [5] Yin HP and Ehrlacher A, Variational approach of displacement discontinuity method and application to crack problems, Int J Frac, 63, 135-153, 1993

214


[6] Aliabadi MH and Rooke DP, Numerical Fracture Mechanics, Computational Mechanics Publications: Southampton, 1991 [7] Ju SH, Simulating stress intensity factors for anisotropic materials by the leastsquare method, Int J Frac, 81, 283-397, 1996 [8] Ju SH, Simulating three-dimensional stress intensity factors by the least-square method, Int J Numer Meth Eng, 43, 1437-1451, 1998 [9] Khalil SA, Sun CT and Kwang WC, Application of a hybrid finite element method to determine stress intensity factors in unidirectional composites, Int J Frac, 31, 37-51, 1986 [10] Qin QH, Micromechanics-BE solution for properties of piezoelectric materials with defects, Eng Anal Boundary Elements, 28, 809-814, 2004 [11] Qin QH and Yu SW, Effective moduli of thermopiezoelectric material with microcavities, Int J Solids Struct, 35, 5085-5095, 1998 [12] Hill LR and Farris TN, Three dimensional piezoelectric boundary elements. Proceedings of SPIE, Mathematics and Control in Smart Structures, 3039, 406417, 1997 [13] Khutoryaansky N, Sosa H and Zu WH, Approximate Green’s functions and a boundary element method for electroelastic analysis of active materials, Compu & Struct, 66, 289-299, 1998 [14] Oden JT and Kelley BE, Finite element formulation of general electrothermoelasticity problems, Int J Numer Meth Eng, 3, 161-179, 1971 [15] Anderson JL, Fracture Mechanics: Fundamentals and Applications, CRC Press: Boston, 1991

215

Chapter 7 Trefftz Boundary Element Method 7.1 Introduction In the previous two chapters, we presented boundary element formulation of piezoelectric problem using Green’s function and generalized dislocation solution. The boundary element formulation can also be established using the Trefftz function approach [1-4]. The name ‘Trefftz function’ here indicates that the regular trial functions, in contrast to the singular Green’s function, satisfy all governing differential equations. To distinguish the above two types of boundary element formulations, we refer to the former as Somigliana boundary element formulation (SBEF) and the latter as Trefftz boundary element method (TBEM). The TBEM can, in general, be classified as either indirect and direct formulations. In the indirect formulation [1,2], which is thought to be the original one presented by Trefftz, the solution of the problem is approximated by the superposition of the functions satisfying the governing differential equation, and then the unknown parameters are determined so that the approximate solution satisfies the boundary condition by means of the collocation, least square or Galerkin method. In the direct formulation, a relatively new formulation presented by Cheung and his co-workers [5,6], the weighted residual expression of the governing equation is derived by taking the regular Trefftz functions satisfying the governing equation as the weighting function, and then the boundary integral equation is obtained by applying the Gauss divergence formula to it twice. The resulting boundary integral equation, as in the boundary element method, is discretized and solved for the boundary unknowns. It should be mentioned that the basic distinction between the Trefftz and the conventional (sometimes known as Somigliana) BEM, presented in the previous two chapters, is that while Trefftz methods are based on the use of Trefftz functions which are regular functions, Somigliana methods make use of Green’s functions which are either singular or hyper-singular functions. As a consequence, whenever an integration is carried out, it is always simpler and more economical in the Trefftz method than in the Somigliana method. This is an important advantage that makes the TBEM so powerful when compared with conventional BEM. In this chapter, the two alternative techniques (indirect and direct formulations) are described briefly in order to provide an introduction to the TBEM. 7.2 Anti-plane piezoelectric problems In this section the application of the TBEM to anti-plane electroelastic problems is described. In particular, both direct and indirect methods with domain decomposition are considered. The TBEM discussed here is based on a weighted residual formulation presented in [1]. 7.2.1 Trefftz functions To show how the Trefftz functions of anti-plane piezoelectric problems can be generated, consider an anti-plane shear deformation involving only out-of-plane displacement uz and in-plane electric fields as defined by Eqs (2.278)-(2.285). The boundary conditions corresponding to the boundary value problem (2.278)-(2.285) can be written: uz = uz

on Gu

(7.1)

t = s3 j n j = t

on Gt

(7.2)


216

Dn = Di ni = - qs = Dn

on GD

(7.3)

f=f

on Gf

(7.4)

where u , t , Dn and f are, respectively, prescribed boundary displacement, traction force, surface charge and electric potential, an overhead bar denotes prescribed value, and G=Gu+ Gt =GD +Gf is the boundary of the solution domain W. It is well known that the solutions of the Laplace equation (2.285) may be found using the method of variable separation. By this method, the Trefftz functions corresponding to Eq (2.285) are obtained as [7] ¥

u z (r , q) = ¦ r m (a m cos mq + bm sin mq)

(7.5)

m =0 ¥

f(r , q) = ¦ r m (c m cos mq + d m sin mq)

(7.6)

m =0

for a bounded region and ¥

u z (r , q) = a 0* + a 0 ln r + ¦ r - m (a m cos mq + bm sin mq) ,

(7.7)

m =1 ¥

f(r , q) = c0* + c0 ln r + ¦ r - m (c m cos mq + d m sin mq)

(7.8)

m =1

for an unbounded region, where r and q are a pair of polar coordinates. Thus, the associated Trefftz function sets of Eqns (7.5)-(7.8) can be expressed in the form N = {1, rcosq, rsinq,..., r m cos mq, r m sin mq,...} = {N i } ,

(7.9)

N = {1, ln r , r -1 cos q, r -1 sin q,..., r - m cos mq, r - m sin mq,...} = {N i }

(7.10)

7.2.2 Special solution set for a subdomain containing an angular corner It is well known that singularities induced by local defects such as angular corners, cracks, and so on, can be accurately accounted for in the conventional FE or BEM model by way of appropriate local refinement of the element mesh. However, an important feature of the Trefftz method is that such problems can be far more efficiently handled by the use of particular solutions [7]. In this sub-section we show how particular solutions can be constructed to satisfy both the Laplace equation (2.285) and the traction-free boundary conditions on angular corner faces (Fig. 7.1). The derivation of such functions is based on the general solution of the twodimensional Laplace equation: ¥

¥

n =1

n =1

u z (r , q) = a 0 + ¦ (a n r l n + c n r -l n ) cos(l n q) + ¦ (d n r l n + g n r -l n ) sin(l n q) , (7.11) ¥

¥

n =1

n =1

f(r , q) = b0 + ¦ (bn r ln + f n r -ln ) cos(l n q) + ¦ (en r ln + hn r -ln ) sin(l n q)

(7.12)

Appropriate trial functions for a sub-domain containing a singular corner are obtained by considering an infinite wedge (Fig. 7.1) with particular boundary conditions prescribed along the sides q=±q0 forming the angular corner. The boundary conditions

217

Trefftz boundary element method

on the upper and lower surfaces of the wedge are free of surface traction and surface charge: s zq = c 44

¶u z ¶u ¶f ¶f =0, Dq = e15 z - k11 =0 + e15 r¶q r¶q r¶q r¶q

(7.13)

This leads to ¶u z = 0, ¶q

¶f =0 ¶q

(for q = ±q 0 )

(7.14)

x2 r Free edge

D

D

q

q0 q0

x1

Fig. 7.1 Typical subdomain containing a singular corner

Considering the symmetry of the free boundary condition on the x1 -axis of the singular corner and the different properties of sin- and cos- functions, which should depend on different variables in order to satisfy the same boundary conditions, introduce a set of new constants b n and rewrite the general solutions (7.11) as ¥

¥

n =1

n =1

u z (r , q) = a0 + ¦ (a n r l n + c n r -l n ) cos(l n q) + ¦ (d n r b n + g n r -b n ) sin(b n q) (7.15)

where l n and b n are two sets of constants which are assumed to be greater than zero. Differentiating the solution (7.15) and substituting it into Eq (7.14) yields ¶u z ¶q

¥

q= ± q0

= - ¦ l n (a n r l n + c n r -l n ) sin(± l n q 0 ) n =1

¥

+ ¦ b n (d n r bn + g n r -bn ) cos(±b n q 0 ) = 0

(7.16)

n =1

Since the solution must be limited for r = 0, we should specify cn = g n = 0 From Eq (7.16) it can be deduced that

sin(±l n q 0 ) = 0,

cos(±b n q 0 ) = 0,

(7.17)

(7.18)


218

leading to l n q 0 = np ,

(n =1,2,3,¼),

(7.19)

2b n q 0 = np ,

(n =1,3,5,¼)

(7.20)

Thus, for an element containing an edge crack (in this case q0 = p ), the solution can be written in the form ¥

u z (r , q) = a 0 + ¦ a n r n cos(nq) + n =1

¥

n

¥ ¥ n d n r 2 sin( q) = ¦ a n rn* + ¦ d n f n* ¦ 2 n =1, 3, 5 n =1 n =1, 3, 5

(7.21)

n

n where rn* = r n cos(nq) and f n* = r 2 sin( q). 2 It is obvious that the displacement function (7.21) includes the term proportional to r1/2, whose derivative is singular at the crack tip. The solution for the second equation of (7.14) can be obtained similarly and denoted as ¥

f(r , q) = b0 + ¦ bn r n cos(nq) + n =1

¥

n

¥ ¥ n * 2 e r sin( q ) = b r + ¦n ¦ ¦ en f n* n n 2 n =1, 3, 5 n =1 n =1, 3, 5

(7.22)

Thus, the associated T-complete sets of Eqs (7.21) and (7.22) can be expressed in the form 1

q N = {1, r cos q, r 2 sin( ),..., r m cos mq, r 2

2 m -1 2

sin(

2m - 1 q),...} = {N i } 2

(7.23)

7.2.3 Stress intensity factor Generally, stress intensity factors (SIF) can be evaluated by analysing stress and displacement fields near crack-tips using various numerical methods such as conventional FEM and BEM. These procedures are usually complicated and timeconsuming as they cannot calculate the SIF directly from basic variables like the coefficients di and ei. But in the light of the special purpose function for the crack-tip element, local field distributions can be easily obtained in crack problems, such as stress and electric displacement fields. Thus, the high efficiency in solving singular problems by the HTBE approach creates the attractive possibility of straightforwardly evaluating SIF K III and KD from di and ei, which are associated with the singular factors in particular solutions (7.21) and (7.22). To show this, considering the r-1/2 type of stress singularity, the corresponding SIF K III can be defined as s32 =

q q K III cos K III = lim 2pr s32 / cos 0 r ® 2 2 2pr

(7.24)

and when q = 0 K III = lim 2pr s32 r ®0

Substituting Eqs (2.281), (2.282), and (7.21) into Eq (7.25), we have

(7.25)

219

Trefftz boundary element method ¥ ¥ ¶r * ¶f * ½ K III = lim 2pr ®¦ (c 44 a n + e15 bn ) n + ¦ (c 44 d n + e15 en ) n ¾ r ®0 ¶y n =1,3,5 ¶y ¿ ¯ n =1

(7.26)

When the crack tip is defined at the origin of the polar coordinate system (see Fig. 7.1), Eq (7.26) can be written as K III = lim r ®0

2pr r

¥ § ¥ ¶r * ¶f * · ¨ ¦ (c 44 a n + e15 bn ) n + ¦ (c 44 d n + e15 en ) n ¸ ¨ ¶q n =1,3,5 ¶q ¸¹ © n =1

(7.27)

Substituting Eq (7.27) into Eq (7.26), we can obtain the expression of the stress singular factor K III =

p (c44 d1 + e15 e1 ) 2

(7.28)

In general, when q 0 ¹ p , the singularity becomes the type r l -1 , where l = 1 - p / 2q0 , and the general expression of stress intensity factors corresponding to stress singularity is defined as K III

º ª ( 2 p) 1 2 = lim « 1- l s32 (r ,0)» r ®0 »¼ «¬ r

(7.29)

and it can be also written as: K III = 2p (c 44 d1 + e15 e1 )

p 2q 0

(7.30)

Similarly, the singularity factor KD can also be written as K D = 2p (e15 d1 - k11e1 )

p 2q 0

(7.31)

7.2.4. Indirect formulation In the indirect method, the unknown displacement uz and electric potential f are approximated by the expansions as u ½ m ª N 1 j u = ® z¾ = ¦« ¯ f ¿ j =1 ¬ 0

º c j ½ ª N1 º c = Nc ¾= ® N 2( m + j ) »¼ ¯c m + j ¿ «¬N 2 »¼ 0

(7.32)

where Nij is taken from Eq (7.9) for subdomains without cracks and from Eq (7.23) for the rest, and c denotes the unknown vector. Using the definitions (2.280), (2.282), (7.2) and (7.3), the generalized boundary force and electric displacements can be given by t ½ s 3 j n j ½ m ª Q1 j T=® ¾=® ¾ = ¦« ¯ Dn ¿ ¯ D j n j ¿ j =0 ¬Q2 j

Q1( m + j ) º c j ½ ª Q1 º c = Qc ¾= ® Q2 ( m + j ) »¼ ¯c m + j ¿ «¬Q 2 »¼

(7.33)

With the expressions above, the indirect formulation corresponding to the anti-plane problem can be expressed by

³

Gu

(u z - u z ) w1 ds + ³ ( f - f) w2 ds + ³ (t - t ) w3 ds + ³ ( Dn - Dn ) w4 ds = 0 Gf

Gt

GD

(7.34)


220

where wi (i=1-4) are arbitrary weighting functions and uz, f, t, Dn have the series representations (7.32) and (7.33). (a) Galerkin method If we use the Galerkin method, the weighting functions are chosen as arbitrary variations of the expressions (7.32) and (7.33), that is: w1 = Q1dc , w2 = Q 2 dc , w3 = - N 1dc , w4 = -N 2 dc

(7.35)

Substituting Eq (7.35) into Eq (7.34), yields Kc = f

(7.36)

where K=³

Gu

f =³

Gu

Q1T N1ds - ³

Gt

Q1T u z ds - ³

Gt

N1T Q1ds + ³ N1T t ds + ³

QT2 N 2 ds - ³

Gf

Gf

QT2 fds - ³

GD

GD

NT2 Q 2 ds

NT2 Dn ds

(7.37) (7.38)

It is noted that the formulation above applies only to the solution domain containing one semi-infinite crack, when the particular solution (7.23) is used as the weighting function. For multi-crack problems, the domain decomposition approach is required. In this case, the solution domain is divided into several sub-domains (Fig. 7.2). For example, a domain containing two cracks can be divided into four sub-domains (Fig. 7.2), In the figure, Wi (i=1-4) denote the sub-domains, G the outer boundary, and GIij the inner boundaries between sub-domains. For each sub-domain, the indirect method leads to K ici = fi

(i=1-4)

(7.39)

W3 GI 23

W1

GI 34

W2 GI 12

GI 24

W4

G

Fig. 7.2 Four sub-domain problem

221


On the inner boundary GIij, the continuity conditions provide u zIi = u zIj , f iI = f Ij , t Ii = -t Ij , DnIi = - DnIj

(7.40)

where the subscript I stands for the inner boundary, and superscript i (or j) means the ith (or jth) subdomain. Eqs (7.39) and (7.40) can be used to solve multiple crack problems. (b) Point-collocation formulation The collocation technique is obtained when the four weighting functions in Eq (7.34) are chosen as the Dirac delta function: w j = d( P - Pi ) (j=1-4)

(7.41)

where Pi is the collocation point. Substituting Eq (7.41) into Eq (7.34) yields u z ( Pi ) = N 1 ( Pi )c = u z ( Pi ) ,

( Pi Î Gu , i = 1,2, " M 1 )

(7.42)

f( Pi ) = N 2 ( Pi )c = f( Pi ) ,

( Pi Î G f , i = 1, 2," M 2 )

(7.44)

t ( Pi ) = Q1 ( Pi )c = t ( Pi ) ,

( Pi Î G t , i = 1, 2," M 3 )

(7.43)

Dn ( Pi ) = Q 2 ( Pi )c = Dn ( Pi ) ,

( Pi Î GD , i = 1,2, " M 4 )

(7.45)

where M1-M4 are the numbers of the collocation points placed on Gu , Gt , G f and GD , respectively. Using matrix notation, Eqs (7.42)-(7.45) can be written as K ij c j = f i or Kc = f

(7.46)

where the unknown c j represents the constant coefficient of the jth term in the expansion (7.32), and Kij and fi are respectively given by N1 j ( Pi ) ° N (P ) ° 2j i K ij = ® ° Q1 j ( Pi ) °¯ Q2 j ( Pi )

u z ( Pi ) ° f( P ) ° i fi = ® t ( P i) ° °¯ Dn ( Pi )

if Pi Î G u , if Pi Î G f , if Pi Î G t ,

(7.47)

if Pi Î G D

if Pi Î Gu , if Pi Î G f , if Pi Î Gt ,

(7.48)

if Pi Î G D

(c) Least square method The least square method formulation can be obtained by considering the residual function R (c) = ³ ( N 1c - u z ) 2 ds + a1 ³ ( N 2 c - f) 2 ds Gu

Gf

+ a 2 ³ (Q1c - t ) 2 ds + a 3 ³ (Q 2 c - Dn ) 2 ds Gt

GD

(7.49)


222

where a i (i=1,2,3) are three weighting parameters which preserve the numerical equivalence between the first and the remaining terms on the right hand side of the above equation. In the least square method, the derivative of the residual function with respect to c is forced to vanish:

R(c) = 2³ N1T (N1c - u z )ds + 2a1 ³ N T2 (N 2 c - f)ds Gu Gf ¶c + 2a 2 ³ Q1T (Q1c - t ) 2 ds + 2a 3 ³ QT2 (Q 2c - Dn )ds = 0 Gt

GD

(7.50)

Rearranging the above equations, we have T ª N T N ds + a Q 1T Q1 ds + a 3 ³ Q T2 Q 2 ds ºc 1 ³ N 2 N 2 ds + a 2 ³ «¬ ³ Gu 1 1 »¼ Gf Gt GD

=³

Gu

(7.51)

N 1T u z ds + a 1 ³ N 1T fds + a 2 ³ Q 1T t ds + a 3 ³ Q T2 Dn ds Gf

Gt

GD

or in matrix form K ij c j = f i

or Kc = f

(7.52)

where

K ij = ³ N1i N1 j ds + a1 ³ N 2i N 2 j ds + a 2 ³ Q1iQ1 j ds + a3 ³ Q2iQ2 j ds Gu

fi = ³

Gf

Gu

Gt

GD

N1iu ds + a1 ³ N 2i fds + a 2 ³ Q1iu ds + a3 ³ Q2i Dn ds Gf

Gt

GD

(7.53) (7.54)

In this formulation, it is important to choose the proper value of a i . How to choose the value of a i is, however, still an open question, and there is no general rule for that choice at present. Generally, the optimal value of a i for a given type of problem should be found by numerical experimentation. (d) Modified Trefftz formulation Another indirect formulation called the ‘modified Trefftz formulation’ appears to be due to Oliveira [7] and Patterson and Sheikh [8,9]. In this approach, the approximate solutions u and qn are expressed in terms of a linear combination of the singular fundamental solution, and then the unknown parameters are determined so that the approximate solution satisfies the boundary conditions by using the collocation method. It is well known that the fundamental solution of the two-dimensional Laplace equation (2.285) u *z (rPQ ) =

1 §¨ 1 ln 2p ¨© rPQ

§ · ¸ , f * (rPQ ) = 1 ln¨ 1 ¸ 2p ¨© rPQ ¹

· ¸ ¸ ¹

(7.55)

has a singularity at P=Q, where P and Q stand for the observation point and source point, respectively. To avoid this singularity, the source points Qi (i=1,2,…,N) are placed on an imaginary boundary outside the solution domain and u(P) is approximated as follows:

223

Trefftz boundary element method * 0 º c j ½ ª N1* º u ( P)½ m ªu z (rPQ j ) c = N *c u( P ) = ® z »® ¾= ¾ = ¦« * f (rPQ j )»¼ ¯c m + j ¿ «¬N *2 »¼ ¯ f( P) ¿ j =1 «¬ 0

(7.56)

where m is the total number of the source points, and rPQi = ( x P - x Qi ) 2 + ( y P - y Qi ) 2 .

(7.57)

Differentiating eqn (10.27) with respect to the normal direction n, we have * t ½ s 3 j n j ½ m ª Q1 j = T=® ¾=® ¾ ¦« * ¯ Dn ¿ ¯ D j n j ¿ j =0 ¬«Q2 j

Q1*( m + j ) º c j ½ ªQ1* º * »® ¾ = « »c = Q c Q2*( m + j ) ¼» ¯c m + j ¿ ¬Q *2 ¼

(7.58)

The collocation method leads to the following equations: u z ( Pi ) = N 1* ( Pi )c = u z ( Pi ) ,

( Pi Î Gu , i = 1,2, " M 1 )

(7.59)

t ( Pi ) = Q1* ( Pi )c = t ( Pi ) ,

( Pi Î Gt , i = 1,2, " M 2 )

(7.60)

f( Pi ) = N *2 ( Pi )c = f( Pi ) ,

( Pi Î Gf , i = 1,2, " M 3 )

(7.61)

Dn ( Pi ) = Q *2 ( Pi )c = Dn ( Pi ) ,

( Pi Î GD , i = 1,2, " M 4 )

(7.62)

or in matrix form Kc = f

(7.63)

where K and f are determined in a way similar to the treatment in Eqs (7.46)-(7.48), and N =M1+M2+M3+M4. In this approach, it is important to select the appropriate points Qi (i=1,2,…N). 7.2.5 Direct formulation The Trefftz direct formulation is obtained by considering [1]

³³

W

[Ñ 2 u z v1 + Ñ 2 fv 2 ]dW = 0

(7.64)

Performing the integration by parts and taking the expression (7.32) as weighting function, that is: v ½ m ª N1 j v = ® 1¾ = ¦« ¯v 2 ¿ j =1 ¬ 0

0 º c j ½ ª N1 º c = Nc ® ¾= N 2 j »¼ ¯c m + j ¿ «¬N 2 »¼

(7.65)

we have cT ³

G

(N 1T t - Q1T u z + N T2 Dn - Q T2 f)ds = 0

(7.66)

Since the equation is valid for arbitrary vectors c, we have

³

G

(N 1Ti t - Q1Ti u z + N T2i Dn - Q T2i f)ds = 0

(7.67)

The analytical results for Eq (7.67) are, in general, impossible, and therefore a numerical procedure must be used to solve the problem. As in the conventional BEM, the boundary G is divided into k linear elements, for which uz, t, f, and Dn are approximated by


224 k

k

k

k

i =1

i =1

i =1

i =1

u z ( s ) = ¦ u zi Fi ( s ) , t ( s ) = ¦ t i Fi ( s ) , f( s ) = ¦ f i Fi ( s ) , Dn ( s ) = ¦ Dni Fi ( s ) (7.68)

where u zi , t i , f and Dn are, respectively, their values at node i, s is a length coordinate defined in Fig. 6.2, Fi(s) is defined in Eqs (6.10)-(6.12) and is a global shape function associated with the ith-node. Fi(s) is zero-valued over the whole mesh except within two elements connected to the ith-node (see Fig. 6.2). Following the abovementioned discretization, Eq (7.67) becomes Gu = Ht

(7.69)

Applying the boundary conditions, we have

[G 1

G2 G3

t ½ u z ½ ° ° °u ° ° ° ° t ° G 4 ]® z ¾ = [H 1 H 2 H 3 H 4 ]® ¾ °ĳ° ° Dn ° °¯ ĳ °¿ °¯D n °¿

(7.70)

or simply

Kx = f

(7.71)

The direct formulation above is only suitable for single crack problems. For multicrack problems, as in Section 7.2.4, the domain decomposition approach is used to convert them into several single crack problems. For a particular single crack problem with sub-domain i (see Fig. 7.2), Eq (7.71) becomes K i xi = fi

in Wi

(7.72)

while on the inner boundary GIij, the continuity condition is again defined in Eq (7.40). 7.3 Plane piezoelectricity The application of the Trefftz boundary element method to plane piezoelectricity was discussed in [2-4,11,12]. Sheng and Sze’s formulation was based on the multi-region approach and a special set of Trefftz functions that satisfy the traction-free and charge-free conditions at free surfaces such as crack faces [4]. Sheng et al [3] presented the same TBEM as in [4] but used a different approach to derive the Trefftz function. Yao and Wang [11] and Li and Yao [12] developed a modified TBEM which is based on the concept of an imaginary boundary and Green’s functions of plane piezoelectricity. 7.3.1 Governing equations For convenience, the basic equations of plane piezoelectricity presented in Chapter 1 are briefly summarized as follows: s11,1 + s12,2 + b1 = 0, s12,1 + s 22,2 + b2 = 0, D1,1 + D2,2 + be = 0 ,

e i, j =

1 (u i , j + u j ,i ) , E i = -f ,i 2

(in W )

(7.73)

(in W )

(7.74)

225


s11 ½ ª c11 c12 °s ° « c °° 22 °° « 12 c22 0 ® s12 ¾ = « 0 °D ° « 0 0 ° 1° « « ¯° D2 ¿° ¬e21 e22 ui = ui ti = sij n j = ti

0 0 c33

0 0 e13

e13

-k11

0

0

on Gu ½ ¾, on Gt ¿

e21 º e11 ½ e22 »» °° e 22 °° ° ° 0 » ® 2e12 ¾ , » 0 » ° - E1 ° ° ° -k 22 ¼» ¯°- E2 ¿°

(in W )

f=f Dn = Di ni = - qs

on G f ½ ¾ on G D ¿

(7.75)

(7.76)

7.3.2 Trefftz functions For the boundary value problem (7.73)-(7.76), its Trefftz functions can be constructed based on the solutions (1.160) and (1.163). It should be noted that the constants w0, u0, v0 in Eq (1.163) represent rigid body displacements, and f0 is a reference potential. These constants are ignored in the following development. Using the solutions (1.160) and (1.163), the plane strain piezoelectric problem is reduced to the one of finding three complex potentials, F1 , F 2 and F 3 , in some region W of the material. Each potential is a function of a different generalized complex variable zk = x1 + pk x2 . The discussion in this subsection is from the development in [3,13]. (a) Trefftz functions for domain without defects For an interior domain problem without any flux singularities, F1 , F 2 and F 3 can be represented by the Talor series as [3,13] ¥

F k ( zk ) = ¦ (a (kn ) + ib(kn ) ) zkn (k=1,2,3)

(7.77)

n =1

where i = -1 , and a’s and b’s are real constants to be determined. By substituting the expression (7.77) into Eq (1.163) and using the polar coordinate systems (rk , qk ) defined by zk = x1 + pk x2 = rk (cos qk + i sin qk ) ,

(7.78)

the basic set of Trefftz functions for interior domain problems can be obtained as Suint = {Re(D1 )r1n cos nq1 - Im(D1 )r1n sin nq1 , - Re( D1 )r1n sin nq1 - Im( D1 )r1n cos nq1 , Re(D2 )r2n cos nq2 - Im(D2 )r2n sin nq2 , - Re(D2 )r2n sin nq2 - Im(D2 )r2n cos nq2 , Re(D3 )r3n cos nq3 - Im(D3 )r3n sin nq3 , - Re(D3 )r3n sin nq3 - Im(D3 )r3n cos nq3 } for n=1,2,3…

(7.79)

which corresponds to the coefficients a (kn ) and b(kn ) , where Dk = { pk* , qk* , tk*}T , and the superscript u indicates that the basic solution set is for the vector of primary variables u1, u2, and f. In the case of exterior domain problems without flux singularity, Sheng et al [3] expressed the potential functions Fk in terms of the following Laurent series as ¥

F k ( zk ) = ¦ (a (k- n ) + ib(k- n ) ) zk- n n =1

(k=1,2,3)

(7.80)


226

Substituting Eq (7.80) into Eq (1.163), the basic set of Trefftz functions for exterior domain problems can be obtained as Suext = {Re(D1 ) Re( z1- n ) - Im( D1 ) Im( z1- n ), - Re(D1 ) Im( z1- n ) - Im(D1 ) Re( z1- n ), Re(D2 ) Re( z2- n ) - Im(D2 ) Im( z2- n ), - Re(D2 ) Im( z2- n ) - Im(D2 ) Re( z2- n ), (7.81) Re(D3 ) Re( z3- n ) - Im(D3 ) Im( z3- n ), - Re(D3 ) Im( z3- n ) - Im(D3 ) Re( z3- n )} which corresponds to the unknown coefficients a (k- n ) and b(k- n ) . x2¢ (poling direction)

x2

r x1

x q

x1¢ crack-tip

Fig. 7.3 An arbitrarily oriented crack (b) Trefftz functions for domain with an impermeable crack Consider a domain with an arbitrarily oriented impermeable crack as shown in Fig. 7.3. In the figure, the system ( x1¢, x2¢ ) represents the principal material coordinate system, and (x1,x2) is the coordinate system defined on the crack-tip with the x1-axis along the crack line. x stands for the angle between the crack line and the x1¢ -axis. The special set of Trefftz functions can be constructed by considering the following complex potential functions

F k ( zk ) = ¦ (a (kh) + ib(kh) ) zk( h) = ¦ (a (kh) + ib(kh) )rkh (cos hqk + i sin hqk ) h

(7.82)

h

satisfying the boundary conditions on crack faces s 22 q=± p = s12 q=± p = D2 q=± p = 0

(7.83)

where h’s are eigenvalues to be determined. Substituting Eq (7.82) into Eqs (1.160) and (1.163), we have [3] 3

f = 2¦¦ rkh (a (kh) [ Atk cos hqk - Btk sin hqk ] - b(kh) [ Atk sin hqk + Btk cos hqk ]) (7.84) k =1 h

3

s 22 = 2¦¦ hrkh-1 (a (kh) cos(h - 1)qk - b(kh) sin(h - 1)qk ) k =1 h

(7.85)

227

Trefftz boundary element method 3

s12 = 2¦¦ hrkh-1 (a (kh) [ Apk cos(h - 1)qk - B pk sin(h - 1)qk ] k =1 h

(7.86)

( h) k

- b [ Apk sin(h - 1)qk + B pk cos(h - 1)qk ]) 3

D2 = 2¦¦ hrkh-1 (a (kh) [ Avk cos(h - 1)qk - Bvk sin(h - 1)qk ] k =1 h

(7.87)

( h) k

- b [ Avk sin(h - 1)qk + Bvk cos(h - 1)qk ])

where Atk = Re(tk* ), Btk = Im(tk* ), Apk = Re( pk ), B pk = Im( pk ), Avk = Re(v k ), and Bvk = Im(v k ) . Noting that qk = ±p and rk = r at q = ±p , substituting Eqs (7.85)(7.87) into Eq (7.83) leads to X(h)Gq = 0

(7.88)

X(h) = diag[cos hp,sin hp, cos hp,sin hp, cos hp,sin hp]

(7.89)

where

a1( h) ½ ° ( h) ° ° b1 ° °°a ( h) °° q = ® (2h) ¾ , ° b2 ° °a3( h) ° ° ( h) ° °¯ b3 °¿

ª 1 « 0 « «A G = « p1 « B p1 « Av1 « «¬ Bv1

0

1

0

1

1

0

1

0

- B p1 Ap1

Ap 2 Bp 2

- Bp 2 Ap 2

Ap 3 Bp3

- Bv1

Av 2

- Bv 2

Av 3

Av1

Bv 2

Av 2

Bv 3

0 º 1 »» - Bp3 » » Ap 3 » - Bv 3 » » Av 3 »¼

(7.90)

As was pointed out in [3], G is in full rank for most practical piezoelectric materials. In this case, the non-trivial solution of q is determined by setting the determinant of X(h) to zero. Thus, the eigenvalues h’s are solved to be h= n/2,

for n = 0,1, 2,3,"

(7.91)

As was done in [3], taking a1( h) , h1( h) , and a (2h) as independent coefficients, the following relationships can be found from Eq (7.88) as b(2h) = J1a1( h) + J 2b1( h) + J 3a (2h) , a3( h) = J 4 a1( h) + J 5b1( h) + J 6 a (2h) , b

( h) 3

( h) 1

= J 7a

( h) 8 1

+J b

+ J 9a

(7.92)

( h) 2

where J1 = ( Bv 3 B p1 - Bv1 B p 3 ) / J 0 ,

J 2 = [ Bv 3 ( Ap1 - Ap 3 ) - B p 3 ( Av1 - Av 3 )] / J 0 , (7.93)

J 3 = ( Bv 3 B p 2 - Bv 2 B p 3 ) / J 0 ,

J 4 = [ Bv1 ( Ap 2 - Ap 3 ) - B p1 ( Av 2 - Av 3 )] / J 0 , (7.94)

J 5 = [ Ap1 ( Av 3 - Av 2 ) + Ap 2 ( Av1 - Av 3 ) + Ap 3 ( Av 2 - Av1 )] / J 0 ,

(7.95)

J 6 = [ Bv 2 ( Ap 2 - Ap 3 ) - B p 2 ( Av 2 - Av 3 )] / J 0 ,

J 7 = - J1 ,

(7.96)

J 8 = [ Bv 3 ( Ap1 - Ap 2 ) - B p 3 ( Av1 - Av 2 )] / J 0 ,

J9 = - J3 ,

(7.97)


228

J 0 = Bv 3 ( Ap 3 - Ap 2 ) - B p 3 ( Av 3 - Av 2 )

(7.98)

for odd n, i.e. h = 1/ 2,3 / 2,5 / 2," , and J1 = [ Bv 3 ( Ap1 - Ap 3 ) - B p 3 ( Av1 - Av 3 )] / J 0 , J 3 = [ Bv 3 ( Ap 2 - Ap 3 ) - B p 3 ( Av 2 - Av 3 )] / J 0 , J 7 = [ Bv 2 ( Ap 3 - Ap1 ) - B p 2 ( Av 3 - Av1 )] / J 0 ,

J 2 = ( Bv1 B p 3 - Bv 3 B p1 ) / J 0 ,

(7.99)

J 4 = J 6 = -1, J 5 = 0,

(7.100)

J 8 = ( Bv 2 B p1 - Bv1 B p 2 ) / J 0 , (7.101)

J 9 = [ Bv 2 ( Ap 3 - Ap 2 ) - B p 2 ( Av 3 - Av 2 )] / J 0 ,

J 0 = Bv 3 B p 2 - Bv 2 B p 3 ,

(7.102)

for even n, i.e. h = 0,1, 2,3," . Incorporating Eqs (7.82), (7.91), and (7.92) into Eq (1.163), the following special solution set for impermeable crack-tip fields can be obtained as Sucrack -im = {S a( n1 / 2) + J1Sb( n2 / 2) + J 4S a( n3/ 2) + J 7Sb( n3 / 2) , Sb( n1 / 2) + J 2Sb( n2 / 2) + J 5S a( n3/ 2) + J 8Sb( n3 / 2) , S (an2/ 2) + J 3Sb( n2 / 2) + J 6S a( n3/ 2) + J 9Sb( n3 / 2) },

for n = 0,1, 2,3," (7.103) ( n / 2) 1

which corresponds to the independent coefficients {a

( n / 2) 1

,b

,a

( n / 2) 2

} , where

n n S (ank/ 2) = 2 Re(Dk )rkn / 2 cos qk - 2 Im(Dk )rkn / 2 sin qk , 2 2 n n Sb( nk / 2) = -2 Re(Dk )rkn / 2 sin qk - 2 Im(Dk )rkn / 2 cos qk 2 2

(7.104)

(c) Trefftz functions for domain with a permeable crack In addition to the impermeable boundary condition, Sheng et al [3] also consider a domain with a permeable crack. In the case of a permeable crack, the boundary conditions along the crack faces (7.83) become s 22 q=± p = s12 q=± p = 0, f q=p = f q=-p , D2

q=p

= D2 q=-p

(7.105)

Making use of Eqs (7.84)-(7.87) and (7.105), an equation analogous to Eq (7.88) can be obtained. The pertinent eigenvalues h’s are again solved to be Eq (7.91). However, the eigenfunctions are different: (i) For odd n, i.e. h = 1/ 2,3 / 2,5 / 2," , and with a1( h) and b1( h) taken as independent coefficients, the following relationships can be found as a (2h) = J11a1( h) + J12b1( h) ,

b(2h) = J13a1( h) + J14b1( h) ,

a3( h) = J15a1( h) + J16b1( h) ,

b3( h) = J17 a1( h) + J18b1( h)

(7.106)

where J11 = [( Ap 3 - Ap 2 )( Bv 3 Bt1 - Bv1 Bt 3 ) + ( Av 3 - Av 2 )( B p1 Bt 3 - B p 3 Bt1 ) + ( At 3 - At 2 )( Bv1 B p 3 - Bv 3 B p1 )] / J10

(7.107)

J12 = [( Ap 3 - Ap 2 )( Bv 3 At1 - Av1 Bt 3 ) + ( Ap 3 - Ap1 )( Av 2 Bt 3 - Bv 3 At 2 ) + ( Ap 2 - Ap1 ) ´ ( Bv 3 At 3 - Av 3 Bt 3 ) + Av 3 B p 3 ( At 2 - At1 ) + Av 2 B p 3 ( At1 - At 3 ) + Av1 B p 3 ( At 3 - At 2 )] / J10

(7.108)

229


J13 = [ Bv 3 ( B p 2 Bt1 - B p1 Bt 2 ) + Bv 2 ( B p1 Bt 3 - B p 3 Bt1 ) + Bv1 ( B p 3 Bt 2 - B p 2 Bt 3 )] / J10 (7.109) J14 = [( Ap 3 - Ap1 )( Bv 3 Bt 2 - Bv 2 Bt 3 ) + ( Av 3 - Av1 )( B p 2 Bt 3 - B p 3 Bt 2 ) + ( At 3 - At1 )( Bv 2 B p 3 - Bv 3 B p 2 )] / J10 J15 = [( Ap 3 - Ap 2 )( Bv1 Bt 2 - Bv 2 Bt1 ) + ( Av 3 - Av 2 )( B p 2 Bt1 - B p1 Bt 2 ) + ( At 3 - At 2 )( Bv 2 B p1 - Bv1 B p 2 )] / J10

(7.110)

(7.111)

J16 = [( At 2 - At1 )( Bv 2 Ap 3 - B p 2 Av 3 ) + ( At 3 - At 2 )( Ap1 Bv 2 - B p 2 Av1 ) + ( At 3 - At1 ) ´ ( B p 2 Av 2 - Bv 2 Ap 2 ) + Ap1 Bt 2 ( Av 2 - Av 3 ) + Ap 2 Bt 2 ( Av 3 - Av1 ) + Ap 3 Bt 2 ( Av1 - Av 2 )] / J10

(7.112) J17 = - J13

(7.113)

J18 = [( Ap 2 - Ap1 )( Bv 2 Bt 3 - Bv 3 Bt 2 ) + ( Av 2 - Av1 )( B p 3 Bt 2 - B p 2 Bt 3 ) + ( At 2 - At1 )( Bv 3 B p 2 - Bv 2 B p 3 )] / J10 J10 = ( Ap 3 - Ap 2 )( Bv 2 Bt 3 - Bv 3 Bt 2 ) + ( Av 3 - Av 2 )( B p 3 Bt 2 - B p 2 Bt 3 ) + ( At 3 - At 2 )( Bv 3 B p 2 - Bv 2 B p 3 )

(7.114)

(7.115)

(ii) for even n, i.e. h = 0,1, 2,3," and with a1( h) , b1( h) , a (2h) , and b(2h) taken as independent coefficients, we have a3( h) = -a1( h) - a (2h) ,

b3( h) = J19 a1( h) + J 20b1( h) + J 21a (2h) + J 22b(2h)

(7.116)

where J19 =

( Ap1 - Ap 3 ) Bp3

, J 20 = -

B p1 Bp3

, J 21 =

( Ap 2 - Ap 3 ) Bp3

, J 22 = -

Bp 2 Bp3

(7.117)

Incorporating Eqs (7.82), (7.91), (7.106), and (7.117) into Eq (1.163) yields S ucrack - p = S un1 È S un2

(7.118)

where Sun1 = {S (an11 / 2) - S a( n31 / 2) + J19Sb( n31 / 2) , Sb( n11 / 2) + J 20Sb( n31 / 2) , S (an21 / 2) - S a( n31 / 2) + J 21Sb( n31 / 2) , Sb( n21 / 2) + J 22Sb( n31 / 2) }, for n1 = 0, 2, 4," Sun2 = {S (an12 / 2) + J11S a( n22 / 2) + J13Sb( n22 / 2) + J15S a( n32 / 2) + J17Sb( n32 / 2) , Sb( n1 2 / 2) + J12S a( n22 / 2) + J14Sb( n22 / 2) + J16S a( n32 / 2) + J18Sb( n32 / 2) }, for n2 = 1,3,5, "

(7.119)

(7.120)

and S’s are defined by Eq (7.104). (d) Trefftz functions for domain with an impermeable elliptic hole Consider a piezoelectric plate containing an impermeable elliptic hole as shown in Fig. 7.4. To apply the hole boundary conditions conveniently, the conformal mappings defined by Eq (2.162) are used and rewritten as


230

zk =

c - ipk b c + ipk b 1 zk + 2 2 zk

(7.121)

where a and b are the semi-axis of the elliptic hole (Fig. 7.4). The inverse mappings of z k are given by z k1,2 =

zk ± ( zk2 - a 2 - pk2b 2 )1/ 2 a - ipk b

(7.122)

in which the sign of the square root ( ± ) is chosen in such a way that z k ³ 1. In terms of the variables, Eqs (1.160) and (1.163) are rewritten as

pk* ½ u1 ½ 3 ° *° ° ° ®u2 ¾ = 2 Re ¦ ® qk ¾F k (z k ) k =1 ° * ° °f° ¯ ¿ ¯ tk ¿

pk2 ½ s11 ½ 3 ° ° F¢k (z k ) ° ° , ®s 22 ¾ = 2 Re ¦ ® 1 ¾ zk¢ (z k ) k =1 ° °s ° ° ¯ 12 ¿ ¯- pk ¿

(7.123)

3 v k pk ½ F¢k (z k ) D1 ½ ® ¾ = 2 Re ¦ ® ¾ k =1 ¯ -v k ¿ zk¢ (z k ) ¯ D2 ¿

(7.124)

x2¢ (poling direction) x2

r x

x1 q x1¢

2b 2a

Fig. 7.4 Coordinate systems for elliptic hole

For the traction-free and charge-free conditions along the hole boundary, we have 3

3

3

k =1

k =1

k =1

Re ¦ F k (z k ) = 0, Re ¦ pk F k (z k ) = 0, Re ¦ v k F k (z k ) = 0, for z k = 1 (7.125) which can be expressed in matrix form [3,13] ª 1 1 1 º F1 ½ ª 1 1 1 º F1 ½ « p p p » °F ° = « p p p » °F ° 2 3»® 2¾ 2 3»® 2¾ « 1 « 1 «¬ v1 v 2 v3 ¼» ¯° F 3 ¿° ¬«v1 v 2 v3 ¼» ¯° F 3 ¿°

(7.126)

231


Eq (7.126) can be further written as F1 ½ ª E11 E12 E13 º F1 ½ ° ° « »° ° ®F 2 ¾ = « E21 E22 E23 » ®F 2 ¾ °F ° « E E E » °F ° ¯ 3 ¿ ¬ 31 32 33 ¼ ¯ 3 ¿

(7.127)

where ª E11 E12 E13 º ª 1 1 1 º «E E E » = « p p p » 2 3» « 21 22 23 » « 1 «¬ E31 E32 E33 »¼ «¬ v1 v 2 v3 »¼

-1

ª1 1 1º «p p p » 2 3» « 1 «¬v1 v 2 v3 »¼

(7.128)

The complex potential functions F k (z k ) for the elliptic hole problem can thus be chosen in the following form [3,13] ¥

F k ( zk ) = ¦ [(a (kn ) + ib(kn ) )z kn + (a (k- n ) + ib(k- n ) )z k- n ]

for k=1,2,3

(7.129)

n =1

It is noted that at any point along the hole boundary, z k ’s can be expressed as z1 = z 2 = z 3 = e i q

(7.130)

where q Î [-p, p]. By incorporating Eqs (7.129) and (7.130) into Eq (7.127), six constraints on the 12 real coefficients a’s and b’s can be established. Taking a (kn ) ’s and b(kn ) ’s as the independent coefficients and substituting Eq (7.129) into Eq (7.123) yields the special set of Trefftz functions in the form [3] Suhole = {Ĭ (an1) , Ĭb( n1 ) , Ĭ a( n2) , Ĭb( n2 ) , Ĭ (an3) , Ĭb( n3 ) } for n = 0,1, 2,3,"

(7.131)

where 3

3

i =1

i =1

Ĭ (ank) = Ȥ a( nk) + ¦ Re( Eik )Ȥ a( -i n ) - ¦ Im( Eik )Ȥ b( -i n ) , 3

Ĭ

(n) bk

=Ȥ

(n) bk

+ ¦ Im( Eik )Ȥ i =1

(-n) ai

3

- ¦ Re( Eik )Ȥ i =1

(7.132) (-n) bi

,

with Ȥ (ank) = 2 Re(Dk ) Re(z nk ) - 2 Im(Dk ) Im(z nk ), Ȥ (a-kn ) = 2 Re(Dk ) Re(z -k n ) - 2 Im(Dk ) Im(z k- n ), Ȥ b( nk ) = -2 Re(Dk ) Im(z nk ) - 2 Im(Dk ) Re(z nk ), Ȥ b( -k n ) = 2 Re(Dk ) Im(z -k n ) - 2 Im(Dk ) Re(z -k n )

7.3.3 Indirect formulations In the indirect techniques the generalized displacement u is approximated as

(7.133)


232

u1 ½ N1 ½ u1 ½ ° ° ° ° ° ° u = ®u2 ¾ = ® N 2 ¾ c + ®u2 ¾ = Nc + u ° f ° °N ° ° f ° ¯ ¿ ¯ 3¿ ¯ ¿

(7.134)

where cj stands for undetermined coefficient (denoted as a’s and b’s in the last subsection), Nj are the Trefftz functions extracted from the basic or special solution sets given in Section 7.3.2, and u (= {u1 , u2 , f}T ) are known special solutions due to generalized body force. If the governing differential equation (7.73) is rewritten in a general form Âu(x) + b(x) = 0

(7.135)

where Â stands for the differential operator matrix for Eq (7.73), x for the position vector, b (= {b1 , b2 , be }T ) for the known generalized body force term. Then u = u(x) and N = N(x) in eqn (7.134) have to be chosen such that Âu + b = 0 and ÂN = 0 (7.136) Using the above definitions the generalized boundary forces and electric displacements can be derived from Eqs (8), (9) and (7.134), and denoted as t1 ½ s1 j n j ½ t1 ½ Q1 ½ ° ° ° ° ° ° ° ° T = ® t2 ¾ = ®s2 j n j ¾ = ® t2 ¾ + ®Q 2 ¾ c = Qc + T (7.137) ° D ° ° D n ° ° D ° °Q ° ¯ n¿ ¯ j j ¿ ¯ n¿ ¯ 3¿ where t i and Dn are derived from u . Since the displacement u is approximated by Trefftz functions that satisfy Eq (7.136), the weighted residual statement for the plane piezoelectric problem can be written as

³

Gu

(ui - ui ) w1i ds + ³ (f - f ) w2 ds + ³ ( ti - ti ) w3i ds + ³ ( Dn - Dn ) w4 ds = 0 (7.138) Gf

Gt

Gt

(a) Galerkin formulation The Galerkin formulation is obtained when both the weighting and trial functions are chosen from Eqs (7.134) and (7.137) such that w1i = Qi dc, w2 = Q3dc, w3i = - N i dc, w4 = -N 3dc ui = N i c + ui , f = N 3c + f, ti = Qi c + ti , Dn = Q3c + Dn Substituting these expressions into Eq (7.138), we have T T ³ Gu Qi (Nic + ui - ui )ds + ³ Gf Q3 (N3c + f - f) w2 ds - ³ NTi (Q i c + ti - ti ) ds - ³ NT3 (Q 3c + Dn - Dn ) ds = 0 Gt

(7.139) (7.140)

(7.141)

Gt

where the repeated index i represents the conventional summation from 1 to 2. Eq (7.141) can be conveniently written as the matrix form Kc = f with

(7.142)

233


K = ³ QTi N i ds - ³ NTi Qi ds + ³ QT3 N 3 ds - ³ Gu

Gt

Gf

GD

NT3 Q3 ds

f = ³ QTi (ui -ui )ds - ³ NTi ( ti - ti )ds + ³ QT3 ( f - f)ds - ³ Gu

Gt

Gf

GD

(7.143)

NT3 ( Dn - Dn )ds (7.144)

(b) Point-collocation technique The point collocation formulation is obtained when the weighting functions wij and wi in Eq (7.138) are chosen to be the Dirac delta function: w1i = w3i = w2 = w4 = d( P - Pi ) Substituting Eq (7.145) into Eq (7.138) yields u j ( Pi ) = N j ( Pi )c + u j ( Pi ) - u j ( Pi ) = 0 , (for Pi Î Gu , i = 1, 2," M 1 ) f( Pi ) = N 3 ( Pi )c + f( Pi ) - f ( Pi ) = 0 ,

t j ( Pi ) = Q j ( Pi )c + t j ( Pi ) - t j ( Pi ) = 0 ,

(for Pi Î G f , i = 1, 2," M 2 ) (for Pi Î Gf , i = 1,2," M 3 )

Dn ( Pi ) = Q3 ( Pi )c + Dn ( Pi ) - Dn ( Pi ) = 0 , (for Pi Î G D , i = 1, 2," M 4 )

(7.145)

(7.146) (7.147) (7.148) (7.149)

The above two equations may be written in matrix form as K ij c j = fi

or

Kc = f

(7.150)

in which Kij and fi are given by if j £ M 1 and Pi Î Gu N1 j ( Pi ) °N (P ) if j > M 1 and Pi Î Gu ° 2j i if Pi Î G f °° N 3 j ( Pi ) K ij = ® ° Q1 j ( Pi ) if j £ 2 M 1 + M 2 + M 3 and Pi Î Gt ° Q2 j ( Pi ) if j > 2 M 1 + M 2 + M 3 and Pi Î Gt ° if Pi Î G D °¯ Q3 j ( Pi )

(7.151)

if i £ M 1 and Pi Î Gu u1 ( Pi ) - u1 ( Pi ) ° u ( P ) - u ( P ) if i > M 1 and Pi Î Gu ° 2 i 2 i if Pi Î G f °° f( Pi ) - f( Pi ) fi = ® ° t1 ( Pi ) - t1 ( Pi ) if i £ 2 M 1 + M 2 + M 3 and Pi Î Gt ° t2 ( Pi ) - t2 ( Pi ) if i > 2 M 1 + M 2 + M 3 and Pi Î Gt ° if Pi Î G D °¯ Dn ( Pi ) - Dn ( Pi )

(7.152)

and

The points of collocation can be set at any location where a boundary value is known.


234

(c) Least-square formulation The least-square formulation can be obtained by choosing wij and wi in Eq (7.138) as arbitrary variations of the respective boundary residues, i.e. w1i = d(ui - ui ) = N i dc , w3i = d(ti - ti ) = Qi dc ,

w2 = d(f - f) = N 3dc

(7.153)

w4 = d( Dn - Dn ) = Q3dc

(7.154)

Through substitution of Eqs (7.153) and (7.154) into Eq (7.138), we have dcT [ ³ NTi (N i c + ui - ui )ds + a1 ³ NT3 (N 3c + f - f)ds Gu Gf (7.155) + a 2 ³ QTi (Qi c + ti - ti )ds + a3 ³ QT3 (Q3c + Dn - Dn )ds ] = 0 Gt

GD

which leads to [ ³ NTi N i ds + a1 ³ NT3 N 3 ds + a 2 ³ QTi Qi ds + a 3 ³ QT3 Q3 ds ]c = ³ NTi (ui - ui )ds Gu Gf Gt GD Gu + a1 ³ NT3 (f - f)ds + a 2 ³ QTi (ti - ti )ds + a3 ³ QT3 ( Dn - Dn )ds Gf

Gt

GD

(7.156) where ai again represents the weighting parameters serving the purpose of restoring the homogeneity of the physical dimensions. This system of equations may be written in the same matrix form as Eq (7.150), in which the elements of matrix K are given by K jm =

³

Gu

NTij N im ds + a1 ³ NT3 j N 3m ds + a 2 ³ QTij Q im ds + a 3 ³ QT3 j Q3m ds (7.157) Gf

Gt

GD

and the right-hand term fi is in the form f j = ³ NTij (ui - ui )ds + a1 ³ NT3 j (f - f)ds + a 2 ³ QTij (ti - ti )ds + a 3 ³ QT3 j ( Dn - Dn )ds Gu

Gf

Gt

GD

(7.158) 7.3.4 Direct formulation The derivation of the direct Trefftz formulation is based on the following weighted residual statement

³

W

[(sij , j + bi )ui* + ( Di ,i + be )f* ]d W = 0

(7.159)

Integrating by parts for the term containing sij , j and Di ,i in Eq (7.159), we obtain the following expression:

³

W

(sij , j ui* + Di ,i f* ]d W = ³ (ti ui* + Dn f* )ds - ³ (sij ui*, j + Di f*,i ]d W G

W

(7.160)

Noting that sij ui*, j + Di f*,i = s*ij ui , j + Di*f,i

(7.161)

for linear piezoelectric material and integrating by parts to the second integral on the

235


right-hand side of Eq (7.160), we obtain

³

W

(s*ij ui , j +Di*f,i )d W = ³ (ti*ui + Dn*f)ds - ³ (s*ij , j ui +Di*,i f)d W G

W

(7.162)

Making use of Eqs (7.160) and (7.162) yields

³

(s*ij , j ui +Di*,i f + bi ui* + be f* )d W = ³ (ti*ui + Dn*f - ti ui* - Dn f* )ds

W

(7.163)

G

Notice that the four terms on the right-hand side are integrals on the G surface of the solution domain. Let us now consider that the boundary G is divided into two parts, Gu and Gt ( Gf and G D ), on each of which the boundary conditions (7.76) apply. Hence, Eq (7.163) can be written as

³

W

(s*ij , j ui +Di*,i f + bi ui* + be f* )d W = ³ +³

Gt

(ti*ui - ti ui* )ds + ³

Gf

Gu

(ti*ui - ti ui* )ds

(Dn* f - Dn f* )ds + ³

GD

( Dn*f - Dn f* )ds

(7.164)

which can be used as the starting point for the direct Trefftz boundary element formulation. In the direct technique, the weighting function ui* and f* is chosen as ui* = N i dc , f* = N 3dc , ti* = Qi dc ,

Dn* = Q3dc

(7.165)

Hence, Eq (7.164) reduces to

³

W

(NTi bi + NT3 be )d W = ³ +³

Gt

Gu

(QTi ui - NTi ti )ds

(QTi ui - NTi ti )ds + ³

Gf

(QT3 f - NT3 Dn )ds + ³

GD

(QT3 f - NT3 Dn )ds

(7.166)

which is free of the unknown vector c. Similar to the treatment in Section 7.2.5, the boundary G is divided into a number of elements, for which uz, t, f, and Dn over a particular element, say element e, are approximated by the following interpolation functions: uˆ , uie = N ie ie

(if element e Î Gt )

(7.167)

tˆ , tie = Q ie ie

(if element e Î Gu )

(7.168)

ĳˆ , fe = N 3e e

(if element e Î G D )

(7.169)

D ˆ Dne = Q 3e ne ,

(if element e Î G f )

(7.170)

ˆ are the nodal displacements, tractions, electric potential, and uˆ ie , tˆ ie , ĳˆ e , and D ne are suitable and Q electric displacement over the related elements, and N ie ie interpolation functions, which may be of the following form:

Constant element: N (x) = 1

(7.171)


236

Linear element: N1 (x) =

x2 - x , x 2 - x1

N 2 ( x) =

x - x1 x2 - x1

(7.172)

Quadratic element: N1 (x) =

(x - x 2 )(x - x3 ) , (x1 - x 2 )(x1 - x3 )

(7.173)

N 2 (x) =

(x - x1 )(x - x3 ) , (x 2 - x1 )(x2 - x3 )

(7.174)

N 3 ( x) =

(x - x1 )(x - x 2 ) (x3 - x1 )(x3 - x2 )

(7.175)

where x varies from 0 to 1, and x i is the value of x at node i. Assembling all the elements, we can obtain the global interpolation functions for uˆ and tˆ over the whole boundary such that uˆ , ui = N i i

(if element e Î Gt )

(7.176)

tˆ , ti = Q i i

(if element e Î Gu )

(7.177)

ĳˆ , f=N 3

(if element e Î G D )

(7.178)

D ˆ Dn = Q 3 n,

(if element e Î G f )

(7.179)

and T being the global polynomial expansions defined only on G, and with N i i ˆ standing for vectors of nodal parameters on the boundary. uˆ i , tˆ i , ĳˆ , and D n The final result of the algebraic equation system can be written as

Kx = f

(7.180)

where x is a vector of unknown values, and K and f are given by ds + K = - ³ NTi Q i ³ Gu

Gt

ds - NT Q QiT N i ³ 3 3ds + ³

f = ³ (NTi bi + NT3 be )d W - ³ QTi ui ds + ³ W

Gu

Gf

Gt

Gt

ds Q3T N 3

NTi ti ds - ³ QT3 fds + ³ Gf

Gt

NT3 Dn ds

(7.181) (7.182)

7.3.5 Modified Trefftz formulation The modified Trefftz method presented in [11] is based on the principle of superposition and the basic idea of virtual BEM for elasticity. The use of a virtual boundary can obviate the need for computation of a singular integral on the real boundary. This subsection briefly summarizes the development in [11]. To obtain a weak solution of the boundary value problem (7.73)-(7.76), Yao and Wang [11] assumed that there are L points x0k (k = 1, 2,3," , L) within the domain W, at which there are x- and z-direction point loads and point charge Fk1 , Fk 2 , Fk 3 ,

237


respectively. Suppose also that there are distributed unknown virtual source loads F (x) = {j1 (x), j2 (x), j3 (x)} on the virtual boundary G¢ (see Fig. 7.5), where j1 , j2 , j3 denote x- and z-direction loads and charge at source point x , respectively. N nodes on the real boundary

the real boundary the virtual boundary

G

y o

G'

x

N fictitious source points on the virtual boundary

Fig. 7.5 Illustration of a computational domain and point discretization on real and virtual boundaries In the absence of body forces and free electric volume charge, according to the principle of superposition the elastic displacements, electric potential, stress, and electric displacement at field point x induced by the loads mentioned above can be expressed as L 3 ª 3 º uI (x) = ³ « ¦ uIJ* (x, x)j J (x) »d G¢ + ¦¦ u IJ* (x, x 0k )FkJ G¢ k =1 J =1 ¬ J =1 ¼

(7.183)

L 3 ª 3 * º * (x, x)j K (x) »d G¢ + ¦¦ S MiJ (x, x 0k ) FkM P iJ (x) = ³ « ¦ S KiJ G¢ k =1 J =1 ¬ J =1 ¼

(7.184)

* was where i=1,2,3, uIJ* was explained in the paragraph just after Eq (5.29), and S KiJ defined by Eq (5.45). To evaluate the boundary integral in Eq (7.183) numerically, divide the virtual boundary G¢ into n elements. Assume that the total number of nodes is N and there are m nodal points on a particular element, say element p. Therefore, the value of the distributed loads and charge at any point on the element p can be expressed as m

j J = ¦ N l (x)jlJ

(7.185)

l =1

where N l (x) are the conventional interpolation functions, x is the dimensionless coordinate explained after Eq (7.175), and jlJ represents the value of loads and charge at the lth nodal point. Substituting Eq (7.185) into Eqs (7.183) and (7.184) yields


238

n L 3 ª 3 º uI (x) = ¦ ³ « ¦ uIJ* (x, x)j J (x) »d G¢ + ¦¦ uIJ* (x, x0k )FkJ ¢ Gp p =1 k =1 J =1 ¬ J =1 ¼ n m 3 L 3 = ¦¦¦ ª ³ uIJ* (x, x) N l (x)d G¢ºjlJ + ¦¦ uIJ* (x, x0k )FkJ « G¢ »¼ p =1 l =1 J =1 ¬ p k =1 J =1

(7.186)

n L 3 ª 3 * º * P iJ (x) = ¦ ³ « ¦ S KiJ (x, x)j K (x) »d G¢ + ¦¦ S MiJ (x, x0k ) FkM G¢p p =1 k =1 J =1 ¬ J =1 ¼ n m 3 L 3 * * = ¦¦¦ ª ³ S KiJ (x, x) N l (x)d G¢ºjlK + ¦¦ S MiJ (x, x0k ) FkM « » G¢p ¬ ¼ p =1 l =1 J =1 k =1 J =1

(7.187)

To determine the 3N unknowns jlJ , the N collocation points x q (q=1,2,3,…,N) on the real boundary are selected. Then, using Eqs (7.186) and (7.187), the generalized displacement and stress at point x q can be expressed as n

m

L

3

3

u I (x q ) = ¦¦¦ hiJlpq (x q )jlJ + ¦¦ u IJ* (x q , x 0k )FkJ p =1 l =1 J =1

n

m

(7.188)

k =1 J =1 L

3

3

pq * P iJ (x p ) = ¦¦¦ g KiJl (x p )jlK + ¦¦ S MiJ (x q , x 0k ) FkM

(7.189)

hijlpq = ³ u IJ* (x, x) N l (x)d G¢

(7.190)

pq * = ³ S KiJ g KiJl ( x, x ) N l ( x ) d G ¢

(7.191)

p =1 l =1 J =1

k =1 J =1

where G¢p

G¢p

A linear equation system for the unknowns jlJ can thus be obtained by enforcing the given boundary conditions at x q (q=1,2,3,…,N) as

HA = Y

(7.192)

where A = {j11 , j12 , j13 , j12 ," , j3N }T , Y designates a vector with 3N components, which is formed according to the given boundary conditions at collocation points x q (q=1,2,3,…,N) on the real boundary, and H stands for the influence matrix of 3N ´ 3N. As was noted in [14], the effectiveness of the modified Trefftz method depends strongly on following three aspects: --- shape of the virtual boundary --- distance between the virtual and physical boundaries --- investigation of the conditioning number of the solution matrix Firstly, any shape of virtual boundary can, theoretically, be used in the calculation. However, due to inherent limitations of computer precision, the shape of the virtual boundary may influence the numerical accuracy of the output results. It has been proved that a circular virtual boundary [15] and a similar virtual boundary [16] are suitable for the modified Trefftz method. Based on these two schemes, for a rectangular domain for example, the shape of a virtual boundary can be chosen as either a rectangle or a circle (see Fig. 7.6). The effectiveness of the two schemes is assessed in [14].

239

Trefftz boundary element method l a2 + b2

la

lb

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