Green's function and boundary elements of multifield materials

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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)

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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)

Green’s function and boundary elements of multifield materials

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

Green’s function and boundary elements of multifield materials

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 << T0 . The electric field Ei is defined by

E i = -f , i ,

(1.49)

in which f is electric potential, a comma followed by arguments denotes partial differentiation with respect to the arguments. From the exact differential

dg = s ij de ij - D m dE m - sdT

(1.50)

where sij and eij are respectively stress and strain, while eij is defined by

1 (u i , j + u j ,i ) , 2 in which ui is elastic displacement, we obtain e ij =

ª ¶g º s = -« » , ¬ ¶T ¼ e,E

ª ¶g º sij = « » , ¬« ¶eij ¼»T ,E

(1.51)

ª ¶g º Dm = - « » ¬ ¶Em ¼ e,T

(1.52)

When the function g is expanded with respect to T, eij and Em within the scope of linear interactions, we have

g=

¶ ¶ 1 §¨ ¶ + e ij + Em T ¨ 2 © ¶T ¶e ij ¶E m

·§ ¶ ¶ ¶ · ¸¨ T ¸g + e kl + En ¸¨ ¶T ¶e kl ¶E n ¸¹ ¹©

(1.53)

The following constants can then be defined for developing a linear constitutive relation corresponding to the functional g: ª ¶2g (T , E ) =« c ijkl «¬ ¶e ij ¶e kl

( e, E ) º ª ¶2g º ª¶2g º rC v ( e ,T ) = -« 2 » , » , k nm = - « » , T0 »¼ T , E ¬ ¶T ¼ e , E ¬ ¶E n ¶E m ¼ e ,T

Green’s function and boundary elements of multifield materials

8

ª ¶2g º ª ¶2g º (e) l(ijE ) = - « » , c m = -« » «¬ ¶e ij ¶T »¼ E ¬ ¶T¶E m ¼ e

ª ¶2g º (T ) = -« emij » , «¬ ¶e ij ¶E m »¼ T

(1.54)

(T , E ) where c ijkl are the elastic moduli measured at a constant electric field and

temperature, k (nme ,T ) the dielectric constants measured at a constant strain and temperature, r mass density, C v

(e, E )

(T ) specific heat per unit mass, e mij the

piezoelectric coefficients measured at a constant temperature, l(ijE ) the thermal-stress coefficients measured at a constant electric field, and c (me ) the pyroelectric coefficients measured at a constant strain. When the function g is differentiated according to Eq (1.50), and the above constants are used, we find s=

rCv T + l ij eij + c m Em , T0

sij = -l ijT + cijkl e kl - emij Em ,

(1.55)

Dn = c nT + enij eij + k mn Em

A set of these three equations is the constitutive relation in the coupled system. It should be noted that the superscripts “e”, “T ” and “E ” appearing in Eq (1.54) have been dropped here. To simplify subsequent writing, we omit them in the remaining part of this book. Using the notation defined above, the Gibbs function per unit volume can now be expressed as [9]: g=

rC v 2 1 1 T - e ijk E i e jk - c m TE m - l ij Te ij c ijkl e ij e kl - k ij E i E j 2 2 2T0

(1.56)

Having defined the material constants, the related divergence equations and boundary conditions can be derived by considering the modified Biot’s variational principle [9,10] § dT · d ³ ( B + 2 F )d W - ³ (bi dui + be df) d W - ³ ¨ ti dui - qs df - hn ¸ d G = 0 (1.57) W W G T0 ¹ © where d is the variational symbol, W and G are the domain and boundary of the material, bi and be are the body force per unit volume and electric charge density, ti ,

qs and hn are the applied surface traction, applied surface charge, and prescribed surface heat flow, respectively, B and F are Biot’s generalized free energy density and the dissipation function, which are respectively defined by [10,11] as

B = U - T0 s = g + E i Di + Ts, F =

1 k ij T,i T, j 2T0

(1.58)

in which kij is the heat conduction coefficient. The variational principle (1.57) provides the following results: sij , j + bi = 0,

Di ,i + be = 0,

h i.i = -T0 Ds ,

(1.59)

Introduction

9

sij n j = ti , Di ni = - qs , hi ni = hn

(1.60)

and the constitutive equations (1.55), where ni is the outer unit normal vector to G, and hi is heat flow. Eqs (1.59) are the elastic equilibrium equations, Gauss’ law of electrostatics, and the heat conduction equation, respectively; Eqs (1.60) are boundary conditions and Eq (1.55) the constitutive equations. In this discussion we have shown how a thermal system, as a third system, affects to a greater or lesser degree the elastic, dielectric, and piezoelectric constants. It is therefore necessary to specify the thermal condition, i.e. whether it is isothermal or adiabatic. Since most electromechanical measurements are made under an alternating field or stress, the observed constants are adiabatic. On the other hand, discussion of phase transformation in solid-state physics requires knowledge of isothermal constants. In fact, the distinction between isothermal and adiabatic constants is rarely mentioned because electrical-to-thermal and mechanical-to-thermal couplings are rather weak, except in a few special cases. Therefore, it is worthwhile to pay attention to an electromechanically coupled system. At a constant temperature, Eqs (1.55) are reduced to

sij = cijkl e kl - emij Em ,

Dn = enij eij + k mn Em

(1.61)

In addition to the constitutive relation (1.61) above, three other forms of constitutive representation are commonly used in the stationary theory of linear piezoelectricity to describe the coupled interaction between elastic and electric variables. Each type has its own different set of independent variables and corresponds to a different thermodynamic function, as listed in Table 1.1. Although all equations are actually tonsorial, the indices have been omitted for brevity. It should be pointed out that an alternative derivation of formulae is merely a transformation from one type of relation to another. Some relationships between various constants occurring in the four types are as follows: bnp k pm = dnm , benm - bsnm = g nkl hnkl , ksnm - kenm = d nkl emkl , D E E cijkl - cijkl = emij hmkl , f ijklE - f ijklD = d mij g mkl , d nij = ksnm g mij = enkl f klij , e nm mij

enij = k h

E nkl klij

s nm

D klij

e nm mij

= d c , g nij = b d mij = hnkl f , hnij = b e

(1.62) D nkl klij

=g c

Table 1.1 Four types of fundamental electroelastic relations Independent variables

Constitutive relations ­s = c E e - eT E ® e ¯ D = ee + k E

Thermodynamic functional Electric Gibbs energy

e, D

­ s = c D e - hT D ® e ¯E = -he + b D

Helmholtz free energy g1 = g 0 +ED

s,E

­e = f E s + dT E ® s ¯ D = ds + k E

Gibbs free energy g 2 = g 0 -se

s, D

­ e = f D s + gT D ® s ¯E = -gs + b D

Elastic Gibbs energy g 3 = g 0 + ED-se

e, E

g0 =

1 2

1 2

cEe2- keE2-eeE

10

Green’s function and boundary elements of multifield materials

The material constants can be reduced by the following consideration. According to definition (1.51) we may write eij=eji. It follows that

cijkm = cijmk

(1.63)

cijkm = c jikm , ekij = ekji

(1.64)

Further, from sij = s ji we have

In view of these properties, it is useful to introduce the so-called two-index notation or compressed matrix notation [12]. Two-index notation involves replacing ij or km by p or q, i.e. cijkm=cpq, eikm=eiq, sij= sp, where i, j, k, m take the values 1-3, and p, q assume the values 1-6 according to the replacements 11 ® 1, 22 ® 2, 33 ® 3, 23 or 32 ® 4, 13 or 31 ® 5, 12 or 21 ® 6. With the two-index system, the constitutive relations (1.61) then become s p = c pq e q - e kp E k , Di = eiq e q + k ik E k ,

(1.65)

­ eij , when i = j , ep = ® ¯2eij , when i ¹ j

(1.66)

in which

In addition, the elastic, piezoelectric and dielectric constants can now be written in matrix form since they all are described by two indices. The arrays for an arbitrarily anisotropic material are ªc11 c12 c13 c14 c15 c16 º «c » « 12 c 22 c 23 c 24 c 25 c 26 » «c c 23 c 33 c 34 c 35 c 36 » (1.67) c = « 13 », «c14 c 24 c 34 c 44 c 45 c 46 » «c15 c 25 c 35 c 45 c 55 c 56 » « » «¬c16 c 26 c 36 c 46 c 56 c 66 »¼ ª e11 e = ««e21 «¬e31

e12

e13

e14

e15

e22 e32

e23 e33

e24 e34

e25 e35

ª k11 k = «« k12 ¬« k13

k12 k 22 k 23

k13 º k 23 »» k 33 ¼»

e16 º e26 »» , e36 »¼

(1.68)

(1.69)

It can be seen that there are 21+18+6=45 independent constants for this material type. The matrices for a material with monoclinic symmetry, with x1 the diagonal axis (Class 2), are

Introduction

11

ª c11 «c « 12 «c c = « 13 «c14 «0 « ¬« 0 ªe11 e = «« 0 «¬ 0

c12

c13

c14

0

c 22 c 23

c 23 c 33

c 24 c 34

0 0

c 24 0 0

c 34 0 0

c 44 0 0

0 c 55 c 56

e12

e13

e14

0

0

0

0

e 25

0

0

0

e 35

ª k 11 k = «« 0 «¬ 0

0

0º 0 »» 0» », 0» c 56 » » c 66 ¼»

(1.70)

0º e 26 »» , e36 »¼

(1.71)

0 º k 23 »» k 33 »¼

k 22 k 23

(1.72)

This type of material symmetry reduces the number of independent constants to 25. For a transversely isotropic material with x3 in the poling direction (Class C6v=6 mm), the related material matrices are ªc11 «c « 12 «c13 c=« 0 « «0 « «0 ¬

c12 c 22

c13 c 23

c 23 0 0 0

ª0 e = «« 0 ¬«e 31

c 33 0 0

0 0 0

0 0 0

c 44 0

0 c 44

0

0

0

0 0 e 31

0 0 e33

0 e15 0

ª k 11 k = «« 0 ¬« 0

0 k 11 0

0 0 0

º » » » », 0 » » 0 » 1 (c11 - c12 )» 2 ¼ e15 0 0

0º 0»» , 0»¼

0 º 0 »» k 33 ¼»

(1.73)

(1.74)

(1.75)

Thus it is clear that a material with this type of symmetry is described by ten independent material constants. This category of material is important because the corresponding polarized ceramics have high piezoelectric coupling. Finally, an isotropic dielectric material has arrays which are similar to the arrays for transversely isotropic materials, except that there are some additional relations among the material constants. They are

eip = 0

for all values of i and p ,

c12 = c13 = l, c11 = c 33 = l + 2G, c 44 = c 66 = G, k 11 = k 33

(1.76) (1.77)

12

Green’s function and boundary elements of multifield materials

where G=E/2(1+u) is the shear modulus of elasticity, l=2Gu/(1-2u) is the Lamé constant and E, u are Young’s modulus and Poisson’s ratio, respectively. In the MKS system the material constants and the variables mentioned above are measured in the following units: [cij ] = Nm -2 , [eij ] = N(Vm) -1 = Cm -2 , [ kij ] = C2 N -1m -2 = NV -2 , [sij]= Nm -2 , [eij]= mm -1 , [Di]= Cm -2 = N(Vm) -1 , [Ei]= NC -1 = Vm -1 , [f]=V. For poled barium-titanate (BaTiO3) and lead-zirconate-titanate, these physical constants are of the orders: cij ~1011 Nm -2 , eij ~10 Nm -2 , kij ~ 10 -8 NV -2 . Substitution of Eqs (1.49) and (1.51) into Eq (1.65), and later into Eq (1.59), results in 1 1 c11u1,11 + (c11 + c12 )u2,12 + (c13 + c44 )u3,13 + (c11 - c12 )u1,22 2 2 + c44u1,33 + (e31 + e15 )f,13 + b1 = 0,

(1.78)

1 1 c11u2,22 + (c11 + c12 )u1,12 + (c13 + c44 )u3,23 + (c11 - c12 )u2,11 2 2 + c44u2,33 + (e31 + e15 )f,23 + b2 = 0,

(1.79)

c44u3,11 + (c44 + c13 )(u1,31 + u2,32 ) + c44u3,22 + c33u3,33 + e15 (f,11 + f,22 ) + e33f,33 + b3 = 0, e15 (u3,11 + u3,22 ) + (e15 + e31 )(u1,31 + u2,32 ) + e33u3,33 - k11 (f,11 + f,22 ) - k33f,33 + be = 0,

(1.80)

(1.81)

for transversely isotropic materials (Class C6v=6 mm) with x3 as the poling direction and the x1-x2 plane as the isotropic plane. This type of material is adopted in the most of following chapters. 1.4 Linear theory of magnetoelectroelastic materials

1.4.1 Basic equations of general anisotropy In the previous section, the linear theory of piezoelectricity was presented. Extension of the theory to include magnetic effect is described in this section. For a linear magnetoelectroelastic solid, the governing equations of the mechanical and electric fields are in the same form as those of Eqs (1.49), (1.51), and (1.59)1,2. The magnetic field is governed by the following equations Bi ,i + bm = 0 ; H i = -y ,i

(1.82)

where bm is the body electric current; Bi , H i , and y are the magnetic induction, magnetic field, and magnetic potential, respectively. Eqs (1.49), (1.51), and (1.59)1,2 are coupled to Eq (1.82) through following constitutive relations [13] sij = cijkl e kl - elij El - elij H l ½ ° Di = eikl e kl + kil El + a il H l ¾ Bi = eikl e kl + a il El + mil H l °¿

(1.83)

where e~lij , a ij , and m ij are respectively piezomagnetic constants, electromagnetic

constants, and magnetic permeabilities. Let G be the boundary of the solution domain W of the magnetoelectroelastic solid considered, then the boundary conditions

Introduction

13

can be given in the form [13] ui = ui

(on Gu )

(1.84)

ti = sij n j = ti

(on Gt )

(1.85)

f=f

(on G f )

(1.86)

Dn = Di ni = -qs (on G D )

(1.87)

y=y

(on G y )

(1.88)

Bn = Bi ni = m

(on G B )

(1.89)

Where a bar over a variable indicates that the variable is prescribed, and G = Gu È Gt = G f È G D = G y È G B . Obviously, the first four equations (1.84)-(1.87) are the electric and mechanical boundary conditions which are the same as those presented in Section 1.3, and the remaining two are for magnetic field. Eqs (1.49), (1.51), and (1.59)1,2, (1.82)-(1.89) constitute the complete set of equations of a linear magnetoelectroelastic solid. This set of equations is coupled among magnetic, electric, and mechanical fields. It should be mentioned that there are eight equivalent constitutive representations commonly used in the stationary theory of linear magnetoelectroelastic solid to describe the coupled interaction among elastic, electric, and magnetic variables [14]. Table 1.2 lists the eight forms of constitutive relations, and the corresponding independent variables and generalized Gibbs energy functionals. In Table 1.2, c and s are elastic stiffness and compliance tensors, k and b are permittivity and impermittivity tensors, m and n are permeability and reluctivity tensors, a, l, h, and z are magnetoelectric constants, e, h, d, and g are piezoelectric constants, and ~e , ~ ~ h, d and ~ g are piezomagnetic constants.

1.4.2 Transversely isotropic simplification If the magnetoelectroelastic solid considered is transversely isotropic, the equations described in Section 1.4.1 can be further simplified [15]. In a fixed rectangular coordinate system (x,y,z), the constitutive equations of a transversely isotropic magnetoelectroelastic solid with the isotropic plane perpendicular to z axis can be expressed in the following form s xx = c11u1, x + c12u2, y + c13u3, z + e31f, z - e31y , z s yy = c12u1, x + c11u2, y + c13u3, z + e31f, z - e31y , z s zz = c13u1, x + c13u2, y + c33u3, z + e33f, z - e33y , z s yz = c44 (u2, z + u3, y ) + e15f, y - e15y , y

(1.90)

s xz = c44 (u1, z + u3, x ) + e15f, x - e15y , x s xy = c66 (u1, y + u2, x )

Dx = e15 (u1, z + u3, x ) - k11f, x - a11y , x Dy = e15 (u2, z + u3, y ) - k11f, y - a11y , y Dz = e31 (u1, x + u2, y ) - e33u3, z - k33f, z - a 33y , z

(1.91)

Green’s function and boundary elements of multifield materials

14

Table 1.2 Eight types of constitutive models[14] Independent variables e, E,H

e, D,H

e, E,B

e, D,B

s, D,B

s, E,B

s, D,H

s, E,H

Constitutive relations

­ı = cİ - eT E - e T H ° ® D = eİ + țE + ĮH ° B = eİ  + ĮE + ȝH ¯

Thermodynamic functional

1 P1 = (cİ 2 - țE2 - ȝH 2 ) 2  - ĮEH - eİE - eİH

­ı = cİ - hT D - e T H ° ® E = -hİ + ȕD - ȗH ° B = eİ  + ȗD + ȝH ¯

P 2 = P1 + DE

­ ı = cİ - eT E - h T B ° ® D = eİ + țE + ȘB °H = -hİ  - ȘE + ȞB ¯ ­ı = cİ - hT D - h T B ° ® E = -hİ + ȕD - ȜB ° H = hİ  - ȜD + ȞB ¯ ­İ = sı + gT D + g T B ° ®E = -gı + ȕD - ȜB ° H = gı  - ȜD + ȞB ¯

P 3 = P1 + BH

P 4 = P1 + DE + BH

P 5 = P1 + DE + BH - ıİ

­ İ = sı + dT E + g T B ° ® D = dı + țE + ȘB °H = -gı  - ȘE + ȞB ¯ ­İ = sı + gT D + d T H ° ® E = -gı + ȕD - ȗH ° B = dı  + ȗD + ȝH ¯ ­İ = sı + dT E + d T H ° ® D = dı + țE + ĮH ° B = dı  + ĮE + ȝH ¯

P 6 = P1 + BH - ıİ

P 7 = P1 + DE - ıİ

P 8 = P1 - ıİ

Bx = e15 (u1, z + u3, x ) + a11f, x - m11y , x By = e15 (u2, z + u3, y ) + a11f, y - m11y , y

(1.92)

Bz = e31 (u1, x + u2, y ) + e33u3, z + a33f, z - m33y , z The governing equations (1.59)1,2 and (1.82)1 are now replaced by 1 1 c11u1, xx + (c11 - c12 )u1, yy + (c11 + c12 )u2, xy + (c13 + c44 )u3, xz 2 2 + c44u1, zz + (e31 + e15 )f, xz - (e15 + e31 )y , xz + b1 = 0,

(1.93)

Introduction

15

1 1 c11u2, yy + (c11 - c12 )u2, xx + (c11 + c12 )u1, xy + (c13 + c44 )u3, yz 2 2 + c44u2, zz + (e31 + e15 )f, yz - (e15 + e31 )y , yz + b2 = 0,

c44 (u3, xx + u3, yy ) + (c44 + c13 )(u1, xz + u2, yz ) + c33u3,33 + e15 (f, xx + f, yy ) + e33f, zz - e33y , zz - e15 (y , xx + y , yy ) + b3 = 0, e15 (u3, xx + u3, yy ) + (e15 + e31 )(u1, xz + u2, yz ) + e33u3, zz - k11 (f, xx + f, yy ) - k33f, zz - a11 (y , xx + y , yy ) - a 33y , zz + be = 0, e15 (u3, xx + u3, yy ) + (e15 + e31 )(u1, xz + u2, yz ) + e33u3, zz + a11 (f, xx + f, yy ) + a 33f, zz - m11 (y , xx + y , yy ) - m33y , zz + bm = 0,

(1.94)

(1.95)

(1.96)

(1.97)

Equations (1.90)-(1.97) are used in later chapters. 1.4.3 Extension to include thermal effect The equations presented in Section 1.4.1 can be extended to include thermal effects if the temperature field does not fully couple with the magnetoelectroelastic field, that is, if the magnetoelectroelastic field can be affected by the temperature field through constitutive relations but the temperature field is not affected by the magnetoelectroelastic field. Under such assumptions the governing equations of thermomagnetoelectroelastic problem can be expressed as [16]

sij , j + bi = 0 ˈ Di ,i + be = 0, Bi ,i + bm = 0

(1.98)

sij = cijkl ekl - elij El - elij H l - l ijT ½ ° Di = eikl ekl + k il El + ail H l + ciT ¾ Bi = eikl e kl + ail El + mil H l - niT ¿°

(1.99)

1 eij = (ui , j + u j ,i ) ˈ Ei = -f,i , H i = -y ,i 2

(1.100)

hi ,i = 0,

hi = - kijT, j

(1.101)

where ni are pyromagnetic coefficients. The boundary conditions of the thermomagnetoelectroelastic problem are still defined by Eqs (1.84)-(1.89) together with the boundary conditions for the thermal field. 1.4.4 Variational formulation Taking P 1 (İ, E, H ) (see Table 1.2) as an example, we now present a variational principle for a magnetoelectroelastic solid in the domain W bounded by G. First the explicit expression of P 1 (İ, E, H ) in terms of e ij , Ei and H i is presented: 1 1 1 P1 = cijkl eij e kl - kij Ei E j - mij H i H j - eijk Ei e jk - eijk H i e jk - a ij Ei H j (1.102) 2 2 2 which results in the following constitutive relations

Green’s function and boundary elements of multifield materials

16

sij =

¶P1 ¶P ¶P , Di = - 1 , Bi = - 1 ¶eij ¶Ei ¶H i

(1.103)

Then, based on the variational functional P 1 (İ, E, H ) and the basic equations (1.49), (1.51), and (1.59)1,2, (1.82)-(1.89), a variational principle can be constructed in the form d ª ³ (P1 - bi ui - be f - bm y )d W - ³ ti ui d G - ³ Dn fd G - ³ myd G º = 0 (1.104) Gt GD GB ¬« W ¼» in which Eqs (1.83), (1.84), (1.86), and (1.88) are assumed to be satisfied, a priori. We now proceed to show that Eqs (1.59)1,2, (1.82)1, (1.85), (1.87), and (1.89) can be derived from Eq (1.104) for the independent variables du i , df, and dy. Noting the variation of P 1 expressed by

dP1 = cijkl e kl deij - eijk ( Ek deij + e jk dEi ) - k ij E j dEi - mij H j dH i - eijk ( H k deij + e jk dH i ) - aij ( E j dH i + H j dEi )

(1.105)

=sij deij - Di dEi - Bi dH i , Eq (1.104) can be further written as

³

W

(sij deij - Di dEi - Bi dH i - bi dui - be df - bm dy )d W - ³ ti dui d G - ³ Gt

Dn dfd G - ³ mdyd G = 0

GD

(1.106)

GB

Making use of Eqs (1.49), (1.51), and (1.82)2, the variables can be expressed as 1 deij = (du j ,i + dui , j ), dEi = -df,i , dH i = -dy ,i 2

(1.107)

By substituting Eq (1.107) into Eq (1.106) and employing the chain rule of differentiation, integration by parts, the divergence theorem, and the boundary conditions (1.84), (1.86), (1.88), the first three terms of Eq (1.106), with the substitution of Eq (1.103), become

³

W

sij deij d W = ³ sij n j dui d G - ³ sij , j du j d G Gt

W

= ³ ti dui d G - ³ sij , j du j d G Gt

- ³ Di dEi d W = ³ W

=³ W

=³

W

GD

GD

- ³ Bi dH i d W = ³ GD

(1.108)

Di ni dfd G - ³ Di ,i dfd G W

Dn dfd G - ³ Di ,i dfd G

(1.109)

W

GD

Bi ni dyd G - ³ Bi ,i dyd G W

Bn dyd G - ³ Bi ,i dyd G W

Then, with substitution of Eqs (1.108)-(1.110) into Eq (1.106), we have

(1.110)

Introduction

³

17

W

[(sij , j + bi )dui + ( Di ,i + be )df + ( Bi ,i + bm )dy ]d W

- ³ (ti - t )i dui d G - ³ Gt

GD

( Dn - Dn )dfd G - ³ ( Bn - m)dyd G = 0

(1.111)

GB

The above equation must be satisfied for independent variables du i , df, and dy. Hence, the volume integrals lead to Eqs (1.59)1,2 and (1.82)1, and the surface integrals yield boundary conditions (1.85), (1.87), and (1.89). Similarly, we can present variational principles for the other seven thermodynamic functionals ( P 2 ~ P 8 ) in a straightforward way. 1.5 Two solution procedures in anisotropic multifield materials For two-dimensional deformations in a general anisotropic multifield material, in which all fields depend on x1 and x2 (or x3) only, there are two powerful solution procedures in the literature. One is Lekhnitskii’s approach [17], which starts with equilibrated stress functions, followed by compatibility equations. This approach is discussed in Section 1.5.2. Another is Stroh’s formalism [18], which starts with the displacements, electric potential, and magnetic potential, followed by equilibrium equations. The equivalence of these two formalisms has been discussed by Suo [19]. The details of Stroh’s formulation are given in the following section. 1.5.1 Solution with Stroh formalism To treat a multifield variable system (elastic, electric, and magnetic fields in piezoelectric/piezomagnetic materials) on an equal footing and reduce the amount of writing, the shorthand notation given by Barnett and Lothe [20] is employed here. With this shorthand notation, the governing equation (1.59)1,2, (1.82)1 and the constitutive relationship (1.83) can be rewritten as

P iJ ,i + bJ = 0 ,

(1.112)

P iJ = E iJKmU K , m ,

(1.113)

where b4=be and b5=bm, and ­sij , J £ 3, ° P iJ = ® Di , J = 4, ° B , J = 5, ¯ i

EiJMn

­cijmn , °e , ° nij ° enij , ° ° eimn , ° = ® -kin , ° -a , ° in ° eimn , ° -a , ° in °¯ -min

­um , M £ 3, ° U M = ® f, M = 4, ° y, M = 5, ¯

(1.114)

J , M £ 3, J £ 3, M = 4, J £ 3, M = 5, J = 4, M J = 4, M J = 4, M J = 5, M

£ 3, = 4, = 5, £ 3,

(1.115)

J = 5, M = 4, J = 5, M = 5,

Denoting U = {u1 u2 u3 f y}T, where the superscript T stands for the transpose, a

18

Green’s function and boundary elements of multifield materials

general solution can be obtained by considering an arbitrary function of the form [20] U = af (z ) ,

(1.116)

z = x1 + px 2

(1.117)

where p and a are determined by inserting Eq (1.116) into Eq (1.113), and later into Eq (1.112), f(z) is an arbitrary function of z. In the absence of any body force, body electric current, and free charge distribution, that is, bJ=0 (J=1-5), we have [Q + p (R + R T ) + p 2 T]a = 0

(1.118)

where Q, R and T are 5´5 real matrices whose components are Q IK = E1IK 1 , R IK = E1IK 2 , T IK = E 2 IK 2

(1.119)

The stress, magnetic induction, and electric displacement (SMED) obtained by substituting Eq (1.116) into Eq (1.113) can be written in terms of a SMED function j as

P1J = -j J , 2 , P 2 J = j J ,1 ,

(1.120)

j = bf (z ) ,

(1.121)

where

b = (R T + pT)a = - p -1 (Q + pR )a (1.122) The second equality in Eq (1.122) follows from Eq (1.118). It suffices therefore to consider only the SMED function j because the stresses sij, magnetic induction Bi and the electric displacement Di can be obtained by differentiation of j. There are ten eigenvalues p from Eq (1.118) which consist of five pairs of complex conjugates [20]. If pJ , a J ( J = 1, 2,", 10) are the eigenvalues and the associated eigenvectors, let

Im pJ > 0, pJ + 5 = pJ , a J + 5 = aJ , b J + 5 = b J ,

(J=1-5)

(1.123)

where “Im” stands for the imaginary part of a complex number and the overbar denotes a complex conjugate. Assuming that pJ are distinct, the general solutions for U and j obtained by superposing ten solutions of the form of Eqs (1.116) and (1.121) are 5

U = ¦ {a J f J ( z J ) + a J f J +5 ( z J )} ,

(1.124)

J =1 5

j = ¦{b J f J ( z J ) + b J f J + 5 ( z J )}

(1.125)

J =1

In Eqs (1.124) and (1.125), fJ (J = 1, 2,", 10) are arbitrary functions of their argument and z J = x1 + pJ x2 In most applications fJ assume the same functional form, so that we may write

(1.126)

Introduction

19

f J ( z J ) = qJ f ( z J ),

f J +5 ( z J ) = qJ f ( z J ),

(J=1-5)

(1.127)

where qJ are complex constants to be determined. The second equation of Eq (1.127) is used for obtaining real solutions for U and j. Expressions (1.124) and (1.125) can then be written in a compact form: U = 2 Re{Af ( z )} = 2 Re{A f ( z a ) q} ,

(1.128)

j = 2 Re{Bf ( z )} = 2 Re{B f ( z a ) q} ,

(1.129)

in which “Re” stands for the real part of a complex number, f(z)={f1(z1) f2(z2) f3(z3) f4(z4) f5(z5)}T, A, B are 5´5 complex matrices defined by A = [a1 a 2 a 3 a 4 a 5 ], B = [b 1 b 2 b 3 b 4 b 5 ] and

f ( za )

(1.130)

is a diagonal matrix

f ( za ) = diag [ f ( z1) f ( z2 ) f ( z3 ) f ( z4 ) f ( z5 )]

(1.131)

For a given problem, it is clear that all that is required is to determine the unknown function f (zJ) and the complex constant vector q. In the following, we present some identities concerning matrices A and B which will be used in later chapters. For this purpose, Eq (1.122) is rewritten as ª -RT « ¬ -Q

I º ­a ½ ª T 0º ­a ½ »® ¾ = p« »® ¾ 0 ¼ ¯b ¿ ¬ R I ¼ ¯b ¿

(1.132)

where I is the identity matrix. Since T -1 exists, we can reduce Eq (1.132) to

Nx = px

(1.133)

ªN N º ­a ½ N = « 1 T2 » , x = ® ¾ ¯b ¿ ¬ N 3 N1 ¼

(1.134)

N1 = -T-1RT , N 2 = T-1 = NT2 , N 3 = RT-1RT - Q = NT3

(1.135)

where

The 10×10 matrix N is the generalization of the 6×6 matrix N for anisotropic elastic materials. Eq (1.133) is a standard eigenrelation in the ten-dimensional space. The vector x in Eq (1.133) is a right eigenvector. The left eigenvector h is defined by hT N = phT ,

N T h = ph ,

(1.136)

and can be shown to be [20] ­b ½ h= ® ¾ ¯a ¿

Normalization of x I and hJ (which are orthogonal to each other) gives

(1.137)

20

Green’s function and boundary elements of multifield materials

hTJ x K = d JK

(1.138)

where dJK is the Kronecker delta. Making use of Eqs (1.123), (1.128), (1.129), (1.134)2 and (1.137), Eq (1.138) can be written as ª BT « T ¬B

A T º ª A A º ª I 0º » « »= A T ¼ ¬ B B ¼ «¬ 0 I »¼

(1.139)

This is the orthogonal relation. The two 10´10 matrices on the left-hand side of Eq (1.139) are the inverse of each other. Their product commutes so that ª A A º ª BT « » « T ¬ B B ¼ ¬B

A T º ª I 0º »=« » AT ¼ ¬0 I ¼

(1.140)

This is the closure relation and is equivalent to ABT + ABT = BA T + BA T = I, AA T + AA T = BBT + BBT = 0

(1.141)

Equations (1.141)1,2 tell us that the real part of ABT is I/2 while Eq (1.141)3,4 imply that AAT and BBT are purely imaginary. Hence, the three Barnett-Lothe tensors S, H, L, defined by S = i (2ABT - I), H = 2iAA T , L = -2iBBT ,

(1.142)

are real. It is clear that H and L are symmetric. 1.5.2 Solution with Lekhnitskii formalism The mathematical method known as Lekhnitskii formalism was originally developed to solve two-dimensional problems in anisotropic elastic materials [17]. The evolution of the method and a number of extensions to electroelastic problems were described in [22-25]. In this section the Lekhnitskii formalism used in linear piezoelectricity is briefly summarized. For a complete derivation and discussion, the reader is referred to [9, 17, 22-25]. Consider a two-dimensional piezoelectric plate, where the material is transversely isotropic and coupling between in-plane stresses and in-plane electric fields takes place. For a Cartesian coordinate system Oxyz, choose the z-axis as the poling direction, and denote the coordinates x and z by x1 and x2 in order to generate a compacted notation. If a plain strain problem is considered, it requires that e33 = e 23 = e13 = E3 = 0 By substitution of Eq (1.143) into Eq (1.61), we have

(1.143)

Introduction

21

­ s11 ½ ª c11 c12 °s ° « c °° 22 °° « 12 c22 0 ® s12 ¾ = « 0 °D ° « 0 0 ° 1° « « ¯° D2 ¿° ¬e21 e22

0 0 c33 e13

0 0 e13 -k11

0

0

e21 º ­ e11 ½ e22 »» °° e 22 °° ° ° 0 » ® 2e12 ¾ , » 0 » ° - E1 ° ° ° -k 22 ¼» ¯°- E2 ¿°

(1.144)

g 21 º ­ s11 ½ g 22 » ° s22 ° » °° °° 0 » ® s12 ¾ » 0 » ° D1 ° ° ° -b22 »¼ ¯° D2 ¿°

(1.145)

or inversely ­ e11 ½ ª f11 °e ° «f °° 22 °° « 12 ® 2e12 ¾ = « 0 ° -E ° « 0 ° 1° « ¯° - E2 ¿° «¬ g31

f12

0

0

f 22

0 f 33 g13

0 g13 -b11

0

0

0 0 g33

in which fij is the elastic compliance tensor of the material, gij is the piezoelectric tensor and bij is the dielectric impermeability tensor. In the constitutive equations (1.144) and (1.145), -Ei is used instead of Ei because it allows the construction of a symmetric generalized linear response matrix which will prove to be advantageous. From the constitutive equations (1.145), we observe that D2 produces normal strains e11 and e22, while the stress component s12 induces an electric field E1, and s11 and s22 produce E2. Equation (1.144) constitutes a system of five equations in ten unknowns. Additional equations are provided by elastic equilibrium and Gauss’ law

s11,1 + s12,2 = 0, s12,1 + s22,2 = 0, D1,1 + D2,2 = 0 ,

(1.146)

in which the absence of body forces and free electric volume charge has been assumed, and by one elastic and one electric compatibility condition e11,22 + e22,11 - 2e12,12 = 0, E1,2 - E2,1 = 0

(1.147)

Having formulated the electroelastic problem, we seek a solution to Eqs (1.145)-(1.147) subjected to a given loading and boundary condition. To this end, the well-known Lekhnitskii stress function F and induction function V satisfying the foregoing equilibrium equations are introduced as follows [9,22]: s11 = F,22 , s22 = F,11 , s12 = - F,12 , D1 = V,2 , D2 = -V,1

(1.148)

Inserting Eq (1.148) into Eq (1.145), and later into Eq (1.147) leads to L4 F - L3V = 0,

L3 F + L2V = 0

(1.149)

where L4 = f 22

¶4 ¶4 ¶4 f (2 f f ) , + + + 11 12 33 ¶x14 ¶x24 ¶x12¶x22

¶3 ¶3 L3 = g 22 3 + ( g 21 + g13 ) , ¶x1 ¶x1 ¶x22

¶2 ¶2 L2 = b 22 2 + b11 2 ¶x1 ¶x2

(1.150)

22

Green’s function and boundary elements of multifield materials

Note that if the problem was purely mechanical, L4 would be the only nonzero operator and its form would coincide with the plane anisotropic case discussed, among others, by Lekhnitskii [17]. In order to solve Eq (1.149) we reduce the system to a single partial differential equation of order six in either F or V. Choosing F, we obtain ( L4 L2 + L3 L3 ) F = 0

(1.151)

As discussed in [17] within the framework of anisotropic elasticity, Eq (1.151) can be solved by assuming a solution of F(z) such that

p = a + ib

F ( z ) = F ( x1 + px2 ) ,

(1.152)

where a and b are real numbers. By introducing Eq (1.152) into Eq (1.151), and using the chain rule of differentiation, an expression of the form {} × F (6) = 0 is obtained. A nontrivial solution follows by setting the characteristic equation (i.e., L4L2+L3L3) equal to zero, namely 2 f11b11 p 6 + ( f11b22 + f 33b11 + 2 f12b11 + g 21 + g132 + 2 g 21 g13 ) p 4 2 + ( f 22b11 + 2 f12b22 + f 33b22 + 2 g 21 g 22 + 2 g13 g 22 ) p 2 + f 22b 22 + g 22 =0

(1.153)

Owing to the particular material symmetry of the piezoelectric solid under investigation, the polynomial is expressed in terms of even powers of p. This allows us to solve Eq (1.153) analytically, rendering p1 = ib1 , p2 = a2 + ib 2 , p3 = -a2 + ib 2 , p4 = p1 , p5 = p2 , p6 = p3

(1.154)

where b1 , a2 and b2 depend on the material constants. Once the roots pj, j=1, 2, 3 are known, the solution for stress function F is written as 3

F ( x1 , x2 ) = 2 Re ¦ F j (z j )

(1.155)

j =1

The next step is to find the function V using one of Eqs (1.149). If we consider L3F=-L2V, assuming solutions of the form F(zk) and V(zk), we have Vk¢¢( zk ) = v k ( pk ) Fk¢¢¢ ( zk )

(1.156)

where primes indicate differentiation with respect to the related argument, and v k ( pk ) = -

( g 21 + g13 ) pk2 + g 22 b11 pk2 + b22

(1.157)

Integration of Eq (1.156) leads to Vk ( zk ) = v k ( pk ) Fk¢( zk )

(1.158)

It should be noted here that the arbitrary constants of integration could be set equal to zero [22]. Thus, the solution for the induction function can be expressed as follows:

Introduction

23 3

3

j =1

j =1

V ( x1 , x2 ) = 2 Re ¦V j (z j ) = 2 Re ¦ v j F j¢(z j )

(1.159)

With the aid of Eqs (1.155) and (1.159) we can obtain expressions for the stress and electric displacement components. Using Eqs (1.148), (1.155) and (1.159), we obtain

­ pk2 ½ ­ s11 ½ 3 ° ° ° ° ® s22 ¾ = 2 Re ¦ ® 1 ¾F¢k ( zk ), k =1 ° °s ° ° ¯ 12 ¿ ¯ - pk ¿

3 ­v k pk ½ ­ D1 ½ ® ¾ = 2 Re ¦ ® ¾F¢k ( zk ) k =1 ¯ -v k ¿ ¯ D2 ¿

(1.160)

where F k ( zk ) = Fk¢( zk ) . Finally, using the constitutive equations in conjunction with Eq (1.160) allows us to find expressions for the strain and electric field. They are

ª pk* º ­ e11 ½ 3 3 ª tk* º ­ E1 ½ ° ° « *» ¢ 2 Re ( ), q z e = F = ® 22 ¾ ® ¾ ¦ ¦ « * » F¢k ( zk ) « k» k k E2 ¿ k =1 k =1 ¬ vk ¼ ¯ * ° 2e ° « rk » ¯ 12 ¿ ¬ ¼

(1.161)

where pk* = f11 pk2 + f12 - g 21v k , qk* = ( f12 pk2 + f 22 - g 22v k ) / pk , rk* = ( g13v k - f 33 ) pk , tk* = -( g13 + b11v k ) pk , vk* = g 21 pk2 + g 22 + b22v k

(1.162)

Substitution of Eqs (1.56) and (1.58) into Eq (1.161), and then integration of the normal strains and the electric field E = -grad f produces ­ pk* ½ ­ u1 ½ ­ w0 x2 + u0 ½ 3 ° ° ° *° ° ° ®u2 ¾ = 2 Re ¦ ® qk ¾F k ( zk ) + ® -w0 x1 + v0 ¾ 1 k = °f° ° t* ° ° ° f0 ¯ ¿ ¯ ¿ ¯ k¿

(1.163)

where the constants w0, u0, v0 represent rigid body displacements, and f0 is a reference potential. Recapitulating, based on the procedure above, the plane strain piezoelectric problem is reduced to 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 . References [1] Roach GF, Green’s functions: Introductory theory with applications, London:Van Nostrand Reinhold, 1970 [2] Barton G, Elements of Green's functions and propagation: potentials, diffusion, and waves, Oxford: Clarendon Press, 1989 [3] Greenberg MD, Application of Green’s functions in science and engineering, Englewood Cliffs: Prentice-Hall, 1971 [4] Lighthill MJ, Introduction to Fourier analysis and generalized functions, Cambridge: Cambridge University Press, 1958 [5] Schwartz L, Mathematics for the physical sciences, Paris: Hermann, 1966 [6] Cady WG, Piezoelectricity, vols. 1 & 2, New York: Dover Publishers, 1964

24

[7]

Green’s function and boundary elements of multifield materials

Tiersten HF, Linear piezoelectric plate vibrations, New York: Plenum Press, 1964 [8] Parton VZ and Kudryavtsev BA, Electromagnetoelasticity, piezoelectrics and electrically conductive solids, New York: Gordon and Breach Science Publishers, 1988 [9] Qin QH, Fracture mechanics of piezoelectric materials, Southampton: WIT Press, 2001 [10] Biot MA, Thermoelasticity and irreversible thermodynamics, J Appl Phys, 27, 240-253, 1956 [11] Mindlin RD, Equations of high frequency vibrations of thermopiezo-electric crystal plates, Int J Solids Struct, 10, 625-637, 1974 [12] Nye JF, Physical properties of crystals: their representation by tensors and matrices, Oxford: Clarendon Press, 1972 [13] Lee PCY, A variational principle for the equations of piezoelectromagnetism in elastic dielectric crystals, J Appl Phys, 6, 7470-7473, 1991 [14] Soh AK and Liu JX, On the constitutive equations of magnetoelectroelastic solids, J Intelligent Mat Sys Struc, 16, 597-602, 2005 [15] Hou PF, Ding HJ and Chen JY, Green’s function for transversely isotropic magnetoelectroelastic media, Int J Eng Sci, 43, 826-858, 2005 [16] Gao CF and Noda N, Thermal-induced interfacial cracking of magnetoelectroelastic materials, Int J Eng Sci, 42, 1347-1360, 2004 [17] Lekhnitskii SG, Theory of elasticity of an anisotropic elastic body, San Francisco: Holden-Day, Inc., 1963 [18] Stroh AN, Dislocations and cracks in anisotropic elasticity, Phil Mag, 3, 625-646, 1958 [19] Suo Z, Singularities, interfaces and cracks in dissimilar anisotropic media. Proc R Soc Lond, A427, 331-358, 1990 [20] Barnett DM and Lothe J, Dislocations and line charges in anisotropic piezoelectric insulators, Phys Stat Sol(b), 67, 105-111, 1975 [21] Chung MY and Ting TCT, The Green function for a piezoelectric piezomagnetic magnetoelectric anisotropic elastic medium with an elliptic hole or rigid inclusion, Phil Mag Letters, 72, 405-410, 1995 [22] Sosa H, Plane problems in piezoelectric media with defects, Int J Solids Struct, 28, 491-505, 1991 [23] Qin QH, A new solution for thermopiezoelectric solid with an insulated elliptic hole, Acta Mechanica Sinica, 14, 157-170, 1998 [24] Qin QH and Mai YW, A new thermoelectroelastic solution for piezoelectric materials with various openings, Acta Mechanica, 138, 97-111, 1999 [25] Wang SS and Choi I, The interface crack between dissimilar anisotropic composite materials, J Appl Mech, 50, 169-178, 1983

25

Chapter 2 Green’s function of electroelastic problem 2.1 Introduction Green’s function plays an important role in the solution of numerous problems in the mechanics and physics of solids. It is at the heart of many analytical and numerical methods such as singular integral equation methods, boundary element methods, eigenstrain approaches, and dislocation methods [1-3]. Extensive studies have been carried out on static Green’s functions in anisotropic piezoelectric solids. Benveniste [4] studied three-dimensional (3D) solutions in piezoelectric solids using the Fourier transform. Chen [5] and Chen and Lin [6] expressed the infinite body Green’s functions and their derivatives as the contour integrals over the unit circle using 3D Fourier transforms. Dunn [7] obtained explicit Green’s functions for transversely isotropic piezoelectric solids using the Radon transform, coordinator transformation, and evaluation of residues in sequence. However, the expressions of this solution are very complicated, so that it is difficult to verify and inconvenient to use. Lee and Jiang [8] obtained a two-dimensional (2D) fundamental solution for an infinite plane by using double Fourier transform. Sosa and Castro [9] obtained the solutions to the problem of concentrated loads acting at the boundary of a 2D half-plane by means of a state-space method in conjunction with the Fourier transform. Using the potential function approach, Dunn and Wienecke [10] and Ding et al [11,12] presented closed form solutions of transversely isotropic piezoelectric materials where displacements and electric potential are derivable from two potential functions. Pan [13] gave expressions for 2D piezoelectric Green’s functions and their boundary integral equations for dealing with fracture problems. Gao and Fan [14] obtained the Green’s functions of a piezoelectric plane containing an elliptic hole. Norris [15] discussed the derivation of dynamic Green’s functions for problems dealing with 2D dynamic piezoelectricity. Khutoryansky and Sosa [16] further examined the dynamic Green’s function of piezoelectric materials and gave a general representation formula of the governing equations of transient piezoelectricity through a generalization of the reciprocal theorem and the plane wave transform method. Qin [17] presented Green’s functions for 2D piezoelectric materials with holes of various shapes and applied them to establish boundary singular integral equations. Qin and Mai [18] also derived explicit Green’s functions for an interface crack subject to an edge dislocation in various piezoelectric bimaterial combinations. Pan and Yuan [19] gave 3D Green’s functions for anisotropic piezoelectric bimaterials. They showed that the 3D bimaterial Green’s function can be expressed in terms of a full-space part or the Kelvin-type solution and a complementary part or the Mindlin-type part. Studies in [6,20] suggested a numerical algorithm to compute the derivatives of the piezoelectric Green’s function. Green’s functions for piezoelectric materials can also be found in [21,22] for half-plane; [23,24] for semi-infinite crack; [25] for two-phase piezoelectric composites; [26] for anti-plane arc-crack; [27,28] for an impermeable elliptic hole or crack; [14,29] for a permeable elliptic hole; [30] for the case of collinear permeable cracks; and [31,32] for straight permeable interface cracks between two dissimilar piezoelectric media. In this chapter, we begin with an introduction to free space Green’s function of piezoelectricity with some typical approaches including the Radon transform method, the potential function approach, and the Fourier transform scheme. This is followed by discussion about extension of the results to problems with half-plane, bimaterials, interface crack, elliptic hole and inclusion, arbitrarily shaped hole, semi-infinite crack, and anti-plane problems. Finally, applications of the Green’s function approach to

Green’s function and boundary elements of multifield materials

26

dynamic problems are described. 2.2 Green’s function by Radon transforms Applications of the Radon transform to deriving Green’s function of electroelastic problems have been studied by Dunn [7] and Pan and Tonon [20] for transversely isotropic piezoelectricity, Hill and Farris [33] for three-dimensional piezoelectricity where the resulting solution is represented by a line integral that is evaluated numerically using standard Gaussian quadrature, Denda et al [34] for time-harmonic problems of 2D piezoelectricity, and Ma and Wang [35] and Wang and Zhang [36] for 3D dynamic problems. In this section we follow the results given in [7]. Consider a 3D electroelastic problem of a piezoelectric solid, where the field quantities depend on the coordinates x1 and x2 only. Using the shorthand notation described in Section 1.5, the governing equations (1.59)1,2 and constitutive relation (1.65) can be written as

EiJMnU J ,in + bM = 0,

P iJ = EiJMnU M ,n

(2.1)

where the generalized displacement UJ, stress P iJ , stiffness constant EiJMn are different from those in Eqs (1.114) and (1.115) which were initially for magnetoelectroelastic problems. These constants are now defined by ­s , J £ 3, P iJ = ® ij ¯ Di J = 4,

EiJMn

­u ,, J £ 3, UJ = ® J ¯ f J = 4,

J , M £ 3, ­ cijmn , ° e , J £ 3, M = 4, ° = ® nij ° eimn , J = 4, M £ 3, °¯ -kin , J = 4, M = 4

(2.2)

(2.3)

By comparing Eq (1.33), the Green’s function corresponding to Eq (2.1) is defined by the following differential equation   EiJMnGMR ,in (x - x) + d JRd (x - x) = 0 (2.4)  where the components GIJ (x - x) represent the elastic displacement at x in the xI-direction (for I=1,2,3) or the electric potential (for I=4) due to a unit point force at   x in the xJ-direction (for J=1,2,3), or a unit point charge at x (for J=4), and d JR is  the generalized Kronecker delta. The boundary conditions on GIJ (x - x) are that  they and their first spatial partial derivative must vanish as x - x ® ¥ .  The Green’s function GIJ (x - x) can be obtained by applying the Radon transform (see Appendix A for details) to Eq (2.4)[7]:

K JM (n)

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

(2.5)

where K JM (n) = EiJMl ni nl , n and w are variables in the transformed space, and Gˆ is the Radon transform of G defined in Appendix A.

27

Green’s function of electroelastic problem

 Gˆ MR (n, w - n × x) =

³³

 GMR (x - x)dS (x)

(2.6)

n×x =w

Here the integration is to be performed over the infinite plane n × x = w . The inverse transform of Eq (2.6) yields 1  GMR (x - x) = 2 8p

ª ¶ 2Gˆ MR (n, w - n × x ) º dS (n) » ³³2 « ¶w2 ¼ n×x =w S ¬

(2.7)

where S 2 represents the surface of the unit sphere n = 1 . Making use of Eq (2.7), the inversion of Eq (2.5) followed by some manipulation yields  GMR (x - x) =

1  8p 2 x - x

³³ K

-1 MR

(n)d(n × t )dS (n)

(2.8)

S2

 -1 (n) = d JR and t is defined as a unit vector in the direction of x - x where K JM (n) K MR (see Fig. 2.1). The integral expression (2.8) for the extended Green's displacement components is similar to the expressions obtained by Chen [5] and Deeg [37]. The surface integral in Eq (2.8) can be converted to a contour integral by considering the coordinate system defined by the three mutually orthogonal vectors t-m-d (see Fig. 2.1) and utilizing the well-known properties of the Dirac delta function to yield [7]: 1  -1 GMR (x - x) = 2  K MR (n)dS (2.9) 8p x - x S³1  where S 1 is the contour n = 1 which lies in the m-d plane normal to x - x. Equation (2.9) is analogous to the contour integral obtained by Synge [38] for the Green’s function for anisotropic elasticity. As mentioned in [7], the contour integral in Eq (2.9) can be evaluated numerically for arbitrary anisotropy and explicitly for the case of transversely isotropic material symmetry. Such explicit expression can be found in [7,20].

m

n =1

d

 x-x t

Fig. 2.1 Definition for the mutually orthogonal unit vectors m, d and t 2.3 Green’s function by the potential function approach The Green’s function described in Section 2.2 for a transversely isotropic

28

Green’s function and boundary elements of multifield materials

piezoelectricity can also be solved by the potential function approach, which is based on expressing the four unknowns u, v, w, f in terms of two potential functions g(x, y, z) and v (x, y, z), in such a way that [10] ª ¶2 ¶4 º ¶v Ñ 2 + [(c44 + c13 )e33 - c33 (e15 + e31 )] u = « (c13e15 - c44 e31 ) g3» ¶x¶z ¶x¶z ¼ ¶y ¬

(2.10)

ª ¶2 ¶4 º ¶v Ñ 2 + [(c44 + c13 )e33 - c33 (e15 + e31 )] v = «(c13e15 - c44 e31 ) g+ 3» ¶y¶z ¶y¶z ¼ ¶x ¬

(2.11)

ª º ¶4 ¶2 w = « -c11e15Ñ 2Ñ 2 - c44 e33 4 + [c13 (e15 + e31 ) + c44 e31 - c11e33 ] 2 Ñ 2 » g ¶z ¶z ¬ ¼

(2.12)

ª º ¶4 ¶2 f = « c11c44Ñ 2Ñ 2 + c44 c33 4 + (c11c33 - 2c44 c13 - c132 ) 2 Ñ 2 » g ¶ ¶ z z ¬ ¼

(2.13)

where Ñ 2 is the 2D Laplace’s operator defined in Eq (1.29). Substituting Eqs (2.10)-(2.13) into Eqs (1.78)-(1.81) and after some manipulation, we have § ¶2 · ¨ Ñ2 + 2 2 ¸ v = 0 , v0 ¶z ¹ ©

§ ¶2 · § ¶2 · § ¶2 · ¨ Ñ2 + 2 2 ¸ ¨ Ñ2 + 2 2 ¸ ¨ Ñ2 + 2 2 ¸ g = 0 v1 ¶z ¹ © v2 ¶z ¹ © v3 ¶z ¹ ©

(2.14)

in which all force terms bi are assumed to be zero for simplicity, and v0 = (c66 / c44 )1/ 2 , vi2 are the three roots of the characteristic equation:

av 6 + bv 4 + cv 2 + d = 0

(2.15)

with a = c11 ( k11c44 + e152 ) 2 b = c11 ( k11c33 + 2e15e33 ) - k11c13 (c13 + 2c44 ) + c44 ( k33c11 + e31 ) - 2e15c13 (e31 + e15 )

c = c33 [ k11c44 + k33c11 + e31 (e31 + e15 )] - c13 k33 (c13 + 2c44 ) + (e31 + e15 )(c33e15 - 2c13e33 ) + e33 (c11e33 - 2c44 e31 ) 2 d = c44 ( k33c33 + e33 )

The three roots vi2 are easily obtained using the well-known solution for a cubic equation. Either all three roots are real or one is real and the other two are complex conjugates. Based on the general solution (2.10)-(2.13), Dunn and Wienecke [10] presented a complete set of Green’s functions by considering following two problems: (i) a point charge and a point force directed along the z-axis, both acting at the origin; and (ii) a point force at the origin directed along the x-axis. In the following we present some results given in [10] in order to provide a brief introduction to the potential function approach of Green’s function. The reader may refer to [10-12] for more detail. (i) Green’s function for a point charge and point force in the z-direction

29

Green’s function of electroelastic problem

In this case the problem is axially symmetric about the z-axis and the potential functions g and v can be assumed in the form [10]: v=0 3

3

i =1

i =1

g = ¦ Ai f (xi ) = ¦ Ai [Q1 (xi ) zi ln Ri* + Q2 (xi ) Ri + Q3 (xi ) zi ]

(2.16)

where Ai are unknown constants, xi = ( x, y, zi ) , and Ri = x 2 + y 2 + zi2 , Ri* = Ri + zi , zi = vi z

Q j (xi ) = q j 0 + q j1 x + q j 2 y + q j 3 zi + q j 4 xy + q j 5 xzi + q j 6 yzi + q j 7 x 2 + q j 8 y 2 + q j 9 zi2

( j = 1, 2,3)

(2.17) (2.18)

The linear relation between the functions g and f in Eq (2.16) requires that f also satisfy the z-weighted harmonic: § 2 ¶2 · ¨Ñ + 2 ¸ f = 0 ¶zi ¹ ©

(2.19)

To find the explicit form of f the thirty qij are determined using Eq (2.19). After substantial computation, Dunn and Wienecke [10] found that f can be expressed in the form f (xi ) = 3(3Ri2 - 5 zi2 ) zi ln Ri* - (4 Ri2 - 15 zi2 ) Ri - 6 zi3

(2.20)

Substituting Eq (2.20) into (2.16), and later into Eqs (2.10)-(2.13), yields u = ¦ Ai l uv i

3

3 y x , v = ¦ Ai l uv , i * * R Ri Ri i =1 i Ri

3

3

i =1

1 w = ¦ Ai l , Ri i =1

1 f = ¦ Ai l Ri i =1

w i

(2.21)

f i

where 3 l uv i = [(c13 + c44 )e33 - c33 (e31 + e15 )]vi + (c44 e31 - c13 e15 )vi

l iw = -c44 e33vi4 - [e31 (c13 + c44 ) - e33c11 + e15c13 ]vi2 - e15c11 f i

4 44 33 i

(2.22)

2 i

l = c c v + [c13 (c13 + 2c44 ) - c11c33 ]v + c11c44

There are three unknowns Ai in Eq (2.21). We then need to find three conditions to determine the three unknowns. The fact that displacements u and v are bounded on the z-axis leads to 3

¦ Al i

uv i

=0

(2.23)

i =1

The remaining two equations can be established by considering the requirements of the force and charge balance. These requirements are enforced by integrating the traction and normal component of the electric displacement over the surface of a small spherical cavity centered at the origin, and requiring these to balance the point

30

Green’s function and boundary elements of multifield materials

force Pz and the point charge Q. These conditions lead to nia P = z , 2 2p i -1

3

¦A v i

i =1

3

nie Q = 2 2p i -1

¦A v i

i =1

(2.24)

where 2 w f nia = 2[l uv i (c13 + c44 vi ) + vi l i (c44 - c33 ) + vi l i (e15 - e33 )] f w 2 nie = 2[-l uv i (e15 vi + e31 ) + vi l i (e33 - e15 ) + vi l i ( k11 - k33 )]

(2.25)

The Green’s functions can then be obtained by setting Pz and Q to be unity and solving Eqs (2.23) and (2.24) for Ai. (ii) Green’s function for a point force Px in the x-direction As with the treatment for the point force in z-direction, the potential functions g and v are assumed in the form: 3 ª º xz g = ¦ Di «Q4 (xi ) x ln Ri* + Q5 ( xi ) *i + Q6 (xi ) x » Ri i =1 ¬ ¼ - D0 y v= R0*

(2.26)

where Qj (j=4-6) are quadratic polynomials defined by Eq (2.18). With a treatment similar to that in (i) above, we can express the function g in the form [10] 3 ª º xz g = ¦ Di «3( Ri2 - 5 zi2 ) x ln Ri* + (13Ri2 - 15 zi2 ) *i + 4 xzi2 » Ri i =1 ¬ ¼

(2.27)

Substitution of Eqs (2.26)2 and (2.27) into Eqs (2.10)-(2.13) leads to 3 ª1 y2 º 3 x2 º x uv ª 1 u = D0 « * D l , w = ¦ Di l iw , ¦ i i « * *2 » *2 » * R R R R R R R i =1 i =1 i i ¼ i Ri 0 0 ¼ ¬ 0 ¬ i 3 3 xy xy x uv , v = D0 D Di l if + l f = ¦ ¦ i i *2 *2 R0 R0 Ri Ri Ri Ri* i =1 i =1

(2.28)

There are four unknowns Di (i=0-3) in Eq (2.28), which can be determined by enforcing both the requirement that the solution be bounded on the z-axis and the force balance condition. The fact that the displacements u and v are bounded on the z-axis leads to 3

D0 v0 + ¦ Di vi luv i =0

(2.29)

i =1

Requiring that w and f be bounded on the z-axis leads to the following two equations: 3

¦Dl i

i =1

3

w i

= 0,

¦Dl i

f i

=0

(2.30)

i =1

The single remaining equation can be obtained by considering the force balance in the x-direction:

31

Green’s function of electroelastic problem 3

D0 v0 c44 + ¦ Di i =1

nit P = x 2 vi - 1 2p

(2.31)

where nit = vi l iuv (c44 - c11 ) + l iw (c13vi2 + c44 ) + l if (e31vi2 + e15 )

(2.32)

The unknowns Di can thus be determined from Eqs (2.30)-(2.32). Having determined all the unknowns Ai and Di, the fundamental solutions (2.10)-(2.13) can be explicitly expressed in terms of material constants and generalized coordinates in the form [22] ­ u ½ ª G11 G12 ° v ° «G ° ° « 21 G22 ® ¾= ° w° «G31 G32 « ¯° f ¿° ¬G41 G42

G13 G23 G33 G43

G14 º ­ Px ½ G24 »» °° Py °° ® ¾ G34 » ° Pz ° » G44 ¼ ¯°-Q ¿°

(2.33)

where Gij, defined in Eq (2.4), are given by G11 =

2 2 3 x 2 D0 uv y + zi + zi Ri D l , ¦ i i R0 R0*2 i =1 Ri Ri*2 3

G13 = G31 = ¦ Bi l uv i i =1

3

G14 = G41 = -¦ Ai l uv i i =1

G12 = G21 =

3 xyD0 xy + Di l uv , ¦ i *2 Ri Ri*2 R0 R0 i =1

3 x x = Di l iw , ¦ * Ri Ri* Ri Ri i =1

2 2 3 3 x x y 2 D0 f uv x + zi + zi Ri = l D G = D l , , ¦ i i R R* 22 R R*2 ¦ i i Ri Ri* i =1 Ri Ri*2 i =1 i i 0 0 3

G23 = G32 = ¦ Bi l uv i i =1

3 y y = ¦ Di l iw * * , R Ri Ri i =1 i Ri

3

G24 = G42 = -¦ Ai l uv i i =1

3

G33 = ¦ Bi l iw i =1

3 y y = ¦ Di l if * * , R Ri Ri i =1 i Ri

3 3 1 1 w 1 = ¦ Bi l if , , G34 = G43 = -¦ Ai l i R Ri R i =1 i =1 i i 3

G44 = -¦ Ai l if i =1

1 Ri

(2.34)

with A1 =

1 (v12 - 1)(v22 - 1)(v32 - 1) 4pg e v1 (v12 - v22 )(v12 - v32 )

B1 =

1 2 (v12 - 1)[n2e l 3uv (v32 - 1) - n3e l uv 2 (v2 - 1)] 4pg a

D0 =

1 , 4pc44 v0

D1 =

(2.35)

1 l f2 l3w - l3fl 2w . 4pg t c44

The constants A2, B2, D2, A3, B3, and D3 are obtained from A1, B1, and D1 by cyclically

32

Green’s function and boundary elements of multifield materials

permuting the indices 1, 2, and 3. The constants g a , g e , and g t appearing in Eq (2.35) are a e a e 2 uv a e a e g a = (v12 - 1)l1uv (n2a n3e - n3a n2e ) + (v22 - 1)l uv 2 ( n3 n1 - n1 n3 ) + (v3 - 1)l 3 ( n1 n2 - n2 n1 ) (2.36)

g e = ( k11 - k33 )[c11 (c44 - c33 ) + c44 (c33 + 2c13 ) + c132 ] + c11 (e33 - e152 ) 2 + c33 (e31 + e15 ) 2 - c44 (e33 + e31 ) 2 + 2c13 [e15 (e15 + e31 - e33 ) - e33e31 ] f w f w uv f w f w g t = v1l1uv (l 3f l 2w - l f2 l 3w ) + v2 luv 2 (l1 l 3 - l 3 l1 ) + v3 l 3 (l 2 l1 - l1 l 2 )

(2.37) (2.38)

2.4 Green’s function by Fourier transforms In this section, we briefly examine the application of Fourier transform techniques to the fundamental solution of piezoelectric materials. Lee and Jiang [8] used Fourier transform techniques to derive Green’s functions of 2D piezoelectric material. They began with considering a plain strain problem of a transversely isotropic piezoelectric solid with the symmetry of a hexagonal class. The solid is subjected to a point force Fi and a point electric charge Y at the source point x. Note that Fi and Y are unity, but are used here as a bookkeeping device [8]. If the x2 - x3 plane is the isotropic plane, one can employ either the x1 - x2 or x1 - x3 plane for the study of plane electromechanical phenomena. Choosing the former, the plain strain problem is defined by Eqs (1.144) and (1.145). Recently, Li and Wang [39] extended Lee and Jiang’s formulation to the case of three-dimensional problem. The discussion below focus on the results presented in [8] and [39]. Making use of Eqs (1.49), (1.(51), (1.144), and noting that w=0 or constant in the case of plane strain, Eq (1.59)1,2 can be written in the form ª ¶2 ¶2 c c + « 11 2 33 2 ¶x1 ¶x2 « « ¶2 « (c12 + c33 ) ¶x1¶x2 « « ¶2 « -(e21 + e13 ) ¶x1¶x2 ¬«

(c12 + c33 )

¶2 ¶x1¶x2

¶2 º » ¶x1¶x2 » ­u ½ ­ F1d(ȟ, x) ½ ¶2 » ° ° ° ° + e22 2 » ® v ¾ + ® F2 d(ȟ, x) ¾ = 0 (2.39) ¶x2 » ° ° ° ° ¯f ¿ ¯ Yd(ȟ, x) ¿ ¶2 » + k 22 2 » ¶x2 ¼»

(e21 + e13 )

c33

¶2 ¶2 c + 22 ¶x12 ¶x22

e13

¶2 ¶x12

-e13

¶2 ¶2 e 22 ¶x12 ¶x22

k11

¶2 ¶x12

After applying the double Fourier transform which is defined by:

1 ¥ fˆ (l1 , l 2 ) = 2p ³ -¥ ³

¥ -¥

f ( x1 , x2 )e - i ( l1x1 +l2 x2 ) dx1dx2

(2.40)

- A13 º ­ Fˆ1 ½ °° °° - A23 »» ® Fˆ2 ¾ ˆ° - A33 ¼» ° Y °¯ ¿°

(2.41)

Eq (2.39) may be further written as [8] ­uˆ ½ ª A11 ° ° 1 « ® vˆ ¾ = « A12 ° ˆ ° 2p « A ¬ 13 ¯f ¿ where

A12 A22 A23

33

Green’s function of electroelastic problem

A11 = (a11l14 + b11l12 l 22 + g11l 24 ) N -1 , A12 = (a12 l13l 2 + b12 l1 l 32 ) N -1 , A13 = (a13l13l 2 + b13l1 l32 ) N -1 , 4 23 1

2 23 1

2 2

4 2

A22 = (a 22 l14 + b22 l12 l 22 + g 22 l 24 ) N -1 , -1

4 33 1

2 33 1

2 2

4 2

(2.42)

-1

A23 = (a l + b l l + g 23l ) N , A33 = (a l + b l l + g 33l ) N ,

N (l1 , l 2 ) = l 6 + k1l14 l 22 + k 2 l12 l 42 + k3l 62

(2.43)

In Eq (2.42), constants aij , bij , g ij , and ki are given by 2 a11 = (e132 + c33 k11 ) P, b11 = (2e13e22 + c22 k11 + c33 k 22 ) P, g11 = (e22 + c22 k 22 ) P,

a12 = -(c12 k11 + c33 k11 + e132 + e13e21 ) P, b12 = -(c12 k 22 + c33 k 22 + e13e22 + e22 e21 ) P, a13 = (c12 e13 - c33e21 ) P, b13 = (c12 e22 + c33e22 - e13c22 - e21c22 ) P, 2 a 22 = c11k11 P, b22 = (2e13e21 + c11k 22 + e132 + e21 + c33 k11 ) P, g 22 = c33 k 22 P,

a 23 = -c11e13 P, b23 = (c12 e13 - c11e22 + e21c12 + e21c33 ) P, g 23 = -c33e22 P, a33 = -c11c33 P, b33 = (c122 - c11c22 + 2c33c12 ) P, g 33 = -c33c22 P, k1 = (c11c22 k11 + 2c11e13e22 - 2c12 e132 + c11c33 k 22 - 2c12 c33 k11 2 - 2c12 e21e13 + e21 c33 - c122 k11 ) P,

k 2 = (2e13e21c22 - 2c33e21e22 - 2c12e22 e13 - c122 k 22 - 2c12c33 k 22 2 2 + c22 c33 k11 - 2c12 e22 e21 + e132 c22 + c11c22 k 22 + e22 c11 + e21 c22 ) P, 2 k3 = (c33e22 + c33c22 k 22 ) P,

(2.44)

P = (c11e132 + c11c33 k11 ) -1

(2.45)

where

Applying the Fourier inversion defined by f ( x1 , x2 ) =

1 ¥ 2p ³ -¥ ³

¥ -¥

fˆ (l1 , l 2 )ei ( l1x1 +l2 x2 ) d l1d l 2

(2.46)

to Eq (2.41), followed by some manipulation, yields the Green’s function in the original space:

G11 (ȟ, x) =

1 1 (a11 I 4 + b11 I 2 + g11 I 0 ), G12 (ȟ, x) = G21 (ȟ, x) = (a12 I 3 + b12 I1 ), 2p 2p

G22 (ȟ, x) =

1 1 (a 22 I 4 + b22 I 2 + g 22 I 0 ), G31 (ȟ, x) = -G13 (ȟ, x) = (a13 I 3 + b13 I1 ), (2.47) 2p 2p

G33 (ȟ, x) = where

1 1 (a33 I 4 + b33 I 2 + g 33 I 0 ), G32 (ȟ, x) = -G23 (ȟ, x) = (a 23 I 4 + b23 I 2 + g 23 I 0 ) 2p 2p

34

Green’s function and boundary elements of multifield materials

Im =

1 ¥ ¥ Ȝ1m l 42- m i ( l1x1 +l2 x2 ) e d l1d l 2 2p ³ -¥ ³ -¥ N (l1 , l 2 )

(2.48)

and the components Gij (x, ȟ ) represent the elastic displacement at x in the xi-direction (for i=1,2), or the electric potential (for i=3) due to a unit point force at x in the xj-direction (for j=1,2), or a unit point charge at x (for j=3). It is now clear that all that is required is to evaluate the integral (2.48). In what follows we describe Lee and Jiang’s approach presented in [8] for calculating the integral Im. They indicated that the explicit expression of Im depends on the roots of the equation N (l1 , l 2 ) = 0 . In general there are two independent sets of roots: 1) l1 = ±ip0 l 2 ,

l1 = ±( p1 ± iq1 )l 2 ;

2) l1 = ±ip0l 2 ,

l1 = ±( p1 ± q1 )l 2

(2.49)

where p0 , p1 and q1 are real positive numbers. Taking the first set of roots as an example, N (l1 , l 2 ) can be factored into the following form:

N (l1 , l 2 ) = ( l12 + ( p0 l 2 ) 2 )( (l1 - p1l 2 ) 2 + (q1l 2 ) 2 )( (l1 + p1l 2 ) 2 + (q1l 2 ) 2 ) (2.50) The integral (2.48) can then be evaluated using the residue theorem for the variable l1 and by considering the so-called Hadamard’s finite part [40]. As a result, Im may be written in the form 6

I m = ¦ Rmi Li , (m = 0,1, 2,3, 4) i =1

in which R01 = Q, R02 = 0, R03 = R05 = -

Q 1 , R04 = - R06 = ( p02 - q12 + 3 p12 )Q, 2 4 p1

R11 = 0, R12 = R21 = - p02Q, R13 = - R15 =

1 ( p02 - q12 - p12 )Q, 4 p1

1 1 R14 = R16 = (q12 + p12 )Q, R22 = 0, R23 = R25 = p02Q, 2 2

R24 = - R26 = -

1 ( p02 q12 + p02 p12 - p14 - 2 p12 q12 - q14 )Q, R31 = 0, 4 p1

R32 = p04Q, R33 = - R35 =

1 (3 p02 p12 - p02 q12 + p14 + 2 p12 q12 + q14 )Q, 4 p1

1 R34 = R36 = - (q12 + p12 ) p02Q, R41 = p04Q, R42 = 0, 2

1 R43 = R45 = (2 p02 p12 - 2 p02 q12 + p14 + 2 p12 q12 + q14 )Q, 2 1 (2 p02 p12 q12 - p02 q14 + 3q12 p14 + 3 p12 q14 + q16 + p16 )Q, R44 = - R46 = 4 p1

(2.51)

35

Green’s function of electroelastic problem

L1 = -

§p x · 1 1 ln p02 x12 + x22 , L2 = - 2 tan -1 ¨ 0 1 ¸ , p0 p0 © x2 ¹

L3 = -

L4 =

1 ln ( p1 x1 + x2 ) 2 + (q1 x1 ) 2 , q1

2 2 § hx + p1 x2 · º § p1 ª« p · §q · - ln ¨ x1 + 1 x2 ¸ + ¨ 1 x2 ¸ + tan -1 ¨ 1 ¸» , q1h « q1 x2 ¹ » h ¹ ©h ¹ © © ¬ ¼

L5 = -

1 ln ( x2 - p1 x1 ) 2 + (q1 x1 ) 2 , q1

2 2 § hx - p1 x2 · º § p1 ª« p1 · § q1 · L6 = ln ¨ x1 - x2 ¸ + ¨ x2 ¸ - tan -1 ¨ 1 ¸» q1h « q1 x2 ¹ » h ¹ ©h ¹ © © ¬ ¼

(2.52)

with x1 = x1 - x1 , x2 = x 2 - x2 , h = p12 + q12 Q=

1 4 (q1 - 2 p02 q12 + 2 p12 q12 + p04 + 2 p01 p12 + p14 ) -1 2p

(2.53)

The Green’s function obtained above is suitable for two-dimensional piezoelectric material only. Extension of the formulation to three-dimensional material is straightforward [39]. To this end, taking the Fourier transform in the x1 -, x2 - and x3 - directions for GMR (x) in Eq (2.4), we have [39] ¥ Gˆ MR (ȟ ) = ³ GMR (x)eix×ȟ dx

(2.54)

-¥

where ȟ = {x1 , x 2 , x3 } , and the inverse of the Fourier transform (2.54) gives GMR (x) =

1 8p

3

¥

³

-¥

Gˆ MR (ȟ )e -ix×ȟ dȟ

(2.55)

Then Eq (2.4) turns into a set of algebraic equations EiJMl Gˆ MR (ȟ )xixl = d JR

(2.56)

-1 -1 (ȟ )d JR = K RM (ȟ ) Gˆ MR (ȟ ) = K JM

(2.57)

Solving Eq (2.56) provides

Taking the Fourier inverse transform (2.57), we obtain GMR (x) =

1 8p 3

³

¥

-¥

-1 (ȟ )e -ix×ȟ dȟ = K MR

1 8p 3

³

¥

-¥

AMR (ȟ ) - ix×ȟ e dȟ D (ȟ )

(2.58)

Replacing x in Eq (2.58) with -x, we have[39] GMR (x) =

1 8p 3

³

¥

-¥

AMR (ȟ ) ix×ȟ e dȟ D (ȟ )

(2.59)

36

Green’s function and boundary elements of multifield materials

where D(x) and AMR(x) are the determinant and adjoint of matrix KMR(x), respectively. Li and Wang [39] provided further expressions of GMR(x) by defining the volume element dx as dx=x2dxdS, where ȟ = (xi xi )1/ 2 and S is the surface element on the unit sphere S2 in the x-space, centred at the origin of the coordinates xi (see Fig. 2.2). x3

x

x

q dS o

r

j x2

x1 S1 S2

Fig. 2.2 The unit sphere S2 in the x-space; S1 is the unit circle on S2 intersected by the plane perpendicular to x; r is a local polar coordinate [39]

By adding Eqs (2.58) and (2.59), we obtain under the sphere coordinate system GMR (x) =

1 16p3

³

¥

-¥

dȟ

³

S

AMR (ȟ 0 ) i x ȟ x0 ×ȟ0 e dS (ȟ 0 ) D (ȟ 0 )

2

(2.60)

where ȟ0 =

ȟ , ȟ

x0 =

x , x

x = ( xi xi )1/ 2

(2.61)

Integrating Eq (2.60) with respect to x leads to GMR (x) =

1 8p 2

³

S

d( x x 0 × ȟ 0 )

2

AMR (ȟ 0 ) dS (ȟ 0 ) D(ȟ 0 )

(2.62)

Denoting the angle between x and x by q, we have

ȟ 0 × x0 = cos q, d (ȟ 0 × x0 ) = - sin qd q, dS (ȟ 0 ) = sin qd qd j

(2.63)

where j is defined on the plane perpendicular to x. Substituting Eq (2.63) into Eq (2.62), we obtain

GMR (x) =

1 8p2

³

1

-1

d (x0 × ȟ 0 )d( x x 0 × ȟ 0 ) ³

2p

0

AMR (ȟ 0 ) 1 dj = 2 D(ȟ 0 ) 8p x

v³

S

1

AMR (ȟ 0 ) dj D(ȟ 0 )

(2.64)

Thus GMR(x) can be expressed by a line integral along S1 which lies on the plane perpendicular to x. Eq (2.64) can be further simplified by considering ȟ 0 = eij as GMR (x) =

1 4p2 x

³

p/ 2

-p / 2

AMR (eij ) dj D ( e ij )

(2.65)

37

Green’s function of electroelastic problem

Defining j = arctan z and R = 1 + z 2 (see Fig. 2.3), and substituting them into Eq (2.65), we have GMR (x) =

1 4p 2 x

³

¥

-¥

AMR ( Rei arctan z ) 1 dz = 2 i arctan z D( Re ) 4p x

³

¥

-¥

AMR (p + zq) dz D(p + zq)

(2.66)

where p and q are two unit vectors and p ^ q (see Fig. 2.3).

q

p o

j

z

R

Fig. 2.3 The unit vector p and q on the plane where the unit circle S1 is located

Considering that Eq (2.4) are elliptic and the coefficients in D(x) are all real, D(x) has no real root and all imaginary roots occur in complex conjugate pairs. Therefore the eight-order polynomial equation of z

D(p + zq) = 0

(2.67)

has eight roots, four of them being the conjugate of the remainder. With these roots, D(p + zq) can be expressed as 8

4

k =0

m =1

D(p + zq) = ¦ ak +1z k = a9 Õ (z - z m )(z - z m )

(2.68)

where Im(z m ) > 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

Green’s function and boundary elements of multifield materials

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

Green’s function and boundary elements of multifield materials

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

Green’s function and boundary elements of multifield materials

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 x2<0, where q (1)* = A (1)T q 0 + B (1)T b, and qb , qb are unknown constant vectors which are determined by substituting (2.95)-(2.98) into (2.94). Following the derivation in Section 2.5, we obtain

A (1)qb(1) + A (2) qb(2) = A (1) I bq (1)* ,

B (1)qb(1) + B (2) qb(2) = B (1) I bq (1)* .

(2.99)

Solving Eq (2.99) yields qb(1) = B (1) -1[I - 2(M (1) -1 + M (2)-1 ) -1 L(1) -1 ]B (1) I bq (1)* ,

(2.100)

qb(2) = 2B (2) -1 (M (1)-1 + M (2) -1 ) -1 L(1) -1B (1) I bq (1)* ,

(2.101)

where M ( j ) = -iB ( j ) A ( j ) -1 is the surface impedance matrix. 2.7 Bimaterial solid with an interface crack In the previous section we presented the Green’s function for a bimaterial solid without cracks. This section discusses the Green’s function for a bimaterial system with an interface crack subjected to a line force-charge q 0 and a generalized line

43

Green’s function of electroelastic problem

dislocation b. Material combinations of the bimaterial system here may be piezoelectric-piezoelectric (PP), piezoelectric-anisotropic conductor (PAC), piezoelectric-isotropic dielectric (PID), or piezoelectric-isotropic conductor (PIC). The focus is on the fundamental solution of a bimaterial solid with an interface crack subjected to loadings q 0 and b [2].

za·

x2 (q 0 , b)

material 1

· z0

0 2c interface

x1

material 2

Fig. 2.6 Geometry of the interface crack system

The geometrical configuration analysed is depicted in Fig. 2.6. A crack of length 2c is shown lying along the interface between two dissimilar materials, subjected to loadings q 0 and b at z0 in material 1. In the following, Green’s functions satisfying the traction-charge free conditions on the faces of the interface crack are derived by way of the Stroh formalism and the solution for bimaterials due to loadings q 0 and b. It is clear from Eq (1.128) that the Green’s function of U for the cracked medium subject to a line force-charge q 0 and a generalized line dislocation b can be determined if the solution of f(z) in Eq (1.128) is known. Therefore our focus here is to determine the function f(z). (a) Green’s function in a cracked homogeneous material (HM) Consider the problem of a two-dimensional infinite piezoelectric medium (see Fig. 2.6, in which material 1 is the same as material 2 in this case). If the generalized loadings q 0 and b are located at z0(x10, x20), the electroelastic solution f(z) in Eq (1.128) is [2]

f 0 ( z ) = f 0 ( za ) =

q* ln( za - za 0 ) 2pi

(2.102)

where q* is defined in Eq (2.86). The expression for f0(z) is obtained by imposing q 0 net traction-charge in a circuit around the source point z0, and by demanding that the jumps in U be given by b = {Du1 Du2 Du3 Dj}T , where Du is the jump for the quantity u across the dislocation line, and the generalized line force q 0 = {q01 q02 q03 q04 }T , in which q0i (i £ 3) represents components of a line

44

Green’s function and boundary elements of multifield materials

-distributed force vector and q04 stands for a line electric charge. In the presence of an interface crack, however, the solution for a line force-charge q 0 and a generalized line dislocation b is no longer given by f0(z) alone. In addition to the conditions imposed at z0, the boundary conditions for the interface crack must be satisfied. This can be accomplished by evaluating the traction-charge on the crack faces due to the generalized loadings and introducing an appropriate function fI(z) to cancel the traction-charge on the crack faces. Substituting for f0(z) from Eq (2.102) into Eq (1.129), and then into Eq (1.120), the traction-charge on the crack surface induced by q 0 and b alone at z0 is of the form t ( x) = Bf0¢ ( x) + Bf0¢ ( x) =

1 1 1 1 B q* B q* 2pi x - z0 a 2pi x - z0 a

(2.103)

The prescribed traction-charge, -t(x1) acting on the crack faces, leads to the following Hilbert problem [44]:

t ( x)dx c( z ) c + cc( z ) 2pi ³ -c c + ( x)( x - z ) where c(z) is the basic Plemelj function defined as Bf I¢ ( z ) = -

c( z ) = ( z 2 - c 2 ) -1/ 2

(2.104)

(2.105)

Using contour integration one is finally led to

f I¢ ( z ) = -

1 1 -1 Bq* F* ( z , z0 )q* + B BF* ( z , z0 )q* , c = 2pi 2pi 2pi

(2.106)

where

F* ( z , z0 ) =

1 1 2 za - z0 a

ª c ( za ) º «1 » ¬ c( z0 a ) ¼

(2.107)

The functions defined in Eqs (2.102) and (2.106) together provide a Green’s function for the crack problem:

f ¢( z ) = f0¢ ( z ) + f I¢ ( z ) =

½° º 1 ­° ª 1 - F* ( z , z0 ) » q* + B -1BF* ( z , z0 )q* ¾ ®« 2pi ¯° ¬ za - z0 a ¼ ¿°

(2.108)

It is obvious that the singularity is contained in f0(z) and the crack interactions are accounted for by fI(z). The integration of Eq (2.108) with respect to z provides f(z), and then substituting the result of f(z) into Eqs (1.128) and (1.129), neglecting the integration constants which correspond to rigid body motion, yields the Green’s functions for U and j. (b) Green’s function in a PP bimaterial The solutions due to loadings q 0 and b for this bimaterial group (in this case, materials 1 and 2 are both piezoelectric, but the material constants are different) were provided in the previous section (Section 2.6). Using those solutions, the functions

45

Green’s function of electroelastic problem

f0 ( z ( i ) ) are given by f0(1) ( z (1) ) =

4 1 ­ (1) (1) (1)* (1) (1) * (1)* ½ ® ln( z* - z0* ) q + ¦ ln( z* - z0 k ) B k q ¾ , 2pi ¯ k =1 ¿

(2.109)

for material 1 (x2>0), 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 (x2<0), where q (1)* is the same as in Eq (2.95) and

B*k = B (1) -1[I - 2(M (1) -1 + M (2) -1 ) -1 L(1) -1 ]B (1) I k ,

(2.111)

(2) -1 B** (M (1) -1 + M (2) -1 ) -1 L(1) -1B (1) I k . k =B

(2.112)

The interface traction-charge induced by the generalized loading q (1)* is then of the form (2.113) t ( x) = B (2)f0¢(2) ( x) + B (2) f0¢(2) ( x) = F** ( x)q (1)* + F** ( x)q (1)* where

F** ( x) =

1 4 B (2) B** k ¦ 2pi k =1 x - z0(1)k

(2.114)

This traction-charge vector, t(x), can be removed by superposing a solution with the traction-charge, -t(x), induced by the potential fI(z). The solution for fI(z) can be obtained by setting ­ B (1)f I¢(1) ( z ), in material 1, h( z ) = ® -1 (2) (2) ¯H HB f I¢ ( z ), in material 2,

(2.115)

where H = -M (1) - M (2) . The function h(z) defined above, together with prescribed traction-charge, -t(x), on the crack faces, yields the following non-homogeneous Hilbert problem:

h + ( x) + H -1Hh - ( x) = -t ( x)

(2.116)

Writing the quantities h(x) and t(x) in their components in the sense of the eigenvectors w, w3 and w4 as [45]:

h( x) = h1 ( x)w + h2 ( x)w + h3 ( x)w 3 + h4 ( x)w 4 , t ( x ) = t ( x ) w + t ( x ) w + t3 ( x ) w 3 + t 4 ( x ) w 4 , we obtain

(2.117)

46

Green’s function and boundary elements of multifield materials

c( z ) c t ( x)dx + c1c( z ), + ³ c 2pi c ( x)( x - z ) c( z ) c t ( x)dx h2 ( z ) = + c2 c( z ), + ³ c 2pi c ( x)( x - z ) c ( z ) c t3 ( x)dx h3 ( z ) = - 3 + c3c( z ), 2pi ³ -c c3+ ( x)( x - z ) h1 ( z ) = -

h4 ( z ) = -

(2.118)

c 4 ( z ) c t4 ( x)dx + c4 c( z ) 2pi ³ -c c +4 ( x)( x - z )

where w, w3 and w4 are eigenvectors and have been defined in [2], and c( z ) = ( z + c) -1/ 2-ie ( z - c) -1/ 2+ie , c3 ( z ) = ( z + c) -1/ 2-k ( z - c) -1/ 2+k , -1/ 2 +k

(2.119)

-1/ 2 -k

c 4 ( z ) = ( z + c) ( z - c) The functions hi and ti are evaluated by taking inner products with w, w3 and w4, respectively, i.e. h1 ( x) =

wT Hh( x) wT Hh( x) w T Hh( x) , h2 ( x) = , h3 ( x) = 4T , T T w Hw w Hw w 4 Hw 3

w T Hh( x) w T Ht ( x) wT Ht ( x) w T Ht ( x) h4 ( x) = 3T , t= T , t3 = 4T , t4 = 3T w 3 Hw 4 w Hw w 4 Hw 3 w 3 Hw 4 The contour integration of Eq (2.118) provides wT H 4 h1 ( z ) = T ¦ [F(e, c, z, z0(1)k , B(2) , B**k q(1)* ) +F(e, c, z, z0(1)k , B(2) , B**k q(1)* )], w Hw k =1 h2 ( z ) =

h3 ( z ) =

h4 ( z ) =

(2.120)

(2.121)

wT H 4 (1)* (1)* [F(-e, c, z , z0(1)k , B (2) , B** ) + F(-e, c, z , z0(1)k , B (2) , B** )], (2.122) ¦ k q k q T w Hw k =1

w T4 H 4 ¦ [F(-ik, c3 , z, z0(1)k , B(2) , B**k q(1)* ) +F(-ik, c3 , z, z0(1)k , B(2) , B**k q(1)* )], wT4 Hw 3 k =1 (2.123) wT3 H wT3 Hw 4

4

¦ [F(ik, c , z, z 4

(1) 0k

(1)* (1)* , B (2) , B** ) +F(i k, c 4 , z , z0(1)k , B (2) , B** )], k q k q

k =1

(2.124) X* wT H w T H X* , c2 = - T , T -2 pe w Hw 1 + e w Hw 1 + e 2 pe wT H wT H X* X* c3 = - T 4 , c4 = - T 3 2 ipk w 4 Hw 3 1 + e w 3 Hw 4 1 + e -2ipk c1 = -

where

(2.125)

47

Green’s function of electroelastic problem

F(r , c, z , z0 k , X, Y) = -

XY ª c( z ) º 1 , «1 » (1 + e -2 pr ) ¬ c( z0 k ) ¼ z - z0 k

4

X* = ¦ ( B B q (2)

** k

(1)*

(2)

** k

+B B q

(1)*

(2.126)

).

k =1

Once hi(z) has been obtained, the functions f I¢( j ) ( z ) can be evaluated through use of Eqs (2.115) and (2.117). (c) Green’s function in a PAC bimaterial In (b) the solutions for a PP bimaterial due to loadings q 0 and b were presented. Here we extend these solutions to the case of a PAC bimaterial. When one of the two materials, say material 2, is an anisotropic conductor, the interface condition (2.94) should be changed to (1)

(1)

(2)

(2)

­ j1 ½ ­ j1 ½ °j ° °j ° ° 2° ° 2° ® ¾ =® ¾ . ° j3 ° ° j3 ° °¯j4 ¿° ¯° 0 ¿°

­ u1 ½ ­ u1 ½ °u ° °u ° ° 2° ° 2° ® ¾ =® ¾ , °u3 ° °u3 ° °¯ f ¿° ¯° 0 ¿°

(2.127)

Further inside the conductor, we have Di(2) = Ei(2) = 0 .

(2.128)

The formulations presented in (b) are still available if the following modifications are made: ªA A (2) = « (3´3) ¬ 0

0º , 1 »¼

ªB B (2) = « (3´3) ¬ 0

(2) (2) p4(2) = i, qb(2) = {qb(2) 1 qb 2 qb 3 0} ,

0º , 1 »¼ (b=1, 2, 3, 4)

(2.129) (2.130)

where A(3´3) and B(3´3) are well-defined material matrices for anisotropic elastic materials (see [43], for example). (d) Green’s function in a PID bimaterial In this subsection we consider a PID bimaterial for which the upper half-plane (x2>0) is occupied by a piezoelectric material (material 1), and the lower half-plane (x2<0) is occupied by an isotropic dielectric material (material 2). For simplicity a plane strain deformation only is considered here. Thus the related interface condition is reduced to (1)

(2)

(1)

(2)

­ u1 ½ ­ u1 ½ ­ j1 ½ ­ j1 ½ ° ° ° ° ° ° ° ° at x2=0 (2.131) ®u2 ¾ = ®u2 ¾ , ®j2 ¾ = ®j2 ¾ , °f° °f° °j ° °j ° ¯ ¿ ¯ ¿ ¯ 3¿ ¯ 3¿ (d1) General solutions for isotropic dielectric material The general solutions for an isotropic dielectric material can be assumed in the form

48

Green’s function and boundary elements of multifield materials

ui = 2 Re[vi f ( z )], ji = 2 Re[hi f ( z )], z = x1 + px2

(2.132)

where v(={vi}) and h(={hi}) are determined by [Q + p (R + RT ) + p 2 T]v = 0,

(2.133)

h = (R T + pT) v

For an isotropic dielectric material, the 3´3 matrices Q, R and T are in the form 0 º 0 0 º ª l + 2G 0 ª 0 l 0º ªG « » « » « G 0 » , R = «G 0 0 » , T = « 0 l + 2G 0 »» (2.134) Q=« 0 «¬ 0 «¬ 0 0 0 »¼ «¬ 0 0 -e0 »¼ 0 -e 0 »¼

where e0 is the dielectric constant of the isotropic dielectric material, and l and G are the Lamé constants [46] defined by l=

Em , (1 + m)(1 - 2m)

G=

E 2(1 + m)

(2.135)

Since p=i is a double root, a second independent solution can be written as

­¶ ½ ­¶ ½ ui = 2 Re ® [vi f ( z )]¾ , ji = 2 Re ® [hi f ( z )]¾ , ¶ ¶ p p ¯ ¿ ¯ ¿

(2.136)

where v = vp =

1 {1 i 0}, h = h p = {i 1 0} 2G

(2.137)

for plane elastic deformation, and v = ve =

-1 {0 0 1}, h = h e = {0 0 i} e0

(2.138)

for electric field. Following the method of Ting and Chou [46], the complete general solutions can be expressed as z-z z-z v p q2 f ¢( z ), j = Bqf ( z ) + 2i 2i 0 º ª1 (1 - 2k )i ªi 1 « » , B = « -1 2 0 A= i k » « 2G « «¬0 «¬ 0 0 -2Ge0-1 »¼

U = Aqf ( z ) +

h p q2 f ¢( z ),

(2.139)

0 0º -i 0 »» 0 i »¼

(2.140)

where q={q1 q2 q3}T are constants to be determined, and k=

l + 2G l+G

(2.141)

49

Green’s function of electroelastic problem

(d2) Green’s function for homogeneous isotropic materials For simplicity and conciseness of the following discussion, we consider here a line dislocation b, rather than a generalized loading q* , applied to an infinite plate at the point z0 (= x10 + ix20 ) . The function f(z) in Eq (2.139), in this case, can be assumed as f ( z ) = ln( z - z0 )

(2.142)

The condition

³

C

dU = b ,

for any closed curve C enclosing the point z0

(2.143)

yields q = q 0 = {q10 q20 q30 }T =

1 * A b, 2pi

(2.144)

where

A* =

0 ª -4kG -(2 - 4k )iG º 1 « » 2 2 0 iG G » 1 - 4k « «¬ 0 0 -e0 (1 - 4k ) »¼

(2.145)

(d3) Green’s function for PID bimaterials (d3.1) z0 located in isotropic dielectric material (material 2). Consider a PID bimaterial for which a line dislocation b is applied at z0 (= x10 + ix20 , x20<0). The interface condition (2.94) suggests that the solution can be assumed in the form: U (1) =

1 Im[ A (1) ln( za - z0 q1 ], p

j (1) =

1 Im[B (1) ln( za - z0 q1 ] p

(2.146)

for material 1 in x2>0 and

U (2) = j

(2)

p( z - z ) v p q20 º 1 ª (2) * + A (2)q 2 ln( z - z0 ) » , Im « A A b ln( z - z0 ) + p ¬ ( z - z0 ) ¼

p( z - z )h p q20 º 1 ª = Im «B 2 A*b ln( z - z0 ) + + B (2)q 2 ln( z - z0 ) » p ¬ ( z - z0 ) ¼

(2.147)

for material 2 in x2<0, where q1 and q2 are unknowns to be determined. The substitution of Eqs (2.146) and (2.147) into Eq (2.94), yields A (1)q1 + A (2)q 2 = A (2) A*b, B (1) q1 + B (2)q 2 = B (2) A*b

(2.148)

Solving Eq (2.148), we have q1 = B a b,

q 2 = Bbb ,

(2.149)

50

Green’s function and boundary elements of multifield materials

where B a = (B (2) -1B (1) - A (2) -1A (1) ) -1 (B (2) -1B (2) - A (2) -1A (2) ) A* , Bb = (B (1) -1B (2) - A (1) -1A (2) ) -1 (B (1) -1B (2) - A (1) -1A (2) ) A*

(2.150)

(d3.2) z0 located in piezoelectric material (material 1). When the single dislocation b is applied at z0 with x20 > 0 , the electroelastic solution can be assumed in the form U (1) =

3 1 1 Im[ A (1) ln( za - z0 a B (1)T b] + Im ¦ [A (1) ln( za - z0b qb(1) ] , p p b=1

(2.151)

3 1 1 Im[B (1) ln( za - z0 a B (1)T b] + Im ¦ [B (1) ln( za - z0b qb(1) ] p p b=1

(2.152)

j (1) =

for material 1 in x2>0 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<0. Substituting Eqs (2.151)-(2.154) into Eq (2.94), we find the solution has the same form as that given in (b) except that all matrices here are 3´3 matrices. The interface traction-charge vector induced by the dislocation b is, then, of the form t ( x) = j,1(1) = 2 Re[B (1)f0¢(1) ] =

º 1 ª 1 Im « B (1) B a b » p ¬ x - z0 ¼

(2.155)

for z0 in material 2 and

t ( x) = j,1(2) = 2 Re[B (2)f0¢(2) ] =

º 1 ª 3 (2) 1 Im « ¦ B qb(2) » = [F** ( x) + F** ( x)]b p ¬« b=1 x - z0 b »¼ (2.156)

for z0 in material 1. Similar to the treatment in (b), the traction-charge t(x) given in Eq (2.155) [or Eq (2.156)] can be removed by superposing a solution with the traction-charge, -t(x), induced by the potential fI(z). The solution for fI(z) can also be obtained by setting ­ B (1)f I¢(1) ( z ), in material 1, h( z ) = ® -1 (2) (2) ¯H HB f I¢ ( z ), in material 2

The subsequent derivation is the same as that in (b) except that

(2.157)

51

Green’s function of electroelastic problem c( z ) = ( z + c ) -1 / 2 - ie ( z - c ) -1 / 2 + ie ,

(2.158)

c3 ( z ) = ( z + c ) -1 / 2 ( z - c ) -1 / 2

where e is determined in a similar manner to that of Suo et al [45]. (e) Green’s function in a PIC bimaterial As treated in (c), the solutions for a PIC bimaterial due to a single dislocation can be obtained based on the results given in (d). When one of the two materials, say material 2, is an isotropic conductor, the interface condition (2.94) should be changed to (1)

(2)

­ u1 ½ ­ u1 ½ ° ° ° ° u = ® 2¾ ®u2 ¾ , °j° °0° ¯ ¿ ¯ ¿

(1)

(2)

­ f1 ½ ­ f1 ½ ° ° ° ° f = ® 2¾ ®f 2 ¾ , °f ° °0° ¯ 3¿ ¯ ¿

at x2=0.

(2.159)

The formulations presented in subsection (d) are still available if the following modification is made: q 2 = {q21 q22 0}

(2.160)

2.8 Elliptic hole and inclusion The problem of determining Green’s functions in a piezoelectric plate with an elliptic hole has been studied by Gao and Fan [14] and Lu et al [29] for a permeable elliptic hole, Liu et al [27] and Lu and Williams [28] for an impermeable elliptic hole or a crack, and Cheung and Ting [47] for an inclusion or elliptic hole in an infinite piezoelectric plate. More recently, Zhou et al [48] presented Green’s function for elliptic hole and small crack problems. Here we follow the results given in [28,48]. 2.8.1 Mapping of an ellipse to a unit circle When the boundary of a linear piezoelectric plate has the shape of an ellipse, it is convenient to transform the ellipse to a unit circle before solving the problem. Let us consider an infinite two-dimensional piezoelectric material containing an elliptic hole, where the material is transversely isotropic and coupling between in-plane stress and in-plane electric fields takes place. The geometric equation of an ellipse G (see Fig. 2.7) can be expressed as x1 = a cos q,

x2 = b sin q, r(q) = (a 2 cos 2 q + b 2 sin 2 q)1/ 2 , T

T

­ dx dx ½ ­ dx dx ½ n = ® 1 , 2 , 0 ¾ , m = ®- 2 , 1 , 0 ¾ , ds = r(q)d q ds ds ¯ ¿ ¯ ds ds ¿

(2.161)

where q is a real parameter and a, b are the half-length of the major and minor axes respectively of the ellipse (see Fig. 2.7), m and n are the unit vectors tangential and normal to the elliptic boundary respectively, and s is the infinitesimal arc length of the ellipse. To transform the ellipse to a unit circle, consider the mapping [49] zk = ck z k + d k z -k 1

(2.162)

52

Green’s function and boundary elements of multifield materials

where ck and d k are constants to be chosen such that the ellipse is mapped into a unit circle. Thus, when x1 and x2 are given by Eq (2.161), z k = eiq . Hence zk = x1 + pk x2 = a cos q + pk b sin q = ck eiq + d k e- iq = ck (cos q + i sin q) + d k (cos q - i sin q)

(2.163)

from which we obtain 1 1 ck = (a - ipk b), d k = (a + ipk b) 2 2

(2.164)

One of the two solutions for z in Eq (2.162) is z k1 =

zk + ( zk2 - 4ck d k )1/ 2 2ck

(2.165)

This solution has the property that z k 1 ® ¥ as zk ® ¥ . The other solution is given by zk 2 =

zk - ( zk2 - 4ck d k )1/ 2 2ck

(2.166)

This solution has the property that z k 2 ® 0 as zk ® ¥ [49].

(q 0 , b) ·z

x2

0

m

n

w

q

a

x1

G0

Wc

b

G

W

Fig. 2.7 Geometry of the elliptic hole problem

The mapping (2.162) is one-to-one mapping for points outside the ellipse in the (x1,x2)-plane to points outside the circle in the z k -plane if the roots of dzk / d z k = ck - d k z k-2 = 0

(2.167)



are located inside the unit circle, i.e. z k < 1. Let the roots be ±z k , where



zk =

dk ck

(2.168)

53

Green’s function of electroelastic problem



The inequality z k = (d k / ck )1/ 2 < 1 can be proved by considering pk = a + ib, where a and b are real. It is evident that z k zk =

d k d k (a - bb) 2 + (ba) 2 = <1 ck ck (a + bb) 2 + (ba) 2

(2.169)



Therefore ±z k are inside the unit circle, which ensures that the mapping is one-to-one mapping. However, the mapping in not conformal [49]. The angle between any two line elements in the (x1,x2)-plane is in general different from the angle between the corresponding elements in the z k -plane. 2.8.2 Green’s functions for a piezoelectric plate with an elliptic hole Consider an infinite piezoelectric plate 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 filled with a homogeneous gas of dielectric constant k0, and is free of mechanical traction. On the hole boundary the normal component of electric displacement and the electric potential are continuous. We therefore have boundary conditions on the contour G: s nn = snm = sn 3 = 0, Dn = -k0

¶fc = Dnc , f = fc ¶m

(2.170)

where n and m respectively represent the normal and tangent to the hole boundary (Fig. 2.7). snn and snm are the normal and shear stresses along the hole boundary, Dn is the normal component of electric displacement vector, superscript c indicates the quantity associated with the hole medium. Similar to the treatment in Section 2.5 for half-plane problems, this problem can also be divided into two subproblems [48]: (i) a homogeneous infinite piezoelectric medium is subjected to the generalized line dislocation b and the traction-charge q 0 both at the point z0(x10,x20); and (ii) the boundary of the elliptic hole is subjected to loadings which make the hole boundary traction free and normal electric displacement and electric potential to be continuous across rim of the hole. For the case in which the generalized line dislocation b and the traction-charge q 0 are applied at a point z0(x10,x20) in a homogeneous infinite piezoelectric medium, the equilibrium conditions of the force and the single-valued conditions of the generalized displacement are still defined by Eq (2.77). In order to satisfy the jump conditions defined by Eq (2.77), the arbitrary function in Eqs (1.128) and (1.129) is chosen to be f ( za ) = ln(z a - z a 0 )

(2.171)

where z a and za are related by Eq (2.162) and z a0 =

za 0 + ( za2 0 - 4ck d k )1/ 2 , za 0 = x10 + pa x20 2ck

(2.172)

Substituting Eq (2.171) into Eqs (1.128) and (1.129), and later into Eq (2.77), we obtain the first basic solution [47] U I = 2 Re{A ln( za - za 0 ) q f },

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

(2.173)

54

Green’s function and boundary elements of multifield materials

where q f is given in Eq (2.80). The Green’s function can be easily obtained if the surface traction and the normal component of the electric displacement vanish on the hole boundary. This means that j = 0 on the hole boundary. The general Green’s function solution in this case can be assumed in the form [50] U = U I + U II =

4 1 1 Im{A ln(z a - z a 0 ) q*} + ¦ Im{A ln(z a-1 - zb 0 ) qb } , (2.174) p b=1 p

j = j I + j II =

4 1 1 Im{B ln(z a - z a 0 ) q*} + ¦ Im{B ln(z a-1 - zb 0 ) qb } (2.175) p b=1 p

in which q* is given in Eq (2.86). Making use of the condition j = 0 and the expression z a = eiq on the hole boundary, we have ­ 4 ½ Im{B ln(eiq - z a 0 )q* } + Im ®¦ ln(e - iq - zb 0 )Bqb }¾ = 0 ¯ b=1 ¿

(2.176)

The first term can be replaced by the negative of its complex conjugate, that is Im{B ln(eiq - z a 0 )q* } = - Im{B ln(e - iq - z a 0 )q* }

(2.177)

­ ½ = - Im ®¦ ln(e -iq - zb 0 )BI bq* ¾ ¯ b=1 ¿ 4

Substituting Eq (2.177) into (2.176) yields qb = B -1BI bq*

(2.178)

If the ellipse is a rigid inclusion with vanishing elastic displacement and electric potential, we have U=0. A similar derivation gives [50] qb = A -1AIbq*

(2.179)

If the hole is assumed to be filled with a homogeneous gas of dielectric constant k0, the boundary condition on the hole boundary is defined by Eq (2.170). In this case, the induced electric potential fc inside the hole should be considered and can be expressed as [48]: fc ( z ) = 2 Re[ f c ( z )], Eic = -f,ci , z = x1 + ix2

(2.180)

The electric potential fc in the hole can be solved from the differential equation f,cii = 0

(2.181)

To solve the problem above, consider the single-valued mapping function

z = c0 z 0 + d 0 z 0-1 , c0 = The roots of dz / d z 0 = 0 are z 01,2 = ±

a+b a -b , d0 = 2 2

(2.182)

1- e , where e=b/a. Thus, mapping of the 1+ e

55

Green’s function of electroelastic problem

region of the hole, denoted by Wc, can be achieved by excluding a straight line G0 along x1 and of length 2a 1 - e 2 from the ellipse (Fig. 2.7). In this case, the mapping function (2.182) will transform G and G0 (Fig. 2.7) into the ring of outer and inner 1- e circles with radii rout = 1 and rin = , respectively. 1+ e Considering the differential equation (2.181) and the mapping function (2.182), c f ( z ) in Eq (2.180), defined in the annular ring mentioned above, can be expressed by Laurent’s expansion as [48]

f c (z 0 ) =

¥

¦a

0k

z 0k

(2.183)

k =-¥

where a0k is a complex constant to be determined. To ensure the function f c is single-valued inside the hole and along the straight line G0, the following condition must be satisfied [2]

f c (rin eiq ) = f c (rin e- iq )

(2.184)

from which the relations between a0 k and a-0 k in Eq (2.183) can be determined as c a = r 2 k a . Therefore f ( z ) can be further written as -0 k

in

0k

¥

f c (z 0 ) = ¦ a0 k (z 0k + rin2 k z 0- k )

(2.185)

k =1

Substituting (2.185) into Eq (2.170)2 and noting that z 0 = eiq on the hole boundary, we have Dnc = G

2k0 rin

¥

¦ {[-k (1 + r ) Im a 2k in

0k

k =1

}

]sin(nq) + [k (1 - rin2 k ) Re a0 k ]cos(nq)

(2.186)

In order to provide analytical solutions outside the ellipse, ¥

f II (z ) = ¦ z a- k ( AT g k + BT h k ), k =1

¥

U II = 2¦ Re ª¬ A z a- k ( AT g k + BT h k ) º¼ ,

(2.187)

k =1

¥

j II = 2¦ Re ¬ªB z a- k ( AT g k + BT h k ) ¼º k =1

are assumed for problem (ii) defined in the second paragraph after Eq (2.170), where g k and h k are real constants to be determined. Noting that z a = eiq on the hole boundary, Eq (2.187) can be further written as U II j II PnII

¥

G

= ¦ [cos(k q)h k - sin( k q)hˆ k ],

G

= ¦ [cos(k q)g k - sin( k q)gˆ k ],

k =1

¥

G

k =1

=-

1 ¥ ¦{k[sin(k q)g k + cos(k q)gˆ k ]} r(q) k =1

(2.188)

56

Green’s function and boundary elements of multifield materials

where hˆ k = Sh k + Hg k , gˆ k = ST g k - Lh k

(2.189)

where S, H, and L are defined in Eq (1.142). In addition, in order to obtain g k and h k from the boundary condition (2.170), the solution U I and j I should also be expressed in terms of sin(k q) and cos(k q) . In doing so, Eq (2.173)2 is rewritten as jI =

ª § d /c 1 ­° Im ®B ln(z a - z a 0 ) + ln «ca ¨1 - a a p ° ¬ © zaza0 ¯

· º * ½° ¸» q ¾ ¹¼ °¿

(2.190)

in which the relation

ª § d / c ·º ln( za - za 0 ) = ln(z a - z a 0 ) + ln «ca ¨ 1 - a a ¸ » ¬ © zaza0 ¹¼

(2.191)

has been used. Along the unit circle boundary, it follows that jI

G

=

½° ª § d /c ·º 1 ­° Im ®B ln(eiq - z a 0 ) + ln «ca ¨1 - a a e - iq ¸ » q* ¾ p ° za0 ¹¼ ¬ © °¿ ¯

(2.192)

Making use of the series representation ¥ 1 ln(1 - x) = -¦ x k , k =1 k

(2.193)

Eq (2.192) can be further written as jI

G

=

k k ­ ½ 1 ° 1 ¥ ª § e i q · § d a e - iq · º * ° » Im ®B ln(-ca z a 0 ) - ¦ «¨ + q ¾ ¸ ¨ ¸ k k =1 «© z a 0 ¹ © ca z a 0 ¹ » p ° °¿ ¬ ¼ ¯

(2.194)

where eik q = cos(k q) + i sin(k q) . Therefore the traction-charge vector Pn on the elliptic boundary G can obtained as k k ­ ¥ ª § d a e - iq · § eiq · º * ½° 1 ° I I Im ®iB ¦ «¨ Pn = (j, s ) = (2.195) ¸ -¨ ¸ » q ¾ G G pr(q) ° k =1 «© ca z a 0 ¹ © z a 0 ¹ ¼» ¬ ¯ ¿° To ensure that the Green’s functions satisfy the traction-free and electric displacement conditions as well as electric potential continuous condition on the interface of the elliptic hole and its outer infinite piezoelectric solid of the original problem, a generalized surface traction t n must be applied to the elliptic boundary in problem (ii), where t n = - PnI - PnII + Dnc i 4 , i 4 = {0, 0, 0,1}T G

G

G

(2.196)

The substitution of Eqs (2.186), (2.188)3, and (2.195) into Eq (2.196) provides [48]

57

Green’s function of electroelastic problem (2) ˆ ˆ (1) ˆ (2) g k = g (1) k + gk , gk = gk + gk ,

ª 1 Im « B kp « ¬ ª 1 Re « B = kp « ¬

k

k

k

k

g (1) k =

§ da · § 1 · ¨ ¸ +¨ ¸ © ca z a 0 ¹ © z a 0 ¹

gˆ (1) k

§ da · § 1 · ¨ ¸ -¨ ¸ © ca z a 0 ¹ © z a 0 ¹

º q* » , » ¼ º q* » , » ¼

k g (2) k = -2 k 0 (1 + ( d 0 / c0 ) Im[ a0 k ]i 4 , k gˆ (2) k = 2 k 0 (1 - ( d 0 / c0 ) Re[ a0 k ]i 4 ,

(2.197)

in which a0k in (2.197)4,5 is still unknown, which can be determined by imposing the condition of electric potential continuity across the elliptic hole boundary. It should be mentioned that g k and gˆ k are divided into two parts: the first part is associated with the traction-free condition, and the second with the conditions of electric potential and electric displacement continuity across the elliptic hole boundary. Making use of Eqs (2.187)3, (2.190), and combining them with Eqs (2.191) and (2.197), we can finally obtain the generalized stress function outside the ellipse as j=

4 1 1 Im{B ln(z a - z a 0 ) q*} + ¦ Im{B ln(z a-1 - zb 0 ) B -1BIbq*} p p b=1 ¥

(2.198)

+ 2k 0 ¦ Im[B z a- k B -1{a0 k - (d 0 / c0 ) k a0 k }i 4 ] k =1

in which following relations have been used: ¥

¦z

-k a

zb-0k / k = - ln(1 - z a-1zb-01 ), if z a-1zb-01 < 1,

k =1

T

T

-1 T

-1

T

(2.199)

-1

2( A + B L S ) = B , 2B L = iB The constants a0k in Eq (2.198) can be determined in the following way. In order to use the condition of electric potential continuity f = fc across the elliptic hole boundary, fc in Eq (2.180) is rewritten in terms of q as f

c

ª ­° § d · k ½° º ­° § d · k ½° 0 « = 2 Re[ f (e )] = 2 Re ¦ a0 k ®1 + ¨ ¸ ¾ cos(k q) + i ®1 - ¨ 0 ¸ ¾ sin(k q) » (2.200) c « ¯° © c0 ¹ ¿° » k =1 ¯° © 0 ¹ ¿° ¬ ¼ c

G

iq

¥

Comparing Eqs (2.173)1 and (2.188)1 with Eq (2.200) and using the single-valued condition yields (h k ) 4 = 2(1 + ( d 0 / c0 ) k ) Re[ a0 k ], (hˆ k ) 4 = 2(1 - ( d 0 / c0 ) k ) Im[ a0 k ],

for k ³ 1. Substituting Eqs (2.197) and (2.201) into Eq (2.189), we have

(2.201)

58

Green’s function and boundary elements of multifield materials

2[(1 + (d 0 / c0 ) k ) + k0 (1 - (d 0 / c0 ) k ) L-441 ]Re[a0 k ] + 2k0 (1 + (d 0 / c0 ) k ) L-41i SiT4 ]Im[a0 k ] = C1k 2[(1 - (d 0 / c0 ) k ) + k 0 (1 + (d 0 / c0 ) k ) L-441 ]Im[a0 k ]

(2.202)

- 2k 0 (1 - (d 0 / c0 ) k ) L-41i SiT4 ]Re[ a0 k ] = C2 k in which C1k and C2 k are real constants given by C1k =

ª 1 + (d a / ca ) k * º ª 1 - (d a / ca ) k * º ½° L-41j ­° T q » + Re «B q »¾ , ® S ji Im «B k k p °¯ za0 z ak 0 ¬ ¼ ¬ ¼ °¿

C2 k =

ª 1 - (d a / ca ) k * º ½° L-41j ­° ª 1 + (d a / ca ) k * º T B q q »¾ Im Re S ® « » «B ji k p ¯° ¬ z ak 0 z ak 0 ¼ ¬ ¼ °¿

(2.203)

Solving Eq (2.202) for a0 k , we have a0 k = a k / bk ,

Ck = C1k + iC2 k

1 a k = [Ck (d 0 / c0 ) k (1 - k 0 L-441 + i k 0 L-41j S Tj 4 ) - Ck (1 + k0 L-441 + i k 0 L-41j S Tj 4 )], 2 bk = {(1 - (d 0 / c0 ) 2 k [1 - ( k 0 L-441 ) 2 - ( k 0 L-41j S Tj 4 ) 2 ] - 2 k 0 (1 + (d 0 / c0 ) 2 k ) L-441 },

(2.204)

where Lij are components of matrix L defined in Eq (1.142). 2.8.3 Green’s function of piezoelectric matrix with an inclusion In Section 2.8.2, Green’s function for a piezoelectric solid with an elliptic hole was presented. We now extend the results to the case of a piezoelectric solid with a piezoelectric inclusion. The discussion follows the results presented in [28]. When the hole considered in Section 2.8.2 becomes an inclusion, the condition (2.170) is replaced by

UM = U I , jM = jI ,

on G

(2.205)

provided that the matrix and the elliptic inclusion are bonded perfectly along the bonded interface, where the superscripts M and I label the quantities relating to the matrix and inclusion, respectively.

59

Green’s function of electroelastic problem x2

h

b

W2

1

za = ca z a + d a z a-1

W1

rin

a

W0

W2

x

x1

G0

W1

G0

G

G

( a ) z a - plane

(b) z-plane

Fig. 2.8 Geometry of elliptic inclusion problem Let a generalized line dislocation b and a line force-charge q 0 be applied at a point z0(x10,x20) in the region of matrix (in W1), as shown in Fig. 2.8. The Green’s functions for such a problem may be expressed in the following form [28] U M = 2 Re[ A M {f0 (ȗ M ) + f1 (ȗ M )}], j M = 2 Re[B M {f0 (ȗ M ) + f1 (ȗ M )}]

(2.206)

for the matrix material, and U I = 2 Re[ A I f 2 (ȗ I )], j I = 2 Re[B I f 2 (ȗ I )]

(2.207)

for the inclusion, where ȗ x = {z1x , z 2x , z 3x , z 4x }T , (x=M,I), and the relation between z ax and zax ( zax = x1 + pax x2 ) are defined by Eq (2.162). f0 represents the singular solution for homogeneous media for the generalized line dislocation b and a line force-charge q 0 , which was obtained in Section 2.5, and can be written as

f0 (ȗ M ) = f 0 (z aM q f =

1 ln(z aM - z aM0 ) q M * 2pi

(2.208)

where q M * = ( A M )T q 0 + (B M )T b . f1 and f2 are the functions corresponding to the perturbed field, due to the existence of the inclusion, of matrix and inclusion, respectively. They are holomorphic in the regions W1 and W 2 , respectively (see Fig. 2.8). In the z-plane, W1 is the region outside the unit circle while W 2 is the region of the annular ring between the unit circle and the circle of radius rin =

a + ibpaI . a - ibpaI

Noting that f2 is holomorphic in the annular ring W 2 mentioned above, it can be represented by Laurent’s series as f 2 (ȗ I ) =

¥

¦ (z ) I a

j

cj

(2.209)

j =-¥

whose coefficients cj can be related by means of the condition (2.184) in the following

60

Green’s function and boundary elements of multifield materials

manner: * j

c- j = G c j ,

* j

G = G

§ a + ibpaI · = ¨ I ¸ © a - ibpa ¹

* ja

j

.

(2.210)

Therefore, Eq (2.209) can be further written as ¥

f 2 (ȗ I ) = ¦ ( z aI ) + G*ja ( z aI ) j

-j

(2.211)

cj

j =1

in which the constant c0, which represents the rigid body motion, is neglected. Substituting Eq (2.211) into Eq (2.207)2 and using the continuouity condition (2.205)2 and z aM = z aI = eiq = s along the interface, yields G

G

¥

B f1 (s) + B M f0 (s) - ¦ {B I ck + B I ī*k c k }s - k M

k =1

¥

(2.212)

= -B f1 (s) - B f0 (s) + ¦ {B c k + B ī c }s M

M

I

I

* k k

k

k =1

One of the important properties of holomorphic functions used in the method of analytic continuation is that if the function f(z) is holomorphic in W1, the region outside the ellipse, (or W0 + W 2 ), then f (1/ z ) is holomorphic in W0 + W 2 (or W1), W0 denoting the region inside the circle of radius rin. According to this property, we may introduce a function which is holomorphic in the entire domain including the interface boundary, i.e. [28] ¥ ­ M M B f ( ȗ ) + B f (1/ ȗ ) [B I c j + B I ī*j c j ]ȗ - j , ȗ Î W1 , ¦ 1 0 ° j =1 ° w(ȗ) = ® ¥ °-B M f (1/ ȗ ) - B M f (ȗ) + [B I c + B I ī* c ]ȗ j , ȗ Î W + W , ¦ 1 0 j j j 0 2 °¯ j =1

(2.213)

where the function w(z) is holomorphic and single valued in the whole plane. By Liouville’s theorem, we have w(z)=constant. However, constant function f corresponds to rigid body motion, which may be neglected. Thus, by letting w(z)=0, we have ¥

¦ [B c I

j

+ B I ī*j c j ]ȗ - j = B M f1 (ȗ) + B M f0 (1/ ȗ ),

ȗ Î W1 ,

j =1

(2.214)

¥

¦ [B I c j + B I ī*j c j ]ȗ j = B M f1 (1/ ȗ ) + B M f0 (ȗ),

ȗ Î W0 + W 2

j =1

Similarly, the continuity condition (2.205)1 gives ¥

¦ [A c I

j

+ A I ī*j c j ]ȗ - j = A M f1 (ȗ) + A M f0 (1/ ȗ ),

ȗ Î W1 ,

j =1

(2.215)

¥

¦ [A I c j + A I ī*j c j ]ȗ j = A M f1 (1/ ȗ ) + A M f0 (ȗ), j =1

The equations (2.214)2 and (2.215)2 provide

ȗ Î W0 + W 2

61

Green’s function of electroelastic problem ¥

f0 (ȗ) = ¦ i ( A M )T {(M M + M I ) A I c k + (M M - M I ) A I ī*k ck }ȗ k

(2.216)

k =1

where

M x = -iB x ( A x ) -1 ,

(x=I,M)

(2.217)

f 0( k ) (0) , k!

(2.218)

With the use of series representation ¥

f 0 ( x) = ¦ ek x k , ek = k =1

of f0 from Eq (2.208), and comparing the coefficient ck of the corresponding terms in Eq (2.216), we have

c k = (G 0 - G k G 0-1G k ) -1 (t k - G k G 0-1 tk ), k = 1, 2," , ¥,

(2.219)

where G 0 = (M M + M I ) A I , G k = (M M - M I ) A I ī*k , tk =

1 1§ 1 · ( A M ) -T ¨ ¸ k © za ¹ 2p

k

qM *

(2.220)

With the solution obtained for ck, f1 can be obtained from Eqs (2.214)1 and (2.215)1. The functions f0, f1 and f2 can then be further written as 1 ln(z aM - z aM0 q M * , f0 (ȗ M ) = 2pi ¥

f1 (ȗ M ) = ¦ (B M ) -1 (z aM ) - k (B I ck + B I ī*k c k ) - (B M ) -1 B M f0 (1/ ȗ M ),

(2.221)

k =1 ¥

f 2 (ȗ I ) = ¦ { (z aI ) k + (z aI ) - k ī*k }c k k =1

The Green’s functions Ux and jx (x=I,M) can thus be obtained by substituting the results (2.221) into Eqs (2.206) and (2.207). 2.9 Arbitrarily shaped hole In this section, we present Green’s functions for holes of various shapes embedded in an infinite plane piezoelectric plate subjected to a line force-charge q 0 and generalized line dislocations b (Fig. 2.9). The Green’s functions satisfying traction free and exact electric boundary conditions along the hole boundary are derived using Lekhnitskii’s formalism and the technique of one-to-one mapping. A detailed discussion of the critical points for the mapping function of a piezoelectric plate with polygonal holes is also presented. Study shows that the transformation function (2.223) below is nonsingle-valued. A simple approach is presented to treat such a situation. This section details the development given in [17].

62

Green’s function and boundary elements of multifield materials n

m

(q 0 , b) ·z

0

x2

q

W Wc

x1 G

Fig. 2.9 Geometry of a particular hole (a=1, e=1, m=4, g=0.1) 2.9.1 One-to-one mapping Consider an infinite two-dimensional piezoelectric material containing a hole of various contours, where the material is transversely isotropic and coupling between in-plane stress and in-plane electric fields takes place. The constitutive relation used here is defined by Eq (1.144). The contour of the hole considered, say G (see Fig. 2.9), in this section is represented by x1 = a (cos q + gem1 cos mq), x2 = ae(sin q - gem1 sin mq)

(2.222)

where q is a real parameter (see Fig. 2.9), eij = 1 if i ¹ j; eij = 0 if i = j , 0<e£1, m is an integer and the same value exists for subscripts and for the argument of functions. By appropriate selection of the parameters e, m and g, we can obtain various special kinds of hole, such as ellipse (m=1), circle (m=e=1), triangle (m=2), square (m=3) and pentagon (m=4). Since complex mapping is a fundamental tool used to find the solution of hole problems, the transformation [2] zk = a(a1k z k + a2 k z -k 1 + em1a3k z mk + em1a4 k z -k m ) ,

(2.223)

in which a1k = (1 - ipk e) / 2,

a2 k = (1 + ipk e) / 2,

a3k = g (1 + ipk e) / 2,

a4 k = g (1 - ipk e) / 2

(2.224)

is used to map the contour of the hole onto a unit circle in the z-plane. For a particular value of zk, there exist 2m roots for zk in Eq (2.223). Numerical study of Eq (2.223) shows that half of the roots are located outside the unit circle and the remaining roots are inside it. Thus the transformation (2.223) will be single-valued for m=1 (ellipse) since only one root locates outside the unit circle. For m>1, however, the transformation (2.223) is multi-valued as there are m roots located outside the unit circle. The question is, which root should be chosen. To determine this, some numerical investigation has been performed and typical results are listed in Tables 2.1-2.3 for a=e=1 and g=0.1. In these tables, pk= -0.2291853 +1.003833i, which is 2 one of the eigenvalues of material BaTiO3 used in [2], z = zz , and zki is the ith

63

Green’s function of electroelastic problem

root of zk for Eq (2.223). It is found from the tables that the magnitudes among the m-roots are obviously different from each other. Numerical study in [2] showed that the root, say z*k , whose magnitude has the minimum value among the m-roots, can provide acceptable results. So we choose z*k as the solution for Eq (2.223) in our analysis. Hence, the entire zk-plane is now mapped onto part of the zk-plane with a one-to-one transformation. Moreover, it is interesting to see from Tables 2.1-2.3 that the real and imaginary parts of z*k always retain the same symbol as those of zk, i.e., sign[Im(zk)]= sign[Im( z*k )] and sign[Re(zk)]= sign[Re( z*k )], where sign(x) is defined by ­ 1 if x >0 ° (2.225) Sign( x) = ® 0 if x =0 °-1 if x <0 ¯ which means that z*k and zk are always situated in the same quadrant of a rectangular coordinate system. Next, to find a single-valued mapping of the region Wc (see Fig. 2.9), occupied by vacuum(or air), let (2.226) z = a (a1c z + a2c z -1 + em1a3c z m + em1a4 c z - m ) where z = x1 + ix2 , and a1c = (1 + e) / 2, a2 c = (1 - e) / 2, (2.227) a3c = g (1 - e) / 2, a4c = g (1 + e) / 2 Table 2.1: The properties of solution x kj (j=1-m) for m=2 zk

5+5pk

-5+5pk

-5-5pk

5-5pk

xk1

7.51-91.75i 92.06

16.35-92.73i 94.16

16.30-82.84i 84.43

5.156-81.59i 81.74

3.97+4.52i 6.016

-4.885+5.4757i 7.338

-4.85-4.41i 6.555

6.294-5.65i 8.458

x k1

xk2 xk 2

Table 2.2: The properties of solution x kj (j=1-m) for m=3 zk

5+5pk

-5+5pk

-5-5pk

5-5pk

xk1

6.02-8.63i 10.52

8.96-8.07i 12.06

9.39-5.05i 10.66

-8.96+8.07i 12.06

-9.39+5.05i 10.66

-7.73+0.784i 7.770

-6.02+8.63i 10.52

7.73-0.784i 7.770

3.38+3.60i 4.938

-1.221+7.274i 7.376

-3.38-3.60i 4.938

x k1

xk2 xk2

xk3 xk3

1.221-7.274i 7.376

64

Green’s function and boundary elements of multifield materials

Table 2.3: The properties of solution x kj (j=1-m) for m=4 zk

5+5pk

-5+5pk

-5-5pk

5-5pk

xk1

-4.69-3.22i 5.689 -2.02+4.43i 4.869 3.71-3.61i 5.177 3.01+2.41i 3.856

4.58-2.98i 5.464 1.94+4.68i 5.066 -3.52-4.15i 5.442 -3.49+2.45i 4.264

5.23-1.53i 5.449 -4.43-0.16i 4.433 0.436+5.58i 5.597 -1.255-3.90i 4.097

-5.31-1.88i 5.633 4.78+0.30i 4.789 -0.94+5.75i 5.826 1.46-4.15i 4.399

x k1

xk2 xk2

xk3 xk3

xk4 xk 4

The roots of dz / dz = 0 represent the critical points for transformation (2.226) and can be obtained by solving a1c - a2 c z -2 + mem1a3c z m -1 - mem1a4 c z - m -1 = 0

(2.228)

The analytical solution of Eq.(2.228) is, in general, not easy to find for m>1. For simplicity, we consider following two cases only: (i) m=1(ellipse). Solving Eq.(2.228), we have z 1, 2 = ±

1- e 1+ e

(2.229)

Thus, the mapping of the region Wc can be performed by excluding a straight line G0 along x1 and of length 2 a 1 - e2 from the ellipse. In this case the function z = a (a1cz + a2 cz -1 ) (2.230) will transform G and G0 (see Fig. 2.8) into the ring of outer and inner circles with radii 1- e rout = 1 and rin = , respectively. 1+ e (ii) m>1 and e=1. In this case the mapping function (2.226) becomes z = a(a1c z + a4 c z - m )

(2.231)

The roots of equation (2.228) can then be easily found as z 0 = m +1 mg

(2.232)

In practice, g is a small number. If we assume mg<1, the function (2.231) will transform the lines G and G0 (a straight line along x1-axis and of length 1

a (mg ) m +1 (1 + m -1 ) ) into an annular circle with inner and outer radii rin = m +1 mg and rout=1, respectively.

2.9.2 Green’s function for electroelastic field Consider an infinite piezoelectric plate containing a hole subjected to a line force-charge q0 and a generalized line dislocation b, both located at z0(x10,x20). The hole surface is assumed to be free of traction, but the normal component of electric

65

Green’s function of electroelastic problem

displacement and the electric potential are continuous along the hole boundary. We therefore have the following boundary conditions: along the hole boundary: ¶j c s nn = 0, s nm = 0, Dn = - e 0 , j = jc (2.233) ¶n at infinity:

s ij ® 0, Di ® 0

(2.234)

where n and m, respectively, stand for the normal and tangent to the hole boundary (Fig. 2.9). snn and snm are the normal and shear stresses along the boundary, e0 is the dielectric constant of vacuum, superscript c indicates the quantity associated with the hole medium. It can be shown that the electroelastic boundary conditions (2.233) can be expressed in terms of complex potentials in the form [2]: 3

3

k =1

k =1

2 Re ¦ F k = 0, 2 Re ¦ { pk F k } = 0, 3 ¶jc 2 Re ¦ {v k F k } = -k0 ³ ds, 2 Re ¦ {tk*F k } = jc , 0 ¶n k =1 k =1 3

(2.235)

s

where s is arc length of G, and v k and tk* are defined by Eqs (1.157) and (1.162). Noting that [10] ¶j c ( x1 + ix 2 ) ¶j c ( x1 + ix 2 ) = -i ¶n ¶m

(2.236)

Eq (2.235)3 can be further simplified as 3

2 Re ¦ {v k F k } = 2 Re{i k0 F ( z )}

(2.237)

jc(z)= F ( z ) + F ( z )

(2.238)

k =1

where

Conditions (2.234) and (2.235) suggest that the complex potentials Fk be chosen as F k ( z k ) = g k ln(z k - z k 0 ) + F k 0 (z k )

(2.239)

where ¥

F k 0 (z k ) = a k 0 + ¦ a ki z -k i

in W

(2.240)

i =1

while the function F(z) may be assumed as ¥

F (z ) = b0 + ¦ bk z - k

(2.241)

k =1

for the case of a crack, and ¥

F (z ) =

¦b z k

k =-¥

k

(2.242)

66

Green’s function and boundary elements of multifield materials

for the other holes in which the coefficients have the following relation [11]: b- k = rin2 k bk

(2.243)

Here gk are complex constants which can be determined from following conditions:

³

A*d F = b and

C

³

C

B*d F = q 0 for any closed curve C enclosing the point z0 (2.244)

where z 0 = {z 10 z 20 z 30 }T , F = {F 1 F 2 F 3 }T ª p1* p2* p3* º « » A* = « q1* q2* q3* » , « t1* t2* t3* » ¬ ¼

ª - p1 - p2 - p3 º B* = «« 1 1 1 »» «¬ -v1 -v 2 -v 3 »¼

(2.245)

(2.246)

Substituting Eq. (2.239) into Eq. (2.244), leads to g = {g1 g 2 g3 }3 =

1 ( A*-1A* - B*-1B* ) -1 ( A*-1b - B*-1q 0 ) 2pi

(2.247)

Using the solution (2.247) and expression (2.239), the boundary conditions (2.235) can be further written as ¥

3

¦ [F

k0

k =1

k

k =1

k

k0

(s) + pk F k 0 (s) + ¦ ( pk g k ekm sm + pk g k ekm s- m )] = 0

(2.249)

m =1

¥

3

¦ [v F

(2.248)

m =1 ¥

3

¦[ p F

( s) +F k 0 ( s) + ¦ ( g k e km s m + g k e km s - m )] = 0

k0

k =1

(s) + v k F k 0 (s) + ¦ (v k g k ekm s m + v k g k ekm s- m )] = ik 0 [ F (s) - F (s)] (2.250) m =1

¥

3

¦ [t F * k

k0

k =1

(s) + tk* F k 0 (s) + ¦ (tk* g k ekm sm + tk* g k ekm s - m )] = F (s) + F (s)

(2.251)

m =1

where s = e iq denotes the point located on the unit circle. During the derivation of Eqs. (2.248)-(2.251), the series representation for function ln(z k - z k 0 ) has been employed. In fact, for a function f(x), the series representation may be written in the form: ¥

f ( x) = ¦ ek x k , ek = k =1

f ( k ) (0) 1 f ( x) = dx ³ C 2pi k! x k +1

(2.252)

Thus, the function ln(z k - z k 0 ) can be expressed as ¥

ln( z k - z k 0 ) = ¦ e km z m ,

e km = -

m =1

z k-0m m

(2.253)

The substitution of Eqs. (2.240)~(2.242) into Eqs. (2.248)~(2.251), yields 3

¦ (a

ki

k =1

where

3

+ g k eki ) = 0,

¦(p a k

k =1

3

ki

+ pk g k eki ) = 0,

¦ (c k =1

a + tki aki ) = li

ki ki

(2.254)

67

Green’s function of electroelastic problem 3

tki = v k + i k0tk* , c ki = 0, li = ¦ (v k - i k0tk* )g k eki

(2.255)

k =1

3

bi = ¦ (tk* g k eki - tk*aki )

(2.256)

k =1

for cracks, and tki = v k -

ik0 2i k0 rin2i tk* (1 + rin4i ) tk* , c ki = , 4i 1 - rin 1 - rin4i

ik0 li = ¦ {v k g k eki + [2rin2i tk* g k eki - (1 + rin4i )tk* g k eki ]}, 1 - rin4i k =1 3

bi =

(2.257)

3

1 1 - rin4i

¦ [t g e * k

k ki

- tk*aki - rin2i ( tk* g k eki - tk* aki )]

(2.258)

k =1

for the other holes. Solving Eqs. (2.254), one obtains a i = E i g + Fi g

(2.259)

a i = {a1i a 2 i a 3i }T

(2.260)

E i = ( Pi - Q i Pi-1 Q i ) -1 ( E *i - Q i Pi-1 Fi* )

(2.261)

Fi = ( Pi - Q i Pi-1 Q i ) -1 ( Fi* - Q i Pi-1 E i* )

(2.262)

where

ª1 1 1º Pi = «« p1 p2 p3 »» , ¬«c1i c 2i c3i »¼ ª0 0 0º E = «« 0 0 0 »» , «¬b1*i b2*i b3*i »¼ * i

ª0 0 0º Qi = «« 0 0 0 »» ¬« t1i t2i t3i »¼

(2.263)

ª -e1i -e2i -e3i º F = «« -e1i p1 -e2i p2 -e3i p3 »» «¬ b1**i b2**i b3**i »¼

(2.264)

* i

in which bki* = (v k - i k0tk* )eki , bki** = 0

(2.265)

for cracks, and bki* = [v k -

i k0 (1 + rin4i ) * 2i k0 rin2i * ** t e b tk eki ] , = k ki ki 1 - rin4i 1 - rin4i

(2.266)

for other holes. Using the expression (2.259), the potential solution, F, can further be written as ¥

F = ln(z a - z a 0 ) g + ¦ z a- i ( E i g + Fi g )

(2.267)

i =1

where

f a = diag[ f 1 f 2 f 3 ] represents a diagonal matrix. Therefore, the Green’s

function of U can be written as

68

Green’s function and boundary elements of multifield materials ¥

U = {u1 , u2 , f}T = 2 Re[ A*{ ln(z a - z a 0 ) g + ¦ z a- i (Ei g + Fi g )}]

(2.268)

i =1

2.10 Semi-infinite cracks Consider an infinite piezoelectric solid with a semi-infinite crack Lc along the negative direction of the x1-axis, as shown in Fig. 2.10. The infinite piezoelectric solid is loaded by a line force-charge q0 and a generalized line dislocation b, both located at z0(x10,x20)[51]. The crack faces are assumed to be free of mechanical forces and external electric charge. Therefore, along the crack faces, we have

jL =0

(2.269)

c

We introduce the mapping function [51] za = z a2

(2.270)

which maps the crack faces q = ±p in the z-plane into the imaginary axis in the z-plane (see Fig. 2.10). Based on the mapping function (2.270), the corresponding Green’s function of U and j may be assumed in the following form [2,50] 4 1 1 U = Im{A ln(z a - z a 0 ) q*} + ¦ Im{A ln(-z a - zb 0 ) qb } , (2.271) p b=1 p 4 1 1 j = Im{B ln(z a - z a 0 ) q*} + ¦ Im{B ln(-z a - zb 0 ) qb } (2.272) p p b=1 where q* is defined by Eq (2.86).

(q 0 , b) ·z

h

za = z a2

0

x2

Lc

Lc o

(a ) z a - plane

x1

(q 0 , b) · z0

o

x

(b) z-plane

Fig. 2.10 Geometry of the semi-infinite crack system

69

Green’s function of electroelastic problem

The substitution of Eq (2.272) into Eq (2.269) leads to 4

Im{B ln(z c - z a0 ) q*} + ¦ Im{ln(-z c - zb0 )Bqb } = 0

(2.273)

b=1

where z c is the value of z a along the imaginary axis [51]. Noting that zc = -z c holds along the imaginary axis, the first term in Eq (2.273) can be replaced by the negative of its complex conjugate: Im{B ln(z c - z a 0 )q* } = - Im{B ln(-z c - z a 0 )q* } ­ 4 ½ = - Im ®¦ ln(-z c - zb 0 )BIbq* ¾ ¯ b=1 ¿

(2.274)

Equation (2.273) then yields qb = B -1BI bq*

(2.275)

which is in the same form as that in Eq (2.178). Therefore the Green’s function can be written in terms of za as U=

4 1 1 Im{A ln( za - za 0 ) q*} + ¦ Im{A ln(- za - za 0 ) B -1BI bq*} , (2.276) p b=1 p

j=

4 1 1 Im{B ln( za - za 0 ) q*} + ¦ Im{B ln(- za - za 0 ) B -1BI bq*} (2.277) p p b=1

2.11 Anti-plane problems in a cracked solid In this section, Green’s function for anti-plane problems in an piezoelectric solid with a crack is presented based on the complex variable method and the perturbation technique. The discussion is based on the results given in [52].

2.11.1 Basic equations of anti-plane problem In the case of anti-plane shear deformation involving only out-of-plane displacement uz and in-plane electric fields, whose variables are independent of the longitudinal coordinate z, u x = u y = 0, u z = u z ( x, y ), f = f( x, y )

(2.278)

The differential governing equations (1.59)1,2 are thus reduced to c44Ñ 2u z + e15Ñ 2 f = 0,

with the constitutive equations ­s xz ½ ªc 44 °s ° « ° yz ° « 0 ® ¾= ° D x ° « e15 °¯ D y °¿ «¬ 0 or

e15Ñ 2u z - k11Ñ 2 f = 0

0

- e15

c 44

0

0 e15

k11

0

0 º ­ g xz ½ ° ° - e15 »» °g yz ° ® ¾ 0 »° Ex ° »° ° k11 ¼ ¯ E y ¿

in W

(2.279)

(2.280)

70

Green’s function and boundary elements of multifield materials

­ g xz ½ ª f 44 °g ° « ° yz ° « 0 ® ¾= ° E x ° «- g15 °¯ E y °¿ «¬ 0

0 f 44

g 15

0

b11

- g 15

0

0 º ­s xz ½ ° ° g15 »» °s yz ° ® ¾ 0 » ° Dx ° »° ° b11 ¼ ¯ D y ¿

0

(2.281)

where Ñ 2 is the 2D Laplace’s operator defined by Eq (1.29), g xz , g yz and E x , E y are, respectively, shear strains and electric fields given by g xz =

¶u z , ¶x

g yz =

¶u z ¶f , Ex = - , ¶x ¶y

Ey = -

¶f ¶y

(2.282)

The constants f44, g15 and b11 are defined by the relations: f 44 =

k11 , D

g15 =

e15 , D

b11 =

c 44 , D

D = c 44 k11 + e152

(2.283)

It is obvious from Eq (2.279) that it requires c 44 k11 + e152 ¹ 0

(2.284)

to have a non-trivial solution for the out-of-plane displacement and in-plane electric fields. With the relation (2.284), Eq (2.279) results in Ñ 2u z = 0 ,

Ñ 2f = 0

(2.285)

If we let the harmonic functions uz and f be the imaginary parts of some complex potentials of the variable z=x1+ix2: u z = Im[u ( z )],

f = Im[y ( z )],

(2.286)

then the generalized strain and stress fields can be expressed as g 23 + i g13 = W ( z ), E2 + iE1 = -F ( z ), t23 + it13 = c44W ( z ) + e15F ( z ), D2 + iD1 = e15W ( z ) - k11F ( z )

(2.287)

where W ( z ) = u ¢( z ), F ( z ) = f¢( z ), and the prime denotes the derivative with respect to the argument z. The mechanical boundary conditions on the crack faces are t23 ( x) + = t23 ( x) - = 0,

x
u z ( x) - = u z ( x) - , x ³ a

(2.288) (2.289)

where the superscripts ‘+’ and ‘-’ refer, respectively, to the upper and lower crack faces. The electric boundary conditions over the crack surface are given by [52] E2 ( x) + = E2 ( x) - = Cr E0 ( x),

x < a,

f( x) + = f( x) - ,

x ³ a,

(2.290)

where E0 ( x) is the boundary value of Ey(z) on the crack faces due to external

71

Green’s function of electroelastic problem

loading without the disturbance of the crack. Cr is referred to as the electric crack condition parameter [52].

y

(q 0 , b ) ·

z0

o x

2a

Fig. 2.11 Anti-plane solid with a finite crack

2.11.2 Green’s function of the anti-plane problem To obtain the Green’s function of the anti-plane problem described above, assume that a screw dislocation bz, a line force f, a line charge q, and an electric potential jump bf are located at a point z0(x0,y0) in an infinite piezoelectric plate (see Fig. 2.11). Using the concept of perturbation technique, the complex potential W and F can be assumed in the form [52] W ( z ) = W0 ( z ) + W1 ( z ),

F( z ) = F 0 ( z ) + F1 ( z ),

(2.291)

where W1(z) and F1(z) are two analytical functions corresponding to the perturbed field due to the existence of the crack. W0(z) and F0(z) are associated with the unperturbed fields without the crack, and can be written as W0 ( z ) =

( A1 + iA2 ) , z - z0

F0 ( z) =

( B1 + iB2 ) , z - z0

(2.292)

where A1, A2, B1, and B2 are constants which can be determined from the following conditions:

³

C

t3 j n j ds = - f ,

³

¶u z ds = bz , ¶s

³

³

C

C

C

D j n j ds = q,

(2.293)

¶f ds = bf ¶s

(2.294)

Substituting Eq (2.286) into Eqs (2.293) and (2.294) yields A1 =

b e15 q - k11 f c44 q + e15 f bz , A2 = , B1 = f , B2 = , 2 2p 2p(c44 k11 + e15 ) 2p 2p(c44 k11 + e152 )

(2.295)

Making use of Eq (2.287)3, the condition (2.288) yields c44 [W ( x) + + W ( x) - ] + e15 [F ( x) + + F ( x) - ] = 0,

x
(2.296)

72

Green’s function and boundary elements of multifield materials

and c44 [W ( x) - + W ( x) + ] + e15 [F ( x) - + F ( x) + ] = 0,

x
(2.297)

The subtraction and addition of Eqs (2.296) and (2.297), respectively, provides c44 {[W ( x) - W ( x)]+ - [W ( x) - W ( x)]- }

+ e15{[F ( x) - F ( x)]+ - [F( x) - F( x)]- } = 0,

x
(2.298)

and c44 {[W ( x) + W ( x)]+ + [W ( x) + W ( x)]- }

+ e15{[F( x) + F( x)]+ + [F( x) + F ( x)]- } = 0,

x
(2.299)

Substituting Eq (2.291) into Eqs (2.298) and (2.299), we have c44 {[W1 ( x) - W1 ( x)]+ - [W1 ( x) - W1 ( x)]- }

+ e15{[F1 ( x) - F1 ( x)]+ - [F1 ( x) - F1 ( x)]- } = 0,

x
c44 {[W1 ( x) + W1 ( x)]+ + [W1 ( x) + W1 ( x)]- }

+ e15{[F1 ( x) + F1 ( x)]+ + [F1 ( x) + F1 ( x)]- } = -2 F0 ( x),

x
(2.300)

(2.301)

where § A + iA2 A1 - iA2 · § B1 + iB2 B1 - iB2 · F0 ( x) = c44 ¨ 1 + + ¸ + e15 ¨ ¸ x - z0 ¹ x - z0 ¹ © x - z0 © x - z0

(2.302)

The solution for Eqs (2.300) and (2.301) can be written as [44] c44W1 ( z ) + e15F1 ( z ) = -c44 Fw ( z ) - e15 Ff ( z ),

(2.303)

in which Fw ( z ) =

Ff ( z ) =

2 2 · A1 + iA2 A - iA2 A + iA2 § z0 - a ¨ + 1 - 1 + 1¸ ¸ 2( z - z0 ) 2( z - z0 ) 2 z 2 - a 2 ¨ z - z0 © ¹ 2 2 · A - iA2 § z0 - a ¨ - 1 + 1¸ , ¸ 2 z 2 - a 2 ¨© z - z0 ¹

(2.304)

2 2 · B1 + iB2 B - iB2 B + iB2 § z0 - a ¨ + 1 - 1 + 1¸ 2 2 ¸ 2( z - z0 ) 2( z - z0 ) 2 z - a ¨ z - z0 © ¹ 2 2 · B - iB2 § z0 - a ¨ - 1 + 1¸ ¸ 2 z 2 - a 2 ¨© z - z0 ¹

(2.305)

In a similar way, the condition (2.290) provides F1 ( z ) = (Cr - 1) Ff ( z ) Substituting Eq (2.306) into Eq (2.303), we have

(2.306)

73

Green’s function of electroelastic problem

W1 ( z ) = - Fw ( z ) -

Cr e15 Ff ( z ) c44

(2.307)

Chen et al [52] indicated that the solutions (2.306) and (2.307) correctly recover both the impermeable and permeable crack solutions as limiting cases. The impermeable crack solution is obtained by letting Cr=0, while the permeable crack solution is achieved by setting Cr=1. The Green’s function of uz and f are obtained by integrating Eqs (2.306) and (2.307) with respect to z, i.e. u z ( z ) = Im[( A1 + iA2 ) ln( z - z0 ) - ³ {Fw ( z ) +

Cr e15 Ff ( z )}dz ] c44

f( z ) = Im[( B1 + iB2 ) ln( z - z0 ) + (Cr - 1) ³ Ff ( z )dz ]

(2.308) (2.309)

in which the integral constants representing rigid body motion have been ignored. 2.12 Dynamic Green’s function 2.12.1 Dynamic anti-plane problem [53] Green’s function of the dynamic anti-plane problem for a piezoelectric plate subjected to a time harmonic line force feiwt and a line charge qeiwt at the origin of the plate, with w being the circular frequency, is presented in this section. Since the anti-plane problem contains only the out-of-plane displacement uz and electric potential f, which are functions of the plane coordinates x1 and x2, the governing differential equations can be written as c44Ñ 2u z + e15Ñ 2 f + rw2u z = - f d( x1 )d( x2 ), e15Ñ 2u z - k11Ñ 2 f = qd( x1 )d( x2 )

(2.310)

where r is the mass density of the piezoelectric plate. When c44 k11 + e152 ¹ 0, Eq (2.310) may be further written as Ñ 2u z +

e q - k11 f rw2 k11 u z = 15 d( x1 )d( x2 ), 2 c44 k11 + e15 c44 k11 + e152

rw2 e15 -e f - c44 q u z = 15 Ñ 2f + d( x1 )d( x2 ) 2 c44 k11 + e15 c44 k11 + e152

(2.311)

Making use of the new function [53] m = f-

e15 uz k11

(2.312)

Eq (2.311) can be converted to the following equivalent system: Ñ 2u z + k 2 u z =

e15 q - k11 f d( x1 )d( x2 ), c44 k11 + e152

q Ñ 2m = d( x1 )d( x2 ) k11 where k is the wave number given by

(2.313)

74

Green’s function and boundary elements of multifield materials

k =w

rk11 c44 k11 + e152

(2.314)

It is evident from Eq (2.313) that the original problem is equivalent to the determination of the Green’s functions for the 2D Helmholtz equation as well as for the 2D Laplace equation, which are well-known in the literature [53] as u z = -i

e15 q - k11 f q ln r , H 0(1) (kr ), m = 2 4(c44 k11 + e15 ) 2pk11

(2.315)

where H 0(1) is the zeroth order Hankel function of the first kind. From Eq (2.312) one obtains the explicit expression of f as f=-

e (e q - k11 f ) (1) q ln r - i 15 15 H 0 (kr ) 2pk11 4k11 (c44 k11 + e152 )

(2.316)

2.12.2 3D Dynamic Green’s function In this section, the Radon transform is used to derive 3D dynamic Green’s functions. The discussion follows the results presented in [54]. Consider a homogeneous piezoelectric solid whose motion in the Euclidean space is described in terms of the independent variables x = {xi} and t. The piezoelectric solid is subjected to an impulsive mechanical point force and an impulsive electric point charge at the origin x=0. The Green’s function of this problem is governed by the following differential equations: EiJKl GKM ,il (x, t ) - rd*JK  GKM (x, t ) = -d JM d(x)d(t ),

(2.317)

and the initial conditions GKM (x, t ) = 0,

for t < 0,

(2.318)

where the dots over a variable denote temporal derivatives, GKM denotes Green’s function of displacement in the xk-direction (K=1,2,3) or electric potential (K=4) at the field point x due to an impulsive point force in the xM-direction (M=1,2,3) or an impulsive electric charge (M=4) at the source point x=0, and d*JK is the generalized Kronecker delta defined by ­d , d*JK = ® JK ¯ 0,

J , K = 1, 2,3, otherwise

(2.319)

The Green’s function GIJ (x, t ) can be obtained by applying the Radon transform defined by (2.6) to Eqs (2.317) and (2.318) [7] K JK (n)

¶ 2Gˆ KM (w, t ) ¶ 2Gˆ KM (w, t ) = -d JM d(w)d(t ), - rd*JK 2 ¶w ¶t 2 Gˆ KM (w, t ) = 0,

for t < 0,

(2.320) (2.321)

where w is the parameter of the radon transform which is defined by w = n × x with n being a unit vector, and K IJ (n) is the generalized Christoffel tensor used in Eq (2.5)

75

Green’s function of electroelastic problem

and is rewritten here for convenience K JK (n) = EiJKl ni nl

(2.322)

We consider, as the first step, the Green’s function due to a mechanical point load. Eq (2.320) can be, in this case, rewritten as K jk

¶ 2Gˆ km (w, t ) ¶ 2Gˆ 4 m (w, t ) ¶ 2Gˆ km (w, t ) + = -d jm d(w)d(t ), K rd 4 j jk ¶w2 ¶w2 ¶t 2

(2.323)

¶ 2Gˆ km (w, t ) ¶ 2Gˆ 4 m (w, t ) + =0 K 44 ¶w2 ¶w2

(2.324)

K4k

Solving Eq (2.324) for Gˆ 4 m,ww , and then substituting it back to Eq (2.323), we have L jk

¶ 2Gˆ km (w, t ) ¶ 2Gˆ km (w, t ) = -d jm d(w)d(t ), - rd jk 2 ¶w ¶t 2

(2.325)

where L jk = K jk - K j 4 K 4 k / K 44 . It is obvious that the solution of Eq (2.325) depends on the eigenvalues of the matrix Ljk. The eigenvalues, denoted by l q , are the roots of the characteristic equation [54] det( L jk - ld jk ) = 0

(2.326)

If l q are all distinct, Ljk can be expressed in terms of l q as 3

L jk = ¦ l q Pjkq ,

(2.327)

q =1

where Pjkq are the projection operators of the matrix Ljk with the following properties [55] 3

Pjip Pikq = d pq Pjkq ,

¦P

q ji

= d ji

(2.328)

q =1

They are given by [54] Pjkq =

Aqjk Aiiq

, Aqjk = adj( L jk - l q d jk )

(2.329)

where ‘adj’ represents the adjoint matrix of a square matrix. If l q is not a simple root, the projectors can be determined indirectly by using Eq (2.328). For example, when l1 ¹ l 2 = l3 , Pjk1 is given by Eq (2.329), while Pjk2 = 1 - Pjk1 . By multiplying both sides of Eq (2.325) by Pijq and using the properties of the projection operators (2.327) and (2.328), we have § ¶2 ¶2 · q ˆ q ¨ l q 2 - r 2 ¸ Pij G jm (w, t ) = - Pim d(w)d(t ), ¶w ¶ t © ¹

(2.330)

It should be mentioned that in Eq (2.330) and in the sequel the summation convention does not apply to the suffix q whenever l q appears. For each fixed q, i and m, Eq (2.330) represents a one-dimensional wave equation which, together with Eq (2.321),

76

Green’s function and boundary elements of multifield materials

can be solved by using the well-known d’Alembert solution. The result is Pijq Gˆ jm (w, t ) = -

H (t ) Pimq [ H (w + cq t ) - H (w - cq t )] 2rcq

(2.331)

where H(t) is defined in Eq (1.46) and cq = l q / r . Using Eq (2.328), we obtain [54] Q Pq Gˆ jm (w, t ) = H (t )¦ im [ H (w + cq t ) - H (w - cq t )] q =1 2rcq

(2.332)

where Q is the number of distinct roots of Eq (2.326). Making use of Eq (A7) in Appendix A, the inversion of Eq (2.332) followed by some manipulation yields Gim (x, t ) = -

Q Pimq  H (t ) [d(cq t + n × x) + d(cq t - n × x)]dn 2 ³ ¦ 16p n =1 q =1 l q

(2.333)

Since the two d-function terms in the integration above yield the same contribution to Gim, Eq (2.243) can be further written as Gim (x, t ) = -

Q Pimq  H (t ) d(cq t + n × x)dn ¦ 2 ³ 8p n =1 q =1 l q

(2.334)

Substituting Eq (2.332) into Eq (2.334), we have Q K Pq Gˆ 4 m (w, t ) = - H (t )¦ 4 k km [ H (w + cq t ) - H (w - cq t )] q =1 2r cq K 44

(2.335)

Applying the inverse Radon transform to Eq (2.335) we obtain G4 m (x, t ) =

Q K 4i Pimq  H (t ) d(cq t + n × x)dn ¦ 8p2 n³=1 q =1 K 44l q

(2.336)

Eq (2.334) and (2.336) represent the Green’s functions due to a mechanical point force. Following essentially the same procedure above, the corresponding Green’s functions due to an electric point charge can be obtained. The results are Gm 4 (x, t ) =

G44 (x, t ) = -

Q K k 4 Pmkq  H (t ) d(cq t + n × x)dn, ¦ 2 ³ 8p n =1 q =1 K 44 l q

Q K 4i Pimq K m 4 d(t ) H (t ) d(cq t + n × x)dn , ¦ 2 2 ³ 8p n =1 q =1 K 44 l q 4pR L

(2.337)

(2.338)

where L = det[ kij ], R = kij-1 xi x j ,

(2.339)

with kij-1 being the inverse components of the dielectric permittivity tensor kij . Eqs (2.334), (2.336)-(2.338) can be rewritten in a unified form as GIJ (x, t ) = -

Q PIJq d d d(t ) H (t ) d(cq t + n × x)dn - 4 I 4 J ¦ 2 ³ 8p n =1 q =1 l q 4pR L

(2.340)

77

Green’s function of electroelastic problem

where ­ ° Pijq I , J = 1, 2,3, ° q °° K P PIJq = ® - k 4 ik I = 1, 2,3, J = 4, K 44 ° ° K Pq K ° 4 k kl2 l 4 I = J = 4. °¯ K 44

(2.341)

2.13 Green’s functions for an infinite medium with a crack In this section, Green’s functions for an infinite medium containing a conducting crack or an impermeable crack are presented based on extended Lekhnitskii’s formalism and distributed dislocation modelling. The discussion fellows the results presented in [56]. 2.13.1 Green’ function for line force/charge Consider a concentrated line force/charge P ( P1 , P2 , P3 ), i.e., horizontal force P1, vertical force P2, and electric charge P3, applied at z 0 ( x0 , y0 ). The complete Green’s functions of an infinite medium were given in Stroh formalism in Section 2.5. With extended Lekhnitskii’s formalism, the corresponding expressions of Green’s functions can be obtained by substituting the following potentials [56] F k ( zk ) =

1 ln( zk - zk0 ) ski* Pi 2pi

(2.342)

into Eqs (1.160) and (1.163), where F k ( zk ) is defined in Eq (1.160), and [ sij* ] = [ gij* ]-1 g1*j = p j + pi fij* , g 2* j = 1 + f 2*j , g3* j = v j + vi f ij* f1*j = [q*j ( p2* t3* - p3* t2* ) - p*j (q2* t3* - q3* t2* ) - t *j ( p2*q3* - p3*q2* )] / D1* f 2*j = [ p*j (q1* t3* - q3* t1 * ) - q*j ( p1* t3* - p3* t1 * ) - t *j ( p3*q1* - p1*q3* )] / D1*

(2.343)

f3*j = [q*j ( p1* t2* - p2* t1 * ) - p*j (q1* t2* - q2* t1 * ) - t *j ( p1*q2* - p2*q1* )] / D1* D1* = p1* (q2* t3* - q3* t2* ) - p2* (q1* t3* - q3* t1 * ) + p3* (q1* t2* - q2* t1 * ) where overbar denotes the complex conjugate of a complex-valued quantity, pi , vi , pi* , qi* , and ti* are defined in Eqs (1.154), (1.157) and (1.162). 2.13.2 Green’s function for a solid with a crack [56] In this subsection, Green’s functions for an infinite medium containing either an impermeable crack or a conducting crack are presented based on extended Lekhnitskii’s formalism. Since this is a linear problem it can be decomposed into two

78

Green’s function and boundary elements of multifield materials

simple subproblems as shown in Fig. 2.12 (b) and (c). The solution to the subproblem (b)

P

P 2a

y

y

y

x

+

=

x

x

(c)

(b)

(a)

bj

Fig. 2.12 Decomposition used to derive Green’s function with a crack[56] was given by Eq (2.342). For the subproblem in Fig. 2.12(c), a continuous generalized dislocation field with densities b is utilized to model the crack. The resultant electroelastic fields on the crack faces due to the line force/charge and distributed dislocation field should satisfy the corresponding boundary conditions. Let us consider first the case of an impermeable crack. The boundary conditions on the crack faces in this case are s yy = 0, s xy = 0,

Dy = 0,

(-a £ x £ a)

(2.344)

The solution for problem (c) satisfying the boundary condition of Eq (2.344) is obtained based on the solution (2.70) for a single generalized edge dislocation. The dislocation field b is then obtained by enforcing the boundary conditions of Eq (2.344). The final results for the potential functions F k ( zk ) corresponding to the subproblem in Fig 2.12(c) are obtained as

F k ( zk ) =

3 ª ( z 0j ) 2 - a 2 z 2j - a 2 + z 0j z j - a 2 * º ski Im ¦ «v ji ln s jn Pn » 2pi z j - z 0j » j =1 « ¬ ¼

where sij and vij are given by [56]

(2.345)

79

Green’s function of electroelastic problem

[ sij ] = [ gij ]-1 g1 j = p*j + pi* f ij , g 2 j = q*j + qi* f ij , g3 j = t *j + ti * fij f1 j = [ p3 v 2 - p2 v3 + p j v3 - p j v 2 - v j p3 + v j p2 )] / D1 f 2 j = [ p1 v3 - p3v1 - p j v3 + p j v1 - v j p1 + v j p3 )] / D1

(2.346)

f3*j = [ p2 v1 - p1v 2 - p j v1 + p j v 2 - v j p2 + v j p1 )] / D1 D1 = p1 (v 2 - v3 ) + p2 (v3 - v1 ) + p3 (v1 - v 2 ) vij = u j1 - pi u j 2 - vi u j 3

(2.347)

[uij ] = [rij ]-1

(2.348)

with

and 3

{r1 j , r2 j , r3 j }T = Im ¦ {1, - pn , -v n }T snj

(2.249)

n =1

Superposition of Eqs (2.342) and (2.345) yields the complex potential functions corresponding to Green’s functions for an impermeable crack. Similar, for an infinite medium containing a conducting crack whose boundary conditions are defined by s yy = 0, s xy = 0,

Ex = 0,

(-a £ x £ a) ,

(2.350)

the potential functions F k ( zk ) corresponding to the subproblem in Fig 2.12(c) can be obtained as [56] 3 ª ( z 0j ) 2 - a 2 z 2j - a 2 + z 0j z j - a 2 * º sˆki « F k ( zk ) = Im ¦ vˆ ji ln s jn Pn » 2pi z j - z 0j » j =1 « ¬ ¼

(2.351)

where coefficients are given by [ sˆij ] = [ gˆ ij ]-1 gˆ1 j = p*j + pi* fˆij , gˆ 2 j = q*j + qi* fˆij , gˆ 3 j = v j + vi fˆij fˆ1 j = [ p3 t2* - p2 t3* + p j t3* - p j t2* - t *j p3 + t *j p2 )] / Dˆ 1 f 2 j = [ p1 t3* - p3 t1 * - p j t3* + p j t1 * - t *j p1 + t *j p3 )] / Dˆ 1 f3*j = [ p2 t1 * - p1t2* - p j t1 * + p j t2* - t *j p2 + t *j p1 )] / Dˆ 1 Dˆ 1 = p1 ( t2* - t3* ) + p2 ( t3* - t1 * ) + p3 ( t1 * - t2* )

(2.352)

80

Green’s function and boundary elements of multifield materials

vˆij = uˆ j1 - pi uˆ j 2 - ti*uˆ j 3

(2.353)

[uˆij ] = [rˆij ]-1

(2.354)

with

and 3

{rˆ1 j , rˆ2 j , rˆ3 j }T = Im ¦ {1, - pn , -tn*}T sˆnj

(2.355)

n =1

Superposition of Eqs (2.342) and (2.351) yields the complex potential functions corresponding to the Green’s functions for a conducting crack. 2.14 Approximate expressions of Green’s function In this section we present a review of the approximate expressions of Green’s function developed in [57]. In that paper, the authors concentrated on developing approximate Green’s functions for two-dimensional transversely isotropic materials subjected to line force/charge. The constitutive relation and differential equations are, in this case, governed by Eqs (1.144) and (1.146). The corresponding Green’s function for the two-dimensional electroelastic problem are obtained by solving Eq (2.39). For the convenience of the following derivation, Eq (2.39) is rewritten as

L(Ñ x )G (ȟ - x) = d(ȟ - x)I 3

(2.356)

where I3 is the 3´3 unit matrix, and ª ¶2 ¶2 « c11 2 + c33 2 ¶x1 ¶x2 « « ¶2 L(Ñ x ) = - « (c12 + c33 ) ¶x1¶x2 « « ¶2 « -(e21 + e13 ) ¶x1¶x2 «¬

(c12 + c33 )

¶2 ¶x1¶x2

¶2 º » ¶x1¶x2 » ¶2 » + e22 2 » (2.357) ¶x2 » ¶2 » + k 22 2 » ¶x2 »¼

(e21 + e13 )

c33

¶2 ¶2 + c22 2 2 ¶x1 ¶x2

e13

¶2 ¶x12

-e13

¶2 ¶2 e 22 ¶x12 ¶x22

k11

¶2 ¶x12

In [57], the authors indicated that Eq (2.356) can be solved by the plane wave decomposition approach [16]. That is, G is represented over the unit circle n = 1 in terms of its transformed function g as follows

G (ȟ - x) = ³

2p

n =1

g(n, w)d b = ³ g (n, w)d b 0

(2.358)

where w = (ȟ - x) × n in which x1 - x1 = ȟ - x cos q, x 2 - x2 = ȟ - x sin q, n=(n1,n2) =(cosb, sinb). Substituting Eq (2.358) into Eq (2.356) and using the chain rule leads to 2p

2p

0

0

L(Ñ x ) ³ g(n, w)d b = ³ L(n) where

¶2 g(n, w)d b = d(ȟ - x)I 3 ¶w2

(2.359)

81

Green’s function of electroelastic problem

ª A11 A12 A13 º L(n) = - «« A12 A22 A23 »» «¬ A13 A23 A33 »¼

(2.360)

with A11 = c11n12 + c33 n22 , A12 = (c12 + c33 )n1n2 , A13 = (e21 + e13 ) n1n2 , A22 = c33 n12 + c22 n22 , A23 = e13 n12 + e22 n22 ,

A33 = k11n12 + k 22 n22

(2.361)

For obtaining explicit expression of the function g (n, w) in Eq (2.359), consider the function 2p

f (ȟ - x) = a ³ ln (ȟ - x) × n d b

(2.362)

ª ȟ-x º ln (ȟ - x) × n = ln « ȟ - x × n » = ln ȟ - x + ln cos(b - q) , ȟ - x »¼ ¬«

(2.363)

0

Since

it follows that 2p

f (ȟ - x) = 2p ln (ȟ - x) + a ³ ln cos(b - q) d b 0

(2.364)

where the second term is a constant. Using the identity [57] d(ȟ - x) =

1 Ñ 2 ln (ȟ - x) 2p

(2.365)

¶2 ¶2 + , ¶x12 ¶x22

(2.366)

where Ñ2 =

and then by applying Ñ 2 to Eq (2.362) and taking a = 1/ 4p2 , we have d(ȟ - x) = Ñ 2 f (ȟ - x) =

1 4p2

³

2p

0

¶2 ln w d b ¶w2

(2.367)

By comparing Eq (2.367) with Eq (2.359)2, we obtain 1 ln w I 3 4p 2

(2.368)

1 -1 L (n) ln w 4p 2

(2.369)

L(n)g(n, w) = Consequently g(n, w) =

To simplify the calculation of g (n, w) in Eq (2.369) and avoid making use of numerical integration in Eq (2.358), express the coefficients L-ij1 as a Fourier series in terms of the parameter b such that [57]

82

Green’s function and boundary elements of multifield materials

L-ij1 (b) =

aij(0) 2

¥

+ ¦ [aij( k ) cos(2kb) +bij( k ) sin(2kb)]

(2.370)

k =1

where aij( k ) =

1 2 p -1 Lij (b) cos(2kb)d b p ³0

(k=1,2,3,…)

(2.371)

bij( k ) =

1 2 p -1 Lij (b) sin(2kb)d b p ³0

(k=1,2,3,…)

(2.372)

It should be mentioned that the periodicity property of Lij (b) = Lij (b + p) has been used in Eq (2.370). Substituting Eq (2.370) into Eq (2.369) and subsequently into Eq (2.358) and integrating yields Gij (ȟ - x) =

¥ ½ 1 ­ (0) (-1) k +1 ( k ) a ȟ x ln + [aij cos(2kb) + bij( k ) sin(2kb)]¾ + const. (2.373) ® ij ¦ k 4p ¯ k =1 ¿

where the constant term is associated with rigid displacement or a reference potential and can be omitted in engineering analysis. As indicated by Khutoryansky et al [57], the approximate Green’s function has the following features: (a) The coefficients aij( k ) and bij( k ) need to be calculated only once for each integral independently of the value x-x. (b) L-ij1 (b) is a rational function of cos2b and sin2b with the denominator being a cubic polynomial of cos2b. Using the roots of this polynomial one can calculate exactly all the coefficients aij( k ) and bij( k ) . (c) For most piezoelectric materials these coefficients decrease very rapidly as k ® ¥ . Therefore, instead of Eq (2.373) we can use the following approximate Green’s functions Gij (ȟ - x) »

m ½ 1 ­ (0) (-1) k +1 ( k ) [aij cos(2kb) + bij( k ) sin(2kb)]¾ (2.374) ®aij ln ȟ - x + ¦ k 4p ¯ k =1 ¿

where m should be chosen to satisfy the required accuracy. 2.15 Green’s functions for time-dependent thermo-piezoelectricity Applications of Radon transform technique to deriving Green’s functions of time-dependent thermo-piezoelectricity are discussed in this section. The presentation here is from the development in [58]. For a thermo-piezoelectric solid under quasi-static conditions, its differential equations governing mechanical and electric behaviour were defined in Eq (1.59) and constitutive equations were given in Eq (1.55). In terms of entropy balance, the heat conduction equation (1.59)3 is rewritten as hi ,i + T0 s + bt = 0

(2.375)

in order to include the effect of heat source density bt, where T = T a - T0 << T0 is assumed.

83

Green’s function of electroelastic problem

The Fourier law for heat conduction of anisotropic solid is hi = kij F j

(2.376)

where F j = -T, j is the temperature gradient. 2.15.1 Green’s functions Consider a solid which is subjected to a point force Fi, a point electric charge Fe,  and a point heat source Ft at the source point x . Note that Fi, Fe, and Ft are unity, but are used here as a bookkeeping device [58]. Making use of Eqs (1.49), (1,51), (1.55), and (2.376), Eqs (1.55)1,2 and (2.375) can be written in the form [58] T0 l ij l kl · T0 l ij cl · T0 l ij § §  s, j + Fi d(x - x) H (t ) = 0 (2.377) ¨ cijkl + ¸ uk ,lj + ¨ eijl ¸ f,lj rcv ¹ rcv ¹ rcv © © T0 l ij cl § ¨ elij rcv ©

· ¸ ui , jl ¹

§ Tcc - ¨ kik - 0 i k rcv ©

· T0 ci  s,i + Fe d(x - x) H (t ) = 0 (2.378) ¸ f,ik + r c v ¹  kijT,ij - T0 s + Ft d(x - x)d(t ) = 0 (2.379)

where H(t) is the Heaviside step function. Applying the Radon transform to both sides of Eqs (2.377) and (2.378) leads to T0 l ij l kl · T0 l ij cl · T0 l ij § § n j sˆ¢ + Fi d(w) H (t ) = 0 ¨ cijkl + ¸ n j nl uˆk¢¢ + ¨ eijl ¸ n j nl fˆ ¢¢ rcv ¹ rcv ¹ rcv © © (2.380) T0 l ij cl § ¨ elij rcv ©

· § T0 ci c k ¸ nl n j uˆi¢¢ - ¨ kik rcv ¹ ©

· T0 ci ni sˆ¢ + Fe d(w) H (t ) = 0 ¸ ni nk fˆ ¢¢ + rcv ¹

(2.381)

where w = n × x, w and n are defined in Eq (A1) and Fig. 2.1. Solving the above two equations simultaneously for uˆk¢¢ and fˆ ¢¢ in terms of sˆ¢ yields uˆk¢¢ = -ck sˆ¢ - hki Fi d(w) H (t ) - d k Fe d(w) H (t )

(2.382)

fˆ ¢¢ = -esˆ¢ - ri Fi d(w) H (t ) - qFe d(w) H (t )

(2.383)

[hik ] = [bik ]-1

(2.384)

where

T0 l ij l kl § bik = ¨ cijkl + rcv ©

2 º · § T0 · 1ª « n n + e e ¸ j l ¨ ¸ l ij l kn cl c m » n j nn nm nl ijl mnk a« »¼ ¹ © rcv ¹ ¬

§ Tcc a = ¨ kik - 0 i k rcv ©

· ¸ ni nk ¹

§ 1 · T T ck = ¨ - eijl c m n j nl nm + 0 l ij cl c m n j nl nm - 0 l ij n j ¸ hki a a r c r c v v © ¹

(2.385)

(2.386)

(2.387)

84

Green’s function and boundary elements of multifield materials

Tl c 1§ di = - ¨ ekml - 0 kl m a© rcv e=

· ¸ nm nl hki ¹

(2.388)

· ck § T0 T l kl ci ¸ ni nl + 0 ci ni ¨ eilk + rcv a© arcv ¹

(2.389)

T0 l kl c m · 1§ ¨ emlk + ¸ nm nl hki a© rcv ¹

(2.390)

º · T0 1ª § l kl ci ¸ ni nl d k » «1 + ¨ eilk + a ¬« © rcv ¹ ¼»

(2.391)

ri =

q=

Applying the Radon transform to Eq (1.55)1 and taking the derivative with respect to w twice, we have Tˆ ¢¢ =

(

T0 sˆ¢¢ - l ij n j ui¢¢¢+ ni c i fˆ ¢¢¢

)

rc v

(2.392)

Making use of Eqs (2.382) and (2.383), the relation between Tˆ ¢¢ and sˆ¢¢ can be obtained as Tˆ ¢¢ = gsˆ¢¢ + (vi Fi + wFe )d¢(w) H (t )

(2.393)

where g=

T0 (1 + l ij ci n j - c i ni e) rc v

(2.394)

vi =

T0 (l km hki nm - c j n j ri ) rc v

(2.395)

w=

T0 (l km d k nm - c j n j q) rc v

(2.396)

Applying the Radon transform to Eq (2.379) leads to kij ni n jTˆ ¢¢ - T0 sˆ + Ft d(w)d(t ) = 0

(2.397)

To eliminate Tˆ ¢¢ from Eq (2.397), substituting Eq (2.393) into Eq ((2.397) above, we have F sˆ - Asˆ¢¢ = t d(w)d(t ) + ( Bi Fi + CFe )d¢(w) H (t ) T0

(2.398)

where A= Bk =

k ij ni n j g T0 k ij ni n j v k T0

(2.399)

(2.400)

85

Green’s function of electroelastic problem

k ij ni n j w

C=

(2.401)

T0

In order to obtain Green’s functions for each loading condition, consider first the  effect of a unit force in the ith direction acting at the source point x and beginning at time zero. Eq (2.398), in this case, becomes sˆ - Asˆ¢¢ = Bi d¢(w) H (t )

(2.402)

Its solution is well established in the literature [59]: sˆ(w, t ) = -

Bi § w · erf ¨ ¸ 2A © 2 At ¹

(2.403)

where erf represents the error function defined by erf ( x ) =

2 p

§ y2 · exp ³0 ¨¨© - 2 ¸¸¹dy x

(2.404)

Substituting Eq (2.403) into Eqs (2.382), (2.383) and (2.392) leads to the second-order derivative for displacements, electric potential, and temperature in the transform domain as § w2 · exp ¨ ¸ - h ji d(w) H (t ) 4 tA3 © 4 At ¹

(2.405)

§ w2 · Bf fˆ ¢¢i = i exp ¨ ¸ - qi d(w) H (t ) 4 tA3 © 4 At ¹

(2.406)

§ w2 · B gw Tˆi ¢¢= i exp ¨ ¸ + vi d¢(w) H (t ) 8 t 3 A5 © 4 At ¹

(2.407)

Bi c j

uˆij¢¢ =

where f =

Tl c · Tcn 1§ ¨¨ eijk + 0 kj i ¸¸ni n j ck + 0 i i a© rc v rcv ¹

qm =

Tl c 1§ ¨¨ eijk + 0 kj i a© rc v

vi =

· ¸¸ni n j hkm ¹

T0 (l jk nk h ji - c j n j qi ) cv

(2.408)

(2.409)

(2.410)

 Similarly, Green’s functions for a unit point charge acting at the point x and beginning at time zero can be obtained as uˆ4¢¢i =

§ w2 · exp ¨ ¸ - di d(w) H (t ) 4 tA3 © 4 At ¹

(2.411)

fˆ ¢¢4 =

§ w2 · exp ¨ ¸ - qd(w) H (t ) 4 tA3 © 4 At ¹

(2.412)

Cci

Ce

86

Green’s function and boundary elements of multifield materials

§ w2 · Cg w Tˆ4¢¢ = exp ¨ ¸ + wd¢(w) H (t ) 8 t 3 A5 © 4 At ¹

(2.413)

 Finally, the infinite space response to a unit pulse heat source at the point x and beginning at time zero is presented. Eq (2.398), in this case, becomes 1 sˆ - Asˆ¢¢ = d(w)d(t ) T0

(2.414)

The solution to this equation is well known and can be found in many text books [59]: sˆ(w, t ) =

1 2T0

§ w2 · exp ¨ ¸ pAt © 4 At ¹

(2.415)

As a result, the derivatives of Green’s function components in the transform domain are expressed in the following forms § w2 · exp ¨ ¸ pt 3 A3 © 4 At ¹

(2.416)

§ w2 · exp ¨ ¸ pt 3 A3 © 4 At ¹

(2.417)

§ w2 · (w2 - 2 At ) g exp ¨ Tˆ5¢¢ = ¸ 8T0 pt 5 A5 © 4 At ¹

(2.418)

uˆ5¢¢i = fˆ ¢¢5 =

wci 4T0

we 4T0

In order to obtain Green’s functions uij , fi , and Ti in the space domain, it is necessary to apply the inverse Radon transform defined in Eq (A3) to Eqs (2.405)-(2.407), (2.411)-(2.413) and (2.416)-(2.418). As a result, the responses to the above three loading cases are obtained as  Bi ( n)c j ( n) § [n × ( x - x )]2 · H (t ) 2 p 1  ¸¸ds( n) + 2 ³ h ji dy uij ( x - x ) = - 2 ³ exp¨¨ 3 n = 1 8p 4tA( n) ¹ 8p r 0 4 tA ( n) ©  § [n × ( x - x )]2 · Bi ( n) f H (t ) 2 p 1  ¨ ¸¸ds( n) + 2 ³ qi dy exp¨ fi ( x - x ) = - 2 ³ 3 = 1 n 8p 4tA( n) ¹ 8p r 0 4 tA ( n) ©   § [n × ( x - x )]2 · Bi ( n) gn × ( x - x ) 1 H (t )  ¸¸ds( n) + 2 2 Ti ( x - x ) = - 2 ³ exp¨¨ n p r 8p n =1 8 t 3 A5 ( n) 4 tA ( ) 8 ¹ ©

³

2p

0

¶vi dy ¶b

 Cc j ( n) § [n × ( x - x )]2 · 1 H (t ) 2 p  ¸¸ds( n) + 2 ³ d i dy u4i ( x - x ) = - 2 ³ exp¨¨ 3 = 1 n 8p 4tA( n) ¹ 8p r 0 4 tA ( n) ©  § [n × ( x - x )]2 · 1 eC H (t ) 2 p  ¸¸ds ( n) + 2 ³ qdy exp¨¨ f4 ( x - x ) = - 2 ³ 3 = 1 n 8p 4tA( n) ¹ 8p r 0 4 tA ( n) ©   § [n × ( x - x )]2 · 1 Cgn × ( x - x ) H (t )  ¸¸ds( n) + 2 2 T4 ( x - x ) = - 2 ³ exp¨¨ 3 5 = 1 n 8p 4tA( n) ¹ 8p r 8 t A ( n) ©

³

2p

0

¶w dy ¶b

88

Green’s function and boundary elements of multifield materials

ei =

xi and r = x r

(2.424)

In the e-d plane the unit vector n is decomposed into n = ad + be

(2.425)

a 2 + b2 = 1

(2.426)

where

Therefore, w = n × x = rb . It is noted that b ranges from -1 reflecting the negative x direction to 1. Furthermore, d may be expressed in terms of another parameter y (see Figs. 2.13 and A1). On the plane normal to e, we have

d = cos yp + sin yq

(2.427)

where p, q, and e form a right-hand coordinate system. Hence, n=

· xx xx x x2 + x2 r 1 - b 2 § x2 cos y - 1 2 3 sin y, - 1 cos y - 2 2 3 sin y, 1 2 2 sin y ¸ ¨ 2 2 r r r r r x1 + x2 © ¹ (2.428)

x · §x x + b¨ 1 , 2 , 2 ¸ r ¹ ©r r As a result, the surface integral can be expressed in terms of b and y as follows f ( x) = -

1 8p 2

³

2p

0

1 d y ³ fˆ (rb, n(b, y))db -1

(2.429)

Using the property of the Dirac delta function, d(w) = d(rb) = d(b) / r , we have f ( x) = -

1 1 2p fˆ (w, n)d(w)ds (n) = - 2 ³ fˆ (0, n(0, y ))d y 2 ³ n =1 8p 8p r 0

(2.430)

If fˆ (w, n) = 1 in Eq (2.430), we have

³

n =1

2p

1

0

-1

d(w)ds (n) = ³ d y ³ d(rb)db =

2p . r

(2.431)

References [1] Mura T, Micromechanics of Defects in Solids, 2nd Edition, Dordrecht: Martinus Nijhoff Publishers, 1987 [2] Qin QH, Fracture mechanics of piezoelectric materials, Southampton: WIT Press, 2001 [3] Qin QH and Mai YW, BEM for crack-hole problems in thermopiezoelectric materials, Eng Frac Mech, 69, 577-588, 2002 [4] Benveniste Y, The determination of the elastic and electric-fields in a piezoelectric inhomogeneity, J Appl Phys, 72, 1086-1095, 1992 [5] Chen T, Green’s functions and the non-uniform transformation problem in a piezoelectric medium. Mech Res Commun, 20, 271-278, 1993

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[6] Chen T and Lin FZ, Numerical evaluation of derivatives of the anisotropic piezoelectric Green’s function, Mech Res Commun, 20, 501-506, 1993 [7] 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 [8] Lee JS and Jiang LZ, Boundary integral formulation and 2D fundamental solution for piezoelectric media, Mech Res Commun, 22, 47ˉ54, 1994 [9] Sosa HA and Castro MA, On concentrated load at boundary of a piezoelectric half-plane, J Mech Phys Solids, 42, 1105-1122, 1994 [10] Dunn ML and Wienecke HA, Green’s functions for transversely isotropic piezoelectric solids, Int J Solids Struct, 33, 4571-4581, 1996 [11] Ding HJ, Wang GQ and Chen WQ, A boundary integral formulation and 2D fundamental solutions for piezoelectric media, Comput Meth Appl Mech Eng, 158, 65-80, 1998 [12] Ding HJ, Wang GQ and Jiang AM, Green’s function and boundary element method for transversely isotropic piezoelectric materials, Eng Anal Boun Elements, 28, 975-987, 2004 [13] Pan, E. A BEM analysis of fracture mechanics in 2D anisotropic piezoelectric solids, Eng Anal Boun Elements, 23, 67-76, 1999 [14] Gao CF and Fan WX, Green’s functions for generalized 2D problems in piezoelectric media with an elliptic hole, Mech Res Commun, 25, 685-693, 1998 [15] Norris AN, Dynamic Green’s functions in anisotropic, thermoelastic, and poroelastic solids, Proc R Soc Lond, A447, 175-188, 1994 [16] Khutoryansky NM and Sosa HA, Dynamic representation formulas and fundamental solutions for piezoelectricity, Int J Solids Struct, 32, 3307-3325, 1995 [17] Qin QH, Green’s function and its application for piezoelectric plate with various openings, Arch Appl Mech, 69, 133-144, 1999 [18] Qin QH and Mai YW, Crack branch in piezoelectric bimaterial system, Int J Eng Sci, 38, 673-693, 2000 [19] Pan E and Yuan FG, Three-dimensional Green’s functions in anisotropic piezoelectric bimaterials, Int J Eng Sci, 38, 1939-1960, 2000 [20] Pan E and Tonon F, Three-dimensional Green’s functions in anisotropic piezoelectric solids, Int J Solids Struct, 37, 943-958, 2000 [21] Gao CF and Fan WX, Green’s functions for the plane problem in a half-infinite piezoelectric medium, Mech Res Commun, 25, 69-74, 1998 [22] Dunn ML and Wienecke HA, Half-space Green’s function for transversely isotropic piezoelectric solids, J Appl Mech, 66, 675-679, 1999 [23] Gao CF and YU JH, Two-dimensional analysis of a semi-infinite crack in piezoelectric media, Mech Res Commun, 25, 695-700, 1998 [24] Chen BJ, Xiao ZM and Liew KM, On the interaction between a semi-infinite anti-crack and a screw dislocation in piezoelectric solid, Int J Eng Sci, 42, 1-11, 2004 [25] Ding HJ, Chen B and Liang J, On the green’s functions for two-phase transversely isotropic piezoelectric media, Int J Solids Struct, 34, 3041-3057, 1997 [26] Gao CF, Kessler H and Balke H, Green’s function for anti-plane deformations of a circular arc-crack at the interface of piezoelectric materials, Arch Appl Mech, 73, 467-480, 2003 [27] Liu JX, Wang B and Du SY, Line force, charge and dislocation in anisotropic

90

[28] [29]

[30] [31] [32] [33] [34] [35] [36]

[37] [38]

[39] [40] [41] [42] [43] [44] [45] [46] [47] [48]

[49]

Green’s function and boundary elements of multifield materials

piezoelectric materials with an elliptic hole or a crack, Mech Res Commun, 24, 399-405, 1997 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 Lu P, Tan MJ and Liew KM, A further investigation of Green’s functions for a piezoelectric media with a cavity or a crack, Int J Solids Struct, 37, 1065-1078, 2000 Gao CF and Fan WX, A general solution for the plane problem in piezoelectric media with collinear cracks, Int J Eng Sci, 37, 347-363, 1999 Gao CF and Wang MZ, Green’s function of an interface crack between two dissimilar piezoelectric media, Int J Solids Struct, 38, 5323-5334, 2001 Gao CF and Wang MZ, General treatment on mode III interfacial crack problems in piezoelectric materials. Arch Appl Mech, 71, 296-306, 2001 Hill LR and Farris TN, Three-dimensional piezoelectric boundary element method, AIAA J, 36, 102-108, 1998 Denda M, Araki Y and Yong YK, Time-harmonic BEM for 2-D piezoelectricity applied to eigenvalue problems, Int J Sloids Struct, 41, 7241-7265, 2004 Ma H and Wang B, The scattering of electroelastic waves by an ellipsoidal inclusion in piezoelectric medium, Int J Sloids Struct, 42, 4541-4554, 2005 Wang CY and Zhang Ch, 3-D and 2-D Dynamic Green's functions and time-domain BIEs for piezoelectric solids, Eng Anal Boun Elements, 29, 454-465, 2005 Deeg, WF, The analysis of dislocation, crack, and inclusion problems in piezoelectric solids, PhD dissertation, Stanford University, 1980 Synge JL, The hypercircle in mathematical physics: a method for the approximate solution of boundary value problems, Cambridge: Cambridge University Press, 1957 Li XY and Wang MZ, Three-dimensional Green’s functions for infinite anisotropic piezoelectric media, Int J Solids Struct (in press) Hadamard J, Lectures on Cauchy’s problem in partial differential equations, New Haven: Yale University Press, 1923 Liu JX, Wang B and Du S,Electro-elastic Green's functions for a piezoelectric half-space and their application, Appl Mathe Mech, 18, 1037-1043,1997 Ding HJ, Wang GQ and Chen WQ, Green's functions for a piezoelectric half plane, Science in China Ser E, 41, 70-75, 1998 Ting TCT, Image singularities of Green’s functions for anisotropic elastic half-spaces and bimaterials, Q J Mech Appl Math, 45, 119-139, 1992 Muskhelishvili NI, Some Basic Problems of Mathematical Theory of Elasticity, Noordhoff: Groningen, 1954 Suo Z, Kuo CM, Barnett DM and Willis JR, Fracture mechanics for piezoelectric ceramics, J Mech Phys Solids, 40, 739-765, 1992 Ting TCT and Chou SC, Edge singularities in anisotropic composites, Int J Solids Struct, 17, 1057-1068, 1981 Chung MY and Ting TCT, Piezoelectric solid with an elliptic inclusion or hole, Int J Solids Struct, 33, 3343-3361, 1996 Zhou ZD, Zhao SX and Kuang ZB, Stress and electric displacement analyses in piezoelectric media with an elliptic hole and a small crack, Int J Solids Struct, 42, 2803-2822, 2005 Ting TCT, Green's functions for an anisotropic elliptic inclusion under

Green’s function of electroelastic problem

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generalized plane strain deformations, Q J Mech Appl Math, 49, 1-18, 1996 [50] Chung MY and Ting TCT, The Green’s function for a piezoelectric piezomagnetic magnetoelectric anisotropic elastic medium with an elliptic hole or rigid inclusion, Phil Mag Letters, 72, 405-410, 1995 [51] 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 [52] 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 [53] 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 [54] Wang CY and Zhang C, 3-D and 2-D Dynamic Green's functions and time-domain BIEs for piezoelectric solids, Eng Anal Boun Elements, 29, 454-465, 2005 [55] Broida J and Williamson SG, A comprehensive introduction to linear algebra, California: Addison-Wesley, 1989 [56] Rajapakse RKND and Xu XL, Boundary element modeling of cracks in piezoelectric solids, Eng Anal Boun Elements, 25, 771-781, 2001 [57] Khutoryansky NM, Sosa HA and Zu WH, Approximate Green’s functions and a boundary element method for electroelastic analysis of active materials, Computers & Structures, 66, 289-299, 1998 [58] Jiang LZ, Integral representation and Green’s functions for 3D time-dependent thermo-piezoelectricity, Int J Solids Structures, 37, 6155-6171, 2000 [59] Carslaw HS and Jaeger JC, Conduction of heat in solids, 2nd ed., Clarendon Press, Oxford, 1959 [60] Wang CY and Achenbach JD, A new method to obtain 3D Green’s functions for anisotropic solids, Wave Motion, 18, 273-289, 1993

92

Chapter 3 Green’s function for thermoelectroelastic problems 3.1 Introduction In the previous chapter we described Green’s functions of piezoelectric material without or with defects such as cracks, holes, bimaterial interface, half-plane boundary, and inclusions. However, the widespread use of piezoelectric materials in structural applications has generated renewed interest in thermoelectroelastic behaviour. In particular, information about thermal stress concentrations around material or geometrical defects in piezoelectric solids has wide application in composite structures. Unlike in the cases of anisotropic elasticity and piezoelectricity, relatively little work has been reported regarding Green’s functions in thermoelastic and thermopiezoelectric solids. Using Stroh’s formalism and one-to-one mapping methods, Qin [1] obtained Green’s functions in closed form for an infinite piezoelectric plate with an elliptic hole induced by temperature discontinuity. Qin and Mai [2,3] also investigated thermoelectroelastic Green’s functions for half-plane and bimaterial problems. The studies in [4-6] further investigated thermoelectroelastic Green’s functions for piezoelectric materials with various openings or an elliptic inclusion. In this chapter, we begin with a discussion of Green’s functions in free space of thermopiezoelectricity. The results are then extended to cases of infinite thermopiezoelectricity with a half-plane boundary, bimaterial interface, holes of various shapes, and an elliptic inclusion. 3.2 Green’s function in free space 3.2.1 General solution in Stroh formalism Consider an anisotropic piezoelectric solid in which all fields are assumed to depend on in-plane coordinates x1 and x2 only. The general solutions of stress and electric displacement (SED) Pi, elastic displacement and electric potential (EDEP) vector U, temperature T, and heat flux hi in the solid can be expressed in terms of complex analytic functions as follows [3]: T = 2 Re[ g ¢( zt )],

h1 = -J,2 , h2 = J,1 ,

U = 2 Re[ A f ( za ) q + cg ( zt )], P1 = -j,2 , P2 = j,1

(3.1)

where J = 2k Im[ g ¢( zt )], j = 2 Re[B f ( za ) q + dg ( zt )],

zt = x1 + p1* x2 , za = x1 + pa x2 ,

(3.2)

where J is the heat flux function, p1* represents the eigenvalue of heat conduction equation [7],

f ( za ) is defined in Eq (1.131), but it is a 4´4 matrix in this chapter

rather than the original 5´5 matrix, an overbar denotes the complex conjugate, a prime represents the differentiation with respect to the argument, q is a constant vector to be determined by the boundary conditions, i = -1 , k = k11k22 - k122 , kij are the coefficients of heat conduction, and f(zi) and g(zt) are arbitrary functions with complex arguments zi and zt, respectively. A, B, c and d are associated with material constants and are well defined in the literature (see [7], for example). 3.2.2 Green’s function in free space

Green’s function for thermoelectroelastic problems

93

For an infinite space subjected to a line heat source h* and a line temperature discontinuity Tˆ both located at zt0(=x10+ p1* x20), the function g ¢( zt ) can be chosen in the form [3]

g ¢( zt ) = q0 ln( zt - zt 0 )

(3.3)

where q0 is a complex constant which can be determined from the conditions

³

C

³

C

dT = Tˆ

for any closed curve C enclosing the point zt0,

(3.4)

d J = - h*

for any closed curve C enclosing the point zt0

(3.5)

and

Substitution of Eq (3.3) into Eqs (3.1)1 and (3.2)1, and later into Eqs (3.4) and (3.5), yields q0 = Tˆ / 4pi - h* / 4pk

(3.6)

The function g(zt) in Eq (3.1)4 or (3.2)2 can be obtained by integrating Eq (3.3) with respect to zt, which yields

g ( zt ) = f * ( zt - zt 0 )q0

(3.7)

where f * ( x) = x(ln x - 1). Substituting Eq (3.7) into Eqs (3.1)4 and (3.2)2 and noting that q = 0 for the current thermal problem, the Green’s function for electroelastic field in free space is derived as U = 2 Re[cf * ( zt - zt 0 )q0 ], j = 2 Re[df * ( zt - zt 0 )q0 ]

(3.8)

3.3 Half-plane solid In this section the Green’s function in a half-plane piezoelectric solid subjected to loadings h* and Tˆ , both located at zt0(=x10+ p1* x20), are presented based on the concept of perturbation [8]. 3.3.1 Green’s function for thermal fields If the infinite straight boundary of the half-plane solid is in the thermally insulated condition, we have J=0

(3.9)

To satisfy the thermally insulated condition (3.9), the general solution of the thermal field can be assumed in the form from the concept of perturbation [8], T = 2 Re[ f 0 ( zt ) + f1 ( zt )],

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

(3.10)

The function f0 in Eq (3.10) can be chosen to represent the solutions associated with the unperturbed thermal fields (or the solution for an infinite homogeneous solid), and f1 is the function corresponding to the perturbed field of the solid due to defects (a half-plane boundary in this Section). f1=0 if there is no defect in a homogeneous solid. Therefore, f0 is given by Eq (3.3) as f 0 ( zt ) = g ¢( zt ) = q0 ln( zt - zt 0 )

(3.11)

94

Green’s function and boundary elements of multifield materials

Making use of Eq (3.10)2, the thermally insulated condition on the surface x2 = 0, i.e. J = 0, provides f1 ( zt ) = q0 ln( zt - zt 0 )

(3.12)

Further, if T= 0, rather than J = 0, on the surface x2 = 0, we have f1 ( zt ) = - q0 ln( zt - zt 0 )

(3.13)

Having obtained the solutions of f 0 and f1 , the function g ¢( zt ) can now be written as g ¢( zt ) = q0 ln( zt - zt 0 ) ± q0 ln( zt - zt 0 )

(3.14)

where the symbol “+” before q0 stands for the condition J=0 on the x1-axis, and “-” stands for the condition T=0 on x2=0. For the sake of clarity, we always use the symbol “+” in the following derivation. It should be understood that the symbol “+” is replaced by “-” if the boundary condition T=0 on x2=0 is involved. Substitution of Eq (3.14) into Eq (3.10) yields the Green’s functions of temperature and heat flux: T ( zt ) = 2 Re[q0 ln( zt - zt 0 ) + q0 ln( zt - zt 0 )] ,

(3.15)

J( zt ) = 2k Im[q0 ln( zt - zt 0 ) + q0 ln( zt - zt 0 )]

(3.16)

3.3.2 Green’s function for electroelastic fields For linear thermopiezoelectric problems, the solution (3.2)2 can be assumed to consist of a particular part j p , atributable to thermal loading, and a modified part j m , to ensure satisfaction of the half-plane boundary condition, as j = j m + j p . j p can be determined from the function g(zt) defined Eq (3.14). Therefore, the remaining task is to find the modified part j m , which is related to j p through the traction-charge condition on the boundary x2=0: j = j p + jm = 0

on x2=0

(3.17)

On the basis of the expressions (3.1)4 and (3.2)2, the particular solutions Up and j p for electroelastic fields can be written as U p = 2 Re[cg ( zt )],

j p = 2 Re[dg ( zt )]

(3.18)

where subscript “p” refers to the particular solution. The function g(zt) in Eq (3.18) can be obtained by integrating Eq (3.14) with respect to zt, which yields g ( zt ) = q0 f * ( zt - zt 0 ) + q0 f * ( zt - zt 0 )

(3.19)

The particular solution j p induced by g(zt) [see Eq (3.18)] does not, generally, satisfy the traction-charge free condition (3.17) along the axis x2 = 0 . We therefore need to find a modified isothermal solution j m for a given problem such that when it is superposed on the particular electroelastic solution the traction-charge free condition along the axis x2 = 0 will be satisfied. Owing to the fact that f(zk) and g(zt) have the same rule affecting the traction-charge in Eq (3.2)2, possible function forms come from the partition of g(zt). This is

95

Green’s function for thermoelectroelastic problems

j m = 2 Re[B f ( za ) q ]

(3.20)

f ( za ) = q0 f * ( za - zt 0 ) + q0 f * ( za - zt 0 )

(3.21)

where

Substitution of Eqs (3.18)2 and (3.20) into Eq (3.17) yields q = - B -1d

(3.22)

q = - A -1c

(3.23)

Likewise, if U=0 on x2 = 0 , then

Thus, the resulting Green’s function for the half-plane problem can be written as T = 2 Re{q0 ln yt + q0 ln yt*} ,

(3.24)

U = 2 Re{- A f ( za ) B -1d + c[q0 f * ( yt ) + q0 f * ( yt* )]} ,

(3.25)

j = 2 Re{-B f ( za ) B -1d + d[q0 f * ( yt ) + q0 f * ( yt* )]}

(3.26)

where f ( za ) is defined by Eq (3.21) and yt = zt - zt 0 ,

yt* = zt - zt 0

(3.27)

3.4 Bimaterial solid Consider a bimaterial solid in which the upper half-plane ( x2 > 0) is occupied by material 1 and the lower half-plane ( x2 < 0) by material 2 (see Fig. 2.5). They are rigidly bonded together so that T (1) = T (2) ,

U (1) = U (2) ,

J(1) = J(2) , j (1) = j (2) ,

at x2 = 0

(3.28)

at x2 = 0

(3.29)

where the superscripts (1) and (2) label the quantities relating to materials 1 and 2, respectively. In the rest of this section we present Green’s functions of the bimaterial solid described above due to a line heat source h* and a line temperature discontinuity Tˆ both located at zt(1)0 (= x10 + p1*(1) x20 ) in material 1. 3.4.1 Green’s function for temperature fields On the basis of the concept of perturbation given by Stagni [8], the general solution for the bimaterial solid can be assumed in the form [7] T (1) = 2 Re[ f 0 ( zt(1) ) + f1 ( zt(1) )], T (2) = 2 Re[ f 2 ( zt(2) )],

J(1) = 2k (1) Im[ f 0 ( zt(1) ) + f1 ( zt(1) )] ,

J(2) = 2k (2) Im[ f 2 ( zt(2) )] ,

x2 > 0 , (3.30) x2 < 0 (3.31)

where the function f0 is defined by Eq (3.11), f1 and f2 are functions corresponding to the perturbed field of the solid due to the existence of the bimaterial interface, and zt(1) = x1 + p1*(1) x2 ,

zt(2) = x1 + p1*(2) x2 ,

k (1) = k11(1) k22(1) - ( k12(1) ) , 2

k (2) = k11(2) k22(2) - ( k12(2) )

2

(3.32)

Green’s function and boundary elements of multifield materials

96

To satisfy the interface conditions (3.28), the functions f1 and f 2 can be assumed in the form f1 ( zt(1) ) = q1 ln y2(1) ,

f 2 ( zt(2) ) = q2 ln y1(2)

(3.33)

(2) where y2(1) = zt(1) - zt(1) = zt(2) - zt(1)0 , and q1 and q2 are two constants to be 0 , y1 determined. Substitution of Eq (3.33) into Eqs (3.30) and (3.31), and later into Eq (3.28), yields

q1 = -b1q0 ,

q2 = b2 q0 ,

b1 =

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

b2 =

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

(3.34)

Therefore the Green’s function for thermal fields can be written as T (1) = 2 Re[ q0 ln( zt(1) - zt(1)0 ) + q1 ln y2(1) )], J(1) = 2k (1) Im[q0 ln( zt(1) - zt(1)0 ) + q1 ln y2(1) )],

T (2) = 2 Re[q2 ln y1(2) ],

J(2) = 2k (2) Im[q2 ln y1(2) ] ,

x2 > 0

(3.35)

x2 < 0

(3.36)

3.4.2 Green’s function for electroelastic fields To use the interfacial condition (3.29) we first consider the particular solution due to the thermal field. This can be done by substituting Eq (3.33) into Eqs (3.30) and (3.31), and then integrating the result with respect to zt, yielding the function g(zt), which is required when determining U and j in Eqs (3.1) and (3.2), in the form g1 ( zt(1) ) = q0 f * ( y1(1) ) - b1q0 f * ( y2(1) )

(3.37)

g 2 ( zt(2) ) = b2 q0 f * ( y1(2) )

(3.38)

for x2 > 0 , and

for x2 < 0 , where y1(1) = zt(1) - zt(1)0 . The particular solution for electroelastic fields can then be given by (1) (1) (1) (1) * (1) * (1) U (1) p ( zt ) = 2 Re[c g1 ( zt )] = 2 Re{c [ q0 f ( y1 ) - b1q0 f ( y2 )]} ,

(3.39)

(1) (1) (1) (1) * (1) * (1) j (1) p ( zt ) = 2 Re[d g1 ( zt )] = 2 Re{d [ q0 f ( y1 ) - b1q0 f ( y2 )]}

(3.40)

for x2 > 0 , and (2) (2) U (2) g 2 ( zt(2) )] = 2b2 Re[c (2) q0 f * ( y1(2) )] , p ( zt ) = 2 Re[c

(3.41)

(2) (2) j (2) g 2 ( zt(2) )] = 2b2 Re[d (2) q0 f * ( y1(2) )] p ( zt ) = 2 Re[d

(3.42)

for x2 < 0 . The solutions (3.39)-(3.42) do not generally satisfy the interface condition (3.29). Therefore, development of a modified solution is needed such that when it is superposed on the particular solutions (3.39)-(3.42) the interface condition (3.29) will be satisfied. Owing to the fact that f(zi) and g(zt) have the same rule affecting U and j in Eqs (3.1)4 and (3.2)2, possible function forms come from the partition of solution g(zt). This is

97

Green’s function for thermoelectroelastic problems

f ( za( j ) ) = diag[ f ( y1*( j ) ) f ( y2*( j ) ) f ( y3*( j ) ) f ( y2*( j ) )]

(3.43)

where yi*( j ) = zi( j ) - zt(1)0 ,

i=1-4, j=1, 2

(3.44)

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

U (1) = 2 Re{A (1) f ( za(1) ) q1 + c(1) [q0 f ( y1(1) ) - b1q0 f ( y2(1) )]},

(3.45)

j (1) = 2 Re{B (1) f ( za(1) ) q1 + d (1) [q0 f ( y1(1) ) - b1q0 f ( y2(1) )]}

(3.46)

for x2 > 0 , and U (2) = 2 Re[ A (2) f ( za(2) ) q 2 + b2 q0c(2) f ( y1(2) )],

(3.47)

j (2) = 2 Re[B (2) f ( za(2) ) q 2 + b2 q0d (2) f ( y1(2) )]

(3.48)

for x2 < 0 . The substitution of Eqs (3.45)-(3.48) into Eq (3.29) yields q1 = q0 [B (1) - B (2) A (2)-1A (1) ]-1 ´ {[b2d (2) + b1d (1) - d (1) ] - B (2) A (2) -1[b2c(2) + b1 c (1) - c(1) ]},

q 2 = q0 [B(1) A (1)-1A (2) - B(2) ]-1 ´ {[b2d (2) + b1d (1) - d (1) ] - B(1) A (1)-1[b2c (2) + b1c (1) - c (1) ]}

(3.49)

(3.50)

Thus, the thermoelectroelastic Green’s functions can be obtained by substituting Eqs (3.49) and (3.50) into Eqs (3.45)-(3.48). 3.5 Elliptic hole problems 3.5.1 Green’s function of thermal field for insulated hole problems Consider an infinite piezoelectric plate containing an elliptic hole subjected to loadings h* and Tˆ at zt0, as shown in Fig. 2.7 (it is noted that the loading vector here is ( h* , Tˆ ) rather than (q 0 , b) ). If the hole is thermally insulated, the boundary conditions can be written as J=0,

hi ® 0 ,

(3.51) at infinity,

(3.52)

in addition to the balance conditions (3.4) and (3.5). The geometric equation of the elliptic hole G (see Fig. 2.7) is still defined by Eq (2.161), and the mapping function is given by Eq (2.162). Following the treatment in Section 2.8, a suitable function satisfying the boundary conditions (3.51) and (3.52) can be given in the form [7]: J = 2k Im[ g ¢( zt )] = 2k Im[q0 ln(z t - z t 0 ) + q1 ln(z t-1 - zt 0 )]

(3.53)

where z t and z t 0 are related to zt and zt0 by zt = ct z t + d t z t-1 , zt 0 = ct z t 0 + d tz t-01 with

(3.54)

Green’s function and boundary elements of multifield materials

98

ct = (a - ip1*b) / 2, d t = (a + ip1*b) / 2

(3.55)

It should be mentioned that the function ln( z - zt 0 ) is single valued since z -t 1 is -1 t

always located inside the circle and zt 0 is a point outside the unit circle. When x1 and x2 are given by Eq (2.161)1,2, z t = eiq , and angle ș is defined in Fig. 2.7. Hence, the substitution of Eq (3.53) into Eq (3.51) yields J = 2k Im[ g ¢( zt )] = 2k Im[q0 ln(eiq - z t 0 ) + q1 ln( e - iq - zt 0 )] = 0

(3.56)

The first term in Eq (3.56) can be replaced by the negative of its complex conjugate, that is, Im[ q0 ln(eiq - z t 0 )] = - Im[q0 ln( e -iq - zt 0 )]

(3.57)

Substituting Eq (3.57) into (3.56), we have q1 = q0

(3.58)

3.5.2 Green’s functions for electroelastic fields Noting Eqs (3.1)4 and (3.2)2, the particular solution for the electroelastic field induced by the line heat source h* and the line temperature discontinuity Tˆ can be written as u p = 2 Re[cg ( zt )],

j p = 2 Re[dg ( zt )]

(3.59)

The function g(zt) in Eq (3.59) above can be obtained by integrating g ¢( zt ) in Eq (3.53) with respect to zt, which yields g ( zt ) = q0{ct F1 (z t ) + d t F2 (z t )} + q0 {ct F4 (z t ) + d t F3 (z t )}

(3.60)

where F1 ( s ) = ( s - z t 0 )[ln( s - z t 0 ) - 1], F2 ( s ) = ( s -1 - z t-01 ) ln( s - z t 0 ) + z t-01 ln s, F3 ( s ) = ( s -1 - zt 0 )[ln( s -1 - zt 0 ) - 1], F4 ( s ) = ( s - zt-01 ) ln( s -1 - zt 0 ) - zt-01 ln s

(3.61)

For a zero traction-charge hole, the electroelastic boundary condition will be satisfied if [7] j=0

(3.62)

To satisfy condition (3.62) along the hole boundary, possible function forms of f(za) come from the partition of g(zt), for the same reason as that stated for Eq (3.43). They are fi ( za ) = Fi (z a ),

(i=1-4)

(3.63)

where z a is related to za by Eq (2.162). The Green’s functions for the electroelastic field can thus be chosen as 4

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

(3.64)

k =1

4

j = 2 Re ¦ [B f k (z a ) q k + dg (z t )] k =1

(3.65)

Green’s function for thermoelectroelastic problems

99

Noting that z a = z t = eiq on the hole boundary, we have

q1 = -q0ct B -1d, q 2 = -q0 d tB -1d,

q3 = -q0ct B -1d, q 4 = -q0 d tB -1d

(3.66)

Substituting Eq (3.66) into Eqs (3.64) and (3.65), the Green’s functions can then be further written as U = -2 Re[ A{q0ct F1 ( z a ) + q0d t F2 ( z a ) + q0ct F4 ( z a ) + q0d t F3 ( z a ) }B -1d ] (3.67) +2Re[cq0 {ct F1 ( z t ) + d t F2 ( z t )} + cq0 {ct F4 ( z t ) + d t F3 ( z t )}] j = -2 Re[B{q0ct F1 ( z a ) + q0d t F2 (z a ) + q0ct F4 ( z a ) + q0d t F3 ( z a ) }B -1d ]

+2Re[dq0 {ct F1 ( z t ) + d t F2 ( z t )} + dq0{ct F4 ( z t ) + d t F3 ( z t )}]

(3.68)

3.6 Arbitrarily shaped hole problems In Section 2.9, applications of Green’s function to a piezoelectric plate containing an arbitrarily shaped hole were presented. Extension of the procedure to include thermal effects is described in this section. Green’s functions in closed form for an infinite thermopiezoelectric plate with various holes induced by thermal loads are derived using Stroh formalism. The loads may be a point heat source or a temperature discontinuity. 3.6.1 Basic equations As was treated in Section 2.9, we consider a plane strain analysis where the material is transversely isotropic and where coupling occurs between in-plane stresses and in-plane electric fields. For a Cartesian coordinate system Oxyz, choose the z-axis as the poling direction, and denote the coordinates x and z by x1 and x2 in order to obtain a compacted notation. The plane strain constitutive equations are expressed in matrix form as hi = k ij q j , ­ s11 ½ ª c11 °s ° «c °° 22 °° « 12 ®s12 ¾ = « 0 ° D1 ° « 0 ° ° « ¯° D2 ¿° ¬«e21

c12

0

c 22 0 0 e22

0 c33 e13 0

0 e21 º ­ e11 ½ ­ l 11 ½ 0 e22 »» °° e 22 °° °l 22 ° ° ° ° ° e13 0 » ® 2e12 ¾ - ® 0 ¾T - k11 0 » ° - E1 ° ° 0 ° »° ° ° ° 0 - k 22 ¼» ¯°- E 2 ¿° ¯ c 2 ¿

(3.69)

(3.70)

or inversely q i = r ij h j , ­ e11 ½ ª f 11 °e ° «f °° 22 °° « 12 ® 2e12 ¾ = « 0 ° - E1 ° « 0 ° ° « ¯°- E 2 ¿° ¬« g 21

f 12

0

f 22 0 0 g 22

0 f 33 g13 0

0 g 21 º ­ s11 ½ ­ a 11 ½ 0 g 22 »» °°s 22 °° °a 22 ° ° ° ° ° g13 0 » ®s12 ¾ + ® 0 ¾T - b11 0 » ° D1 ° ° 0 ° »° ° ° ° 0 - b 22 ¼» ¯° D2 ¿° ¯ l 2 ¿

(3.71)

(3.72)

where c2 and l2 are pyroelectric constants, lij and aij are stress-temperature constants and thermal expansion constants respectively, qj and hj are heat intensity and heat flux,

Green’s function and boundary elements of multifield materials

100

kij and rij the coefficients of heat conductivity and heat resistivity. The boundary conditions at the rim of the hole can be written as J = j = 0,

(3.73)

if the hole is thermal-insulated, traction- and charge-free along the hole boundary. Besides, the infinite boundary condition h ® 0, P ® 0

at infinity

(3.74)

and the balance conditions

³

C

dT = Tˆ

and

³ dJ = -h C

*

for any closed curve C enclosing the point z t 0

(3.75)

should also be satisfied. 3.6.2 Green’s function for thermal fields Based on the one-to-one mapping described in Section 2.9 and the concept of perturbation [8], 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 )] ,

(3.76)

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

(3.77)

where z t and z t are related by the mapping function zt = a ( a1tz t + a2 tz t-1 + em1a3tz tm + em1a4 tz t- m )

(3.78)

with a1t = (1 - ip1* e) / 2, a 2 t = (1 + ip1* e) / 2 , a 3t = g (1 + ip1* e) / 2, a 4 t = g (1 - ip1* e) / 2

(3.79)

For an infinite piezoelectric plate containing a hole subjected to loadings h* and Tˆ at z t 0 (see Fig. 2.9), the function f0 can be chosen in the form f 0 (z t ) = q0 ln(z t - z t 0 )

(3.80)

where q0 is given in Eq (3.6). Thus the boundary condition (3.73)1 requires that f1 (z t ) = q0 ln(z t-1 - z t 0 )

(3.81)

The function g in Eq (3.2)2 can thus be obtained by integrating the functions f0 and f1 with respect to zt, which leads to g ( zt ) = aa1t [ q0 F1 ( z t , z t 0 ) + q0 F2 ( z t-1 , zt 0 )] + aa2 t [ q0 F2 (z t , z t 0 ) + q0 F1 ( z t-1 , zt 0 )] + ae j1{a3t [ q0 F3 ( z t , z t 0 ) + q0 F4 ( z t-1 , zt 0 )] + aa4 t [ q0 F4 ( z t , z t 0 ) + q0 F3 ( z t-1 , zt 0 )]} (3.82)

101

Green’s function for thermoelectroelastic problems

where F1 (z t , z t 0 ) = (z t - z t 0 )[ln(z t - z t 0 ) - 1], F2 (z t , z t 0 ) = (z

-1 t

-1 t0

(3.83) -1 t0

- z ) ln(z t - z t 0 ) + z ln z t ,

m 1§ z F3 (z t , z t 0 ) = (z tm - z tm0 ) ln(z t - z t 0 ) - z tm0 ¦ ¨¨ t n =1 n © z t 0

F4 (z t , z t 0 ) = (z

-m t

-z

-m t0

) ln(z t - z t 0 ) + z

-m t0

(3.84) n

· ¸ , ¸ ¹

ln z t - z

-m t0

(3.85) m -1

1 § zt0 ¨ ¦ ¨ n =1 n © z t

· ¸¸ ¹

n

(3.86)

3.6.3 Green’s functions for electroelastic fields. From Eqs (3.1)4 and (3.2)2 the particular solution for electroelastic fields induced by thermal loading can be written as

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

(3.87)

To satisfy the condition (3.73)2 along the hole boundary, possible function forms of f(zk) should come from the partition of g(zt), for the same reason as that stated for Eq (3.43). They are

f1 ( z k ) = a[ q0 F1 (z k , z t 0 ) + q0 F2 (z k , z t 0 ) + q 0 F1 (z k-1 , z t 0 ) + q0 F2 (z k-1 , z t 0 )] / 2 + e j1 ag[q 0 F3 (z k , z t 0 ) + q 0 F4 (z k , z t 0 ) + q0 F3 (z k-1 , z t 0 ) + q0 F4 (z k-1 , z t 0 )] / 2, (3.88) f 2 ( z k ) = ip k ae[- q0 F1 (z k , z t 0 ) + q0 F2 (z k , z t 0 ) + q0 F1 (z -k 1 , z t* ) - q 0 F2 (z -k 1 , z t* )] / 2 + ie j1 p k aeg[q 0 F3 (z k , z t 0 ) - q0 F4 (z k , z t 0 ) - q 0 F3 (z -k1 , z t 0 ) + q 0 F4 (z -k1 , z t 0 )] / 2 (3.89) where functions f1 and f2 represent two different possible functions. The Green’s functions for electroelastic fields can thus be chosen as ­ 2 ½ ­ 2 ½ U = 2 Re ®¦ [A f k ( za ) q k ] + cg ( zt ) ¾ , j = 2 Re ®¦ [B f k ( za ) q k ] + dg ( zt ) ¾ ¯ k =1 ¿ ¯ k =1 ¿

(3.90) Substitution of Eqs (3.88) and (3.89) into Eq (3.90)2 and later into Eq (3.73)2 leads to q 1 = -B -1 d , q 2 = -P -1 B -1 d p1* .

(3.91)

Substituting Eq (3.91) into Eq (3.90), the Green’s functions can then be written as U = 2 Re{- A[ f 1 ( z a ) + f 2 ( z a ) P -1 p1* ]B -1 d + cg ( z t )} ,

(3.92)

Green’s function and boundary elements of multifield materials

102

j = 2 Re{- B[ f1 ( za ) + f 2 ( za ) P -1 p1* ]B -1d + dg ( zt )}

(3.93)

3.7 Elliptic inclusion problems In this section fundamental solutions for thermal loading applied outside, inside or on the surface of an elliptic piezoelectric inclusion in an infinite piezoelectric matrix are presented. By combining the method of Stroh’s formalism, the technique of one-to-one mapping, the concept of perturbation, and the method of analytical continuation, Green’s functions are derived for an elliptic piezoelectric inclusion embedded in an infinite piezoelectric matrix subjected to a line heat source h* and a line temperature discontinuity Tˆ (Fig. 3.1).

x2

x2

(h , Tˆ ) *

W1

·z

b

W2

W2

a x1

G0

b

W1

t0

(h* , Tˆ ) ·

zt0

a x1

G0

G

G

(a)

(b)

x2

W1

b

(h* , Tˆ ) ·

W2

zt0

a x1

G0 G

(c)

Fig. 3.1 Three loading cases: (a) outside the inclusion; (b) inside; (c) on the interface To make the derivation tractable, consider again the one-to-one mapping defined by Eq (2.162). It will map the region outside the elliptic inclusion onto the exterior of a unit circle in the zk-plane. Further, the transformation (2.162) has been proved in Section 2.8 to be single valued outside the ellipse, since the roots of equation (2.167) are located inside the unit circle z k < 1 , where z k is related to zk by Eq (2.162). However, the mapping (2.162) is not single valued inside the ellipse, because the roots of Eq (2.167) are located inside the unit circle, as indicated in Section 2.8. To

103

Green’s function for thermoelectroelastic problems

circumvent this problem, the mapping of W2 (see Fig. 2.8) is done by excluding a slit, G0, which represents a circle of radius mk (= a2(2)k / a1(2) k ) in the zk-plane, from the ellipse [9]. In this case the function (2.162) will transform G and G0 into a ring of outer and inner circles with radii rout = 1 and rin = mk , respectively. Besides, anywhere inside the ellipse and on the slit G0 a function f (zk) must satisfy the condition [6] f [ m k s( q)] = f [ m k e 2iqk / s( q)]

(3.94)

to ensure that the field is single valued [9], where s(q) = e iq stands for a point located on the unit circle in the zk-plane, and q is a polar angle defined in Fig. 2.7. 3.7.1 Green’s functions of thermal loading applied outside the inclusion Consider an elliptic inclusion embedded in an infinite piezoelectric matrix subjected to loadings h* and Tˆ at a point (x10, x20) which is outside the inclusion (see Fig. 3.1a). If the inclusion and matrix are assumed to be perfectly bonded along the interface, the temperature, heat flow (hn), elastic displacements, electric potential, stress and electric displacement(Pn) across the interface should be continuous, i.e., T (1) = T (2) , J(1) =J(2) , U (1) =U (2) , j (1) =j (2) ,

(3.95)

along the interface. 3.7.1.1 Green’s functions for thermal fields. On the basis of the one-to-one mapping (3.54) and the concept of perturbation given by Stagni [8], the general solution for temperature and heat-flow function can be assumed in the form T (1) = f 0 (z t(1) ) + f 0 (z t(1) ) + f1 (z t(1) ) + f1 (z t(1) ) (1)

(1) t

(1) t

(1) t

z (1) t Î W1 ,

(1) t

J = -ik1 f 0 (z ) + ik1 f 0 (z ) - ik1 f1 (z ) + ik1 f1 (z )

T (2) = f 2 (z t(2) ) + f 2 (z t(2) ) (2)

J

(2) t

z (2) t Î W2 .

(2) t

= -ik2 f 2 (z ) + ik2 f 2 (z )

(3.96)

(3.97)

When an infinite space is subjected to loadings h* and Tˆ at (x10, x20), the function f0 can be chosen in the form (1) (1) f 0 ( z (1) t ) = q0 ln( z t - z t 0 )

(3.98)

where z (1) and z (1) are related to the complex arguments zt(1) and zt(1) 0 ( = x10 t t0 + p1*(1) x20 ) through the following transformation functions: z t( i ) =

zt( i ) + zt( i )2 - a 2 - p1*( i )2b 2 a - ip1*( i )b

, z t( i0) =

zt(0i ) + zt(0i )2 - a 2 - p1*(i )2b 2 a - ip1*( i )b

, (i=1,2)

(3.99)

and q0 has the same form as that defined in Eq (3.6) in which k is replaced by k(1). As for the function f2, noting that it is holomorphic in the annular ring (see Fig. 2.8), it can be represented by Laurent’s series,

Green’s function and boundary elements of multifield materials

104

f 2 (z t(2) ) =

¥

¦cz j

(2) j t

,

(3.100)

j =-¥

whose coefficients c-j can be related to cj by means of Eq (3.94) in the following manner: j

* j

c- j = G c j ,

§ a + ibp1*(2) · G =¨ . *(2) ¸ © a - ibp1 ¹ * j

(3.101)

Inserting Eqs (3.98) and (3.100) into Eqs (3.96) and (3.97) and later into Eqs (3.95)1,2 yields ¥

¥

j =1

j =1

f1 ( s) + f 0 (s) - ¦ [c j + G*j c j ]s - j = ¦ [c j + G*j c j ]s j - f1 ( s) - f 0 ( s) ,

- f1 (s) + f 0 (s) -

k (2) k (1)

¥

¦ [c

-j * =j - G j c j ]s

j =1

k (2) k (1)

¥

¦ [c

j

(3.102)

- G*j c j ]s j + f 0 (s) - f1 (s) (3.103)

j =1

One of the important properties of holomorphic functions used in the method of analytic continuation is that if the function f(z) is holomorphic in W1 (or W0 + W 2 ), then f (1/ z ) is holomorphic in W0 + W 2 (or W1), W0 denoting the region inside the circle of radius

mt

. Hence, put

¥ ­ * -j ° f1 ( z ) + f 0 (1/z ) - ¦ [c j + G j c j ]z , z Î W1 , j =1 ° w( z ) = ® ¥ ° - f (1/ z ) - f ( z ) + [c + G* c ]z j , z Î W + W , ¦ 1 0 j j j 0 2 °¯ j =1

(3.104)

where the function w(z) is holomorphic and single valued in the whole plane. By Liouville’s theorem, we have w(z) = constant. However, constant function f does not produce stress and electric displacement (SED), which may be neglected. Thus, letting w(z)=0, we have ¥

¦ [c

j

+ G*j c j ]z - j = f1 (z ) + f 0 (1/ z ),

z Î W1 ,

j =1

¥

¦ [c

j

+ G*j c j ]z j = f1 (1/ z ) + f 0 ( z ),

z Î W0 + W 2 .

(3.105)

j =1

It should be mentioned that the superscripts (1) and (2) are omitted in Eqs (3.104) and (3.105). To further simplify subsequent writing, we shall omit them again in the related expressions when the distinction is unnecessary. Similar to the procedures in Eq (3.102), we can derive from Eq (3.103) that

105

Green’s function for thermoelectroelastic problems

k (2) k (1) k (2) k (1)

¥

¦ [c

j

- G*j c j ]z - j = - f1 (z ) + f 0 (1/ z ),

z Î W1 ,

j =1

(3.106)

¥

¦ [c

j

j

* j

- G c j ]z = - f1 (1/ z ) + f 0 (z ),

z Î W0 + W 2 .

j =1

The equations (3.105)2 and (3.106)2 provide f 0 (z ) =

1 ¥ ¦ [(1 + k (2) / k (1) )c j + (1 -k (2) / k (1) )G*j c j ]z j . 2 j =1

(3.107)

With the use of the series representation ¥

f 0 ( x ) = ¦ ek x k , ek = k =1

f 0( k ) (0) 1 f0 ( x) = dx , 2pi ³ C x k +1 k!

(3.108)

the function f0(z) given in Eq (3.98) can be expressed as ¥

f 0 (z ) = ¦ e j z j ,

ej = -

j =1

q0 z t(1)0 - j j

(3.109)

where dij=1, when i=j; dij=0, when i¹j. Comparing the coefficients of corresponding terms in Eqs (3.107) and (3.109), one obtains c j = (G0 - G jG j / G0 ) -1 (e j - G j e j / G0 ),

(j = 1, 2,", ¥)

(3.110)

where G0 = (1 + k2 / k1 ) / 2, G j = (1 - k2 / k1 ) G*j / 2 . With the solution obtained for ck, the functions f1, g1¢ and g 2¢ can be further written as ¥

-j f1 ( z ) = ¦ [c j + G*j c j ]z (1) - q0 ln( z t(1)-1 - zt(1) t 0 ),

(3.111)

j =1

¥

(1) (1) -1 * (1) - j - zt(1) g1¢( z t(1) ) = q0 ln( z (1) , t - z t 0 ) - q0 ln( z t 0 ) + ¦ [ c j + G j c j ]z t

(3.112)

j =1

¥

g 2¢ ( z t(2) ) = ¦ c j [z t(2) j + G*j z t(2)- j ] .

(3.113)

j =1

Substitution of Eqs (3.112) and (3.113) into Eqs(3.1)1 and (3.2)1 yields Green’s functions for thermal fields. 3.7.1.2 Green’s functions for electroelastic fields. From Eqs (3.1)4 and (3.2)2, the particular piezoelectric solution induced by the thermal loadings h* and Tˆ 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),

(3.114)

The function g(zt) in Eq (3.114) can be obtained by integrating Eqs (3.112) and (3.113) with respect to zt,, which yields

Green’s function and boundary elements of multifield materials

106

(1) (1) (1) (1) -1 (1) (1) (1) g1 ( z (1) , zt(1) t ) = a1t [ q0 F1 ( z t , z t 0 ) - q0 F2 ( z t 0 )] + a2 t [ q0 F2 ( z t , z t 0 ) ¥

* (1) (1) (1) - j -1 - q0 F1 ( z (1) , zt(1) , t 0 )] + ( c1 + G1 c1 ) a1t ln z t + ¦ G1 j z t

(3.115)

j =1

¥

-j (1) j g 2 ( z (2) + G3 j z (2) ], t ) = ¦ [G2 j z t t

(3.116)

j =1

where F1 ( z t , z t 0 ) = ( z t - z t 0 )[ln( z t - z t 0 ) - 1], F2 ( z t , z t 0 ) = (z t-1 - z t-01 ) ln( z t - z t 0 ) + z t-01 ln z t , a1(tk ) = ( a - ip1*( k )b) / 2, a2( kt ) = ( a + ip1*( k )b) / 2, * j +1 j +1

(1) 1t

G1 j = -[( c j +1 + G c )a (2) 1t

(1) 2t

(k = 1, 2),

* j -1 j -1

- a ( c j -1 + G c ) s j1 ] / j,

(3.117)

(2) 2 t j +1

G2 j = ( a c j -1s j1 - a c ) / j, * (2) * G3 j = -( a1(2) t G j +1c j +1 - a2 t G j -1c j -1s j1 ) / j ,

and sij = 1 for i ¹ j, sij = 0 for i = j . To satisfy the condition (3.95)3,4 along the interface, the function f(zm) can be assumed in the form [6]: f1(mj ) (z (mj ) ) = a[q0 F1 (z (mj ) , z t( 0j ) ) + q0 F2 (z (mj ) , z t( 0j ) ) - q0 F1 (z (mj ) -1 , zt(0j ) ) - q0 F2 (z (mj ) -1 , zt(0j ) )] / 2, f 2(mj ) (z (mj ) ) = ipm( j )b[-q0 F1 (z (mj ) , z t( 0j ) ) + q0 F2 (z (mj ) , z t( 0j ) )

(j=1, 2),

(3.118)

- q0 F1 (z (mj ) -1 , zt(0j ) ) + q0 F2 (z (mj ) -1 , zt(0j ) )] / 2, f3(mj ) (z (mj ) ) = a1(mj ) ln z (mj ) , ¥

¥

k =1

k =1

f 4( j ) (z ( j ) ) = ¦ z (aj ) k rk( j ) , f5( j ) (z ( j ) ) = ¦ z (aj ) - k s(k j ) ,

(j=1, 2), (3.119) (j=1, 2)

(3.120)

where f k( j ) are four component vectors, and rk( j ) and s(k j ) are constant vectors with four components to be determined. It should be pointed out that the vector s (k j ) is different from the symbol sij appearing in Eq (3.117). The Green’s functions for the electroelastic fields can thus be chosen as 5 ­ 3 ½ U ( j ) = 2 Re ®¦ A ( j ) f k(aj ) (z a( j ) ) q (k j ) + ¦ A ( j )f k( j ) (z ( j ) ) + c( j ) g j (z t( j ) ) ¾ , k =4 ¯ k =1 ¿

(3.121)

5 ­ 3 ½ j( j ) = 2 Re ®¦ B ( j ) f k(aj ) (z (aj ) ) q (k j ) + ¦ B ( j )f k( j ) (z ( j ) ) + d ( j ) g j (z t( j ) ) ¾ k =4 ¯ k =1 ¿

(3.122)

The above two expressions, together with the interface condition (3.95)3,4, provide

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

(3.123)

107

Green’s function for thermoelectroelastic problems

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

(3.124)

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

(3.125)

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

q

(1) 3

(1) -1 1a

= a

q3(2) = a1(2) a

-1

(1) 1t

(3.126)

* 1 1

(1) 1

a ( c1 + G c )q ,

(3.127)

* (2) a1(1) t (c1 + G1 c1 )q1 ,

(3.128)

¥

A (1)f5(1) (s) + A (1) f 4(1) (s) - A (2)f5(2) (s) - A (2) f 4(2) (s) + ¦ (c(1)G1 j - c (2)G2 j - c(2)G3 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)G1 j - c(2)G2 j - c (2)G3 j )s j j =1

(3.129) ¥

B (1)f5(1) (s) + B (1) f 4(1) (s) - B (2)f5(2) (s) - B (2) f 4(2) (s) + ¦ (d (1)G1 j - d (2)G2 j - d (2)G3 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)G1 j - d (2)G2 j - d (2)G3 j )s j j =1

(3.130) where X1 = ( A (2) -1A (1) - B (2)-1B (1) ) -1 , X 2 = ( A (1)-1A (2) - B (1)-1B (2) ) -1 .

(3.131)

Therefore, by Liouville’s theorem, Eqs (3.129) and (3.130) yield ¥

A (1)f5(1) (s) + A (1) f 4(1) (s)=A (2)f5(2) (s) + A (2) f 4(2) (s) - ¦ (c(1)G1 j - c (2)G2 j - c(2)G3 j )s - j , j =1

(3.132) ¥

A (1)f 4(1) (s) + A (1) f5(1) (s)=A (2)f 4(2) (s) + A (2) f5(2) (s) - ¦ (c (1)G1 j - c(2)G2 j - c (2)G3 j )s j , j =1

(3.133) ¥

B (1)f5(1) (s) + B (1) f 4(1) (s)=B (2)f5(2) (s) + B (2) f 4(2) (s) - ¦ (d (1)G1 j - d (2)G2 j - d (2)G3 j )s - j , j =1

(3.134) (1) (1) 4

(1) (1) 5

(2) (2) 4

B f (s) + B f (s)=B f

(2)

(s) + B f

(2) 5

¥

(s) - ¦ (d G1 j - d G2 j - d G3 j )s j (1)

(2)

(2)

j =1

(3.135) The above four equations are not completely independent. For example, Eqs (3.132) and (3.134) can be obtained from Eqs (3.133) and (3.135). Thus only two of the equations are independent. However, there are four sets of constant vectors, i.e., rk( j ) and s (k j ) ( j = 1, 2) , to be determined. We need two more equations to make the solution unique. Through use of the relation (3.94) and Eqs (3.121) and (3.122), the

Green’s function and boundary elements of multifield materials

108

unknown vectors r j(2) and s (2) appearing in f 4(2) and f5(2) can be determined as j follows: (2) (2) -1 (2) r j(2) = ( A (2) -1 G (2) - B (2) -1 G (2) ) [ A (2) -1 (c(2)G3 j - G (2) ja A ja B ja c G2 j ) (2) - B (2) -1 (d (2)G3 j - G (2) ja d G2 j )],

( j = 1," , ¥),

(3.136) (2) -1 (2) (2) s (2) [ G (2) + d (2)G2 j ) - d (2)G3 j ] j =B ja (B r j

where

§ a + ibpa( i ) · G (jia) = ¨ (i ) ¸ © a - ibpa ¹

( j = 1," , ¥) , (3.137)

j

.

are obtained, the unknown vectors Once the constant vectors r j(2) and s (2) j r (1) and s(1) appearing in functions f4(1) and f5(1) can be determined from Eqs (3.132) j j

and (3.134) [or(3.133) and (3.135)]. They are r j(1) = iA (1)T {(M 2 + M1 ) A (2)r j(2) + (M1 - M 2 ) A (2) s j(2) - M1 ( c (1)G1 j - c(2)G2 j - c (2)G3 j ) + i (d (1)G1 j - d (2)G2 j - d (2)G3 j )}, (1)T (2) (2) s (1) {(M 2 + M1 ) A (2)s (2) rj j = iA j + (M1 - M 2 ) A

- M1 (c(1)G1 j - c (2)G2 j - c(2)G3 j ) + i (d (1)G1 j - d (2)G2 j - d (2)G3 j )}

(3.138)

(3.139)

where M k = -iB ( k ) A ( k ) -1 = H ( k ) -1 (I + iS ( k ) ), H ( k ) = 2iA ( k ) A ( k )T , S ( k ) = i (2A ( k ) B ( k )T - I ) , (k=1, 2)

(3.140)

3.7.2 Green’s functions for thermal loads applied inside the inclusion 3.7.2.1 Green’s functions for thermal fields. For the case of thermal loads h* and Tˆ located at a point (x10, x20) inside the inclusion, the general solution for temperature and heat-flow function can be assumed in the form [5] T (1) = f 0 (z t(1) ) + f1 (z t(1) ) + f 0 (z t(1) ) + f1 (z t(1) ),

(3.141)

J(1) = -ik (1) f 0 (z t(1) ) - ik (1) f1 (z t(1) ) + ik (1) f 0 (z t(1) ) + ik (1) f1 (z t(1) ) T (2) = f 0* (z t(2) ) + f 2 (z t(2) ) + f 0* (z t(2) ) + f 2 (z t(2) ),

(3.142)

J(2) = -ik (2) f 0* (z t(2) ) - ik (2) f 2 (z t(2) ) + ik (2) f 0* (z t(2) ) + ik (2) f 2 (z t(2) )

Here, f 0 and f 0* can be chosen to represent the solutions associated with the unperturbed thermal fields caused by thermal load. f1 and f2 are the functions corresponding to the perturbed field of matrix and inclusion, respectively. Since (x10, x20) is inside the inclusion, there is a branch cut extending from (x10, x20) to infinity. Thus the choice of f 0 and f 0* should take into account the discontinuity across this branch cut. A proper form of f 0 and f 0* may be chosen as [10]:

109

Green’s function for thermoelectroelastic problems (1) f 0 (z (1) t ) = d ln z t ,

f 0* (z (t 2 ) ) = q0 [ln( z t( 2 ) - z t(02 ) ) - ln a1(t2 ) ]

(3.143)

¥

f 2 (z (t 2 ) ) =

¦c z j

(2) j t

j =-¥

, c - j = G *j c j

(3.144)

where a1(tk ) is defined in Eq (3.117), d and cj are unknowns to be determined, *(2) z t( i ) and z t(02 ) are related to the complex arguments z t(i ) and zt(2) x20 ) by 0 ( = x10 + p1 Eq (3.99), and q0 is a complex number which can be determined from the conditions dT = Tˆ for any closed curve C enclosing the point z ( 2 ) (3.145)

³

³

C

t0

C

d J = - h*

for any closed curve C enclosing the point z (t 02 )

(3.146)

Substitution of Eqs (3.143) and (3.144) into (3.141) and (3.142) and later into (3.145) and (3.146) yields q0 = T0 / 4pi - h* / 4pk (2)

(3.147)

As for the unknowns d, cj and the function f1, they can be derived in the following way. Noting the relation ln[( z t - z t 0 ) / a1t ] = ln(z t - z t 0 ) + ln(1 -

a 2 t / a1t ) ztzt0

(3.148)

where a1t = (a - ip1*b) / 2, a2 t = (a + ip1*b) / 2 , we have the series representation ¥

f 0* ( z t( 2 ) ) = q0{ln z t( 2 ) - ¦ e j z (t 2 )- j }

for z (t 2 ) > 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

Green’s function and boundary elements of multifield materials

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)

Green’s function and boundary elements of multifield materials

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

Green’s function for thermoelectroelastic problems

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)

Green’s function and boundary elements of multifield materials

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

Green’s function for thermoelectroelastic problems

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

Green’s function and boundary elements of multifield materials

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

Green’s function for thermoelectroelastic problems

(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

Green’s function and boundary elements of multifield materials

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

Green’s function and boundary elements of multifield materials

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

Green’s function for magnetoelectroelastic problems

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)

Green’s function and boundary elements of multifield materials

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

Green’s function for magnetoelectroelastic problems

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)

Green’s function and boundary elements of multifield materials

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)

Green’s function for magnetoelectroelastic problems

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,

Green’s function and boundary elements of multifield materials

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

Green’s function for magnetoelectroelastic problems

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]

Green’s function and boundary elements of multifield materials

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 q (1)* , qb , and qb have the same forms as those in Eqs (2.95), (2.100) and (2.101), but now each of these three vectors has five components instead of four.

4.6.2 Green’s function for magnetoelectroelastic bimaterial with vertical interface. For a magnetoelectroelastic bimaterial plate with a vertical interface subjected to loadings q 0 and b in the left half-plane at z0(x10, x20) (Fig. 4.4), the solution may be

131

Green’s function for magnetoelectroelastic problems

assumed, in a way similar to that for half-plane problem, in the form U (1) =

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

(4.77)

ij (1) =

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

(4.78)

for material 1 in the region x1 < 0 and 5 1 U ( 2) = ¦ Im{A ( 2 ) ln( z a*( 2) - z b*(01) ) q b( 2) } , p b =1

(4.79)

5 1 (2) ij (2) = ¦ Im{B (2) ln( za*(2) - zb*(1) (4.80) 0 ) qb } b=1 p (1) (1) for material 2 in the region x1 > 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].

Green’s function and boundary elements of multifield materials

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

Green’s function for magnetoelectroelastic problems

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)

Green’s function and boundary elements of multifield materials

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

Green’s function for magnetoelectroelastic problems

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

Green’s function and boundary elements of multifield materials

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 (x1<0) is occupied by material 1, and the right half-plane (x1>0) 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

Green’s function for magnetoelectroelastic problems

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)

Green’s function and boundary elements of multifield materials

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

Green’s function for magnetoelectroelastic problems

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)

Green’s function and boundary elements of multifield materials

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

Green’s function for magnetoelectroelastic problems

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

Green’s function and boundary elements of multifield materials

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

Green’s function for magnetoelectroelastic problems

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)

Green’s function and boundary elements of multifield materials

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

Green’s function for magnetoelectroelastic problems

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)

Green’s function and boundary elements of multifield materials

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

Green’s function for magnetoelectroelastic problems

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

Green’s function and boundary elements of multifield materials

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

Green’s function for magnetoelectroelastic problems

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

Green’s function and boundary elements of multifield materials

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)

Green’s function and boundary elements of multifield materials

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)

Green’s function and boundary elements of multifield materials

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

Boundary element method for piezoelectricity

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

Green’s function and boundary elements of multifield materials

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

Boundary element method for piezoelectricity

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)

Green’s function and boundary elements of multifield materials

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

Boundary element method for piezoelectricity

   -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

Green’s function and boundary elements of multifield materials

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

Boundary element method for piezoelectricity

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

Green’s function and boundary elements of multifield materials

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

Boundary element method for piezoelectricity

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

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 (x, x, t - t)uI (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

Green’s function and boundary elements of multifield materials

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

Boundary element method for piezoelectricity

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

Green’s function and boundary elements of multifield materials

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

Boundary element method for piezoelectricity

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

Green’s function and boundary elements of multifield materials

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

Boundary element method for piezoelectricity

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)

Green’s function and boundary elements of multifield materials

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

Boundary element method for piezoelectricity

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

Green’s function and boundary elements of multifield materials

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

Boundary element method for piezoelectricity

(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

Green’s function and boundary elements of multifield materials

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

Boundary element method for piezoelectricity

 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

Green’s function and boundary elements of multifield materials

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

Boundary element method for piezoelectricity

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)

Green’s function and boundary elements of multifield materials

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

Boundary element method for piezoelectricity

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

Green’s function and boundary elements of multifield materials

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

Boundary element method for piezoelectricity

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 (D<1). Following this scheme the discontinuities required for each side can be established separately. To emphasize again, the geometry is still interpolated with respect to the unshifted nodes. To obtain the r-dependent discontinuous shape functions for displacement interpolation Ni(U ) Wippler and Kuna [36] assumed N i(U ) (r , t ) = a1i + a2i r + a3i t + a4i r + a5i t 2 + a6i t r + a7i rt + a8i t 2 r

(5.203)

where r =1+s, and a ij are determined by the condition [37]

N i ( s j , t j ) = dij (i, j=1,2,…8)

(5.204)

N i ( s, t ) = a1i + a2i s + a3i t + a4i s 2 + a5i t 2 + a6i st + a7i s 2t + a8i t 2 s

(5.205)

with

These crack front elements possess r behavior of displacement and electric potential with respect to one element side. Depending on the side which is adjacent to the crack front, r can also be (1-s), (1+t), or (1-t). To incorporate the r -1/ 2 behavior of the generalized tractions at the ligament ahead of the crack front, Eq (5.203) is divided by r , giving the shape function N i(T ) [36]:

Green’s function and boundary elements of multifield materials

182

N i(T ) (r , t ) =

a1i r

+ a2i +

a3i t r

+ a4i r + r + a6i t + a7i t r + a8i t 2

(5.206)

If symmetry planes and free faces intersect the crack front, a new element is required with the r-dependence being with respect to one corner node only [36]: N i( SU ) (r , a) = a1i + a2i r + a3i a + a4i r + a5i a 2 + a6i a r + a7i r a + a8i a 2 r

(5.207)

where r = (1 + s ) 2 + (1 + t ) 2 and a = arctan[(1 + t ) /(1 + s )] , the superscript S here standing for symmetry plane or surface. The corresponding generalized traction can be approximated by the following shape functions N i( SU ) (r , a) =

a1i r

+ a2i +

a3i a r

+ a4i r +

a5i a 2 r

+ a6i a + a7i a r + a8i a 2

(5.208)

where a is the angle around the node of singularity.

5.9 Mixed BEM-homogenization method The BE formulation described in the previous sections can be used to predict effective material properties of piezoelectric composites by means of the homogenization approach. The homogenization model is used to establish the relationship between overall material properties and boundary fields for composites with inhomogeneities, and introducing the BE formulation can predict the numerical results of the boundary fields. 5.9.1 Basic concept of effective material properties Let us consider a piezoelectric composite in which the inclusions or holes are of cylindrical shape. In this case both the matrix and the inclusion can be viewed as transversely isotropic, and coupling occurs between in-plane stresses and in-plane electric fields. For a Cartesian coordinate system Oxyz, choose the z-axis as the poling direction, and denote the coordinates x and z by x1 and x2 to achieve a compact notation. The plane strain constitutive equations are expressed by Eqs (1.144) and (1.145). These two equations can be written in matrix form as follows [38]

Ȇ = CZ, Z = FȆ

(5.209)

where F=C-1, and

Ȇ = {P i }T = {s11s 22 s12 D1 D2 }T , Z = {Z i }T = {e11e 22 2e12 - E1 - E 2 }T (5.210)

S

G (a) RVE with an inclusion

(b) RVE with a void

Fig. 5.8 Two RVE models

183

Boundary element method for piezoelectricity

The homogenization formulation is established by considering a representative volume element (RVE) W which is chosen so as to be statistically representative of the two-phase composite. In particular, the characteristic size of the heterogeneities needs to be small with respect to the dimension of the RVE, which in turn needs to be small compared to the wavelength of the macroscopic structure. To understand the key point of the homogenization procedure, consider a RVE consisting of the matrix material and inclusion phase (see Fig. 5.8). As the RVE is comprised of different materials, the micro-constitutive law that governs each material or phase in a RVE is given by the standard constitutive law. On the other hand, the stress and electric displacement (SED) and strain and electric field (SEF) on the macro-level are directly associated with the global analysis of a two-phase composite. On the macro-level, the RVE is regarded as just a point with a homogenized constitutive law. The macro SED, P i , is usually defined as the volume average of SED in a RVE, P i , as follows Pi = P i

W

=

1 V

³

W

P i dW

(5.211)

where W represents the domain of the RVE and V is its volume. Similarly, the volume average of SEF Z i and the volume average of free energy density W in a RVE is defined by

Zi = Zi

=

W

1 V

³

W

Z i dW

1 1 P i Z i dW WdW = ³ W 2V ³W V 1 1 = C ij Z i Z j dW = Fij P i P j dW ³ W 2V 2V ³W

(5.212)

W =

(5.213)

where (C)ij = Cij and (F)ij = Fij are, respectively, local stiffness and local compliance coefficients which are different from phase to phase. Moreover, the macroscopic strain energy should satisfy W =

1 Pi Zi 2

(5.214)

The effective properties represented by effective stiffness C ij* or compliancy Fij* of the piezoelectric composites can be defined by the average SED and SEF as P i = C ij* Z j , Z i = Fij* P j

(5.215)

or by the equivalence of the free strain energy 1 1 Pi Zi = 2V 2

³

1 1 C ij Z i Z j = 2 2V

³

W

P i Z i dW

(5.216)

C ij Z i Z j dW

(5.217)

or W

The linearity of the stress-strain relation for elastic bodies leads to

Green’s function and boundary elements of multifield materials

184

C ij =

¶ 2W ¶Z i ¶Z j

(5.218)

Then an explicit form of the effective stiffness components can be evaluated as below. 5.9.2 Homogenization model The homogenization method for a composite with defects has been discussed in [39,40]. For the reader’s convenience we describe the method here briefly. For piezoelectric materials with inclusions or microcavities, the homogenization theory may be applied based on some fundamental results in the theory of two-phase linear piezoelectric media. In the case of two-phase materials, the volume average of SED and SEF tensors is defined by [38] Ȇ = v (1) Ȇ (1) + v ( 2 ) Ȇ ( 2 ) , Z = v (1) Z (1) + v ( 2) Z ( 2) (5.219) where superscripts (1) and (2) denote the matrix and inclusion phases, v (1) and v ( 2 ) their volume fractions. Substituting Eq (5.219) into Eq (5.215) and noting that Ȇ (i ) = C (i ) Z (i ) , we have

C * = C (1) + (C ( 2) - C (1) ) A ( 2 ) v ( 2 ) , F * = F (1) + (F ( 2 ) - F (1) )B ( 2 ) v ( 2 )

(5.220)

in which the symmetric tensors A(2) and B(2) are defined by the linear relations

Z ( 2) = A ( 2) Z 0 ,

Ȇ ( 2) = B ( 2) Ȇ 0

(5.221)

with Z0 and P0 being remote SEF and SED fields applied on the effective medium. The interpretation of Z ( 2 ) in Eq (5.221) follows from the average strain theorem

Z ij( 2 ) =

1 2W 2

³

¶W 2

{[1 + H (i - 3)]U i n j + U j ni )dW

(5.222)

where W2 and ˜W2 are the total volume and boundary of the inclusion or void, H(i) is the Heaviside step function, n={n1,n2,0}T is the normal local to inclusion surface, and {Z 11 , Z 22 ,2 Z 12 , Z 31 , Z 32 } = {e11 , e 22 ,2e12 ,- E1 ,- E 2 },

{U i } = {u1 , u 2 , f} (5.223)

We now consider the case when inclusions become voids which are thought of as being filled with air. This implies that C ( 2 ) ® 0, F ( 2 ) ® ¥. Thus we assume that C(2)=0, where C(2) stands for the stiffness constants of the void-phase. Then Eq (5.220) become

C* = C (1) (I - A ( 2 ) v ( 2) ) , F * = F (1) (I + B ( 0) v ( 2 ) )

(5.224)

where I is the unit tensor and B(0) is defined by

Z ( 2 ) = F (1) B ( 0 ) Ȇ 0

(5.225)

Therefore, the estimation of integral (5.222) and thus A(2) is the key to predicting the effective electroelastic moduli C* . The calculation of integral (5.222) through use of BEM is the subject of the next section.

185

Boundary element method for piezoelectricity

5.9.3 BE equations Noting that Eq (5.222) contains unknown variables on the boundary only, the BE equation (5.116) can be used to evaluate the displacements and electric potential on the boundary ˜W. The two subdomains of the RVE shown in Fig. 5.8 are separated by the interfaces S between fiber and matrix (see Fig. 1). Each subdomain can be separately modeled by Eq (5.116). Global assembly of the BE subdomains is then performed by enforcing continuity of the DEP and SED at the subdomain interface. In a two-phase piezoelectric composite, the BE formulation (5.116) takes the form N N   C( a ) (xi )U ( a ) (xi ) + ¦ H ij( a ) U ( a ) j = ¦ Gij( a ) T( a ) j (5.226) j =1

j =1

where no generalized body force is assumed, the superscript (a) stands for the quantity associated with the ath phase, (a=1 being matrix and a=2 being fiber), and ­1 if ȟ Î W ( a ) ­S + G a = 1 ( a ) ° c (ȟ ) = ®0.5 if ȟ Î S ( a ) ( S ( a ) smooth) S (a ) = ® (5.227) S a = 2 ¯ ° 0 if ȟ Ï W ( a ) È S ( a ) ¯ in which G and S are the boundaries of the RVE and inclusions, respectively (see Fig. 5.8a). When the inclusion in Fig. 5.8a becomes a hole, the boundary integral equation (5.226) still holds true if one takes a=1 only. In this case the interfacial continuity condition is replaced by the hole boundary condition: T j = 0 along the boundary S (Fig. 5.8b).

5.9.4 Algorithms for self-consistent and Mori-Tanaka approaches (a) Self-consistent BEM approach. As stated in [41,42], in the self-consistent method, for each inclusion (or hole), the effect of inclusion (or hole) interaction is taken into account approximately by embedding each inclusion (or hole) in the effective medium whose properties are unknown. In this case, the material constants appearing in the BE formulation (5.226) are unknown. Consequently, a set of initial trial values of the effective properties is needed and an iteration algorithm is required. The algorithm is described in detail here. (a) Assume initial values of material constants C *( 0 ) (b) Solve Eq (5.226) for U (i ) using the values of C *( i -1) , where the subscript “(i)” stands for the variable associated with the ith iterative cycle. (c) Calculate A ((i2)) in Eq (5.221) by way of Eq (5.222) and using the current values of U (i ) , and then determine C *( i ) by way of Eq (5.224). (d) If e (i ) = C*(i ) - C*(i -1) / C*( 0) £ e , where e is a convergent tolerance, terminate

the iteration; otherwise take C *( i ) as the initial value and go to step (b). (b) Mori-Tanaka-BEM approach The key assumption in Mori-Tanaka theory [40] is that the concentration matrix 2) 2) A (MT (here we use A (MT , rather than A(2) to distinguish it from A(2) in Section 5.8.2) is given by the solution for a single inclusion (or void) embedded in an intact solid subject to an applied strain field equal to the as yet unknown average field in the

186

Green’s function and boundary elements of multifield materials

composite, which means that the introduction of inclusions in the composite results in a value of Z ( 2 ) given by 2) Z ( 2 ) = A (DIL Z (1)

(5.228)

2) is the concentration matrix related to the dilute model, which can be where A (DIL calculated by way of Eqs (5.221), (5.222) and (5.226). In this case, the material constants appearing in the BE formulation (5.226) are all known. As such, it is easy to prove that [40,41] 2) 2) 2 ) -1 A (MT = A (DIL (v1I + v 2 A (DIL )

(5.229)

It can be seen from Eq (5.229) that the Mori-Tanaka approach provides explicit expressions for effective constants of a defective piezoelectric solid. Therefore, no iteration is required with the Mori-Tanaka-BEM. 5.10 Numerical assessments To illustrate the application of the element model described above, three examples are presented. The first example deals with a piezoelectric column subjected to tension at the two ends of the column, the second treats an infinite piezoelectric solid with a horizontal finite crack, and the third illustrates the behavior of crack tip fields in a skew-cracked rectangular panel. 5.10.1 A piezoelectric column under uniaxial tension [6] In this example, a piezoelectric prism under simple extension is considered (see Fig. 5.9). The size of the prism is 2a´2a´2b. The corresponding boundary conditions are given by s z = p, s xz = s yz = Dz = 0, for z = ±b , f = 0, for z = 0 s x = s xz = Dx = 0, for x = ± a s y = s yz = Dy = 0, for y = ± a

y D

·

C· 2b

O

··

A

B

x

2a Fig. 5.9 Geometry of the piezoelectric prism

In the calculation, 2a=3m, 2b=10m, and p = 100Nm 2 are assumed and 32 elements

187

Boundary element method for piezoelectricity

are used [6]. The material considered is PZT-4 whose material parameters are c11 = 13.9 ´1010 Nm -2 , c12 = 7.78 ´1010 Nm -2 , c13 = 7.43 ´1010 Nm -2 c33 = 11.5 ´1010 Nm -2 , c44 = 2.56 ´1010 Nm -2 , e15 = 12.7Cm -2 e31 = -5.2Cm -2 , e33 = 15.1Cm -2 , k11 = 730k 0 , k 33 = 635k 0 where k 0 = 8.854 ´ 10 -12 C 2 / Nm 2 . Table 5.1 lists the displacements and electric potential at points A(2,0), B(3,0), C(0,5), and D(0,10) using BEM, and comparison is made with analytical results. It is found that the BEM results are in good agreement with the analytical results even when only six boundary elements are used in the calculation [6]. Table 5.1 u1 , u 2 and f of BEM results and comparison with exact solution [6] Point BEM

Exact [6]

u1 ´ 1010 (m) u 2 ´ 10 9 (m) f(V) u1 ´ 1010 (m) u 2 ´ 10 9 (m) f(V)

A(2,0) B(3,0) -0.9665 -1.4502

C(0,5) 0

D(0,10) 0

0 0 0 0 -0.9672 -1.4508 0 0 0 0

0.4997 0.6879 0 0.5006 0.6888

1.0004 1.3762 0 1.0011 1.3775

5.10.2 A horizontal finite crack in an infinite piezoelectric solid [11] The second example is a finite horizontal crack along the x-direction in an infinite PZT-4 medium under a uniform far-field stress or electric displacement. Pan [11] used 20 discontinuous quadratic elements as described in Section 5.7.2 to discretize the crack surface which has a length of 2a (=1m). Tables 5.2 and 5.3 list the GRCD caused by a far-field stress s yy (=1N/m2) and a far-field electric displacement Dy (=1C/m2), and comparison is made with analytical results. It is obvious that a far-field stress induces a non-zero Df even though the corresponding KIV is zero. Similarly, a far-field electric displacement can induce a non-zero Du y . 5.10.3 A rectangular piezoelectric solid with a central inclined crack [13] The third example is a rectangular piezoelectric solid with a central crack (a =0.1m) inclined q = 45D with respect to the positive x-direction. The ratios of crack length to width and of height to width are a/w = 0.2 and h/w = 2, respectively (see Fig. 5.10). The analysis is carried out for the rectangle loaded by a uniform tension and electric displacement applied in the y-direction. Table 5.4 lists the normalized GSIF for the two loading conditions considered and the results obtained are compared with those given by Pan [11], who used 10 discontinuous quadratic elements on the crack surfaces and 32 quadratic elements on the outside boundaries in his analysis. Tables 5.5 and 5.6 give the corresponding GRCD. In Table 5.4, D* is a nominal electric displacement expressed in the unit of C/m2 with its amplitude equal to that of s yy expressed in N/m2, and s* is a nominal stress expressed in the unit of N/m2 with its amplitude equal to that of Dy expressed in C/m2.

Green’s function and boundary elements of multifield materials

188

Table 5.2 GRCD caused by a far-field s yy (=1N/m2)[11] Du y (10-12 m)

x(m) 0.492 0.425 0.358 0.292 0.225 0.158 0.092 0.025

BEM 0.032 0.094 0.124 0.144 0.158 0.168 0.174 0.177

Df(10-1 V)

analytical 0.032 0.093 0.124 0.144 0.158 0.168 0.174 0.177

BEM 0.040 0.116 0.154 0.179 0.197 0.210 0.217 0.221

analytical 0.040 0.116 0.154 0.179 0.197 0.210 0.217 0.221

Table 5.3 GRCD caused by a far-field Dy (=1C/m2) [11] Du y (10-1 m)

Df(108 V) x(m) 0.492 0.425 0.358 0.292 0.225 0.158 0.092 0.025

BEM 0.161 0.466 0.616 0.717 0.789 0.838 0.868 0.882

analytical 0.160 0.465 0.616 0.717 0.789 0.838 0.868 0.882

BEM 0.040 0.116 0.154 0.179 0.197 0.210 0.217 0.221

analytical 0.040 0.116 0.154 0.179 0.197 0.210 0.217 0.221

Table 5.4 GSIF for cracked rectangle loaded by sy or Dy Loaded by sy Ref[11] Ref[13] Loaded by Dy Ref[11] Ref[13]

K I / s y pa

K II / s y pa

K IV / D* pa

0.5303 0.5292 K I / s* pa

0.5151 0.5163 K II / s* pa

-2.97´10-12 -2.79´10-12 K IV / Dy pa

-1.42´106 -1.44´106

1.69´105 1.64´105

-0.7278 -0.7283

It can be seen from Table 5.4 that an electric load of Dy (=1C/m2) can produce very large mechanical stress intensity factors. Conversely, the electric displacement intensity factor due to mechanical loads is usually negligible. This phenomenon indicates clearly that crack initiation criteria based on a single SIF cannot be simply extended to the piezoelectric case. It is also found from Tables 5.5 and 5.6 that a mechanical load can induce relative crack electric potential and, vice versa, an electric load can give rise to relative crack displacement.

189

Boundary element method for piezoelectricity

Table 5.5 GRCD for cracked rectangle caused by a far-field s yy (=1N/m2) [13] x=y(10-1m)

Du x (10-13 m)

Du y (10-11 m)

0.477 0.424 0.371 0.318 0.265 0.212 0.159 0.106 0.053

0.256 0.279 0.299 0.315 0.328 0.339 0.347 0.352 0.356

-0.190 -0.206 -0.219 -0.230 -0.239 -0.246 -0.251 -0.255 -0.257

Df(10-2 V) 0.232 0.252 0.268 0.281 0.292 0.300 0.307 0.312 0.314

Table 5.6 GRCD for cracked rectangle caused by a far-field Dy (=1C/m2) [13] x=y(10-1m)

Du x (10-5 m)

0.477 0.424 0.371 0.318 0.265 0.212 0.159 0.106 0.053

0.353 0.371 0.385 0.391 0.406 0.410 0.416 0.420 0.431

Du y (10-2 m)

0.232 0.252 0.268 0.281 0.292 0.300 0.307 0.312 0.314

Df(108 V) 0.093 0.100 0.107 0.112 0.116 0.120 0.123 0.125 0.126

y

2a 2h

q

x

2w

s yy or Dy Fig. 5.10 Finite rectangular solid with an inclined crack under uniform tension or electric displacement in the y-direction

Green’s function and boundary elements of multifield materials

190

5.10.4 A magneto-electro-elastic column [21] The example taken from Ding and Jiang [21] is a magneto-electro-elastic column of size a ´ b under uniform axial tension, electric displacement or magnetic induction. The problem is treated as a plane-strain one and three load cases are considered. The material properties used in the numerical calculation are as follows c11 = 16.6 ´1010 Nm -2 , c12 = 7.7 ´1010 Nm -2 , c13 = 7.8 ´1010 Nm -2 c33 = 16.2 ´ 1010 Nm -2 , c44 = 4.3 ´1010 Nm -2 , c66 = 4.45 ´1010 Nm -2 , e15 = 11.6 Cm -2 , e31 = -4.4 Cm -2 , e33 = 18.6 Cm -2 , e15 = 550 N/(A m) , e31 = 580.3 N/(A m) , e33 = 699.7 N/(A m) , k11 = 1.12 ´ 10-8 C/(V m) , k33 = 1.26 ´10-8 C/(V m), a11 = 5.0 ´10-12 N s/(V C) , a33 = 3.0 ´10-12 N s/(V C), m11 = 5.0 ´10-6 N s 2 /C2 , m33 = 10 ´10-6 N s 2 /C2 , and the boundary conditions of this problem can be written as z = ±b / 2 : s zz = P0 , Dz = 0, Bz = 0, (for load case 1) s zz = 0, Dz = D0 , Bz = 0, (for load case 2) s zz = 0, Dz = 0, Bz = B0 , (for load case 3) x = ±a / 2 : s xx = s xz = Dx = Bx = 0 (for all load cases) In the numerical analysis, the solution domain is modeled by 28 linear boundary elements. The values of a, b, P0 , D0 , and B0 are assumed to be: a = 0.6 m, b = 0.02 m, P0 = 10 Pa, D0 = 10-10 C m -2 , B0 = 10-8 N/(A m) The numerical results at the central point (a/2, b/2) obtained from the boundary formulation described in Section 5.2.3 are listed in Table 5.7 and comparison is made with the exact results. Table 5.7 Numerical and analytical results of column variables 12

u ´10 w ´1012 f ´104 y ´105

Load case 1 BEM Exact[21] -9.501 -9.500 0.5680 0.5683 9.495 9.495 2.138 2.139

2 BEM -0.2107 0.0095 -0.6289 0.0257

3 Exact[21] -0.2108 0.0095 -0.6289 0.0257

BEM 0.5077 0.0214 0.2564 -0.7520

Exact[21] 0.5077 0.0214 0.2567 -0.7521

5.10.5 A simply supported beam of magneto-electro-elastic material [21] This example is also taken from [21] in which a simply supported rectangular beam of length L, height h and width b is considered. It is subjected to uniformly distributed loads on the upper and lower surfaces. The problem is treated as a plane-stress one and the boundary conditions are z = h / 2 : s zz = q0 sin(px / L), s xz = Dz = Bz = 0,

191

Boundary element method for piezoelectricity

z = - h / 2 : s zz = s xz = Dz = Bz = 0, x = 0, L : s xx = w = f = y = 0. The material properties are the same as those used in Section 5.9.4, and other parameters are assumed to be L=0.2 m, h =0.02 m, b =0.001 m, and q0 = -10 Pa. Eight linear elements are used to model this problem. Table 5.8 lists the boundary element results at four reference points and comparison is made with the analytical results. Coordinates of the four points are, respectively, A(0.1, 0), B(0.125, 0), and C(0.15, 0). Table 5.8 Numerical and analytical results of rectangular beam

w ´10 f ´102 y ´103 s zz

point A BEM Exact[21] -1.977 -2.000 -2.300 -2.314 -1.794 -1.808 -4.9969 -5.0000

BEM -1.796 -2.123 -1.588 -4.6168

Exact[21] -1.847 -2.138 -1.671 -4.6190

BEM -1.378 -1.625 -1.258 -3.5336

Exact[21] -1.414 -1.637 -1.279 -3.5351

Dz ´1011

-1.113

-1.114

-1.032

-1.029

-0.7896

-0.7875

10

-2.254

-2.282

-2.114

-2.109

-1.619

-1.614

2

2.796

2.818

2.583

2.604

1.976

1.993

variables 9

Bz ´10

s xx ´10

B

C

References [1] Lee JS and Jiang, LZ, A boundary integral formulation and 2D fundamental solution for piezoelectric media, Mech Res Comm, 21, 47-54, 1994 [2] Lee JS, Boundary element method for electroelastic interaction in piezoceramics, Eng Analysis Boun Elements, 15, 321-328, 1995 [3] Denda M and Mansukh M, Upper and lower bounds analysis of electric induction intensity factors for multiple piezoelectric cracks by the BEM, Eng Analysis Boun Elements, 29, 533-550, 2005 [4] Sanz JA, Ariza MP and Dominquez J, Three-dimensional BEM for piezoelectric fracture analysis, Eng Analysis Boun Elements, 29, 586-596, 2005 [5] Lu P and Mahrenholtz O, A variational boundary element formulation for piezoelectricity, Mech Res Comm, 21, 605-611, 1994 [6] Ding HJ, Wang GP and Chen WQ, Boundary integral formulation and 2D fundamental solutions for piezoelectric media, Comp Meth Appl Mech Eng, 158, 65-80, 1998 [7] Rajapakse RK, Boundary element methods for piezoelectric solids, Procs of SPIE, Mathematics and Control in Smart Structures, 3039, 418-428, 1997 [8] Xu XL and Rajapakse RKND, Boundary element analysis of piezoelectric solids with defects, Composites Part B: Engineering, 29, 655-669, 1998 [9] Rajapakse RKND and Xu XL, Boundary element modeling of cracks in piezoelectric solids, Eng Analysis Boun Elements, 25, 771-781, 2001 [10] Liu YJ and Fan H, On the conventional boundary integral formulation for piezoelectric solids with defects or of thin shapes, Eng Analysis Boun Elements, 25, 77-91, 2001 [11] Pan E, A BEM analysis of fracture mechanics in 2D anisotropic piezoelectric

192

Green’s function and boundary elements of multifield materials

solids, Eng Analysis Boun Elements, 23, 67-76, 1999 [12] Denda M and Lua J, Development of the boundary element method for 2D piezoelectricity, Composites: Part B, 30, 699-707, 1999 [13] Davi G and Milazzo A, Multidomain boundary integral formulation for piezoelectric materials fracture mechanics, Int J Solids Srtuc, 38, 7065-7078, 2001 [14] Groh U and Kuna M, Efficient boundary element analysis of cracks in 2D piezoelectric structures, Int J Solids Srtuc, 42, 2399-2416, 2005 [15] Zhao MH, Fang PZ and Shen YP, Boundary integral-differential equations and boundary element method for interfacial cracks in three-dimensional piezoelectric media, Eng Analysis Boun Elements, 28, 753-762, 2004 [16] Liew KM and Liang J, Modeling of 3D transversely piezoelectric and elastic bimaterials using the boundary element method, Computa Mech, 29, 151-162, 2002 [17] Ding HJ, Chen B and Liang J, On the Green’s functions for two-phase transversely isotropic piezoelectric media, Int J Solids Srtuc, 34, 3041-3057, 1997 [18] 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 [19] Jiang LZ, Integral representation and Green’s functions for 3D time-dependent thermo-piezoelectricity, Int J Solids Struc, 37, 6155-6171, 2000 [20] Kogl M and Gaul L, A boundary element method for transient piezoelectric analysis, Eng Analysis Boun Elements, 24, 591-598, 2000 [21] Ding HJ and Jiang AM, A boundary integral formulation and solution for 2D problems in magnetoelectroelastic media, Comput Struct, 82, 1599-1607, 2004 [22] Chen T and Lin FZ, Boundary integral formulations for three-dimensional anisotropic piezoelectric solids, Computa Mech, 15, 485-496, 1995 [23] Ding HJ and Jiang J, The fundamental solution for transversely isotropic piezoelectricity and boundary element method, Comput Struct, 71, 447-455, 1999 [24] Ding HJ, Chen WQ and Jiang AM, Green’s functions and boundary element method for transversely isotropic piezoelectric materials, Eng Analysis Boun Elements, 28, 975-987, 2004 [25] Chen MC, Application of finite part integrals to the three-dimensional fracture problems for piezoelectric media, Part I: hypersingular integral equation and theoretical analysis. Int J Frac, 121, 133-148, 2003 [26] Chen MC, Application of finite part integrals to the three-dimensional fracture problems for piezoelectric media, Part II: numerical analysis. Int J Frac, 121, 149-161, 2003 [27] Liew KM and Liang J, Three-dimensional piezoelectric boundary element analysis of transversely isotropic half-place, Computa Mech, 32, 29-39, 2003 [28] Garcia-Sanchez F, Saez A and Dominguez J, Anisotropic and piezoelectric materials fracture analysis by BEM, Comput Struct, 83, 804-820, 2005 [29] Schclar NA, Anisotropic analysis using boundary elements, Computational Mechanics Publications, Southampton, 1994 [30] Tanaka K and Tanaka M, A boundary element formulation in linear piezoelectric problems, J Appl Math Phys (ZAMP), 31, 568-580, 1980 [31] Wang CY and Zhang Ch, 3-D and 2-D dynamic Green’s functions and time domain BIEs for piezoelectric solids, Eng Analysis Boun Elem, 29, 454-465, 2005

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193

[32] Khutoryaansky N and Sosa H, Dynamic representation formulas and fundamental solutions for piezoelectricity, Int J Solids Structures, 32, 3307-3325, 1995 [33] Courant R and Hilbert D, Methods of Mathematical Physics, vol II. New York, Interscience, 1962 [34] Qin TY and Noda NA, Application of hypersingular integral equation method to a three-dimensional crack in piezoelectric materials, JSME International, Series A, 47, 173-180, 2004 [35] Qin TY and Noda NA, Analysis of a three-dimensional crack terminating at an interface using a hypersingular integral equation method, J Appl Mech, 69, 626631, 2002 [36] Wippler K and Kuna M, Crack analyses in three-dimensioanl piezoelectric structures by the BEM, Computational Materials Science (2006, in press) [37] Mi Y and Aliabadi MH, Discontinuous crack-tip elements: Application to 3D boundary element method, Int J Fracture, 67, R67-R71, 1994 [38] Qin QH, Material Properties of Piezoelectric Composites by BEM and Homogenization Method, Composite Structures, 66, 295-299, 2004 [39] Hashin Z, Analysis of composite materials—A survey, J Appl Mech, 50, 481505, 1983 [40] Qin QH, Mai YW and Yu SW, Effective moduli for thermopiezoelectric materials with microcracts, Int J Fracture, 91, 359-371, 1998 [41] Nemat-Nasser S and Hori M. Micromechanics: Overall properties of heterogeneous materials. Amsterdam, North Holland Publ, 1993 [42] Qin QH and Yu SW, Effective moduli of thermopiezoelectric material with microcavities, Int J Solids Struct, 35, 5085-5095, 1998

194

Chapter 6 Boundary element method for discontinuity problems 6.1 Introduction In the previous chapter we described the BE formulations derived based on reciprocity theorem. These formulations are useful for electroelastic problems in which no discontinuity of load is involved. This chapter deals with applications of BEM to thermoelectroelastic problems with discontinuous loadings, especially discontinuity of thermal loading. Although the BEM has been widely used for modeling various engineering problems, analysis of discontinuity problems with this approach has been very limited [1-4]. Qin [1] and Qin and Lu [2] proposed a BE formulation for fracture analysis of thermo-piezoelectricity with a half-plane boundary and an inclusion, respectively, based on the dislocation method and the potential variational principle. Later Qin and Mai [3] and Qin [4] extended this procedure to the discontinuity problem of thermopiezoelectricity with holes and thermomagnetoelectroelastic problems. The developments in [1-4] are briefly described in this chapter. 6.2 BEM for thermopiezoelectric problems Consider a two-dimensional thermoelectroelastic solid inside of which there exists a number of defects such as cracks and holes. The numerical approach to such a problem will involve two major steps: (i) first solve a heat transfer problem to obtain the steady-state temperature field; (ii) calculate the stress and electric displacement (SED) caused by the temperature field, then calculate an isothermal solution to satisfy the corresponding mechanical and electric boundary conditions, and finally, solve the modified problem for elastic displacement and electric potential (EDEP) and SED fields. In what follows, we begin by deriving a variational principle for temperature discontinuity and then extend it to the case of thermoelectroelastic problems. 6.2.1 BEM for temperature discontinuity problems Let us consider a finite region W1 bounded by G, as shown in Fig. 6.1(a). The heat transfer problem to be considered is stated as in W1 (6.1) kij T,ij = 0

hn = hi ni = hn

on Gh

(6.2)

T =T

on GT

(6.3)

on L

(6.4)

hi ni = 0

where ni is the normal to the boundary G, hn and T are the prescribed values of heat flow and temperature, which act on the boundaries Gh and GT , respectively. For simplicity, we define Tˆ = T + - T - on L (=L++L-), where T is the temperature L

L

discontinuity, L is the union of all cracks, and L+ and L- are defined in Fig. 6.1b. Further, if we let W2 be the complementary region of W1 (i.e., the union of W1 and W2 forms the infinite region W) and Tˆ = T G+ - T G- = T , the problem shown in Fig. 6.1a can be extended to the infinite case (see Fig. 6.1b). Here G = G + + G - , in which G + and G - stand for the boundaries of W1 and W2, respectively (see Fig. 6.1b).

195

Boundary element method for discontinuity problems W2

L+

L

-

G

G+

G-

L

W1

W1

(a)

W2

(b)

Fig. 6.1 Configuration of piezoelectric plate for BEM analysis 6.2.1.1 Potential variational principle In a manner similar to that of Yin and Ehrlacher [5], the total generalized potential energy for the thermal problem defined above can be given as 1 ˆ P (T , Tˆ ) = ³ kijT,iT, j d W + ³ hnTdL G 2 W

(6.5)

By transforming the region integral in Eq (6.5) into a boundary integral, we have

1 ˆ P (T , Tˆ ) = - ³ J(T )Tˆ, s ds + ³ hnTds G 2 L

(6.6)

in which the relations hi = -kijT, j ,

³

L

ˆ = [(JTˆ ) - JTˆ ]ds hnTds ,s ,s ³ L

(6.7)

have been used and the temperature discontinuity is assumed to be continuous over L and zero at the ends of L. Moreover, temperature T in Eq (6.6) can be expressed in terms of Tˆ through use of Green’s function presented in Chapter 3. Thus, the potential energy can be further written as 1 ˆ P (Tˆ ) = - ³ J(Tˆ )Tˆ, s ds + ³ hnTds G 2 L

(6.8)

6.2.1.2 Boundary element formulation Analytical results for the minimum potential (6.8) are not, in general, possible, and therefore a numerical procedure must be used to solve the problem. As in conventional BEM, the boundaries G and L are divided into M G and M L linear boundary elements, for which the temperature discontinuity may be approximated by the sum of elemental temperature discontinuities: M

Tˆ ( s ) = ¦ Tˆm Fm ( s )

(6.9)

m =1

where Tˆm is the temperature discontinuity at node m, M= M G + M L +N, and N is the number of cracks. It should be mentioned that the appearance of N is due to the number of nodes being one more than the number of elements in each crack. s is a length coordinate (s>0 in the element located at the right of the node m, s<0 in the element located at the left of the node), Fm(s) is a global shape function associated with the node m. Fm(s) is zero-valued over the whole mesh except within two

Green’s function and boundary elements of multifield materials

196

elements connected to the node m (see Fig. 6.2). Since Fm(s) is assumed to be linear within each element, it has three possible forms: Fm ( s ) = (lm -1 + s ) / lm -1

(6.10)

for a node located at the left end of a line (see Fig. 6.2a), or Fm ( s ) = (lm - s ) / lm

(6.11)

for a node located at the right end of a line (see Fig. 6.2b), or

­(l + s) / lm -1 , if s Î element lm -1 , Fm ( s) = ® m -1 if s Î element lm , ¯ (lm - s) / lm ,

(6.12)

where lm and lm-1 are the lengths of the two elements connected to the mth node, lm being to the right and lm -1 being to the left (see Fig. 6.2c). While

­ 0, at node m, ° s = ® lm , at node m + 1, °-l , at node m - 1 ¯ m -1

Fm (s )

Fm (s ) 1

s

m - 1( s =- l m-1 )

(6.13)

1

m(s=0)

m(s=0)

m + 1( s = l m )

s (b)

(a) Fm (s )

Fm (s ) 1

m - 1( s = - l m-1 )

m(s=0) s

m + 1( s = l m )

(c) Fig. 6.2 Definition of Fm ( s )

With the use of Green’s functions presented in Chapter 3 and the approximation (6.9), the temperature and heat-flow function at point zt (or zt (zt) can be given as M

T (z t ) = ¦ Im[am (z t )]Tˆm

(6.14)

m =1

M

J(z t ) = - k ¦ Re[am (z t )]Tˆm

(6.15)

m =1

where am (z t ) has a different form for different problems. For example, we have for the problems of

197

Boundary element method for discontinuity problems

(i) a bimaterial solid (see Fig. 2.5 and formulation in Section 3.4.1): l +s 1 m -1) m -1) {ln( zt(1) - zt(1)( ) + b1 ln( zt(1) - zt(1)( )} m -1 am ( zt(1) ) = ds 0 0 ³ l 2p m-1 lm -1 l -s 1 m) m) {ln( zt(1) - zt(1)( ) + b1 ln( zt(1) - zt(1)( )} m + ds; 0 0 ³ l m lm 2p

(6.16)

(ii) a plate containing an elliptic hole (see Fig. 2.7 and formulation in Section 3.5): am (z t ) =

l +s 1 ln(z t - z t( 0m -1) ) - ln(z t-1 - zt(0m -1) ) m -1 ds ³ l 2p m-1 lm -1

{

}

l -s 1 ln(z t - z t( 0m ) ) - ln(z t-1 - zt(0m ) ) m + ds ³ l m 2p lm

{

(6.17)

}

where

z t = f ( zt ) =

zt + zt2 - a 2 - p1*2b 2 *2 1

a - ip b

, z t( 0m -1) = f ( zt(0m -1) ), z t( 0m ) = f ( zt(0m ) ) ; (6.18)

(iii) a half-plane plate (see Fig. 2.4 and formulation in Section 3.3): l +s 1 {ln( zt - zt(0m -1) ) - ln( zt - zt(0m -1) )} m -1 ds ³ l m 1 2p lm -1

am ( zt ) = +

l -s 1 {ln( zt - zt(0m ) ) - ln( zt - zt(0m ) )} m ds; 2p ³ lm lm

(6.19)

(iv) a plate containing a hole of various shapes (see Fig. 2.9 and formulation in Section 3.7):

am (z t ) =

l +s 1 {ln(z t - z t( 0m -1) ) - ln(z t-1 - zt(0m -1) )} m -1 ds ³ l 2p m-1 lm -1

l -s 1 {ln(z t - z t( 0m ) ) - ln(z t-1 - zt(0m ) )} m + ds ³ l m 2p lm where z t and zt are related by Eq (3.78);

(6.20)

(v) a plate with a wedge boundary (see Fig. 4.5 and formulation in Section 4.8.5) am ( zt ) =

l +s 1 {ln( ztl - zt(0m -1) l ) - ln(- ztl - zt(0m -1) l )} m -1 ds 2p ³ lm-1 lm -1 l -s 1 {ln( ztl - zt(0m ) l ) - ln( - ztl - zt(0m ) l )} m + ds ³ l 2p m lm

(6.21)

where l is defined in Eq (4.170). In the formulations listed above zt(0m ) is defined as zt(0m -1) = dtm + s (cos a m -1 + p1* sin a m -1 ), zt(0m ) = dtm + s (cos a m + p1* sin a m ) (6.22) and dtm = x1m + p1* x2 m , ( x1m , x2 m ) represent the coordinates at node m, a m-1 is the angle between the element in the left of node m and the x1 -axis, with a m defined

Green’s function and boundary elements of multifield materials

198

similarly. It should be pointed out that the superscript “(1)” for variables z t and zt has been dropped in Eq (6.17) in order to simplify the writing. Hereafter, we again omit the superscript “(1)” when the distinction is unnecessary. In particular the temperature at node j can be written as M

T j [ f (d j )] = ¦ Im{am [ f (d j )]}Tˆm

(6.23)

m =1

where d j = x1 j + p1* x2 j , ( x1 j , x2 j ) are the coordinates at node j. For the total potential energy (6.8), substituting Eqs (6.9) and (6.15) into it yields M ªM § 1 º · P (Tˆ ) » ¦ « ¦ ¨ - K mjTˆmTˆj ¸ + G jTˆj » 2 ¹ j =1 ¬ m =1 © ¼

(6.24)

where Kmj is the so-called stiffness matrix and Gj the equivalent nodal heat flow vector, with the form K mj = -

k l j -1

³

l j -1

Re[am ( x ( j -1) )]ds + Gj = ³

l j -1 + l j

k lj

³

lj

Re[am ( x ( j ) )]ds ,

h*0 F j ( s )ds ,

(6.25) (6.26)

where h*0 = h0 when s Î G h , and h*0 = 0 for other cases, x ( k ) = zt(0k ) for the cases (i), (iii) and (v), and x ( k ) = z t( 0k ) for the remaining two cases. The minimization of P(Tˆ ) yields M

¦

K mjTˆj = Gm

(6.27)

j =1

The final form of linear equation to be solved is obtained by selecting the appropriate equation from Eqs (6.23) and (6.27). Equation (6.23) will be chosen for those nodes at which the temperature is prescribed, and Eq (6.27) will be chosen for the remaining nodes. After the nodal temperature discontinuities have been calculated, the displacement and stress at any point in the region can be evaluated through use of Eqs (3.1) and (3.2). They are M

M

M

j =1

j =1

j =1

U = ¦ u jTˆj , P 1 = ¦ x jTˆj , P2 = ¦ y jTˆj

(6.28)

where uj, xj and yj have different forms for different problems. They are (i) bimaterial problems uj =

l j -1 + s 1 Im ³ {A (1) f ( za(1) ) q1* + c(1) [ f ( y1(1) ) + b1 f ( y2(1) )]} ds l j -1 2p l j -1 lj - s 1 Im ³ {A (1) f ( za(1) ) q1 + c(1) [ f ( y1(1) ) + b1 f ( y2(1) )]} + ds, lj 2p lj

(6.29)

199

Boundary element method for discontinuity problems

xj = -

-

l +s 1 ds Im ³ {B (1) pa(1) f ¢( za(1) ) q1* + d (1) p1*(1) [ f ¢( y1(1) ) + b1 f ¢( y2(1) )]} j -1 l j -1 l j -1 2p

lj - s 1 ds, Im ³ {B (1) pa(1) f ¢( za(1) ) q1* + d(1) p1*(1) [ f ¢( y1(1) ) + b1 f ¢( y2(1) )]} l j lj 2p (6.30)

yj =

l +s 1 Im ³ {B (1) f ¢( za(1) ) q1* + d (1) [ f ¢( y1(1) ) + b1 f ¢( y2(1) )]} j -1 ds l j -1 2p l j -1 +

lj - s (1) 1 Im ³ {B f ¢( za(1) ) q1* + d (1) [q0 f ¢( y1(1) ) + b1 f ¢( y2(1) )]} ds lj 2p lj

(6.31)

where q1* = q1 / q0 , q1 being defined in Eq (3.49), and all other functions and constants used are defined in Section 3.4. (ii) a plate containing an elliptic hole and multiple cracks uj =

1 ª 4 ºl +s Im ³ « ¦ A f k (z a ) B -1dqk + cg* (z t ) » j -1 ds l j -1 2p ¬ k =1 ¼ l j -1 +

xj = -

1 ª 5 º lj - s Im ³ « ¦ A f k (z a ) B -1dqk + cg* (z t ) » ds, lj 2p ¬ k =1 ¼ lj

¶z ¶z º l j -1 + s 1 ª 4 Im ³ « ¦ B pa f k¢ (z a ) a B -1dqk + cp1* g*¢ (z t ) t » ds l j -1 2p ¶za ¶zt ¼ l j -1 ¬ k =1

¶z ¶z º l - s 1 ª 4 Im{³ « ¦ B pa f k¢ (z a ) a B -1dqk + cp1* g*¢ (z t ) t » j ds, lj 2p ¶za ¶zt ¼ l j ¬ k =1 yj =

¶z ¶z º l j -1 + s 1 ª 4 Im ³ « ¦ B f k¢ (z a ) a B -1dqk + cg*¢ (z t ) t » ds l j 1 2p ¶za ¶zt ¼ l j -1 ¬ k =1 ¶z ¶z º l - s 1 ª 4 Im{³ « ¦ B f k¢ (z a ) a B -1dqk + cg*¢ (z t ) t » j ds, + l j 2p ¶za ¶zt ¼ l j ¬ k =1

where fk is difined in Eq (3.63), and q1 = -q4 = -ct q2 = - q3 = - d t g* (z t ) = ct F1 (z t ) + d t F2 (z t ) - ct F4 (z t ) - d t F3 (z t ) .

(6.32)

(6.33)

(6.34)

(6.35) (6.36)

with Fi being defined in Eq (3.61); (iii) half-plane problems with multiple cracks uj =

l j -1 + s 1 Im ³ [ A f* ( za ) (-B -1d) + cf* ( zt )] ds l j -1 2p l j -1 lj - s 1 Im ³ [ A f* ( za ) (-B -1d) + cf* ( zt )] + ds, l j 2p lj

(6.37)

Green’s function and boundary elements of multifield materials

200

xj =

l +s 1 Im ³ [B pa f*¢( za ) (B -1d) - dp1* f*¢( zt )] j -1 ds l j -1 2p l j -1 lj - s 1 Im ³ [B pa f*¢( za ) (B -1d) - dp1* f*¢( zt )] ds, + l j 2p lj

yj = -

l +s 1 Im ³ [B f*¢( za ) (B -1d) - df*¢( zt )] j -1 ds l j -1 2p l j -1

lj - s 1 Im ³ [B f*¢( za ) (B -1d) - df*¢( zt )] ds l j 2p lj

(6.38)

(6.39)

where f* ( z x ) = ( z x - zt 0 )[ln( z x - zt 0 ) - 1] - ( z x - zt 0 )[ln( z x - zt 0 ) - 1]

(6.40)

(iv) a plate containing multiple cracks and a hole of various shapes (see Fig. 2.9 and formulation in Section 3.7) uj = -

l +s 1 Im ³ [A( f1* (z a + p1* pa f 2* (z a )B -1d - cg* ( zt )] j -1 ds l j -1 2p l j -1

lj - s 1 Im ³ [ A( f1* (z a + p1* pa f 2* (z a )B -1d - cg* ( zt )] ds, l j 2p lj xj =

l j -1 + s ¶z 1 Im ³ [B { f1*¢ (z a ) + p1* pa f 2*¢ (z a )} pa a B -1d - cp1* g*¢ ( zt )] ds l j -1 2p l j -1 ¶za l -s ¶z 1 Im ³ [B { f1*¢ (z a ) + p1* pa f 2*¢ (z a )} pa a B -1d - cp1* g*¢ ( zt )] j ds, + l j 2p lj ¶za

yj = -

l j -1 + s ¶z 1 Im ³ [B { f1*¢ (z a ) + p1* pa f 2*¢ (z a )} a B -1d - cg*¢ ( zt )] ds l j 1 2p l j -1 ¶za

l -s ¶z 1 Im ³ [B { f1*¢ (z a ) + p1* pa f 2*¢ (z a )} a B -1d - cg*¢ ( zt )] j ds, l j 2p lj ¶za

(6.41)

(6.42)

(6.43)

where

f1* (z a ) = a[ F1 (z a , z t 0 ) + F2 (z a , z t 0 ) - F1 (z a-1 , zt 0 ) - F2 (z a-1 , zt 0 )] / 2 +e j1a g[ F3 (z a , z t 0 ) + F4 (z a , z t 0 ) - F3 (z a-1 , zt 0 ) - F4 (z a-1 , zt 0 )] / 2, f 2* (z a ) = ipa ae[- F1 (z a , z t 0 ) + F2 (z a , z t 0 ) - F1 (z a-1 , zt* ) + F2 (z a-1 , zt* )] / 2 + ie j1 pa aeg[ F3 (z a , z t 0 ) - F4 (z a , z t 0 ) + F3 (z a-1 , zt 0 ) - F4 (z a-1 , zt 0 )] / 2 g* ( zt ) = aa1t [ F1 (z t , z t 0 ) - F2 (z t-1 , zt 0 )] + aa2 t [ F2 (z t , z t 0 ) - F1 (z t-1 , zt 0 )] + ae j1{a3t [ F3 (z t , z t 0 ) - F4 (z t-1 , zt 0 )] + aa4 t [ F4 (z t , z t 0 ) - F3 (z t-1 , zt 0 )]} (v) a plate containing multiple cracks and a wedge boundary (see Fig. 4.5 and formulation in Section 4.8.5)

201

Boundary element method for discontinuity problems

uj =

l +s 1 ds} Im{³ [- A( f1 ( za ) - f 2 ( za ) )B -1d + cg* ( zt )] j -1 l j -1 l j -1 2p

lj - s 1 Im{³ [- A( f1 ( za ) - f 2 ( za ) )B -1d + cg* ( zt )] + ds} l j 2p lj xj =

+

l +s 1 ds} Im{³ [B( pa f1¢( za ) - pa f 2¢( za ) )B -1 - p1* g*¢ (z t )]d j -1 l j -1 l j -1 2p lj - s 1 Im{³ [B( pa f1¢( za ) - pa f 2¢( za ) )B -1 - p1* g*¢ (z t )]d ds} lj lj 2p

yj =

l +s 1 ds} Im{³ [-B( f1¢( za ) - f 2¢( za ) )B -1 + g*¢ (z t )]d j -1 l j -1 l j -1 2p

lj - s 1 Im{³ [-B( f1¢( za ) - f 2¢( za ) )B -1 + g*¢ (z t )]d ds} + l j lj 2p

(6.44)

(6.45)

(6.46)

where f1 and f2 are defined in Eq (4.185), and g* ( zt ) = fˆ1 ( zt ) - fˆ2 ( zt ) . Thus, the SED and EDEP on a boundary induced by temperature discontinuity are of the form M

M

t qn ( s ) = P i ni = ¦ (x j n1 + y j n2 )Tˆj ,

UTq ( s ) = ¦ u jTˆj .

j =1

(6.47)

j =1

In general, t qn ( s ) ¹ 0 and UTq ( s) ¹ 0. To eliminate these quantities on the corresponding boundaries, we must superimpose a solution of the corresponding isothermal problem with a SED (or an EDEP) equal and opposite to those of Eq (6.47). The details are given in the following sub-section. 6.2.2 BEM for displacement and potential discontinuity problems Consider again the domain W1, in which the governing equation and its boundary conditions are described as follows:

tni

L+

= -tni

L-

P ij , j = 0 ,

in W1,

(6.48)

tni = P ij n j = ti - (t qn )i ,

on Gt,

(6.49)

ui = ui - (UTq )i ,

on Gu,

(6.50)

= -(t qn )i ,

bi = ui

L+

- ui

L-

- (UTq )i

L+

+ (UTq )i

L-

, on L

(6.51)

where Gt and Gu are the boundaries on which the prescribed values of stress ti and displacement ui are imposed. Similarly, the related potential energy for the elastic problem can be given as 1 P (b) = ³ [j(b) × b, s + 2 tn × b]ds - ³ ( tn - t qn ) × bds (6.52) G 2 L

Green’s function and boundary elements of multifield materials

202

where the electroelastic solutions of functions j(b) and U (b) appearing later have been discussed in Chapter 2 and are listed for the readers’ convenience: (i) bimaterial problems For a bimaterial plate subjected to a line dislocation b located in the upper half-plane at z0(x10, x20), the solution is given by U (1) =

4 1 1 Im{A (1) ln( za(1) - za(1)0 ) BT }b + ¦ Im{A (1) ln( za(1) - zb(1)0 ) qb(1) } , (6.53) p b=1 p

j (1) =

4 1 1 Im{B (1) ln( za(1) - za(1)0 ) BT }b + ¦ Im{B(1) ln( za(1) - zb(1)0 ) qb(1) } (6.54) p b=1 p

for material 1 in x2>0 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).

Green’s function and boundary elements of multifield materials

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

Boundary element method for discontinuity problems

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)

Green’s function and boundary elements of multifield materials

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

Boundary element method for discontinuity problems

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

Green’s function and boundary elements of multifield materials

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)

Boundary element method for discontinuity problems

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

Green’s function and boundary elements of multifield materials

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

Boundary element method for discontinuity problems

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

Green’s function and boundary elements of multifield materials

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

Boundary element method for discontinuity problems

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

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Green’s function and boundary elements of multifield materials

[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)

Green’s function and boundary elements of multifield materials

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)

Green’s function and boundary elements of multifield materials

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 Ÿ 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)

Green’s function and boundary elements of multifield materials

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

Trefftz boundary element method

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)

Green’s function and boundary elements of multifield materials

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

Green’s function and boundary elements of multifield materials

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

Trefftz boundary element method

­ 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)

Green’s function and boundary elements of multifield materials

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)

Green’s function and boundary elements of multifield materials

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

Trefftz boundary element method

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

Green’s function and boundary elements of multifield materials

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

Trefftz boundary element method

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)

Green’s function and boundary elements of multifield materials

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  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

Trefftz boundary element method

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.

Green’s function and boundary elements of multifield materials

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

Trefftz boundary element method

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)

Green’s function and boundary elements of multifield materials

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

Trefftz boundary element method

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

Green’s function and boundary elements of multifield materials

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

<

Center

a

b

Fig. 7.6 Rectangular and circular virtual boundaries of a simple rectangular domain The distance between fictitious source points and the physical boundary also has an important effect on the accuracy of the modified Trefftz method. To determine the location of the virtual boundary, a parameter l representing the ratio between the characteristic lengths of the virtual and physical boundaries is introduced [16,17], defined by

l =

characteristic length of virtual boundary characteristic length of physical boundary

(7.193)

From the computational viewpoint, the accuracy of the numerical results will deteriorate if the distance between the virtual and physical boundaries is too small (i.e., the similarity ratio l is close to one), as that may cause problems due to singularity of the fundamental solutions. Conversely, round-off error in C/Fortran floating point arithmetic may be a serious problem when the source points are far from the real boundary. In that case, the coefficient matrix of the system of equations is nearly zero [18]. The similarity ratio l is generally selected to be in the range of 1.8-4.0 for internal problems and 0.6-0.8 for external problems in practical computation [16,17]. The condition number of the solution matrix may be influenced by the location and shape of the virtual boundary as well as by the number of fictitious source points. In general, th coefficient matrix formed by using the virtual boundary collocation method has a large condition number. In this case, the standard Gauss elimination or LU decomposition (i.e. a procedure for decomposing a matrix A into a product of a lower triangular matrix L and an upper triangular matrix U) solutions may be invalid and the singular value decomposition is recommended to keep results stable for illconditioned systems [19].

References [1] Qin QH, Mode III fracture analysis of piezoelectric materials by Trefftz BEM, Struc Eng Mech, 20, 225-239, 2005

240

Green’s function and boundary elements of multifield materials

[2] Jin WG, Sheng N, Sze KY, and Li J, Trefftz indirect methods for plane piezoelectricity, Int J Numer Meth Eng, 63, 2113-2138, 2006 [3] Sheng N, Sze KY and Cheung YK, Trefftz solutions for piezoelectricity by Lekhnitskii’s formalism and boundary-collocation method, Int J Numer Meth Eng, 65, 139-158, 2005 [4] Sheng N and Sze KY, Multi-region Trefftz boundary element method for fracture analysis in plane piezoelectricity, Computational Mech, 37, 381-393, 2006 [5] Cheung YK, Jin WG and Zienkiewicz OC, Direct solution procedure for solution of harmonic problems using complete, non-singular, Trefftz functions, Commun Appl Numer Meth, 5, 159-169, 1989 [6] Jin WG, Cheung YK and Zienkiewicz OC, Application of the Trefftz method in plane elasticity problems, Int J Numer Meth Eng, 30, 1147-1161, 1990 [7] Qin QH, The Trefftz Finite and Boundary Element Method, WIT Press, Southampton, UK, 2000 [8] Oliveira ERA, Plane stress analysis by a general integral method, J Eng Mech Div, Proc ASCE, 94, 79-101, 1968 [9] Patterson C and Sheikh MA, On the use of fundamental solutions in Trefftz method for potential and elasticity problems, in: Boundary Element Methods in Engineering, Proc. 4th Int. Conf. on BEM, ed. C.A. Brebbia, Springer, pp. 43-57, 1982 [10] Patterson C and Sheikh MA, A modified Trefftz method for three dimensional elasticity, in: Boundary Elements, Proc. 5th Int. Conf. on BEM, eds. C.A. Brebbia, T. Futagami & M. Tanaka, Compu Mech Pub/Springer, pp. 427-437, 1983 [11] Yao WA and Wang H, Virtual boundary element integral method for 2-D piezoelectric media, Finite Elements in Analysis and Design, 41, 875-891, 2005 [12] Li XC and Yao WA, Virtual boundary element integral collocation method for the plane magnetoelectroelastic solids, Eng Analysis Bound Elements, 30, 709717, 2006 [13] Wang XW, Zhou Y and Zhou WL, A novel hybrid finite element with a hole for analysis of plane piezoelectric medium with defects, Int J Solids Struct, 41, 71117128, 2004 [14] Wang H and Qin QH, A meshless method for generalized linear or nonlinear Poisson-type problems, Eng Analy Boun Elements, 30, 515-521, 2006 [15] Bogomolny A, Fundamental solutions method for elliptic boundary value problems. SIAM J Num Aanl, 22, 644-669, 1985 [16] Wang H, Qin QH and Kang YL, A new meshless method for steady-state heat conduction problems in anisotropic and inhomogeneous media, Arch Appl Mech. 74, 563-579, 2005 [17] Sun HC and Yao WA, Virtual boundary element-linear complementary equations for solving the elastic obstacle problems of thin plate, Finite Elements in Analysis and Design, 27, 153-161, 1997 [18] Mitic P and Rashed YF, Convergence and stability of the method of meshless fundamental solutions using an array of randomly distributed sources, Eng Analy Boun Elements, 28, 143-153, 2004 [19] Ramachandran PA, Method of fundamental solutions: singular value decomposition analysis, Commun. Numer. Meth. Engng, 18, 789–801, 2002

241

Appendix A: Radon Transform In this Appendix, the Radon transform of f on Euclidean space is discussed. The discussion includes 2D and 3D Radon transform, their inverse representation, and some important properties of Radon transform. This appendix follows the relevant results presented in [1,2]. A.1 2D Radon transform Let x=(x,y) and n=(cosy,siny) be vectors in a 2D space (see Fig. A1), and consider an arbitrary function f(x) defined on some domain W (the domain W may include the entire plane or some region of the plane, as shown in Fig. A1). If L is any line in the plane defined by

w = x × n = x cos y + y sin y ,

(A1)

then the mapping defined by the projection or line integral of f along all possible line lines L is the 2D Radon transform of f provided the integral exists. Based on the statement above, the 2D Radon transform may be written as an integral over W by allowing the Dirac delta function to select the line w = x × n from W[2]: fˆ (w, n) = R[ f (x)] =

f (x)d(w - x × n)dx

³

(A2)

w= x×n

Keep in mind that the unit vector n defines the direction in terms of the angle y, as shown in Fig. A1. y

· (x,y) n

y w

W

x L

Fig. A1 The line L and coordinates w and y The inverse Radon transform is an integration in the n-space over the surface W containing the origin, which can be given by[1,2]

-1 f (x) = R*[ fˆ (w, n)] = 2 4p Example: Let f ( x, y ) = e - x

2

- y2

³

2p 0

dy ³

¶fˆ (w, n) / ¶w dw -¥ w-n×x ¥

(A3)

. The Radon transform is then

¥ fˆ (w, n) = R[ f (x)] = ³

-¥

³

¥ -¥

e- x

2

- y2

d ( w - n × x ) dx

(A4)

Green’s function and boundary elements of multifield materials

242

Using the orthogonal linear transformation[1] ­u ½ ª n1 ® ¾=« ¯ v ¿ ¬ - n2

n2 º ­ x ½ ® ¾ n1 ¼» ¯ y ¿

(A5)

We have from Eq (A4) ¥ fˆ (w, n) = ³

-¥

³

¥ -¥

e-u

2

-v2

2

d(w - u )dudv = e -w

³

¥ -¥

2

2

e - v dv = pe -w

(A6)

A.2 3D extension The generalization of Eq (A2) to 3D is accomplished by letting x be a vector in 3D, x=(x,y,z) and dx=dxdydz. The vector n is still a vector, but in 3D (Fig. A2). The line L is now replaced by a plane, as shown in Fig. A2. The plane is defined by w = n × x = n1 x + n2 y + n3 z

(A7)

The corresponding Radon transform is exactly in the same form as that of Eq (A2), but the integrations are over planes rather than lines. Geometrically, as indicated in Fig. A2, w is the distance from the origin to the plane and n is a unit vector along w that defines the orientation of the plane.

y

w n x

z

Fig. A2 Grometry for the Radon transform in 3D Inverse of 3D Radon transform is defined by[1] -1 ¶ 2 fˆ (w, n) f (x) = R*[ fˆ (w, n)] = 2 ³ dn 8p n =1 ¶w2

(A8)

A.3 Basic properties of Radon transform[1,2] A brief discussion of some basic properties treated here follows the results presented in [1]. From the definition (A2), we have following properties of Radon transform: (i) Homogeneity

fˆ ( sw, sn) =

³

w= x×n

(ii)Linearity

f (x)d( sw - x × sn)dx = s

-1

fˆ (w, n)

(A9)

243

Appendix A

R[cf + dg ] = cfˆ + dgˆ

(A10)

R[ f (x + a)] = fˆ (w + n × a, n)

(A11)

(iii) Shifting property

(iv)Transform of a linear transformation R[ f ( A -1x)] = det A fˆ (w, AT n)

(A12)

where det A ¹ 0 , and ‘det’ represents the determinant. (v)Transform of derivatives ª ¶ 2 f ( x) º ª ¶f (x) º ¶fˆ (w, n) ¶ 2 fˆ (w, n) R« , R« » = ni n j » = ni ¶w ¶w2 ¬ ¶xi ¼ ¬« ¶xi ¶x j ¼» (vi) Derivatives of the transform ¶ ¶ ¶2 ¶2 R [ f ( x) ] = R [ xi f (x) ] , R [ f (x) ] = 2 R ª¬ xi x j f (x) º¼ ¶ni ¶w ¶ni ¶n j ¶w

(A13)

(A14)

(vii) Transform of convolution Let f be the convolution of g and h,

f (x) = g * h = ³ g (y )h(x - y )dy ,

(A15)

R[ f (x)] = R[ g * h] = gˆ * hˆ

(A16)

then

References

[1] Dears SR, The Radon transform and some of its applications, New York: WileyInterscience Publication, 1983 [2] Pan E and Tonon F, Three-dimensional Green’s functions in anisotropic piezoelectric solids, Int J Solids Struct, 37, 943-958, 2000

244

Appendix B: The constants a j , s j , and bmj appeared in Section 4.3

(i) The constants a j is given by[1,2] a j = - n1 + n2 s 2j - n3 s 4j ,

(B1)

where[2] 2 n1 = (c13 + c44 )( k11m11 - a11 ) + (e15 + e31 )(e15m11 - a11e15 ) - (e15 + e31 )(e15a11 - k11e15 ),

n2 = (c13 + c44 )( k11m33 + k33m11 - 2a11a 33 ) + (e15 + e31 )(e15m33 + e33m11 - a11e33 - a 33e15 ) - (e15 + e31 )(e15a 33 + e33a11 - k11e33 - k33e15 ), 2 n3 = (c13 + c44 )( k33m33 - a33 ) + (e15 + e31 )(e33m33 - a 33e33 ) - (e15 + e31 )(e33a 33 - k33e33 ), (B2)

and s0 = (c66 / c44 )1/ 2 , s j ( j = 1 - 4) are four characteristic roots of the following equation a1s8 - a2 s 6 + a3 s 4 - a4 s 2 + a5 = 0 with 2 2 a1 = c44 [c33 ( k 33m33 - a33 ) - 2e33a33e33 + m 33e33 + k 33e332 ], 2 a2 = c11a1 / c44 + c44 [c44 ( k 33m 33 - a33 ) + c33 ( k11m33 + k 33m11 - 2a11a33 ) 2 - 2e15a33e33 - 2e33 ( a11e33 + a33e15 ) + m11e33 + 2m33e15e33 + k11e332 + 2k 33e15e33 ] 2 - ( c13 + c44 )[(c13 + c44 )( k 33m 33 - a33 ) + ( e15 + e31 )( e33m33 - a33e33 )

- ( e15 + e31 )( e33a33 - k 33e33 )] - ( e15 + e31 )[( c13 + c44 )( e33m 33 - a33e33 ) - ( e15 + e31 )( c33m 33 + e332 ) + ( e15 + e31 )( c33a33 + e33e33 )] - ( e15 + e31 )[( c13 + c44 )( k 33e33 - e33a33 ) + ( e15 + e31 )( c33a33 + e33e33 ) 2 - ( e15 + e31 )( c33k 33 + e33 )], 2 a3 = c11[c44 ( k33m33 - a33 ) + c33 ( k11m33 + k33m11 - 2a11a33 ) - 2e15a 33e33 2 - 2e33 (a11e33 + a33e15 ) + m11e33 + 2m33e15e33 + k11e332 + 2 k33e15e33 ] 2 + c44 [c44 ( k11m33 + k33m11 - 2a11a 33 ) + c33 ( k11m11 - a11 ) - 2e15 (a11e33 + a33e15 )

- 2e33a11e15 + 2m11e15e33 + m33e152 + 2k11e15e33 + k33e152 ] - (c13 + c44 )[(c13 + c44 )( k11m33 + k33m11 - 2a11a33 ) + (e15 + e31 )(e15m33 + e33m11 - e15a 33 - e33a11 ) - (e15 + e31 )(a 33e15 + a11e33 - e15 k33 - e33 k11 )] - (e15 + e31 )[(c13 + c44 )(m33e15 + m11e33 - a11e33 - e15a33 ) -( e15 + e31 )( c44m 33 + c33m11 + 2e15e33 ) + (e15 + e31 )( c44a33 + c33a11 + e15e33 + e15e33 )] - (e15 + e31 )[(c13 + c44 )( k11e33 + k 33e15 - a33e15 - a11e33 ) +( e15 + e31 )(c44a33 + c33a11 + e15e33 + e15e33 ) -( e15 + e31 )( c44 k 33 + c33k11 + 2e15e33 )],

(B3)

245

Appendix B 2 a4 = c11[c44 ( k11m33 + k33m11 - 2a11a33 ) + c33 ( k11m11 - a11 ) - 2e15 (a11e33 + a33e15 )

- 2e33a11e15 + 2m11e15e33 + m33e152 + 2 k11e15e33 + k33e152 ] 2 ) - 2a11e15e15 + m11e152 + k11e152 ] + c44 [c44 ( k11m11 - a11 2 ) + (e15 + e31 )(e15m11 - e15a11 ) - (c13 + c44 )[(c13 + c44 )( k11m11 - a11

- (e15 + e31 )(a11e15 - k11e15 )] - (e15 + e31 )[(c13 + c44 )(m11e15 - e15a11 ) - (e15 + e31 )(c44m11 + e152 ) + (e15 + e31 )(c44 a11 + e15e15 )] - (e15 + e31 )[(c13 + c44 )( k11e15 - e15a11 ) + (e15 + e31 )(c44 a11 + e15e15 ) - (e15 + e31 )(c44 k11 + e152 )], 2 a5 = c11[c44 ( k11m11 - a11 ) - 2a11e15e15 + m11e152 + k11e152 ]

(B4)

(ii) The constants b mj are defined by b mj = - n4 m + n5m s 2j - n6 m s 4j + n7 m s 6j ,

(m=1-3)

(B5)

where 2 n41 = c11 ( k11m11 - a11 ), n42 = c11 (e15m11 - e15a11 ), n43 = c11 ( k11e15 - e15a11 ), 2 n51 = c11 ( k11m33 + k 33m11 - 2a11a33 ) + c44 ( k11m11 - a11 ) + m11 (e15 + e31 ) 2

+ k11 ( e15 + e31 ) 2 - 2a11 (e15 + e31 )(e15 + e31 ), n52 = c11 ( e15m33 + e33m11 - a11e33 - a33e15 ) + c44 ( e15m11 - a11e15 ) - ( e15 + e31 )[m11 ( c13 + c44 ) + e15 ( e15 + e31 )] + ( e15 + e31 )[a11 ( c13 + c44 ) + e15 ( e15 + e31 )], n53 = c11 ( k11e33 + k 33e15 - e15a33 - e33a11 ) + c44 ( k11e15 - e15a11 ) + ( e15 + e31 )[a11 ( c13 + c44 ) + e15 ( e15 + e31 )] - (e15 + e31 )[ k11 ( c13 + c44 ) + e15 ( e15 + e31 )], 2 n61 = c11 ( k 33m33 - a33 ) + c44 ( k11m33 + k 33m11 - 2a11a33 ) + m33 ( e15 + e31 ) 2

+ k 33 ( e15 + e31 ) 2 - 2a33 ( e15 + e31 )(e15 + e31 ), n62 = c44 (e15m 33 + e33m11 - a11e33 - a33e15 ) + c11 (e33m33 - a33e33 ) - ( e15 + e31 )[m 33 ( c13 + c44 ) + e33 ( e15 + e31 )] + ( e15 + e31 )[a33 ( c13 + c44 ) + e33 ( e15 + e31 )], n63 = c44 ( k11e33 + k 33e15 - e15a33 - e33a11 ) + c11 ( k 33e33 - e33a33 ) + ( e15 + e31 )[a33 ( c13 + c44 ) + e33 ( e15 + e31 )] - ( e15 + e31 )[ k 33 ( c13 + c44 ) + e33 (e15 + e31 )], 2 n71 = c44 ( k 33m 33 - a33 ), n72 = c44 ( e33m33 - a33e33 ), n73 = -c44 ( e33a33 - k 33e33 )

(B6)

References

[1] Hou PF, Ding HJ and Chen JY, Green’s function for transversely isotropic magnetoelectroelastic media, Int J Eng Sci, 43, 826-858, 2005 [2] Hou PF, Leung YTA and Ding HJ, The elliptical Hertzian contact of transversely isotropic magnetoelectroelastic bodies, Int J Solids Struct, 40, 28332850, 2003

246

Appendix C: Numerical Integration C.1 Introduction In this Appendix, guidelines for numerical computation of the element and its boundary integrals are presented. Since Gaussian integration formulae are general, simple and very accurate for a given number of points, attention is given to this type of numerical integration procedure. C.2 One dimensional Gaussian quadrature [3] The integrals in this case can be written as

I1 = ³

1 -1

n

f (x)d x = ¦ f (xi ) wi + En

(C1)

i =1

where xi is the coordinate of the ith integration point, wi is the associated weighting factor, and n is the total number of integration points. They are listed in Table C.1. Notice that xi values are symmetric with respect to x = 0 , wi being the same for the two symmetric values. En is the error defined as [1] En =

22 n +1 (n !) 4 d 2 n f (x) , (2n + 1)(2n !)3 d x 2 n

(-1< x <1)

(C2)

Formula (C1) is based on the representation of f (x) by means of Legendre polynomials Pn (x) . The xi value is the coordinate at a point i where Pn is zero and for which the weights are given by 2

ª dP (x) º wi = 2 /(1 - x ) « n » ¬ d x ¼ x=xi 2 i

(C3)

C.3 Two- and three-dimensional quadrature for rectangles and rectangular hexahedra Two- and three-dimensional formulae are obtained by combining expression (C1) in the form I2 = ³ I3 = ³

1 -1

1

1

-1

-1

³ ³

1 -1

³

1 -1

n

n

f (x, h)d xd h @ ¦¦ f (xi , h j ) wi w j

(C4)

j =1 i =1 n

n

n

f (x, h, z )d xd hd z @ ¦¦¦ f (xi , h j , z k ) wi w j wk

(C5)

k =1 j =1 i =1

where the integration point coordinates and weighting factors are listed in Table C.1.

247

Appendix C

Table C1. Coefficients ±xi and wi in eqn (C1). ±xi

±xi

wi

wi

n=8 0.18343 46424 95650 0.36268 37833 78362 0.52553 24099 16329 0.31370 66458 77887 0.79666 64774 13627 0.22238 10344 53374 0.96028 98564 97536 0.10122 85362 90376

n =2 0.57735 02691 89626 1.00000 00000 00000 n=3 0.00000 00000 00000 0.88888 88888 88888 0.77459 66692 41483 0.55555 55555 55555

n=9 0.00000 00000 00000 0.33023 93550 01260 0.32425 34234 03809 0.31234 70770 40003 0.61337 14327 00590 0.26061 06964 02935 0.83603 11073 26636 0.18064 81606 94857 0.96816 02395 07626 0.08127 43883 61574

n=4 0.33998 10435 84856 0.65214 51548 62546 0.86113 63115 94053 0.34785 48451 37454 n=5 0.00000 00000 00000 0.56888 88888 88889 0.53846 93101 05683 0.47862 86704 99366 0.90617 98459 38664 0.23692 68850 56189

n=10 0.14887 43389 81631 0.29552 42247 14753 0.43339 53941 29247 0.26926 67193 09996 0.67940 95682 99024 0.21908 63625 15982 0.86506 33666 88985 0.14945 13491 50581 0.97390 65285 17172 0.06667 13443 08688

n=6 0.23861 91860 83197 0.46791 39345 72691 0.66120 93864 66265 0.36076 15730 48139 0.93246 95142 03152 0.17132 44923 79170

n=12 0.12523 34085 11469 0.24914 70458 13403 0.36783 14989 98180 0.23349 25365 38355 0.58731 79542 86617 0.20316 74267 23066 0.76990 26741 94305 0.16007 83285 43346 0.90411 72563 70475 0.10693 93259 95318 0.98156 06342 46719 0.04717 53363 86512

n=7 0.00000 00000 00000 0.41795 91836 73469 0.40584 51513 77397 0.38183 00505 05119 0.74153 11855 99394 0.27970 53914 89277 0.94910 79123 42759 0.12948 49661 68870

x2

2 x 3 = 1 - x1 - x 2

· (0,1,0)

x1 = 1 - x 2

1 · (1,0,0)

3

· (0,0,1)

x1

Fig. C1: Definition of triangular coordinates (x1 , x 2 ). C.4 Triangular domain [1] Numerical integration over a triangle can be carried out using triangular coordinates as shown in Fig. C1. This gives

I =³

1 0

(³

1-x2 0

)

n

f (x1 , x 2 , x3 )d x1 d x2 = ¦ wi f (x1i , xi2 , x3i )

(C6)

i =1

where n is the number of integration points, x1i , xi2 and x3i are the coordinates of the ith integration point and wi the associated weighting factor. Values of x1i , xi2 , x3i and wi compiled from Hammer [2] are given in Table C2.

Green’s function and boundary elements of multifield materials

248

By combining expression (C6) with (C1), numerical integration formulae for three-dimensional pentahedral cells can be obtained as before. Table C2. Coefficients xij (j = 1, 2,3) and wi in Eq (C6). n i 1(linear) 1 3(quadratic) 1 2 3 1 4(cubic) 2 3 4 7(quintic) 1 2 3 4 5 6 7

x 1i

1/3 1/2 0 1/2 1/3 3/5 1/5 1/5 0.333 333 33 0.797 426 99 0.101 286 51 0.101 286 51 0.059 715 87 0.470 142 06 0.470 142 06

x i2

1/3 1/2 1/2 0 1/3 1/5 3/5 1/5 0.333 333 33 0.101 286 51 0.797 426 99 0.101 286 51 0.470 142 06 0.059 715 87 0.470 142 06

x i3

1/3 0 1/2 1/2 1/3 1/5 1/5 3/5 0.333 333 33 0.101 286 51 0.101 286 51 0.797 426 99 0.470 142 06 0.470 142 06 0.059 715 87

2 wi

1 1/3 1/3 1/3 -9/16 25/48 25/48 25/48 0.225 000 00 0.125 939 18 0.125 939 18 0.125 939 18 0.132 394 15 0.132 394 15 0.132 394 15

References

Brebbia C.A. & Dominguez J. Boundary Elements: An Introductory Course, 2nd Edition, Comp. Mech. Pub./McGraw-Hill Inc.: Southampton & New York, 1992. [2] Hammer, P.C., Marlowe, O.J. & Stroud, A.H. Numerical Integration over simplexes and cones, Math. Tables and Other Aids to Computation, Vol. 10, 1956. [3] Stroud, A.H. & Secrest, D. Gaussian Quadrature Formulas, Prentice-Hall: New York, 1966. [1]

0080451349-Index.fm Page 249 Friday, December 29, 2006 8:21 PM

249

Index analytical continuation method, 110 anisotropic multifield materials, two solution procedures in, 17–23 with Lekhnitskii formalism, 20–23 with Stroh formalism, 17–20 anti-plane piezoelectric problems, 215–224 arbitrarily shaped hole problems, 61–68, 99–102, see also under thermoelectroelastic problems electroelastic field, Green’s function for, 64–68 one-to-one mapping, 61–64 auxiliary boundary value problem, 2 Barnett-Lothe tensors, 20 BE formulation, 164–169, see also under magnetoelectroelastic problems bimaterial problems, 130–131 magnetoelectroelastic bimaterial with horizontal interface, 130 magnetoelectroelastic bimaterial with vertical interface, 130–131 bimaterial problems, solution of, 41–42 bimaterial solid with an interface crack, 42–51, 95–97, 197, see also under thermoelectroelastic problems cracked homogeneous material (HM), Green’s function in, 43–44 interface crack system geometry, 43 PAC bimaterial, Green’s function in, 47 PID bimaterial, Green’s function in, 47–51, see also under piezoelectric-isotropic dielectric (PID) PP bimaterial, Green’s function in, 44–47 Biot’s variational principle, modified, 8 boundary element method (BEM) for piezoelectricity, 151–191, see also under piezoelectricity boundary element method for discontinuity problems, 194–215, see also under discontinuity problems boundary integral equation, 151–164 boundary integral equation for piezoelectricity, 152–163 reciprocity theorem method, 156–163, see also individual entry variational method, 152–153 weighted residual method, 153–154 governing equations, 151–152 boundary conditions, 152 constitutive relations, 151

elastic strain-displacement and electric field-potential relations, 152 equilibrium equations, 151 for magnetoelectroelastic solid, 163–164 boundary shape coefficients, 159 Cauchy’s residue theorem, 126–127 Christoffel tensor, 74 constitutive models, types, 14 coordinate transformation, 127–128 crack problems, boundary integral equation for, 156–160 cracked homogeneous material (HM), 43–44 cracked piezoelectricity, effective properties of, 206–210 and concentration matrix P, 206–209 self-consistent and Mori-Tanaka approaches, algorithms for, 209–210 cracked solid, anti-plane problems in, 69–73 anti-plane solid with a finite crack, 71 basic equations, 69–71 Green’s function of, 71–73 crack-inclusion system geometry, 211, 213 d’Alembert solution, 76 derivative property, of Dirac delta function, 6 dielectric constants, in matrix form, 10 Dirac delta function, 2, 5–7 Green’s functions derivation using, 6 derivative property, 6 integral property, 6 physical dimension of, 6 sifting property, 6 unit impulse property, 6 three-dimensional, 5 discontinuity problems, boundary element method for, 194–215 BEM for thermopiezoelectric problems, 194–205, see also under thermopiezoelectric problems (d-m-t) system, 120–121 dynamic boundary integral equation, 160–163 dynamic Green’s function, 73–77 3D dynamic Green’s function, 74–77 dynamic anti-plane problem, 73–74 eigenvector, 19 elastic constants, in matrix form, 10 elastic displacement and electric potential (EDEP), 92, 194

0080451349-Index.fm Page 250 Friday, December 29, 2006 8:21 PM

250 elastic strain-displacement and electric field-potential relations, 152 electroelastic field, Green’s function for, 25–88, 94–99 anti-plane problems in a cracked solid, 69–73, see also under cracked solid approximate expressions of Green’s function, 80–82 features, 82 arbitrarily shaped hole, 61–68, see also individual entry bimaterial solid with an interface crack, 42–51, see also individual entry dynamic Green’s function, 73–77, see also individual entry elliptic hole and inclusion, 51–61, see also individual entry by Fourier transforms, 32–38 half-plane problem, 38–41 infinite medium with a crack, Green’s functions for, 77–80, see also individual entry Lee and Jiang’s approach, 34 by the potential function approach, 27–32 by Radon transforms, 26–27 semi-infinite cracks, 68–69 solution of bimaterial problems, 41–42 for time-dependent thermo-piezoelectricity, 82–88, see also under thermopiezoelectricity electroelastic relations, types, 9 electromechanically coupled system, 9 elliptic hole and inclusion, 51–61 elliptic hole problem, geometry of, 52 geometry of elliptic inclusion problem, 59 mapping of an ellipse to a unit circle, 51–53 piezoelectric matrix with an inclusion, 58–61 piezoelectric plate with an elliptic hole, Green’s functions for, 53–58 elliptic hole problems, 97–99, see also under elliptic hole and inclusion; thermoelectroelastic problems elliptic inclusion problems, 102–117, see also under thermoelectroelastic problems equilibrium equations, 151 Euclidean space, 74 field point (observation point), 2 Fourier transform, 5, 33 Green’s functions derivation using, 4, 32–38 fracture analysis, BEM application to, 179–182 8-node crack element, 181–182 alternative crack-tip element, 180

Index conformal and non-conformal boundary elements, 181 one quarter-point quadratic crack-tip element, 179–180 quarter-point element, configuration, 180 free space, Green’s function in, 92–93 Stroh formalism, 92–93 fundamental solution coefficients, 159 Galerkin method, 220, 232 Gauss’ law, 21 generalized relative crack displacement (GRCD), 179 generalized stress intensity factors (GSIF), 179 Gibbs function per unit volume, 8 Green’s function in a cracked homogeneous material (HM), 43–44 of electroelastic problem, 25–88, see also under electroelastic problem foundation of, 1–5 for magnetoelectroelastic problems, 118–150, see also under magnetoelectroelastic problems for a point charge and point force in the z-direction, 28–30 for a point force Px in the x-direction, 30–32 for thermoelectroelastic problems, 92–117, see also under thermoelectroelastic problems Hadamard’s finite part, 34 half-plane problem, 38–41 half-plane solid, 93–95, see also under thermoelectroelastic problems Heaviside step function, 6, 83, 147, 184 Helmholtz equation, 4, 74 Hilbert problem, 44 homogeneous isotropic materials, Green’s function for, 49 homogenization model, 184 horizontal finite crack in an infinite piezoelectric solid, 187 hypersingular integrals, 160, 171–172 infinite medium with a crack, Green’s functions for, 77–80 decomposition used, 78 for line force/charge, 77 for a solid with a crack, 77–80 integral property, of Dirac delta function, 6 interface crack system, 42–51, see also under bimaterial solid isotropic dielectric material, solutions for, 47–48

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Index Kronecker delta, 20, 26, 74 Lamé constant, 12 Laplace equation/operator, 4, 28, 74, 147, 216 Laurent’s expansion/series, 55, 59, 103, 133, 225 least square method, 222–223, 234 Lee and Jiang’s approach, 34 Lekhnitskii’s approach, 17, 20–23, 61, 77 Liouville’s theorem, 60, 104 magnetoelectroelastic materials, linear theory, 12–17 extension to include thermal effect, 15 general anisotropy, basic equations, 12–13 transversely isotropic simplification, 13–15 variational formulation, 15–17 magnetoelectroelastic media, dynamic Green’s functions of, 142–149 dynamic potentials, 140–146 magnetoelectroelastic problems, Green’s function for, 118–150 3D Green’s functions by potential function approach, 120–124 3D magnetoelectroelastic solids, Green’s function for, 121–124 magnetoelectroelastic solids, general solutions for, 120–121 3D Green’s functions by Radon transform, 118–120 explicit expression, 119–120 integral expressions, 118–119 bimaterial problems, 130–131, see also individual entry elliptic hole or a crack, problems with, 131–134 hole problem, basic formulation, 132 hole problem, Green’s function for, 132–134 half-plane problems, 127–130 horizontal half-plane boundary, 127 magnetoelectric problems, Green’s functions for, 124–127 thermomagnetoelectroelastic problems, Green’s functions for, 134–142, see also individual entry vertical half-plane boundary, 127–130 coordinate transformation, 127–128 magnetoelectroelastic plate with vertical half-plane boundary, 128–129 for a solid with an arbitrarily oriented half-plane boundary, 129–130 mixed BEM-homogenization method, 182–186 BE equations, 185

251 effective material properties, basic concept, 182–184 homogenization model, 184 RVE models, 182 self-consistent and Mori-Tanaka approaches, 185–186 Mori-Tanaka-BEM approach, 185–186 self-consistent BEM approach, 185 Mori-Tanaka approaches, 185–186 Mori-Tanaka-BEM approach, 185–186 in thermopiezoelectric problems, 209–210 mutually orthogonal unit vectors, definition, 27 8-node crack element, 181–182 non-hypersingular integrals, 170–171 one-to-one mapping, 61–64, 100 perturbation technique, 71, 100 physical dimension, of Dirac delta function, 6 piezoelectric-anisotropic conductor (PAC), 43 Green’s function in, 47 piezoelectric-isotropic conductor (PIC), 43 Green’s function in, 51 piezoelectric-isotropic dielectric (PID), 43 Green’s function in, 47–51 for homogeneous isotropic materials, 49 isotropic dielectric material, solutions for, 47–48 for PID bimaterials, 49–51 z0 located in isotropic dielectric material, 49–50 z0 located in piezoelectric material, 50 piezoelectricity, see also plane piezoelectricity; thermopiezoelectric problems basic equations of, 7–12 boundary element method for, 151–191, see also individual entry boundary integral equation for, 152–163, see also under boundary integral equation for magnetic effect, 12–13 piezoelectric column under uniaxial tension, 186–187 piezoelectric constants, in matrix form, 10 piezoelectric matrix with an inclusion, 58–61 piezoelectric plate with an elliptic hole, Green’s functions for, 53–58 piezoelectricity, boundary element method for, 151–191 BE formulation, 164–169 8-node isoparametric element, 167

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252 BEM application to fracture analysis, 179–182, see also under fracture analysis boundary integral equation, 151–164, see also individual entry crack tip singularity by evaluating hypersingular integrals, 174–178 main-part analytical method, 175–176 mixed BEM-homogenization method, 182–186, see also individual entry multi-domain problems, 178–179 numerical assessments, 186–191 horizontal finite crack in an infinite piezoelectric solid, 187 magneto-electro-elastic column, 190 piezoelectric column under uniaxial tension, 186–187 rectangular piezoelectric solid with a central inclined crack, 187–190 simply supported beam of magnetoelectro-elastic material, 190–191 singular integral in time-domain boundary integral equation, 173–174 singular integrals, evaluation, 170–174 hypersingular integrals, 171–172 non-hypersingular integrals, 170–171 self-adaptive subdivision method, 172–173 weakly singular integrals, 170 piezoelectric-piezoelectric (PP), 43–47 Green’s function in, 44–47 plane piezoelectricity, 224–239 direct formulation, 234–236 constant element, 235 linear element, 236 quadratic element, 236 governing equations, 224–225 indirect formulations, 231–234 Galerkin formulation, 232 least-square formulation, 234 point-collocation technique, 233 modified Trefftz formulation, 236–239 Trefftz functions, 225–231 coordinate systems for elliptic hole, 230 for domain with a permeable crack, 228 for domain with an impermeable crack, 226 for domain with an impermeable elliptic hole, 229–230 for domain without defects, 225 Plemelj function, 44 point-collocation formulation, 221, 233 Poisson’s ratio, 12 potential function approach, Green’s function by, 27–32

Index Radon transform, 74 Green’s function by, 26–27, 118 inverse Radon transform, 76 evaluation, 87–88 reciprocity theorem method, 156–163 for crack problems, boundary integral equation for, 156–160 dynamic boundary integral equation, 160–163 in electroelastic problems, 156 static boundary integral equation, 156–160 representative volume element (RVE), 182–183 Riemann convolution, 173 self-adaptive subdivision method, 172–173 self-consistent approaches, 185–186 in thermopiezoelectric problems, 209–210 semi-infinite cracks, 68–69 geometry, 68 sifting property, of Dirac delta function, 6 singular integral equation method (SIEM), 212 Somigliana boundary element formulation (SBEF), 215 Somigliana’s identity, 154 stiffness matrix, 198 strain and electric field (SEF), 183 stress and electric displacement (SED), 92, 104, 183, 194 stress intensity factors (SIF), 218–219 stress, magnetic induction, and electric displacement (SMED), 18 Stroh formalism, 17–20, 38, 77, 92–93 thermal loading applied outside the inclusion, 103–108 electroelastic fields, 105–108 thermal fields, 103–105 thermal loads applied inside the inclusion, 108–112 for electroelastic fields, 111–112 for thermal fields, 108–111 thermoelectroelastic problems, Green’s function for, 92–117 arbitrarily shaped hole problems, 99–102 basic equations, 99–100 for electroelastic fields, 101–102 for thermal fields, 100–101 bimaterial solid, 95–97 electroelastic fields, 96–97 temperature fields, 95–96 for electroelastic fields, 98–99 elliptic hole problems, 97–99 thermal field for insulated hole problems, 97–98

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Index elliptic hole, 114–117 boundary conditions, 115 cracks, 116–117 for holes, 115–116 elliptic inclusion problems, 102–117 loading cases, 102 thermal loading applied outside the inclusion, 103–108, see also individual entry in free space, 92–93, see also under free space half-plane solid, 93–95 for electroelastic fields, 94–95 for thermal fields, 93–94 thermal loads applied inside the inclusion, 108–112, see also individual entry thermal loads applied on the interface, 112–114 for thermal fields, 112–114 thermomagnetoelectroelastic problems, Green’s functions for, 134–142 basic equation for, 134–135 bimaterial problems, Green’s function for, 136–138 for magnetoelectroelastic field in bimaterial solids, 137–138 for thermal field in bimaterial solids, 136–137 elliptic hole problems, Green’s function for, 138–140 wedge or semi-infinite crack, 140–142, see also under wedge or semiinfinite crack half-plane problems, Green’s function for, 135–136 Green’s function for thermal field, 135–136 magnetoelectroelastic field, Green’s function for, 136 magnetoelectroelastic media, dynamic Green’s functions of, 142–149, see also individual entry thermopiezoelectric problems, BEM for, 194–205 boundary element formulation, 195–201 bimaterial problems, 198–199 bimaterial solid, 197 half-plane plate, 197 half-plane problems with multiple cracks, 199 plate containing a hole of various shapes, 197 plate containing an elliptic hole and multiple cracks, 199 plate containing an elliptic hole, 197

253 plate containing multiple cracks and a hole of various shapes, 200 plate containing multiple cracks and a wedge boundary, 200–201 plate with a wedge boundary, 197 cracked piezoelectricity, effective properties of, 206–210, see also individual entry Dm function problems, 203–204 bimaterial problems, 203 half-plane problems, 203 plate containing an elliptic hole, 203 wedge problems, 204 for displacement and potential discontinuity problems, 201–205 bimaterial problems, 202 half-plane problems, 202 plate containing a hole of various shapes, 202 plate containing an elliptic hole and multiple cracks, 202 wedge problems, 202 piezoelectric plate configuration, 195 potential variational principle, 195 in SED intensity factors determination, 205–206 SED function and SED intensity factors, relation between, 205–206 simulating K by BEM and least-square method, 206 for temperature discontinuity problems, 194–201 thermo-piezoelectricity, time-dependent, Green’s functions, 82–88 inverse Radon transform, evaluation, 87–88 three-dimensional formulation, of linear piezoelectricity, 7–12 time-domain boundary integral equation, 173–174 transversely isotropic simplification, 13–15, 26 Trefftz boundary element method (TBEM), 215–248 anti-plane piezoelectric problems, 215–224 Trefftz functions, 215–216 indirect formulation, 219–223 direct formulation, 223–224 Galerkin method, 220 least square method, 222–223 modified Trefftz formulation, 222–223 point-collocation formulation, 221 plane piezoelectricity, 224–239, see also individual entry stress intensity factors (SIF), 218–219

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254 subdomain containing an angular corner, solution set for, 216–218 two-index notation, 10 two-point Green’s function, 2 uniform strain, 207 uniform traction, 207 unit impulse property, 6 unit point force, 1

Index variational formulation/method, 15–17, 152–153 wedge or semi-infinite crack, Green’s function for, 140–142 magnetoelectroelastic fields, 141–142 thermal field, general solution for, 140–141 weighted residual method, 153–154 Young’s modulus, 12