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 Ω1 bounded by Γ, as shown in Fig. 6.1(a). The heat transfer problem to be considered is stated as k Tij ij, = 0 in Ω1 (6.1)

n i i nh h n h= = on Γh (6.2)

T T= on ΓT (6.3)

0=iinh on L (6.4)

where ni is the normal to the boundary Γ, and nh T are the prescribed values of heat flow and temperature, which act on the boundaries Γ Γh T and , respectively. For simplicity, we define

ˆ

L LT T T+ −= − on L (=L++L-), where T is the temperature

discontinuity, L is the union of all cracks, and L L+ −and are defined in Fig. 6.1b. Further, if we let Ω2 be the complementary region of Ω1 (i.e., the union of Ω1 and

Ω2 forms the infinite region Ω) and T̂ T T T+ −Γ Γ= − = , the problem shown in Fig.

6.1a can be extended to the infinite case (see Fig. 6.1b). Here + −Γ = Γ +Γ , in which and + −Γ Γ stand for the boundaries of Ω1 and Ω2, respectively (see Fig. 6.1b).

195

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

, ,

1ˆ ˆ( , )2 ij i j nP T T k T T d h TdL

Ω Γ= Ω+∫ ∫ (6.5)

By transforming the region integral in Eq (6.5) into a boundary integral, we have

,

1ˆ ˆ ˆ( , ) ( )2 s nL

P T T T T ds h TdsΓ

= − ϑ +∫ ∫ (6.6)

in which the relations

, , , ˆ ˆ ˆ, [( ) ]i ij j n s sL L

h k T h Tds T T ds= − = ϑ −ϑ∫ ∫ (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 T̂ through use of Green’s function presented in Chapter 3. Thus, the potential energy can be further written as

,

1ˆ ˆ ˆ ˆ( ) ( )2 s nL

P T T T ds h TdsΓ

= − ϑ +∫ ∫ (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 Γ and L are divided into MΓ and LM linear boundary elements, for which the temperature discontinuity may be approximated by the sum of elemental temperature discontinuities:

1

ˆ ˆ( ) ( )M

m mm

T s T F s=

=∑ (6.9)

where mT̂ is the temperature discontinuity at node m, M= MΓ + LM +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

LΓ

Ω1

(a)

Ω1 Ω 2

Γ+ Γ −

Ω 2

L+

L−

(b)

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:

1 1( ) ( ) /m m mF s l s l− −= + (6.10)

for a node located at the left end of a line (see Fig. 6.2a), or

( ) ( ) /m m mF s l s l= − (6.11)

for a node located at the right end of a line (see Fig. 6.2b), or

1 1 1( ) / , if element ,( )

( ) / , if element ,m m m

mm m m

l s l s lF s

l s l s l− − −+ ∈⎧

= ⎨ − ∈⎩ (6.12)

where 1 and m m-l l are the lengths of the two elements connected to the mth node, ml being to the right and 1ml − being to the left (see Fig. 6.2c). While

1

0, at node , , at node 1,

, at node 1m

m

ms l m

l m−

⎧⎪= +⎨⎪− −⎩

(6.13)

m(s=0)

(a)

1

s

(b)m(s=0)

1

s

(c)

m(s=0)

1

s

)(1 mlsm =+

)(1 mlsm =+

)(1 =−− 1−mlsm

)(sFm)(sFm

)(sFm )(sFm

=(1 − )1−mlsm −

Fig. 6.2 Definition of ( )mF 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 ζt (zt) can be given as

1

ˆ( ) Im[ ( )]M

t m t mm

T a T=

ζ = ζ∑ (6.14)

1

ˆ( ) Re[ ( )]M

t m t mm

k a T=

ϑ ζ = − ζ∑ (6.15)

where ( )m ta ζ has a different form for different problems. For example, we have for the problems of

197

(i) a bimaterial solid (see Fig. 2.5 and formulation in Section 3.4.1):

1

(1) (1) (1)( 1) (1) (1)( 1) 10 1 0

1

(1) (1)( ) (1) (1)( )0 1 0

1( ) {ln( ) ln( )}2

1 {ln( ) ln( )} ;2

m

m

m m mm t t t t tl

m

m m mt t t tl

m

l sa z z z b z z dsl

l sz z b z z dsl

−

− − −

−

+= − + −

π

−+ − + −

π

∫

∫ (6.16)

(ii) a plate containing an elliptic hole (see Fig. 2.7 and formulation in Section 3.5):

{ }

{ }1

( 1) 1 ( 1) 10 0

1

( ) 1 ( )0 0

1( ) ln( ) ln( )2

1 ln( ) ln( )2

m

m

m m mm t t t t tl

m

m m mt t t tl

m

l sa dsl

l s dsl

−

− − − −

−

−

+ζ = ζ −ζ − ζ − ζ

π

−+ ζ − ζ − ζ − ζ

π

∫

∫ (6.17)

where

2 2 *2 2

1*21

( ) , t tt t

z z a p bf z

a ip b+ − −

ζ = =−

( 1) ( 1) ( ) ( )0 0 0 0( ), ( )m m m m

t t t tf z f z− −ζ = ζ = ; (6.18)

(iii) a half-plane plate (see Fig. 2.4 and formulation in Section 3.3):

1

( 1) ( 1) 10 0

1

( ) ( )0 0

1( ) {ln( ) ln( )}2

1 {ln( ) ln( )} ;2

m

m

m m mm t t t t tl

m

m m mt t t tl

m

l sa z z z z z dsl

l sz z z z dsl

−

− − −

−

+= − − −

π

−+ − − −

π

∫

∫ (6.19)

(iv) a plate containing a hole of various shapes (see Fig. 2.9 and formulation in Section 3.7):

1

( 1) 1 ( 1) 10 0

1

( ) 1 ( )0 0

1( ) {ln( ) ln( )}2

1 {ln( ) ln( )}2

m

m

m m mm t t t t tl

m

m m mt t t tl

m

l sa dsl

l s dsl

−

− − − −

−

−

+ζ = ζ − ζ − ζ − ζ

π−

+ ζ − ζ − ζ − ζπ

∫

∫ (6.20)

where and t tzζ are related by Eq (3.78); (v) a plate with a wedge boundary (see Fig. 4.5 and formulation in Section 4.8.5)

1

( 1) ( 1) 10 0

1

( ) ( )0 0

1( ) {ln( ) ln( )}2

1 {ln( ) ln( )}2

m

m

m m mm t t t t tl

m

m m mt t t tl

m

l sa z z z z z dsl

l sz z z z dsl

−

λ − λ λ − λ −

−

λ λ λ λ

+= − − − −

π

−+ − − − −

π

∫

∫ (6.21)

where λ is defined in Eq (4.170). In the formulations listed above ( )

0m

tz is defined as

( 1) * ( ) *0 1 1 1 0 1(cos sin ), (cos sin )m m

t tm m m t tm m mz d s p z d s p−− −= + α + α = + α + α (6.22)

and *1 1 2 , tm m md x p x= + 1 2( , )m mx x represent the coordinates at node m, 1m−α is the

angle between the element in the left of node m and the 1x -axis, with mα defined

198

similarly. It should be pointed out that the superscript “(1)” for variables tζ 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

1

ˆ[ ( )] Im{ [ ( )]}M

j j m j mm

T f d a f d T=

=∑ (6.23)

where *1 1 2j j jd x p x= + , 1 2( , )j jx x are the coordinates at node j.

For the total potential energy (6.8), substituting Eqs (6.9) and (6.15) into it yields

1 1

1ˆ ˆ ˆ ˆ( )2

M M

mj m j j jj m

P T K T T G T= =

⎡ ⎤⎛ ⎞≈ − +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦∑ ∑ (6.24)

where Kmj is the so-called stiffness matrix and Gj the equivalent nodal heat flow vector, with the form

1

( 1) ( )

1

Re[ ( )] Re[ ( )]j j

j jmj m ml l

j j

k kK a x ds a x dsl l−

−

−

= − +∫ ∫ , (6.25)

1

*0 ( )

j jj jl l

G h F s ds− +

= ∫ , (6.26)

where *0 0h h= when hs∈Γ , and *0 0h = for other cases, ( ) ( )0

k ktx z= for the cases (i),

(iii) and (v), and ( ) ( )0

k ktx = ζ for the remaining two cases.

The minimization of ˆ( )P T yields

1

ˆM

mj j mj

K T G=

=∑ (6.27)

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

1 21 1 1

ˆ ˆ ˆ, , M M M

j j j j j jj j j

T T T= = =

= = =∑ ∑ ∑U u x yΠ Π (6.28)

where uj, xj and yj have different forms for different problems. They are (i) bimaterial problems

1

1(1) (1) (1) (1) (1)1* 1 1 2

1

(1) (1) (1) (1) (1)1 1 1 2

1 Im { ( ) [ ( ) ( )]}2

1 Im { ( ) [ ( ) ( )]} ,2

j

j

jj l

j

j

lj

l sf z f y b f y ds

l

l sf z f y b f y ds

l

−

−α

−

α

+= + +

π

−+ + +

π

∫

∫

u A q c

A q c

(6.29)

199

1

1(1) (1) (1) (1) *(1) (1) (1)1* 1 1 1 2

1

(1) (1) (1) (1) *(1) (1) (1)1* 1 1 1 2

1 Im { ( ) [ ( ) ( )]}2

1 Im { ( ) [ ( ) ( )]} ,2

j

j

jj l

j

j

lj

l sp f z p f y b f y ds

l

l sp f z p f y b f y ds

l

−

−α α

−

α α

+′ ′ ′= − + +

π

−′ ′ ′− + +

π

∫

∫

x B q d

B q d

(6.30)

1

1(1) (1) (1) (1) (1)1* 1 1 2

1

(1) (1) (1) (1) (1)1* 0 1 1 2

1 Im { ( ) [ ( ) ( )]}2

1 Im { ( ) [ ( ) ( )]}2

j

j

jj l

j

j

lj

l sf z f y b f y ds

l

l sf z q f y b f y ds

l

−

−α

−

α

+′ ′ ′= + +

π

−′ ′ ′+ + +

π

∫

∫

y B q d

B q d (6.31)

where 1* 1 0/ q=q q , 1q 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

1

411

* 1 1

51

* 1

1 Im ( ) ( )2

1 Im ( ) ( ) ,2

j

j

jj k k tl

k j

jk k tl

k j

l sf q g ds

l

l sf q g ds

l

−

−−α

= −

−α

=

+⎡ ⎤= ζ + ζ⎢ ⎥π ⎣ ⎦

−⎡ ⎤+ ζ + ζ⎢ ⎥π ⎣ ⎦

∑∫

∑∫

u A B d c

A B d c (6.32)

1

411 *

1 * 1 1

41 *

1 * 1

1 Im ( ) ( )2

1 Im{ ( ) + ( ) ,2

j

j

jtj k k tl

k t j

jtk k tl

k t j

l sp f q p g ds

z z l

l sp f q p g ds

z z l

−

−−αα α

= α −

−αα α

= α

+⎤∂ζ ∂ζ⎡ ′ ′= − ζ + ζ ⎥⎢π ∂ ∂⎣ ⎦−⎤∂ζ ∂ζ⎡ ′ ′− ζ ζ ⎥⎢π ∂ ∂⎣ ⎦

∑∫

∑∫

x B B d c

B B d c (6.33)

1

411

* 1 1

41

* 1

1 Im ( ) ( )2

1 Im{ ( ) + ( ) ,2

j

j

jtj k k tl

k t j

jtk k tl

k t j

l sf q g ds

z z l

l sf q g ds

z z l

−

−−αα

= α −

−αα

= α

+⎤∂ζ ∂ζ⎡ ′ ′= ζ + ζ ⎥⎢π ∂ ∂⎣ ⎦−⎤∂ζ ∂ζ⎡ ′ ′+ ζ ζ ⎥⎢π ∂ ∂⎣ ⎦

∑∫

∑∫

y B B d c

B B d c (6.34)

where fk is difined in Eq (3.63), and 1 4 2 3 q q c q q dτ τ= − = − = − = − (6.35)

* 1 2 4 3( ) ( ) ( ) ( ) ( )t t t t tg c F d F c F d Fτ τ τ τζ = ζ + ζ − ζ − ζ . (6.36)

with Fi being defined in Eq (3.61); (iii) half-plane problems with multiple cracks

1

11* *

1

1* *

1 Im [ ( ) ( ) ( )]2

1 Im [ ( ) ( ) ( )] ,2

j

j

jj tl

j

jtl

j

l sf z f z ds

l

l sf z f z ds

l

−

−−α

−

−α

+= − +

π

−+ − +

π

∫

∫

u A B d c

A B d c

(6.37)

200

1

11 ** 1 *

1

1 ** 1 *

1 Im [ ( ) ( ) ( )]2

1 Im [ ( ) ( ) ( )] ,2

j

j

jj tl

j

jtl

j

l sp f z p f z ds

l

l sp f z p f z ds

l

−

−−α α

−

−α α

+′ ′= −

π

−′ ′+ −

π

∫

∫

x B B d d

B B d d (6.38)

1

11* *

1

1* *

1 Im [ ( ) ( ) ( )]2

1 Im [ ( ) ( ) ( )]2

j

j

jj tl

j

jtl

j

l sf z f z ds

l

l sf z f z ds

l

−

−−α

−

−α

+′ ′= − −

π

−′ ′− −

π

∫

∫

y B B d d

B B d d (6.39)

where

* 0 0 0 0( ) ( )[ln( ) 1] ( )[ln( ) 1]x x t x t x t x tf z z z z z z z z z= − − − − − − − (6.40)

(iv) a plate containing multiple cracks and a hole of various shapes (see Fig. 2.9 and formulation in Section 3.7)

1

1* 11* 1 2* *

1

* 11* 1 2* *

1 Im [ ( ( ( ) ( )]2

1 Im [ ( ( ( ) ( )] ,2

j

j

jj tl

j

jtl

j

l sf p p f g z ds

l

l sf p p f g z ds

l

−

−−α α α

−

−α α α

+= − ζ + ζ −

π

−− ζ + ζ −

π

∫

∫

u A B d c

A B d c (6.41)

1

1* 1 *1* 1 2* 1 *

1

* 1 *1* 1 2* 1 *

1 Im [ { ( ) ( )} ( )]2

1 Im [ { ( ) ( )} ( )] ,2

j

j

jj tl

j

jtl

j

l sf p p f p p g z ds

z l

l sf p p f p p g z ds

z l

−

−−αα α α α

α −

−αα α α α

α

+∂ζ′ ′ ′= ζ + ζ −π ∂

−∂ζ′ ′ ′+ ζ + ζ −π ∂

∫

∫

x B B d c

B B d c (6.42)

1

1* 11* 1 2* *

1

* 11* 1 2* *

1 Im [ { ( ) ( )} ( )]2

1 Im [ { ( ) ( )} ( )] ,2

j

j

jj tl

j

jtl

j

l sf p p f g z ds

z l

l sf p p f g z ds

z l

−

−−αα α α

α −

−αα α α

α

+∂ζ′ ′ ′= − ζ + ζ −π ∂

−∂ζ′ ′ ′− ζ + ζ −π ∂

∫

∫

y B B d c

B B d c (6.43)

where 1 1

1* 1 0 2 0 1 0 2 01 1

1 3 0 4 0 3 0 4 0

( ) [ ( , ) ( , ) ( , ) ( , )] / 2

+ [ ( , ) ( , ) ( , ) ( , )] / 2,t t t t

j t t t t

f a F F F F

e a F F F F

− −α α α α α

− −α α α α

ζ = ζ ζ + ζ ζ − ζ ζ − ζ ζ

γ ζ ζ + ζ ζ − ζ ζ − ζ ζ

1 * 1 *2* 1 0 2 0 1 2

1 11 3 0 4 0 3 0 4 0

( ) [ ( , ) ( , ) ( , ) ( , )] / 2

[ ( , ) ( , ) ( , ) ( , )] / 2t t t t

j t t t t

f ip ae F F F F

ie p ae F F F F

− −α α α α α α

− −α α α α α

ζ = − ζ ζ + ζ ζ − ζ ζ + ζ ζ

+ γ ζ ζ − ζ ζ + ζ ζ − ζ ζ

1 1* 1 1 0 2 0 2 2 0 1 0

1 11 3 3 0 4 0 4 4 0 3 0

( ) [ ( , ) ( , )] [ ( , ) ( , )]

{ [ ( , ) ( , )] [ ( , ) ( , )]}t t t t t t t t t

j t t t t t t t t

g z aa F F aa F F

+ae a F F aa F F

− −τ τ

− −τ τ

= ζ ζ − ζ ζ + ζ ζ − ζ ζ

ζ ζ − ζ ζ + ζ ζ − ζ ζ

(v) a plate containing multiple cracks and a wedge boundary (see Fig. 4.5 and formulation in Section 4.8.5)

201

1

111 2 *

1

11 2 *

1 Im{ [ ( ( ) ( ) ) ( )] }2

1 Im{ [ ( ( ) ( ) ) ( )] }2

j

j

jj tl

j

jtl

j

l sf z f z g z ds

l

l sf z f z g z ds

l

−

−−α α

−

−α α

+= − − +

π

−+ − − +

π

∫

∫

u A B d c

A B d c (6.44)

1

11 *1 2 1 *

1

1 *1 2 1 *

1 Im{ [ ( ( ) ( ) ) ( )] }2

1 Im{ [ ( ( ) ( ) ) ( )] }2

j

j

jj tl

j

jtl

j

l sp f z p f z p g ds

l

l sp f z p f z p g ds

l

−

−−α α α α

−

−α α α α

+′ ′ ′= − − ζ

π

−′ ′ ′+ − − ζ

π

∫

∫

x B B d

B B d (6.45)

1

111 2 *

1

11 2 *

1 Im{ [ ( ( ) ( ) ) ( )] }2

1 Im{ [ ( ( ) ( ) ) ( )] }2

j

j

jj tl

j

jtl

j

l sf z f z g ds

l

l sf z f z g ds

l

−

−−α α

−

−α α

+′ ′ ′= − − + ζ

π

−′ ′ ′+ − − + ζ

π

∫

∫

y B B d

B B d (6.46)

where f1 and f2 are defined in Eq (4.185), and * 1 2ˆ ˆ( ) ( ) ( )t t tg z f z f z= − .

Thus, the SED and EDEP on a boundary induced by temperature discontinuity are of the form

1 21 1

ˆ ˆ( ) ( ) , ( )M M

n i i j j j T j jj j

s n n n T s Tθ θ

= =

= Π = + =∑ ∑t x y U u . (6.47)

In general, ( ) 0n sθ ≠t and ( ) 0.T sθ ≠U 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 Ω1, in which the governing equation and its boundary conditions are described as follows:

, 0ij jΠ = , in Ω1, (6.48)

( )ni ij j i n it n t θ= Π = − t , on Γt, (6.49)

( )i i T iu u θ= − U , on Γu, (6.50)

( ) , ( ) ( )ni ni n i i i i T i T iL L L L L Lt t b u u+ − + − + −

θ θ θ= − = − = − − +t U U , on L (6.51)

where and t uΓ Γ are the boundaries on which the prescribed values of stress

it and displacement

iu are imposed. Similarly, the related potential energy for the elastic problem can be given as

,

1( ) [ ( ) 2 ] ( )2 s n n nL

P ds dsθ

Γ= ⋅ + ⋅ − − ⋅∫ ∫b b b t b t t bϕ (6.52)

202

where the electroelastic solutions of functions ( )ϕ 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

4

(1) (1) (1) (1) (1) (1) (1) (1)0 0

1

1 1Im{ ln( ) } Im{ ln( ) }Tz z z zα α α β ββ=

= − + −π π∑U A B b A q , (6.53)

4

(1) (1) (1) (1) (1) (1) (1) (1)0 0

1

1 1Im{ ln( ) } Im{ ln( ) }Tz z z zα α α β ββ=

= − + −π π∑B B b B qϕ (6.54)

for material 1 in x2>0 and

4

(2) (2) (2) (1) (2)0

1

1 Im{ ln( ) }z zα β ββ=

= −π∑U A q , (6.55)

4

(2) (2) (2) (1) (2)0

1

1 Im{ ln( ) }z zα β ββ=

= −π∑ B qϕ (6.56)

where

(1) (1) 1 (1) 1 (2) 1 1 (1) 1 (1)[ 2( ) ] T− − − − −β β= − +q B I M M L B I B b , (6.57)

(2) (2) 1 (1) 1 (2) 1 1 (1) 1 (1)2 ( ) T− − − − −β β= +q B M M L B I B b , (6.58)

with ( ) ( ) ( ) 1j j ji −= −M B A as the surface impedance matrix. (ii) a plate containing an elliptic hole and multiple cracks

4

1 10 0

1

1 1( ) Im[ ln( ] Im[ ln( ]T T− −α α α β β

β=

= ζ − ζ + ζ − ζπ π∑U b A B b A B BI B b (6.59)

4

1 10 0

1

1 1( ) Im[ ln( ] Im[ ln( ]T T− −α α α β β

β=

= ζ − ζ + ζ − ζπ π∑b B B b B B BI B bϕ (6.60)

(iii) half-plane problems

4

10 0

1

1 1( ) Im[ ln( ] Im[ ln( ]T Tz z z z −α α α β β

β=

= − + −π π∑U b A B b A B BI B b (6.61)

4

10 0

1

1 1( ) Im[ ln( ] Im[ ln( ]T Tz z z z −α α α β β

β=

= − + −π π∑b B B b B B BI B bϕ (6.62)

(iv) a plate containing a hole of various shapes For the case of a plate containing a hole of various shapes the expressions of ( )bϕ and ( )U b have the same form as those of Eqs (6.59) and (6.60), but the relation between and zα αζ is defined by Eq (2.223) (v) wedge problems

203

4

10 0

1

1 1( ) Im[ ln( ] Im[ ln( ]T Tk k kz z z zλ λ λ λ −

β ββ=

= − + − −π π∑U b A B b A B BI B b (6.63)

4

10 0

1

1 1( ) Im[ ln( ] Im[ ln( ]T Tk k kz z z zλ λ λ λ −

β ββ=

= − + − −π π∑b B B b B B BI B bϕ (6.64)

As treated earlier, the boundaries L and Γ are divided into a series of boundary

elements, for which the EDEP discontinuity may be approximated through linear interpolation as

1

( ) ( )M

m mm

s F s=

= ∑b b . (6.65)

With the approximation (6.65), the EDEP and SED functions can be expressed in the form

1 1

( ) Im[ ( )] , ( ) Im[ ( )]M M

m m m mm m= =

ζ = ζ ζ = ζ∑ ∑U AD b BD bϕ (6.66)

where Dm has different forms for different problems. The function Dm is given below for five typical problems.

(i) bimaterial problems

1

4( 1) ( 1) * 1

0 0 1 1

4( ) ( ) *

0 0 1

1( ) ln( ) ln( )

1 + ln( ) ln( )

m

m

m T m T mm l

m

m T m T ml

m

l sz z dsl

l sz z dsl

−

− − −αα αβ β

β= −

αα αβ ββ=

⎧ ⎫ += +⎨ ⎬π ⎩ ⎭

⎧ ⎫ −+⎨ ⎬π ⎩ ⎭

∑∫

∑∫

D z B B BI B

B B BI B

(6.67)

where ( ) ( )0 0

m mz z zαα α α= − , ( ) ( )0 0 ,m mz z zαβ α β= − * (1) 1 (1) 1 (2) 1 1 (1) 1[ 2( ) ]− − − − −= − +B B I M M L .

(i) a plate containing an elliptic hole

{1

4( 1) 1 ( 1) 1 1

0 0 1 1

4( ) 1 ( ) 1

0 0 1

1( ) ln( ) ln( )

1 + ln( ) ln( )

m

m

m T m T mm l

m

m T m T ml

m

l s dsl

l s dsl

−

− − − − −α α α β β

β= −

− −α α α β β

β=

⎫ += ζ − ζ + ζ − ζ ⎬π ⎭

⎧ ⎫ −⎪ ζ − ζ + ζ − ζ⎨ ⎬π ⎪ ⎭⎩

∑∫

∑∫

D B B BI B

B B BI B

ζ

(6.68)

(ii) half-plane problems

1

41 1 1 10 0

1 1

41

0 0 1

1( ) ln( ) ln( )

1 + ln( ) ln( )

m

m

m T m T mm l

m

m T m T ml

m

l sz z dsl

l sz z dsl

−

− − − −αα αβ β

β= −

−αα αβ β

β=

⎧ ⎫ += +⎨ ⎬π ⎩ ⎭

⎧ ⎫ −+⎨ ⎬π ⎩ ⎭

∑∫

∑∫

D z B B BI B

B B BI B

(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 αζ is related to zα by Eq (2.223).

204

(v) wedge problems

1

5( 1) ( 1) 1 10 0

1 1

5( ) ( ) 10 0

1

1( ) { ln( ) ln( ) }

1 + { ln( ) ln( ) }

m

m

m T m T mm l

m

m T m T ml

m

l sz z z z dsl

l sz z z z dsl

−

λ λ − λ λ − − −α α α β β

β= −

λ λ λ λ −α α α β β

β=

+= − + − −π

−− + − −

π

∑∫

∑∫

D z B B BI B

B B BI B

(6.70) In particular the displacement at node j is given by

1

[ ( ) ] Im{ [ ( ) ]}M

j m j mm

f d f dα α=

=∑U AD b . (6.71)

Substituting Eq (6.66) into Eq (6.52), we have

1 1

( ) / 2M M

Ti ij j i

i jP

= =

⎡ ⎤⎛ ⎞= ⋅ −⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦∑ ∑b b k b g (6.72)

where

1

10 0

1

1 1Im[ ( ) ] Im[ ( ) ]j j

T j T T j Tij i il l

j j

ds dsl l−

−

−

= −∫ ∫k D B D Bζ ζ (6.73)

1

( )j j

j j jl lF s ds

− += ∫g G (6.74)

and j nθ= −G t when node j is located at the boundary L, 0

j nθ= −G t t for the other

nodes. The minimization of Eq (6.72) leads to a set of linear equations:

1

M

ij j ij=

=∑k b g . (6.75)

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

1 21 1Im[ ( )] , Im[ ( )]

M M

m m m mm m= =

′ ′= − ζ = ζ∑ ∑BPD b BD bΠ Π (6.76)

Therefore, the surface traction-charge vector Πn in a coordinate system local to a particular crack line, say the ith crack, can be expressed in the form

1 2( ){ sin + cos }Tn i i i= Ω α − α αΠ Π Π (6.77)

where ( )iΩ α is defined by

cos sin 0 0sin cos 0 0

( )0 0 1 00 0 0 1

α α⎡ ⎤⎢ ⎥− α α⎢ ⎥Ω α =⎢ ⎥⎢ ⎥⎣ ⎦

(6.78)

Using Eq (6.77) we can evaluate the SED intensity factors by the following definition

205

0( ) { } lim 2 ( )T

II I III D nrc K K K K r r

→= = πK Π (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], least-square 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]

20 ( )k k k kz z w− = ξ = ξ , (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]

2Re[ ( ) ( )], 2Re[ ( ) ( )]t tg z g z= + = +U Af z c Bf z dϕ (6.81)

The EDEP and SED fields in ξ-plane can then be written as

22Re[ ( ) ( )], 2Re[ ( ) / ( ) / ]t t tg g′ ′= ξ + ξ = ξ ξ + ξ ξU Af c Bf dΠ (6.82)

With the usual definition, the vector of SED intensity factor, K, is evaluated by

20 0lim 2 2 2 lim Re[ ( ) ( )]tgξ→ ξ→

′ ′= ξ πΠ = π ξ + ξK Bf d (6.83)

The functions f and g near the crack tip can be assumed as simple polynomials of ξ

2 2

21 1

( ) ( ) , ( )n n

j jj n j t j t

j ji g r+

= =

ξ ≈ + ξ ξ ≈ ξ∑ ∑f s s (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., ϕ =0, substitution of Eq (6.84) into Eq (6.81)2 yields

2

21

2 Re [ ( ) ] 0n

j jj n j j t

ji r+

=

= + ξ + ξ =∑ B s s dϕ (6.85)

Noting that ξj= jtξ =(− x) j/2 along the crack surface, where x is the distance from crack

tip to the point concerned, we have

12 [ Im{ }]n j R I j jr−

+ = − +s B B s d , for j=1, 3, 5, ,… 2n-1, (6.86)

12 [ Re{ }]n j I R j jr−

+ = +s B B s d , for j=2, 4, 6, ,… 2n, (6.87)

206

in which BR=Re(B), BI=Im(B). Substituting Eqs (6.86) and (6.87) into Eq (6.84), and then into Eqs (6.82) and (6.83), yields

2

1

1 11 1 1

[ ( ) ( , )],

2 2 Re[ ( ) Im( ) ]

n

j j j tj

R I Ri i r r=

− −

= ξ + ξ ξ

= π − + +

∑U Q s S

K B I B B s BB d d (6.88)

where 1 1

t( ) [ ] , ( , ) Im[ ] + , ( 1, 3, , 2 1),j j jj R I j t R j ji i r r j n− −ξ = − ξ ξ ξ = ξ ξ = −Q I B B S B d d … (6.89)

1 1t( ) [ ] , ( , ) Re[ ] + , ( 2, 4, , 2 )j j j

j I R j t I j ji i r r j n− −ξ = + ξ ξ ξ = ξ ξ =Q I B B S B d d … (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)

2

1[ ( ) ( , )]

n

k j k j j k tk kj=

= ξ + ξ ξ −∑R Q s S U , (k=1, 2, ,… m) (6.91)

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

{ } [ ] [ ]{ } 2{ } [ ] ({ } { }) terms without { }T T T Tπ = − − +s Q Q s s Q U S s (6.92)

provides

[ ] [ ]{ } [ ] ({ } { })T T= −Q Q s Q U S , (6.93) where 1 2 2{ }={ , , , }T

ns s s s… , 1 2{ } { , , , } ,Tm=U U U U… (6.94)

* * *1 2{ } { , , , }T

m=S S S S… , 2

*

1( , )

n

k j k tkj=

= ξ ξ∑S S , (6.95)

11 21 2 ,1

12 22 2 ,2

1 2 2 ,

[ ]

n

n

m m n m

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

Q Q QQ Q Q

Q

Q Q Q

……

… … … ……

, (6.96)

( )jk j k= ξQ Q (6.97)

Once the unknown vector {s} has been obtained from Eq (6.93), the SED intensity factor K can be evaluated from Eq (6.88)2. In the calculation, an appropriate number m can be set to obtain the required accuracy. 6.4 Effective properties of cracked piezoelectricity 6.4.1 Effective properties and concentration matrix P In Section 5.8 we presented formulations for predicting effective properties of

207

piezoelectricity with inclusions or holes. For piezoelectricity weakened by cracks, the effective material properties can be analyzed by the combination of micromechanics models presented in Section 5.8 and the BE Eqs (6.71) and (6.75). To this end, define the overall elastic, piezoelectric, and dielectric constants of the piezoelectric solid by [10]

EκεeD

EeεCσ**

** )(+=

−= T

(6.98)

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, 0σ , and electric displacement, D0, on boundary Γ of the RVE

(see Fig. 6.3):

00 , DDσσ == (6.99)

b) Uniform strain, ε0, and electric field, E0, on boundary Γ of the RVE:

00 , EEεε == (6.100)

Since the material behavior is linear, the principle of superposition is used to

decompose the load (ε0, E0) into two elementary loadings (ε0, E0=0) and (ε0=0, E0).

For the loading case (ε0, E0=0), Eq (6.98) becomes

0*

0*

εeDεCσ

=

= (6.101)

Using relations (5.219) and (6.101), we have

22121

*22121

*

)(

)(

v

v

Aeeee

ACCCC

−+=

−+= (6.102)

where the concentration tensor A2 is defined by the linear relation

022 εAε = (6.103)

Following the average strain theorem [11], the strain tensor, 2ε , can be expressed as

208

2

2 2

1( ) ( )2ij i j j iu n u n dS

∂Ω= +

Ω ∫ε (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. 0/22 →ΩΩ=v , one obtains

∫∫ Δ+ΔΩ

=Δ+ΔΩ

=→ L ijjiL ijjiijv

dSnunudSnunuv

v

2

2220

)(21)(

2)(lim

2

ε (6.105)

where iuΔ is the jump of displacement across the crack faces, ,21 NlllL ∪∪∪= … li

is the length of the ith crack, N the number of cracks within the RVE under

consideration. For convenience, we define

)(lim 2202

vv

AP→

= (6.106)

where P can be calculated by the relation:

∫ Δ+ΔΩ

==→ L ijjiijvij dSnunuv

220

0 )(21)(lim)(

2

εPε (6.107)

Substituting Eq (6.106) into Eq (6.102) and considering 02 →C and 02 →e when

inclusions become cracks, yields

)(

)(

1*

1*

PIee

PICC

−=

−= (6.108)

On the other hand, for the loading case (ε0=0, E0), Eq (6.98) leads to

0*

0* )(EκD

Eeσ=

−= T

(6.109)

Similar to the treatment in Eq (6.102), we have

22121

*21221

*

)(

)(

v

vT

Bκκκκ

eeBee

−+=

−+= (6.110)

where the tensor B2 is defined by

022 EBE = (6.111)

By comparing Eqs (6.102)2 and (6.110)1, it is evident that

)()( 122212 eeBAee −=− T (6.112)

209

Therefore the effective constitutive law (6.98) is completely defined once the

concentration factor A2 has been determined.

The estimation of integral (6.104) or (6.105) and thus A2 (or P) is the key to

predicting the effective electroelastic moduli *C , *e , and *κ . 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 Δui 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 *)0(C , *

)0(e , and *)0(κ ( *

)0(C =C0,

*)0(e =e0 and *

)0(κ =κ0 are used as initial value in our analysis)

(b) Solve Eqs (6.71) and (6.75) for )(imb (= ( )m iuΔ ) using the values of *)1( −iC , *

)1( −ie ,

and *)1( −iκ , 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

)(imb , and then determine *)(iC , *

)(ie , and *)(iκ by way of Eqs (6.102) and

(6.110).

(d) If ε≤−=ε −*

)0(*

)1(*

)()( / FFF iii , where ε is a convergent tolerance, terminate

the iteration; F may be C, e or κ, otherwise take *)(iC , *

)(ie , and *)(iκ 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 MTP is given by the solution for a single crack

embedded in an intact solid subject to an applied strain field equal to the as yet

210

unknown average field in the solid, which means that the introduction of cracks in the

solid results in a value of 2ε given by

122 εAε DIL= (6.113)

where DIL2A is the concentration matrix related to the dilute model. As such, it is easy

to prove that [11]

1)( −+= DILDILMT PIPP (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

11 12 13 33 44

* 6 1 * 6 1 511 33 3

2 2 231 33 15 11 0

33 0 0

150 GPa, 66 GPa, 66 GPa, 146 GPa, 44 GPa,

8.53 10 K , 1.99 10 K , 0.133 10 N/CK,

4.35 C/m , 17.5 C/m , 11.4 C/m , 1115 ,

1260 ,

c c c c c

e e e

− − − −

= = = = =

α = × α = × λ = ×

= − = = κ = κ

κ = κ κ 12 2 28.85 10 C / Nm permittivity of free space−= × =

(ii) Material constants for Cadmium Selenide

11 12 13 33

6 1 2 6 1 244 11 33

5 1 2 2 23 31 33

15

74.1 GPa, 45.2 GPa, 39.3 GPa, 83.6 GPa,

13.2 GPa, 0.621 10 NK m , 0.551 10 NK m ,

0.294 10 CK m , 0.160 Cm , 0.347 Cm ,

0.138

c c c c

c

g e e

e

− − − −

− − − −

= = = =

= γ = × γ = ×

= − × = − =

= 2 12 2 1 2 12 2 1 211 33 Cm , 82.6 10 C N m , 90.3 10 C N m− − − − − − −κ = × κ = ×

where cij is elastic stiffness, *11α and *

33α are thermal expansion constants, λ3 and g3 are pyroelectric constants, eij is piezoelectric constants, and γij is a piezothermal constant. Since the values of the coefficient of heat conduction for BaTiO3 and Cadmium Selenide could not be found in the literature, the values k33/k11=1.5 for BaTiO3 and k33/k11=1.8 for Cadmium Selenide, k13=0 and k11=1 W/mK are assumed.

In our analysis, plane strain deformation is assumed and the crack line is assumed to be in the x1-x2 plane, i.e., D3=u3=0. Therefore the stress intensity factor vector K*

now has only three components (KI, KII, KD). In the least-square method, the SED intensity factors are affected by the

parameters n, dmax and dmin, where n is the number of terms in Eq (6.91), dmax is the

211

maximum distance from crack tip to the n-point at which the residual vectors are calculated, and dmin is the minimum distance. In our analysis dmin is set to be 0.05c.

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 βi at point A (see Fig. 6.4) vary with dmax when the crack angle 0=α and n=5, 10, 15, respectively, where βi are defined by

0 33 1 1

0 11 2 1

0 3 1

/ ,

/ ,

/

I n

II n

D n D

K h c c k

K h c c k

K h c c k

= π γ β

= π γ β

= π χ β

(6.115)

Table 6.1 The BEM results for coefficients βi vs dmax in Example 1

dmax/c n β1 β2 βD 0.5 5

10 15

1.230 0.328 0.755 1.224 0.323 0.747 1.222 0.322 0.745

1.0 5 10 15

1.225 0.321 0.749 1.221 0.318 0.745 1.220 0.318 0.743

1.5 5 10 15

1.231 0.328 0.757 1.227 0.324 0.752 1.226 0.323 0.750

SIEM 1.207 0.311 0.732

α

2.5b

b

a x1

x2

2b

A

B

212

Fig. 6.5 The coefficients iβ versus crack angle α

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 least-square 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 βi as a function of crack angle α when dmax=c and n=15. It is evident from the figure that all the coefficients βi 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 βi at point A (see Fig. 6.6) are presented as a function of crack orientation angle α. 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 βi are more sensitive to crack orientation than those in Example 1. They reach their peak values at 137 for α = β ,

242 for α = β , and Dβ=α for 50 , 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%.

0 20 40 60 80 1000.2

0.4

0.6

0.8

1

1.2

1.4

213

Fig. 6.6 Configuration of the crack-inclusion system in Example 2 (a=2b)

Fig. 6.7 SED intensity factors versus crack angle α

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 thermo-piezoelectric 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

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0 20 40 60 80 1000

0.5

1

1.5

2

1β

2β

Dβ

α(deg.)

Finite element method

Present method

T=0 T=0

02 =h

02 =h

0hhn =

ab

b

1.5b

α2b

4a

3.5aA

B

0=nh

214

[6] Aliabadi MH and Rooke DP, Numerical Fracture Mechanics, Computational Mechanics Publications: Southampton, 1991

[7] Ju SH, Simulating stress intensity factors for anisotropic materials by the least-square method, Int J Frac, 81, 283-397, 1996

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[15] Anderson JL, Fracture Mechanics: Fundamentals and Applications, CRC Press: Boston, 1991