Bui_CMS_2012

atiin

en, G

Keywords:Piezoelectric materialsDynamic fractureDynamic intensity factorsX-FEMEnrichment techniques

ientlids

developed. The sixfold basis enrichment functions particularly suitable for cracks in piezoelectric mate-

widele to t

become one of the most important research areas, in which theevaluation and characterization of the mechanical and electricalreliability, integrity and durability in engineering applications isof great importance. Study of such complex dynamic fractureproblems in piezoelectric materials often requires us to solve thecorresponding initial-boundary value problems. Because of thelimitations of the analytical solutions and the expensiveness of

tions. Pan [13] has presented a single-domain boundary elementmodel using the complex variable function method to analyzethe fracture parameters in two-dimensional (2D) anisotropic pie-zoelectric materials. Rajapakse and Xu [14] has applied the BEMto linear fracture problems in 2D piezoelectric solids using the ex-tended Lekhnitskiis formalism and distributed dislocation model.Garcia-Sancherz et al. [15,16] proposed a time-domain BEM(TDBEM) using a combination of the strongly singular displace-ment boundary integral equations and the hypersingular tractionboundary integral equations to transient dynamic analysis innite and innite cracked piezoelectric solids, and a time-domain

Corresponding author. Tel.: +49 2717402836; fax: +49 2717404074.E-mail addresses: [email protected] (T.Q. Bui), c.zhang@

Computational Materials Science 62 (2012) 243257

Contents lists available at

M

lseuni-siegen.de (C. Zhang).tween the mechanical and electrical elds. Modeling andsimulation of the defects, e.g., cracks, in such piezoelectric struc-tures and materials have been receiving more and more attentionsin the last few decades [1]. Assessing the relevant dynamic fractureparameters of cracks in piezoelectric materials subjected to thecoupled electromechanical impact loads is essential, which maygain valuable knowledge to enhance and improve the performanceof smart piezoelectric structures and devices in engineeringapplications. As a consequence, the dynamic fracture analysis has

Most of the previous works dealing with fracture problems inpiezoelectric solids are based on the FEM and the BEM. The mostimportant task and also a key factor in such a work is to evaluatethe relevant eld intensity factors efciently and accurately, thusthe singularity of the mechanical and electrical elds at thecrack-tips must be described somehow in the formulations. Kunaand his co-workers [412] have developed special singular crack-tip elements in the FEM to accurately model many crack problemsin piezoelectric materials under static and dynamic loading condi-1. Introduction

Piezoelectric materials have beengent components and structures du0927-0256/$ - see front matter 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.commatsci.2012.05.049rials are adopted to fully capture the singular elds at the crack-tips in piezoelectricity. The governingequations are transformed into a weak-form and the time-dependent system of discrete equations is thenobtained, which is solved by the unconditionally implicit time integration method at each time-step. Toaccurately assess the relevant dynamic mechanical stress and electric displacement intensity factors pre-cisely and efciently, domain-form of the contour integration integral taking the inertial effect intoaccount in conjunction with the asymptotic near crack-tip elds of piezoelectric materials is presented.Four numerical examples for stationary cracks in homogeneous piezoelectric solids with impermeablecrack-face boundary condition under impact loads are considered, respectively. Validation of the presentmethod is made by comparing the present results with reference solutions available in the literature, andvery good agreements are obtained. The effects of different poling directions and combined electrome-chanical impact loads are analyzed in details.

2012 Elsevier B.V. All rights reserved.

y used in many intelli-he coupling effects be-

the experimental works, the numerical methods have thus becomea key numerical tool to accomplish that purpose, and among themthe nite element method (FEM) [2] and the boundary elementmethod (BEM) [3] are the most popular techniques.Accepted 20 May 2012nite element method (X-FEM). To serve this purpose, a dynamic X-FEM computer code using quadrilat-eral elements in conjunction with the level set method to accurately describe the crack geometry isExtended nite element simulation of stin piezoelectric solids under impact load

Tinh Quoc Bui , Chuanzeng ZhangDepartment of Civil Engineering, University of Siegen, Paul-Bonatz Str. 9-11, 57076 Sieg

a r t i c l e i n f o

Article history:Received 30 April 2012Received in revised form 18 May 2012

a b s t r a c t

This work presents a transand linear piezoelectric so

Computational

journal homepage: www.ell rights reserved.onary dynamic cracksg

ermany

dynamic analysis of stationary cracks in two-dimensional, homogeneoussubjected to coupled electromechanical impact loads using the extended

SciVerse ScienceDirect

aterials Science

vier .com/locate /commatsci

eij 2 ui;j uj;i; Ei u;i 3

Matcollocation-Galerkin BEM (TDGBEM) has been implemented byWnsche et al. [17]. The works presented in [18,19] represent inessence the so-called mesh-free methods (i.e. no mesh or elementsare required), Liew et al. [18] has applied a traction boundary inte-gral equation method using integration by parts and the movingleast-squares approximation while Liu et al. [19] has extendedthe approximated spaces by embedding the known enrichmentfunctions to study crack and interface discontinuities in piezoelec-tric solids. Other works in the framework of the meshless methodscan also be found, e.g., [2022], and so on.

Recently, the extended nite element method (X-FEM) pio-neered by Belytschko and his co-workers [23,24] in terms of thepartition of unity [25] has gained considerable attention in the eldof computational methods in engineering applications and mate-rial sciences. A broad range of problems dealt by this approachhas been increasingly expanded in the last decades. Remarkableagreements of the previous studies have successfully illustratedthe high applicability and the effectiveness of the method in solv-ing boundary value problems on domains with discontinuities.However, the application of the X-FEM to study fracture problemsin piezoelectric materials and structures is rather rare. In 2009,Bchet et al. [26] have introduced a new set of sixfold basis enrich-ment functions into the X-FEM to investigate the semi-innitecrack and GriffthIrwin crack in 2D piezoelectric materials withan arbitrary polarization direction. This new set of enrichmentfunctions is derived from Lekhnitskiis formalism and Whlliamseigenfunction expansion approach. They have found that the stan-dard fourfold enrichment functions for isotropic materials [23,24]can be applied to crack problems in piezoelectric materials withno signicant difference in the results compared with that usingtheir own sixfold basis enrichment functions. This may be coherentwith the weak coupling effect among the elastic and electric elds.Obviously, this nding is in fact useful since the much simplerfourfold basis could be used for some complex practical problemswith less implementation efforts. Recently, Bhargava and Sharmahave performed a static study of nite size effects in cracked 2Dpiezoelectric media using the standard fourfold basis [27]. More re-cently, they have presented a new set of six enrichment functionsthat is also based on Lekhnitskiis formalism with a slightly differ-ent sense to analyze two-unequal-collinear cracks [28]. Anothernovel application of the X-FEM to fracture problems in multiphasemagnetoelectroelastic composite materials has presented in [29].In all the aforementioned works on the X-FEM for crack analysisin piezoelectric materials, only static loading is considered. In thecontrary, numerical simulations of the dynamic fracture problemsremain a challenging task, and dynamic loads are frequently pres-ent in many practical engineering problems. As a result, the moti-vation of tackling this dynamic task is due to the fact that theinertia forces in case of the dynamic loads can cause higher stressesand electric displacements in the vicinity of a crack-tip than thestatic ones. To the best knowledge of the authors, none of any tran-sient dynamic studies in cracked piezoelectric solids subjected toimpact loadings using the X-FEM can be found in the literature un-til the present work is being reported.

In this work we present a numerical analysis of stationary dy-namic cracks in transversely isotropic piezoelectric solids usingthe X-FEM with the sixfold basis enrichment functions. The effectsof the polarization directions, mesh sizes, time-steps, combineddynamic electromechanical impact loads, and intensity of the elec-tric impact loading, etc. on the dynamic intensity factors are ana-lyzed. In order to calculate the relevant dynamic intensity factorsefciently, an interaction integral derived from the domain-formof the path-independent electromechanical J-integral taking the

244 T.Q. Bui, C. Zhang / Computationalinertial effect into account is presented. Standard implicit timeintegration scheme is employed for solving the time-dependentsystem of discrete equations. Numerical examples are given andIn the piezoelectric initial-boundary value problem, the primaryeld variables comprising ui and u are yet to be determined andthey must satisfy the essential boundary conditions on the bound-aries Cu and Cu as

uj uj on Cu; u u on Cu 4and on the boundaries Cr and CD, the mechanical stresses and theelectric displacements must satisfy the natural boundary conditionsas

rijnj tmechj on Cr; Djnj telec on CD 5Here, the terms with over-bar stand for the prescribed values.Throughout the study, the crack-faces CC are assumed to be trac-the results obtained by the proposed X-FEM are presented, veriedand discussed in details.

The paper is organized as follows. After the introduction, prob-lem statement and asymptotic crack-tip elds in piezoelectricmaterials are briey reported. The X-FEM formulation particularlydeveloped for stationary dynamic crack problems in piezoelectricsolids is presented in Section 3. Section 4 describes the interactionintegral and the computation of the generalized dynamic intensityfactors in piezoelectric materials. The key steps of the numericalsolution procedure are given in the next section. In Section 6, fournumerical examples are presented and discussed in details. Finally,the essential conclusions drawn from the present study are givenin the last section.

2. Problem statement and asymptotic crack-tip eld

2.1. Problem statement

Let us consider a 2D homogeneous and linear piezoelectric solidcontaining a traction-free crack CC CC [ CC occupied by a do-main X R2 bounded by its boundary C with an outward unitnormal vector with the components ni. The boundary C is sub-jected, respectively, to the essential boundary conditions pre-scribed by the displacements on Cu or the electric potential onCu, and to the natural boundary conditions imposed by the trac-tions on Cr or the electric displacements on CD, so thatCu [ Cr = C or CD [ Cu =C. Under the quasi-electrostaticassumption and in the presence of body forces fmech and electricbody charges felec, the equations of motion for the stresses andthe Gauss law for the electric displacements can be written as

rij;j fmechi qui 0; Di;i f elec 0; on X 1where q is the mass density, and i denotes the second time deriv-ative of the displacements or the acceleration, while rij and Di rep-resent the mechanical stress tensor and the electric displacementvector, respectively.

The generalized constitutive equations for homogeneous andlinear piezoelectric materials are [1,4]

rij Cijklekl elijEl; Di eiklekl jilEl 2where Cijkl and jil represent the elastic stiffness tensor and thedielectric permittivities, whereas elij and hlij are the piezoelectricand piezomagnetic coupling coefcients, respectively. The kine-matic relations among the mechanical strain tensor eij and themechanical displacement vector ui as well as the electric eld vectorEi and the scalar electric potential u, are given by

1

erials Science 62 (2012) 243257tion-free and electrically impermeable, i.e.

rijnj 0; Djnj 0 on CC 6

with ui and _u being the initial displacements and velocities,

Matrespectively.

2.2. Asymptotic crack-tip elds in linear piezoelectric materials

Following [30,412] and using polar coordinates r; ~h with theorigin at the crack-tip, the mechanical stress and electrical dis-placement elds for cracks in homogeneous piezoelectric mediacan be expressed as

rijr; ~h 12pr

pXN

KNf Nij ~h; Dir; ~h 12pr

pXN

KNgNi ~h 8

and the near tip displacement eld and electric potential can be gi-ven by

uir; ~h 2rp

r XN

KNdNi ~h; ur; ~h

2rp

r XN

KNvN~h 9

where i, j = 1, 2, and the summation over N = {II, I, III, IV} comprisesthe fracture modes as denoted by KN = {KII,KI,KIII,KIV}T, but KIII isomitted for the 2D case. The standard angular functions f Nij ~h,gNi ~h; dNi ~h and vN~h depending on the material properties aredetermined by means of the generalized Strohs formalism andsemi-analytical calculations. Finally, they can be expressed in termsof complex material eigenvalues p-, eigenvectors AM- and matricesN-N and MM- as

f Ni1 X4-1

ReMi-N-Np-cos~hp- sin~h

q8>:

9>=>;; f Ni2 X4-1

ReMi-N-N

cos~hp- sin~hq

8>:9>=>;

gN1 X4-1

ReM4-N-Np-cos~hp- sin~h

q8>:

9>=>;; gN2 X4-1

ReM4-N-N

cos~hp- sin~hq

8>:9>=>;10

dNi X4-1

Re Ai-N-Ncos ~h p- sin ~h

q ;

vN X4-1

Re A4-N-Ncos ~h p- sin ~h

q 11

where Re{} denotes the real part of the quantity in brackets. Thefour conjugate pairs of the eigenvalues p- and eigenvectors AM-can be derived by solving the following characteristic eigenvalueequation

Ci1k1 ei11e1k1 j11

Ci2k1Ci1k2 ei21ei12

e2k1e1k2 j12j21

p Ci2k2 ei22

e2k2 j22

p2

AiA4

0

12Only the four eigenvalues having positive imaginary part and thecorresponding eigenvectors are used in Eqs. (10) and (11). The(4 4) matrices N-N and MM- are determined by

MM- N1-N Ci2k1 Ci2k2p-Ak- e1i2 e2i2p-A4-e2k1 e2k2p-Ak- j21 j22p-A4-

13

3. X-FEM for stationary dynamic piezoelectric crack problems

The X-FEM model associated with the level set method used forThe initial conditions at time t = 0 are specied as

uit 0 ui0; _uit 0 _ui0 7

T.Q. Bui, C. Zhang / Computationaldescription of the crack geometry [31,32] and the implicit timeintegration scheme is an efcient numerical tool for solving dy-namic crack problems in piezoelectric materials due to its versatilefeature in treating the discontinuities. Within the X-FEM, the niteelement mesh is independent of the crack and the mesh does notrequire to be conformed to the crack-faces, which avoids re-mesh-ing in crack propagation modeling. Associated with the level settechnique, the crack in 2D case is essentially described by two nor-mal and tangent level set functions. The normal level set functionis dened as a signed distance function to the union of the crackand the tangent extension from its front, whereas the tangent levelset function is also a signed distance function but to the surfacethat passes by the crack boundary and normal to the crack. Thus,the crack-faces are determined as the subset of the zero normal le-vel set, where the tangent function is negative, while the crack-tipis dened as the intersection of the two zero level sets.

The enriched nite element approximations, the weak-form andthe discrete system of algebraic equations, the enrichment func-tions particularly used for piezoelectric materials, and the implicittime integration scheme within the framework of the X-FEM arepresented consistently in the following.

3.1. Enriched nite element approximation

The essential idea of the X-FEM is to use a displacement approx-imation that is able to model arbitrary discontinuities and the nearcrack-tip asymptotic elds using the concept of partition of unity[25]. The standard local displacement approximation around thecrack is enriched with discontinuous jump function across thecrack-faces and the asymptotic crack-tip elds around the crack-tip. When the problem domain is discretized by nite elementswith Ns being the nodal set, the extended nite element approxi-mation for the mechanical displacements and electric potentialcan be written explicitly as

uhxXi2NseNixui X

j2NcuteNjxHf hxHfjajX

l2NtipeNlxX6

k1Fk r;~h;lrek ;l

imk

Fk xl ;lrek ;limk

h ibkl

uhxXi2NseNixui X

j2NcuteNjxHf hxHfjcjX

l2NtipeNlxX6

k1Fk r;~h;lrek ;l

imk

Fk xl ;lrek ;limk

h idkl

14

where x = (x,y) in 2D, f(x) represents an implicit function descrip-tion, i.e. a level set, and Ncut and Ntip denote the sets of the enrichednodes associated with crack-faces and of the enriched nodes associ-ated with the crack-tips, respectively, with Ncut \ Ntip = as de-picted in Fig. 1. Also, eNix represents the shape functionsassociated with the node i that construct the partition of unity, uiand ui are the vectors of the nodal degrees of freedom (DOFs) con-taining the nodal displacements and electric potentials dened inthe conventional nite elements, while ai;b

ki and ci;d

ki are the en-

riched DOFs in the elements containing the crack, H(f(x)) is the gen-eralized Heaviside step function enabling the modeling of a crackthat fully cuts a nite element, i.e.,

Hf 1 if f > 01 otherwise

15

and Fk r; ~h;lrek ;limk

are the asymptotic crack-tip enrichment

functions given by Bchet et al. [26]

Fk r; ~h;lrek ;limk

rp g1~h; g2~h; g3~h; g4~h; g5~h; g6~hn o

16with r; ~h being the polar coordinate system at the crack-tip, wherer denes the amplitude from the crack-tip to an arbitrary pointaround the crack-tip, i.e. r = kx xtipk, while lrek and limk are thereal and imaginary parts of a complex number lk, respectively.

To allow for arbitrary poling directions as depicted in Fig. 2, it is~

erials Science 62 (2012) 243257 245controlled by assigning x h h, with h being the orientation ofthe material axes with respect to the crack. The functions gm~hin Eq. (16) are determined by

Mat246 T.Q. Bui, C. Zhang / Computationalgm~hqm x~h;h;lrem ;limm

coswm x~h;h;lrem ;limm

2

if limm >0

qm x~h;h;lrem ;limm

sinwm x~h;h;lrem ;limm

2

if limm 60

8>>>>>: 17The complex numbers lm lrem ilimm , with i

1

pbeing the

imaginary unit, are the six roots of the characteristic equation(see Appendix A) whose imaginary parts are positive. The modied

angle wm x~h; h;lrem ;limm

and the modied radius

qm x~h; h;lrem ;limm

are determined, respectively, by

wm x~h; h;lrem ;limm

p2 pint x

p

arctan cos x pint

xp

lrem sin x pint xp limm

sin x pint xp

0B@1CA 18

qduTudX deTrdX dETDdX

du f dX du f dX du t dC

du t dC 20

tions, a system of discretized piezoelectric nite element equations

i i

Fig. 1. Selection of the enriched nodes for 2D crack problems in a nite elementmesh. Blank circled nodes stored in the set of nodes Ncut are enriched by thediscontinuity function, whereas the blank squared nodes stored in the set of nodesNtip are enriched by the asymptotic crack-tip functions.

Fig. 2. Notation of the material axes at the crack-tip and the polarization direction.for the non-enriched elements.In the above equations, we have denoted by

U fu /gT;a fai cigT; b fbki dki gT and their detailed compo-nents aswithout damping effect can be derived in compact form as

muuu kuuu kuuu fmech

kuuu kuuu felecor Md Kd F 21

whereM, K and d are the global mass and stiffness matrices and theglobal nodal displacement vector of the system. For the enrichedelements, the elementary consistent mass and stiffness matrices(superscript e) are obtained as

meij mUUij m

Uaij m

Ubij

maUij maaij m

abij

mbUij mbaij m

bbij

26643775; keij

kUUij kUaij k

Ubij

kaUij kaaij k

abij

kbUij kbaij k

bbij

2666437775 22

whereas for the non-enriched elements

meij mUUij ; keij kUUij 23

In Eq. (21), F represents the vector of the external nodal forces, andthe element contribution to the global element force vector is givenby

fei fUi fai fbin oT

24

for the enriched elements, whereas

fe fU 25CD

Substituting the enriched approximated functions in Eq. (14)into the weak-form Eq. (20) involving arbitrary virtual displace-ments and electric potential and after some appropriate manipula-X X CrZTelecX X XZT mech

ZT elec

ZT mechqm x~h; h;lrem ;limm

12

pjlmj2 lrem sin2x jlmj2 1

cos2x4

r19

3.2. Weak-form and discrete equations

When introducing the weighting quantities du (virtual displace-ments) and du (virtual electric potential) by using the principle ofvirtual work, time-dependent discrete equations of the X-FEM forthe coupled electromechanical initial-boundary value problemsas presented in Section 2 are generated from the following weak-formZ Z Z

erials Science 62 (2012) 243257mUUij ZXeqeNTi eNjdX; 26

Matmaaij ZXeqeNiHiTeNjHjdX;

mbbij ZXeqeNiFki TeNjFkj dX;

mUaij maUij ZXeqeNiTeNjHjdX;

mUbij mbUij ZXeqeNiTeNjFkj dX;

mabij mbaij ZXeqeNiHiTeNjFkj dX

krsij ZXeBri TCBsj dX; r; s U;a;b 27

fUi Z@XeeNitdC Z

XeeNifdX 28

fai Z@XeeNiHf hx HfitdC Z

XeeNiHf hx HfifdX 29

fbi Z@XeeNi Fk r; ~h;lrem ;limm Fk xi;lrem ;limm h itdC

ZXeeNi Fkr; ~h;lrem ;limm Fk xi;lrem ;limm h ifdX 30

Here, f and t represent the prescribed extended forces per unit vol-ume and the prescribed extended tractions containing (fmech, felec)and tmech;telec, respectively. The matrices of the derivatives ofthe shape functions BUi and B

ai are dened explicitly by

Bsi

eSi;x 0 00 eSi;y 0eSi;y eSi;x 00 0 eSi;x0 0 eSi;y

2666666664

3777777775; s U;a 31

It is noted that in Eq. (31) the term eS is different for BUi and Bai . Forinstance, eS eN when Bsi BUi , and eS eNHf x Hf xi whenBsi Bai , while the matrix of the derivatives of shape function Bbi isslightly different from those, which comprises of six components gi-ven by

Bbi Bb1i Bb2i Bb3i Bb4i Bb5i Bb6i 32in which each Bbki has the same form as the matrices B

Ui and B

ai in Eq.

(31) but eS eN Fk r; ~h;lrem ;limm Fk xi;lrem ;limm h i is imple-mented instead.

3.3. Implicit time integration scheme

The unconditionally stable implicit Newmark time integrationmethod has been widely used in structural dynamics analysisand it is also adopted in this study to solve the discrete dynamicequilibrium equations of the X-FEM at time t + Dt. Eq. (21) is thusrewritten as follows [2]

MdtDt KdtDt FtDt 33The accelerations in the Newmark method without damping effectare given by

M ~bDt2KdtDt FtDt K dt Dt _dt 1 2~bDt2

2dt

34

in which Dt denotes the time-step and _d represents the velocity

T.Q. Bui, C. Zhang / Computationalvector. Once dtDt is determined by Eq. (34), the corresponding vec-tors of the displacements dt+Dt and the velocities _dtDt at the timet + Dt can then be evaluated by usingdtDt dt Dt _dt 1 2~bDt2

2dt ~bDt2dtDt 35

_dtDt _dt 1 ~cDtdt ~cDtdtDt 36In each time-step of the analysis, the values of the displace-

ments, velocities and accelerations are obtained based on the cor-responding known values from the previous time-step. Thisapproach is an implicit direct integration scheme and the choiceof ~cP 0:5 and ~bP 0:25~c 0:52 guarantees the unconditionalstability with second-order accuracy. In all the numerical examplesgiven in the following sections, the time-step Dt is set consistentlyso that acceptable solutions can be achieved.

4. Interaction integral and generalized dynamic intensityfactors

In this study we use the domain-form of the contour interactionintegral to accurately calculate the generalized dynamic intensityfactors in the piezoelectric materials by taking the inertial effectinto account. The amplitudes of the dynamic fracture parametersare characterized by the mechanical stress intensity factors (DSIFs)KII, KI and the electrical displacement intensity factor (DEDIF) KIV.The interaction integral method is an effective tool for calculatingsuch generalized intensity factors in homogeneous piezoelectricmaterials as shown in [30], and we thus extend the method toour dynamic crack problems. To this end, we apply the path-inde-pendent electromechanical J-integral for a cracked homogeneouspiezoelectric body [5,10,12]

J ZC

Wd1j rij @ui@x1

Dj @u@x1

njdC 37

where the indices i and j vary from 1 to 2 in 2D piezoelectric solid,d1j is the Kronecker delta while nj is the jth component of the out-ward unit vector normal to an arbitrary contour C enclosing thecrack-tip, and W = (rijeij DjEj)/2 is the electric enthalpy densityfor a linear piezoelectric material. It is noted that Eq. (37) is validonly for a crack lying in x1-direction.

In order to evaluate J in the nite element analysis, the contourintegral in Eq. (37) is then transformed into an equivalent domain-form by applying the divergence theorem associated with an arbi-trary smooth weight function ~q. Additionally, the equations of mo-tion and the compatibility equations as well as the assumption ofthe traction-free boundary conditions on the crack-faces are alsotaken into account, and after some mathematical manipulationswe nally arrive at

J ZA

rij@ui@x1

Dj @u@x1

Wd1j

@~q@xj

q @2ui@t2

@ui@x1

~q

" #dA 38

with A being the area inside an arbitrary contour enclosing thecrack-tip, while ~q is an arbitrary smooth weighting function, whichhas a value of unity at the crack-tip, zero along the boundary of thedomainA, and a smooth linear variation in-between.

Let us now consider two independent dynamic equilibriumstates of the cracked body. The rst state corresponds to the actualstate under study, whereas the second one corresponds to an aux-iliary state, which may be selected as the asymptotic crack-tipelds of any fracture modes. Superposition of these two statesleads to another dynamic equilibrium state for which the do-main-form of the J-integral is given by

J J1 J2 M1;2 39

erials Science 62 (2012) 243257 247where J(1) and J(2) represent the electromechanical J-integrals for theactual (1) and the auxiliary (2) states, respectively, and

M1;2 ZA

r1ij@u2i@x1

r2ij@u1i@x1

D1j@u2

@x1D2j

@u1

@x1W 1;2d1j

!@~q@xj

dA

ZAq

@2u1i@t2

@u2i@x1

!~qdA

40

5. Key steps of the numerical solution procedure

Only the key steps of the numerical solution procedure of the X-FEMmodel for the stationary dynamic crack problems in 2D homo-geneous piezoelectric solids are outlined as follows:

e21 e22 e16 j11 j22 q

0 0.01 0.02 0.03 0.040.04

0.03

0.02

0.01

0

0.01

0.02

0.03

0.04

(a)

(b)

Fig. 3. A rectangular piezoelectric plate with a central crack under impact loading(a); a regular ne mesh of 5000 quadrilateral elements (b).

248 T.Q. Bui, C. Zhang / Computational Materials Science 62 (2012) 243257is an interaction integral for the two states, in which

W 1;2 12r1ij e

2ij r2ij e1ij D1j E2j D2j E1j

41

According to [27], the electromechanical J-integral for linear piezo-electric materials under mixed-mode loading conditions can bewritten as

J 12KTNYNMKM 42

where K = {KIIKIKIIIKIV}T is the vector of the four intensity factors,and YNM is the (4 4) generalized Irwin matrix, which depends onlyon the material properties and determined by

YNM ImfAM-N-Ng 43with Im{} being the imaginary part of the quantity in brackets, andAM- and N-N are determined by Eqs. (12) and (13) as presented inSubsection 2.2. From Eq. (42), the J-integral can be applied to anydynamic equilibrium state and in 2D problems it can be reduced to

J 12K2IIY11

12K2I Y22

12K2IVY44 KIKIIY12 KIIKIVY14

KIKIVY24 44Applying Eq. (44) to the two states (1) and (2) as J(1) and J(2) andsubstituting them into Eq. (39), then the interaction integral M(1,2)

can be rewritten as

M1;2 K1II K2II Y11 K1I K2I Y22 K1IV K2IV Y44 K1I K2II K1II K2I Y12 K1II K2IV K1IV K2II

Y14

K1I K2IV K1IV K2I

Y24 45

To extract the individual fracture parameters for the actual state,they are done by judiciously choosing the auxiliary state appropri-ately. For instance, if the auxiliary state is taken for the crack open-ing mode, i.e. K2I 1; K2II K2IV 0, then I(1,I) yieldsM1;I K1I Y22 K1II Y12 K1IV Y24 46Similarly, other modes can be obtained as

M1;II K1I Y12 K1II Y11 K1IV Y14M1;IV K1I Y24 K1II Y14 K1IV Y44

47

As a result, the generalized stress intensity factors are obtained bysimultaneously solving the following system of linear algebraicequations

M1;II

M1;I

M1;IV

0B@1CA Y K

1II

K1IK1IV

0BB@1CCA 48

Table 1Material properties with units: Cij (MPa), eij (C/m2), jij (C/GV m) and q (kg/m3).

C11 C22 C66 C12PZT-5H 126.0 117.0 23.0 84.1 6.BaTiO3 150.0 146.0 44.0 66.0 4.50 23.3 17.0 15.04 13.0 750035 17.5 11.4 9.87 11.2 5800

0* IV Present XFEM

(b

Mat0 1 2 3 4 50.5

0

0.5

1

1.5

2

2.5

t cL/h

K* ITDBEMPresent XFEM

=0

(a)Poling direction: =00

0

0.05

0.1

0.15

0.2

K* IV

TDBEMPresent XFEM

=0

(b)Poling direction: =00

T.Q. Bui, C. Zhang / Computational(1) Dene the problem domain containing cracks and input datadening the specimen, material constants, and loadings.

(2) Discretize the problem domain into a set of elements, inwhich the node coordinates and the element connectivityare thus dened.

(3) Dene the normal and tangent level sets through the crackgeometries. Based on the dened normal and tangent levelsets, the enrichment nodes and enrichment elements aredetected and identied. The non-enrichment elements andnodes are also marked.

(4) Specify the nodal information for the essential boundaryconditions and loadings.

(5) Dene the poling direction and compute the matrix of thematerial constants through the specied poling angles.

(6) Solve the characteristic eigenvalue equations as dened in Eq.(12) to determine the four conjugate pairs of the eigenvaluesp- and eigenvectors AM-, and then obtain the matrices N-NandMM-.

(7) Loop over the elementsa. Loop over the quadrature dened based on the basis

of elements.b. Condition 1: non-enriched elements.

Compute the mass matrix accordingly as dened inEqs. (23) and (26)

Compute the stiffness matrix accordingly asdened in Eqs. (23) and (27)

Compute the force vector as dened in Eqs. (25) and(28)

0 1 2 3 4 5

0.2

0.15

0.1

0.05

t cL/h

Fig. 4. Normalizeddynamic stress intensity factor (a) andnormalizeddynamic electricaldisplacement intensity factor (b)versusdimensionless timeforapuremechanical impact.0.5

TDBEMD0=+1.0(8)(9)

(10)

(11)

(12)

(13)

Fig. 5.electricelectrict cL/h

1

1.5)0 1 2 3 4 52

1.5

1

0.5

0

0.5

1

1.5

2

K* I

TDBEMPresent XFEM

D0=1.0

D0=1.0

D0=+1.0

D0=+1.0

(a)Poling direction: =00

erials Science 62 (2012) 243257 249c. Condition 2: enriched elements. Compute the mass matrix accordingly as dened in Eqs.

(22) and (26) Compute the stiffness matrix accordingly as dened in

Eqs. (22) and (27) Compute the force vector as dened in Eqs. (24) and (28),

(29), (30)

d. Assemble the static stiffness matrix, mass matrix andload vector into the global static stiffness matrix, glo-bal mass matrix, and force vector.

e. End the loop over the quadrature.f. End the loop over the elements.

Imposing the boundary conditions.Solve the system of linear algebraic equations to obtain thenodal mechanical displacements, electric potentials, andevaluate the mechanical strains, stresses, electric elds andelectric displacements if necessary.Specify the integration parameters of Newmark algorithm,i.e. ~c; ~b in Subsection 3.3.Calculate some other integration constants of time integra-tion scheme.Form the effective stiffness matrix based on Newmarkalgorithm.Specify the initial conditions for displacement, velocity andacceleration vectors.

0 1 2 3 4 51.5

1

0.5

t cL/h

KD0=1.0 Poling direction: =00

Normalized dynamic stress intensity factor (a) and normalized dynamical displacement intensity factor (b) versus dimensionless time for a pureal impact.

0 1 2 3 4 5

0.5

0

0.5

1

1.5

2

2.5

t cL/h

K* I

=0.0

=0.25

=0.5

=0.0

=0.5

=1.0

=1.0

=0.25

=0.25=0.5

Poling direction: =00

0

0.2

0.4

0.6

0.8

=0.5

=0.25

=0.0

(b) Poling direction: =00

Mat0 1 2 3 4 5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

t cL/h

K* I

TDBEMFEM (ANSYS)Present XFEM

=1.0

(a)

Poling direction: =00

1

0.95

0.9(b) Poling direction: =00

250 T.Q. Bui, C. Zhang / Computational(14) Loop over time-stepsa. Calculate the effective load vector based on Newmark

algorithm.b. Solve for the nodal mechanical displacement vector and

electric potential at each time-step.c. Compute the J-integral and then determine the relevant

dynamic intensity factors including the mechanical stressintensity factors KII, KI and the electrical displacement inten-sity factor KIV.

(15) Visualization and post-processing of the numerical results.

6. Numerical examples

Four benchmark numerical examples for stationary dynamiccracks in piezoelectric solids are presented in the following to illus-trate the accuracy of the developed X-FEM. The accuracy is numer-ically conrmed through the comparison of the normalizeddynamic intensity factors (NDIFs) obtained by the X-FEM withthose available in the literature. The impact loadings includingthe mechanical, the electrical and their combination are consid-ered throughout the study. Numerical calculations in the followingare carried out for two different piezoelectric materials, whoseconstants are given in Table 1 [1517]. Plane-strain conditionand the impermeable crack-face boundary condition are assumed,as well as the piezoelectric material PZT-5H is used throughout thestudy unless stated otherwise. In addition, only regular ne

0 1 2 3 4 51.25

1.2

1.15

1.1

1.05

t cL/h

K* IV

TDBEMFEM (ANSYS)Present XFEM

=1.0

Fig. 6. Comparison of the normalized dynamic stress intensity factor (a) and thenormalized dynamic electrical displacement intensity factor (b) versus dimension-less time for a coupled electromechanical impact among the X-FEM, the FEM andthe BEM.3

3.5TDBEMPresent XFEM =0.25

=0.5(a)

erials Science 62 (2012) 243257meshes of quadrilateral elements are used to ensure the accuracyof the solutions. For numerical integration of the weak form, wemerely adopt the sub-division technique [23,24] for conduct thistask throughout the study. The implicit Newmark time integrationis unconditionally stable but a sufciently small time-step is used

0 1 2 3 4 51.2

1

0.8

0.6

0.4

0.2

t cL/h

K* IV

TDBEMPresent XFEM

=0.25

=0.5

=1.0

Fig. 7. Normalized dynamic stress intensity factor (a) and normalized dynamicelectrical displacement intensity factor (b) versus dimensionless time for differentloading parameter k.

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

t cL/h

K* I

TDGBEMPresent XFEM

=00

=300

=900

=600

Fig. 8. Comparison of the normalized dynamic mode-I stress intensity factor versusdimensionless time for different ration angles h obtained by the TDGBEM [17] andthe X-FEM.

6.1. A central crack in a nite piezoelectric plate

The specimen contains a central crack of length 2a in a homoge-neous and linear piezoelectric plate as depicted in Fig. 3a withh = 40.0 mm and a = 2.4 mm. Three different loadings are consid-ered in the study include (a) an impact tensile mechanical loadingr (t) = r22 = r0H(t), (b) an impact electrical loading D(t) =D2 = D0H(t), or (c) a combination of both impact mechanical andelectrical loadings, where r0 and D0 are the loading amplitudeswhile H(t) denoting the Heaviside step function. The problem issolved by using a regular ne mesh of 50 100 = 5000 quadrilat-eral elements as depicted in Fig. 3b.

6.1.1. Pure mechanical impact loadingWe rst consider the plate subjected to an impact mechanical

loading as the case (a) (i.e. k = 0.0). In this case, the dynamic inten-0 1 2 3 4 51

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

t c /h

K* II

TDGBEMPresent XFEM

=00

=900

=600=300

T.Q. Bui, C. Zhang / Computational Materials Science 62 (2012) 243257 251L

Fig. 9. Comparison of the normalized dynamic mode-II stress intensity factorfor all numerical calculations throughout the study to ensure theaccuracy of the solutions.

0 1 2 3 4 51.2

1

0.8

0.6

0.4

0.2

0

t cL/h

K* IV

TDGBEMPresent XFEM

=300

=00

=900

=600

Fig. 10. Comparison of the normalized dynamic mode-IV electric displacementintensity factor versus dimensionless time for different ration angles h obtained bythe TDGBEM [17] and the X-FEM.

Fig. 11. Scattered elastic waves at four different norm

versus dimensionless time for different ration angles h obtained by the TDGBEM[17] and the X-FEM.sity factors are normalized by

KI KIKstI

; KII KIIKstI

and KIV e22j22

KIVKstI

49

with KstI r0pa

pand a denotes the half-length of the crack. The

numerical results for the normalized dynamic intensity factors(NDIFs) against the dimensionless time t = tcL/h are presented in

Fig. 4, with cL C22 e222=j22=q

qbeing the velocity of the longi-

tudinal wave along the second principal material axis. In our timeintegration algorithm, a very small time-step Dt = 1.02 106 s isused for instance. Fig. 4 presents a comparison of the X-FEM resultsfor the normalized dynamic factors KI and K

IV with those obtained

by the time-domain boundary element method (TDBEM) [15]. TheX-FEM results match well with the TDBEM solutions, and mostimportantly it can be conrmed here that a pure mechanical impactcauses an electrical eld in the considered piezoelectric solids.

6.1.2. Pure electrical impact loadingThe plate in this case is now subjected to an impact electrical

loading as the case (b). The dynamic intensity factors are normal-ized by

KI j22e22

KIKstIV

and KIV KIVKstIV

50

where KstIV D0pa

p. The computed results for the NDIFs are pre-

sented in Fig. 5 in comparison with the TDBEM solutions [15],which shows very good agreement with each other. Here, twoimportant points arising from the numerical results can be ob-served. First, the amplitude of the NDIFs is the same when changingalized time-steps t for the poling angle h = 0.

the direction in the electrical loading controlled by the loadingamplitude D0, and the only change is their sign. Second, the essen-tial difference observed from the numerical results of mechanicaland electrical impact loadings is the tendency when the normalizedtime t? 0. The NDIFs tend to zero as t? 0 for a pure mechanicalimpact loading (see Fig. 4), whereas they tend to nite values whent? 0 for a pure electrical impact (see Fig. 5). Additionally, the KIfactor is negative in some small time ranges, which happens inthe behavior of the KIV factor in the case of an impact mechanicalloading. Once again and most importantly, it is worth noting thata pure electrical impact also induces a dynamic stress intensity

investigation of the effects induced by the electrical loading onthe dynamic fracture parameters. We here consider the sameexample but now the plate is subjected simultaneously to a com-bined mechanical and electrical impact load as the case (c) above.However, the following loading parameter is additionally denedto measure the intensity of the electrical impact

k e22j22

D0r0

51

1.5

2

2.5

K* I

TDBEMPresent XFEM

=1.0

=0.5

=0.25

=0.0(a)

Fig. 12. Scattered elastic waves at four different normalized time-steps t for the poling angle h = 30.

252 T.Q. Bui, C. Zhang / Computational Materials Science 62 (2012) 243257factor.

6.1.3. Combined mechanical and electrical impactsAs well-known that the most important and interesting issue in

studying the fracture behavior of piezoelectric materials is theFig. 13. An edge crack in a nite piezoelectric plate subjected to an impact load.0 1 2 3 4 50

0.5

1

t cL/ht cL/h0 1 2 3 4 5

1.2

1

0.8

0.6

0.4

0.2

0

0.2

0.4

K* IV

TDBEMPresent XFEM

=0.5

=0.25

=0.0

=1.0

(b)

Fig. 14. Normalized dynamic stress intensity factor (a) and normalized dynamicelectrical displacement intensity factor (b) versus dimensionless time for differentloading parameter k.

t c /h

Mat0 1 2 3 4 50

0.5

1

1.5

2

2.5

t cL/h

K* ITDBEM501003070204020201020

(a)

=0

0.5

0.6

0.7TDBEM501003070204020201020

(b)

T.Q. Bui, C. Zhang / ComputationalWe rst consider the intensity of the electrical impact k = 1.0,and the computed results for the NDIFs are presented in Fig. 6 incomparison with the ones obtained by the TDBEM [15] and theFEM using ANSYS software [15]. A very good agreement amongthem is found, which further conrms the high accuracy of thepresent X-FEM. Similar to the TDBEM solutions, it is also seen thatthe X-FEM results contain some peaks and small spikes, which maybe induced by the reected and the scattered elastic waves fromthe top and the bottom boundaries as well as the crack-faces.

Next, we analyze the effects of the intensity of the electrical im-pact loading on the NDIFs. By doing that, the loading parameter k isthus varied and taken as 0.5, 0.25, 0.0, 0.25, 0.5 and 1.0, and theNDIFs are evaluated individually and then depicted in Fig. 7 includ-ing the TDBEM solutions [15]. The comparison shows an excellentagreement between each other for each value of the loadingparameter. The global behaviors of the inuences of the intensityof the electrical impact on the NDIFs obtained by the X-FEM aresimilar to those of the TDBEM. The maximum values of the NDIFsare reduced with increasing k, and it is seen in Fig. 7 that the elec-trical impact affects the KI -factor signicantly. The peak values ofthe NDIFs are decreased with increasing the electrical loading. Ifonly a pure mechanical loading is applied (i.e. k 0:0;KI 0 untilthe mechanical wave impinges on the crack at the normalized timearound t = 1.0. In this case, the elastic waves induced by themechanical impact require some time to reach and open the crack.In contrast, if an electric loading is applied, the variation of the KI

t cL/h0 1 2 3 4 5

0

0.1

0.2

0.3

0.4

K* IV =0

Fig. 15. Normalized dynamic stress intensity factor (a) and normalized dynamicelectrical displacement intensity factor (b) versus dimensionless time for differentmesh sizes.L

0.3

0.35

0.4

0.45(b)0 1 2 3 4 50

0.5

1

1.5

2

2.5

K* I

TDBEMt=1.02106

t=1.02105

t=1.02104

t=1.02103

=0

(a)

erials Science 62 (2012) 243257 253starts from t = 0 due to the quasi-electrostatic assumption for theelectrical eld, which means that the cracked plate is immediatelysubjected to an electrical impact and the crack thus opens at t = 0.The mode-IV factor seems weakly dependent on the time, which isalso a consequence of the quasi-electrostatic assumption of theelectrical eld. As a result, it leads to a strong dependence on theload parameter k.

6.1.4. Poling direction effectThe inuence of the orientation of the material poling direction

with respect to the y-axis on the NDIFs is now analyzed. The inves-tigation is, respectively, carried out for four different polarizationangles such as 0, 30, 60 and 90. It is noted here that the platesize is reset to h = 20.0 mm, and the velocity of the longitudinalwave to cL

C22=q

p, so that the results computed can be com-

pared with those based on the time-domain collocation-Galerkinboundary element method (TDGBEM) [17]. Figs. 810 show a com-parison of the NDIFs derived from both methods, where an excel-lent agreement for all the considered angles is obtained. Asfound by Wnsche et al. in [17] that the normalized static stressintensity factors do not change for different rotation angles,whereas the normalized dynamic stress intensity factors (see Figs.8 and 9) have a signicant dependence on the poling angle h, whichmay be induced by the scattered wave eld. The same behavior isobtained by both methods as observed in Fig. 10 for the normalizedelectrical displacement intensity factor, which implies that when

t cL/h0 1 2 3 4 5

0

0.05

0.1

0.15

0.2

0.25

K* IV

TDBEMt=1.02106

t=1.02105

t=1.02104

t=1.02103 =0

Fig. 16. Normalized dynamic stress intensity factor (a) and normalized dynamicelectrical displacement intensity factor (b) versus dimensionless time for differenttime steps.

Mat(a)

254 T.Q. Bui, C. Zhang / Computationalincreasing the poling angle the electrical displacement intensityfactors increase, and they are equal to zero when h = 90 sincethe piezoelectric effect vanishes for a crack parallel to the polingdirection.

Additionally, for better views Figs. 11 and 12 present, respec-tively, the scattered elastic waves at four different dimensionlesstime-steps, e.g., t = 0.5, 1.16, 2.5 and 4.3, for two different poling

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

0.015

0.01

0.005

0

0.005

0.01

0.015(b)

Fig. 17. A nite piezoelectric plate with a slanted edge crack subjected to an impactload (a); A regular ne mesh of 5000 quadrilateral elements (b).

0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

t cL/h

K* I

TDBEMPresent XFEM

=1.0

=0.5

=0.25

=0.0

=0.5=0.25

=0.0

=1.0

Fig. 18. Normalized dynamic mode-I stress intensity factor versus dimensionlesstime for different loading parameter k.0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t cL/h

K* II

TDBEMPresent XFEM

=1.0

=0.5

=0.25

=0.0

=1.0

=0.5=0.25=0.0

Fig. 19. Normalized dynamic mode-II stress intensity factor versus dimensionlesstime for different loading parameter k.

erials Science 62 (2012) 243257angles h = 0 and h = 30. Basically, the pictures show the propaga-tion of the elastic waves induced by the combined mechanical andelectrical impact loading, due to which the elastic waves start,reach and open the crack and then reect, respectively.

6.2. An edge crack in a nite piezoelectric plate

As the second example, we consider an edge crack parallel tothe top and the bottom boundary of a nite homogeneous piezo-electric plate as depicted in Fig. 13. The geometry of the plate is gi-ven by h = 20.0 mm and the crack-length a = 2.4 mm. The problemis solved by using a regular ne mesh of 50 100 = 5000 quadrilat-eral elements. To take into account the effects of the intensity ofthe electrical impact on the NDIFs, four values of the loadingparameter k = 0, 0.25, 0.5 and 1.0 are examined, respectively, andthe gained results for the NDIFs are presented in Fig. 14 in compar-ison with the TDBEM solutions [15]. Here again, the agreement be-tween both sets of the numerical results is very good and it isfound again that increasing the intensity of the impact electricalloading leads to a decrease of the maximum NDIFs.

In other words, the effects of mesh sensitivity and time-step onthe NDIFs are additionally analyzed, and their calculated results ofthe NDIFs are presented in Figs. 15 and 16, respectively, accounted

0 2 4 6 8 101.4

1.2

1

0.8

0.6

0.4

0.2

0

0.2

t cL/h

K* IV

TDBEMPresent XFEM

=0.25

=0.0

=0.5

=1.0

Fig. 20. Normalized dynamic mode-IV electrical displacement intensity factorversus dimensionless time for different loading parameter k.

1.5(b)

Mat0.005

0.01

0.015(b)

(a)

T.Q. Bui, C. Zhang / Computationalfor the case k = 0. Very good convergences of the NDIFs with re-spect to the mesh can be seen in Fig. 15, and less accuracy onthe DNIFs is found for the coarse meshes as compared with the ref-erence solutions. A similar manner is also found for large time-steps, which essentially reduce the accuracy of the DNIFs as de-picted in Fig. 16. As a consequence, it generally reveals a rigorousrequirement that in order for gaining an acceptable solution anadequately small time-step and a ne mesh must be used in thepresent X-FEM formulation.

6.3. A slanted edge crack in a nite piezoelectric plate

Next, the third example deals with a mixed-mode problem witha slanted edge crack of length a in a homogeneous and linear pie-zoelectric plate as depicted in Fig. 17a. The geometrical parametersof the cracked plate are given by h = 22.0 mm, w = 32.0 mm,c = 6.0 mm and a = 22.63 mm. The crack has an inclination angleof 45 with respect to the vertical plate boundary as shown inthe gure. Similarly, a regular ne mesh of 100 50 = 5000 quad-rilateral elements is applied (see Fig. 17b).

As considered in the previous example, four different values ofthe loading parameter k = 0, 0.25, 0.5 and 1.0 are considered to seehow the intensity of the electrical impact alters the NDIFs in thismixed-mode crack problem. Once again, the computed NDIFs aspresented in Figs. 1820 match well with those derived from theTDBEM [17]. The KI and K

II factors are, however, independent on

the intensity of the electrical impact in two small time ranges as

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040.015

0.01

0.005

0

Fig. 21. A nite piezoelectric plate with two equal collinear cracks subjected to animpact load (a). A regular ne mesh of 6000 quadrilateral elements (b).0.5

1

TDGBEMPresent XFEM

=1.00 1 2 3 4 52

1

0

1

2

3

4

t cL/h

K* I

TDGBEMPresent XFEM

=1.0

=1.0

=0.0

=0.0

=1.0

=1.0

(a)

erials Science 62 (2012) 243257 255observed in the gures. It is slightly different from the behaviorof the curves in the previous example for the KI , whereas the K

IV

factor in both cases is similar.

6.4. Two equal cracks in a nite piezoelectric plate

The last example considers a nite piezoelectric plate with twoequal cracks subjected to a coupled impact tensile and electricalloading on the top and bottom boundary of the plate. The plateis made of BaTiO3 (see Table 1 for its constants) and the geometryis shown in Fig. 21a with h = 16.0 mm, w = 20.0 mm, d = 12.0 mmand a = 2.0 mm. In this example, a regular ne mesh of100 60 = 6000 quadrilateral elements is used, see Fig. 21b. Again,different loading parameters k = 0, 1.0 and 1.0 are examined,respectively, and the corresponding NDIFs at tip B are then pre-sented in Fig. 22. The obtained NDIFs are compared with theTDGBEM solutions [17] and the agreement is very good. As statedin [17], it is again found in the present X-FEM results that the glo-bal behavior of different curves is not much different for the ap-plied loading, but has a signicant jump in the peak values.

Furthermore, Fig. 23 additionally shows the scattered elasticwaves at four different dimensionless time-steps as in the previousexample, i.e., t = 0.5, 1.16, 2.5 and 4.3. Only the poling angle h = 0is considered here for the visualization and a very similar behaviorto those as shown in the rst example (Fig. 11) is observed.

t cL/h0 1 2 3 4 5

1.5

1

0.5

0K* IV

=0.0

=1.0

Fig. 22. Normalized dynamic stress intensity factor (a) and normalized dynamicelectrical displacement intensity factor (b) versus dimensionless time for differentloading parameter k.

nor

Mat7. Conclusions

In this work, transient stationary dynamic crack analysis in 2Dhomogeneous and linear piezoelectric solids is presented. A dy-namic X-FEM integrated with the sixfold enrichment functions as

Fig. 23. Scattered elastic waves at four different

256 T.Q. Bui, C. Zhang / Computationalwell as the implicit time integration scheme is developed for thispurpose. To extract the relevant dynamic intensity factors, aninteraction integral for linear piezoelectric materials utilizing thedomain-form taking the inertial effect into account is imple-mented. To verify the accuracy of the present X-FEM, numerical re-sults for the NDIFs are presented and compared with the TDBEM[15], TDGBEM [17] and FEM [15]. The effects of the combinedmechanical and electrical impacts, polarization direction, meshsensitivity, time-step, etc. on the NDIFs are analyzed and discussedin details. From the numerical results for the NDIFs derived fromthe proposed X-FEM, it can be concluded that the present X-FEMis stable and accurate, and the agreement of the present numericalresults with other available reference solutions is very good. As aresult, the present X-FEM is general and has no limitations onthe crack geometry and loading conditions. As future researchworks, crack growth problems, other electric crack-face boundaryconditions [34], crack-face contact, and multiple cracks in piezo-electric solids under dynamic impact loading conditions wouldbe very interesting and should be simulated by using X-FEM.

In other words, the computational efciency of the X-FEMdeveloped for the dynamic problem is almost dependent on thetime that we specify in the time integration scheme. Just estimat-ing the computational times of solving the equations systems ofthe stiffness and mass matrices as well as the force vector doesnot make too much sense. It is because the meshing tasks of a com-plicated domain by rigorously requiring a conforming mesh to thecrack-faces and re-meshing in crack growth are those that costmost of the human labors and time-consuming works in the con-ventional FEM. Contradictorily, the X-FEM is dominant over theFEM in this particular case due to the mesh independence of thecrack geometry. Nonetheless, further information and other issuesregarding the superior advantages, robustness, convergences, ef-ciency, etc. of the standard or improved X-FEM fashions, one canreach, e.g., see [3538], and many others available in the literature.

Acknowledgment

malized time-steps tfor the poling angle h = 0.

erials Science 62 (2012) 243257The nancial support of the German Research Foundation (DFG)under the Project No. ZH 15/14-1 is gratefully acknowledged.

Appendix A. Derivation of the characteristic equation

In the present work, we restrict our analysis to the plane-straincondition and the constitutive equations are thus expressed asfollows

exx

eyy

cxy

Ex

Ey

8>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>=>>>>>>>>>>>;

a11 a12 0 0 b21

a12 a22 0 0 b22

0 0 a33 b13 0

0 0 b13 d11 0

b21 b22 0 0 d22

266666666664

377777777775

rxx

ryy

sxy

Dx

Dy

8>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>=>>>>>>>>>>>;A:1

in which the coefcients aij, bij and dij indicate the reduced materialconstants and more details can be found in [26].

The complex potential functions U(x) and v(x) are introducedby applying the extended Lekhnitskiis formalism to the piezoelec-tric materials, which are related to the mechanical stresses andelectrical displacements by

rxx @2Ux@y2

; ryy @2Ux@x2

; sxy @2Ux@x@y

A:2

Dx @vx@y

; Dy @vx@x

It should be noted that the equilibrium Eq. (1) without the iner-tial term are automatically satised by Eq. (A.2). Using the consti-tutive equations, in which the stresses and the electric

displacements are expressed through the two complex potentialfunctions U(x) and v(x), the compatibility equations can be re-duced to a sixth order differential equation for U(x) [26,33]

L4L2Ux L3L3Ux 0 A:3where

L2 d22 @2

@x2 d11 @

2

@y2; L3 b22 @

3

@x3 b12 b13 @

3

@x@y2

L4 a22 @4

@x4 a11 @

4

@y4 2a12 a33 @

4

@x2@y2

A:4

The solution U(x) is given by

Ux Ux ly with l lre ilim A:5Substituting the solution Eq. (A.5) into Eq. (A.3), the characteristicequation of the differential equation (A.3) may be expressed interms of l as

a11d11l6 a11d22 2a12 a33d11 b12b12 2b13 b213l4 a22d11 2a12 a33d22 2b22b12 b13l2

a22d22 b222 0 A:6

References

[1] Y.E. Pak, Int. J. Fract. 54 (1992) 79100.[2] T. Hughes, The Finite Element Method Linear Static and Dynamic Finite

Element Analysis, Prentice Hall, Englewood Cliffs, New Jersey, 1987.[3] G. Beer, I. Smith, C. Duenser, The Boundary Element Method with

[9] F. Shang, M. Kuna, M. Abendroth, Eng. Fract. Mech. 70 (2003) 143160.[10] M. Enderlein, A. Ricoeur, M. Kuna, Int. J. Fract. 134 (2005) 191208.[11] O. Gruebner, M. Kamlah, D. Munz, Eng. Fract. Mech. 70 (2003) 13991413.[12] M. Abendroth, U. Groh, M. Kuna, A. Ricoeur, Int. J. Fract. 114 (2002) 359378.[13] E. Pan, Eng. Anal. Bound. Elem. 23 (1999) 6776.[14] R.K.N.D. Rajapakse, X.L. Xu, Eng. Anal. Bound. Elem. 25 (2001) 771781.[15] F. Garcia-Sanchez, Ch. Zhang, A. Saez, Comput. Methods Appl. Mech. Eng. 197

(2008) 31083121.[16] F. Garcia-Sanchez, Ch. Zhang, J. Sladek, V. Sladek, Comput. Mater. Sci. 39 (2007)

179186.[17] M. Wnsche, F. Garcia-Sanchez, A. Saez, Ch. Zhang, Eng. Anal. Bound. Elem. 34

(2010) 377387.[18] K.M. Liew, Y. Sun, S. Kitipornchai, Int. J. Numer. Methods Eng. 69 (2007) 729

749.[19] C.W. Liu, E. Taciroglu, Int. J. Numer. Methods Eng. 67 (2006) 15651586.[20] J. Sladek, V. Sladek, Ch. Zhang, P. Solek, L. Starek, CMES Comput. Model. Eng.

Sci. 19 (2007) 247262.[21] J. Sladek, V. Sladek, Ch. Zhang, P. Solek, E. Pan, Int. J. Fract. 145 (2007) 313326.[22] J. Sladek, V. Sladek, P. Solek, A. Saez, Int. J. Solids Struct. 45 (2008) 45234542.[23] T. Belytschko, T. Black, Int. J. Numer. Methods Eng. 45 (1999) 601620.[24] N. Moes, J. Dolbow, T. Belytschko, Int. J. Numer. Methods Eng. 46 (1999) 131

150.[25] J.M. Melenk, I. Babuska, Comput. Methods Appl. Mech. Eng. 139 (1996) 289

314.[26] E. Bchet, M. Scherzer, M. Kuna, Int. J. Numer. Methods Eng. 77 (2009) 1535

1565.[27] R.R. Bhargava, K. Sharma, Comput. Mater. Sci. 50 (2011) 18341845.[28] R.R. Bhargava, K. Sharma, Int. J. Mech. Des. (2012), http://dx.doi.org/10.1007/

s10999-012-9182-x.[29] R. Rojas-Diaz, N. Sukumar, A. Saez, F. Garcia-Sanchez, Int. J. Numer. Methods

Eng. 88 (2011) 12381259.[30] B.B. Rao, M. Kuna, Int. J. Solids Struct. 45 (2008) 52375257.[31] M. Stolarska, D.L. Chopp, N. Moes, T. Belytschko, Int. J. Numer. Methods Eng. 51

(2001) 943960.[32] G. Ventura, E. Budyn, T. Belytschko, Int. J. Numer. Methods Eng 58 (2003)

15711592.

T.Q. Bui, C. Zhang / Computational Materials Science 62 (2012) 243257 257Programming For Engineers and Scientists, Springer-Verlag/Wien,Germany, 2008.

[4] M. Kuna, Eng. Fract. Mech. 77 (2010) 309326.[5] M. Kuna, Arch. Appl. Mech. 76 (2006) 725745.[6] L. Janski, M. Scherzer, P. Steinhorst, M. Kuna, Int. J. Numer. Methods Eng. 81

(2010) 14921513.[7] M. Scherzer, M. Kuna, Int. J. Fract. 127 (2004) 6199.[8] M. Kuna, Comput. Mater. Sci. 13 (1998) 6780.[33] H.A. Sosa, Y.E. Pak, Int. J. Solids Struct. 26 (1990) 115.[34] J. Sladek, V. Sladek, Ch. Zhang, M. Wunsche, CMES Comput. Model. Eng. Sci.

68 (2010) 158220.[35] T.P. Fries, T. Belytschko, Int. J. Numer. Methods Eng. 84 (2010) 253304.[36] T. Belytschko, R. Gracie, G. Ventura, Model. Simul. Mater. Sci. Eng. 17 (2009)

043001.[37] A. Yazid, N. Abdelkader, H. Abdelmadjid, Appl. Math. Model. 33 (2009) 4269

4282.[38] Y. Abdelaziz, A. Hamouine, Comput. Struct. 86 (2008) 11411151.

Extended finite element simulation of stationary dynamic cracks in piezoelectric solids under impact loading1 Introduction2 Problem statement and asymptotic crack-tip field2.1 Problem statement2.2 Asymptotic crack-tip fields in linear piezoelectric materials

3 X-FEM for stationary dynamic piezoelectric crack problems3.1 Enriched finite element approximation3.2 Weak-form and discrete equations3.3 Implicit time integration scheme

4 Interaction integral and generalized dynamic intensity factors5 Key steps of the numerical solution procedure6 Numerical examples6.1 A central crack in a finite piezoelectric plate6.1.1 Pure mechanical impact loading6.1.2 Pure electrical impact loading6.1.3 Combined mechanical and electrical impacts6.1.4 Poling direction effect

6.2 An edge crack in a finite piezoelectric plate6.3 A slanted edge crack in a finite piezoelectric plate6.4 Two equal cracks in a finite piezoelectric plate

7 ConclusionsAcknowledgmentAppendix A Derivation of the characteristic equationReferences