Research Article A Cell-Based Smoothed XFEM for Fracture ......is paper presents a cell-based...

Research ArticleA Cell-Based Smoothed XFEM for Fracture inPiezoelectric Materials

Li Ming Zhou, Guang Wei Meng, Feng Li, and Shuai Gu

School of Mechanical Science and Engineering, Jilin University, Changchun 130025, China

Correspondence should be addressed to Shuai Gu; [email protected]

Received 29 July 2015; Revised 23 November 2015; Accepted 9 December 2015

Academic Editor: Bert Blocken

Copyright © 2016 Li Ming Zhou et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper presents a cell-based smoothed extended finite element method (CS-XFEM) to analyze fractures in piezoelectricmaterials.Themethod, which combines the cell-based smoothed finite element method (CS-FEM) and the extended finite elementmethod (XFEM), shows advantages of both methods. The crack tip enrichment functions are specially derived to represent thecharacteristics of the displacement field and electric field around the crack tip in piezoelectric materials. With the help of thesmoothing technique, integrating the singular derivatives of the crack tip enrichment functions is avoided by transforming interiorintegration into boundary integration. This is a significant advantage over XFEM. Numerical examples are presented to highlightthe accuracy of the proposed CS-XFEM with the analytical solutions and the XFEM results.

1. Introduction

Because of their inherent coupling of electric andmechanicalbehaviors, piezoelectric materials have been widely used insensors, actuators, signal transmitters and surface acousticwave devices, aerospace panels, and civil structures. In thoseapplications, piezoelectric materials may experience highmechanical stresses and electric field concentrations. As aresult they may fail due to dielectric breakdown or fractures.These materials are usually inhomogeneous and brittle, withlow ultimate tensile strength and fracture toughness. There-fore, defects such as cracks and voids should be detected toensure reliability and durability of the piezoelectric struc-tures. Numerical simulation of fractures in piezoelectricceramics was conducted in [1], using finite element method(FEM) [2, 3], boundary element method (BEM) [4, 5],meshless method [6], and extended finite element method(XFEM) [7–9].

The theoretical fundamentals of piezoelectric fracturemechanics for cracks were presented in [10, 11].The analyticalwork to investigate the fracture mechanics of piezoelectricstructures was based on Lekhnitskii and Stroh formalism[12]. An elliptic hole with a major axis perpendicular tothe polarization direction inside piezoelectric ceramic andthe field variables around the cavity were studied in [13].

An overview and critical discussion about the present statein the field of piezoelectric fracture mechanics were givenin [14]. A survey on using numerical methods of crackanalyses in piezoelectric medium along with FEM to solvefracture parameters is presented in [15]. Recently, XFEMwas applied to analyze the 2D crack problems and fullycoupled piezoelectric effect of piezoelectric ceramics [16].The M integral and J integral were used to solve the stressintensity factors and electric displacements intensity factors[17, 18]. The newly developed crack tip enrichment functionsof XFEM were found suitable for cracks in piezoelectricmaterials [19]. An extension of XFEM for dynamic fracturein piezoelectric materials was presented in [20].

In recent years, the smoothed FEMhas been well adoptedto solve fracture mechanics problem [21–27]. Based ongeneralized gradient smoothing technique a lot of noveland powerful numerical methods have been developed [28–36]. A cell-based smoothed finite element method (CS-FEM) has been developed with the smoothing domainsconstructed based on cell of the elements. In the method lineintegration was used along the boundaries of the smoothingcells instead of area integration. Moreover, CS-FEM does notneed mapping and derivatives. It also showed that the resultsare less sensitive to distorted elements.

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2016, Article ID 4125307, 14 pageshttp://dx.doi.org/10.1155/2016/4125307

2 Advances in Materials Science and Engineering

In terms of the advantages, smoothing technique wasincorporated into the extended FEM [37–39]. An edge-basedsmoothed XFEM was developed to combine the advantagesof the edge-based smoothed FEM and the XFEM [40]. Anode-based smoothed XFEM was applied to linear elasticfracture mechanics [41].

Recently, extended finite element method, collocationboundary element, cell-based smoothed finite elementmethod, and so forth were used to solve the problemof fracture in piezoelectric materials. The extended finiteelement method focuses on the definition of new enrichmentfunctions suitable for cracks in piezoelectric structures andgeneralized domain integrals are used for the determinationof crack tip parameters [42]. The collocation boundaryelement with subdomain technique is developed, wherebythe fundamental solutions are computed by a fast numer-ical algorithm applying Fourier series [43]. The cell-basedsmoothed finite element method and VCCT have been usedto simulate the fracture mechanics of piezoelectric materials.A piezoelectric element tailored for VCCT was used to studythe crack of piezoelectricmaterials. CS-FEM andVCCTwereintroduced into fracture mechanics of piezoelectric materialsand CSFEM-VCCT for piezoelectric material with cracks wasput forward [44]. In this paper cell-based smoothed XFEM(CS-XFEM) is extended to simulate flaws in piezoelectricstructures. CS-XFEM combining characteristics of extendedfinite element method and smoothed finite element methodcan improve the accuracy of XFEM. Nodal enrichment canmodel crack propagation without remeshing. CS-XFEM hasadvantages that the crack tip element does not need finedivision; the shape function is simple and no derivatives ofshape functions are needed.

This paper is outlined as follows. In Section 2, the gov-erning equations of piezoelectric materials are introduced.Section 3 focuses on the formulation of cell-based smoothedextended finite element method. Section 4 presents the elec-tromechanical 𝐽-integral for 2D crack analysis. In Section 5,numerical examples with the assumption of impermeablecrack face boundary conditions are presented to demonstratethe accuracy and efficiency of CS-XFEM. Section 6 is theconclusion.

2. Governing Equations

The electroelastic response of a piezoelectric body of volumeΩ and regular boundary surface 𝑆 is governed by themechanical and electrostatic equilibrium equations:

𝜎𝑖𝑗,𝑗+ 𝑓

𝑖= 0 in Ω,

𝐷𝑖,𝑖− 𝑡 = 0 in Ω,

(1)

where 𝑓𝑖is mechanical body force, 𝑡 is electric body charge,

𝜎𝑖𝑗is the symmetric Cauchy stress tensor, and 𝐷

𝑖is electric

displacement vector components.

⊕

⊕

⊕⊕

⊕⊕

⊕

Q

F

Γn

ΓnC+

C−

Ω

Γeu = u

𝜙 = 𝜙

Figure 1: Piezoelectric domain with a crack.

The constitutive equations for a two-dimensional piezo-electric material in the 𝑥-𝑧 plane can be expressed in terms ofthe stress and the electric field:

𝜎𝑖𝑗= 𝐶

𝑖𝑗𝑘𝑙𝜀𝑘𝑙− 𝑒

𝑘𝑖𝑗𝐸𝑘,

𝐷𝑖= 𝑒

𝑖𝑘𝑙𝜀𝑘𝑙+ 𝜅

𝑖𝑘𝐸𝑘,

(2)

where 𝜀𝑘𝑙, 𝐷

𝑖, and 𝐸

𝑘are the strain tensor, the electric

displacement vector, and the electric field vector, respectively;𝐶𝑖𝑗𝑘𝑙

, 𝑒𝑘𝑖𝑗, and 𝜅

𝑖𝑘denote elastic stiffness at constant electric

field, piezoelectric constants, and dielectric permittivity atconstant strain, respectively.

The strains tensor are related to displacements by

𝜀𝑖𝑗=(𝑢𝑖,𝑗+ 𝑢

𝑗,𝑖)

2

(3)

and the electric field vector is related to electric potential by

𝐸𝑖= −𝜙

,𝑖. (4)

The piezoelectric bodyΩ could be subjected to the followingessential and natural boundary conditions (Figure 1).

Essential Boundary Conditions. Consider

𝑢 = 𝑢

or 𝜙 = 𝜙

on Γ𝑒

(5)

Natural Boundary Conditions. Consider

𝜎𝑖𝑗𝑛𝑗= f

𝑖

(or) 𝐷𝑖𝑛𝑖 = −g

on Γ𝑛,

(6)

where 𝑢, 𝜙, f𝑖, g, and 𝑛

𝑖are mechanical displacement, electric

potential, surface force components, surface charge, and

Advances in Materials Science and Engineering 3

outward unit normal vector components, respectively. Thecrack facesC+ andC− are considered traction-free.The cracksare assumed to be electrically impermeable. Nevertheless,extension to limited permeable cracks is possible.

The two-dimensional matrix form of the mechanical andelectrical constitutive equations can be given by [15]

[[

[

𝜎𝑥

𝜎𝑧

𝜏𝑥𝑧

]]

]

=[[

[

C11

C130

C31

C330

0 0 C55

]]

]

[[

[

𝜀𝑥

𝜀𝑧

𝛾𝑥𝑧

]]

]

−[[

[

0 e31

0 e33

e150

]]

]

[𝐸𝑥

𝐸𝑧

] ,

[𝐷𝑥

𝐷𝑧

] = [0 0 e

15

e31

e330][[

[

𝜀𝑥

𝜀𝑧

𝛾𝑥𝑧

]]

]

+ [𝜅110

0 𝜅33

][𝐸𝑥

𝐸𝑧

] ,

(7)

where C𝑖𝑗are the elastic compliance constants, e

𝑖𝑗are piezo-

electric constants, and 𝜅𝑖𝑖are the dielectric constants.

3. Cell-Based Smoothed Extended FiniteElement Method

In CS-XFEM, the approximation of displacement and electricpotential field in a piezoelectric material are given by

uℎ (x) = ∑𝐼∈𝑁

CS-FEM

𝑁𝑢

𝐼(x) u𝐼

+ ∑

𝐽∈𝑁CS-𝑐

𝑁𝑢

𝐽(x) (𝐻 (x) − 𝐻 (x𝐽)) a𝐽

+ ∑

𝐾∈𝑁CS-𝑓

𝑁𝑙

𝐾(x)

4

∑

𝑙=1

(F𝑙 (x) − F𝑙 (x𝐾)) b

𝑙

𝐾,

(8)

Φℎ(x) = ∑

𝐼∈𝑁CS-FEM

𝑁𝜙

𝐼(x)Φ𝐼

+ ∑

𝐽∈𝑁CS-𝑐

𝑁𝜙

𝐽(x) (𝐻 (x) − 𝐻 (x𝐽))𝛼𝐽

+ ∑

𝐾∈𝑁CS-𝑓

𝑁𝜙

𝐾(x)

4

∑

𝑙=1

(F𝑙 (x) − F𝑙 (x𝐾))𝛽

𝑙

𝐾,

(9)

where 𝑁𝑢𝐼(x), 𝑁𝑢

𝐽(x), and 𝑁𝑙

𝐾(x) are shape functions of

the nodal displacement, while u𝐼are the nodal degrees

of freedom associated with node 𝐼, and a𝐽and b

𝐾are

additional nodal degrees of freedom corresponding to theHeaviside function𝐻(x) and the near-tip functions, {F

𝑙}1≤𝑙≤4

,respectively.𝑁𝜙

𝐼(x),𝑁𝜙

𝐽(x), and𝑁𝜙

𝐾(x) are shape functions of

the nodal electric potential, while Φ𝐼are the nodal degrees

of freedom associated with node 𝐼, and 𝛼𝐽and 𝛽

𝐾are

additional nodal degrees of freedom corresponding to theHeaviside function𝐻(x) and the near-tip functions, {F

𝑙}1≤𝑙≤4

,respectively.

Nodes in set𝑁CS-𝑐 have supports split by crack and nodesin set𝑁CS-𝑓 which belong to the smoothing domains containa crack tip. These nodes are enriched with the Heavisideand asymptotic branch function fields depicted with squares

Crack

Split element

Standard element

Split-blending element

Centroid

Normal node

Crack tip

Tip elementTip-blending element

Node enrichment with H function

Node enrich with F

Figure 2: Classification of smoothing domains in CS-XFEM.

and circles, respectively. The support domain of 𝑁CS-FEM isassociated with nodes of CS-FEM, shown in Figure 2.

As illustrated in Figure 2, four-node quadrilateral cells areused for cell-based strain smoothing operation.The meshingcharacteristics of CS-XFEM are consistent with XFEM. Thecomplex structure can adopt the fine mesh. For CS-FEM,the number of subcells (SC) would affect the performanceof the results. In the case that the solution of SFEM (SC= 1) is overestimated to the exact solution, there exists oneoptimal value SC > 1 (normally SC = 4) which gives the bestresults as compared to the exact ones. In the case that thesolution of SFEM (SC = 1) is underestimated to the exactsolution, it is suggested that we should use SC = 2 to obtainthe solution.This solution will be stable and have the smallestdisplacement and energy norms. In practical calculation, wecan use SC = 4 for all problems. The results will always bebetter than that of standard FEM. And in many cases (notall) this solution (SC = 1) is closest to the exact solution [21].So we can use four subcells for each quadrilateral.

In CS-XFEM, Heaviside enriched degrees of freedom areadded to nodes in𝑁CS-𝑐 whose support domain is split by thecrack and tip enriched degrees of freedom are added to nodesin set𝑁CS-𝑓 whose support domain contains the crack tip. Inorder to keep the convergence rate as high as possible, a so-called geometric enrichment which is independent from thediscretization is used [45–47]:

𝐻(x) ={

{

{

1 (x − x∗) ⋅ n ≥ 0

−1 otherwise,(10)

where x∗ is a point on the crack surface; see Figure 3.For piezoelectric problems, it is advisable to use the

regular enrichment functions stemming from the isotropicelasticity. It should be mentioned that similar results havebeen obtained independently with alternative enrichmentfunctions for cracks in confined plasticity problems [42].The


S

x

enes

x∗

Figure 3: Normal and tangential coordinates for a crack.

Crack

Crack tip

x

r

e2

e1𝜃

Figure 4: Polar coordinate system associated with a crack tip.

near-tip enrichment consists of functions which incorporatethe radial and angular behaviors of the two-dimensionalasymptotic crack tip displacement field [37, 48]:

{F𝑙}1≤𝑙≤4

= √𝑟{sin(𝜃2) cos(𝜃

2) cos (𝜃) sin(𝜃

2) sin (𝜃) cos(𝜃

2)} ,

(11)

where 𝑟 and 𝜃 are polar coordinates in the local crack tipcoordinate system; see Figure 4.

Employing the strain smoothing operation, the smoothedstrain in the domain Ω

𝑐from the displacement approxima-

tion in (8) can be written in the following matrix form:

𝜀𝑘= ∑

𝐼∈𝑁CS-FEM

B𝑢𝐼(x𝑘) u

𝐼

+ ∑

𝐽∈𝑁CS-𝑐

B𝑎𝐽(x𝑘) (𝐻 (x) − 𝐻 (x𝐽)) a𝐽

+ ∑

𝐾∈𝑁CS-𝑓

B𝑏𝐾(x𝑘)

4

∑

𝑙=1

(F𝑙 (x) − F𝑙 (x𝐾)) b

𝑙

𝐾,

(12)

where B𝑢𝐼(x𝑘) is the smoothed strain gradient matrix for the

standard CS-FEM part; B𝑎𝐽(x𝑘) and B𝑏

𝐾(x𝑘) correspond to the

enriched parts of the smoothed strain gradient matrix asso-ciated with the Heaviside and branch functions, respectively.

When employing the electric field smoothing operation,the smoothed electric field in the domain Ω

𝑐from the

displacement approximation in (9) can be written in thefollowing matrix form:

E𝑘= ∑

𝐼∈𝑁CS-FEM

B𝜙𝐼(x𝑘) u

𝐼

+ ∑

𝐽∈𝑁CS-𝑐

B𝛼𝐽(x𝑘) (𝐻 (x) − 𝐻 (x𝐽))𝛼𝐽

+ ∑

𝐾∈𝑁CS-𝑓

B𝛽𝐾(x𝑘)

4

∑

𝑙=1

(F𝑙 (x) − F𝑙 (x𝐾))𝛽

𝑙

𝐾,

(13)

where B𝜙𝐼(x𝑘) is the smoothed electric field gradient matrix

for the standardCS-FEMpart;B𝛼𝐽(x𝑘) andB𝛽

𝐾(x𝑘) correspond

to the enriched parts of the smoothed electric field gradientmatrix associated with the Heaviside and branch functions,respectively. These matrixes can be written as follows:

B𝑟𝐼(x𝑘) =[[[

[

𝑏𝑟

𝐼𝑥(x𝑘) 0

0 𝑏𝑟

𝐼𝑦(x𝑘)

𝑏𝑟

𝐼𝑦(x𝑘) 𝑏

𝑟

𝐼𝑥(x𝑘)

]]]

]

, 𝑟 = 𝑢, 𝑎, 𝑏,

B𝑡𝐼(x𝑘) = [

𝑏𝑡

𝐼𝑥(x𝑘)

𝑏𝑡

𝐼𝑦(x𝑘)] , 𝑡 = 𝜙, 𝛼, 𝜙,

(14)

where

𝑏𝑢

𝐼ℎ(x𝑘) =1

𝐴𝑠𝑘

∫Γ𝑠

𝑘

𝑛ℎ(x𝑘)𝑁

𝑢

𝐼𝑑Γ𝑠=

𝑁seg

∑

𝑚=1

(

𝑁gau

∑

𝑛=1

𝑛ℎ(x𝑚,𝑛)

⋅ 𝑁𝑢

𝐼(x𝑚,𝑛) 𝑤

𝑚,𝑛) , ℎ = 𝑥, 𝑧,

𝑏𝑎

𝐼ℎ(x𝑘) =1

𝐴𝑠𝑘

∫Γ𝑠

𝑘

𝑛ℎ(x𝑘)𝑁

𝑢

𝐼(𝐻 (x) − 𝐻 (x𝐼)) 𝑑Γ

𝑠

=

𝑁seg

∑

𝑚=1

(

𝑁gau

∑

𝑛=1

𝑛ℎ(x𝑚,𝑛)𝑁

𝑢

𝐼(x𝑚,𝑛) (𝐻 (x

𝑚,𝑛) − 𝐻 (x

𝐼))

⋅ 𝑤𝑚,𝑛) , ℎ = 𝑥, 𝑧,

𝑏𝑏

𝐼ℎ(x𝑘) =1

𝐴𝑠𝑘

∫Γ𝑠

𝑘

𝑛ℎ(x𝑘)𝑁

𝑢

𝐼

4

∑

𝑙=1

(F𝑙 (x) − F𝑙 (x𝐾)) 𝑑Γ

𝑠

=

𝑁seg

∑

𝑚=1

(

𝑁gau

∑

𝑛=1


𝑢

𝐼(x𝑚,𝑛) (F

𝑙(x𝑚,𝑛) − F

𝑙(x𝐼))

⋅ 𝑤𝑚,𝑛) , ℎ = 𝑥, 𝑧, 𝑙 = 1, 2, 3, 4,

𝑏𝜙

𝐼ℎ(x𝑘) =1

𝐴𝑠𝑘

∫Γ𝑠

𝑘

𝑛ℎ(x𝑘)𝑁

𝐼𝑑Γ𝑠=

𝑁seg

∑

𝑚=1

(

𝑁gau

∑

𝑛=1

𝑛ℎ(x𝑚,𝑛)

⋅ 𝑁𝜙

𝐼(x𝑚,𝑛) 𝑤

𝑚,𝑛) , ℎ = 𝑥, 𝑧,

𝑏𝛼

𝐼ℎ(x𝑘) =1

𝐴𝑠𝑘

∫Γ𝑠

𝑘

𝑛ℎ(x𝑘)𝑁

𝜙

𝐼(𝐻 (x) − 𝐻 (x𝐼)) 𝑑Γ

𝑠

=

𝑁seg

∑

𝑚=1

(

𝑁gau

∑

𝑛=1


𝜙

𝐼(x𝑚,𝑛) (𝐻 (x

𝑚,𝑛) − 𝐻 (x

𝐼))

⋅ 𝑤𝑚,𝑛) , ℎ = 𝑥, 𝑧,


𝑏𝛽

𝐼ℎ(x𝑘) =1

𝐴𝑠𝑘

∫Γ𝑠

𝑘

𝑛ℎ(x𝑘)𝑁

𝜙

𝐼

4

∑

𝑙=1

(F𝑙 (x) − F𝑙 (x𝐾)) 𝑑Γ

𝑠

=

𝑁seg

∑

𝑚=1

(

𝑁gau

∑

𝑛=1


𝜙

𝐼(x𝑚,𝑛) (F

𝑙(x𝑚,𝑛) − F

𝑙(x𝐼))

⋅ 𝑤𝑚,𝑛) , ℎ = 𝑥, 𝑧, 𝑙 = 1, 2, 3, 4,

(15)

where 𝑁seg is the number of segments of the boundary Γ𝑠

𝑘,

𝑁gau is the number of Gauss points used in each segment,𝑤𝑚,𝑛

is the corresponding Gauss weights, 𝑛𝑥and 𝑛

𝑧are the

outward unit normal components to each segment on thesmoothing domain boundary, and 𝑥

𝑚,𝑛is the 𝑛th Gaussian

point on the𝑚th segment of the boundary Γ𝑠𝑘.

The standard discrete system of equations is obtained:

[[[

[

K𝑢𝑢𝐼𝐽

K𝑢𝑎𝐼𝐽

K𝑢𝑏𝐼𝐽

K𝑎𝑢𝐼𝐽

K𝑎𝑎𝐼𝐽

K𝑎𝑏𝐼𝐽

K𝑏𝑢𝐼𝐽

K𝑏𝑎𝐼𝐽

K𝑏𝑏𝐼𝐽

]]]

]

{{

{{

{

uab

}}

}}

}

+

[[[[

[

K𝑢𝜙𝐼𝐽

K𝑢𝛼𝐼𝐽

K𝑢𝛽𝐼𝐽

K𝑎𝜙𝐼𝐽

K𝛼𝛼𝐼𝐽

K𝛼𝛽𝐼𝐽

K𝑏𝜙𝐼𝐽

K𝑏𝛼𝐼𝐽

K𝑏𝛽𝐼𝐽

]]]]

]

{{

{{

{

Φ

𝛼

𝛽

}}

}}

}

=

{{{

{{{

{

ff𝑎

f𝑏

}}}

}}}

}

,

[[[

[

K𝜙𝑢𝐼𝐽

K𝜙𝑎𝐼𝐽

K𝜙𝑏𝐼𝐽

K𝛼𝑢𝐼𝐽

K𝛼𝑎𝐼𝐽

K𝛼𝑏𝐼𝐽

K𝛽𝑢𝐼𝐽

K𝛽𝑎𝐼𝐽

K𝛽𝑏𝐼𝐽

]]]

]

{{

{{

{

uab

}}

}}

}

−

[[[[

[

K𝜙𝜙𝐼𝐽

K𝜙𝛼𝐼𝐽

K𝜙𝛽𝐼𝐽

K𝛼𝜙𝐼𝐽

K𝛼𝛼𝐼𝐽

K𝛼𝛽𝐼𝐽

K𝛽𝜙𝐼𝐽

K𝛽𝛼𝐼𝐽

K𝛽𝛽𝐼𝐽

]]]]

]

{{

{{

{

Φ

𝛼

𝛽

}}

}}

}

=

{{{

{{{

{

gg𝑎

g𝑏

}}}

}}}

}

,

(16)

where

K𝑢𝑢𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝑢𝐼)𝑇

CB𝑢𝐽𝑑Ω =

𝑁𝑠

∑

𝑘=1

(B𝑢𝐼)𝑇

CB𝑢𝐽𝐴𝑠

𝑘,

K𝑢𝑎𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝑢𝐼)𝑇

CB𝑎𝐽𝑑Ω =

𝑁𝑠

∑

𝑘=1

(B𝑢𝐼)𝑇

CB𝑎𝐽𝐴𝑠

𝑘

= (K𝑎𝑢𝐼𝐽)𝑇

,

K𝑢𝑏𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝑢𝐼)𝑇

CB𝑏𝐽𝑑Ω =

𝑁𝑠

∑

𝑘=1

(B𝑢𝐼)𝑇

CB𝑏𝐽𝐴𝑠

𝑘

= (K𝑏𝑢𝐼𝐽)𝑇

,

K𝑎𝑎𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝑎𝐼)𝑇

CB𝑎𝐽𝑑Ω =

𝑁𝑠

∑

𝑘=1

(B𝑎𝐼)𝑇

CB𝑎𝐽𝐴𝑠

𝑘,

K𝑎𝑏𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝑎𝐼)𝑇

CB𝑏𝐽𝑑Ω =

𝑁𝑠

∑

𝑘=1

(B𝑎𝐼)𝑇

CB𝑏𝐽𝐴𝑠

𝑘

= (K𝑏𝑎𝐼𝐽)𝑇

,

K𝑏𝑏𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝑏𝐼)𝑇

CB𝑏𝐽𝑑Ω =

𝑁𝑠

∑

𝑘=1

(B𝑏𝐼)𝑇

CB𝑏𝐽𝐴𝑠

𝑘,

K𝑢𝜙𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝑢𝐼)𝑇

eB𝜙𝐽𝑑Ω =

𝑁𝑠

∑

𝑘=1

(B𝑢𝐼)𝑇

eB𝜙𝐽𝐴𝑠

𝑘

= (K𝜙𝑢𝐼𝐽)𝑇

,

K𝑢𝛼𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝑢𝐼)𝑇

eB𝛼𝐽𝑑Ω =

𝑁𝑠

∑

𝑘=1

(B𝑢𝐼)𝑇

eB𝛼𝐽𝐴𝑠

𝑘

= (K𝛼𝑢𝐼𝐽)𝑇

,

K𝑢𝛽𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝑢𝐼)𝑇

eB𝛽𝐽𝑑Ω =

𝑁𝑠

∑

𝑘=1

(B𝑢𝐼)𝑇

eB𝛽𝐽𝐴𝑠

𝑘

= (K𝛽𝑢𝐼𝐽)𝑇

,

K𝑎𝜙𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝑎𝐼)𝑇

eB𝜙𝐽𝑑Ω =

𝑁𝑠

∑

𝑘=1

(B𝑎𝐼)𝑇

eB𝜙𝐽𝐴𝑠

𝑘

= (K𝜙𝑎𝐼𝐽)𝑇

,

K𝑎𝛼𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝑎𝐼)𝑇

eB𝛼𝐽𝑑Ω =

𝑁𝑠

∑

𝑘=1

(B𝑎𝐼)𝑇

eB𝛼𝐽𝐴𝑠

𝑘

= (K𝑢𝑎𝐼𝐽)𝑇

,

K𝑎𝛽𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝑎𝐼)𝑇

eB𝛽𝐽𝑑Ω =

𝑁𝑠

∑

𝑘=1

(B𝑎𝐼)𝑇

eB𝛽𝐽𝐴𝑠

𝑘

= (K𝛽𝑎𝐼𝐽)𝑇

,

K𝑏𝜙𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝑏𝐼)𝑇

eB𝜙𝐽𝑑Ω =

𝑁𝑠

∑

𝑘=1

(B𝑏𝐼)𝑇

eB𝜙𝐽𝐴𝑠

𝑘

= (K𝜙𝑏𝐼𝐽)𝑇

,

K𝑏𝛼𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝑏𝐼)𝑇

eB𝛼𝐽𝑑Ω =

𝑁𝑠

∑

𝑘=1

(B𝑏𝐼)𝑇

eB𝛼𝐽𝐴𝑠

𝑘

= (K𝑢𝑏𝐼𝐽)𝑇

,

K𝑏𝛽𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝑏𝐼)𝑇

eB𝛽𝐽𝑑Ω =

𝑁𝑠

∑

𝑘=1

(B𝑏𝐼)𝑇

eB𝛽𝐽𝐴𝑠

𝑘

= (K𝛽𝑏𝐼𝐽)𝑇

,


K𝜙𝜙𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝜙𝐼)𝑇

𝜅B𝜙𝐽𝑑Ω =

𝑁𝑠

∑

𝑘=1

(B𝜙𝐼)𝑇

𝜅B𝜙𝐽𝐴𝑠

𝑘,

K𝜙𝛼𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝜙𝐼)𝑇

𝜅B𝛼𝐽𝑑Ω =

𝑁𝑠

∑

𝑘=1

(B𝜙𝐼)𝑇

𝜅B𝛼𝐽𝐴𝑠

𝑘

= (K𝛼𝜙𝐼𝐽)𝑇

,

K𝜙𝛽𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝜙𝐼)𝑇

𝜅B𝛽𝐽𝑑Ω =

𝑁𝑠

∑

𝑘=1

(B𝜙𝐼)𝑇

𝜅B𝛽𝐽𝐴𝑠

𝑘

= (K𝛽𝜙𝐼𝐽)𝑇

,

K𝛼𝛼𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝛼𝐼)𝑇

𝜅B𝛼𝐽𝑑Ω =

𝑁𝑠

∑

𝑘=1

(B𝛼𝐼)𝑇

𝜅B𝛼𝐽𝐴𝑠

𝑘,

K𝛼𝛽𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝛼𝐼)𝑇


𝑁𝑠

∑

𝑘=1

(B𝛼𝐼)𝑇


𝑘

= (K𝛽𝛼𝐼𝐽)𝑇

,

K𝛽𝛽𝐼𝐽=

𝑁𝑠

∑

𝑘=1

∫Ω𝑠

𝑘

(B𝛽𝐼)𝑇


𝑁𝑠

∑

𝑘=1

(B𝛽𝐼)𝑇


𝑘.

(17)

4. Electromechanical 𝐽-Integral

According to research by Rice [49] and Eshelby [50] on puremechanical applications,

F𝑘= ∮

𝑠

(𝐻𝛿𝑘𝑗− 𝜎

𝑖𝑗𝑢𝑖,𝑘+ 𝐷

𝑗𝐸𝑘) 𝑛

𝑗𝑑𝑠, (18)

𝐻 =1

2(𝜎𝑖𝑗𝜀𝑖𝑗− 𝐷

𝑖𝐸𝑖) . (19)

Equation (18) describes thematerial forcewhen an electrome-chanically loaded domain is virtually displaced by the vector𝛿𝑥𝑗. The term in brackets is called the piezoelectric energy

momentum tensor:

𝑄𝑘𝑗= 𝐻𝛿

𝑘𝑗− 𝜎


𝑗𝐸𝑘, (20)

analogous to the energy momentum tensor, which wasintroduced by Eshelby [50]. If the integration path 𝑆 containsneither defects nor source terms 𝑏

𝑖and 𝜔

Ω, the force vanishes

and F𝑘≡ 0. Now we consider a path Γ

𝜀enclosing the tip of a

crack; see Figure 5.The electromechanical 𝐽em𝑘

-integral vectoris the material force associated with the crack tip singularity.It is defined as the limit when Γ

𝜀is shrunk towards 𝑟 → 0

𝐽em𝑘= lim𝑟→0

∫Γ𝜀

𝑄𝑘𝑗𝑛𝑗𝑑𝑠

= lim𝑟→0

∫Γ𝜀



𝑗𝐸𝑘) 𝑛

𝑗𝑑𝑠.

(21)

r

x

z

nj

nj

Γ

Γ+

Γ−

Γ𝜀

Ω

Crack face (+)

Crack face (−)

Figure 5: Combined integration path around a crack tip.

For the numerical analysis of the 𝐽em𝑘

-integral, an equiv-alent domain integral is easier to handle than a line integral.The condition for the transformation into a domain integralapplying the Gaussian integral theorem, a closed integrationpath 𝑆 = Γ − Γ

𝜀+ Γ

++ Γ

− with an outward pointing normalvector is given, as can be seen in Figure 5. Then (21) can bewritten as follows:

𝐽em𝑘= −∫

𝑠

𝑄𝑘𝑗𝑛𝑗𝑑𝑠 + ∫

Γ+Γ++Γ−

𝑄𝑘𝑗𝑛𝑗𝑑𝑠. (22)

Now a weighting function 𝑞 is introduced, which should becontinuous and satisfy the condition

𝑞 ={

{

{

0 on Γ

1 on Γ𝜀.

(23)

The calculation of the 𝐽-integral is carried out in a ring ofelements surrounding the crack tip. The elements within theringmove as a rigid body. 𝑞 is a constant in these elements; sothe derivative of 𝑞 with respect to 𝑥

𝑗is zero. For the elements

outside the ring, 𝑞 is zero, and again the derivative of 𝑞 is zero.For the elements belonging to the ring, the vector 𝑞 is 1. If theweighting function 𝑞 is introduced in (22), the contributionof the integral along Γ disappears and

𝐽em𝑘= −∫

𝑠

𝑄𝑘𝑗𝑛𝑗𝑞 𝑑𝑠 + ∫

Γ++Γ−

𝑄𝑘𝑗𝑛𝑗𝑞 𝑑𝑠. (24)

The transformation into an equivalent domain integral leadsto

𝐽em𝑘= −∫

Ω

(𝑄𝑘𝑗𝑞),𝑗𝑑Ω + ∫

Γ++Γ−

𝑄𝑘𝑗𝑛𝑗𝑞 𝑑𝑠. (25)

If we use the expression of 𝑄𝑘𝑗

for (20), we get the 2-dimensional 𝐽em

𝑘-integral as a domain integral:

𝐽em𝑘= −∫

Ω



𝑗𝐸𝑘) 𝑞

,𝑗𝑑Ω

− ∫Ω

(𝐻,𝑘

exp − 𝑏𝑖𝑢𝑖,𝑘 + 𝜔Ω𝐸𝑘) 𝑞 𝑑Ω

+ ∫Γ++Γ−

(𝐻𝑛𝑘− 𝑇

𝑖𝑢𝑖,𝑘− 𝜔

𝑠𝐸𝑘) 𝑞 𝑑𝑠.

(26)


If the material is homogenous, no volume forces or chargesand no crack face tractions or charges are applied, and thecrack face normal points to the 𝑧-directions; that is,

𝐻,𝑘

exp = 0,

𝑏𝑖= 0,

𝜔Ω= 0,

𝑇𝑖= 0,

𝜔𝑠= 0,

𝑛𝑘= (0, ±1)

𝑇.

(27)

Equation (26) can be reduced as

𝐽em𝑘= −∫

Ω



𝑗𝐸𝑘) 𝑞

,𝑗𝑑Ω

+ ∫Γ++Γ−

(𝐻𝑛𝑘) 𝑞 𝑑𝑠.

(28)

The 𝑥-component of the electromechanical 𝐽em𝑘

-integral vec-tor has a physical meaning of the energy release rate𝐺 = 𝐽em

1.

The mechanical stresses and the electric fluxes behavesingular as 1/√𝑟, whereas the electric potential and themechanical displacements show a parabolic shape ∼√𝑟.The angular functions 𝑓

𝑖𝑗, 𝑔

𝑖𝑗, 𝑑

𝑗, and V

𝑗depend only on

material constants. The coefficients 𝐾I,𝐾II, and 𝐾III are thewell-known mechanical stress intensity factors, which arecomplemented by the new forth “electric intensity factor”𝐾D, that characterizes the electric field singularity. Mutualinterdependence between mechanical and electrical crack tipparameters can be given by

𝜎𝑖𝑗 (𝑟, 𝜃) =

1

√𝑟[𝐾I𝑓

I𝑖𝑗(𝜃) + 𝐾II𝑓

II𝑖𝑗(𝜃) + 𝐾III𝑓

III𝑖𝑗(𝜃)

+ 𝐾D𝑓D𝑖𝑗(𝜃)] ,

𝑢𝑖𝑗 (𝑟, 𝜃) =

1

√𝑟[𝐾I𝑔

I𝑖𝑗(𝜃) + 𝐾II𝑔

II𝑖𝑗(𝜃) + 𝐾III𝑔

III𝑖𝑗(𝜃)

+ 𝐾D𝑔D𝑖𝑗(𝜃)] ,

𝐷𝑗 (𝑟, 𝜃) =

1

√𝑟[𝐾I𝑑

I𝑗(𝜃) + 𝐾II𝑑

II𝑗(𝜃) + 𝐾III𝑑

III𝑗(𝜃)

+ 𝐾D𝑑D𝑗(𝜃)] ,

𝜑 (𝑟, 𝜃) =1

√𝑟[𝐾IV

I𝑗(𝜃) + 𝐾IIV

II𝑗(𝜃) + 𝐾IIIV

III𝑗(𝜃)

+ 𝐾DVD𝑗(𝜃)] .

(29)

If the limit to an infinitesimal crack growth is consideredtaking place inside the near-tip solution, its relationship withthe intensity factors can be found in [6]. Consider

𝐺 =1

2{K}𝑇 [Y (𝐶, 𝑒, 𝜅)] {K} ,

{K} = {𝐾II 𝐾I 𝐾III 𝐾D} .(30)

− − − − − − − − − − − −

+ + + + + + + + + + + +

Polarization

𝜎 D

𝜎 D

2a2w

2w

Figure 6: Electromechanical Griffith crack (uniaxial load).

The generalized Irwin-matrix [Y] depends on the elastic,piezoelectric, and dielectric material constants and the rel-ative orientation of the crack with respect to material’s axesand polarization vectors. For a special case when the cracklies perpendicular to the polarization, the crack tip field andthe coefficients of [Y] were determined in [10, 11]. Then 𝐺 isreduced to the following expression, where,𝐶

𝐿,𝐶𝑇,𝐶𝐴, 𝑒, and

𝜅 are material constants:

𝐺 = 𝐺I𝑚+ 𝐺

II𝑚+ 𝐺

III𝑚+ 𝐺D

=1

2{𝐾2

I𝐶𝑇

+𝐾2

II𝐶𝐿

+𝐾2

III𝐶𝐴

+𝐾I𝐾D𝑒−𝐾2

D𝜅+𝐾I𝐾D𝑒} .

(31)

The energy release rate consists of the mechanical terms (foreach opening mode) and the electric contribution.

5. Numerical Examples

5.1. Electromechanical Griffith Crack (Uniaxial Load). Inorder to test the accuracy of the CS-XFEM for crack analysis,the methods were applied to a crack in a plane subjected tonormal uniaxial tensionwhen 𝜎 = 1.0MPa and electrical flux𝐷 = 0.001C/m2. Figure 6 shows the polarization directionas 𝑃. The distance of the central crack along the 𝑥

1-direction

is 2𝑎 and side length of the plate is 10m. At the crack faces,impermeable electric boundary conditions are prescribed.Piezoelectric materials PZT-4, P7, and PZT-H5 were adoptedfor numerical simulation.Thematerial parameters are shownin Table 1.


Table 1: Material constant.

Material Elastic constants/(1010N/m2) Piezoelectric constant/(C/m2) Dielectric constant/(1010 C/Vm)

c11

c12

c13

c44

c55

e31

e33

e15

d11

d33

PZT-4 13.9 7.78 7.43 2.56 11.3 −6.98 13.84 13.44 60.0 54.7P-7 13.0 8.3 8.3 2.5 11.9 −10.3 14.7 13.5 171.0 186.0PZT-H5 12.6 7.95 8.41 2.3 11.7 −6.5 23.3 17.44 150.3 130.0

(I) 2288 elements (II) 1521 elements

Figure 7: Mesh of the piezoelectric plate.

The exact analytical solution for the crack in the infiniteplane under far-field loads 𝜎∞

33, 𝜎∞13, and𝐷∞

3was given by Pak

[11]. Consider

𝐾I = 𝜎∞

33√𝜋𝑎,

𝐾II = 𝜎∞

13√𝜋𝑎,

𝐾D = 𝐷∞

3√𝜋𝑎.

(32)

In order to verify the reliability of CS-XFEM, we set cracklength as 2𝑎 = 2m and use two grid models ((I) 2288elements, (II) 1521 elements) before the fracture starts, asshown in Figure 7.There are five types of smoothing elements((i) split smoothing element: there is one Gauss point oneach boundary segment for split smoothing element; (ii)split-blending smoothing element: one Gauss point on eachboundary segment is sufficient; (iii) tip smoothing element:five Gauss points on a segment of smoothing element aresufficient; (iv) tip-blending smoothing element: five Gausspoints on each boundary segment are sufficient; (v) standardsmoothing element: one Gauss point on each boundarysegment is sufficient) being used for numerical integration asmentioned in [41]. In the simulation every 4-node calculationgrid adopts four smoothing elements. The results are com-pared with those of the XFEM and the theoretical solutions.

The results of stress intensity factors and electric displace-ments factors produced by CS-XFEM and XFEM for threeloading cases are listed in Table 2. It can be seen that theresults of CS-XFEM are closer to the analytical solution thanthose of XFEMwhen using the samemesh.This confirms that

combining cell-based smoothing technique with the XFEMcan improve accuracy.

The normalized mechanical and electrical intensity fac-tors of PZT-4 and P7 with different length of cracks andmodel (I) grids are listed in Table 3. It is obvious that theCS-XFEM can produce more accurate results than XFEMwhen using the same number of nodes, which indicates thatthe smoothing technique adopted in this work improve thecalculation of normalized mechanical and electrical intensityfactors for fracture in piezoelectric materials.

5.2. Electromechanical Griffith Crack (Shear Load). In thesecond example of straight electromechanical crack in theplane, the normal mechanical load is replaced by a shearload. The far-field loadings are 𝜎∞

33= 0, 𝜎∞

13= 1MPa, and

𝐷∞

3= 0.001C/m2; see Figure 8.From Table 4, we can observe that the result of the

mechanical and electrical intensity factors of PZT-H5 has ahigh precision in two models. The accuracy of CS-XFEMis higher than that of XFEM. The results prove that theCS-XFEM can decrease stiffness of the system and improvesolution accuracy. Also model (I) of PZT-H5 was used underthe CS-XFEM and XFEM using gauss integral calculationefficiency. CS-XFEM takes 46.247 seconds, while XFEM takes48.252 second with the following CPU setup: Intel Core i5-3470 3.20GHz, RAM: 8G. The efficiency of CS-XFEM hasbeen improved but is not obvious.

The normalized mechanical and electrical intensity fac-tors of PZT-4 and P7 with different length of cracks whenusing model (I) grids are listed in Table 5. It is obvious that


Table2:Com

paris

onof

XFEM

results

andanalyticalsolutio

nsfore

lectromechanicalG

riffith

crack(uniaxialload).

Loading

Analyticalsolutio

nMesh(I)

Mesh(II)

CS-X

FEM

XFEM

CS-X

FEM

XFEM

𝐾I/(MNm−3/2)𝐾

D/(CM

−3/2)𝐾

I/(MNm−3/2)𝐾

D/(CM

−3/2)𝐾

I/(MNm−3/2)𝐾

D/(CM

−3/2)𝐾

I/(MNm−3/2)𝐾

D/(CM

−3/2)𝐾

I/(MNm−3/2)𝐾

D/(CM

−3/2)

Mechanical

1.7724×10−1

01.7

505×10−1

01.7

385×10−1

01.7494×10−1

01.7

365×10−1

0Electrical

01.7

724×10−4

01.7

569×10−4

01.7479×10−4

01.7

5481×10−4

01.7459×10−4

Mechanical+

electrical

1.7724×10−1

1.7724×10−4

1.7505×10−1

1.7569×10−4

1.7385×10−1

1.7479×10−4

1.7494×10−1

1.75481×10−4

1.7365×10−1

1.7459×10−4


Table 3: Normalized intensity factors under different crack lengths for electromechanical Griffith crack (uniaxial load).

𝑎 Normalized intensity factors PZT-4 P-7CS-XFEM XFEM CS-XFEM XFEM

0.5m 𝐾I/𝜎∞

33√𝜋𝑎 0.989 0.965 0.978 0.967

𝐾D/𝐷∞

33√𝜋𝑎 0.994 0.974 0.986 0.978

1.0m 𝐾I/𝜎∞

33√𝜋𝑎 0.941 0.932 0.948 0.925

𝐾D/𝐷∞

33√𝜋𝑎 0.963 0.940 0.956 0.948

1.5m 𝐾I/𝜎∞

33√𝜋𝑎 0.983 0.966 0.988 0.978

𝐾D/𝐷∞

33√𝜋𝑎 0.987 0.973 0.979 0.979

2.0m 𝐾I/𝜎∞

33√𝜋𝑎 0.991 0.969 0.993 0.970

𝐾D/𝐷∞

33√𝜋𝑎 0.993 0.987 0.984 0.978

− − − − − − − − − − − −

+ + + + + + + + + + + +

Polarization

D

D

𝜏

𝜏

2a

2w

2w

Figure 8: Electromechanical Griffith crack (uniaxial load).

the CS-XFEM can producemore accurate results than XFEMusing the same number of nodes.

5.3. Piezoelectric Model with a Hole with Cracks. A piezo-electric model, with a center circular hole 𝑅 = 10mm andhorizontal cracks 𝑎 on the left and right, is subjected tounidirectional uniform tensile 𝜎∞ = 10MPa and electricdisplacement 𝐷∞ = 10−3 C/m2 at infinity. The geometricmodel is expressed as in Figure 9, the length of side 𝑙 =200mm, 𝑃 is the direction of polarization, and material isPZT-H5.The situation of meshing when 𝑎 = 4mmwas givenby Figure 10.

The stress intensity factor and electric displacementintensity factor on the left side of the crack tip obtained byCS-XFEM and XFEM in different crack lengths are shownin Table 6. The maximum error of calculation in CS-XFEMis 2.9%, and the maximum error of calculation in XFEM is4.1%. The results verified the accuracy of CS-XFEM.

P

D

A B

C

Raa

l=200

mm

l = 200mm

x1

x3

𝜎∞

𝜎∞ D∞

−D∞

Figure 9: Piezoelectric model with a hole with cracks.

Figure 10: Discrete model.


Table4:Com

paris

onof

XFEM

results

andanalyticalsolutio

nsfore

lectromechanicalG

riffith

crack(shear

load).

Loading

Analyticalsolutio

nMesh(I)

Mesh(II)

CS-X

FEM

XFEM

CS-X

FEM

XFEM

𝐾II/(MNm−3/2)𝐾

D/(Cm

−3/2)𝐾

II/(MNm−3/2)𝐾

D/(Cm

−3/2)𝐾

II/(MNm−3/2)𝐾

D/(Cm

−3/2)𝐾

II/(MNm−3/2)𝐾

D/(Cm

−3/2)𝐾

II/(MNm−3/2)𝐾

D/(Cm

−3/2)

Mechanical

1.7724×10−1

01.7452×10−1

01.7

315×10−1

01.744

1×10−1

01.7

295×10−1

0Electrical

01.7

724×10−4

01.7

534×10−4

01.7

392×10−4

01.7

513×10−4

01.7

372×10−4

Mechanical+

electrical

1.7724×10−1

1.7724×10−4

1.7452×10−1

1.7534×10−4

1.7315×10−1

1.7392×10−4

1.744

1×10−1

1.7513×10−4

1.7295×10−1

1.7372×10−4


Table 5: Normalized intensity factors under different crack lengths for electromechanical Griffith crack (shear load).

𝑎 Normalized intensity factors PZT-4 P-7CS-XFEM XFEM CS-XFEM XFEM

0.5m 𝐾II/𝜏∞

13√𝜋𝑎 0.988 0.979 0.977 0.964

𝐾D/𝐷∞

33√𝜋𝑎 0.994 0.972 0.985 0.971

1.0m 𝐾II/𝜏∞

13√𝜋𝑎 0.968 0.948 0.957 0.937

𝐾D/𝐷∞

33√𝜋𝑎 0.986 0.961 0.975 0.958

1.5m 𝐾II/𝜏∞

13√𝜋𝑎 0.958 0.940 0.948 0.935

𝐾D/𝐷∞

33√𝜋𝑎 0.973 0.955 0.963 0.940

2.0m 𝐾II/𝜏∞

13√𝜋𝑎 0.994 0.987 0.981 0.972

𝐾D/𝐷∞

33√𝜋𝑎 0.998 0.991 0.996 0.989

Table 6: Intensity factor under different crack lengths.

𝑎 Intensity factor CS-XFEM XFEM Analyticalsolution

1mm 𝐾I/107× Pa⋅mm0.5 0.2995 0.2950 0.3076

𝐾D/10−9× C⋅mm−1.5 0.2986 0.2983 0.3076

2mm 𝐾I/107× Pa⋅mm0.5 0.5322 0.5263 0.5416

𝐾D/10−9× C⋅mm−1.5 0.5300 0.5230 0.5416

4mm 𝐾I/107× Pa⋅mm0.5 0.8513 0.8368 0.8681

𝐾D/10−9× C⋅mm−1.5 0.8454 0.8351 0.8681

6mm 𝐾I/107× Pa⋅mm0.5 1.0517 1.0560 1.0800

𝐾D/10−9× C⋅mm−1.5 1.0603 1.0432 1.0800

6. Conclusions

With the growing applications of piezoelectric structuresin innovative technical areas, problems of strength andreliability become important and have to be carefully inves-tigated. In order to quantitatively assess fracture and fatigue,sophisticated analysis of cracks’ electromechanical propertiesis needed. The fracture mechanics approach for crack-likedefects in piezoelectric materials reveals coupled electricaland mechanical field singularities. Effective numerical meth-ods are needed to evaluate fracture behavior of cracks inarbitrary piezoelectric structures subjected to combined elec-tromechanical loading.

In this paper CS-XFEM with crack tip enrichment func-tions was presented for electromechanical crack analyses.Two examples are used to verify the accuracy of CS-XFEM.Through the numerical simulation conclusions can be drawnas follows:

(1) The cell-based smoothing technique is extended intoXFEM to combine the advantages of XFEM andCS-FEM. Thus, CS-XFEM has high accuracy andconvergence rate. The present method also simpli-fies the integration of discontinuous approximationby transforming interior integration into boundaryintegration.More importantly, no derivatives of shapefunctions are needed to compute the stiffness matrix.

(2) The examples show thatCS-XFEMcanperformbetterin accuracy and convergence rate compared with

XFEM in terms of stress intensity factors and electricdisplacements factors.

(3) CS-XFEM is superior to XFEM in regard to thecalculation of the stiffness matrix and singularity inthe integrand. It also avoids the mapping process,which will increase the complexity of the calculation.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work was financially supported by the National NaturalScience Foundation of China (Grant no. 51305157), theNational Key Scientific Instrument and Equipment Develop-ment Projects, China (Grant no. 2012YQ030075), and JilinProvincial Department of Science and Technology FundProject (Grant no. 20130305006GX).

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