-
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
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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
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4 Advances in Materials Science and Engineering
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
𝑛ℎ(x𝑚,𝑛)𝑁
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
𝑛ℎ(x𝑚,𝑛)𝑁
𝜙
𝐼(x𝑚,𝑛) (𝐻 (x
𝑚,𝑛) − 𝐻 (x
𝐼))
⋅ 𝑤𝑚,𝑛) , ℎ = 𝑥, 𝑧,
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Advances in Materials Science and Engineering 5
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
𝑛ℎ(x𝑚,𝑛)𝑁
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
,
-
6 Advances in Materials Science and Engineering
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
𝜅B𝛽𝐽𝑑Ω =
𝑁𝑠
∑
𝑘=1
(B𝛼𝐼)𝑇
𝜅B𝛽𝐽𝐴𝑠
𝑘
= (K𝛽𝛼𝐼𝐽)𝑇
,
K𝛽𝛽𝐼𝐽=
𝑁𝑠
∑
𝑘=1
∫Ω𝑠
𝑘
(B𝛽𝐼)𝑇
𝜅B𝛽𝐽𝑑Ω =
𝑁𝑠
∑
𝑘=1
(B𝛽𝐼)𝑇
𝜅B𝛽𝐽𝐴𝑠
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
𝑖𝑗𝑢𝑖,𝑘+ 𝐷
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
(𝐻𝛿𝑘𝑗− 𝜎
𝑖𝑗𝑢𝑖,𝑘+ 𝐷
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
(𝐻𝛿𝑘𝑗− 𝜎
𝑖𝑗𝑢𝑖,𝑘+ 𝐷
𝑗𝐸𝑘) 𝑞
,𝑗𝑑Ω
− ∫Ω
(𝐻,𝑘
exp − 𝑏𝑖𝑢𝑖,𝑘 + 𝜔Ω𝐸𝑘) 𝑞 𝑑Ω
+ ∫Γ++Γ−
(𝐻𝑛𝑘− 𝑇
𝑖𝑢𝑖,𝑘− 𝜔
𝑠𝐸𝑘) 𝑞 𝑑𝑠.
(26)
-
Advances in Materials Science and Engineering 7
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
(𝐻𝛿𝑘𝑗− 𝜎
𝑖𝑗𝑢𝑖,𝑘+ 𝐷
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.
-
8 Advances in Materials Science and Engineering
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
-
Advances in Materials Science and Engineering 9
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
-
10 Advances in Materials Science and Engineering
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.
-
Advances in Materials Science and Engineering 11
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
-
12 Advances in Materials Science and Engineering
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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