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Finite element analysis of piezoelectric corner configurations and cracks accounting for different electrical permeabilities
Engineering Fracture Mechanics 74 (2007) 1511–1524 www.elsevier.com/locate/engfracmech
Finite element analysis of piezoelectric corner configurations and cracks accounting for different electrical permeabilities Meng-Cheng Chen *, Xue-Cheng Ping School of Civil Engineering, East China Jiaotong University, Nanchang, Jiangxi 330013, PR China Received 24 January 2006; received in revised form 27 July 2006; accepted 7 August 2006 Available online 1 November 2006
Abstract Based on eigenfunctions of asymptotic singular electro-elastic fields obtained from a kind of ad hoc finite element method [Chen MC, Zhu JJ, Sze KY. Finite element analysis of piezoelectric elasticity with singular inplane electroelastic fields. Engng Fract Mech 2006;73(7):855–68], a super corner-tip element model is established from the generalized Hellinger–Reissner variational functional and then incorporated into the regular hybrid-stress finite element to determine the coefficients of asymptotic singular electro-elastic fields near a corner-tip. The focus of this paper is not to discuss the well-known behavior of electrically impermeable and permeable (usually it means fully permeable, hereinafter the same) cracks but analyze the limited permeable crack-like corner configurations embedded in the piezoelectric materials, i.e., study the influence of a dielectric medium inside the corner on the singular electro-elastic fields near the corner-tip. The boundary conditions of the impermeable or permeable corner can be considered as simple approximations representing upper and lower bounds for the electrical energy penetrating the corner. Benchmark examples on the piezoelectric crack problems show that present method yields satisfactory results with fewer elements than existing finite element methods do. As application, a piezoelectric corner configuration accounting for the limited permeable boundary condition is investigated, and it is found that the limited permeable assumption is necessary for corners with very small notch angles. 2006 Published by Elsevier Ltd. Keywords: Piezoelectric elasticity; Corner; Singular electro-elastic field; Finite element; Limited permeable
1. Introduction Piezoelectric corner configurations with different notch angles (see Fig. 1) exist often in intelligent structures. Subsequently, electro-elastic field singularities exist around the corner-tip o. When the intelligent structures are exposed to internal or external mechanical and/or electric loadings, high mechanical and electric field concentrations may arise and lead to abnormal working states or structure failure. Therefore, the optimum design and reliable service lifetime predictions of the intelligent structures inevitably requires a thorough understanding of their fracture behavior under mechanical and electric loadings. *
Corresponding author. Tel.: +86 791 7046582. E-mail address: [email protected] (M.-C. Chen).
0013-7944/$ - see front matter 2006 Published by Elsevier Ltd. doi:10.1016/j.engfracmech.2006.08.008
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y
α2 Ω2
r
θ
α
x
o
α1
R
Ω1
Fig. 1. The corner-tip domain of a piezoelectric corner configuration.
Since Parton analyzed the fracture mechanics of piezoelectric materials from a theoretical stand point of view in 1976 [2], great progresses have been made on the studies of piezoelectric crack problems. Until now, many researchers have reported their work [3–11]. However, in these studies there is a common deficiency that it is assumed that the permittivity of the dielectric medium interior to the crack is zero, in contrast to the reality that electric fields can permeate free space and any gas occupying it. Therefore, much attention has been paid on the electric boundary conditions such as limited permeable, permeable and conducting [12–18]. Even though a large amount of work dealing with the piezoelectric crack problems has been carried out as mentioned above, comparatively, studies on the piezoelectric fracture mechanics of corner-tip are rare. There are a few exploring investigations on piezoelectric corner configurations accounting for different electrical permeabilities [19–21], but these studies are only focused on the discussions of the orders of electro-elastic singularities. Recently, Scherzer and Kuna [22] analyzed interface corner and crack configurations embedded in smart composite materials. However, in their paper the electrical boundary condition inside the corners was still restricted to impermeable one. We have not been aware of any reports on the electro-elastic fields near the corner-tips of piezoelectric corner configurations accounting for the limited permeable boundary conditions. 2. Ad hoc finite element formulation for sectorial domains For the piezoelectric corner configuration, the limited permeable boundary condition is given as Dpiezo Dmedium Dpiezo Dmedium jh¼a1 ¼ 0; jh¼a2 ¼ 0 h h h h
ð1Þ
in which superscripts ‘piezo’ and ‘medium’ denote the components in the piezoelectric corner configuration and the permeable medium, respectively. Dh is the hoop electric displacement. a1 and a2 denote the corner angles (see Fig. 1). Obviously, the influence of corner angle of the magnitude a has been included in Eq. (1). By taken Eq. (1) as the precondition, an ad hoc finite element formulation can be developed to discretize the corner-tip domain composed of piezoelectric material and dielectric medium. The finite element formulation is written as [1] " e # i X dqem T 2 X h T 2 q e e e ðk P þ kQ þ R Þ me dqee ðk Pe þ kQe þ Re Þqee ¼ 0 þ ð2Þ e dqe qe epiezo emedium in which ‘e’ denotes the element components; qem and qee are vectors including the nodal displacement and potential components respectively; Matrices Pe , Qe , Re , Pe, Qe and Re are defined as follows:
M.-C. Chen, X.-C. Ping / Engineering Fracture Mechanics 74 (2007) 1511–1524
e
P ¼
Z
h
he
hs
Qe ¼
Z
T Be1 Ce Be1
h
he
hs e
P ¼ j0
Z
he
hs
Qe ¼ j0
Z
he
e T
2ðN Þ H
e
Be1
i
e
R ¼
dh;
Z
h
he
1513
i T T Be0 Ce Be0 2ðNe Þ He Be0 dh;
hs
e T
B1 h
i T Ce Be0 þ Be0 Ce Be1 2ðNe ÞT He ðBe0 þ Be1 Þ dh;
T Be1 Be1
h
e T
2ðN Þ H
e
Be1
i
e
R ¼ j0
dh;
Z
he
h
i T T Be0 Be0 2ðNe Þ He Be0 dh;
hs
T T i T Be1 Be0 þ Be0 Be1 2ðNe Þ He Be0 þ Be1 dh:
hs
in which, hs and he define the bounds of an 02 " # 1 T T T he a hCh B6 e e C ¼ ; H ¼ @4 0 aehT ajaT 0 2 0 Nj 0 0 Ns 0 6 0 0 Nj 0 Ne ¼ 4 0 N s N s 0 0 Nj
N j 0
arbitrary element. j0 is the permittivity of dielectric medium. 31 0 0 0 0 7C 0 1 0 0 5ACe ; 0 0 1 0 3 Nt 0 0 7 0 Nt 0 5; 0 0 N t 3 Nt 0 0 0 0 0 7 7 7 0 Nt 0 7 7; 7 0 0 N t 5
0 Ns
0 0
6 0 6 6 Be1 ¼ 6 6 0 6 4 0
0
0
0
0
0
Ns
0
0
Nj
0
0
N s
0
0
N j
0
0
0
0 0
0 0
2
2 Be0
0 Ns
6 N 6 s 6 ¼6 6 N s;h 6 4 0 0
0 0
0
0
0 Nj
N s;h
0
Nj
N j;h
0
0
N j;h
0
N s
0
N s;h
0
0 0
0
Nt
N t;h
0
0
N t;h
0
0
0
N j
0
0
7 7 7 0 7 7; 7 N t 5
0
0
N j;h
0
0
N t;h
Ne ¼ ½ N s e
H ¼ ½1
N j N t ; N s N j N t e 0 ; B1 ¼ ; 0 0 0
Be0
¼
0
3
Nt
0
N s
N j
N t
N s;h
N j;h
N t;h
;
where C, e and j are matrices containing the elastic, piezoelectric and permittivity constants of piezoelectric materials respectively [3,7,20], i.e., 2 3 2 3 0 e31 c11 c13 0 j11 0 : C ¼ 4 c13 c33 0 5; e ¼ 4 0 e33 5; j ¼ 0 j33 0 0 c44 e15 0 a and h are transformation matrices between polar coordinates and Cartesian ones, i.e., 2 2 3 c s2 2cs c s a¼ ; h ¼ 4 s2 c2 2cs 5; c ¼ cosðhÞ; s ¼ sinðhÞ: s c cs cs c2 s2 Ns, Nj and Ne are shape functions defined for the ad hoc finite element method [1,20], and Nj,h = dNj/dh. The impermeable boundary condition and the permeable boundary condition are actually upper and lower bounds of the limited permeable boundary conditions. For the impermeable corner, the boundary conditions on the corner surfaces are supposed to shield the electric displacement completely, i.e.,
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Dh jh¼a1 ¼ Dh jh¼a2 ¼ 0:
ð3Þ
On the other hand, the permeable boundary conditions read Dh jh¼a1 ¼ Dh jh¼a2 ;
/jh¼a1 ¼ /jh¼a2
ð4Þ
in which / is the electric potential. 3. Generalized Hellinger–Reissner variational functional In terms of linear elastic theory, the strain-displacement relations are eðx; yÞ ¼ rm uðx; yÞ
ð5Þ T
T
in which e(x, y) = {exx, eyy, cxy} are the strain tensors with components exx, eyy and exy. u(x,y) = {ux, uy} are the displacement tensors with components ux and uy. (x, y) denote Cartesian coordinates where electro-elastic field components are located. The electric field-electric potential relations read ð6Þ
Eðx; yÞ ¼ re /ðx; yÞ T
in which, E(x, y) = {Ex, Ey} are the electric field tensors with components Ex and Ey. $m and $e are matrices composed of differential operators, i.e., 2 3 o=ox 0 o=ox 6 7 o=oy 5 and re ¼ rm ¼ 4 0 : ð7Þ o=oy o=oy o=ox From Eqs. (5)–(7), the linear piezoelectric constitutive equations can be collected as below: eðx; yÞ rðx; yÞ C eT ¼ Eðx; yÞ Dðx; yÞ e j
ð8Þ
in which r(x, y) = {rxx, ryy, sxy}T are the stress tensors with components rxx, ryy and sxy. D(x, y) = {Dx,Dy}T are the electric displacement tensors with components Dx and Dy. If there are no body forces and free charges, the stress equilibrium equation and the charge conservation equation read as rTm r ¼ 0 and
rTe D ¼ 0:
ð9Þ
In addition, the boundary conditions on the corner surfaces are defined as nm r ¼ 0
and
ne D ¼ 0;
ð10Þ
where nm ¼
nx
0
ny
0
ny
nx
and
ne ¼ ½ nx
ny
ð11Þ
in which (nx, ny) constitute the outward normal of the domain boundary, and uðx; yÞ ¼ uðx; yÞ and
yÞ /ðx; yÞ ¼ /ðx;
ð12Þ
herein the components with upper bars are ones prescribed on the boundary. With Eqs. (5)–(12), a generalized Hellinger–Reissner variational functional for plane piezoelectric material can be obtained as [11] Z T T P¼ hm þ rðx; yÞ rm uðx; yÞ þ Dðx; yÞ re /ðx; yÞ dV Vn Z h Z h i i T T yÞg ds fnm rðx; yÞg fuðx; yÞ uðx; yÞg ds fne Dðx; yÞg f/ðx; yÞ /ðx; ð13Þ oV n
oV n
M.-C. Chen, X.-C. Ping / Engineering Fracture Mechanics 74 (2007) 1511–1524
1515
in which Vn denotes the domain of the two-dimensional body. oVn denotes the boundary of the domain. hm is the electrical enthalpy density, i.e., T
T
hm ¼ ðrðx; yÞ eðx; yÞ Dðx; yÞ Eðx; yÞÞ=2:
ð14Þ
4. Asymptotic singular electro-elastic fields Generally speaking, the asymptotic singular electro-elastic fields near the corner-tip can be written approximatively as follows: n N þM X ~ uðr; hÞ u ðhÞ bn rkn þ1 ~ n ð15Þ ¼ lim r!0 / ðhÞ /ðr; hÞ n¼1 ( ) N þM X ~n ðhÞ r rðr; hÞ kn bn r ¼ lim ; ð16Þ e n ðhÞ r!0 Dðr; hÞ D n¼1 ~ n ðhÞ, where bn’s are referred to as coefficients; kn’s denote the orders of electro-elastic singularities; ~un ðhÞ, / n n e ðhÞ are angular variation functions corresponding to eigenvalue kn’s; N and M indicate the num~ ðhÞ and D r ber of complex and real eigenvalues truncated, respectively. In Eq. (2), it has been arranged as kn 6 kn+1 (herein equal means multiple roots). In the scope of fracture mechanics, Re(k) < 1 should be excluded in the series. The orders of singularities and the angular variation functions can be obtained from Eq. (2). In order to examine influences of external loadings and corner geometry on the electro-elastic singular fields, coefficient bn’s must be solved. In the following section, a corner-tip element will be established to determine coefficient bn’s. 5. Establishment of the super corner-tip element Sound variational basis and high coarse mesh accuracy of crack-tip and corner-tip hybrid elements for conventional materials have been discussed [23–28]. Although the application of special finite elements to piezoelectric fracture mechanics has been discussed in several papers [3,4,22], hybrid element method has not been reported to solve coefficient bn’s in Eqs. (15) and (16). In this study, we will further generalize the super corner-tip hybrid element model of Chen and Sze [24] into piezoelectric fracture mechanics. Here, a nine-node super corner-tip element (see Fig. 2) will be developed to establish a finite element model. Through integration by part and the divergence theorem, Eq. (13) can be expressed in the following boundary integration form as:
y 7
8
6
p r
9
α
θ
5
o
x
1
2
3
4
Fig. 2. Definition of a nine-node super corner-tip element.
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M.-C. Chen, X.-C. Ping / Engineering Fracture Mechanics 74 (2007) 1511–1524
P¼
1 2
Z
nm
0
0
ne
oV n
rðx; yÞ
T
uðx; yÞ
/ðx; yÞ
Dðx; yÞ
ds þ
Z oV n
nm
0
0
ne
rðx; yÞ
T
Dðx; yÞ
uðx; yÞ yÞ /ðx;
ds: ð17Þ
The finite element form of Eq. (17) should be 1 P ¼ 2 e
Z oV n
nm
0
0
ne
re ðx; yÞ
T
De ðx; yÞ
ue ðx; yÞ
ds þ
/e ðx; yÞ
Z
nm
0
0
ne
oV n
re ðx; yÞ De ðx; yÞ
T
ue ðx; yÞ e ðx; yÞ /
ds ð18Þ
in which the element components ue(x,y), /e(x, y), re(x, y) and De(x, y), can be given with Eqs. (15) and (16), i.e., # " e e U2ð2N MÞ uðr; hÞ u ðx; yÞ ¼ b ð19Þ ¼ ½Td Ue1ð2N MÞ /ðr; hÞ /e ðx; yÞ # " e e S3ð2N MÞ rðr; hÞ r ðx; yÞ ¼ b; ð20Þ ¼ ½Ts Te2ð2N MÞ Dðr; hÞ De ðx; yÞ where coefficient matrix b = [b1R,b1I, bNR,bNI, b1, ,bM]T. Subscript ‘‘R’’ and ‘‘I’’ mean real part and imaginary part, respectively. Ue, Ue, Se and Te are self-defined matrix which can be calculated from Eq. (2). [Td] and [Ts] are transformation matrices between polar coordinates and Cartesian coordinates (see Fig. 1), i.e., 2 3 cosðhÞ sinðhÞ 0 6 7 ½Td ¼ 4 sinðhÞ cosðhÞ 0 5; 2
0
0
1 sin2 ðhÞ cos2 ðhÞ
2
cos ðhÞ sin2 ðhÞ
6 6 6 ½Ts ¼ 6 6 2 cosðhÞ sinðhÞ 6 4 0
cosðhÞ sinðhÞ cosðhÞ sinðhÞ
0 0
2 cosðhÞ sinðhÞ cos2 ðhÞ sin2 ðhÞ 0 0 0 cosðhÞ
0
0
0
sinðhÞ
0 0
3
7 7 7 7: 0 7 7 sinðhÞ 5 cosðhÞ
e ðx; yÞg T on the boundary of the corner-tip element in Eq. (18) can be expressed The components f ue ðx; yÞ; / T e with its nodal vector fU ðx; yÞ; Ue ðx; yÞg as
ue ðx; yÞ e ðx; yÞ /
(
¼ ½L
Ue ðx; yÞ Ue ðx; yÞ
) ;
ð21Þ
where the matrix [L] is set up by one-dimensional Lagrange linear interpolation method, in which the interelement displacement and electric potential compatibilities are satisfied automatically. The displacement and the potential in the element in Fig. 2 are assumed to be linear along the boundaries, so the interpolation shape function [L] between two adjacent nodes can be expressed as ½L ¼
1 sl I3
s I l 3
;
where s is the distance measured from node p, and l is the length between the two nodes, I3 is the third order identity matrix. Substituting Eqs. (19)–(21) into Eq. (18) yields ( ) 1 T e Ue ðx; yÞ e T e P ¼ b ½H b þ b ½G ð22Þ 2 Ue ðx; yÞ
M.-C. Chen, X.-C. Ping / Engineering Fracture Mechanics 74 (2007) 1511–1524
1517
in which 1 ½H ¼ 2 e
e
½G ¼
Z oV n
Z oV n
nm
0
0
ne
nm
0
0
ne
" "
Se3ð2N þMÞ Te2ð2N þMÞ Se3ð2N þMÞ Te2ð2N þMÞ
#T
Ue ðx; yÞ Ue ðx; yÞ
þ
Ue ðx; yÞ Ue ðx; yÞ
T
nm
0
0
ne
"
# Se3ð2N þMÞ ds; Te2ð2N þMÞ
#! T ½L ds:
The stationary value of the functional Pe of Eq. (22) reads ( ) Ue ðx; yÞ e 1 e b ¼ ð½H Þ ½G : Ue ðx; yÞ Inserting Eq. (23) into Eq. (22) leads to ( )T ( ) 1 Ue ðx; yÞ Ue ðx; yÞ e e P ¼ ½K w ; 2 Ue ðx; yÞ Ue ðx; yÞ
ð23Þ
ð24Þ
where [Kw]e = ([G]e)T([H]e)1[G]e denotes the element matrix of the super corner-tip hybrid element for the piezoelectric corner configuration. Finally, the nine-node corner-tip element is incorporated into standard four-node quadrilateral hybrid elements [11], and the final finite element algebraic equations for the whole piezoelectric domain including the corner configuration is set up as Uðx; yÞ ½K ¼ fFg; ð25Þ Uðx; yÞ where [K] is the global stiffness matrix assembled from the 9-node super corner-tip element matrix [Kw]e and the standard four-node quadrilateral hybrid element matrix; [U(x, y) U(x, y)]T denotes the global displacement and electric potential and {F} the global external mechanical and electric loading vector. By solving Eq. (25), the global displacement U(x, y) and the electric potential U(x, y) are determined. Consequently, coefficients b can be retrieved by Eq. (23). To satisfy the LBB conditions (see Tong et al. [28]), the number of stress and electric displacement parameters should be chosen greater than or equal to the total number of DOF minus the rigid body modes of the hybrid element. In the two-dimensional case, the total rigid body modes are three. 6. Fracture mechanics parameters Based on the numerical solutions for the orders of electro-elastic singularities ks and the singular stress as well as the electric displacement components, the fracture mechanics parameters such as stress and electric displacement intensity factors, energy release rates and so on can be numerically evaluated. For the crack problems (notch angle a ! 0), the stress intensity factors (SIFs) and electric displacement intensity factors (EDIFs) are expressed from [10] as pffiffiffiffiffiffiffi ½K I ; K II ; K IV ¼ lim 2pr½rhh ðr; hÞ; rrh ðr; hÞ; Dh ðr; hÞjh¼0 ; ð26Þ r!0
where rhh(r, h), rrh (r, h) and Dh(r, h) can be calculated from Eq. (16). I, II and IV are corresponding to different fracture mode. rhh(0, 0), rrh(0, 0) and Dh(0, 0) in Eq. (26) are not available except for some particular cases, and hence SIFs and EDIFs can only be calculated numerically. In all actual numerical computations, r in Eq. (26) is set to be 0.0001 instead of zero. Further more, the total energy release rate can be expressed as [3]
1 K 2I K 2II K I K IV K 2IV K I K IV þ þ þ J ¼ JM þ JE ¼ ; ð27Þ 2 cT cL e j e where JM is the mechanical term and JE is the electric contribution. cT, cL, e and j are constants depending on the elastic, piezoelectric and dielectric material constants (see Appendix).
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For the corner problems, according to Tan and Meguid [29], fracture mechanics parameters called as generalized stress intensity parameters (GSIPs) can be defined as Z L0 Z L0 Z L0 1 1 1 rhh ðr; 0Þ dr; sxyL0 ¼ srhL0 ðr; 0Þ dr; DyL0 ¼ Dh ðr; 0Þ dr ð28Þ ryyL0 ¼ L0 0 L0 0 L0 0 in which L0 is characteristic length along the ligament of corner-tip. L0 = 0.01a is chosen herein, and ‘a’ denotes the depth of the notch. ryyL0 , sxyL0 and DyL0 account for the effect of loading, corner geometry on the singular electro-elastic fields. 7. Numerical results In all computational examples, PZT4 polarized along the y-axis is adopted and its material constants are listed in Appendix. In Section 7.1and 7.2, all the notch angles a of the corner-tip elements are set to be 0.001 to simulate the crack problems. 7.1. Evaluation of the central crack problem In order to evaluate the accuracy of the present method, a central crack problem of an infinite PZT4 panel is first considered. As shown in Fig. 3(a), the panel is subjected to normal uniaxial tension r1 yy ¼ 10 MPa and 2 electric flux D1 ¼ 10000 pC=mm . Due to the symmetry, only the right half panel is considered. The element y division is shown in Fig. 3(b), 404 four-node hybrid elements and one corner-tip element are used. The nodal electric potential along positive x-axis (i.e. x > 0) is set to zero. Twenty-one to twenty-seven coefficients b in Eqs. (19) and (20) are used to properly evaluate the numerical solutions of singular electro-elastic fields. The energy release rates obtained through Eq. (27) are depicted in Fig. 4. It can be seen that in most cases except larger electric loading ones present solutions are in good agreement with the traditional finite element solutions of Gruebner et al. [12], and the maximum relative errors of J and JM are respectively lower than 11.2% and 12.5%. It should be pointed out that the number of elements used here are far few from that of Gruebner et al. [12], and the refined mesh division near the crack tip is also avoided successfully. 7.2. Evaluation of a finite panel using a 3PB specimen A 3PB specimen as shown in Fig. 5(a) is considered. The sizes of this specimen are the same as that of Park and Sun [30]. The mesh division is shown in Fig. 5(b), and the size of the corner-tip element is 0.5 mm · 0.5 mm.
σ y∞
Polarisation
y
x
Dy∞ Fig. 3. Geometry and mesh division for modeling the central crack problem in a piezoelectric panel.
M.-C. Chen, X.-C. Ping / Engineering Fracture Mechanics 74 (2007) 1511–1524
1519
Fig. 4. Comparison of our findings (line) with the exist solutions (mark) [12] for J and JM under various external electric loading and constant mechanical loading r1 yy ¼ 10 MPa.
F
4 mm
o
PZT4 y
Polarisation
5 mm
x
19.1 mm
17.1 mm U Fig. 5. (a) The 3PB piezoelectric specimen subjected to electric and mechanical loadings, (b) mesh division by one nine-node crack-tip element and 308 standard four-node quadrilateral element.
In impermeable case, if F = 2 kN and E = 0, according to Eq. (26), the stress and electric displacement intensity factors by the present FEM method are K I ¼ 486:178 N=mm3=2 ;
K IV ¼ 42675:1 pC=mm3=2 ;
and likewise, if F = 0 and E = 100 V/mm, the stress and electric displacement intensity factors are K I ¼ 0:0028 N=mm3=2 ;
K IV ¼ 2480:35 pC=mm3=2 :
Due to the linear assumption, the stress and electric intensity factors under coupling mechanical and electric loadings can be superimposed from those generated by single mechanical and single electric loadings. Therefore, according to Eq. (27), JM under coupling mechanical and electric loadings can be expressed as J M ¼ 7:70275 1012 E2 þ 6:6915 108 EF þ 5:73986 107 F 2 :
ð29Þ
By substituting fracture loadings F = 110 N and E = 0 (Park and Sun [30]) into Eq. (29), it can be shown that the critical mechanical energy release rate JMc is equal to 0.006945 N/mm. Let JM = JMc, the fracture loadings F versus electric fields can be expressed as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð30Þ F ¼ 0:05829 ð 0:99605E2 þ 3:56124 106 EÞ
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M.-C. Chen, X.-C. Ping / Engineering Fracture Mechanics 74 (2007) 1511–1524 160 140 120
F [N]
100 80 60 40 20
Park and Sun [30] Present result
0 -600 -400 -200
0
200
400
600
800 1000 1200
E [V/mm]
Fig. 6. Comparison of the F–E relation between present results based on the fracture criterion JM = JMc and the experiment results of Park and Sun [30].
Fig. 7. Energy release rates J and their mechanical part JM under various external electric fields and corresponding fracture loadings F.
Eq. (30) is plotted in Fig. 6. It can be seen that the present F–E curve exists some differences from the experiment results of Park and Sun [30] but reflects variation trends of the experiment results qualitatively. The reason for the overestimation of the influence of the electric field may be due to the impermeable modeling of the crack, which is physically limited permeable. The F–E curve will be used to calculate the critical fracture parameters in the following text. The total energy release rates J versus electric fields based on three boundary conditions are shown in Fig. 7(a). In impermeable case, J decreases no matter if electric field is increased in or against the poling direction, which is contradicts with the experiment of Park and Sun [30]. In contrast to this, the J values of the other crack permeabilities are positive. However, according to what discussed in Gruebner et al. [12], even for these more realistic electric boundary conditions the use of J as a fracture criterion dose not seem to be advisable because of the large range of variation for different electric loadings. The mechanical energy release rates JM at fracture is presented in Fig. 7(b). In impermeable case, JM keep a constant and can explain the experiment phenomenon. The use of JM in limited permeable and permeable cases do not describe the increasing toughness of the material due to negative applied electric fields. Anyhow, JM in impermeable case leads to better prediction of critical loading F than the use of JM = JMc in limited
M.-C. Chen, X.-C. Ping / Engineering Fracture Mechanics 74 (2007) 1511–1524
1521
permeable and permeable cases, even though the latter seems to be physically more convincing concept at first sight. 7.3. A piezoelectric corner configuration problem As shown in Fig. 8(a), a single edge notch exists in a PZT4 panel. The mesh division is shown in Fig. 8(b). One corner-tip element and 178 four-node standard quadrilaterals hybrid elements are sufficient to obtain satisfactory results. The nodal electric potential along x-axis is set to be zero.
σ y∞ Polarisation
y x
D y∞ Fig. 8. Geometry and mesh division for modeling the single-edge notch of a infinite PZT4 panel.
Table 1 The numerical results of GSIPs for a PZT4 notch (a = 0.01) under various external electric loading D1 y and constant mechanical loading r1 yy ¼ 10 MPa 2
2 D1 y ðpC=mm Þ
Elec. b. c.
ryyL0 ðN=mm2 Þ
DyL0 ðpC/mm Þ
10000
Impermeable Lim. permeable Impermeable Lim. permeable Impermeable Lim. permeable Impermeable Lim. permeable Impermeable Lim. permeable
1.65660E2 1.63250E2 1.65576E2 1.63676E2 1.65491E2 1.64013E2 1.65407E2 1.64529E2 1.65323E2 1.64956E2
1.34215E5 2.64151E4 6.41789E4 3.23478E4 5.85730E3 3.82805E4 7.58935E4 4.42123E4 1.45930E5 5.01495E4
5000 0 5000 10000
Table 2 2 The numerical results of the orders of singularities and GSIPs under constant external electric loading D1 y ¼ 10000 pC=mm and constant mechanical loading r1 ¼ 10 MPa yy Notch angles
Elec. b. c.
kI
ryyL0 ðN=mm2 Þ
kIV
DyL0 ðpC=mm2 Þ
0.01
Impermeable Lim. permeable Impermeable Lim. permeable Impermeable Lim. permeable Impermeable Lim. permeable Impermeable Lim. permeable
0.5000000 0.5000000 0.5000000 0.5000000 0.4999999 0.4999999 0.4999979 0.4999979 0.4999735 0.4999735
1.65323E2 1.64956E2 1.64993E2 1.64634E2 1.63896E2 1.64372E2 1.63828E2 1.63765E2 1.63641E2 1.63590E2
0.49998414 0.02784010 0.49984141 0.22789639 0.49840957 0.44761571 0.49519848 0.47706607 0.48865484 0.48072298
1.45930E5 5.01495E4 1.45417E5 6.33633E4 1.44874E5 1.16168E5 1.43702E5 1.32289E5 1.41513E5 1.36758E5
0.1 1 3 7
1522
M.-C. Chen, X.-C. Ping / Engineering Fracture Mechanics 74 (2007) 1511–1524
Fig. 9. Angular distributions of the electric displacements around the PZT4 notch tip for Mode IV fracture (a) notch angle a = 0.01; (b) notch angle a = 7.
Fig. 10. Distributions of the hoop electric displacements along the ligament of the PZT4 notch tip for Mode IV fracture (a) notch angle a = 0.01; (b) notch angle a = 7.
Table 1 gives the results of GSIPs for the notch with notch angle a = 0.01 under various external electric 1 loading D1 y and constant mechanical loading ryy ¼ 10 MPa. It shows that, under the limited permeable condition, DyL0 s are always positive even if negative electric loading is exerted. It is rather different from those under impermeable condition. Furthermore, it can also be seen that ryyL0 s depend weakly on the electric loading and the electric boundary conditions. The orders of electro-elastic singularities and the numerical solution of GSIPs for notches with the notch angles of 0.01 to 7 are given in Table 2. It is noticed that only the results for Mode IV, i.e., DyL0 s, have obvious relations with the choice of electric boundary conditions. The differences between the impermeable boundary conditions and the limited permeable ones increase remarkably with the decrease of the notch angle a. When the notch angle a is close to 7, the influences of boundary conditions begin to vanish. On the other hand, the angular variations of electric displacements and the hoop electric displacements based on radius from the singular point o are plotted in Fig. 9 and Fig. 10 respectively for r1 yy ¼ 10 MPa and 2 D1 y ¼ 10000 pC=mm . According to these Figures, it is proved once more that the differences between impermeable and permeable boundary conditions exist only for those corners with small notch angles.
M.-C. Chen, X.-C. Ping / Engineering Fracture Mechanics 74 (2007) 1511–1524
1523
8. Conclusions A kind of super corner-tip element accounting for the limited permeable boundary condition is developed to analyze numerically the singular electro-elastic fields near the corner-tips of piezoelectric corner configurations. The benchmark example in Section 7.1 shows that the present effective, and the computational costs is reduced effectively through using fewer elements. The famous 3PB experiment is investigated, according to the numerical results of J and JM, it is shown that JM in impermeable case is more appropriate to be regarded as the fracture criterion, which is further conformed the experiment phenomenon of the Park and Sun [30]. The effects of the limited permeable electric boundary conditions in piezoelectric corner configurations are considered herein. It is found that, as the notch angle a is very small, the GSIP DyL0 s in permeable case present themselves quite different the trends of variation from those in impermeable case. This phenomenon is similar to that of crack problems discussed in [11]. However, with the increase of the notch angle (a 7), the influences of the limited permeable boundary condition on the electro-elastic fields begin to vanish. Therefore, for those larger corners, dielectric mediums inside their gaps are needless to be considered. On the other hand, it is shown that the electric loading affects weakly on the GSIP ryyL0 of the piezoelectric corner configurations. This kind of new corner-tip element has a strong applicability and can be used to deal with more complex piezoelectric corner configurations. Acknowledgements The authors acknowledge the support of National Natural Science Foundation of China through grant no. 10362002, the Jiangxi Provincial Natural Science Foundation of China through grant no. 0350062, the Fund for Returned Overseas Chinese Students and the Project of Training Plan for authorized academic and technical Pioneers. We also thank referees for their detailed comments and helpful suggestions. Appendix The polarized piezoelectric materials (PZT-4) and one dielectric medium (air) are used in the numerical examples. The relevant non-zero material properties are given below: PZT-4 polarized along 3-axis: c11 ¼ 139:0;
c12 ¼ 77:8;
c13 ¼ 74:3;
e15 ¼ 13:44;
e31 ¼ 6:98;
c33 ¼ 113:0;
c44 ¼ 25:6 103 N=mm2 ;
e33 ¼ 13:84 106 pC=mm2 ;
j33 ¼ 5470:0 106 pC=Nmm2 :
j11 ¼ 6000:0; Air:
j11 ¼ j33 ¼ 8:854 106 pC=Nmm2 : The constants in Eq. (27), i.e., cT, cL, e and j , can be computed by cT ¼
3 X
gi ti1
i¼1
cL ¼
3 X
3 X
li ti2
i¼1
3 X
gi ti2
i¼1
3 X
!, li ti1
i¼1
3 X i¼1
! li ti2 ;
i¼1
vi xi1 ; cA ¼ v0 x01 ;
i¼1
e¼
3 X
gi ti1
3 X i¼1
li ti2
3 X i¼1
gi ti2
3 X
!, li ti1
i¼1
all symbols above are defined in [31].
3 X i¼1
! gi ti2 ; j ¼
3 X i¼1
gi ti1
3 X i¼1
li ti2
3 X i¼1
gi ti2
3 X i¼1
!, li ti1
3 X i¼1
! gi ti1 ;
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M.-C. Chen, X.-C. Ping / Engineering Fracture Mechanics 74 (2007) 1511–1524
References [1] Chen MC, Zhu JJ, Sze KY. Finite element analysis of piezoelectric elasticity with singular inplane electroelastic fields. Engng Fract Mech 2006;73(7):855–68. [2] Parton VZ. Fracture mechanics of piezoelectric materials. Acta Astronaut 1976;3:671–83. [3] Kuna M. Finite element analyses of crack problems in piezoelectric structures. Comput Mater Sci 1998;13:67–80. [4] Kuna M, Ricoeur A. Theoretical investigations on the cracking of ferroelectric ceramic. In: Lynch C, editor. Smart structures and Materials 2000: Active Materials: Behavior and Mechanics (Proceedings of SPIE). 2000. p. 185–96. [5] Kuo CM, Barnett DM. Stress singularities of interfacial cracks in bonded piezoelectric half-spaces. In: Wu JJ, Ting TCT, Barnett DM, editors. Modern theory of anisotropic elasticity and applications. Philadelphia: SIAM; 1991. p. 33–50. [6] Narita F, Shindo Y. The interface crack problem for bonded piezoelectric and orthotropic layers under antiplane shear loading. Int J Fract 1999;98:87–101. [7] Pak YE. Linear electro-elastic fracture mechanics of piezoelectric materials. Int J Fract 1992;54:79–100. [8] Park SB, Sun CT. Effect of electric field on fracture of piezoelectric. Int J Fract 1995;70:203–16. [9] Sosa HA. On the fracture mechanics of piezoelectric solids. Int J Solids Struct 1992;21:2613–22. [10] Suo Z, Kuo CM, Barnett DM, Willis JR. Fracture mechanics for piezoelectric ceramics. J Mech Phys Solids 1992;40:739–65. [11] Wu CC, Sze KY, Huang YQ. Numerical solutions on fracture of piezoelectric materials by hybrid element. Int J Solids Struct 2001;38:4315–29. [12] Gruebner O, Kamlah M, Munz D. Finite element analysis of cracks in piezoelectric materials taking into account the permittivity of the crack medium. Engng Fract Mech 2003;70:1399–413. [13] Govorukha V, Kamlah M. Asymptotic fields in the finite element analysis of electrically permeable interface cracks. Archive Appl Mech 2004;74:92–101. [14] Hao TH, Shen ZY. A new boundary conditions on the fracture mechanics of piezoelectric solids. Engng Fract Mech 1994;47:793–802. [15] McMeeking RM. A J-integral for the analysis of electrically induced mechanical stress at cracks in elastic dielectrics. Int J Engng Sci 1990;18:605–13. [16] McMeeking RM. Crack tip energy release rate for a piezoelectric compact tension specimen. Engng Fract Mech 1999;64:217–44. [17] Wipper K, Ricoeur A, Kuna M. Towards the computation of electrically permeable cracks in piezoelectrics. Engng Fract Mech 2004;71:2567–87. [18] Heyer V, Schneider GA, Balke H, Drescher J, Bahr HA. A fracture criterion for conducting cracks in homogeneously poled piezoelectric PZT-PIC151 ceramics. Acta Mater 1998;46:6615–22. [19] Chue CH, Chen CD. Decoupled formulation of piezoelectric elasticity under generalized plane deformation and its application to wedge problems. Int J Solids Struct 2002;39:3131–58. [20] Sze KY, Wang HT, Fan H. A finite element approach for computing edge singularities in piezoelectric materials. Int J Solids Struct 2001;38:9233–52. [21] Xu XL, Rajapakse RKND. On singularities in composite piezoelectric wedges and junctions. Int J Solids Struct 2000;37:3253–75. [22] Scherzer M, Kuna M. Combined analytical and numerical solution of 2D interface corner configurations between dissimilar piezoelectric materials. Int J Fract 2004;127:61–99. [23] Atluri SN, Kobayashi AS, Nakagaki M. An assumed displacement hybrid finite element method for fracture mechanics. Int J Fract 1975;11:257–71. [24] Chen MC, Sze KY. A novel finite element analysis of bimaterial wedge problems. Engng Fract Mech 2001;68:1463–76. [25] Freitas JA, Ji ZY. Hybrid-Trefftz equilibrium model for crack problems. Int J Numer Methods Engng 1996;39:569–84. [26] Lee J, Gao HJ. A hybrid finite element analysis of interface cracks. Int J Numer Methods Engng 1995;38:2465–82. [27] Pian THH, Sumihara K. Rational approach for assumed stress finite elements. Int J Numer Methods Engng 1984;20:1685–95. [28] Tong P, Pian THH, Lasry SJ. A hybrid element approach to crack problems in plane elasticity. Int J Numer Methods Engng 1973;7:297–380. [29] Tan MA, Meguid SA. Analysis of bimaterial wedges using a new singular finite element. Int J Fract 1997;88:373–91. [30] Park S, Sun CT. Fracture criteria for piezoelectric ceramics. J Am Ceramic Soc 1995;78(6):1475–80. [31] Chen MC. Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media, Part I: Hypersingular integral equation and theoretical analysis. Int J Fract 2003;121:133–48.
Finite element analysis of piezoelectric corner configurations and cracks accounting for different electrical permeabilities Meng-Cheng Chen *, Xue-Cheng Ping School of Civil Engineering, East China Jiaotong University, Nanchang, Jiangxi 330013, PR China Received 24 January 2006; received in revised form 27 July 2006; accepted 7 August 2006 Available online 1 November 2006
Abstract Based on eigenfunctions of asymptotic singular electro-elastic fields obtained from a kind of ad hoc finite element method [Chen MC, Zhu JJ, Sze KY. Finite element analysis of piezoelectric elasticity with singular inplane electroelastic fields. Engng Fract Mech 2006;73(7):855–68], a super corner-tip element model is established from the generalized Hellinger–Reissner variational functional and then incorporated into the regular hybrid-stress finite element to determine the coefficients of asymptotic singular electro-elastic fields near a corner-tip. The focus of this paper is not to discuss the well-known behavior of electrically impermeable and permeable (usually it means fully permeable, hereinafter the same) cracks but analyze the limited permeable crack-like corner configurations embedded in the piezoelectric materials, i.e., study the influence of a dielectric medium inside the corner on the singular electro-elastic fields near the corner-tip. The boundary conditions of the impermeable or permeable corner can be considered as simple approximations representing upper and lower bounds for the electrical energy penetrating the corner. Benchmark examples on the piezoelectric crack problems show that present method yields satisfactory results with fewer elements than existing finite element methods do. As application, a piezoelectric corner configuration accounting for the limited permeable boundary condition is investigated, and it is found that the limited permeable assumption is necessary for corners with very small notch angles. 2006 Published by Elsevier Ltd. Keywords: Piezoelectric elasticity; Corner; Singular electro-elastic field; Finite element; Limited permeable
1. Introduction Piezoelectric corner configurations with different notch angles (see Fig. 1) exist often in intelligent structures. Subsequently, electro-elastic field singularities exist around the corner-tip o. When the intelligent structures are exposed to internal or external mechanical and/or electric loadings, high mechanical and electric field concentrations may arise and lead to abnormal working states or structure failure. Therefore, the optimum design and reliable service lifetime predictions of the intelligent structures inevitably requires a thorough understanding of their fracture behavior under mechanical and electric loadings. *
Corresponding author. Tel.: +86 791 7046582. E-mail address: [email protected] (M.-C. Chen).
0013-7944/$ - see front matter 2006 Published by Elsevier Ltd. doi:10.1016/j.engfracmech.2006.08.008
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y
α2 Ω2
r
θ
α
x
o
α1
R
Ω1
Fig. 1. The corner-tip domain of a piezoelectric corner configuration.
Since Parton analyzed the fracture mechanics of piezoelectric materials from a theoretical stand point of view in 1976 [2], great progresses have been made on the studies of piezoelectric crack problems. Until now, many researchers have reported their work [3–11]. However, in these studies there is a common deficiency that it is assumed that the permittivity of the dielectric medium interior to the crack is zero, in contrast to the reality that electric fields can permeate free space and any gas occupying it. Therefore, much attention has been paid on the electric boundary conditions such as limited permeable, permeable and conducting [12–18]. Even though a large amount of work dealing with the piezoelectric crack problems has been carried out as mentioned above, comparatively, studies on the piezoelectric fracture mechanics of corner-tip are rare. There are a few exploring investigations on piezoelectric corner configurations accounting for different electrical permeabilities [19–21], but these studies are only focused on the discussions of the orders of electro-elastic singularities. Recently, Scherzer and Kuna [22] analyzed interface corner and crack configurations embedded in smart composite materials. However, in their paper the electrical boundary condition inside the corners was still restricted to impermeable one. We have not been aware of any reports on the electro-elastic fields near the corner-tips of piezoelectric corner configurations accounting for the limited permeable boundary conditions. 2. Ad hoc finite element formulation for sectorial domains For the piezoelectric corner configuration, the limited permeable boundary condition is given as Dpiezo Dmedium Dpiezo Dmedium jh¼a1 ¼ 0; jh¼a2 ¼ 0 h h h h
ð1Þ
in which superscripts ‘piezo’ and ‘medium’ denote the components in the piezoelectric corner configuration and the permeable medium, respectively. Dh is the hoop electric displacement. a1 and a2 denote the corner angles (see Fig. 1). Obviously, the influence of corner angle of the magnitude a has been included in Eq. (1). By taken Eq. (1) as the precondition, an ad hoc finite element formulation can be developed to discretize the corner-tip domain composed of piezoelectric material and dielectric medium. The finite element formulation is written as [1] " e # i X dqem T 2 X h T 2 q e e e ðk P þ kQ þ R Þ me dqee ðk Pe þ kQe þ Re Þqee ¼ 0 þ ð2Þ e dqe qe epiezo emedium in which ‘e’ denotes the element components; qem and qee are vectors including the nodal displacement and potential components respectively; Matrices Pe , Qe , Re , Pe, Qe and Re are defined as follows:
M.-C. Chen, X.-C. Ping / Engineering Fracture Mechanics 74 (2007) 1511–1524
e
P ¼
Z
h
he
hs
Qe ¼
Z
T Be1 Ce Be1
h
he
hs e
P ¼ j0
Z
he
hs
Qe ¼ j0
Z
he
e T
2ðN Þ H
e
Be1
i
e
R ¼
dh;
Z
h
he
1513
i T T Be0 Ce Be0 2ðNe Þ He Be0 dh;
hs
e T
B1 h
i T Ce Be0 þ Be0 Ce Be1 2ðNe ÞT He ðBe0 þ Be1 Þ dh;
T Be1 Be1
h
e T
2ðN Þ H
e
Be1
i
e
R ¼ j0
dh;
Z
he
h
i T T Be0 Be0 2ðNe Þ He Be0 dh;
hs
T T i T Be1 Be0 þ Be0 Be1 2ðNe Þ He Be0 þ Be1 dh:
hs
in which, hs and he define the bounds of an 02 " # 1 T T T he a hCh B6 e e C ¼ ; H ¼ @4 0 aehT ajaT 0 2 0 Nj 0 0 Ns 0 6 0 0 Nj 0 Ne ¼ 4 0 N s N s 0 0 Nj
N j 0
arbitrary element. j0 is the permittivity of dielectric medium. 31 0 0 0 0 7C 0 1 0 0 5ACe ; 0 0 1 0 3 Nt 0 0 7 0 Nt 0 5; 0 0 N t 3 Nt 0 0 0 0 0 7 7 7 0 Nt 0 7 7; 7 0 0 N t 5
0 Ns
0 0
6 0 6 6 Be1 ¼ 6 6 0 6 4 0
0
0
0
0
0
Ns
0
0
Nj
0
0
N s
0
0
N j
0
0
0
0 0
0 0
2
2 Be0
0 Ns
6 N 6 s 6 ¼6 6 N s;h 6 4 0 0
0 0
0
0
0 Nj
N s;h
0
Nj
N j;h
0
0
N j;h
0
N s
0
N s;h
0
0 0
0
Nt
N t;h
0
0
N t;h
0
0
0
N j
0
0
7 7 7 0 7 7; 7 N t 5
0
0
N j;h
0
0
N t;h
Ne ¼ ½ N s e
H ¼ ½1
N j N t ; N s N j N t e 0 ; B1 ¼ ; 0 0 0
Be0
¼
0
3
Nt
0
N s
N j
N t
N s;h
N j;h
N t;h
;
where C, e and j are matrices containing the elastic, piezoelectric and permittivity constants of piezoelectric materials respectively [3,7,20], i.e., 2 3 2 3 0 e31 c11 c13 0 j11 0 : C ¼ 4 c13 c33 0 5; e ¼ 4 0 e33 5; j ¼ 0 j33 0 0 c44 e15 0 a and h are transformation matrices between polar coordinates and Cartesian ones, i.e., 2 2 3 c s2 2cs c s a¼ ; h ¼ 4 s2 c2 2cs 5; c ¼ cosðhÞ; s ¼ sinðhÞ: s c cs cs c2 s2 Ns, Nj and Ne are shape functions defined for the ad hoc finite element method [1,20], and Nj,h = dNj/dh. The impermeable boundary condition and the permeable boundary condition are actually upper and lower bounds of the limited permeable boundary conditions. For the impermeable corner, the boundary conditions on the corner surfaces are supposed to shield the electric displacement completely, i.e.,
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M.-C. Chen, X.-C. Ping / Engineering Fracture Mechanics 74 (2007) 1511–1524
Dh jh¼a1 ¼ Dh jh¼a2 ¼ 0:
ð3Þ
On the other hand, the permeable boundary conditions read Dh jh¼a1 ¼ Dh jh¼a2 ;
/jh¼a1 ¼ /jh¼a2
ð4Þ
in which / is the electric potential. 3. Generalized Hellinger–Reissner variational functional In terms of linear elastic theory, the strain-displacement relations are eðx; yÞ ¼ rm uðx; yÞ
ð5Þ T
T
in which e(x, y) = {exx, eyy, cxy} are the strain tensors with components exx, eyy and exy. u(x,y) = {ux, uy} are the displacement tensors with components ux and uy. (x, y) denote Cartesian coordinates where electro-elastic field components are located. The electric field-electric potential relations read ð6Þ
Eðx; yÞ ¼ re /ðx; yÞ T
in which, E(x, y) = {Ex, Ey} are the electric field tensors with components Ex and Ey. $m and $e are matrices composed of differential operators, i.e., 2 3 o=ox 0 o=ox 6 7 o=oy 5 and re ¼ rm ¼ 4 0 : ð7Þ o=oy o=oy o=ox From Eqs. (5)–(7), the linear piezoelectric constitutive equations can be collected as below: eðx; yÞ rðx; yÞ C eT ¼ Eðx; yÞ Dðx; yÞ e j
ð8Þ
in which r(x, y) = {rxx, ryy, sxy}T are the stress tensors with components rxx, ryy and sxy. D(x, y) = {Dx,Dy}T are the electric displacement tensors with components Dx and Dy. If there are no body forces and free charges, the stress equilibrium equation and the charge conservation equation read as rTm r ¼ 0 and
rTe D ¼ 0:
ð9Þ
In addition, the boundary conditions on the corner surfaces are defined as nm r ¼ 0
and
ne D ¼ 0;
ð10Þ
where nm ¼
nx
0
ny
0
ny
nx
and
ne ¼ ½ nx
ny
ð11Þ
in which (nx, ny) constitute the outward normal of the domain boundary, and uðx; yÞ ¼ uðx; yÞ and
yÞ /ðx; yÞ ¼ /ðx;
ð12Þ
herein the components with upper bars are ones prescribed on the boundary. With Eqs. (5)–(12), a generalized Hellinger–Reissner variational functional for plane piezoelectric material can be obtained as [11] Z T T P¼ hm þ rðx; yÞ rm uðx; yÞ þ Dðx; yÞ re /ðx; yÞ dV Vn Z h Z h i i T T yÞg ds fnm rðx; yÞg fuðx; yÞ uðx; yÞg ds fne Dðx; yÞg f/ðx; yÞ /ðx; ð13Þ oV n
oV n
M.-C. Chen, X.-C. Ping / Engineering Fracture Mechanics 74 (2007) 1511–1524
1515
in which Vn denotes the domain of the two-dimensional body. oVn denotes the boundary of the domain. hm is the electrical enthalpy density, i.e., T
T
hm ¼ ðrðx; yÞ eðx; yÞ Dðx; yÞ Eðx; yÞÞ=2:
ð14Þ
4. Asymptotic singular electro-elastic fields Generally speaking, the asymptotic singular electro-elastic fields near the corner-tip can be written approximatively as follows: n N þM X ~ uðr; hÞ u ðhÞ bn rkn þ1 ~ n ð15Þ ¼ lim r!0 / ðhÞ /ðr; hÞ n¼1 ( ) N þM X ~n ðhÞ r rðr; hÞ kn bn r ¼ lim ; ð16Þ e n ðhÞ r!0 Dðr; hÞ D n¼1 ~ n ðhÞ, where bn’s are referred to as coefficients; kn’s denote the orders of electro-elastic singularities; ~un ðhÞ, / n n e ðhÞ are angular variation functions corresponding to eigenvalue kn’s; N and M indicate the num~ ðhÞ and D r ber of complex and real eigenvalues truncated, respectively. In Eq. (2), it has been arranged as kn 6 kn+1 (herein equal means multiple roots). In the scope of fracture mechanics, Re(k) < 1 should be excluded in the series. The orders of singularities and the angular variation functions can be obtained from Eq. (2). In order to examine influences of external loadings and corner geometry on the electro-elastic singular fields, coefficient bn’s must be solved. In the following section, a corner-tip element will be established to determine coefficient bn’s. 5. Establishment of the super corner-tip element Sound variational basis and high coarse mesh accuracy of crack-tip and corner-tip hybrid elements for conventional materials have been discussed [23–28]. Although the application of special finite elements to piezoelectric fracture mechanics has been discussed in several papers [3,4,22], hybrid element method has not been reported to solve coefficient bn’s in Eqs. (15) and (16). In this study, we will further generalize the super corner-tip hybrid element model of Chen and Sze [24] into piezoelectric fracture mechanics. Here, a nine-node super corner-tip element (see Fig. 2) will be developed to establish a finite element model. Through integration by part and the divergence theorem, Eq. (13) can be expressed in the following boundary integration form as:
y 7
8
6
p r
9
α
θ
5
o
x
1
2
3
4
Fig. 2. Definition of a nine-node super corner-tip element.
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M.-C. Chen, X.-C. Ping / Engineering Fracture Mechanics 74 (2007) 1511–1524
P¼
1 2
Z
nm
0
0
ne
oV n
rðx; yÞ
T
uðx; yÞ
/ðx; yÞ
Dðx; yÞ
ds þ
Z oV n
nm
0
0
ne
rðx; yÞ
T
Dðx; yÞ
uðx; yÞ yÞ /ðx;
ds: ð17Þ
The finite element form of Eq. (17) should be 1 P ¼ 2 e
Z oV n
nm
0
0
ne
re ðx; yÞ
T
De ðx; yÞ
ue ðx; yÞ
ds þ
/e ðx; yÞ
Z
nm
0
0
ne
oV n
re ðx; yÞ De ðx; yÞ
T
ue ðx; yÞ e ðx; yÞ /
ds ð18Þ
in which the element components ue(x,y), /e(x, y), re(x, y) and De(x, y), can be given with Eqs. (15) and (16), i.e., # " e e U2ð2N MÞ uðr; hÞ u ðx; yÞ ¼ b ð19Þ ¼ ½Td Ue1ð2N MÞ /ðr; hÞ /e ðx; yÞ # " e e S3ð2N MÞ rðr; hÞ r ðx; yÞ ¼ b; ð20Þ ¼ ½Ts Te2ð2N MÞ Dðr; hÞ De ðx; yÞ where coefficient matrix b = [b1R,b1I, bNR,bNI, b1, ,bM]T. Subscript ‘‘R’’ and ‘‘I’’ mean real part and imaginary part, respectively. Ue, Ue, Se and Te are self-defined matrix which can be calculated from Eq. (2). [Td] and [Ts] are transformation matrices between polar coordinates and Cartesian coordinates (see Fig. 1), i.e., 2 3 cosðhÞ sinðhÞ 0 6 7 ½Td ¼ 4 sinðhÞ cosðhÞ 0 5; 2
0
0
1 sin2 ðhÞ cos2 ðhÞ
2
cos ðhÞ sin2 ðhÞ
6 6 6 ½Ts ¼ 6 6 2 cosðhÞ sinðhÞ 6 4 0
cosðhÞ sinðhÞ cosðhÞ sinðhÞ
0 0
2 cosðhÞ sinðhÞ cos2 ðhÞ sin2 ðhÞ 0 0 0 cosðhÞ
0
0
0
sinðhÞ
0 0
3
7 7 7 7: 0 7 7 sinðhÞ 5 cosðhÞ
e ðx; yÞg T on the boundary of the corner-tip element in Eq. (18) can be expressed The components f ue ðx; yÞ; / T e with its nodal vector fU ðx; yÞ; Ue ðx; yÞg as
ue ðx; yÞ e ðx; yÞ /
(
¼ ½L
Ue ðx; yÞ Ue ðx; yÞ
) ;
ð21Þ
where the matrix [L] is set up by one-dimensional Lagrange linear interpolation method, in which the interelement displacement and electric potential compatibilities are satisfied automatically. The displacement and the potential in the element in Fig. 2 are assumed to be linear along the boundaries, so the interpolation shape function [L] between two adjacent nodes can be expressed as ½L ¼
1 sl I3
s I l 3
;
where s is the distance measured from node p, and l is the length between the two nodes, I3 is the third order identity matrix. Substituting Eqs. (19)–(21) into Eq. (18) yields ( ) 1 T e Ue ðx; yÞ e T e P ¼ b ½H b þ b ½G ð22Þ 2 Ue ðx; yÞ
M.-C. Chen, X.-C. Ping / Engineering Fracture Mechanics 74 (2007) 1511–1524
1517
in which 1 ½H ¼ 2 e
e
½G ¼
Z oV n
Z oV n
nm
0
0
ne
nm
0
0
ne
" "
Se3ð2N þMÞ Te2ð2N þMÞ Se3ð2N þMÞ Te2ð2N þMÞ
#T
Ue ðx; yÞ Ue ðx; yÞ
þ
Ue ðx; yÞ Ue ðx; yÞ
T
nm
0
0
ne
"
# Se3ð2N þMÞ ds; Te2ð2N þMÞ
#! T ½L ds:
The stationary value of the functional Pe of Eq. (22) reads ( ) Ue ðx; yÞ e 1 e b ¼ ð½H Þ ½G : Ue ðx; yÞ Inserting Eq. (23) into Eq. (22) leads to ( )T ( ) 1 Ue ðx; yÞ Ue ðx; yÞ e e P ¼ ½K w ; 2 Ue ðx; yÞ Ue ðx; yÞ
ð23Þ
ð24Þ
where [Kw]e = ([G]e)T([H]e)1[G]e denotes the element matrix of the super corner-tip hybrid element for the piezoelectric corner configuration. Finally, the nine-node corner-tip element is incorporated into standard four-node quadrilateral hybrid elements [11], and the final finite element algebraic equations for the whole piezoelectric domain including the corner configuration is set up as Uðx; yÞ ½K ¼ fFg; ð25Þ Uðx; yÞ where [K] is the global stiffness matrix assembled from the 9-node super corner-tip element matrix [Kw]e and the standard four-node quadrilateral hybrid element matrix; [U(x, y) U(x, y)]T denotes the global displacement and electric potential and {F} the global external mechanical and electric loading vector. By solving Eq. (25), the global displacement U(x, y) and the electric potential U(x, y) are determined. Consequently, coefficients b can be retrieved by Eq. (23). To satisfy the LBB conditions (see Tong et al. [28]), the number of stress and electric displacement parameters should be chosen greater than or equal to the total number of DOF minus the rigid body modes of the hybrid element. In the two-dimensional case, the total rigid body modes are three. 6. Fracture mechanics parameters Based on the numerical solutions for the orders of electro-elastic singularities ks and the singular stress as well as the electric displacement components, the fracture mechanics parameters such as stress and electric displacement intensity factors, energy release rates and so on can be numerically evaluated. For the crack problems (notch angle a ! 0), the stress intensity factors (SIFs) and electric displacement intensity factors (EDIFs) are expressed from [10] as pffiffiffiffiffiffiffi ½K I ; K II ; K IV ¼ lim 2pr½rhh ðr; hÞ; rrh ðr; hÞ; Dh ðr; hÞjh¼0 ; ð26Þ r!0
where rhh(r, h), rrh (r, h) and Dh(r, h) can be calculated from Eq. (16). I, II and IV are corresponding to different fracture mode. rhh(0, 0), rrh(0, 0) and Dh(0, 0) in Eq. (26) are not available except for some particular cases, and hence SIFs and EDIFs can only be calculated numerically. In all actual numerical computations, r in Eq. (26) is set to be 0.0001 instead of zero. Further more, the total energy release rate can be expressed as [3]
1 K 2I K 2II K I K IV K 2IV K I K IV þ þ þ J ¼ JM þ JE ¼ ; ð27Þ 2 cT cL e j e where JM is the mechanical term and JE is the electric contribution. cT, cL, e and j are constants depending on the elastic, piezoelectric and dielectric material constants (see Appendix).
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For the corner problems, according to Tan and Meguid [29], fracture mechanics parameters called as generalized stress intensity parameters (GSIPs) can be defined as Z L0 Z L0 Z L0 1 1 1 rhh ðr; 0Þ dr; sxyL0 ¼ srhL0 ðr; 0Þ dr; DyL0 ¼ Dh ðr; 0Þ dr ð28Þ ryyL0 ¼ L0 0 L0 0 L0 0 in which L0 is characteristic length along the ligament of corner-tip. L0 = 0.01a is chosen herein, and ‘a’ denotes the depth of the notch. ryyL0 , sxyL0 and DyL0 account for the effect of loading, corner geometry on the singular electro-elastic fields. 7. Numerical results In all computational examples, PZT4 polarized along the y-axis is adopted and its material constants are listed in Appendix. In Section 7.1and 7.2, all the notch angles a of the corner-tip elements are set to be 0.001 to simulate the crack problems. 7.1. Evaluation of the central crack problem In order to evaluate the accuracy of the present method, a central crack problem of an infinite PZT4 panel is first considered. As shown in Fig. 3(a), the panel is subjected to normal uniaxial tension r1 yy ¼ 10 MPa and 2 electric flux D1 ¼ 10000 pC=mm . Due to the symmetry, only the right half panel is considered. The element y division is shown in Fig. 3(b), 404 four-node hybrid elements and one corner-tip element are used. The nodal electric potential along positive x-axis (i.e. x > 0) is set to zero. Twenty-one to twenty-seven coefficients b in Eqs. (19) and (20) are used to properly evaluate the numerical solutions of singular electro-elastic fields. The energy release rates obtained through Eq. (27) are depicted in Fig. 4. It can be seen that in most cases except larger electric loading ones present solutions are in good agreement with the traditional finite element solutions of Gruebner et al. [12], and the maximum relative errors of J and JM are respectively lower than 11.2% and 12.5%. It should be pointed out that the number of elements used here are far few from that of Gruebner et al. [12], and the refined mesh division near the crack tip is also avoided successfully. 7.2. Evaluation of a finite panel using a 3PB specimen A 3PB specimen as shown in Fig. 5(a) is considered. The sizes of this specimen are the same as that of Park and Sun [30]. The mesh division is shown in Fig. 5(b), and the size of the corner-tip element is 0.5 mm · 0.5 mm.
σ y∞
Polarisation
y
x
Dy∞ Fig. 3. Geometry and mesh division for modeling the central crack problem in a piezoelectric panel.
M.-C. Chen, X.-C. Ping / Engineering Fracture Mechanics 74 (2007) 1511–1524
1519
Fig. 4. Comparison of our findings (line) with the exist solutions (mark) [12] for J and JM under various external electric loading and constant mechanical loading r1 yy ¼ 10 MPa.
F
4 mm
o
PZT4 y
Polarisation
5 mm
x
19.1 mm
17.1 mm U Fig. 5. (a) The 3PB piezoelectric specimen subjected to electric and mechanical loadings, (b) mesh division by one nine-node crack-tip element and 308 standard four-node quadrilateral element.
In impermeable case, if F = 2 kN and E = 0, according to Eq. (26), the stress and electric displacement intensity factors by the present FEM method are K I ¼ 486:178 N=mm3=2 ;
K IV ¼ 42675:1 pC=mm3=2 ;
and likewise, if F = 0 and E = 100 V/mm, the stress and electric displacement intensity factors are K I ¼ 0:0028 N=mm3=2 ;
K IV ¼ 2480:35 pC=mm3=2 :
Due to the linear assumption, the stress and electric intensity factors under coupling mechanical and electric loadings can be superimposed from those generated by single mechanical and single electric loadings. Therefore, according to Eq. (27), JM under coupling mechanical and electric loadings can be expressed as J M ¼ 7:70275 1012 E2 þ 6:6915 108 EF þ 5:73986 107 F 2 :
ð29Þ
By substituting fracture loadings F = 110 N and E = 0 (Park and Sun [30]) into Eq. (29), it can be shown that the critical mechanical energy release rate JMc is equal to 0.006945 N/mm. Let JM = JMc, the fracture loadings F versus electric fields can be expressed as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð30Þ F ¼ 0:05829 ð 0:99605E2 þ 3:56124 106 EÞ
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F [N]
100 80 60 40 20
Park and Sun [30] Present result
0 -600 -400 -200
0
200
400
600
800 1000 1200
E [V/mm]
Fig. 6. Comparison of the F–E relation between present results based on the fracture criterion JM = JMc and the experiment results of Park and Sun [30].
Fig. 7. Energy release rates J and their mechanical part JM under various external electric fields and corresponding fracture loadings F.
Eq. (30) is plotted in Fig. 6. It can be seen that the present F–E curve exists some differences from the experiment results of Park and Sun [30] but reflects variation trends of the experiment results qualitatively. The reason for the overestimation of the influence of the electric field may be due to the impermeable modeling of the crack, which is physically limited permeable. The F–E curve will be used to calculate the critical fracture parameters in the following text. The total energy release rates J versus electric fields based on three boundary conditions are shown in Fig. 7(a). In impermeable case, J decreases no matter if electric field is increased in or against the poling direction, which is contradicts with the experiment of Park and Sun [30]. In contrast to this, the J values of the other crack permeabilities are positive. However, according to what discussed in Gruebner et al. [12], even for these more realistic electric boundary conditions the use of J as a fracture criterion dose not seem to be advisable because of the large range of variation for different electric loadings. The mechanical energy release rates JM at fracture is presented in Fig. 7(b). In impermeable case, JM keep a constant and can explain the experiment phenomenon. The use of JM in limited permeable and permeable cases do not describe the increasing toughness of the material due to negative applied electric fields. Anyhow, JM in impermeable case leads to better prediction of critical loading F than the use of JM = JMc in limited
M.-C. Chen, X.-C. Ping / Engineering Fracture Mechanics 74 (2007) 1511–1524
1521
permeable and permeable cases, even though the latter seems to be physically more convincing concept at first sight. 7.3. A piezoelectric corner configuration problem As shown in Fig. 8(a), a single edge notch exists in a PZT4 panel. The mesh division is shown in Fig. 8(b). One corner-tip element and 178 four-node standard quadrilaterals hybrid elements are sufficient to obtain satisfactory results. The nodal electric potential along x-axis is set to be zero.
σ y∞ Polarisation
y x
D y∞ Fig. 8. Geometry and mesh division for modeling the single-edge notch of a infinite PZT4 panel.
Table 1 The numerical results of GSIPs for a PZT4 notch (a = 0.01) under various external electric loading D1 y and constant mechanical loading r1 yy ¼ 10 MPa 2
2 D1 y ðpC=mm Þ
Elec. b. c.
ryyL0 ðN=mm2 Þ
DyL0 ðpC/mm Þ
10000
Impermeable Lim. permeable Impermeable Lim. permeable Impermeable Lim. permeable Impermeable Lim. permeable Impermeable Lim. permeable
1.65660E2 1.63250E2 1.65576E2 1.63676E2 1.65491E2 1.64013E2 1.65407E2 1.64529E2 1.65323E2 1.64956E2
1.34215E5 2.64151E4 6.41789E4 3.23478E4 5.85730E3 3.82805E4 7.58935E4 4.42123E4 1.45930E5 5.01495E4
5000 0 5000 10000
Table 2 2 The numerical results of the orders of singularities and GSIPs under constant external electric loading D1 y ¼ 10000 pC=mm and constant mechanical loading r1 ¼ 10 MPa yy Notch angles
Elec. b. c.
kI
ryyL0 ðN=mm2 Þ
kIV
DyL0 ðpC=mm2 Þ
0.01
Impermeable Lim. permeable Impermeable Lim. permeable Impermeable Lim. permeable Impermeable Lim. permeable Impermeable Lim. permeable
0.5000000 0.5000000 0.5000000 0.5000000 0.4999999 0.4999999 0.4999979 0.4999979 0.4999735 0.4999735
1.65323E2 1.64956E2 1.64993E2 1.64634E2 1.63896E2 1.64372E2 1.63828E2 1.63765E2 1.63641E2 1.63590E2
0.49998414 0.02784010 0.49984141 0.22789639 0.49840957 0.44761571 0.49519848 0.47706607 0.48865484 0.48072298
1.45930E5 5.01495E4 1.45417E5 6.33633E4 1.44874E5 1.16168E5 1.43702E5 1.32289E5 1.41513E5 1.36758E5
0.1 1 3 7
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Fig. 9. Angular distributions of the electric displacements around the PZT4 notch tip for Mode IV fracture (a) notch angle a = 0.01; (b) notch angle a = 7.
Fig. 10. Distributions of the hoop electric displacements along the ligament of the PZT4 notch tip for Mode IV fracture (a) notch angle a = 0.01; (b) notch angle a = 7.
Table 1 gives the results of GSIPs for the notch with notch angle a = 0.01 under various external electric 1 loading D1 y and constant mechanical loading ryy ¼ 10 MPa. It shows that, under the limited permeable condition, DyL0 s are always positive even if negative electric loading is exerted. It is rather different from those under impermeable condition. Furthermore, it can also be seen that ryyL0 s depend weakly on the electric loading and the electric boundary conditions. The orders of electro-elastic singularities and the numerical solution of GSIPs for notches with the notch angles of 0.01 to 7 are given in Table 2. It is noticed that only the results for Mode IV, i.e., DyL0 s, have obvious relations with the choice of electric boundary conditions. The differences between the impermeable boundary conditions and the limited permeable ones increase remarkably with the decrease of the notch angle a. When the notch angle a is close to 7, the influences of boundary conditions begin to vanish. On the other hand, the angular variations of electric displacements and the hoop electric displacements based on radius from the singular point o are plotted in Fig. 9 and Fig. 10 respectively for r1 yy ¼ 10 MPa and 2 D1 y ¼ 10000 pC=mm . According to these Figures, it is proved once more that the differences between impermeable and permeable boundary conditions exist only for those corners with small notch angles.
M.-C. Chen, X.-C. Ping / Engineering Fracture Mechanics 74 (2007) 1511–1524
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8. Conclusions A kind of super corner-tip element accounting for the limited permeable boundary condition is developed to analyze numerically the singular electro-elastic fields near the corner-tips of piezoelectric corner configurations. The benchmark example in Section 7.1 shows that the present effective, and the computational costs is reduced effectively through using fewer elements. The famous 3PB experiment is investigated, according to the numerical results of J and JM, it is shown that JM in impermeable case is more appropriate to be regarded as the fracture criterion, which is further conformed the experiment phenomenon of the Park and Sun [30]. The effects of the limited permeable electric boundary conditions in piezoelectric corner configurations are considered herein. It is found that, as the notch angle a is very small, the GSIP DyL0 s in permeable case present themselves quite different the trends of variation from those in impermeable case. This phenomenon is similar to that of crack problems discussed in [11]. However, with the increase of the notch angle (a 7), the influences of the limited permeable boundary condition on the electro-elastic fields begin to vanish. Therefore, for those larger corners, dielectric mediums inside their gaps are needless to be considered. On the other hand, it is shown that the electric loading affects weakly on the GSIP ryyL0 of the piezoelectric corner configurations. This kind of new corner-tip element has a strong applicability and can be used to deal with more complex piezoelectric corner configurations. Acknowledgements The authors acknowledge the support of National Natural Science Foundation of China through grant no. 10362002, the Jiangxi Provincial Natural Science Foundation of China through grant no. 0350062, the Fund for Returned Overseas Chinese Students and the Project of Training Plan for authorized academic and technical Pioneers. We also thank referees for their detailed comments and helpful suggestions. Appendix The polarized piezoelectric materials (PZT-4) and one dielectric medium (air) are used in the numerical examples. The relevant non-zero material properties are given below: PZT-4 polarized along 3-axis: c11 ¼ 139:0;
c12 ¼ 77:8;
c13 ¼ 74:3;
e15 ¼ 13:44;
e31 ¼ 6:98;
c33 ¼ 113:0;
c44 ¼ 25:6 103 N=mm2 ;
e33 ¼ 13:84 106 pC=mm2 ;
j33 ¼ 5470:0 106 pC=Nmm2 :
j11 ¼ 6000:0; Air:
j11 ¼ j33 ¼ 8:854 106 pC=Nmm2 : The constants in Eq. (27), i.e., cT, cL, e and j , can be computed by cT ¼
3 X
gi ti1
i¼1
cL ¼
3 X
3 X
li ti2
i¼1
3 X
gi ti2
i¼1
3 X
!, li ti1
i¼1
3 X i¼1
! li ti2 ;
i¼1
vi xi1 ; cA ¼ v0 x01 ;
i¼1
e¼
3 X
gi ti1
3 X i¼1
li ti2
3 X i¼1
gi ti2
3 X
!, li ti1
i¼1
all symbols above are defined in [31].
3 X i¼1
! gi ti2 ; j ¼
3 X i¼1
gi ti1
3 X i¼1
li ti2
3 X i¼1
gi ti2
3 X i¼1
!, li ti1
3 X i¼1
! gi ti1 ;
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