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Damage analysis and fracture criteria for piezoelectric ceramics
International Journal of Non-Linear Mechanics 40 (2005) 1204 – 1213 www.elsevier.com/locate/nlm
Damage analysis and fracture criteria for piezoelectric ceramics Xinhua Yang∗ , Chuanyao Chen, Yuantai Hu, Cheng Wang Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, PR China Received 9 January 2004; received in revised form 31 January 2005; accepted 5 May 2005
Abstract Both the mechanical and the electrical damages are introduced to study fracture mechanics of piezoelectric ceramics in this paper. Two kinds of piezoelectric fracture criteria are proposed by using the damage theory combined with the well-known piezoelectric fracture experiments of Park and Sun [Fracture criteria of piezoelectric ceramics, J. Am. Ceram. Soc. 78 (1995) 1475–1480]. One is based on a critical state of the mechanical damage and the other on a critical value of a proper linear combination of both the mechanical and the electrical damage variables. It is found that the fracture load predicted, which takes the mechanical damage into account only (mode 1), has greater deviation than predicted result by considering a proper linear combination of the mechanical and the electrical damages (mode 2). And the fracture criterion corresponding to mode 2 presented is shown to be superior to mode 1. It is also demonstrated that the mechanical damage has greater effect on fracture than the electrical damage. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Non-linearity; Damage evolution; Fracture criteria; Piezoelectric ceramics
1. Introduction For decades, due to the intrinsic electromechanical coupling, piezoelectric ceramics have been used widely in the modern aeronautical and astronautic engineering, the medical apparatus and the instruments, etc. However, their brittleness might result in damage, such as micro-cracks, in the media subjected to various external mechanical and electrical loads [1]. The damage will be grown gradually, emerge into macro-crack eventually and lead to failure of piezoelectric device. In order to improve reliability and assess lifetime of ∗ Corresponding author.
E-mail address: [email protected] (X. Yang). 0020-7462/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2005.05.008
piezoelectric ceramics, fracture mechanism must be investigated in detail. Based on linear piezoelectric fracture mechanics, some solutions were obtained by Pak [2,3], Sosa and Pak [4], Sosa [5,6], Suo et al. [7], and Suo [8]. However, there has been an unexplained discrepancy between the linear piezoelectric theory and the experimental results obtained for piezoelectric fracture. Based on linear piezoelectric fracture mechanics, stress intensity factors are independent of the applied electric field. But the experimental results [1,15] showed that an existing electric field may impede or accelerate crack propagation, which was related to the direction of the applied electric field. In order to eliminate the discrepancy, the electromechanical
X. Yang et al. / International Journal of Non-Linear Mechanics 40 (2005) 1204 – 1213
non-linearity response near a crack tip must be taken into account carefully. Regarding the piezoelectric ceramics as a class of mechanically brittle and electrically ductile solids, Gao et al. [8] put forward a strip saturation to model electrical non-linearity near a piezoelectric crack-tip. However, this model did not involve mechanical non-linearity so that its generalization to other general piezoelectric materials under different external loads remains to be suspected. The phenomenon of multi-scale regions near a piezoelectric crack tip has been noticed by Gao et al. [9], Shen and Nishioka [10], Sih and Zuo [11], and Zuo and Sih [12], for several years. In general, there exist two different region sizes for mechanical and electrical non-linearity. In our work, the electromechanical non-linearity near a piezoelectric crack-tip was considered from damage analysis [13,14]. As an alternative approach, we proposed to resolve the fundamental discrepancy by introducing a damage constitutive model and establishing a proper piezoelectric fracture criterion from the continuum damage theory. It was shown from the numerical results that the influence of electric field on mechanical damage depends on the positive or negative piezoelectric properties, which was consistent with the existing experimental results [1,15]. For brittle piezoelectric ceramics, onset of crack growth almost inevitably results in failure, so a useful design criterion involves simply the specification of combinations of loads required for a given crack to initiate. While the natural extension of linear elastic fracture mechanics provides an agreeably simple and general criterion [7], it was not supported by experimental studies [1,16]. Arguing that fracture was a mechanical process and should be controlled only by the mechanical part of the energy, Park and Sun [1] used the maximum mechanical strain energy release rate to predict fracture loads under combined mechanical and electrical loads. However, there was no fundamental reason to separate a physical process into an electrical part and a mechanical part. In addition, Shen and Nishioka [10], Sih and Zuo [11], and Zuo and Sih [12] proposed an energy density criterion. Fracture criteria based on damage have been widely used for studying crack growth behaviors in elastic and plastic materials and have already made much progress in these fields [17]. The damage analysis will be extended to piezoelectric fracture mechanics in this
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paper. At first, static damage constitutive equations for piezoelectric materials [13,14] will be briefly introduced and damage evolution models put forward in Section 2. The finite element formulation and solution procedure of non-linear equations for damaged piezoelectricity will be presented in Section 3. And then a proper piezoelectric fracture criterion will be established in Section 4. Finally, a few useful conclusions will be given in Section 5.
2. Damage constitutive relation For a piezoelectric material, electric field might cause deterioration of its mechanical properties, and stress field might also result in degradation of its electrical properties. These changes of material properties are ascribed to occurrence and propagation of damage in material. Generally, damage in piezoelectric media can be divided into three categories: (1) pure mechanical damage such as micro-hole and micro-crack; (2) pure electrical damage such as conducting inclusion, local breakdown, etc; and (3) mechanically and electrically combined damage, for instance, micro-hole and micro-crack full with a conducting liquid or gas. Accordingly, the key of understanding the fracture behavior of piezoelectric ceramics is to establish constitutive equations involving these damages. For a piezoelectric medium without damage, its elastic, dielectric and piezoelectric constants can be denoted as the forth-order tensor c, the second-order tensor , and the third-order tensor e, respectively. While the medium has been damaged, its performance would be degenerated and its material constants may be described by c˜ , ˜ , and e˜ . The new constitutive equations for damaged piezoelectric medium can be thus written as follows:
ij = c˜ij kl kl − e˜kij Ek , Di = e˜ikl kl + ˜ ik Ek ,
(1)
where ij , ij , Di and Ei are the components of the field variables of real stress, real strain, real electrical displacement and real electric field, respectively, for the piezoelectric medium damaged. Based on the continuum damage mechanics theory, the constitutive equations of damaged material can be
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rewritten by the effective state variables ˜ ij , ˜ ij , D˜ i and E˜ i , i.e.
˜ ij = cij kl ˜ kl − ekij E˜ k , D˜ i = eikl ˜ kl + ik E˜ k .
(2)
It is assumed that there are the following transforma˜ E˜ tions between the effective state variables ˜ , ˜ , D, and the real , , D, E:
˜ ij = Mij kl kl , ˜ ij = Nij kl kl , D˜ i = Fij Dj , E˜ i = Gij Ej ,
e˜imn = 21 Gj i (km ln + km ln )ej kl , ˜ il = Gj i Gkl j k .
(7)
Then the mechanical and electrical damage tensors can be introduced respectively as
DijE = ij − Gik Gkj . (3)
Nklij = Mij−1kl , (4)
so that the effect of the mechanical and the electrical damage can be described by only one tensor N and G, respectively. And it was derived that c˜ij mn = Nklij Nopmn cklop , e˜imn = Gj i Nklmn ej kl , ˜ il = Gj i Gkl j k .
c˜ij mn = 41 (ki lj + ki lj )(om pn + om pn )cklop ,
DijM = ij − ik kj ,
where, M, N, F and G are the transformation factors. M and N are called as the mechanical damage influential tensors, and F and G as the electrical damage influential tensors. Now the problem under consideration is transferred into how to determine these transformation factors. Applying the theorem of energy equivalence, i.e. letting the energy density in the effective state identical to that in the real state of damaged material element, the following two relationships were derived by Yang et al. [13] as Gj i = Fij−1 ,
Instituting Eq. (6) into Eqs. (5), yields
(5)
It is easily found from the second and fourth equations of (3) that N is relative to only and G depends on E only too. So the decoupled damage evolution model may be described by the following forms:
ij = ij − ij mn mn , Gij = ij − kij Ek ,
[] = [c][ ˜ ] − [e] ˜ T [E], [D] = [e][ ˜ ] + [˜ ][E]
(10)
with [] = [11 22 33 23 31 12 ]T , [] = [11 22 33 23 31 12 ]T , [D] = [D1 D2 D3 ]T , [E] = [E1 E2 E3 ]T .
Nij kl = 21 (ik j l + ik j l ).
[c] ˜ = [N ]T [c][N ], [e] ˜ = [G]T [e][N ], [˜ ] = [G]T [][G],
Doing so can ensure the effective strain ˜ to have the same symmetry as the real strain .
(9)
where, and are, respectively, the mechanical and the electrical damage parameter tensors which can be obtained from experimental analysis. In general, and are non-linear functions of the strain and the electric field E. In 1994, Yang and Shen [18] analyzed damage and fracture of laminated composites under impact by using the first equation of (9) combined with the experimental data and demonstrated that the method given by Eqs. (9) is practicable. Eqs. (1) can be rewritten in matrix form as
Obviously, Gij = ij for pure mechanical damage, and Nij kl = ik j l for pure electrical damage. is the Kronecker Delta. Referring to the elastic damage analysis, N can be expressed with a second-order tensor as follows [18]: (6)
(8)
(11)
It follows from Eqs. (5)
(12)
X. Yang et al. / International Journal of Non-Linear Mechanics 40 (2005) 1204 – 1213
where
N11 N21 N [N] = 31 N41 N51 N61
N12 N22 N32 N42 N52 N62
N13 N23 N33 N43 N53 N63
N14 N24 N34 N44 N54 N64
N15 N25 N35 N45 N55 N65
N16 N26 N36 N46 N56 N66
11
0 0 [N ] = 0
13
12
0 22 0 23 0 12
1 − 211 − 212 − 213 [D ] = −11 12 − 12 22 − 13 23 −11 13 − 12 23 − 13 33 M
1 − G211 − G212 − G213 [D ] = −G11 G12 − G12 G22 − G13 G23 −G11 G13 − G12 G23 − G13 G33 E
(13)
0 0 33 23 13 0
G11 [G] = G12 G13
G12 G22 G23
G13 G23 . G33
for I, J = 1, 2, 3,
NI J = 21 (NI I kl + NI I lk ) for I = 1, 2, 3; 4, k = 2, l = 3, J = 4, 5, 6 and when J = 5, k = 3, l = 1, 6, k = 1, l = 2, NI J = 2Nij J J
(15)
Eq. (6) can be expressed in matrix form as 1 1 0 1 2 23 1 2 23
1 2 (22 + 33 ) 1 2 12 1 2 13
2
13
0 1 2 13 1 2 12 1 2 (11 + 33 ) 1 2 23
2 12 1 2 12
0 . 1 2 13 1 2 23 1 2 (11 + 22 )
(16)
3. Finite element formulation and iterative procedure
(14)
NI J (I, J = 1, 2, . . . , 6) in Eq. (13) corresponds to Nij kl (i, j, k, l = 1, 2, 3) according to the following laws: NI J = NI I J J
NI J = Nij kl + Nij lk for I, J = 4, 5, 6 4, i = 2, j = 3, and when I = 5, i = 3, j = 1, 6, i = 1, j = 2, 4, k = 2, l = 3, J = 5, k = 3, l = 1, 6, k = 1, l = 2.
DM and DE , which are defined by Eqs. (8), can also be rewritten in matrix form −11 12 − 12 22 − 13 23 −11 13 − 12 23 − 13 33 2 2 2 1 − 12 − 22 − 23 −12 13 − 22 23 − 23 33 , −12 13 − 22 23 − 23 33 1 − 213 − 223 − 233 (17) − G11 G12 − G12 G22 − G13 G23 − G11 G13 − G12 G23 − G13 G33 1 − G212 − G222 − G223 − G12 G13 − G22 G23 − G23 G33 . − G12 G13 − G22 G23 − G23 G33 1 − G213 − G223 − G233 (18)
and
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for I = 4, 5, 6; 4, i = 2, j = 3, J = 1, 2, 3 and when I = 5, i = 3, j = 1, 6, i = 1, j = 2,
To derive the finite element formulation for a piezoelectric structure with damage, the virtual work principle is expressed as follows:
= {[]T [] − [E]T [D] − [u]T [f ]} dV V
T − [u] [P ] dS − w¯ dS = 0, (19) SP
Sw
where V denotes the domain of piezoelectric medium, the variational operator, [u] the displacement, the electrical potential, [f ] the prescribed body force, [P¯ ] the prescribed surface traction on Sp and w¯ the prescribed surface charge density on Sw . The displacement and electrical potential are taken to be the degrees of freedom of node [u] = [Nq ][q],
= [N ][ ]
(20)
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and [] = [L][Nq ][q] = [B][q], [E] = [∇][N ][ ] = [A][ ],
(21)
where Nq and N are the interpolation functions, [q] and [ ] the nodal displacements and the electric potentials separately, and [B] = [L][Nq ],
[A] = [∇][N ].
(22)
The linear differential operator [L] and the gradient operator [∇] can be expressed as follows: j jx1 0 0 [L] = 0 j jx3 j jx 2 j − jx 1 [∇] = 0 0
0
0
0
j jx3 0
j jx1 0
j − jx2 0
0
[K][] = [P ],
(26)
where F q , [P ] = , [ ] = −Q Kuu −Ku [K] = K u K
(27)
in which, [], [P ] and [K] are called as the nodal generalized displacement, the nodal generalized load, and the generalized stiffness matrix, respectively. Inserting Eqs. (25) into the third equation of (27), yields
[B]T [c][B] ˜ −[B]T [e] ˜ T [A] [K] = dV . (28) T ˜ [A]T [˜ ][A] V [A] [e][B]
0 j jx 3 , j jx 2 j jx 1 0
j jx2
where [F ] is the nodal forces and [Q] the nodal charges. The matrices [c], ˜ [e] ˜ and [˜ ] are given in Eqs. (12). Combining the two equations of Eqs. (24), yields
0 . j − jx 3
(23)
The following is to solve Eq. (26) under various boundary conditions or electromechanical loads. Due to the coupling among the damage, the stress and the electric field, Eq. (26) is non-linear and needs to be solved by an iterative procedure. First we choose the initial values of both the mechanical and electrical damages as 1 0 0 (0) (0) [ ] = [G ] = 0 1 0 . (29) 0 0 0
Substituting Eqs. (20) and (21) into Eq. (19), we have
Substituting Eq. (29) into Eqs. (14) and (16), Eq. (12) become
[Kuu ][q] − [Ku ][ ] = [F ], [K u ][q] + [K ][ ] = −[Q]
[c˜(0) ] = [c], (24)
with
[Kuu ] = [Ku ] = [K u ] = [K ] =
V
[B]T [c][B] ˜ dV ,
V
V V
[B]T [e] ˜ T [A] dV , [A]T [e][B] ˜ dV , [A]T [˜ ][A] dV ,
(25)
[e˜(0) ] = [e],
(0)
[˜
] = [ ]
(30)
which makes Eq. (26) linear and easy to be solved to get distributions of the electromechanical quantities with respect to [(0) ] and [G(0) ] in the medium. Second, we re-determine the damage fields [(1) ] and [G(1) ] by Eqs. (9). Substituting [(1) ] and [G(1) ] into Eqs. (14) and (16) again, we get a new set of effective material properties: [c˜(1) ] = [N (1) ]T [c][N (1) ], [e˜(1) ] = [G(1) ]T [e][N (1) ], (1) [˜ ] = [G(1) ]T [][G(1) ]
(31)
X. Yang et al. / International Journal of Non-Linear Mechanics 40 (2005) 1204 – 1213
and a new set of linear equations from Eq. (26). Solving the equations as before yields new [(2) ] and [G(2) ]. We repeat the aforementioned iteration process until the following convergence conditions being satisfied: (m+1)
|ij
(m)
− ij | < ,
(m+1)
|Gij
(m)
− Gij | < ,
(32)
where, is the error tolerance. The superscripts “(m)” and “(m + 1)” denote the mth step and the (m + 1)th step, respectively.
4. Damage calculation and fracture criteria The FEM calculations of damage distribution of test specimens, including the compact tension specimens and the symmetric and unsymmetric three-point bending specimens will be performed based on the theoretical mode presented. Two kinds of specimens were subjected to various mechanical and electrical loads [1]. Figs. 1 and 2, respectively, show the calculating systems of the specimens made of PZT-4. The
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corresponding properties of intact material are given in Table 1. 4.1. Damage at the crack-tips M and D E at crackIt is pointed out that becauseD33 33 tips are dominant for both cases, only these two damM age components can be taken into account. The D33 E vs. E3 and the D33 vs. E3 at the critical mechanical loading are plotted in Figs. 3–6. Fig. 3 shows the critM and D E for the compact tension ical values of D33 33 specimen under the condition of various applied electric fields, and Figs. 4–6 show those for a three-point bending specimen, respectively, with a center crack, a 2 mm off-center crack or a 4 mm off-center crack.
4.2. Fracture criteria Fracture criteria can be constructed by the two following modes. For mode 1, the fracture load predicted f would be obtained by taking the mechanical damage into account only. And the value f would be calculated by considering linear combination of both the mechanical and the electrical damage for mode 2. It is proposed that the fracture load can be determined for mode 1 by f = kD M 33
(33)
and that the formula defining f can be expressed for mode 2 as Fig. 1. Setup of compact tension specimen by Park and Sun [1].
M E f = k(D33 + D33 ),
(34)
where k, and are the parameters to be determined. k for mode 1 can be obtained by
fi k= M , (D33 )i
Fig. 2. Setup of three-point bending specimen by Park and Sun [1].
(35)
in which i = 1, 2, . . . , 6 corresponding to the values −5.1, −3.5, 0, 2.8, 5.3, and 10.5 kV/cm of applied electric field for the compact tension specimen and i = 1, 2, 3, 4 corresponding to the values −5, 0, 5.2, and 10.5 kV/cm for the three-point bending specimen.
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X. Yang et al. / International Journal of Non-Linear Mechanics 40 (2005) 1204 – 1213
Table 1 Undamaged properties for PZT-4 given by Park and Sun [1] Elastic constants (GPa) c11 139
c12 77.8
c13 74.3
c33 113
c44 25.6
Piezoelectric constants (C/m2 )
Dielectric constants (10−9 C/Vm)
e31 −6.98
11
33
6
5.47
e33 13.84
0.7
0.7 Mechanical and electrical damage
Mechanical and electrical damage
M
D33
0.6
E
D33 0.5 0.4 0.3 0.2 0.1 0 -6
-2 0 2 4 6 8 Applied electric field E3(KV/cm)
10
0.6
E
D33 0.5 0.4 0.3 0.2 0.1
12
-6
-4
-2 0 2 4 6 8 Applied electric field E3(KV/cm)
0.7
M
Mechanical and electrical damage
E
D33 0.5 0.4 0.3 0.2 0.1
-4
-2 0 2 4 6 8 Applied electric field E3(KV/cm)
10
12
Fig. 4. Mechanical and electrical damage at the crack-tip in three-point bending specimens with a central crack.
12
M
D33
0.6
10
Fig. 5. Mechanical and electrical damage at the crack-tip in three-point bending specimens with a 2 mm off-center crack.
0.7 Mechanical and electrical damage
M
D33
0
-4
Fig. 3. Mechanical and electrical damage at the crack-tip in compact tension specimens.
0 -6
e15 13.44
D33 0.6
E
D33
0.5 0.4 0.3 0.2 0.1 0
-6
-4
-2 0 2 4 6 8 Applied electric field E3(KV/cm)
10
12
Fig. 6. Mechanical and electrical damage at the crack-tip in three-point bending specimens with a 4 mm off-center crack.
X. Yang et al. / International Journal of Non-Linear Mechanics 40 (2005) 1204 – 1213
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Table 2 Parameters and mean square deviation in the criteria for the compact tension specimen
Table 5 Parameters and mean square deviation in the criteria for the threepoint bending specimen with a 4 mm off-center crack
Ratios and mean square deviation
Mode 1
Mode 2
Ratios and mean square deviation
Mode 1
Mode 2
k
184.9 — — 0.0901
266.4 0.7116 0.2884 0.0252
k
318.8 — — 0.0608
366.1 0.8817 0.1183 0.0497
s
s
Table 3 Parameters and mean square deviation in the criteria for the threepoint bending specimen with a central crack
160
Ratios and mean square deviation
Mode 1
Mode 2
140
k
205.9 — — 0.0629
252.0 0.8311 0.1689 0.0399
s
Table 4 Parameters and mean square deviation in the criteria for the threepoint bending specimen with a 2 mm off-center crack Ratios and mean square deviation
Mode 1
Mode 2
k
235.0 — — 0.0587
285.4 0.8381 0.1619 0.0403
s
The parameters k, and in mode 2 can be defined by
M ) (D i M 2 M E k (D33 (D33 )i − 33 )i (D33 )i E ) (D33 i
M (D33 )i E 2 M E + k (D33 )i (D33 )i (D33 )i − E (D33 )i
M (D )i M E = fi (D33 )i − 33 fi (D33 )i , E ) (D33 i M E k (D33 (D33 )i + k )i = fi ,
Prediction by mode 1 Prediction by mode 2
Fracture load f (N)
Experimental value
120 100 80 60 40 -6
-4
-2 0 2 4 6 8 Applied electric field E3(KV/cm)
10
12
Fig. 7. Comparison of predictions and experimental fracture loads of compact tension specimens.
4.3. Quantitative comparison between criteria
(36)
The fracture loads predicted according to the fracture criteria presented are plotted in Figs. 7–10. For comparison, the experimental values of fracture loads by Park and Sun [1] are also drawn in these figures. The criteria proposed are quantitatively compared by the normalized mean square deviation 1 fi − fci 2 s= , (37) n fci
The values of k, and in the two modes for the different specimens are, respectively listed in Tables 2–5. It is pointed out that the electrical damage values are assumed to have the opposite sign to electric field for mode 2 so that the effect of the direction of applied electric field on fracture can be involved.
in which n = 6 for the compact tension test and n = 4 for the three-point bending tests; fi and fci are respectively the predicted and experimental values of fracture loads. The values of normalized mean square deviation s are listed in Tables 2–5 when different criteria are used for predicting fracture loads of the samples.
+ = 1.
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160
180 Experimental value
140
Experimental value
160
Prediction by mode 1
Prediction by mode 1 Prediction by mode 2
Fracture load f (N)
Fracture load f (N)
Prediction by mode 2
120 100 80
120 100 80
60 40 -6
-4
-2 0 2 4 6 8 Applied electric field E3(KV/cm)
10
12
Fig. 8. Comparison of predictions and experimental fracture loads of three-point bending specimens with a central crack.
Experimental value 160
60 -6
-4
-2 0 2 4 6 8 Applied electric field E3(KV/cm)
10
12
Fig. 10. Comparison of predictions and experimental fracture loads of three-point bending specimens with a 4 mm off-center crack.
damage has greater effect on fracture than electrical damage, because is greater than for mode 2, and the greater / , the greater the prediction deviation by mode 2, and the narrower the gap between the two modes.
180
Fracture load f (N)
140
Prediction by mode 1 Prediction by mode 2
140 120
5. Conclusions 100 80 60 -6
-4
-2 0 2 4 6 8 Applied electric field E3(KV/cm)
10
12
Fig. 9. Comparison of predictions and experimental fracture loads of three-point bending specimens with a 2 mm off-center crack.
By comparison, it is very obvious that the fracture loads predicted according to only the mechanical damage have greater deviation comparing with those by the combined mechanical and electrical damage, so the fracture criterion based on the combined mechanical and electrical damage is more effective. Theoretically, it is reasonable because the effect of electrical damage on the fracture is considered for mode 2. In addition, it can be seen from Tables 2 to 5 that the mechanical
From the results of the present study, the following conclusions have been obtained: (1) The fracture load predicted by using combined mechanical and electrical damage mode has higher accuracy than that by the pure mechanical damage mode, so the fracture criterion based on combined damage mode is more reasonable. (2) The mechanical damage has greater effect on fracture than the electrical damage. When / is very great or when the applied electric field is 2 kV/cm or so, there is little difference for the predicted results from the two modes.
Acknowledgements This work is supported by the National Natural Science Foundation of China (No. 10172036) and by the Scientific Research Foundation for Returned Overseas Chinese Scholars, State Education Ministry.
X. Yang et al. / International Journal of Non-Linear Mechanics 40 (2005) 1204 – 1213
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[11] G.C. Sih, J.Z. Zuo, Multiscale behavior of crack initiation and growth in piezoelectric ceramics, J. Theoret. Appl. Fract. Mech. 34 (2000) 123–141. [12] J.Z. Zuo, G.C. Sih, Energy density theory formulation and interpretation of cracking behavior for piezoelectric ceramics, J. Theoret. Appl. Fract. Mech. 34 (2000) 17–33. [13] X.H. Yang, C.Y. Chen, Y.T. Hu, A static damage constitutive model for piezoelectric materials, in: J.S. Yang, G.A. Maugin (Eds.), The Mechanics of Electromagnetic Materials, Kluwer Academic Publishers, Dordrecht, 2003, pp. 259–272. [14] X.H. Yang, C.Y. Chen, Y.T. Hu, Analysis of damage near a conducting crack in a piezoelectric ceramic, Acta Mech. Solida Sinica 16 (2003) 147–154. [15] V. Heyer, G.A. Schneider, H. Balke, J. Drescher, H.A. Bahr, A fracture criterion for conducting cracks in homogeneously poled piezoelectric PZT-PIC 151 ceramics, Acta Mater. 46 (1998) 6615–6622. [16] S.B. Park, C.T. Sun, Effect of electric field on fracture of piezoelectric ceramics, Int. J. Fracture 70 (1995) 203–216. [17] W. Shen, L.H. Peng, Y.G. Yue, Elastic damage and energy dissipation in anisotropic solid materials, Eng. Fract. Mech. 33 (1989) 273–281. [18] X.H. Yang, W. Shen, An advanced dynamic three-dimensional finite element method to simulate deformation-damage process of laminates under impact, Eng. Fract. Mech. 49 (1994) 631–638.
Damage analysis and fracture criteria for piezoelectric ceramics Xinhua Yang∗ , Chuanyao Chen, Yuantai Hu, Cheng Wang Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, PR China Received 9 January 2004; received in revised form 31 January 2005; accepted 5 May 2005
Abstract Both the mechanical and the electrical damages are introduced to study fracture mechanics of piezoelectric ceramics in this paper. Two kinds of piezoelectric fracture criteria are proposed by using the damage theory combined with the well-known piezoelectric fracture experiments of Park and Sun [Fracture criteria of piezoelectric ceramics, J. Am. Ceram. Soc. 78 (1995) 1475–1480]. One is based on a critical state of the mechanical damage and the other on a critical value of a proper linear combination of both the mechanical and the electrical damage variables. It is found that the fracture load predicted, which takes the mechanical damage into account only (mode 1), has greater deviation than predicted result by considering a proper linear combination of the mechanical and the electrical damages (mode 2). And the fracture criterion corresponding to mode 2 presented is shown to be superior to mode 1. It is also demonstrated that the mechanical damage has greater effect on fracture than the electrical damage. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Non-linearity; Damage evolution; Fracture criteria; Piezoelectric ceramics
1. Introduction For decades, due to the intrinsic electromechanical coupling, piezoelectric ceramics have been used widely in the modern aeronautical and astronautic engineering, the medical apparatus and the instruments, etc. However, their brittleness might result in damage, such as micro-cracks, in the media subjected to various external mechanical and electrical loads [1]. The damage will be grown gradually, emerge into macro-crack eventually and lead to failure of piezoelectric device. In order to improve reliability and assess lifetime of ∗ Corresponding author.
E-mail address: [email protected] (X. Yang). 0020-7462/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2005.05.008
piezoelectric ceramics, fracture mechanism must be investigated in detail. Based on linear piezoelectric fracture mechanics, some solutions were obtained by Pak [2,3], Sosa and Pak [4], Sosa [5,6], Suo et al. [7], and Suo [8]. However, there has been an unexplained discrepancy between the linear piezoelectric theory and the experimental results obtained for piezoelectric fracture. Based on linear piezoelectric fracture mechanics, stress intensity factors are independent of the applied electric field. But the experimental results [1,15] showed that an existing electric field may impede or accelerate crack propagation, which was related to the direction of the applied electric field. In order to eliminate the discrepancy, the electromechanical
X. Yang et al. / International Journal of Non-Linear Mechanics 40 (2005) 1204 – 1213
non-linearity response near a crack tip must be taken into account carefully. Regarding the piezoelectric ceramics as a class of mechanically brittle and electrically ductile solids, Gao et al. [8] put forward a strip saturation to model electrical non-linearity near a piezoelectric crack-tip. However, this model did not involve mechanical non-linearity so that its generalization to other general piezoelectric materials under different external loads remains to be suspected. The phenomenon of multi-scale regions near a piezoelectric crack tip has been noticed by Gao et al. [9], Shen and Nishioka [10], Sih and Zuo [11], and Zuo and Sih [12], for several years. In general, there exist two different region sizes for mechanical and electrical non-linearity. In our work, the electromechanical non-linearity near a piezoelectric crack-tip was considered from damage analysis [13,14]. As an alternative approach, we proposed to resolve the fundamental discrepancy by introducing a damage constitutive model and establishing a proper piezoelectric fracture criterion from the continuum damage theory. It was shown from the numerical results that the influence of electric field on mechanical damage depends on the positive or negative piezoelectric properties, which was consistent with the existing experimental results [1,15]. For brittle piezoelectric ceramics, onset of crack growth almost inevitably results in failure, so a useful design criterion involves simply the specification of combinations of loads required for a given crack to initiate. While the natural extension of linear elastic fracture mechanics provides an agreeably simple and general criterion [7], it was not supported by experimental studies [1,16]. Arguing that fracture was a mechanical process and should be controlled only by the mechanical part of the energy, Park and Sun [1] used the maximum mechanical strain energy release rate to predict fracture loads under combined mechanical and electrical loads. However, there was no fundamental reason to separate a physical process into an electrical part and a mechanical part. In addition, Shen and Nishioka [10], Sih and Zuo [11], and Zuo and Sih [12] proposed an energy density criterion. Fracture criteria based on damage have been widely used for studying crack growth behaviors in elastic and plastic materials and have already made much progress in these fields [17]. The damage analysis will be extended to piezoelectric fracture mechanics in this
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paper. At first, static damage constitutive equations for piezoelectric materials [13,14] will be briefly introduced and damage evolution models put forward in Section 2. The finite element formulation and solution procedure of non-linear equations for damaged piezoelectricity will be presented in Section 3. And then a proper piezoelectric fracture criterion will be established in Section 4. Finally, a few useful conclusions will be given in Section 5.
2. Damage constitutive relation For a piezoelectric material, electric field might cause deterioration of its mechanical properties, and stress field might also result in degradation of its electrical properties. These changes of material properties are ascribed to occurrence and propagation of damage in material. Generally, damage in piezoelectric media can be divided into three categories: (1) pure mechanical damage such as micro-hole and micro-crack; (2) pure electrical damage such as conducting inclusion, local breakdown, etc; and (3) mechanically and electrically combined damage, for instance, micro-hole and micro-crack full with a conducting liquid or gas. Accordingly, the key of understanding the fracture behavior of piezoelectric ceramics is to establish constitutive equations involving these damages. For a piezoelectric medium without damage, its elastic, dielectric and piezoelectric constants can be denoted as the forth-order tensor c, the second-order tensor , and the third-order tensor e, respectively. While the medium has been damaged, its performance would be degenerated and its material constants may be described by c˜ , ˜ , and e˜ . The new constitutive equations for damaged piezoelectric medium can be thus written as follows:
ij = c˜ij kl kl − e˜kij Ek , Di = e˜ikl kl + ˜ ik Ek ,
(1)
where ij , ij , Di and Ei are the components of the field variables of real stress, real strain, real electrical displacement and real electric field, respectively, for the piezoelectric medium damaged. Based on the continuum damage mechanics theory, the constitutive equations of damaged material can be
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rewritten by the effective state variables ˜ ij , ˜ ij , D˜ i and E˜ i , i.e.
˜ ij = cij kl ˜ kl − ekij E˜ k , D˜ i = eikl ˜ kl + ik E˜ k .
(2)
It is assumed that there are the following transforma˜ E˜ tions between the effective state variables ˜ , ˜ , D, and the real , , D, E:
˜ ij = Mij kl kl , ˜ ij = Nij kl kl , D˜ i = Fij Dj , E˜ i = Gij Ej ,
e˜imn = 21 Gj i (km ln + km ln )ej kl , ˜ il = Gj i Gkl j k .
(7)
Then the mechanical and electrical damage tensors can be introduced respectively as
DijE = ij − Gik Gkj . (3)
Nklij = Mij−1kl , (4)
so that the effect of the mechanical and the electrical damage can be described by only one tensor N and G, respectively. And it was derived that c˜ij mn = Nklij Nopmn cklop , e˜imn = Gj i Nklmn ej kl , ˜ il = Gj i Gkl j k .
c˜ij mn = 41 (ki lj + ki lj )(om pn + om pn )cklop ,
DijM = ij − ik kj ,
where, M, N, F and G are the transformation factors. M and N are called as the mechanical damage influential tensors, and F and G as the electrical damage influential tensors. Now the problem under consideration is transferred into how to determine these transformation factors. Applying the theorem of energy equivalence, i.e. letting the energy density in the effective state identical to that in the real state of damaged material element, the following two relationships were derived by Yang et al. [13] as Gj i = Fij−1 ,
Instituting Eq. (6) into Eqs. (5), yields
(5)
It is easily found from the second and fourth equations of (3) that N is relative to only and G depends on E only too. So the decoupled damage evolution model may be described by the following forms:
ij = ij − ij mn mn , Gij = ij − kij Ek ,
[] = [c][ ˜ ] − [e] ˜ T [E], [D] = [e][ ˜ ] + [˜ ][E]
(10)
with [] = [11 22 33 23 31 12 ]T , [] = [11 22 33 23 31 12 ]T , [D] = [D1 D2 D3 ]T , [E] = [E1 E2 E3 ]T .
Nij kl = 21 (ik j l + ik j l ).
[c] ˜ = [N ]T [c][N ], [e] ˜ = [G]T [e][N ], [˜ ] = [G]T [][G],
Doing so can ensure the effective strain ˜ to have the same symmetry as the real strain .
(9)
where, and are, respectively, the mechanical and the electrical damage parameter tensors which can be obtained from experimental analysis. In general, and are non-linear functions of the strain and the electric field E. In 1994, Yang and Shen [18] analyzed damage and fracture of laminated composites under impact by using the first equation of (9) combined with the experimental data and demonstrated that the method given by Eqs. (9) is practicable. Eqs. (1) can be rewritten in matrix form as
Obviously, Gij = ij for pure mechanical damage, and Nij kl = ik j l for pure electrical damage. is the Kronecker Delta. Referring to the elastic damage analysis, N can be expressed with a second-order tensor as follows [18]: (6)
(8)
(11)
It follows from Eqs. (5)
(12)
X. Yang et al. / International Journal of Non-Linear Mechanics 40 (2005) 1204 – 1213
where
N11 N21 N [N] = 31 N41 N51 N61
N12 N22 N32 N42 N52 N62
N13 N23 N33 N43 N53 N63
N14 N24 N34 N44 N54 N64
N15 N25 N35 N45 N55 N65
N16 N26 N36 N46 N56 N66
11
0 0 [N ] = 0
13
12
0 22 0 23 0 12
1 − 211 − 212 − 213 [D ] = −11 12 − 12 22 − 13 23 −11 13 − 12 23 − 13 33 M
1 − G211 − G212 − G213 [D ] = −G11 G12 − G12 G22 − G13 G23 −G11 G13 − G12 G23 − G13 G33 E
(13)
0 0 33 23 13 0
G11 [G] = G12 G13
G12 G22 G23
G13 G23 . G33
for I, J = 1, 2, 3,
NI J = 21 (NI I kl + NI I lk ) for I = 1, 2, 3; 4, k = 2, l = 3, J = 4, 5, 6 and when J = 5, k = 3, l = 1, 6, k = 1, l = 2, NI J = 2Nij J J
(15)
Eq. (6) can be expressed in matrix form as 1 1 0 1 2 23 1 2 23
1 2 (22 + 33 ) 1 2 12 1 2 13
2
13
0 1 2 13 1 2 12 1 2 (11 + 33 ) 1 2 23
2 12 1 2 12
0 . 1 2 13 1 2 23 1 2 (11 + 22 )
(16)
3. Finite element formulation and iterative procedure
(14)
NI J (I, J = 1, 2, . . . , 6) in Eq. (13) corresponds to Nij kl (i, j, k, l = 1, 2, 3) according to the following laws: NI J = NI I J J
NI J = Nij kl + Nij lk for I, J = 4, 5, 6 4, i = 2, j = 3, and when I = 5, i = 3, j = 1, 6, i = 1, j = 2, 4, k = 2, l = 3, J = 5, k = 3, l = 1, 6, k = 1, l = 2.
DM and DE , which are defined by Eqs. (8), can also be rewritten in matrix form −11 12 − 12 22 − 13 23 −11 13 − 12 23 − 13 33 2 2 2 1 − 12 − 22 − 23 −12 13 − 22 23 − 23 33 , −12 13 − 22 23 − 23 33 1 − 213 − 223 − 233 (17) − G11 G12 − G12 G22 − G13 G23 − G11 G13 − G12 G23 − G13 G33 1 − G212 − G222 − G223 − G12 G13 − G22 G23 − G23 G33 . − G12 G13 − G22 G23 − G23 G33 1 − G213 − G223 − G233 (18)
and
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for I = 4, 5, 6; 4, i = 2, j = 3, J = 1, 2, 3 and when I = 5, i = 3, j = 1, 6, i = 1, j = 2,
To derive the finite element formulation for a piezoelectric structure with damage, the virtual work principle is expressed as follows:
= {[]T [] − [E]T [D] − [u]T [f ]} dV V
T − [u] [P ] dS − w¯ dS = 0, (19) SP
Sw
where V denotes the domain of piezoelectric medium, the variational operator, [u] the displacement, the electrical potential, [f ] the prescribed body force, [P¯ ] the prescribed surface traction on Sp and w¯ the prescribed surface charge density on Sw . The displacement and electrical potential are taken to be the degrees of freedom of node [u] = [Nq ][q],
= [N ][ ]
(20)
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and [] = [L][Nq ][q] = [B][q], [E] = [∇][N ][ ] = [A][ ],
(21)
where Nq and N are the interpolation functions, [q] and [ ] the nodal displacements and the electric potentials separately, and [B] = [L][Nq ],
[A] = [∇][N ].
(22)
The linear differential operator [L] and the gradient operator [∇] can be expressed as follows: j jx1 0 0 [L] = 0 j jx3 j jx 2 j − jx 1 [∇] = 0 0
0
0
0
j jx3 0
j jx1 0
j − jx2 0
0
[K][] = [P ],
(26)
where F q , [P ] = , [ ] = −Q Kuu −Ku [K] = K u K
(27)
in which, [], [P ] and [K] are called as the nodal generalized displacement, the nodal generalized load, and the generalized stiffness matrix, respectively. Inserting Eqs. (25) into the third equation of (27), yields
[B]T [c][B] ˜ −[B]T [e] ˜ T [A] [K] = dV . (28) T ˜ [A]T [˜ ][A] V [A] [e][B]
0 j jx 3 , j jx 2 j jx 1 0
j jx2
where [F ] is the nodal forces and [Q] the nodal charges. The matrices [c], ˜ [e] ˜ and [˜ ] are given in Eqs. (12). Combining the two equations of Eqs. (24), yields
0 . j − jx 3
(23)
The following is to solve Eq. (26) under various boundary conditions or electromechanical loads. Due to the coupling among the damage, the stress and the electric field, Eq. (26) is non-linear and needs to be solved by an iterative procedure. First we choose the initial values of both the mechanical and electrical damages as 1 0 0 (0) (0) [ ] = [G ] = 0 1 0 . (29) 0 0 0
Substituting Eqs. (20) and (21) into Eq. (19), we have
Substituting Eq. (29) into Eqs. (14) and (16), Eq. (12) become
[Kuu ][q] − [Ku ][ ] = [F ], [K u ][q] + [K ][ ] = −[Q]
[c˜(0) ] = [c], (24)
with
[Kuu ] = [Ku ] = [K u ] = [K ] =
V
[B]T [c][B] ˜ dV ,
V
V V
[B]T [e] ˜ T [A] dV , [A]T [e][B] ˜ dV , [A]T [˜ ][A] dV ,
(25)
[e˜(0) ] = [e],
(0)
[˜
] = [ ]
(30)
which makes Eq. (26) linear and easy to be solved to get distributions of the electromechanical quantities with respect to [(0) ] and [G(0) ] in the medium. Second, we re-determine the damage fields [(1) ] and [G(1) ] by Eqs. (9). Substituting [(1) ] and [G(1) ] into Eqs. (14) and (16) again, we get a new set of effective material properties: [c˜(1) ] = [N (1) ]T [c][N (1) ], [e˜(1) ] = [G(1) ]T [e][N (1) ], (1) [˜ ] = [G(1) ]T [][G(1) ]
(31)
X. Yang et al. / International Journal of Non-Linear Mechanics 40 (2005) 1204 – 1213
and a new set of linear equations from Eq. (26). Solving the equations as before yields new [(2) ] and [G(2) ]. We repeat the aforementioned iteration process until the following convergence conditions being satisfied: (m+1)
|ij
(m)
− ij | < ,
(m+1)
|Gij
(m)
− Gij | < ,
(32)
where, is the error tolerance. The superscripts “(m)” and “(m + 1)” denote the mth step and the (m + 1)th step, respectively.
4. Damage calculation and fracture criteria The FEM calculations of damage distribution of test specimens, including the compact tension specimens and the symmetric and unsymmetric three-point bending specimens will be performed based on the theoretical mode presented. Two kinds of specimens were subjected to various mechanical and electrical loads [1]. Figs. 1 and 2, respectively, show the calculating systems of the specimens made of PZT-4. The
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corresponding properties of intact material are given in Table 1. 4.1. Damage at the crack-tips M and D E at crackIt is pointed out that becauseD33 33 tips are dominant for both cases, only these two damM age components can be taken into account. The D33 E vs. E3 and the D33 vs. E3 at the critical mechanical loading are plotted in Figs. 3–6. Fig. 3 shows the critM and D E for the compact tension ical values of D33 33 specimen under the condition of various applied electric fields, and Figs. 4–6 show those for a three-point bending specimen, respectively, with a center crack, a 2 mm off-center crack or a 4 mm off-center crack.
4.2. Fracture criteria Fracture criteria can be constructed by the two following modes. For mode 1, the fracture load predicted f would be obtained by taking the mechanical damage into account only. And the value f would be calculated by considering linear combination of both the mechanical and the electrical damage for mode 2. It is proposed that the fracture load can be determined for mode 1 by f = kD M 33
(33)
and that the formula defining f can be expressed for mode 2 as Fig. 1. Setup of compact tension specimen by Park and Sun [1].
M E f = k(D33 + D33 ),
(34)
where k, and are the parameters to be determined. k for mode 1 can be obtained by
fi k= M , (D33 )i
Fig. 2. Setup of three-point bending specimen by Park and Sun [1].
(35)
in which i = 1, 2, . . . , 6 corresponding to the values −5.1, −3.5, 0, 2.8, 5.3, and 10.5 kV/cm of applied electric field for the compact tension specimen and i = 1, 2, 3, 4 corresponding to the values −5, 0, 5.2, and 10.5 kV/cm for the three-point bending specimen.
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X. Yang et al. / International Journal of Non-Linear Mechanics 40 (2005) 1204 – 1213
Table 1 Undamaged properties for PZT-4 given by Park and Sun [1] Elastic constants (GPa) c11 139
c12 77.8
c13 74.3
c33 113
c44 25.6
Piezoelectric constants (C/m2 )
Dielectric constants (10−9 C/Vm)
e31 −6.98
11
33
6
5.47
e33 13.84
0.7
0.7 Mechanical and electrical damage
Mechanical and electrical damage
M
D33
0.6
E
D33 0.5 0.4 0.3 0.2 0.1 0 -6
-2 0 2 4 6 8 Applied electric field E3(KV/cm)
10
0.6
E
D33 0.5 0.4 0.3 0.2 0.1
12
-6
-4
-2 0 2 4 6 8 Applied electric field E3(KV/cm)
0.7
M
Mechanical and electrical damage
E
D33 0.5 0.4 0.3 0.2 0.1
-4
-2 0 2 4 6 8 Applied electric field E3(KV/cm)
10
12
Fig. 4. Mechanical and electrical damage at the crack-tip in three-point bending specimens with a central crack.
12
M
D33
0.6
10
Fig. 5. Mechanical and electrical damage at the crack-tip in three-point bending specimens with a 2 mm off-center crack.
0.7 Mechanical and electrical damage
M
D33
0
-4
Fig. 3. Mechanical and electrical damage at the crack-tip in compact tension specimens.
0 -6
e15 13.44
D33 0.6
E
D33
0.5 0.4 0.3 0.2 0.1 0
-6
-4
-2 0 2 4 6 8 Applied electric field E3(KV/cm)
10
12
Fig. 6. Mechanical and electrical damage at the crack-tip in three-point bending specimens with a 4 mm off-center crack.
X. Yang et al. / International Journal of Non-Linear Mechanics 40 (2005) 1204 – 1213
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Table 2 Parameters and mean square deviation in the criteria for the compact tension specimen
Table 5 Parameters and mean square deviation in the criteria for the threepoint bending specimen with a 4 mm off-center crack
Ratios and mean square deviation
Mode 1
Mode 2
Ratios and mean square deviation
Mode 1
Mode 2
k
184.9 — — 0.0901
266.4 0.7116 0.2884 0.0252
k
318.8 — — 0.0608
366.1 0.8817 0.1183 0.0497
s
s
Table 3 Parameters and mean square deviation in the criteria for the threepoint bending specimen with a central crack
160
Ratios and mean square deviation
Mode 1
Mode 2
140
k
205.9 — — 0.0629
252.0 0.8311 0.1689 0.0399
s
Table 4 Parameters and mean square deviation in the criteria for the threepoint bending specimen with a 2 mm off-center crack Ratios and mean square deviation
Mode 1
Mode 2
k
235.0 — — 0.0587
285.4 0.8381 0.1619 0.0403
s
The parameters k, and in mode 2 can be defined by
M ) (D i M 2 M E k (D33 (D33 )i − 33 )i (D33 )i E ) (D33 i
M (D33 )i E 2 M E + k (D33 )i (D33 )i (D33 )i − E (D33 )i
M (D )i M E = fi (D33 )i − 33 fi (D33 )i , E ) (D33 i M E k (D33 (D33 )i + k )i = fi ,
Prediction by mode 1 Prediction by mode 2
Fracture load f (N)
Experimental value
120 100 80 60 40 -6
-4
-2 0 2 4 6 8 Applied electric field E3(KV/cm)
10
12
Fig. 7. Comparison of predictions and experimental fracture loads of compact tension specimens.
4.3. Quantitative comparison between criteria
(36)
The fracture loads predicted according to the fracture criteria presented are plotted in Figs. 7–10. For comparison, the experimental values of fracture loads by Park and Sun [1] are also drawn in these figures. The criteria proposed are quantitatively compared by the normalized mean square deviation 1 fi − fci 2 s= , (37) n fci
The values of k, and in the two modes for the different specimens are, respectively listed in Tables 2–5. It is pointed out that the electrical damage values are assumed to have the opposite sign to electric field for mode 2 so that the effect of the direction of applied electric field on fracture can be involved.
in which n = 6 for the compact tension test and n = 4 for the three-point bending tests; fi and fci are respectively the predicted and experimental values of fracture loads. The values of normalized mean square deviation s are listed in Tables 2–5 when different criteria are used for predicting fracture loads of the samples.
+ = 1.
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160
180 Experimental value
140
Experimental value
160
Prediction by mode 1
Prediction by mode 1 Prediction by mode 2
Fracture load f (N)
Fracture load f (N)
Prediction by mode 2
120 100 80
120 100 80
60 40 -6
-4
-2 0 2 4 6 8 Applied electric field E3(KV/cm)
10
12
Fig. 8. Comparison of predictions and experimental fracture loads of three-point bending specimens with a central crack.
Experimental value 160
60 -6
-4
-2 0 2 4 6 8 Applied electric field E3(KV/cm)
10
12
Fig. 10. Comparison of predictions and experimental fracture loads of three-point bending specimens with a 4 mm off-center crack.
damage has greater effect on fracture than electrical damage, because is greater than for mode 2, and the greater / , the greater the prediction deviation by mode 2, and the narrower the gap between the two modes.
180
Fracture load f (N)
140
Prediction by mode 1 Prediction by mode 2
140 120
5. Conclusions 100 80 60 -6
-4
-2 0 2 4 6 8 Applied electric field E3(KV/cm)
10
12
Fig. 9. Comparison of predictions and experimental fracture loads of three-point bending specimens with a 2 mm off-center crack.
By comparison, it is very obvious that the fracture loads predicted according to only the mechanical damage have greater deviation comparing with those by the combined mechanical and electrical damage, so the fracture criterion based on the combined mechanical and electrical damage is more effective. Theoretically, it is reasonable because the effect of electrical damage on the fracture is considered for mode 2. In addition, it can be seen from Tables 2 to 5 that the mechanical
From the results of the present study, the following conclusions have been obtained: (1) The fracture load predicted by using combined mechanical and electrical damage mode has higher accuracy than that by the pure mechanical damage mode, so the fracture criterion based on combined damage mode is more reasonable. (2) The mechanical damage has greater effect on fracture than the electrical damage. When / is very great or when the applied electric field is 2 kV/cm or so, there is little difference for the predicted results from the two modes.
Acknowledgements This work is supported by the National Natural Science Foundation of China (No. 10172036) and by the Scientific Research Foundation for Returned Overseas Chinese Scholars, State Education Ministry.
X. Yang et al. / International Journal of Non-Linear Mechanics 40 (2005) 1204 – 1213
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