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1、http:/ 1Electric potential drops across a crack gap in piezoelectrics1 Zhuocheng Ou Yiheng Chen*Department of Engineering Mechanics,Department of Engineering Mechanics,Xian Jiaotong University Xian Jiaotong University Xian City,710049,P.R.China Xian City,710049,P.R.China Abstract This paper presents
2、 a study of electric potential drops across a PKHS crack gap in piezoelectric materials.The PKHS crack was proposed by Parton and Kudryavtsev 1 and Hao and Shen 2,which accounts for the influence induced by the permittivity of medium inside crack gap upon crack tip parameters.Although many efforts h
3、ave been addressed in the past 10 years,the present investigation does provide some novel features and better understandings on electric potential drops across the crack gap.Based on Strohs theory,an alternative technique is proposed in a compact form without any treatment of the mapping technique 3
4、.Different media such as air or vacuum,Silicon oil,and NaCI solution inside the crack gap are considered for comparisons,from which the influences of different permittivities of media inside the crack gap on the NCODJ and EPD across the crack gap respectively in seven kinds of piezoelectric ceramics
5、 are studied in detail.Numerical results of the normal crack opening displacement jump(NCODJ)and the electric potential drop(EPD)against the applied electric field are plotted in figures.It is found that the NCODJ for a central PKHS crack with length 2mm is always extremely small,around 1m or 0.05%o
6、f the crack length when the mechanical loading is taken to be 20MPa and the electric field varies from 1MV/m to 1MV/m,whereas the EPD is very large,around several hundreds or more than thousand Volt.It is also found that the permittivities of media inside the crack gap influence the NCODJ and EPD si
7、gnificantly.Key words:crack;piezoelectric;permittivity.1.Introduction Over the past several decades,piezoelectric materials have been widely used in electromechanical and electronic devices.Commonly used piezoelectric ceramics are very brittle so that it is very important to understand the fracturin
8、g behavior of piezoelectric materials to ensure reliability of smart structures.However,the fracture behavior of a piezoelectric ceramic under combined mechanical and electrical loadings is much more complicated than that for traditional brittle ceramics where the use of linear elastic fracture mech
9、anics suffices 1-22.There exist some controversial results in both experimental observations and theoretical studies for piezoelectric fracture,which have been bothering researchers for a long time.There have been four kinds of variable tendencies of the fracture behavior as influenced by the applie
10、d electric field in different kinds of piezoelectric ceramics under different loading levels using different experimental specimens and different media inside crack gaps(air or vacuum,Silicon oil,and NaCI solution).On one hand,some researchers claimed two kinds of odd functional dependence between t
11、he fracture behavior and the applied electric field.For example,Tobin and Pak 9 and Park and Sun 5,6 found that a positive electric field promotes crack growth and a negative electric field impedes crack growth,whereas Kumar and Singh 7,8 found an opposite feature in their experiments for PZT EC-65
12、that a positive electric field impedes crack growth and a negative electric field promotes crack growth.On the other hand,some researchers claimed two kinds of even functional dependence.For example,Pak 9,Suo et al.10 and many others predicted that both positive and negative electric fields impede c
13、rack growth,whereas more recently,Jiang and Sun 11 found in their indentation tests that both positive and negative electric fields promote crack growth when the indentation mechanical loading became sufficiently larger(49N).Many efforts have been done to explain these controversial results.It is we
14、ll known that there are two main obstacles in understanding piezoelectric fracture 12.One of them is the reasonable selection of the electric boundary condition along crack faces within the framework of the Linear Piezoelectric Fracture Mechanics(LPFM).The other is the near-tip domain 1 Supported by
15、 the Ph.D.Programs Foundation of Ministry of Education of China(20020698017)http:/ 2switching within the framework of the Nonlinear Piezoelectric Fracture Mechanics(NPFM).There exist at least three kinds of the electric boundary condition reported in the literature:the permeable crack,the impermeabl
16、e crack and the PKHS crack model proposed by Parton and Kudryavtsev 1 and Hao and Shen 2 accounting for the permittivity of medium inside crack gap.Adding these three different assumptions respectively in the classical linear elastic fracture mechanics,many theoretical investigations have been perfo
17、rmed in understanding piezoelectric fracture behaviors.Park and Sun 5,6 used the impermeable crack model for PZT-4 and introduced the so-called mechanical strain energy release rate(MSERR)as a fracture criterion,which is,unlike the crack tip ERR,linearly dependent on the electric field implying an o
18、dd functional dependent feature where the positive electric field promotes crack growth and the negative electric field impedes crack growth.In contrast,the crack tip ERR parameter for the impermeable crack model always showed an even functional dependence on the applied electric field,i.e.,it alway
19、s decreases as the strength of the electric loading increases either in the positive or negative direction.Kumar and Singh 7,8 also used the impermeable crack and performed experimental observations and theoretical studies for PZT EC-65.They claimed another odd functional dependence of the electric
20、field on crack growth,i.e.,the positive electric field impedes crack growth and the positive electric field promotes crack growth.Dunn 18 pointed out that the impermeable crack model could lead to significant errors reading the effect of the applied electric field on crack propagation in piezoelectr
21、ic materials subjected to mode III electroelastic loading.Parton and Kudryavtsev 1 and Hao and Shen 2 proposed the PKHS crack model accounting for the permittivity of medium inside crack gap.Up to the date,the fracture behavior of piezoelectric ceramics is still not fully clear and many controversia
22、l results in both experimental observations and theoretical estimations still exist,which need to be clarified in detail.There are no universal fracture criteria that could be used to explain all existing piezoelectric fracture data.Further studies and better understandings are absolutely needed.Nev
23、ertheless,one of the commonly accepted opinions is that the impermeable crack(charge-free crack)is not physically realistic unless the crack opening is sufficient large.In general,the impermeable crack model overestimates the influence induced from the applied electric field on the near-tip mechanic
24、al and electric field.As pointed out recently by McMeeking 19,20,21,the permittivity of air or vacuum,i.e.,8.851012C/Vm,is actually small,about 1/1000 as compared with commercial piezoelectric ceramics,but it is never zero and approaches based on the impermeable crack model are clearly unphysical si
25、nce even a vacuum has finite permittivity and when the gap within the crack is small.He added:an electric field generally can permeate the interior without much dilution or elevation to the electric flux,implying that the electric potential drop across a crack gap full with air could be neglected.Th
26、is opinion is likely contrast to that given earlier by Suo et.al.10 who said：the permeable or conducting boundary conditions cannot be defended on physical grounds.Specifically,for piezoelectric ceramics,the permittivity is 1000 times higher than the environment(e.g.,air or silicon oil).A crack may
27、be thought of as a low-capacitance medium carrying a potential drop.However,no one in the literature had accounted for the magnitude of this electric potential drop across either an impermeable crack gap or a PKHS crack gap when an applied electric field varies from a high negative value to a high p
28、ositive value.It is really not clear that this drop is large or small for a typical central crack in an infinite plane piezoelectric ceramic.It is also not clear that how the applied electric field influence this drop and what happens for this drop when different media are full in the crack gap.Anot
29、her commonly accepted opinion is that the permeable crack is only suitable in some special cases where the crack opening is sufficiently small,i.e.,the so-called sharp crack or slit crack.In general,the permeable crack underestimates the influence induced from the applied electric field since both n
30、ormal electric displacement and electric potential are continuous across the crack gap.Particularly,we noticed that the previous investigations are performed in quite different situations.For example,different piezoelectric ceramics(hard or soft)are used,the experimental situations are quite differe
31、nt(in air or vacuum,in Silicon oil,and with NaCI solution),the levels of applied mechanical and electric loadings are quite different(higher or lower).The electric field applied by some previous researchers was even higher than the coercive field or the poling field of the associated piezoelectric c
32、eramic.It is actually doubt that whether the experimental observations obtained from such different situations could be compared with each other.More recently,Gruebner et.al.22 developed a finite element analysis of cracks in piezoelectric materials taking into account different values of permittivi
33、ty inside the crack gap.They showed that the electrical permeability of the crack has a significant influence on the electromechanical fields especially near the crack tip and in turn on the fracture mechanics parameters.This means that the near-tip mechanical and electric field of the limited perme
34、able cracks with air being crack medium(i.e.,the PKHS crack)cannot be approximated by the near-tip fields calculated from either impermeable or permeable crack models.In this paper,attention is focused on the normal crack opening displacement jump(NCODJ)and the electric potential drop(EPD)at the cen
35、ter of the PKHS crack surfaces.In Section 2 the variable tendencies of the NCODJ across the PKHS crack against the electric field are plotted in figures for http:/ 3seven commercial piezoelectric ceramics:PZT-5H,BaTiO3,PZT-4,PZT-6B,PZT-7A,P-7 and PZT-PIC 151,respectively,when the mechanical loading
36、is 20MPa and the electric field varies from 1MV/m to 1MV/m.Different media such as air or vacuum,Silicon oil,and NaCI solution inside the crack gap are considered for comparisons and detailed discussions are presented to reveal the main features.In Section 3,the EPD across the PKHS crack is studied
37、in the seven commercial piezoelectric ceramics,respectively.The main conclusions are summarized in Section 4.It is found that the NCODJ across a central PKHS crack with length 2mm is always extremely small,around 1m or 0.05%of the crack length,whereas the EPD is very large,around several hundreds or
38、 thousand Volt under the applied electric field of 1MV/m or 1MV/m when the crack gap is full with air or vacuum or Silicon oil.Since the values of the NCODJ across the crack gap in each of the seven piezoelectric ceramics are less than the average particle size of piezoelectric ceramics(210m),the pr
39、esent results reveal that electric discharges are more likely to occur in air or vacuum under so high electric field of 1MV/m or 1MV/m but not in Silicon oil.It is also found that the permittivities of media inside the crack gap influence both NCODJ and EPD significantly in each of the seven piezoel
40、ectric ceramics.Variable tendencies of both NCODJ and EPD show linear dependence with respect to the applied electric field but the slops of the variable curves for different media are quite different.The larger the permittivity of medium is,the smaller the absolute values of the slops of the NCODJ
41、and EPD curves are.Since Silicon oil has the permittivity 2.5 times larger than the permittivity of air or vacuum,it always induces less influence of the applied electric field on both NCODJ and EPD than air or vacuum,whereas NaCI has extremely large permittivity(20000 times air)so that it induces l
42、ittle influence as a permeable crack when the electric field varies significantly.Therefore,the present investigation clarifies the role played by the permittivity in understanding piezoelectric fracture and reveals that previous experimental observations in Silicon oil,in air or vacuum,and with NaC
43、I solution should be quite different and the fracture behaviors could not be compared to each other.2.Normal displacement jump across the central crack gap In order to make a detailed examination and to obtain better understandings for the PKHS crack model,this section deals with the normal crack op
44、ening displacement jump across the central PKHS crack in infinite plane piezoelectric ceramics under combined mechanical and electric loadings.Numerical results of the normal crack opening displacement jump at the center of the crack are plotted in Figures 1-7 for the seven kinds of piezoelectric ce
45、ramics:PZT-5H,PZT-4,BaTiO3,PZT-6B,PZT-7A,P-7 and PZT-PIC 151,respectively.Here,the central crack length is chosen to be 2mm and the applied mechanical loading at infinity denoted by 22 is chosen to be 20MPa,whereas the applied electric field at infinity demoted by 2E varies from 1MV/m to 1MV/m.The m
46、aterial constants of the seven kinds of piezoelectric ceramics are well known in the literature.For making comparisons,we consider the following four kinds of the electric boundary conditions:(i)the permeable crack model with extremely large permittivity(e.g.,for NaCI,a=200000);(ii)the PKHS crack mo
47、del full with air or vacuum(its permittivity is very small,i.e.,a=0,but never zero);(iii)the PKHS crack model full with Silicon oil(its permittivity is 2.5 times the value of air or vacuum,a=2.50,Park and Sun,1995a,b);(iv)the impermeable crack model with extremely small permittivity(a=1080).http:/ 4
48、It is seen from Figures 1-7 that under the fixed mechanical loading 22=20MPa the curves of the electric potential drop with the symbol corresponding to the central permeable crack in each of the seven piezoelectric ceramics are always straight lines parallel to the horizontal axis.The values of the
49、displacement jump are 1.11,0.92,1.31,0.99,1.21,1.47,and 1.62m for a 2mm crack in PZT-5H,BaTiO3,PZT-4,PZT-6B,PZT-7A,P-7 and PZT-PIC 151,respectively.This means that the applied electric field varying from 1MV/m to 1MV/m yields no influence on the normal crack opening displacement jump for the central
50、 permeable crack in all seven piezoelectric ceramics.In other words,the normal crack opening displacement jump denoted by d2 at the center of the permeable crack is determined only by the applied mechanical loading with no regards to the variation of the applied electric field in a wide range betwee