Multiple collinear cracks in a piezoelectric material

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Abstract

In this paper, an exact solution of multiple collinear cracks in piezoelectric material is obtained. The permittivity of air (environment) is considered. Two cases have been studied. In the first case, the permittivity of air is far less than that of piezoelectric material. Therefore, the electric induction in the air is negligible. In the second case, the permittivity of air is comparable with that of piezoelectric material. The problem is deduced into Riemann–Hilbert problem and solved. By the way, the electric boundary conditions are discussed. This result demonstrates that the consideration of air in crack gap reduces the stress intensity factor.

Introduction

From electric engineering to information technology and from ferroelectrics to piezoelectric materials, the electric fracture mechanics is very important. It has been 23 years since Cherepanov (1977) introduced J-integral in the electromagnetic field.

Although many experts Gao et al. (1997); Hao et al. (1996); Sosa (1992); Suo (1991); Suo et al. (1992); Zhang and Tong (1996); Zhang et al. (1997); Zhong et al. (1997); have studied electric fracture mechanics, there are still arguments about electric boundary conditions at crack surfaces. Some authors as Parton (1976), Mikhailov and Parton (1990) considered the following (it is supposed that the crack is located on the Ox1-axis):D2+=D2,φ+,where D2 is the normal component of electric displacement component and φ is the electric potential. The boundary condition (1) has been argued by Pak (1990).

Others have supposed that air (for convenience, the air is used to replace environment because they do not have any mathematical difference) enters when a crack becomes a gap. The permittivity of air is far less than those of piezoelectric materials, so the electric boundary condition can be written as Deeg (1980)D2+=D2=0.

Some times, the crack gap is full of conduction fluid as McMeeking (1987). One hasE1=0orφ=const.

Neither Eq. (1) nor Eq. (2) can avoid their incompletion. If the permittivity of air is not quite smaller than that of the solid, the electric induction in the air (environment) cannot be neglected.

Strictly, the electric boundary condition on crack surfaces can be written in the form (if there is no charge in the crack gap and the large deformation is not considered):[D2]=0,[ϕ]=0,Δϕi=0,where [D2] is the jump of electric displacement component between crack surface and air in deformed crack gap, [ϕ] is the jump of potential ϕ at same place, ϕi is the ϕ in deformed crack gap and Δϕi=0 is ((∂2/∂x12)+(∂2/∂x22))ϕi=0.

Thus, one has to solve Laplace equation in the deformed crack gap. It is too difficult to deal with.

Considering the gap is very small after deformation, Hao et al. (1994) have used the linear change of ϕ along the normal of crack surface to replace the rigorous solution of Laplace equation. In the gap, E2 (for small deformation case, En is replaced by E2) becomes −(ϕ+ϕ)/(u2+u2), and u2 is the displacement component.

Considering D2=εaE2 in air, they obtainD2=−εa+−ϕ)/(u2+−u2),where εa is the permittivity of air. Since in the gap, E2 becomes a constant along the normal, D2 is also a constant along the normal. Then, they obtainD2+=D2=−εa+−ϕ)/(u2+−u2).

It is apparent that Eq. (5) will be reduced to second one of Eq. (1) when u2+u2=0, and to Eq. (2) under the condition εa=0.

Although this boundary condition has considered the air in the crack gap, there is no solution related to this boundary condition except one crack problem. In this paper, the problem of multiple collinear cracks is considered (Fig. 1). Two cases have been dealt with. The first is of boundary condition (2) and the second is of condition (6).

Section snippets

Basic equations

In accordance with Lekhnitskii (1981) and Savin (1961), the constitutive equation is rewritten in the form:εi=aijσj+gkiDk,El=−gljσjlkDk,where the means of εi can be understood by referring to Hao et al. (1994).

When the components of stresses and electric displacement are independent of x3, there are four complex functions fj(zj)(zj=x1jx2). The four constants μj are complex with positive imaginary part. The stresses and electric displacements can be represented by the linear combination of

Boundary conditions

Only the cracks on Ox1 are considered; therefore, on crack surface, it is supposed thatσ246=0D2+=D2=0orD2+=D2=−εa+−ϕ)/(u2+−u2).At infinityσii,Di=Di.Later, the case D2+=D2=m is considered.

It is solved with the substituting method. Firstly, let D2=m and σi=0 in all plane. Therefore, a constant solution is obtained. In this solution, the cracks are not open. Secondly, The electric displacement is D2m at infinity and zero at crack surfaces. Add this solution and constant solution,

Solution to the problem

Using the boundary conditions at infinite ∞ and points on Ox1, this problem is deduced to Riemann–Hilbert problem.

Firstly, the boundary condition (2) (D2+=D2=0 on crack surfaces) is considered and then, the boundary condition (6) are dealt with.

The boundary condition (2) and the mechanical boundary conditions areD2246=0oncracksurfaces.For the linear combination of the four complex variable functions, Eq. (16) becomes2Rej=14kijfj′′(x1)=0,2Rej=14d2jfj′′(x1)=0oncracksurfaceswhere j=1–4, i

Numerical example

In this section it is assumed that the piezoelectric materials is a PZT-4 ceramic with material constants that can be found in Berlincourt et al. (1964).

The plane strain case of a transversely isotropic material is only considered. As Sosa (1991), the isotropic plane is the Oxy plane. The Oxz plane is taken into consideration (if the Oxy plane is dealt with, there is the antiplane case). For convenience, one renames the coordinates such that xx1 and zx2.εi=aijσj+bhiDh,Ef=−bfjσjihDh,where i,

Concluding remarks

An exact solution of collinear multiple cracks in piezoelectric material has been solved by means of complex variable theory. Besides impermeable cases, the permittivity of air (environment) is also dealt with. Further study must focus on the more general method to consider the permittivity of air (environment). It is necessary to discuss the large deformation, as Hao (1990).

Acknowledgements

Thanks to Prof. Shen Ziyuan for his valuable help on English. This work is supported by the National Natural Science Foundation of China.

References (21)

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