Modelling and analysis of the dynamic behaviour of piezoelectric materials containing interacting cracks

doi:10.1016/S0167-6636(00)00043-0

Mechanics of Materials

Volume 32, Issue 12, December 2000, Pages 723-737

https://doi.org/10.1016/S0167-6636(00)00043-0 Get rights and content

Abstract

This study is concerned with the treatment of the dynamic behaviour of piezoelectric materials containing interacting cracks under antiplane mechanical and inplane electric loading. A general electrical boundary condition is used to enable the treatment of both permeable and impermeable conditions along the crack surfaces. The theoretical solution of the problem is formulated using integral transform techniques and an appropriate pseudo-incident wave method. The resulting singular integral equations are solved using Chebyshev polynomials to provide the dynamic stress and electric fields. Numerical examples are provided to show the effect of the geometry of the cracks, the piezoelectric constant of the material and the frequency of the incident wave upon the dynamic stress intensity factors. The results show the significant effect of electromechanical coupling upon local stress distribution.

Introduction

The concept of using a network of piezoelectric actuators and sensors to form self-monitoring and self-controlling smart systems in advanced structure design has drawn considerable attention from the research community. Piezoceramics appears to be the most promising actuation material due to its high degree of linearity, high power density and potential for low cost manufacturability. One of the most fundamental issues surrounding the optimization of the effectiveness and reliability of piezoceramic actuators is that piezoceramic materials are brittle and have a tendency to develop critical cracks during manufacture and service processes. These defects will greatly affect their mechanical integrity and electromechanical behaviour. Another important aspect in characterizing piezoelectric actuators is the evaluation of the inertial effect in cases where high deformation rate is involved, such as the vibration control of smart structures under impact loading and the acoustic control of smart skin systems.

Significant efforts had been made to the study of the quasistatic electromechanical behaviour of piezoelectric materials. For example, Deeg (1980) and Benveniste (1992) conducted research on the single elliptical (ellipsoidal) inhomogeneity in unbounded piezoelectric materials using the Green's function approach; Dunn and Taya (1993) studied the effective properties of piezoelectric composites using different micromechanical models; Pak (1990), Sosa, 1991, Suo et al., 1992, Jain and Sirkis, 1994 and He et al. (1994) studied the fracture and damage of piezoelectric materials. In contrast, relatively few studies have been conducted on the dynamic behaviour of cracked piezoelectric materials. Existing research work regarding this issue has been focussed mainly on simplified single crack problems (Shindo et al., 1996, Li and Mataga, 1996a, Li and Mataga, 1996b for examples).

Two electrical boundary conditions have been commonly used in the treatment of crack surfaces, i.e. permeable and impermeable conditions. These conditions represent two limiting cases where the permittivity of the crack is assumed to be infinite and zero, respectively. For the case where the crack is represented by a line of zero thickness, the electric potential on one surface of the crack will be same as that on the other surface (permeable condition). However, since the dielectric constant of a piezoceramic material is usually much higher than that of the free space inside the crack, the electrical boundary condition is very sensitive to the crack surface opening. It is with this in mind that both permeable and impermeable conditions will be considered in the current study.

The present paper provides a theoretical treatment of the dynamic interaction between cracks in a piezoelectric medium under antiplane mechanical and inplane electrical incident wave. A general electrical boundary condition along the crack surfaces is used to enable the treatment of both permeable and impermeable conditions. The dynamic electromechanical behaviour of a single crack is studied using Fourier transform and solving the resulting singular integral equations using Chebyshev polynomials. The single crack solution is then implemented into a pseudo-incident wave method to account for the interaction between cracks. Numerical examples are provided to show the effect of the geometry of the cracks, the electrical boundary condition, the piezoelectric constant of the material and the frequency of the incident wave upon the dynamic stress intensity factor.

Section snippets

Formulation of the problem

The situation envisaged is that of an infinitely extended piezoelectric material containing N cracks subjected to an antiplane mechanical and inplane electrical incident wave of frequency ω and incident angle Γ, as shown in Fig. 1. The half-length and the orientation angle of crack n are assumed to be a_n and θ_n. A global Cartesian coordinate system (X,Y) and N local systems ( $x_{n},y_{n}) (n=1,2…,N$ ) are employed to characterize different cracks. The position of the centre of crack n is given by $X=X_{n}, Y=Y$

Dynamic electromechanical behaviour of a single crack

Consider now the steady-state dynamic problem of single crack n in a piezoelectric medium subjected to an antiplane mechanical and inplane electrical incident wave. Our interest here is to determine the resulting scattering wave due to the reflection from the crack surfaces. The general solution of the scattering field can be obtained by solving Eq. (3) using Fourier transform and using the continuity conditions along y_n=0, leading to: $w(x_{n},y_{n})= sgn (y_{n})∫^{∞}_{−∞} A(s) e^{−α|y_{n}|−isx_{n}} d s,$ $f(x_{n},y_{n})= sgn (y_{n})∫^{∞}_{−∞}$

The dynamic interaction between cracks

The existence of multiple cracks in a piezoelectric medium subjected to dynamic loading results in complicated reflection of electromechanical waves between cracks. The analysis of such a complicated problem can be greatly simplified by using the previously developed single crack solution as a building block and considering the consistency relation between different cracks. Wang and Meguid (1997) developed a pseudo-incident wave method to treat the dynamic interaction between a crack and an

Results and discussion

In this section, we discuss the fundamental dynamic behaviour of interacting cracks in piezoelectric media. Let us consider the case where the cracks are subjected to a harmonic inplane electrical field (D_y=D₀) and an antiplane shear wave of frequency ω, as shown in Fig. 1. The incident stress field can be given by $τ_{xz}^{(in)} =τ cos (Γ) e^{−ik_{0}(xcosΓ+ysinΓ)}, τ_{yz}^{(in)} =τ sin (Γ) e^{−ik_{0}(xcosΓ+ysinΓ)}$ in which τ is the maximum value of the shear stress corresponding to the incident wave front and Γ is the incident

Concluding remarks

This paper provides a general solution to the dynamic interaction between two cracks in a piezoelectric medium under steady-state inplane electric and antiplane mechanical loads. Both permeable and impermeable boundary conditions are included using a general electrical boundary condition along the crack surfaces. The dynamic interaction analysis is based upon the use of singular integral equations coupled with a pseudo-incident wave method.

Whilst the piezoelectric property shows no effect upon

Acknowledgements

This work was supported by the Research Excellence Envelope Program at the University of Alberta and the Natural Sciences and Engineering Research Council of Canada.

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Modelling and analysis of the dynamic behaviour of piezoelectric materials containing interacting cracks

Abstract

Introduction

Section snippets

Formulation of the problem

Dynamic electromechanical behaviour of a single crack

The dynamic interaction between cracks

Results and discussion

Concluding remarks

Acknowledgements

Eng. Fract. Mech.

Int. J. Solid Struct.

J. Mech. Phys. Solids

J. Mech. Phys. Solids.

Int. J. Solids Struct.

Int. J. Solids Struct.

J. Mech. Phys. Solids

Int. J. Solids Struct.