In-plane Electric Loading Research Articles

Electromechanical Effects of a Screw Dislocation Around a Finite Crack in a Piezoelectric Material

The interaction between a screw dislocation and a finite crack in an unbounded piezoelectric medium is studied in the framework of linear piezoelectric theory. A straight screw dislocation with the Burgers vector, which is normal to the isotropic basal plane, positioned around the tip of a finite crack is considered. In addition to having a discontinuous electric potential across the slip plane, the dislocation is assumed to be subjected to a line force and a line charge at the core. The explicit solution is derived by means of complex variable and conformal mapping methods. All field variables such as stress, strain, electric field, electric displacement near the crack tip, and the forces on a screw dislocation, the field intensity factors, and the energy release rate are determined under the combined out-of-plane mechanical and in-plane electrical loads. Also, the effects of screw dislocation and electrical loads are numerically analyzed.

The electromechanical behavior of a piezoelectric actuator bonded to an anisotropic elastic medium

Significant efforts have been made in the study of electromechanical behavior of piezoelectric structures. However, the success has been mainly confined to the global response of the structures. This paper provides a comprehensive analytical study on the load transfer between a piezoceramic actuator and an anisotropic elastic medium under in-plane mechanical and electrical loading. The actuator is characterized by an electroelastic line model with the poling direction being perpendicular to its length. The electrically induced local stress field is studied in detail by using Fourier transform technique and solving the resulting integral equations in terms of the interfacial shear stress. Typical examples are provided to show the effects of the geometry, the material combination and the material anisotropy of the composite on the load transfer. The study is further extended to treat the interfacial debonding between the actuator and the host material. The load transfer predicted by the current model is found to be in good quantitative accord with the FEM analysis for general orthotropic structures.

An exact solution for the three-phase piezoelectric cylinder model under antiplane shear and its applications to piezoelectric composites

A three-phase piezoelectric cylinder model is proposed and an exact solution is obtained for the model under a far-field antiplane mechanical load and a far-field inplane electrical load. The three-phase model can serve as a fiber/interphase layer/matrix model, in terms of which a lot of interesting mechanical and electrical coupling phenomena induced by the interphase layer are revealed. It is found that much more serious stress and electrical field concentrations occur in the model with the interphase layer than those without any interphase layer. The three-phase model can also serve as a fiber/matrix/composite model, in terms of which a generalized self-consistent approach is developed for predicting the effective electroelastic moduli of piezoelectric composites. Numerical examples are given and discussed in detail.

General treatment of mode III interfacial crack problems in piezoelectric materials

The anti-plane problem of N collinear interfacial cracks between dissimilar transversely isotropic piezoelectric media, which are subjected to piecewise uniform out-of-plane mechanical loading combined with in-plane electric loading at infinity, and also a line loading at an arbitrary point, is addressed by using the complex function method. In comparison with other relevant works, the present study has two features: one is that the analysis is based on the permeable crack model, i.e. the cracks are considered as permeable thin slits, and, thus, both the normal component of electric displacement and the tangential component of electric field are assumed to be continuous across these slits. The other feature is that explicit closed-form solutions are given not only in piezoelectric media, but also inside cracks when the media are subjected to the most general loading. It is shown that the singularities of electric displacement and electric field in the media are always dependent on that of stress for the general case of loading, and all the singularities of field variables are independent of the applied uniform electric loads at infinity. For the interfacial cracks the electric field is square-root singular at the crack tips and shows jumps across the interface, while the normal component of the electric field is linearly variable inside the crack, but the tangential component is square-root singular. However, for a homogeneous medium with collinear cracks, the electric field is always nonsingular in the medium while the electric displacement exhibits square-root singularity. Moreover, in this case, the electric field inside any crack is equal to a constant when uniform loads are applied at infinity.

A generalized self-consistent method for piezoelectric fiber reinforced composites under antiplane shear

A three-phase piezoelectric confocal elliptical cylinder model is proposed, and an exact solution is obtained for the model subjected to antiplane mechanical and inplane electrical loads at infinity by using the conformal mapping integrated with the Laurent series expansion technique. Based on the model and solution, a generalized self-consistent method is developed for predicting the relevant effective electroelastic moduli of piezoelectric fiber reinforced composite, accounting for variations in fiber section shapes and randomness in distribution and orientation. The dilute, self-consistent, differential and Mori–Tanaka micromechanics theories for piezoelectric fiber reinforced composites are also extended to consider randomness in fiber section orientation in a statistical sense. A full comparison is made among these five micromechanics methods and with the Hori and Nemat-Nasser's rigorous upper and lower bounds, which shows that the generalized self-consistent method and Mori–Tanaka method can verify each other's results, whereas other micromechanics methods may lead to significant deviations, or even unacceptable results. Finally, as an application of the proposed generalized self-consistent method, the complex factors that influence the effective piezoelectric modulus are discussed.

Eccentric crack in a rectangular piezoelectric medium under electromechanical loadings

The solutions of an eccentric crack problem in a rectangular piezoelectric ceramic medium under combined anti-plane shear and in-plane electrical loadings are obtained by the continuous electric crack face condition. Fourier transforms and Fourier series are used to reduce the problem to two pairs of dual integral equations, which are then expressed by a Fredholm integral equation of the second kind. Numerical values of the stress intensity factor and the energy release rate are obtained to show the influence of the electric field.

Penny shaped crack in a three-dimensional piezoelectric strip under in-plane normal loadings

The singular mechanical and electric fields in a three-dimensional piezoelectric ceramic strip containing a penny shaped crack under in-plane normal mechanical and electrical loadings based on the continuous electric boundary conditions on the crack surface are considered here. The potential theory and Hankel transforms are used to obtain a system of dual integral equations, which is then expressed as a Fredholm integral equation. All sorts of field intensity factors of Mode I are given, and numerical values for PZT-6B piezoelectric ceramic are graphically shown.

Modelling and analysis of dynamic interaction between piezoelectric actuators

This article provides a comprehensive theoretical treatment of the coupled dynamic electromechanical behaviour of interacting piezoceramic actuators embedded in an elastic medium under inplane electrical load. The actuators are characterized by an electroelastic line model with the poling direction being perpendicular to its length. The theoretical formulations, governing this coupled system, are based upon the use of integral equations and a new pseudo-incident wave method. A new shear stress singularity factor (SSSF) at the tip of the actuator is obtained by solving these integral equations using Chebyshev polynomial expansions. Typical examples are provided to show the effect of the geometry of the actuator, the material combination and the loading frequency upon the SSSF.

Crack problem at interface of piezoelectric strip bonded to elastic layer under anti-plane shear

Using the theory of linear piezoelectricity, the problem of two layered strip with a piezoelectric ceramic bonded to an elastic material containing a finite interface crack is considered. The out-of-plane mechanical and in-plane electrical loadings are simultaneously applied to the strip. Fourier transforms are used to reduce the problem to a pair of dual integral equations, which is then expressed in terms of a Fredholm integral equation of the second kind. The stress intensity factor is determined, and numerical analyses for several materials are performed and discussed.

Interface crack between piezoelectric and elastic strips

An interface crack between piezoelectric and elastic strips is analyzed using the theory of linear piezoelectricity. The combined out-of-plane mechanical and in-plane electrical loads are applied to the layered strip. Fourier transforms are used to reduce the problem to a pair of dual integral equations, which is then expressed in terms of a Fredholm integral equation of the second kind. The stress intensity factor is determined, and numerical analysis is performed and discussed.

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

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.

Analysis of a bi-piezoelectric ceramic layer with an interfacial crack subjected to anti-plane shear and in-plane electric loading

The behaviour of a bi-piezoelectric ceramic layer with a centre interfacial crack subjected to anti-plane shear and in-plane electric loading has been studied. The dislocation density functions and the Fourier integral transform method have been employed to eliminate the problem of singular integral equations. The normalized energy release rate, stress and electrical displacement intensity factors, G/ G 0, K III / K III 0 and K D / K D0 , respectively, were determined for different geometric and property parameters by use of two different crack surface electric boundary conditions, i.e. impermeable and permeable. It has been shown that the effects of the thickness and material constants of the piezoelectric layer on all the three parameters, i.e. G/ G 0, K III / K III 0 and K D / K D0 were significant.

Moving interfacial crack between piezoelectric ceramic and elastic layers

An analysis is performed for the problem of a finite Griffith crack moving with constant velocity along the interface of a two-layered strip composed of a piezoelectric ceramic and an elastic layers. The combined out-of-plane mechanical and in-plane electrical loads are applied to the strip. Fourier transforms are used to reduce the problem to a pair of dual integral equations, which is then expressed in terms of a Fredholm integral equation of the second kind. The dynamic stress intensity factor(DSIF) is determined, and numerical results show that DSIF depends on the crack length, the ratio of stiffness and thickness, and the magnitude and direction of electrical loads as well as the crack speed. In case that the crack moves along the interface of piezoelectric and elastic half planes, DSIF is independent of the crack speed.

Measuring fracture toughness of piezoceramics by vickers indentation under the influence of electric fields

The Vickers indentation technique was used to determine the apparent fracture toughness of PZT-4 piezoceramic under the influence of electric fields. The toughness was measured in terms of the mechanical strain energy release rate. Using the analytical mechanical strain energy release rate for an infinite piezoelectric medium with a center crack and far field in-plane mechanical and electrical loadings, the mechanical strain energy release rate was obtained for Vickers indentation. The effect of the electric field was included. The proposed formula was evaluated with the experimental results. It was found that the fracture toughness depended on the indentation force. Vickers indentation tests were also performed on depoled PZT-4 specimens; these test results indicated that the fracture toughness of depoled PZT-4 was also dependent on the indentation force. To eliminate the indentation force dependence, a modified stress intensity formula based on Vickers indentation was proposed.

Antiplane mechanical and inplane electric time-dependent load applied to two coplanar cracks in piezoelectric ceramic material

This work is concerned with the dynamic response of two coplanar cracks in a piezoelectric ceramic under antiplane mechanical and inplane electric time-dependent load. The cracks are assumed to act either as an insulator or as a conductor. Laplace and Fourier transforms are used to reduce the mixed boundary value problems to Cauchy-type singular integral equations in Laplace transform domain. A numerical Laplace inversion algorithm is used to determine the dynamic stress and electric displacement factors that depend on time and geometry. A normalized equivalent parameter describing the ratio of the equivalent magnitude of electric load to that of mechanical load is introduced in the numerical computation of the dynamic stress intensity factor (DSIF) which has a similar trend as that for the pure elastic material. The results show that the dynamic electric field will impede or enhance crack propagation in a piezoelectric ceramic material at different stages of the dynamic electromechanical load. Moreover, the electromechanical response is greatly affected by the ratio of the crack length to the ligament between the cracks. The stress and electric displacement intensity factor can be combined by the energy density factor or function to address the fracture of piezoelectric materials under the combined influence of electromechanical loading.

Effect of electromechanical coupling on the dynamic interaction of cracks in piezoelectric materials

This article provides a comprehensive treatment of the dynamic interaction between two arbitrarily located and oriented cracks in a piezoelectric medium under steady-state inplane electrical and antiplane mechanical loads. Using an impermeable condition along the crack surfaces, a fundamental dynamic solution was developed for the single crack problem. In this fundamental solution, the single crack problem was treated using Fourier transform and the appropriate singular integral equations. The fundamental solution was then implemented into a pseudo-incident wave method to account for the interaction between the cracks. Numerical examples are provided to show the effect of the geometry of the cracks, the material constants, the frequency of the incident wave and the applied electrical field upon the dynamic stress intensity factors. The results show the significant effect of electromechanical coupling upon the stress intensity factor at the crack tip.

On the electroelastic behaviour of a thin piezoelectric actuator attached to an infinite host structure

In this paper, we examine the coupled electromechanical behaviour of a thin piezoceramic actuator embedded in or bonded to an elastic medium under inplane mechanical and electrical loadings. The actuator is characterized by an electroelastic line model with the poling direction being perpendicular to its length. The theoretical formulations, governing this electromechanically coupled problem, are based upon the use of singular integral equations in terms of an interfacial shear stress. A square-root singularity of the resulting shear stress is found at the tips of the actuator. A new shear stress singularity factor (SSSF) was then obtained by solving these singular integral equations using Chebyshev polynomial expansions. Typical examples are provided to show the effect of the geometry of the actuator, the material combination and interfacial debonding upon the shear stress singularity factor.

604 2 つの円状介在物を有する圧電体の電気弾性問題 : 介在物がき裂を有する場合

In this study, we treat analytically a two-dimensional electroelastic problem of an infinite piezoelectric body with two circular piezoelectric inhomogeneities one of which contains a crack. We consider that the body is subjected to anti-plane shear load and in-plane electric load at infinity. The problem is solved based on Bueckner's principle and is finally reduced to a problem of a singular integral equation of the first kind with respect to the distribution function of screw dislocation. Stress intensity factor is formulated. Some numerical results are shown to investigate the influence of micromechanics parameters on electroelastic field and stress intensity factor.

Interaction Between a Semi-Infinite Crack and a Screw Dislocation in a Piezoelectric Material

The interaction between a semi-infinite crack and a screw dislocation under antiplane mechanical and in-plane electrical loading in a linear piezoelectric material is studied in the framework of linear elasticity theory. A straight dislocation with the Burgers vector normal to the isotropic basal plane near a semi-infinite crack tip is considered. In addition to having a discontinuous electric potential across the slip plane, the dislocation is subjected to a line-force and a line-charge at the core. The explicit solution for the model is derived by means of complex variable and conformal mapping methods. The classical 1/r singularity is observed for the stress, electric displacement, and electric field at the crack tip. The force on a screw dislocation due to the existence of a semi-infinite crack subjected to external electromechanical loads is calculated. Also, the effect of the screw dislocation with the line-force and line-charge at the core on the crack-tip fields is observed through the field intensity factors and the crack extension force. [S0021-8936(00)01501-4]

Coupled Electromechanical Behaviour of Piezoelectric Actuators in Smart Structures

The static and dynamic electromechanical behaviour of piezoceramic actuators attached to an elastic medium under inplane mechanical and electrical loads is studied. An electroelastic line model is used to characterize the actuators. The load transfer from the piezoelectric actuators to the host structure is determined by using Fourier transform and solving the resulting integral equations. The work is further extended to account for the interaction between different actuators using a pseudo-incident wave method. The influences of different pertinent coefficients are investigated to show the effect of the geometry of actuators, the material combination and the loading frequency upon the load transfer between the actuators and the host structure.