Electric Displacement Saturation Research Articles

A Complex Variable Solution Based Analysis of Electric Displacement Saturation for a Cracked Piezoelectric Material

This paper presents an analysis of the effect of electric displacement saturation for a failing piezoelectric ceramic material based on a complex variable solution of a Mode III and a Mode I crack. This particular electric nonlinearity is caused by a reduction of the ionic movement in the material in the presence of high electric fields. Total and strain energy release rates are computed for varying far field stresses, electric displacements, and electric fields and compared for cases without and with full electric displacement saturation to further advance the understanding of failure initiation in piezoelectric ceramics.

Synthesis and characterization of Pb(Yb1/2Nb1/2)O3-based high-Curie temperature piezoelectric ceramics

x Pb(Yb 1/2 Nb 1/2 )O3-(1 − x )Pb(Zr 0.36 Ti 0.64 )O 3 ( x PYN-(1 − x )PZT) piezoelectric ceramics were prepared by the conventional ceramic processing via a B-site oxide mixing route. The synthesized x PYN-(1 − x )PZT ceramics exhibit majority of perovskite structure with slight content of impurity, which exhibit typical tetragonal structure with slight orthorhombic distortion depending on compositions. All the x PYN-(1 − x )PZT ceramics exhibit high Curie temperature ( T C / T m ), higher than 380 °C, and their dielectric behavior above T C / T m can be fitted well by the Curie-Weiss law. The x PYN-(1 − x )PZT ceramics exhibit large resistivity, and excellent ferroelectric and piezoelectric properties, which provide promising for the high-power and high-temperature piezoelectric applications. However, electric energy density of the x PYN-(1 − x )PZT ceramics is small due to their nearly rectangular shape of polarization-electric field ( P-E ) hysteresis loop and early electric displacement saturation, which is not suitable for high energy and power storage applications.

Effect of electric displacement saturation on the hysteretic behavior of ferroelectric ceramics and the initiation and propagation of cracks in piezoelectric ceramics

This paper presents a computational investigation of a proposed simplified account for electric displacement saturation on the hysteretic behavior of initially unpoled ferroelectric ceramics as well as on the initiation and propagation of cracks in poled ferroelectric ceramics within the linear regime of piezoelectricity. For the latter case, experimental observations suggest an odd dependency of the onset of crack initiation in these brittle materials on the orientation of the applied electric field with respect to their poling direction which contradicts theoretical results which propose an even dependency of the energy release rate on the applied electric field within the framework of anisotropic linear piezoelectricity. Electric non-linearities arising at regions of inhomogeneities such as inclusions or at the crack tip are proposed in the literature to avoid this discrepancy. Electric displacement saturation is one such non-linear effect which is investigated in this work. A simplified account of this effect is proposed based on an exponential saturation model of the identified material parameters which can be related to this non-linearity. Its advantage over the superposition of a complex function onto the singular solution of a crack within the framework of linear piezoelectricity lies in the straightforward extension of the proposed approach to problems where no analytical solutions exist. This is outlined based on its incorporation into a rate-dependent ferroelectric model accounting for polarization switching as well as based on its incorporation into a finite element framework capable of simulating the initiation and propagation of cracks in piezoelectric ceramics through strong discontinuities in the displacement field and the electric potential. It is shown that besides the determination of the crack initiation onset also the crack propagation direction is influenced by the appearance of saturation zones arising at the crack tip normal to the polarization direction. The numerically obtained crack paths are found to be close to the experimentally reported results.

Nonlinear fracture of 2D magnetoelectroelastic media: Analytical and numerical solutions

A strip electric–magnetic polarization saturation (SEMPS) model is developed to study the electric and magnetic yielding effects on a crack in magnetoelectroelastic (MEE) media. In this model, the MEE medium is treated as being mechanically brittle, and electrically and magnetically ductile. Analogously to the classic Dugdale model, the electric and magnetic yielding zones in front of the crack are represented for simplicity by two strips. In the electric yielding strip the electric displacement equals the electric displacement saturation and meanwhile in the magnetic yielding zone the magnetic induction equals the magnetic induction saturation. The nonlinear analytical solution of this SEMPS model of crack in an infinite MEE medium is obtained using an integral equation approach. The equivalence between the proposed SEMPS model and the existing strip electric–magnetic breakdown (SEMB) model is demonstrated. To analyze the nonlinear fracture problem in the corresponding finite MEE media, the non-linear hybrid extended displacement discontinuity-fundamental solution (NLHEDD-FS) method is modified, and a multiple iteration approach is adapted to determine the electric and magnetic yielding zones. Comparing with the analytical solution, the applicability and effectiveness of the NLHEDD-FS method is verified. Numerical results based on the SEMPS model for a center crack in infinite and finite MEE strip are presented.

Effects of floating electrodes on the reliability of electrostrictive ceramic multilayeractuators

To investigate the toughness enhancing effect of a floating electrode on an actuator, aconventional actuator and an actuator with a floating electrode are numerically analyzedusing the finite element method. Electrostatic analysis is performed for both typesof actuators based on an assumption of the mathematical equivalence betweenout-of-plane deformation and electrostatics. The electric behavior of a ceramic isidealized by the electric displacement saturation model. The numerical results ofelectric fields and electric displacement fields are obtained from the electrostaticanalysis. For both types of actuators, the self-equilibrating stress fields inducedby a non-uniform distribution of the electric displacement fields are computedusing the finite element method. The stress intensity factors for a flaw-like cracknucleated from the edge of an internal electrode are evaluated for each case. Wefound that the stress intensity factor for the actuator with a floating electrodeis smaller than the factor for the conventional actuator when the length of theflaw-like crack is approximately equal to the grain size. Thus, we conclude thatactuators with floating electrodes have higher reliability than conventional actuators.

Discontinuous crack model for piezoelectric materials and its application to CED evaluation

In order to study the influence of electric displacement saturation on fracture behavior of a piezoelectric material, the electric displacement strip-saturation model by [Gao H, Zhang TY, Tong P. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J Mech Phys Solids 1997;45:491–510] has been applied. However, this model is only applicable to problems such as that of a crack in an infinite plate for which it provides a singular solution. In order to overcome this situation, we developed a crack model for a piezoelectric material named discontinuous crack model that is presented in this paper, to evaluate crack energy density (CED) considering electric yielding and we studied its applicability through finite element analyses. The model is defined and methods to establish its constitutive equation are discussed. Moreover, it is shown that the model by Gao et al. and the ordinary crack model in continuum can be regarded as special cases of the discontinuous crack model. Subsequently, the CED and its derivatives for the discontinuous crack model are defined and their path independent expressions are also derived based on conservation laws. Finally, a finite element formulation is devised and the applicability of the model to the evaluations of CED and its derivatives is studied through finite element analyses of an example.

Modified boundary layer analysis of an electrode in an electrostrictive material

A thin electrode embedded in an electrostrictive material under electric loading is investigated. In order to obtain an asymptotic form of electric fields and elastic fields near the electrode edge, we consider a modified boundary layer problem of an electrode in an electrostrictive material under the small scale saturation condition. The exact electric solution for the electrode is obtained by using the complex function theory. It is found that the shape of the electric displacement saturation zone is sensitive to the transverse electric displacement. A perturbation solution of stress fields induced by incompatible electrostrictive strains for the small value of the transverse electric displacement is obtained. The influence of transverse electric displacement on a microcrack initiation from the electrode edge is also discussed.

A Dielectric Polymer with High Electric Energy Density and Fast Discharge Speed

Dielectric polymers with high dipole density have the potential to achieve very high energy density, which is required in many modern electronics and electric systems. We demonstrate that a very high energy density with fast discharge speed and low loss can be obtained in defect-modified poly(vinylidene fluoride) polymers. This is achieved by combining nonpolar and polar molecular structural changes of the polymer with the proper dielectric constants, to avoid the electric displacement saturation at electric fields well below the breakdown field. The results indicate that a very high dielectric constant may not be desirable to reach a very high energy density.

Asymptotic analysis of an impermeable crack in an electrostrictive material subjected to electric loading

The simple asymptotic problem of an impermeable crack in an electrostrictive ceramic under electric loading is analyzed. Closed form solutions of elastic fields are obtained by using the complex function theory. It is found that the K I-dominant region is very small compared to the electric saturation zone. A fracture parameter for an electrostrictive material subjected to electric loading is discussed. In order to investigate the influence of the transverse electric displacement on fracture behavior under the small-scale conditions, we also consider the modified boundary layer problem of a crack in an electrostrictive material. Analytic solutions of electric displacement fields for the asymptotic problem are obtained based on the nonlinear dielectric theory from a modified boundary layer analysis. The shape of the electric displacement saturation zone is shown to depend on the transverse electric displacement. Stress intensity factors induced by the electrostrictive strains are evaluated using the nonlinear solution of the electric displacements. It is found that the transverse electric displacement affects strongly the variation of the mode mixity.

A Crack with an Electric Displacement Saturation Zone in an Electrostrictive Material

A crack with an electric displacement saturation zone in an electrostrictive material under purely electric loading is analyzed. A strip saturation model is here employed to investigate the effect of the electrical polarization saturation on electric fields and elastic fields. A closed form solution of electric fields and elastic fields for the crack with the strip saturation zone is obtained by using the complex function theory. It is found that the KI-dominant region is very small compared to the strip saturation zone. The generalized Dugdale zone model is also employed in order to investigate the effect of the saturation zone shape on the stress intensity factor. Using the body force analogy, the stress intensity factor for the asymptotic problem of a crack with an elliptical saturation zone is evaluated numerically.

Fracture prediction for piezoelectric ceramics based on the electric field saturation concept

The electrical field saturation model is applied to the fracture prediction of piezoelectric materials containing electrically impermeable cracks. This model is analogously similar to the electric displacement saturation model that available in the literature. An electrical field saturation strip near the crack front is introduced in the analytical model. The stress intensity factor K and the energy release rate G are obtained in closed-form. It is found that fracture predictions based on K and G criteria are identical. Fracture predictions based on the electric field saturation model and the electric displacement model are also found to be the same.

Effect of electric displacement saturation on the stress intensity factor for a crack in a ferroelectric ceramic

A crack in a ferroelectric ceramic with perfect saturation under electric loading is analyzed. The boundary of the electric displacement saturation zone ahead of the crack tip is assumed to be ellipse in shape. The shape and size of ferroelectric domain switching zone near a crack tip is determined based on the nonlinear electric theory. The stress intensity factor induced by ferroelectric domain switching under small-scale conditions is numerically obtained as a function of the electric saturation zone parameter and the ratio of the coercive electric field to the yield electric field. It is found that the stress intensity factor increases as the ratio of the semi-axes of the saturation ellipse increases.