Antiplane Permeable Edge Cracks in a Piezoelectric Strip of Finite Width

A piezoelectric strip with permeable edge cracks normal to the strip boundaries is analyzed. Under uniform antiplane mechanical shear and inplane electric loading, the distribution of the entire electroelastic field in a cracked piezoelectric strip is determined in explicit analytic form via the conformal mapping technique. It is found that the strain and the electric displacement exhibit the same singularity as the stress near the crack tips, while the electric field is always uniform. The field intensity factors and the energy release rate are independent of the applied electric load for prescribed stress, and related to the applied electric load for prescribed strain.     

TRANSIENT RESPONSE OF A CRACKED INFINITE PIEZOELECTRIC STRIP UNDER ANTI-PLANE IMPACT

By using the well-established integral transform methodology, the dynamic response of stress and electric displacement around a finite crack in an infinite piezoelectric strip are investigated under anti-plane impact. The dynamic intensity factors of stress and electric displacement are obtained analytically. The results show that the dynamic electric field will promote or retard the propagation of the crack at different stages of the loading process. On the other hand, the response of the electric field is coherent with the applied electric load and independent of the external mechanical load. The result obtained for the anti-plane impact of a cracked infinite piezoelectric ceramic can be regarded as a special case of the present work when the width of the strip tends to infinity.     

The finite element analysis of elasto-plastic materials by use of dislocation dipole systems

It is shown how the stress field due to any prescribed continuous distribution of dislocation dipoles can be determined. This technique then forms the basis of a general method of solution of elasto-plastic material problems. The presentation is limited to situations which conform to either plane stress or strain conditions. Some results are obtained for relatively simple geometrical and loading configurations and compared with classical plasticity solutions. Finally, the method is applied to the problem of a circular hole in a finite strip under tension.     

压电-压磁板条中反平面裂纹的电-磁-弹性分析

基于线性电磁弹性理论,获得了压电-压磁板条中反平面裂纹尖端附近的奇异应力、电场和磁场。假设裂纹位于和板条边界平行的中心位置,并且裂纹是电磁渗透型的。利用Fourier变换,将裂纹面的混合边值问题化为对偶积分方程,即而归结为第二类Fredholm积分方程。通过渐近分析,得到了裂纹尖端附近应力、应变、电位移、电场、磁场和磁感的封闭表达式。结果表明,对于电磁渗透裂纹,电场强度因子和磁场强度因子总为0;板条的宽度对应力强度因子有显著的影响;能量释放率总为正值。     

The central crack problem for a functionally graded piezoelectric strip

In this paper the dynamic anti-plane problem for a functionally graded piezoelectric strip containing a central crack vertical to the boundary is considered. The crack is assumed to be electrically impermeable or permeable. Integral transforms and dislocation density functions are employed to reduce the problem to Cauchy singular integral equations. Numerical results show the effects of loading combination parameter, material gradient parameter and crack configuration on the dynamic response. With the permeable assumption, the electric impact has no contribution to the crack tip field singularity. With the impermeable assumption, the direction of applied electric impact loading plays a great role in the behavior of dynamic stress intensity factor, and the existence of electric load always enhances the crack propagation. However, the crack is easier to propagate under the negative electric load than that under the positive electric load.     

The strip dielectric breakdown model

This paper reports on the analysis of the strip dielectric breakdown (DB) model for an electrically impermeable crack in a piezoelectric medium based on the general linear constitutive equations. The DB model assumes that the electric field in a strip ahead of the crack tip is equal to the dielectric breakdown strength, which is in analogy with the classical Dugdale model for plastic yielding. Using the Stroh formalism and the dislocation modeling of a crack, we derived the relationship between the DB strip size and applied mechanical and electrical loads, the intensity factors of stresses and electric displacement, and the local energy release rate. Based on the results, we discussed the effect of electric fields on fracture of a transversely isotropic piezoelectric ceramic by applying the local energy release rate as a failure criterion. It is shown that for an impermeable crack perpendicular to the poling direction, a positive electric field will assist an applied mechanical stress to propagate the crack, while a negative electric field will retard crack propagation. However, for an impermeable crack parallel to the poling direction, it is found that the applied electric field does not change the mode I stress intensity factor and the local energy release rate, i.e., the applied electric field has no effect on the crack growth.     

有限高磁电弹性体中双半动态与静态裂纹分析

狭长体中的裂纹是断裂力学中经常采用的研究模型。含有共线无限长裂纹的条形磁电弹性体,当面内的力电磁和反平面的剪应力作用在左边裂纹尖端附近的一段裂纹面上时,往往会产生动态断裂。利用复变函数法中的拱形变换公式,导出了磁电全非渗透型边界条件下左裂纹尖端动态的应力强度因子以及机械应变能释放率的解析解。当运动速度趋于零时退化为静止状态下的解。通过数值算例分析了断裂机理,讨论了静止状态下狭长体和裂纹的几何尺寸、外力、电场和磁场分别对能量释放率的影响,为相关器件的设计与制造提供了帮助。     

Dielectric Breakdown Model For An Electrically Impermeable Crack In A Piezoelectric Material

The present work presents a strip Dielectric Breakdown (DB) model for an electrically impermeable crack in a piezoelectric material. In the DB model, the dielectric breakdown region is assumed to be a strip along the crack's front line. Along the DB strip, the electric field strength is equal to the dielectric breakdown strength. The DB model is exactly in analogy with the mechanical Dugdale model. Two energy release rates emerge from the analysis. An applied energy release rate appears when evaluating J-integral along a contour surrounding both the dielectric breakdown strip and the crack tip, whereas a local energy release rate appears when evaluating J-integral along an infinitesimal contour surrounding only the crack tip. Under small yielding conditions, the local energy release rate, if used as a failure criterion, gives a linear relationship between the applied stress intensity factor and the applied electric intensity factor.     

压电材料中反平面渗透裂纹的解

用复变函数的保角映射法,采用可渗透边界条件,研究了含裂纹的无限大压电材料在平面内电场和反平面荷载作用下的耦合场,得到了精确的解和场强度因子以及能量释放率。结果表明,电场强度在裂尖没有奇异性,应变、应力、电位移具有1/2阶的奇异性,能量释放率总是正的。     

On the plastic zone and residual stresses within a strip and a half plane under a moving load

A quasi-steady solution has been obtained for the determination of the extent of plastic zones and the amplitude of the residual stresses in (1) an elasto-plastic strip lying on a rigid frictionless base and (2) a half-plane, under the first passage of a moving load. The load is assumed to be applied with a semi-elliptic pressure distribution which is of sufficient magnitude to cause plastic yielding in the body. A work-hardening material obeying von-Mises' yield criterion is considered. A numerical solution procedure, based on discretizing the plastic strain field in the coordinate system moving with the load into uniformly strained rectangular or semi-infinite elements has been developed. Numerical results are obtained for a strip with linear work-hardening material properties. The effect on the distribution of residual stresses created by the ratio of load half-width to strip thickness is also discussed.     

Closed-form solution for an eccentric anti-plane shear crack normal to the edges of a magnetoelectroelastic strip

Summary The problem of an anti-plane shear crack embedded in a magnetoelectroelastic strip is investigated. The crack is assumed to be normal to the strip edges. By using the finite Fourier transform, the associated mixed boundary-value problem is reduced to triple series equations, then to singular integral equations. Solving the resulting equations analytically, the field intensity factors and energy release rates at the crack tips can be determined in explicit form. The influences of applied electric and magnetic loadings on the normalized energy release rate and mechanical strain energy release rate are presented graphically. Obtained results reveal that applied electric and magnetic loadings affect crack growth, depending on their directions and adopted fracture criteria. The derived solution is applicable to other cases including two collinear cracks distributed symmetrically in a magnetoelectroelastic strip, and a periodic array of collinear cracks in a magnetoelectroelastic plane.     

Criteria for brittle fracture in biaxial tension

The criteria of maximum tangential stress, maximum tangential principal stress, maximum tangential strain and strain energy density are applied to the problems of slit and elliptical cracks under remote uniform biaxial tension. The predicted direction of crack extension and the critical load are compared with experimental results reported by other investigators. The unstable crack paths are determined. The four criteria differ in the case of unequal tension; the strain energy density criterion is the least satisfactory. The criteria of maximum tangential strain and strain energy density can be modified to give a good prediction of critical load.     

Transient response of a functionally graded piezoelectric strip with a penny-shaped crack under electric time-dependent loading

Summary The dynamic fracture problem for a functionally graded piezoelectric material (FGPM) strip containing a penny-shaped crack parallel to the free boundaries is considered in this study. It is assumed that the electroelastic properties of the strip vary continuously along the thickness direction of the strip, and that the strip is under time-dependent electric load. Integral transform techniques and dislocation density functions are employed to reduce the problem to the solutions of a system of singular integral equations. The stress and electric displacement intensity factors versus time are presented for various values of dimensionless parameters representing the crack size, the crack location and the material nonhomogeneity.     

Stresses in an orthotropic semi-infinite strip due to edge loads

Summary The stresses in an orthotropic elastic semi-infinite strip subject to plane strain are investigated. Symmetrical distributions of surface tractions are prescribed on the sides of the strip, while along the end the boundary conditions are arbitrary. By using an integral transform method the problem is reduced to a singular integral equation. The dependence of the stress singularity and the stress-intensity factors on the orthotropic properties of the strip is investigated. Stress distributions over the strip end are evaluated numerically.With 5 Figures     

An exact thermopiezoelasticity solution for a three-phase composite cylinder

An exact solution is first derived for the stress field in a three-phase piezoelectric composite cylinder induced by a uniform heat flux together with an anti-plane shear and in-plane electric field applied at infinity. Based on the method of analytical continuation in conjunction with the alternating technique, the general expressions of the temperature and stress functions are derived explicitly in each medium of a three-phase composite cylinder. It is discovered that the stress and the electric displacement in the inclusion is always linearly proportional to the coordinate. The numerical results demonstrates that the continuity conditions at the interface are indeed satisfied and shows the effects of material mismatch on the stress and electric displacement fields.     

Investigation into the creeping polarization and strain in PZT-855 under combined mechanical and electrical loadings

The time-dependent remnant polarization and remnant strains evolving with time are measured and modeled on initially unpoled ferroceramic PZT-855 under a compressive stress and an electric field. Experiments reveal that the compression stress significantly inhibits the hysteresis loop curves of the electric displacement and the strain since the two hysteresis loop curves against the applied electric field become narrower and narrower as the compression stress becomes larger and larger. Moreover, the compression stress decreases the coercive electric field significantly when compared to that under an applied purely electric field. The variable tendency of the coercive filed in the present case seems like a power-law curve. It is also seen that the compression stress significantly influences the variable features of the remnant polarization and remnant strains against time although the time-dependent feature still remains. In particular, the larger the compression stress is, the smaller the remnant polarization and remnant strains are. For some extreme cases, the remnant strain transfers from a tensile strain to a compressive strain. The power-law formulation proposed by Liu and Huber (J Eur Ceram Soc 29:2799?C2806, 2006) is extended to model the present experimental data. However, unlike those under purely electric fields, as the mechanical loading varies from 0 to 100 MPa, the predicted saturation of the remnant polarization appears much earlier than the present experimental data. This limits the application of the power-law model. Further theoretical investigations are needed which will be given in a sequel.     

On the effects of an electric field on the fracture toughness of poled piezoelectric ceramics

Central crack specimens have been used to study the effects of an applied electric field on the fracture toughness of poled soft lead zirconate titanate ceramics (PZT-5). The ultrasonic lapping technique was used to machine the central crack of the specimens used. The present experimental study illustrated that changing the field from negative to positive reduced the fracture toughness of a specimen subjected to an applied electric field. These experimental results were in agreement with the observations made by Park and Sun (J. Am. Ceram. Soc. 78 (1995) 1475) and Heyer et al. (Acta Mater. 46 (1998) 6615). Finite element results were employed to compare with the experimental data. Four fracture criteria, i.e. total energy release rate, mechanical energy release rate, local energy release rate and strain energy release rate, were compared with the experimental results. It was found the local energy release rate, which was defined based on a strip electric saturation model, was in broad agreement with the experimental observations. Moreover, it was found that the strain energy density criterion was unable to describe the effect of electric field on fracture toughness when a large electric field was applied.     

Anti‐plane mechanical and in‐plane electrical fields for a finite electrode/punch on a finite piezoelectric layer

The problems of a surface electrode and a rigid punch on a finite piezoelectric layer are considered in this paper. The resultant force and the accumulated electric charge on the electrode/punch are prescribed. Closed‐form solutions for the electromechanical fields at the electrode/punch tip are obtained and are expressed in terms of the applied strain and electric field intensity factors. For infinite layer thickness, the strain and electric field intensity factors are obtained in closed‐form. For finite layer thickness, the strain and electric field intensity factors are obtained numerically by the singular integral equation technique. The effect of layer thickness on the electrode/punch tip fields is discussed. It is found that the field intensities at the electrode/punch tip can be reduced considerably by decreasing layer thickness. In addition to the single electrode/punch problem, this paper also provides a solution technique for two collinear surface electrodes/punches on a finite piezoelectric layer. The effect of the relative distance between the two electrodes/punches on the electromechanical fields in the piezoelectric layer is also discussed.     

狭长磁电弹性体中Ⅲ型半无限裂纹的场强度因子

通过引入合适的保角映射, 利用复变函数方法研究了裂纹面上受反平面剪应力和面内磁电载荷共同作用下狭长磁电弹性体中半无限裂纹的断裂行为, 给出了磁电全非渗透型边界条件下裂纹尖端场强度因子和能量释放率的解析解。当狭长体高度趋于无限大时, 可得到无限大磁电弹性体中半无限裂纹的解析解。若不考虑磁场或电场作用, 所得解可退化为已知解。通过数值算例, 分析了裂纹面上受载长度、狭长体高度以及磁、电和机械载荷对能量释放率的影响。