In this paper, the M -integral is extended for calculating intensity factors for cracked piezoelectric ceramics using the exact boundary conditions on the crack faces. The poling direction is taken at an angle to the crack faces within the plane. Since an analytical solution exists, the problem of a finite length crack in an infinite body subjected to crack face traction and electric flux density is examined. In this case, poling is taken parallel to the crack faces. Numerical difficulties resulting from multiplication of large and small numbers were treated by normalizing the variables. This problem was solved with the M -integral and displacement-potential extrapolation methods. With this example, the superiority of the conservative integral is observed. The values for the intensity factor obtained with the M -integral are found to be more accurate than those found by means of the extrapolation method. In addition, a crack parallel to the poling direction in a four-point bend specimen subjected to both an applied load and an electric field was analyzed and different electric permittivity values in the crack gap were assumed. It is seen that the electric permittivity greatly influences the stress intensity factor K II and the electric flux density intensity factor K IV . The absolute value of these intensity factors increases with an increase in the value of the electric permittivity in the crack. The influence of the permittivity on K I is rather small.

In order to realize a plan for a hypersonic aircraft, development of a smart heat-resisting plate possible to control a thermal stress has been required because the safety of structural members must be secured even if they are exposed to a severe thermal loading beyond an estimated load. In view of such a background, this paper deals with a control problem of a thermal stress in a multilayer composite circular disk consisting of a structural layer and piezoceramic layers with concentrically arranged electrodes. When a heating temperature distribution acts on the structural layer surface, the maximum thermal stress in the structural layer can be suppressed by applying appropriate voltages to the electrodes. This thermo-elastic problem has been theoretically analyzed by employing the potential function techniques. Utilizing the analytical results, the nonlinear optimization problem for determining the applied voltages is transformed into a linear programming problem and then the optimum solution is successfully obtained. Based on the obtained solutions, the structure of a composite disk has been designed in order to demonstrate the function of stress control to the fullest extent possible. Finally, numerical results for the stresses before and after applying the determined voltages as well as for the structure design of the composite disk and the suppression ratio of the maximum thermal stress are shown in graphical and tabular forms. It is seen from the numerical results that the maximum thermal stress can be reduced by about 34% when the structure of the composite disk is designed optimally.

A semi-permeable interface crack in infinite elastic dielectric/piezoelectric bimaterials under combined electric and mechanical loading is studied by using the Stroh complex variable theory. Attention is focused on the influence induced from the permittivity of the medium inside the crack gap on the near-tip singularity and on the energy release rate (ERR). Thirty five kinds of such bimaterials are considered, which are constructed by five kinds of elastic dielectrics and seven kinds of piezoelectrics, respectively. Numerical results for the interface crack tip singularities are calculated. We demonstrate that, whatever the dielectric phase is much softer or much harder than the piezoelectric phase, the structure of the singular field near the semi-permeable interface crack tip in such bimaterials always consists of the singularity r − 1 ∕ 2 and a pair of oscillatory singularities r − 1 ∕ 2 ± i ε . Calculated values of the oscillatory index ε for the 35 kinds of bimaterials are presented in tables, which are always within the range between 0.046 and 0.088. Energy analyses for five kinds of such bimaterials constructed by PZT-4 and the five kinds of elastic dielectrics are studied in more detail under four different cases: (i) the crack is electrically conducting, (ii) the crack gap is filled with air/vacuum, (iii) the crack gap is filled with silicon oil, and (iv) the crack is electrically impermeable. Detailed comparisons on the variable tendencies of the crack tip ERR against the applied electric field are given under some practical electromechanical loading levels. We conclude that the different values of the permittivity have no influence on the crack tip singularity but have significant influences on the crack tip ERR. We also conclude that the previous investigations under the impermeable crack model are incorrect since the results of the ERR for the impermeable crack show significant discrepancies from those for the semi-permeable crack, whereas the previous investigations under the conducting crack model may be accepted in a tolerant way since the results of the ERR show very small discrepancies from those for the semi-permeable crack, especially when the crack gap is filled with silicon oil. In all cases under consideration the curves of the ERR for silicon oil are more likely tending to those for the conducting crack rather than to those for air or vacuum. Finally, we conclude that the variable tendencies of the ERR against the applied electric field have an interesting load-dependent feature when the applied mechanical loading increases. This feature is due to the nonlinear relation between the normal electric displacement component and the applied electromechanical loadings from a quadratic equation.

Four-point-bending V-notched specimens of lead zirconate titanate (PZT) poled parallel to the long axis are fractured under conditions of controlled crack growth in a custom-made device. In addition to the mechanical loading electric fields, up to 500 V ∕ mm are applied parallel and anti-parallel to the poling direction, i.e., perpendicular to the crack surface. To determine the different contributions to the total energy release rate, the mechanical and the piezoelectric compliance, as well as the electrical capacitance of the sample, are recorded continuously using small signal modulation/demodulation techniques. This allows for the calculation of the mechanical, the piezoelectric, and the electrical part of the total energy release rate due to linear processes. The sum of these linear contributions during controlled crack growth is attributed to the intrinsic toughness of the material. The nonlinear part of the total energy release rate is mostly associated to domain switching leading to a switching zone around the crack tip. The measured force-displacement curve, together with the modulation technique, enables us to determine this mechanical nonlinear contribution to the overall toughness of PZT. The intrinsic material toughness is only slightly dependent on the applied electric field (10% effect), which can be explained by screening charges or electrical breakdown in the crack interior. The part of the toughness due to inelastic processes increases from negative to positive electric fields by up to 100%. For the corresponding nonlinear electric energy change during crack growth, only a rough estimate is performed.

This paper is devoted to the linear analysis of a slender homogeneous piezoelectric beam that undergoes tip loading. The solution of the Saint-Venant’s problem presented in this paper generalizes the known solution for a homogeneous elastic beam. The analytical approach in this study is based on the Saint-Venant’s semi-inverse method generalized to electroelasticity, where the stress, strain, and (electrical) displacement components are presented as a set of initially assumed expressions that contain tip parameters, six unknown coefficients, and three pairs of auxiliary (torsion/bending) functions in two variables. These pairs of functions satisfy the so-called coupled Neumann problem (CNP) in the cross-sectional domain. In the limit “elastic” case the CNP transforms to the Neumann problem, for a beam made of a poled piezoceramics the CNP is decomposed into two Neumann problems. The paper develops concepts of the torsion/bending functions, the torsional rigidity and shear center, the tip coupling matrix for a piezoelectric beam. Examples of exact and numerical solutions for elliptical and rectangular beams are presented.

A semi-permeable interface crack in dissimilar piezoelectric materials is studied in detail. Attention is focused on the influence induced from the permittivity of the medium inside the crack gap on the near-tip singularity and the crack tip energy release rate (ERR). The Stroh complex variable theory ( Stroh, A. N., 1958, Philos. Mag. 3, pp. 625–646; Ting, T. C. T., Int. J. Solids Struct., 22, pp. 965–983 ) is used to obtain the solution, from which some useful numerical results for 21 kinds of dissimilar piezoelectric materials are calculated. They are combined from seven kinds of commercial piezoelectric ceramics. The distribution of the normal electric displacement component (NEDC) along the interface crack is assumed to be uniform and the corresponding problem is then deduced to a Hilbert problem with an unknown NEDC. Solving the Hilbert problem and determining the near-tip field for each of the 21 bimaterials, we determine the crack tip singularities and find that the crack-tip singularity for a certain combination of two dissimilar piezoelectric materials can be either oscillatory or nonoscillatory when the poling axes of both piezoelectric materials are perpendicular to the interface crack. Energy analyses for PZT ‐ 4 ∕ Ba Ti O 3 as a typical nonoscillatory class bimaterial and those for PZT - 5 H ∕ Ba Ti O 3 as a typical oscillatory class bimaterial are specially studied in detail under four different conditions: (i) the crack gap is filled with air or vacuum; (ii) the crack gap is filled with silicon oil to avoid discharge; (iii) the crack gap is conducting; and (iv) the electrically impermeable crack. Detailed comparisons are performed among the four cases. We conclude that the different values of the permittivity have no influence on the crack tip singularity but have significant influences on the crack tip ERR under the combined electromechanical loading. We also conclude that the previous investigations under the insulating crack model are incorrect or misleading since the model overestimates the effect of the electric field on the ERR very much and the results of the ERR for the impermeable crack show significant discrepancies from those for the semi-permeable crack. Whereas the previous investigations under the conducting crack model may be accepted in a tolerant, way, the results of the ERR show very small discrepancies from those for the semi-permeable crack model, especially when it filled with silicon oil.

By means of the Hankel transform and dual-integral equations, the nonlinear response of a penny-shaped dielectric crack with a permittivity κ 0 in a transversely isotropic piezoelectric ceramic is solved under the applied tensile stress σ z A and electric displacement D z A . The solution is given through the universal relation, D c ∕ σ z A = K D ∕ K I = M D ∕ M σ , regardless of the electric boundary conditions of the crack, where D c is the effective electric displacement of the crack medium, and K D and K I are the electric displacement and the stress intensity factors, respectively. The proportional constant M D ∕ M σ has been derived and found to have the characteristics: (i) for an impermeable crack it is equal to D z A ∕ σ z A ; (ii) for a permeable one it is only a function of the ceramic property; and (iii) for a dielectric crack with a finite κ 0 it depends on the ceramic property, the κ 0 itself, and the applied σ z A and D z A . The latter dependence makes the response of the dielectric crack nonlinear. This nonlinear response is found to be further controlled by a critical state ( σ c , D z A ) , through which all the D c versus σ z A curves must pass, regardless of the value of κ 0 . When σ z A < σ c , the response of an impermeable crack serves as an upper bound, whereas that of the permeable one serves as the lower bound, and when σ z A > σ c the situation is exactly reversed. The response of a dielectric crack with any κ 0 always lies within these bounds. Under a negative D z A , our solutions further reveal the existence of a critical κ * , given by κ * = − R D z A , and a critical D * , given by D * = − κ 0 ∕ R ( R depends only on the ceramic property), such that when κ 0 > κ * or when ∣ D z A ∣ < ∣ D * ∣ , the effective D c will still remain positive in spite of the negative D z A .

Previous studies assumed that a crack is either impermeable or permeable, which are actually two limiting cases of a dielectric crack. This paper considers the electroelastic problem of a three-dimensional transversely isotropic piezoelectric material with a penny-shaped dielectric crack perpendicular to the poling axis. Using electric boundary conditions controlled by the boundaries of an opening crack, the electric displacements at the crack surfaces are determined. The Hankel transform technique is employed to reduce the considered problem to dual integral equations. By solving resulting equations, the results are presented for the case of remote uniform loading, and explicit expressions for the electroelastic field at any point in the entire piezoelectric body are given in terms of elementary functions. Moreover, the distribution of asymptotic field around the crack front and field intensity factors are determined. Numerical results for a cracked PZT-5H ceramic are evaluated to examine the influence of the dielectric permittivity of the crack interior on the field intensity factors, indicating that the electric boundary conditions at the crack surfaces play an important role in determining electroelastic field induced by a crack, and that the results are overestimated for an impermeable crack, and underestimated for a permeable crack.

Field singularities of collinear and collinear periodic interface cracks between an electrode and a piezoelectric matrix are studied in terms of the Stroh formalism for mixed boundary conditions. In contrast to the relevant work done previously on this subject, the problem is solved based on the assumption that the upper and lower planes embedding the electrode consist of two arbitrary piezoelectric materials, and the cracks are assumed to be permeable. The problem is reduced to an interfacial crack problem equivalent to that in purely elastic media. Explicit expressions are presented for the complex potentials and field intensity factors. All the field variables exhibit oscillatory singularities, and their intensities are dependent on the material properties and the applied mechanical loads, but not on the applied electric loads.

This paper investigates the applicability and effect of the crack-free electrical boundary conditions in piezoelectric fracture. By treating flaws in a medium as notches with a finite width, the results from different electrical boundary condition assumptions on the crack faces are compared. It is found that the electrically impermeable boundary is a reasonable one for engineering problems. Unless the flaw interior is filled with conductive media, the permeable crack assumption may not be directly applied to the fracture of piezoelectric materials in engineering applications.

Recent advances in smart structures technology have lead to a resurgence of interest in piezoelectricity, and in particular, in the solution of fundamental boundary value problems. In this paper, we develop an analytic solution to the axisymmetric problem of an infinitely long, radially polarized, radially orthotropic piezoelectric hollow circular cylinder rotating about its axis at constant angular velocity. The cylinder is subjected to uniform internal pressure, or a constant potential difference between its inner and outer surfaces, or both. An analytic solution to the governing equilibrium equations (a coupled system of second-order ordinary differential equations) is obtained. On application of the boundary conditions, the problem is reduced to solving a system of linear algebraic equations. The stress distribution in the tube is obtained numerically for a specific piezoceramic of technological interest, namely PZT-4. For the special problem of a uniformly rotating solid cylinder with traction-free surface and zero applied electric charge, explicit closed-form solutions are obtained. It is shown that for certain piezoelectric solids, stress singularities at the origin can occur analogous to those occurring in the purely mechanical problem for radially orthotropic elastic materials.

A permeable crack model is proposed to analyze crack growth in a piezoelectric ceramic. In this model, a permeable crack is modeled as a vanishing thin, finite dimension, rectangular slit with dielectric medium inside. A first-order approximation solution is derived in terms of the slit height, h 0 . The main contribution of this paper is that the newly proposed permeable crack model reveals that there exists a realistic leaky mode for electrical field, which allows applied electric field passing through the dielectric medium inside a crack. By taking into account the leaky mode effect, a correct estimation of electrical and mechanical fields in front of a crack tip in a piezoelectric ceramic is obtained. To demonstrate this new finding, a closed-form solution is obtained for a mode III permeable crack under both mechanical as well electrical loads. Both local and global energy release rates are calculated based on the permeable crack solution obtained. It is found that the global energy release rate derived for a permeable crack is in a broad agreement with some known experimental observations. It may be served as a fracture criterion for piezoelectric materials. This contribution reconciles the outstanding discrepancy between experimental observation and theoretical analysis on crack growth problem in piezoelectric materials.

In this paper, the displacement boundary value problem of a piezoelectric material plane with an elliptic hole is studied. As the permittivity of the medium in the hole is far less than that of the piezoelectric material, the electric induction of the medium is negligible. An exact solution is obtained. Its application to the inclusion problem has been given. The components of the stress tensor are discussed.

A new theory is developed to model the hysteresis relation between polarization and electric field of piezoceramics. An explicit formulation governing the hysteresis is obtained by using saturation polarization, remnant polarization, and coercive electric field. A new form of elastic Gibbs energy is proposed to address the coupling relations between electrical field and mechanical field. The nonlinear constitutive relations are derived from the elastic Gibbs energy and are applicable in the case of high stroke actuation. The hysteresis relations obtained using the current model are correlated with experimental results. The static deflection of a cantilever beam with surface-bonded piezoelectric actuators is analyzed by implementing the current constitutive relations. Numerical results reveal that hysteresis is an important issue in the application of piezoceramics.

A thin electrode layer embedded at the interface of two piezoelectric materials represents a common feature of many electroceramic multilayer devices. The analysis of interface cracks between the embedded electrode layer and piezoelectric ceramic leads to a nonstandard mixed boundary value problem which likely prevents a general analytical solution. The present work shows that the associated mixed boundary value problem does indeed admit an exact elementary solution for a special case of major practical interest in which the two piezoelectric half-planes are poled in opposite directions perpendicular to the electrode layer. In this case, it is found that oscillatory singularity disappears, in spite of the unsymmetric characters of the problem, and electroelastic fields exhibit power singularities. Particular emphasis is placed on the near-tip singular stresses along the bonded interface. The results show that tensile stress exhibits the square root singularity along the interface whereas shear stress exhibits the dominant-order nonsquare root singularity. In addition, the present model indicates that a pure electric-field loading could induce the dominant-order singular shear stress directly ahead of the interface crack tip. [S0021-8936(00)00602-4]

The present paper deals with theoretical and computational analysis of quasi-static, normal indentation of a transversely isotropic, linear elastic, piezoelectric half-space by a rigid spherical indenter. The contact is axisymmetric, nonconforming, monotonically advancing with load, frictionless and adhesionless. The indenter was modeled either as perfect conductor or as perfect insulator. The mechanical and electrical fields below the surface were examined. The issues of mechanical and dielectric strength due to indentation were examined using Weibull statistics of surface imperfections. The particular cases of PZT-4, PZT-5A, BaTiO 3 , and Ba 0.917 Ca 0.083 TiO 3 indented by rigid punches having either zero electrical potential or zero electric charge were solved with finite element analysis. [S0021-8936(00)02502-2]