Fracture criteria for piezoelectric materials were investigated. Mode I and mixed mode fracture tests were performed on PZT-4 piezoelectric ceramics to verify the validity of the mechanical strain energy release rate as a fracture criterion. Experimental results indicated that crack extension could be aided or impeded by an electric field, depending on the field direction. Further, the direction of crack extension was studied. A crack closure method, together with finite element analysis, was introduced to calculate the mechanical strain energy release rate. The maximum mechanical strain energy release rate was used to predict fracture loads under combined mechanical and electrical loads. It was found that the mechanical strain energy release rate criterion is superior to other fracture criteria and predicts fracture loads fairly accurately.

Fracture Criteria for Piezoelectric Ceramics

Two difficulties have long troubled the field theory of dielectric solids. First, when two electric charges are placed inside a dielectric solid, the force between them is not a measurable quantity. Second, when a dielectric solid deforms, the true electric field and true electric displacement are not work conjugates. These difficulties are circumvented in a new formulation of the theory in this paper. Imagine that each material particle in a dielectric is attached with a weight and a battery, and prescribe a field of virtual displacement and a field of virtual voltage. Associated with the virtual work done by the weights and inertia, define the nominal stress as the conjugate to the gradient of the virtual displacement. Associated with the virtual work done by the batteries, define the nominal electric displacement as the conjugate to the gradient of virtual voltage. The approach does not start with Newton's laws of mechanics and Maxwell–Faraday theory of electrostatics, but produces them as consequences. The definitions lead to familiar and decoupled field equations. Electromechanical coupling enters the theory through material laws. In the limiting case of a fluid dielectric, the theory recovers the Maxwell stress. The approach is developed for finite deformation, and is applicable to both elastic and inelastic dielectrics. As applications of the theory, we discuss material laws for elastic dielectrics, and study infinitesimal fields superimposed upon a given field, including phenomena such as vibration, wave propagation, and bifurcation.

https://imechanica.org/files/deformable%20dielectrics%20revision%202007%2005%2009.pdf

A nonlinear field theory of deformable dielectrics

Structural reliability concerns of various electromechanical devices call for a better understanding of the mechanisms of fracture in piezoelectric ceramics subjected to combined mechanical and electrical loading. For these materials, due to unexplained discrepancies between theory and experiments, even the basic criterion of fracture remains a point of controversy. A viewpoint adopted in this paper is to model piezoelectric ceramics as a class of mechanically brittle and electrically ductile solids. As a first step toward understanding the effects of electric yielding, a strip saturation model is developed for a finite crack perpendicular or parallel to the poling axis of an infinite poled piezoelectric ceramics medium with electrical polarization reaching a saturation limit along a line segment in front of the crack. This model may be considered as a generalization of the classical Dugdale model for plastic yielding near cracks in thin metal sheets. The essential features of the strip saturation model are analyzed via a simplified electroelasticity formulation. Two energy release rates emerge from this analysis. An “apparent” or global energy release rate appears when evaluating J-integral along a contour surrounding both the electrical yielding strip and the crack tip. Under small scale yielding conditions, this energy release rate is equal to that of a linear piezoelectric crack without electrical yielding. A “local” energy release rate is obtained by evaluating J along an infinitesimal contour near the crack tip. The local energy release rate gives predictions which seem to be in broad agreement with experimental observations. It is also interesting that the local energy release rate is independent of the strength and size of electrical yielding.

Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic

A new electric boundary condition of electric fracture mechanics and its applications

Closed form solutions for all three modes of fracture for an infinite piezoelectric medium containing a center crack subjected to a combined mechanical and electrical loading were obtained. The explicit mechanical and electrical fields near the crack tip were derived, from which the strain energy release rate and the total potential energy release rate were obtained by using the crack closure integral. The suitability in using the stress intensity factor, the total energy release rate, or the mechanical strain energy release rate as the fracture criterion was discussed.

Effect of electric field on fracture of piezoelectric ceramics

We Study cracks either in piezoelectrics, or on interfaces between piezoelectrics and other materials such as metal electrodes or polymer matrices. The projected applications include ferroelectric actuators operating statically or cyclically, over the major portion of the samples, in the linear regime of the constitutive curve, but the elevated field around defects causes the materials to undergo hysteresis locally. The fracture mechanics viewpoint is adopted—that is, except for a region localized at the crack tip, the materials are taken to be linearly piezoelectric. The problem thus breaks into two subproblems: (i) determining the macroscopic field regarding the crack tip as a physically structureless point, and (ii) considering the hysteresis and other irreversible processes near the crack tip at a relevant microscopic level. The first Subproblem, which prompts a phenomenological fracture theory, receives a thorough investigation in this paper. Griffith's energy accounting is extended to include energy change due to both deformation and polarization. Four modes of square root singularities are identified at the tip of a crack in a homogeneous piezoelectric. A new type of singularity is discovered around interface crack tips. Specifically, the singularities in general form two pairs: r1/2±ieand r1/2±ie, where e. and k are real numbers depending on the constitutive constants. Also solved is a class of boundary value problems involving many cracks on the interface between half-spaces. Fracture mechanics are established for ferroelectric ceramics under smallscale hysteresis conditions, which facilitates the experimental study of fracture resistance and fatigue crack growth under combined mechanical and electrical loading. Both poled and unpoled fcrroelectrie ceramics are discussed.

C.-M. Kuo

Papers

Fracture mechanics for piezoelectric ceramics