Andreas Ricoeur

Influence of electric fields on the fracture of ferroelectric ceramics

Abstract This paper deals with the fracture mechanics of piezoelectric solids. All investigations consider a single crack, which is exposed to combined electrical and mechanical loading. The main subject of interest is the influence of electric fields on the fracture toughness of ferroelectric ceramics and the derivation of an appropriate fracture criterion. Numerical techniques are presented, allowing for the calculation of fracture quantities, i.e. stress intensity factors and energy release rates, once the piezoelectric field problem has been solved for arbitrary crack configurations using the finite element method. In order to describe a possible shielding of the crack tip due to ferroelectric/elastic domain switching events, a micromechanical model has been developed, based on a closed form solution of the piezoelectric field problem. In order to verify the theory, fracture experiments on barium titanate DCB specimens have been evaluated and compared to predictions of the model.

Finite element computation of the electromechanical J-Integral for 2-D and 3-D crack analysis

Piezoelectric ceramics find an application in many fields of technology. They may serve as sensors or actuators, mostly beeing exposed to high electric and mechanical loads. Therefore, fracture mechanics of piezoelectrics is an important field preserving strength and reliability under different conditions of application. This paper deals with the calculation of electromechanical energy release rates for arbitrary cracks in spatial piezoelectric structures applying a generalized J-integral. The crack problem is solved using a commercial FEM-code obtaining electric and mechanical field variables in nodes and integration points. These results serve as input data for the numerical computation of the electromechanical J-integral. The results are compared to findings from analytical and alternative numerical methods.

A micromechanical model for the fracture process zone in ferroelectrics

Abstract Piezoelectric and ferroelectric ceramics find an application as actuators, sensors or ultrasonic transducers in many fields of technology. Because of their brittleness, problems of strength and reliability have to be major subjects of investigation. In general, the ceramic material is exposed to combined electromechanical loading conditions. An influence of the electric field upon the fracture toughness has been observed by many researchers. An established fracture criterion for ferroelectrics is not known, though. Our investigations deal with the calculation of ferroelectric/ferroelastic domain switching events near the tip of an electromechanically loaded crack. The calculations are based on a semi-analytical solution of the piezoelectric field problem yielding electric and mechanical fields around a crack tip. By means of a switching criterion, the specific work is related to a threshold value, deciding upon location and species of switching events. The thus determined extension of the fracture process zone is the basis for calculating changes in the fracture toughness due to domain processes. Thereby a fracture criterion is suggested, which requires a pure mechanical stress analysis. The influence of electric fields is taken into account permitting the critical value to be a function of the electric field. Results for one special configuration of poling and electric field directions are compared to experimental findings.

Three-dimensional Green's functions for two-dimensional quasi-crystal bimaterials

Owing to their specific structure, which can neither be classified as crystalline nor amorphous, quasi-crystals (QCs) exhibit properties that are interesting to both material science and mathematical physics or continuum mechanics. Within the framework of a mathematical theory of elasticity, one major focus is on features evolving from the coupling of phonon and phason fields, which is not observed in classical crystalline or amorphous materials. This paper deals with the problems of combinations of point phonon forces and point phason forces, which are applied to the interior of infinite solids and bimaterial solids of two-dimensional hexagonal QCs. By using the general solution of QCs, a series of displacement functions is adopted to obtain the analytical results when the two half-spaces are supposed to be ideally bonded or to be in smooth contact. In the final expressions, we provide three-dimensional Green’s functions for infinite bimaterial QC solids in the closed form, which are very convenient to be used in the study of dislocations, cracks and inhomogeneities of the new solid phase. Furthermore, the paper is concluded by a discussion of some special cases, in which Green’s functions for infinite transversely isotropic solids and Green’s functions for a half-space with free or fixed boundary are given.

The weight function for cracks in piezoelectrics

The weight function in fracture mechanics is the stress intensity factor at the tip of a crack in an elastic material due to a point load at an arbitrary location in the body containing the crack. For a piezoelectric material, this definition is extended to include the effect of point charges and the presence of an electric displacement intensity factor at the tip of the crack. Thus, the weight function permits the calculation of the crack tip intensity factors for an arbitrary distribution of applied loads and imposed electric charges. In this paper, the weight function for calculating the stress and electric displacement intensity factors for cracks in piezoelectric materials is formulated from Maxwell relationships among the energy release rate, the physical displacements and the electric potential as dependent variables and the applied loads and electric charges as independent variables. These Maxwell relationships arise as a result of an electric enthalpy for the body that can be formulated in terms of the applied loads and imposed electric charges. An electric enthalpy for a body containing an electrically impermeable crack can then be stated that accounts for the presence of loads and charges for a problem that has been solved previously plus the loads and charges associated with an unsolved problem for which the stress and electric displacement intensity factors are to be found. Differentiation of the electric enthalpy twice with respect to the applied loads (or imposed charges) and with respect to the crack length gives rise to Maxwell relationships for the derivative of the crack tip energy release rate with respect to the applied loads (or imposed charges) of the unsolved problem equal to the derivative of the physical displacements (or the electric potential) of the solved problem with respect to the crack length. The Irwin relationship for the crack tip energy release rate in terms of the crack tip intensity factors then allows the intensity factors for the unsolved problem to be formulated, thereby giving the desired weight function. The results are used to derive the weight function for an electrically impermeable Griffith crack in an infinite piezoelectric body, thereby giving the stress intensity factors and the electric displacement intensity factor due to a point load and a point charge anywhere in an infinite piezoelectric body. The use of the weight function to compute the electric displacement factor for an electrically permeable crack is then presented. Explicit results based on a previous analysis are given for a Griffith crack in an infinite body of PZT-5H poled orthogonally to the crack surfaces.

Accurate loading analyses of curved cracks under mixed-mode conditions applying the J-integral

In linear elastic fracture mechanics the path-independent J-integral is a loading quantity equivalent to stress intensity factors (SIF) or the energy release rate. Concerning plane crack problems,

Three-dimensional analysis of a spheroidal inclusion in a two-dimensional quasicrystal body

J_k

The Refined Theory of One-Dimensional Quasi-Crystals in Thick Plate Structures

Jk is a 2-dimensional vector with its components