Towards the computation of electrically permeable cracks in piezoelectrics

doi:10.1016/j.engfracmech.2004.03.003

Engineering Fracture Mechanics

Volume 71, Issue 18, December 2004, Pages 2567-2587

https://doi.org/10.1016/j.engfracmech.2004.03.003 Get rights and content

Abstract

The focus of this paper is not the well-known and examined behavior of electrically impermeable and fully permeable cracks but the analysis of limited permeable cracks, i.e. the influence of a dielectric medium inside the crack. The boundary conditions of the impermeable or the fully permeable crack can be considered as simple approximations representing upper and lower bounds for the electrical energy penetrating the crack. In this paper, the accuracy of a known theoretical approach “capacitor analogy” for analyzing limited permeable cracks in piezoelectric ceramics is verified by means of numerical methods and analytical estimations. Different crack configurations in 2D and 3D are analyzed. The cracks are subjected to combined electrical and mechanical loads, which lead to a mixed Mode loading in Mode-I and Mode-IV. The influence of errors committed by the capacitor analogy approaching the crack tip is investigated using piezoelectric crack weight functions and special finite element techniques meshing the medium inside the crack. The numerical results of stress intensity factors, obtained by the crack tip element method which is valid for loaded crack faces, are presented. Finally, the theory is applied to the evaluation of a fracture experiment with a DCB specimen.

Introduction

Nowadays multi-functional piezoelectric materials are used in many fields of engineering. Those materials are integrated into complex structures as sensors, transducers and actuators, where they are exposed to high mechanical and electric loadings. Because of requirements concerning structural strength, reliability and lifetime the behavior of cracks in brittle piezoelectric structures has to be understood and described.

In order to calculate fracture quantities of a crack in a piezoelectric material properly, the electrical properties of the media filling the crack have to be taken into account. In this context impermeable, limited permeable and fully permeable cracks can be distinguished. Especially, the case of a limited permeable crack is supposed to be most interesting because it is closest to the physical nature of the problem. Anyhow, the fully permeable and the impermeable boundary conditions can be considered as theoretical limits. Their application is useful to avoid computational effort. Describing the limited permeable crack in a numerical context, the calculated fracture quantities as stress intensity factors and energy release rates are more accurate compared to those based on permeable or impermeable crack boundary conditions.

This paper critically illuminates a calculation method for limited permeable cracks based on the model of Hao and Shen [1] identifying opposite points on the crack faces as corresponding parallel plates of capacitors of infinitesimal size. The crack surface loads due to the electric displacements inside the crack are determined by iteration. Therefore, in this paper the method is called “iterative capacitor analogy method” (ICA). This approach was also used by Balke et al. [2] and McMeeking [3] performing 2D calculations. Gruebner [4] applied the ICA approach to evaluate CT experiments finding that the crack permeability influences the calculated energy release rates substantially. For a large part of the crack faces the basic assumption of the capacitor analogy, i.e. the parallelism of the opposite surfaces, holds very well. However, this assumption is increasingly violated, if the crack tip is approached. Certainly, considerable errors are restricted to a zone at the crack tip which is very small compared to the total crack length. Nevertheless, fields close to the crack tip have a major influence on the fracture quantities. For the Griffith crack it can be shown that the assumptions the ICA is based on are satisfied all along the crack faces [5]. This holds if poling and loading conditions guarantee that the electric field inside the crack runs perpendicular to the crack plane. In more general cases, when poling and loading directions are oriented arbitrarily with respect to the crack, the requirements to apply the ICA are violated, according to the authors opinion.

A subroutine for the ABAQUS finite element code was developed implementing this ICA approach. It is used for analyzing 2D as well as 3D cracked piezoelectric structures. Fracture quantities are calculated applying the Crack Tip Element method and the Modified Crack Closure Integral. In the latter case the intrinsic charges on the crack faces have to be taken into account. The accuracy of the ICA is validated by the solution of the coupled field problem, meshing the interior of the crack (coupled domain model––CDM). Furthermore, the sensitivity of the field intensity factors towards model based errors near the crack tip is estimated using piezoelectric crack weight functions [6]. Except for the Griffith crack, the applicability of the ICA method to the calculation of fracture quantities for arbitrary limited permeable interior and surface cracks has never been investigated. Therefore, this item plays a major role in this paper. As an application of the numerical tools for limited permeable cracks in piezoelectrics, a fracture experiment with DCB specimens under electromechanical loading is quantitatively investigated.

Section snippets

Basic relations

The phenomenological description of the field problem under quasistatic loading is governed by the balance equations of linear elasticity and electrostatics: $σ_{ij,j} +b_{i} =0$ $D_{i,i} −ω_{V} =0$ In these equations σ_ij denotes the stress tensor, D_i the electric displacement vector, b_i mechanical volume forces and ω_V electrical volume charges. With the unit normal vector n_j of an arbitrary plane within the body or at its boundary, mechanical stresses t_i and electric surface charge densities ω can be written as $σ_{ij} n_{j}$

The Griffith crack

The crack in an infinite plate (Fig. 1) is the first example for developing and validating methods in fracture mechanics. Usually, in numerical calculations this infinite plate is approximated by choosing a crack length to overall length ratio of 10⁻¹ and below.

Fig. 4 illustrates a quarter FE model and the corresponding boundary conditions, i.e. zero displacements u₂ and potentials on the ligament (BC1) and zero displacements u₁ and electric displacements D₁ in the plane of symmetry (BC2). The

Evaluation of fracture experiments applying the ICA method

The implemented and validated ICA is now applied to a practical example, to a double cantilever beam (DCB) specimen. DCB specimens are used for fracture mechanical examinations with stably growing cracks. This specimen type is most suitable to investigate the influence of electric fields on the fracture toughness of ferroelectrics, since the crack growth behavior can be observed in situ when switching on the field. The experiments have been carried out by Förderreuther and Thurn [15], [16], [17]

Conclusions

The paper has presented an implementation of a calculation method for limited permeable cracks in 2D and 3D piezoelectric structures called ICA. The theoretical basis is the capacitor analogy method treating the crack with the interior medium as an accumulation of an infinite number of plane parallel capacitors. This analogy was now extended to 3D problems, too. For analyzing limited permeable cracks, the algorithm uses the electrical potentials and mechanical displacements of the nodes on the

Acknowledgements

This research was funded by the German Research Association (DFG) within the Sino-German Cooperation under contract Ku 929/8.

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Towards the computation of electrically permeable cracks in piezoelectrics

Abstract

Introduction

Section snippets

Basic relations

The Griffith crack

Evaluation of fracture experiments applying the ICA method

Conclusions

Acknowledgements

Engng. Fract. Mech.

Int. J. Solids Struct.

J. Eur. Ceram. Soc.

Comput. Mater. Sci.

Comput. Mater. Sci.

Acta Mater.

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

Engng. Fract. Mech.