Herbert Balke

Crack problems in magnetoelectroelastic solids. Part I: exact solution of a crack

Abstract This paper presents an exact treatment on the problem of an elliptic hole or a crack in a magnetoelectroelastic solid subject to the farfield loadings. First, based on the extended version of Eshelby-Stroh’s formulation, the general solution of an elliptical hole is obtained according to exact boundary conditions at the rim of the hole. Then, when the hole degenerates into a crack, explicit solutions are given for the field intensity factors and electric–magnetic fields inside the crack. It is shown that all the singularities of fields are dependent on the applied mechanical loads, not on the applied electric–magnetic loads. Due to its explicitness, the present solution for a crack can also serve as a benchmark to test the validity of various analysis approaches or assumptions to more complicated crack problems in magnetoelectroelastic media.

Crack problems in magnetoelectroelastic solids. Part II: general solution of collinear cracks

In this paper, we study mechanically traction-free and electromagnetically permeable crack problems in infinite magnetoelectroelastic solids with linear coupling between the elastic and electromagnetic fields. Using the Stroh-formalism, we first obtain the general solution for collinear cracks in a magnetoelectroelastic medium subjected to arbitrary loads. Then, we give specific solutions for several examples: finite or infinite number of collinear crack subjected to arbitrary remote loads, and a single crack subjected to a line load at an arbitrary point. It is found that in the most general cases, the singularity of electric-magnetic field is always dependent on that of stress. Especially when the medium is only loaded by the remote uniform field, the intensity factor of stress is the same as that of isotropic materials, and the electric-magnetic field inside any crack is uniform.

A fracture criterion for conducting cracks in homogeneously poled piezoelectric PZT-PIC 151 ceramics

Abstract A fracture criterion for conducting cracks in piezoelectric ceramics under combined electrical and mechanical loads is proposed in connection with a measured material specific fracture curve of PZT-PIC 151. The characteristic fracture curve of the investigated material is a function of the stress intensity factor, K I , which is known from fracture mechanical concepts, and the recently defined electric field intensity factor, K E , for conducting cracks. A four-point-bending device for fracture tests under combined electrical and mechanical loads serves as the experimental set-up. The measured quantities, force, displacement, applied voltage and crack length, are used to evaluate the experiments. The finite element method (FEM) was used to determine the mechanical stress intensity factor, K I , and the electric field intensity factor, K E , taking into account linear piezoelectric coupling in the material. A qualitative process zone model is proposed for a physical interpretation of the fracture curve.

Cracks in functionally graded materials

Abstract The weight function method is described to analyze the crack growth behavior in functionally graded materials and in particular materials with a rising crack growth resistance curve. Further, failure of graded thermal barrier coatings (TBCs) under cyclic surface heating by laser irradiation is modeled on the basis of fracture mechanics. The damage of both graded and non-graded TBCs is found to develop in several distinct stages: vertical cracking → delamination → blistering → spalling . This sequence can be understood as an effect of progressive shrinkage due to sintering and high-temperature creep during thermal cycling, which increases the energy-release rate for vertical cracks which subsequently turn into delamination cracks. The results of finite element modeling, taking into account the TBC damage mechanisms, are compatible with experimental data. An increase of interface fracture toughness due to grading and a decrease due to ageing have been measured in a four-point bending test modified by a stiffening layer. Correlation with the damage observed in cyclic heating is discussed. It is explained in which way grading is able to reduce the damage.

On the local and average energy release in polarization switching phenomena

Abstract We derive an expression for the energy release during polarization switching in ferroelectric/ferroelastic materials with piezoelectric coupling. The energy release density is proportional to the local changes of the remanent and material properties due to crystal reorientation, and depends on the local electromechanical fields before and after switching. Its equivalent in a representative volume element is applied to constitutive modeling of repolarization and analysis of switching stability in a ferroelectric material.

Fracture analysis of electromagnetic thermoelastic solids

Abstract This paper analyzes the problem of collinear permeable cracks in a magnetoelectroelastic solid subjected to uniform heat flow at infinity. The analysis is conducted according to the extended Stroh formalism. Concise expressions are given for the field intensity factors and the electric–magnetic field inside cracks. It is shown that all the field singularities are independent of the applied electric–magnetic loads, and the electric–magnetic field inside cracks is linearly variable with position along the crack line.

Measurement of the Total Energy Release Rate for Cracks in PZT Under Combined Mechanical and Electrical Loading

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.

A critical investigation of the unloading behavior of sharp indentation

Abstract Finite element simulations of the indentation of rigid indenter into an elastic-plastic half space have been conducted in order to evaluate the influence of material properties on the unloading response. The emphasis of these results has been to determine the load penetration behavior, the development of the contact area and to estimate the hardness and the modulus from load penetration curves. The material parameters are chosen such that the ratio of hardness H and Youngs modulus E varies between 0.01 and 0.1. Most of these materials can be regarded as brittle. Since cracking is excluded from the analysis, the results refer to low load penetration.

Electrically driven cracks in piezoelectric ceramics: experiments and fracture mechanics analysis

Piezoelectric systems like multilayer actuators are susceptible to damage by crack propagation induced by strain incompatibilities. These can arise under electric fields for example between the electroded and external regions. Such incompatibilities have been realised in thin rectangular model specimens from PZT-piezoelectric ceramics with top and bottom electrodes only close to one edge. Under an electric field, controlled crack propagation has been observed in situ in an optical microscope. The crack paths are reproducible with very high accuracy. Small electrode widths lead to straight cracks with two transitions between stable and unstable crack growth regions, while large electrode widths result in curved cracks with four transitions. Fracture mechanics analysis is able to explain the different crack paths. An iteration method is developed to simulate the curved crack propagation also for strong curvature of the crack paths using the finite element method. The computed crack contours exhibit excellent quantitative agreement with the experiment with respect to their shape, the stages of stable and unstable crack propagation and the transitions between them. Finally, also the crack length as a function of the electric field can be predicted.

Fracture analysis of circular-arc interface cracks in piezoelectric materials

The anti-plane problem of N arc-shaped interfacial cracks between a circular piezoelectric inhomogeneity and an infinite piezoelectric matrix is investigated by means of the complex variable method. Cracks are assumed to be permeable and then explicit expressions are presented, respectively, for the electric field on the crack faces, the complex potentials in media and the intensity factors near the crack-tips. As examples, the corresponding solutions are obtained for a piezoelectric bimaterial system with one or two permeable arc-shaped interfacial cracks, respectively. Additionally, the solutions for the cases of impermeable cracks also are given by treating an impermeable crack as a particular case of a permeable crack. It is shown that for the case of permeable interfacial cracks, the electric field is jumpy ahead of the crack tips, and its intensity factor is always dependent on that of stress. Moreover all the field singularities are dependent not only on the applied mechanical load, but also on the applied electric load. However, for the case of a homogeneous material with permeable cracks, all the singular factors are related only to the applied stresses and material constants.