An interface crack with contact zones in a piezoelectric/piezomagnetic bimaterial

An interface crack with a frictionless contact zone at the right crack tip between two semi-infinite piezoelectric/piezomagnetic spaces under the action of a remote mechanical loading, magnetic and electric fluxes as well as concentrated forces at the crack faces is considered. Assuming that all fields are independent on the coordinate x ₂ co-directed with the crack front, the stresses, the electrical and the magnetic fluxes as well as the derivatives of the jumps of the displacements, the electrical and magnetic potentials are presented via a set of analytic functions in the (x ₁, x ₃)-plane with a cut along the crack region. Two cases of magneto-electric conditions at the crack faces are considered. The first case assumes that the crack is electrically and magnetically permeable, and in the second case the crack is assumed electrically permeable while the open part of the crack is magnetically impermeable. For both these cases due to the above-mentioned representation the combined Dirichlet–Riemann boundary value problems have been formulated and solved exactly. Stress, electric and magnetic induction intensity factors are found in a simple analytical form. Transcendental equations and a closed form analytical formula for the determination of the real contact zone length have been derived for both cases of magnetic conditions in the crack region. For a numerical illustration of the obtained results a bimaterial BaTiO₃–CoFe₂O₄ with different volume fractions of BaTiO₃ has been used, and the influence of the mechanical loading and the intensity of the magnetic flux upon the contact zone length and the associated intensity factors as well as the energy release rate has been shown.     

Fracture-mechanical assessment of electrically permeable interface cracks in piezoelectric bimaterials by consideration of various contact zone models

Summary An interface crack with an artificial contact zone at the right-hand side crack tip between two piezoelectric semi-infinite half-planes is considered under remote mixed-mode loading. Assuming the stresses, strains and displacements are independent of the coordinate x ₂, the expression for the displacement jumps and stresses along the interface are found via a sectionally holomorphic vector function. For piezoceramics of the symmetry class 6 mm and for electrically permeable crack faces, the problem is reduced to a combined Dirichlet-Riemann boundary value problem which can be solved analytically. Further, analytical expressions for the stresses, electrical displacements, derivatives of elastic displacement jumps, stress and electrical intensity factors are found at the interface. Real contact zone lengths and the well-known oscillating solution are derived from the obtained solution as well. Analytical relationships between the fracture-mechanical parameters of various models are found, and recommendations are suggested concerning the application of numerical methods to the problem of an interface crack in the discontinuity area of a piezoelectric bimaterial. Received 16 March 1999; accepted for publication 31 May 1999     

On interface crack models with contact zones situated in an anisotropic bimaterial

Summary A boundary value problem for two semi-infinite anisotropic spaces with mixed boundary conditions at the interface is considered. Assuming that the displacements are independent of the coordinate x ₃, stresses and derivatives of displacement jumps are expressed via a sectionally holomorphic vector function. By means of these relations the problem for an interface crack with an artificial contact zone in an orthotropic bimaterial is reduced to a combined Dirichlet-Riemann problem which is solved analytically. As a particular case of this solution, the contact zone model (in Comninou's sense) is derived. A simple transcendental equation and an asymptotic formula for the determination of the real contact zone length are obtained. The classical interface crack model with oscillating singularities at the crack tips is derived from the obtained solution as well. Analytical relations between fracture mechanical parameters of different models are found, and recommendations concerning their implementation are given. The dependencies of the contact zone lengths on material properties and external load coefficients are illustrated in graphical form. The practical applicability of the obtained results is demonstrated by means of a FEM analysis of a finite-sized orthotropic bimaterial with an interface crack. Received 19 October 1998; accepted for publication 13 November 1998     

An electrically conducting interface crack with a contact zone in a piezoelectric bimaterial

A plane problem for an electrically conducting interface crack in a piezoelectric bimaterial is studied. The bimaterial is polarized in the direction orthogonal to the crack faces and loaded by remote tension and shear forces and an electrical field parallel to the crack faces. All fields are assumed to be independent of the coordinate co-directed with the crack front. Using special presentations of electromechanical quantities via sectionally-analytic functions, a combined Dirichlet–Riemann and Hilbert boundary value problem is formulated and solved analytically. Explicit analytical expressions for the characteristic mechanical and electrical parameters are derived. Also, a contact zone solution is obtained as a particular case. For the determination of the contact zone length, a simple transcendental equation is derived. Stress and electric field intensity factors and, also, the contact zone length are found for various material combinations and different loadings. A significant influence of the electric field on the contact zone length, stress and electric field intensity factors is observed. Electrically permeable conditions in the crack region are considered as well and matching of different crack models has been performed.     

On a moving interface crack with a contact zone in a piezoelectric bimaterial

An inplane problem for a crack moving with constant subsonic speed along the interface of two piezoelectric materials is considered. A mechanically frictionless and electrically permeable contact zone is assumed at the right crack tip whilst for the open part of the crack both electrically permeable and electrically insulated conditions are considered. In the first case a moving concentrated loading is prescribed at the crack faces and in the second case an additional electrical charge at the crack faces is prescribed as well. The main attention is devoted to electrically permeable crack faces. Introducing a moving coordinate system at the leading crack tip the corresponding inhomogeneous combined Dirichlet–Riemann problem is formulated and solved exactly for this case. All electromechanical characteristics at the interface are presented in a closed form for arbitrary contact zone lengths, and further, the transcendental equation for the determination of the real contact zone length is derived. As a particular case of the obtained solution a semi-infinite crack with a contact zone is considered. The numerical analysis performed for a certain piezoelectric bimaterial showed an essential increase of the contact zone length and the associated stress intensity factor especially for the near-critical speed region. Similar investigations have been performed for an electrically insulated crack and the same behavior of the above mentioned parameters is observed.     

A closed crack tip model for interface cracks inthermopiezoelectric materials

The interface crack problem of a bimaterial thermopiezoelectric solid was treated byapplying the extended version of Strohs formalism and singular integral equation approach. Theinterface crack considered is subjected to combined thermal, mechanical and electric loads.Under the applied loading, the interface crack is assumed to be partially opened. Formulation ofthe problem results in a set of singular integral equations which are solved numerically. Thestudy shows that the contact zone is extremely small in comparison with the crack length. Basedon the formulation, some physically meaningful quantities of interest such as stress intensityfactors and size of contact zone for a particular material group are analyzed.     

Fracture mechanical assessment of interface cracks with contact zones in piezoelectric bimaterials under thermoelectromechanical loadings II. Electrically impermeable interface cracks

This paper constitutes the second part of a study of interface cracks with contact zones in thermopiezoelectrical bimaterials, and it is concerned with the case of an electrically impermeable interface crack. The principal physical peculiarity of this case in comparison with an impermeable interface crack is connected with the dependencies of the contact zone length and the fracture mechanical parameters on the prescribed electrical flux, and in a mathematical sense the main peculiarity is concerned with the reduction of the problem in question to the joint solution of inhomogeneous combined Dirichlet–Riemann and Hilbert boundary value problems. The exact analytical solutions of the mentioned problems have been found for an arbitrary contact zone length, and the required thermal, mechanical and electrical characteristics at the interface as well as the associated fracture mechanical parameters at the corresponding crack tips are presented. The transcendental equations for the determination of the real contact zone length have been obtained for a general case and for a small contact zone length in an especially simple form. Using the admissible directions of the heat and the electrical fluxes defined in this paper as well, the dependencies of the real contact zone length and the associated fracture and electrical intensity factors on the intensities of the thermal and electrical fluxes are presented in tables and associated diagrams.     

Interface crack with a contact zone in an isotropic bimaterial under thermomechanical loading

A plane problem for a thermally insulated interface crack with a contact zone in an isotropic bimaterial under tension–shear mechanical loading and a temperature flux is considered. The expressions for the stresses and the electrical flux as well as for the derivatives of the displacement and the temperature jumps at the material interfaces via sectionally holomorphic mechanical and thermal potential functions are given. After the solution of the thermal problem the inhomogeneous combined Dirichlet–Riemann boundary value problem is formulated and solved exactly. The stresses at the interface and the stress intensity factors at the singular points are presented in a clear analytical form. Special attention is devoted to the case of a small contact zone when the stress intensity factors can be presented in form similar to the associated presentation for an “open” crack model. A transcendental equation and an asymptotic analytic formula for the determination of the real contact zone length are derived. It is shown that for a certain bimaterial this length as well as the correspondent stress intensity factor are defined by a single parameter which depends on the normal-shear loading and the heat flux.     

Periodic set of limited electrically permeable interface cracks with contact zones

Plane problem for an infinite space composed of two different piezoelectric or piezoelectric/dielectric semi-infinite spaces with a periodic set of limited electrically permeable interface cracks is considered. Uniformly distributed electromechanical loading is applied at infinity. The frictionless contact zones at the crack tips are taken into account. The problem is reduced to the combined Dirichlet–Riemann boundary value problem by means of the electromechanical factors presentation via sectionally analytic functions, assuming that the electric flux is uniformly distributed inside the cracks. An exact solution of the problem is proposed. It permits to find in a closed form all necessary electromechanical characteristics at the interface and to formulate the equation for the determination of the electric flux value. Analysis of this equation confirms the correctness of the assumption concerning the uniform distribution of the electric flux in the crack region.Formulae for stresses, electric displacement vector, elastic displacements and electric potential jump at the interface as well as the intensity factors at the crack tips are given. Equation for the contact zone length determination is presented. Calculations for certain material combinations are carried out. The influence of electric permeability of cracks on electromechanical fields and the fracture mechanical parameters is analyzed.     

An analytically-numerical approach for the analysis of an interface crack with a contact zone in a piezoelectric bimaterial compound

An interface crack with an artificial contact zone at the right-hand side crack tip between two dissimilar finite-sized piezoelectric materials is considered under remote mixed-mode loading. To find the singular electromechanical field at the crack tip, an asymptotic solution is derived in connection with the conventional finite element method. For mechanical loads, the stress intensity factors at the singular points are obtained. As a particular case of this solution, the contact zone model (in Comninou’s sense) is derived. A simple transcendental equation and an asymptotic formula for the determination of the real contact zone length are derived. The dependencies of the contact zone lengths on external load coefficients are illustrated in graphical form. For a particular case of a short crack with respect to the dimensions of the bimaterial compound, the numerical results are compared to the exact analytical solutions, obtained for a piezoelectric bimaterial plane with an interface crack.     

Contact Zone Approach for a Moving Interface Crack in a Piezoelectric Bimaterial under Thermoelectromechanical Loading

An interface crack of a finite length moving with a constant subsonic speed v along an interface of two semi-infinite piezoelectric spaces is considered. It is assumed that the bimaterial compound is loaded by a remote mixed mode mechanical loading and a thermoelectrical field and that a frictionless contact zone arises at the leading crack tip. Electrically permeable and electrically insulated cases of the open part of the crack are involved into the consideration. By introducing a moving coordinate system at the crack tip the problem is reduced to a combined Dirichlet–Riemann boundary value problem which is solved exactly. For both cases of the electrical conditions the transcendental equations are obtained for the determination of the real contact zone length, and moreover, the associated closed form asymptotic formulas are found for small values of this parameter. Variations of the contact zone length and the stress intensity factor with respect to the crack speed and the loading have been investigated both for electrically permeable and electrically insulated cases.     

Asymptotic fields in the finite element analysis of electrically permeable interface cracks in piezoelectric bimaterials

Summary The interface crack problem for a piezoelectric bimaterial based on permeable conditions is studied numerically. To find the singular electromechanical field at the crack tip, an asymptotic solution is derived in connection with the conventional finite element method. For mechanical and electrical loads, the complex stress intensity factor for an interface crack is obtained. The influence of the applied loads on the electromechanical fields near the crack tip is also studied. For a particular case of a short crack with respect to the bimaterial size, the numerical results are compared with the exact analytical solutions, obtained for a piezoelectric bimaterial plane with an interface crack.One author (V.G.) gratefully acknowledges the support provided by the Alexander von Humboldt Foundation of Germany.accepted for publication 7 June 2004     

Two-dimensional piezoelectricity. Part II: general solution,Green’s function and interface cracks

The complete solution space of a piezoelectric material is the direct sum of several orthogonal eigenspaces, one for each distinct eigenvalue. Each one of the 14 different classes of piezoelectric materials has a distinct form of the general solution, expressed in terms of the eigenvectors of the zeroth and higher orders and a kernel matrix containing analytic functions. When these functions are chosen to be logarithmic, one obtains, in a unified way, Green’s function of the infinite space as a single 8 × 8 matrix function G_∞ for the various load cases of concentrated line forces, dislocations, and a line charge. This expression of Green’s function is valid for all classes of nondegenerate and degenerate materials. With an appropriate choice of the parameters, it reduces to the solution of a half space with concentrated (line) forces at a boundary point, and with dislocations in the displacements. As another application, eigenvalues and eigensolutions are obtained for the bimaterial interface crack problem.     

Interaction between a screw dislocation and a viscoelastic piezoelectric bimaterial interface

The complex variable method is employed to derive analytical solutions for the interaction between a piezoelectric screw dislocation and a Kelvin-type viscoelastic piezoelectric bimaterial interface. Through analytical continuation, the original boundary value problem can be reduced to an inhomogeneous first-order partial differential equation for a single function of location z = x + iy and time t defined in the lower half-plane, which is free of the screw dislocation. Once the initial, steady-state and far-field conditions are known, the solution to the first order differential equation can be obtained. From the solved function, explicit expressions are then derived for the stresses, strains, electric fields and electric displacements induced by the piezoelectric screw dislocation. Also presented is the image force acting on the screw dislocation due to its interaction with the Kelvin-type viscoelastic interface. The derived solutions are verified by comparing with existing solutions for the simplified cases, and various interesting features are observed, particularly for those associated with the image force.     

Analysis method of planar interface cracks of arbitrary shape in three-dimensional transversely isotropic magnetoelectroelastic bimaterials

Using the fundamental solutions for three-dimensional transversely isotropic magnetoelectroelastic bimaterials, the extended displacements at any point for an internal crack parallel to the interface in a magnetoelectroelastic bimaterial are expressed in terms of the extended displacement discontinuities across the crack surfaces. The hyper-singular boundary integral–differential equations of the extended displacement discontinuities are obtained for planar interface cracks of arbitrary shape under impermeable and permeable boundary conditions in three-dimensional transversely isotropic magnetoelectroelastic bimaterials. An analysis method is proposed based on the analogy between the obtained boundary integral–differential equations and those for interface cracks in purely elastic media. The singular indexes and the singular behaviors of near crack-tip fields are studied. Three new extended stress intensity factors at crack tip related to the extended stresses are defined for interface cracks in three-dimensional transversely isotropic magnetoelectroelastic bimaterials. A penny-shaped interface crack in magnetoelectroelastic bimaterials is studied by using the proposed method.The results show that the extended stresses near the border of an impermeable interface crack possess the well-known oscillating singularity r^?1/2±iε or the non-oscillating singularity r^?1/2±κ. Three-dimensional transversely isotropic magnetoelectroelastic bimaterials are categorized into two groups, i.e., ε-group with non-zero value of ε and κ-group with non-zero value of κ. The two indexes ε and κ do not coexist for one bimaterial. However, the extended stresses near the border of a permeable interface crack have only oscillating singularity and depend only on the mechanical loadings.     

An electrically impermeable and magnetically permeable interface crack with a contact zone in magnetoelectroelastic bimaterials under a thermal flux and magnetoelectromechanical loads

An interface crack with a frictionless contact zone at the right crack-tip between two dissimilar magnetoelectroelastic materials under the action of a thermal flux and remote magnetoelectromechanical loads is considered. The open part of the crack is assumed to be electrically impermeable and magnetically permeable, and the crack faces are assumed to be heat insulted. The inhomogeneous combined Dirichlet–Riemann and Hilbert boundary value problems are, respectively, formulated and solved analytically. Stress, electrical displacement intensity factors as well as energy release rate are found in analytical forms, and analytical expressions for the contact zone length have been obtained for both the general case and the case of small contact zone length. Some numerical results are presented, which show clearly the effects of thermal and magnetoelectromechanical loads on the contact zone length, stress intensity factor and energy release rate. Results presented in this paper should have potential applications to the design of multilayered magnetoelectroelastic structures and devices.     

Penny-shaped crack at the interface between elastic half-spaces under the action of a shear wave

The three-dimensional problem of elasticity for a bimaterial body with a penny-shaped crack at the interface under the action of a normal harmonic shear wave is solved by the boundary-element method. The distribution of displacements of crack faces and tractions and displacements at the interface is analyzed     

Opening and contact zones of an interface crack in a piezoelectric bimaterial under combined compressive-shear loading

A plane problem for a tunnel electrically permeable interface crack between two semi-infinite piezoelectric spaces is studied. A remote mechanical and electrical loading is applied. Elastic displacements and potential jumps as well as stresses and electrical displacement along the interface are presented using a sectionally holomorphic vector function. It is assumed that the interface crack includes zones of crack opening and frictionless contact. The problem is reduced to a combined Dirichlet–Riemann boundary value problem which is solved analytically. From the obtained solution, simple analytical expressions are derived for all mechanical and electrical characteristics at the interface. A quite simple transcendental equation, which determines the point of separation of open and close sections of the crack, is found. For the analysis of the obtained results, the main attention is devoted to the case of compressive-shear loading. The analytical analysis and numerical results show that, even if the applied normal stress is compressive, a certain crack opening zone exists for all considered loading values provided the shear field is present. It is found that the shear stress intensity factor at the closed crack tip and the energy release rates at the both crack tips depend very slightly on the magnitude of compressive loading.     

Electro-mechanical pre-fracture zones for an electrically permeable interface crack in a piezoelectric bimaterial

A plane strain problem for two piezoelectric half-spaces adhered by a very thin isotropic interlayer with a crack under the action of remote mixed mode mechanical loading and electrical flux is considered. The crack is situated either at an interface or in the interlayer. It is assumed that the substrates are much stiffer than the intermediate layer. Therefore, pre-fracture zones (plastic or damage) arise at the crack continuations. Normal and shear stresses are assumed to be constant in this zones and to satisfy some material equation, which can be taken from theory or derived experimentally. Modeling the pre-fracture zones by the crack continuations with unknown cohesive stresses on their faces reduces the problem to elastic interface crack analysis leading to a Hilbert problem. This problem is solved exactly. The pre-fracture zone lengths and stresses in these zones are found from algebraical and transcendental equations. The latter are derived from the conditions of stress finiteness at the ends of pre-fracture zones and the material equations. The electrical displacement at any point of the pre-fracture zones is found in closed form as well. Particular cases of symmetrical loading and of equivalent properties of the upper and lower bimaterial components are considered. Numerical results corresponding to certain material combinations and interlayer material equations are presented and analysed. In the suggested model, any singularities connected with the crack are eliminated, i.e., all mechanical and electrical characteristics are limited in the near-crack tip region.     

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.