Interfacial edge crack in two bonded dissimilar orthotropic quarter planes under anti-plane shear

An interfacial edge crack in two bonded dissimilar orthotropic quarter planes under antiplane loading is analyzed. The problem is formulated by Fourier integral technique, and reduced to a pair of integral equations. By solving the integral equations, the asymptotic stress field near the crack tip is determined, from which the stress intensity factor is obtained as a closed form. The solution can be used as a Green's function of different loading conditions.     

通电瞬时板内半无限长裂纹尖端域的应力场

以导电弹性体的麦克斯威尔方程为出发点,借助于边界条件和初始条件,推得了在向含半无限长直线裂纹的无限大导电薄板内通入电流的瞬时,裂纹尖端附近电流密度的表达式。在此基础上,得到了裂纹尖端区域处温度和应力的具体表达式。算例表明,在电流所产生的焦耳热源的作用下,裂尖区域处的温度将瞬时升高,并伴有压应力的产生,从而可达到阻止裂纹扩展的目的。     

The thermal effect of piezoelectric medium for anti-plane problem under high electrical impact loading

In present paper, the anti-plane problem of thermal effect near crack tip region of piezoelectric material subjected to electrical impact loading is investigated by means of the integral transforms and the singular integral equations. By introducing the thermal power, the temperature field near crack tip is finally obtained on the basis of the hypotheses of the uncoupling between the thermal field and the electro-mechanical fields and the adiabatic approximation. The numerical results indicate that a high temperature field of small region near crack tip is deduced when high electrical impact load is applied. Moreover, the results show that the temperature field strongly depends on crack size. However, the thermal effect of mechanical impact comparing with electrical impact may almost be neglected.     

The scattering of harmonic elastic anti-plane shear waves by a Griffith crack in a piezoelectric material plane by using the non-local theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material plane under anti-plane shear waves is investigated by using the non-local theory for impermeable crack face conditions. For overcoming the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near the crack tips. By using the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical elasticity solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the circular frequency of incident wave and the lattice parameter. For comparison results between the non-local theory and the local theory for this problem, the same problem in the piezoelectric materials is also solved by using local theory.     

The mode I crack problem for layered piezoelectric plates

The plane strain singular stress problem for piezoelectric composite plates having a central crack is considered. For the case of the crack which is normal to and ends at the interface between the piezoelectric plate and the elastic layer, the order of stress singularity around the tip of the crack is obtained. The Fourier transform technique is used to formulate the problem in terms of a singular integral equation. The singular integral equation is solved by using the Gaus–Jacobi integration formula. Numerical calculations are carried out, and the main results presented are the variation of the stress intensity factor as functions of the geometric parameters, the piezoelectric material properties and the electrical boundary conditions of the layered composites.     

Edge crack in an elastic layer resting on Winkler foundation

The paper deals with the stress analysis near a crack tip in an elastic layer resting on Winkler foundation. The edge crack is assumed to be normal to the lower boundary plane. The upper surface of the layer is loaded by given forces normal to the boundary. The considered problem is solved by using the method of Fourier transforms and dual integral equations, which are reduced to a Fredholm integral equation of the second kind. The stress intensity factor is given in the term of solution of the Fredholm integral equation and some numerical results are presented.     

A boundary integral approach for plane analysis of electrically semi-permeable planar cracks in a piezoelectric solid

A plane electro-elastostatic problem involving arbitrarily located planar stress free cracks which are electrically semi-permeable is considered. Through the use of the numerical Green's function for impermeable cracks, the problem is formulated in terms of boundary integral equations which are solved numerically by a boundary element procedure together with a predictor–corrector method. The crack tip stress and electric displacement intensity factors can be easily computed once the boundary integral equations are properly solved.     

Nanoscale anti-plane cracking of materials with consideration of bulk and surface piezoelectricity effects

The influence of surface effect, including surface elasticity and surface piezoelectricity, on thefracture behavior of piezoelectricmaterials with an anti-plane crack is studied.Based on the coupled surface andinterface elasticity model, the solutions to the problem are obtained by applying the singular integral method. Bycomparing the solutions influenced by the surface piezoelectricity with those affected by the surface elasticity,it is found that the influence of the surface piezoelectricity on the crack opening displacement, the crackelectric potential jump across the crack center, the crack tip stress and electric displacement intensity factorscannot be ignored. Under various electrical boundary conditions, the influence of surface piezoelectricity onthe sliding displacement, crack tip stress and electric displacement intensity factors exhibits the same tendency.Besides, the influence of surface piezoelectricity on the electric displacement intensity factor is independentof the electrical boundary conditions, which is different from the results where only the surface elasticity isconsidered.     

Analysis of the dynamic behavior of two parallel symmetric cracks using the non-local theory

In this paper, the dynamic behavior of two parallel symmetric cracks under harmonic anti-plane shear waves is studied using the non-local theory. For overcoming the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the problem to obtain the stress occurs near the crack tips. The Fourier transform is applied and a mixed boundary value problem is formulated. Then a set of dual integral equations is solved using the Schmidt method. Contrary to the classical elasticity solution, it is found that no stress singularity is present at the crack tip. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the lattice parameter and the distance between two parallel cracks, respectively.     

Numerical Green's functions for some electroelastic crack problems

A plane electroelastic problem involving planar cracks in a piezoelectric body is considered. The deformation of the body is assumed to be independent of time and one of the Cartesian coordinates. The cracks are traction free and are electrically either permeable or impermeable. Numerical Green's functions which satisfy the boundary conditions on the cracks are derived using the hypersingular integral approach and applied to obtain a boundary integral solution for the electroelastic crack problem considered here. As the conditions on the cracks are built into the Green's functions, the boundary integral solution does not contain integrals over the cracks. It is used to derive a boundary element procedure for computing the crack tip stress and electrical displacement intensity factors.     

A study of asymmetrically loaded griffith crack in an infinite anisotropic medium

The two dimensional problem of a Griffith type crack whose surfaces are subjected to asymmetrical loading in an infinite anisotropic elastic medium is studied. The analysis is based on the integral transform method and the finite Hubert transform technique of dual integral equations. Closed form solutions of displacement components along the line of the crack and the formulae for the stress components at a general point are obtained. The near crack tip approximations to stress components are also presented in detail.     

Thermal stresses in a slab containing an annular crack

A linear thermoelastic problem of a slab containing an annular crack is solved. Using integral transform techniques, the problem is reduced to that of solving two singular integral equations of the first kind. The solution of the singular integral equation is obtained in the form of the product of the series of Chebyshev polynomials of the first kind and their weight functions. Thus the essential feature of the singular stress field near the crack is preserved and the crack tip stress intensity factor is easily evaluated. Numerical calculations are also carried out and the variations of the stress intensity factors are plotted against the geometry for various values of physical properties.     

Elastic cracks and screw dislocation pile-ups crossing a bimaterial interface

The concept of continuously distributed dislocations is employed to study the behavior of anti-plane shear cracks crossing a bimaterial interface. The governing equations of the dislocation distribution function are dual singular integral equations. It is noticed that close to the tip, the crack opening displacement behaves as if the crack were imbedded in a homogeneous medium. C onsequendy, the stress in the immediate vicinity of the crack tip varies with the inverse square root of the distance from the tip. Moreover, the stress intensity near the crack tip in the comparatively harder phase is higher than that in the softer phase. The present method of analysis can be applied to the study of screw dislocation pile-ups crossing a phase boundary.     

Dynamic SIF of interface crack between two dissimilar viscoelastic bodies under impact loading*

In this paper, the transient dynamic stress intensity factor (SIF) is determined for an interface crack between two dissimilar half-infinite isotropic viscoelastic bodies under impact loading. An anti-plane step loading is assumed to act suddenly on the surface of interface crack of finite length. The stress field incurred near the crack tip is analyzed. The integral transformation method and singular integral equation approach are used to get the solution. By virtue of the integral transformation method, the viscoelastic mixed boundary problem is reduced to a set of dual integral equations of crack open displacement function in the transformation domain. The dual integral equations can be further transformed into the first kind of Cauchy-type singular integral equation (SIE) by introduction of crack dislocation density function. A piecewise continuous function approach is adopted to get the numerical solution of SIE. Finally, numerical inverse integral transformation is performed and the dynamic SIF in transformation domain is recovered to that in time domain. The dynamic SIF during a small time-interval is evaluated, and the effects of the viscoelastic material parameters on dynamic SIF are analyzed.     

The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.     

Interface crack and nonideal interface concept (Mode III)

Asymptotic behaviour of displacements and stresses in a vicinity of the interface crack tip situated on a nonideal interface between two different elastic materials is investigated. The nonideal interface is described by special transmission conditions along the material bonding. The corresponding modelling boundary value problem is reduced to a singular integral equation with fixed point singularities. It is shown from the solution to the problem that asymptotic behaviour of displacement and stresses near the crack tip essentially depends on the model parameters. Some numerical examples are presented and discussed with respect to the stress singularity exponent and the generalized stress intensity factors.     

Boundary integral equation formulation for Interface cracks in anisotropic materials

We present a boundary integral formulation for anisotropic interface crack problems based on an exact Green's function. The fundamental displacement and traction solutions needed for the boundary integral equations are obtained from the Green's function. The traction-free boundary conditions on the crack faces are satisfied exactly with the Green's function so no discretization of the crack surfaces is necessary. The analytic forms of the interface crack displacement and stress fields are contained in the exact Green's function thereby offering advantage over modeling strategies for the crack. The Green's function contains both the inverse square root and oscillatory singularities associated with the elastic, anisotropic interface crack problem. The integral equations for a boundary element analysis are presented and an example problem given for interface cracking in a copper-nickel bimaterial.     

The effect of the lattice parameter of functionally graded materials on the dynamic stress field near crack tips

In this paper, the effect of the lattice parameter of functionally graded materials on the dynamic stress fields near crack tips subjected to the harmonic anti-plane shear waves is investigated by means of non-local theory. By use of the Fourier transform, the problem can be solved with the help of a pair of dual integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the dual integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularities are present near crack tips. The non-local elastic solution yields a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion in functionally graded materials.     

Boundary integral equation method for conductive cracks in two and three-dimensional transversely isotropic piezoelectric media

Using the fundamental solutions and the Somigliana identity of piezoelectric medium, the boundary integral equations are obtained for a conductive planar crack of arbitrary shape in three-dimensional transversely isotropic piezoelectric medium. The singular behaviors near the crack edge are studied by boundary integral equation approach, and the intensity factors are derived in terms of the displacement discontinuity and the electric displacement boundary value sum near the crack edge on crack faces. The boundary integral equations for two dimensional crack problems are deduced as a special case of infinite strip planar crack. Based on the analogy of the obtained boundary integral equations and those for cracks in conventional isotropic elastic material and for contact problem of half-space under the action of a rigid punch, an analysis method is proposed. As an example, the solution to conductive Griffith crack is derived.     

Boundary element analysis of structures with bridged interfacial cracks

The direct boundary integral equations method has been applied to analyze stresses in a fracture process zone (a crack bridged zone) and to calculate stress intensity factors module for structures with bridged interfacial cracks under mechanical loading. Bridged zones at interfacial cracks are considered as parts of these cracks with assumption that surfaces of interfacial cracks are connected by distributed spring-like bonds with given bond deformation law. For numerical analysis of piecewise structures with bridged interfacial cracks the multi-domain formulation of the boundary elements method is used. The stress intensity factors module evaluation is performed on the basis of displacements and stresses computed at nodal points of special quadratic boundary elements adjoined to a crack tip. The comparative study between the results obtained by the boundary elements method and the results obtained previously by the singular integral–differential equations method is performed and the validity of the presented numerical formulation is demonstrated. The new problem for a bridged circumferential crack between a cylindrical inclusion and a matrix in plate of finite size is also solved. Stresses distributions along the bridged zone and the stress intensity factors modulus dependencies versus the bridged zone length and bonds stiffness are presented and discussed for this problem.