On interface crack propagation with variable velocity for longitudinal shear problems

The antiplane strain problem of straight interface crack propagation between two elastic half-spaces under arbitrary variable loading is considered. The crack edge is specified as an arbitrary smooth function of time. It is assumed that the crack speed is less than the smaller of the shear wave velocities of two media. An integral transform method and factorization technique are used to solve the problem. The solutions are worked out for semi-infinite crack and finite crack problems. The dynamic stress intensity factors at the crack tip of the moving interface crack are given and it is found that the stress intensity factor of the interface crack is slightly higher than that in the homogeneous medium with slower shear wave velocity.     

Time-domain BEM analysis of a rapidly growing crack in a bi-material

Rapid propagation of a matrix crack in a bi-material system is studied with emphasis on the dynamic interaction between the crack and the interface by combining the traditional time-domain displacement boundary element method (BEM) and the non-hypersingular traction BEM. The crack growth is controlled by the fracture criterion based on the maximum circumferential stress, and is modeled by adding new elements to the moving crack tip. Detailed computation is performed for an unbounded bi-material with a crack subjected to incident impact waves and a bounded rectangular bi-material plate under dynamic wedged loading. Numerical results of the crack growth path, speed, dynamic stress intensity factors (DSIFs) and dynamic interface tractions are presented for various material combinations and geometries. The effects of the interface on the crack growth are discussed.     

Elastic interactions of a fatigue crack with a micro-defect by the mixed boundary integral equation method

In this paper, the mixed boundary integral equation method is developed to study the elastic interactions of a fatigue crack and a micro-defect such as a void, a rigid inclusion or a transformation inclusion. The method of pseudo-tractions is employed to study the effect of a transformation inclusion. An enriched element which incorporates the mixed-mode stress intensity factors is applied to characterize the singularity at a moving crack tip. In order to evaluate the accuracy of the numerical procedure, the analysis of a crack emanating from a circular hole in a finite plate is performed and the results are compared with the available numerical solution. The effects of various micro-defects on the crack path and fatigue life are investigated. The results agree with the experimental observations.     

On the numerical evaluation of stress intensity factors for an interface crack of a general shape

A numerical algorithm is presented for the problem of a crack along the interface of an elastic inclusion embedded in an elastic plane subjected to uniform stress at infinity. The algorithm is based on a Fredholm integral equation of the second kind and allows for fast and accurate solutions to geometries of great complexity. In an example crack opening displacement and stress intensity factors are computed for a crack in the interface of an inclusion with nineteen protruding arms. Copyright © 1999 John Wiley & Sons, Ltd.     

A moving interface crack between two dissimilar functionally graded strips under plane deformation with integral equation methods

In this paper a finite interface crack with constant length (Yoffe-type crack) propagating along the interface between two dissimilar functionally graded strips with spatially varying elastic properties under in-plane loading is studied. By utilizing the Fourier transformation technique, the mixed boundary problem is reduced to a system of singular integral equations that are solved numerically. The influences of the geometric parameters, the graded parameter, the strip thickness and crack speed on the stress intensity factors and the probable kink angle are investigated. The numerical results show that the graded parameters, the thicknesses of the functionally graded strips and the crack speed have significant effects on the dynamic fracture behavior of functionally graded material structures.     

The effect of couple-stresses on the dynamic stress concentration around a crack

The dynamic problem is presented for an infinite elastic Cosserat medium weakened by a finite crack where the self-equilibrated system of pressure is varying harmonically in time. There are two sets of mixed boundary values and the problem is reduced to four simultaneous integral equations which are solved by the series method. The numerical examples are carried out to clarify the effect of couple-stresses on the dynamic stress concentration around a crack.     

Numerical simulation of crack deflection and penetration at an interface in a bi-material under dynamic loading by time-domain boundary element method

The hybrid time-domain boundary element method (BEM), together with the multi-region technique, is applied to simulate the dynamic process of crack deflection/ penetration at an interface in a bi-material. The whole bi-material is divided into two regions along the interface. The traditional displacement boundary integral equations (BIEs) are employed with respect to the exterior boundaries; meanwhile, the non-hypersingular traction BIEs are used with respect to the part of the crack in the matrix. Crack propagation along the interface is numerically modelled by releasing the nodes in the front of the moving crack tip and crack propagation in the matrix is modeled by adding new elements of constant length to the moving crack tip. The dynamic behaviours of the crack deflection/penetration at an interface, propagation in the matrix or along the interface and kinking out off the interface, are controlled by criteria developed from the quasi-static ones. The numerical results of the crack growth trajectory for different inclined interface and bonded strength are computed and compared with the corresponding experimental results. Agreement between numerical and experimental results implies that the present time-domain BEM can provide a simulation for the dynamic propagation and deflection of a crack in a bi-material.     

Dynamic stress intensity factor (Mode I) of a penny-shaped crack in an infinite poroelastic solid

This paper considers the transient stress intensity factor (Mode I) of a penny-shaped crack in an infinite poroelastic solid. The crack surfaces are impermeable. By virtue of the integral transform methods, the poroelastodynamic mixed boundary value problems is formulated as a set of dual integral equations, which, in turn, are reduced to a Fredholm integral equation of the second kind in the Laplace transform domain. Time domain solutions are obtained by inverting Laplace domain solutions using a numerical scheme. A parametric study is presented to illustrate the influence of poroelastic material parameters on the transient stress intensity. The results obtained reveal that the dynamic stress intensity factor of poroelastic medium is smaller than that of elastic medium and the poroelastic medium with a small value of the potential of diffusivity shows higher value of the dynamic stress intensity factor.     

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.     

Four co-planar Griffith cracks in an infinite elastic medium

The dynamic in-plane problem of determining the stress and displacement due to four co-planar Griffith cracks moving steadily at a subsonic speed in a fixed direction in an infinite, isotropic, homogeneous medium under normal stress has been treated. The static problem of determining the stress and displacement in an infinite isotropic elastic medium has also been considered. In both cases, employing the Fourier integral transform, the problems have been reduced to solving a set of five integral equations. These integral equations have been solved using the finite Hilbert transform technique to obtain the exact form of crack opening displacement and stress intensity factors which are presented in the form of graphs.     

Sudden twisting of an external circular crack in an infinite medium with a cylindrical inclusion

In this paper the torsional impact response of an external circular crack in an infinite medium bonded to a cylindrical inclusion has been investigated. The infinite medium and cylindrical inclusion are assumed to be of different homogeneous isotropie elastic materials. Laplace and Hankel transforms are used to reduce the problem to the solution of a pair of dual integral equations. These equations are solved by using an integral transform technique and the results are expressed in terms of a Fredhol integral equation of the second kind. By solving Fredholm integral equation of the second kind the numerical results for the dynamic stress-intensity factor are obtained which measure the load transmission on the crack.     

On the stress wave induced curving of fast running cracks — a numerical study by a time-domain boundary element method

Summary Fast crack propagation in dynamically loaded plane structures is investigated. The major point of interest is the evolution of the crack trajectory under the influence of stress waves which are generated and repeatedly reflected at the specimen boundaries. Since these waves may lead to arbitrary mixed-mode and time-dependent loading of the crack tip, both the direction and speed of crack advance are determined from a fracture criterion.Starting point is a system of time-domain boundary integral equations which describes the initial boundary value problem of a linear elastic body containing an arbitrarily growing crack. The unknown displacements and/or tractions on the exterior boundary and the displacement jumps across the crack are computed numerically by a collocation method in conjunction with a time-stepping scheme. Crack growth is modelled by adding new boundary elements of constant length at the running crack tip.The method proves to be of sufficient accuracy when applied to problems treated with other numerical techniques. Moreover, the simulation of dynamic crack propagation under various geometry and loading conditions enables the reproduction and analysis of complex phenomena observed experimentally.     

On the dynamic interaction between a penny-shaped crack and an expanding spherical inclusion in 3-D solid

Three-dimensional stress investigation on the interaction between a penny-shaped crack and an expanding spherical inclusion in an infinite 3-D medium is studied in this paper. The spherical transformation area (the inclusion) expands in a self-similar way. By using the superposition principle, the original physical problem is decomposed into two sub-problems. The transient elastic filed of the medium with an expanding spherical inclusion is derived with the dynamic Green's function. A time domain boundary integral equation method (BIEM) is then adopted to solve the current problem. The numerical scheme applied here uses a constant shape function for elements away from the crack front, and a square root crack-tip shape function for elements near the crack tip to describe the proper behavior of the unknown quantities near the crack front. A collocation method as well as a time stepping scheme is applied to solve the BIEs. Numerical examples for the Mode I stress intensity factor are presented to assess the dynamic effect of the expanding inclusion.     

Three co-planar moving Griffith cracks in an infinite elastic medium

The dynamic in-plane problem of determining the stress and displacement due to three co-planar Griffith cracks moving steadily at a subsonic speed in a fixed direction in an infinite, isotropic, homogeneous medium under normal stress has been treated. The static problem of determining the stress and displacement around three co-planar Griffith cracks in an infinite isotropic elastic medium has also been considered. In both the cases, employing Fourier integral transform, the problems have been reduced to solving a set of four integral equations. These integral equations have been solved using finite Hilbert transform technique and Cook's result [16] to obtain the exact form of crack opening displacement and stress intensity factors which are presented in the form of graphs.     

A numerical Green's function BEM formulation for crack growth simulation

This paper presents a crack growth prediction analysis based on the numerical Green's function (NGF) procedure and on the minimum strain energy density criterion for crack extension, also known as S-criterion. In the NGF procedure, the hypersingular boundary integral equation is used to numerically obtain the Green's function which automatically includes the crack into the fundamental infinite medium. When solving a linear elastic fracture mechanisms (LEFM) problem, once the NGF is obtained, the classical boundary element method can be used to determine the external boundary unknowns and, consequently, the stress intensity factors needed to predict the direction and increment of crack growth. With the change in crack geometry, another numerical analysis is carried out without need to rebuilding the entire element discretization, since only the crack built in the NGF needs update. Numerical examples, contemplating crack extensions for two-dimensional LEFM problems, are presented to illustrate the procedure.     

Numerical analysis of doubly periodic array of cracks/rigid-line inclusions in an infinite isotropic medium using the boundary integral equation method

In this paper, the boundary integral equation approaches are used to study the doubly periodic array of cracks/rigid-line inclusions in an infinite isotropic plane medium. For the doubly periodic rigid-line inclusion problems, the special integral equation containing the axial and shear forces within the rigid-line inclusion is used. The doubly periodic crack problems are dealt with using the displacement discontinuous integral equation approach. Stress intensity factors, effective elastic properties for doubly periodic array of cracks/rigid-line inclusions are calculated and compared with the available numerical solutions.     

The effect of T-stress on plane strain dynamic crack growth in elastic–plastic materials

In this paper, the effects of T‐stress on steady, dynamic crack growth in an elastic–plastic material are examined using a modified boundary layer formulation. The analyses are carried out under mode I, plane strain conditions by employing a special finite element procedure based on moving crack tip coordinates. The material is assumed to obey the J₂ flow theory of plasticity with isotropic power law hardening. The results show that the crack opening profile as well as the opening stress at a finite distance from the tip are strongly affected by the magnitude and sign of the T‐stress at any given crack speed. Further, it is found that the fracture toughness predicted by the analyses enhances significantly with negative T‐stress for both ductile and cleavage mode of crack growth.     

Energy release rate and bifurcation angles of piezoelectric materials with antiplane moving crack

The dynamic fracture problems of the piezoelectric materials with antiplane moving crack are analysed by using function of complex variable in the paper. The results show that the coupled elastic and electric fields inside piezoelectric media depend on the speed of the crack propagation, and have singularity at the crack tip. The stress intensity factor is independent of the speed of the crack propagation, which is identical to the conclusion of purely elasticity. Moreover, independent of the electric loading, the dynamic energy release rate can be expressed by the stress intensity factor and enlarge with the increase of crack speed. High speed of the crack moving could impede the crack growth. At the same time, the crack can be propagated into either curve or bifurcation if the crack speed is higher than the critical speed.     

Multiple interfacial cracks in dissimilar piezoelectric layers under time harmonic loadings

In this study, the dislocation‐based model is developed to study the interaction between time‐harmonic elastic waves and multiple interface cracks in 2 bonded dissimilar piezoelectric layers. In this model, cracks are represented by a distribution of so‐called electro‐elastic dislocations whose density is to be determined by satisfying the boundary conditions. Using the Fourier transform, this formulation leads to 2 singular integral equations, which can be solved numerically for the densities of electro‐elastic dislocations on a crack surface. The formulation is used to determine dynamic field intensity factors for multiple interface cracks without limitation of number of cracks. The dynamic field intensity factors are then calculated for both permeable and impermeable crack, and finally, numerical results are presented to illustrate the variation of these quantities with the electromechanical coupling, crack spacing, and the frequency of loading.     

On a moving dielectric crack in a piezoelectric interface with spatially varying properties

This paper provides a comprehensive theoretical analysis of a finite crack propagating with constant speed along an interface between two dissimilar piezoelectric media under inplane electromechanical loading. The interface is modeled as a graded piezoelectric layer with spatially varying properties (functionally graded piezoelectric materials, i.e., FGPMs). The analytical formulations are developed using Fourier transforms and the resulting singular integral equations are solved with Chebyshev polynomials. Using a dielectric crack model with deformation-dependent electric boundary condition, the dynamic stress intensity factors, electric displacement intensity factor, crack opening displacement (COD) intensity factor, and energy release rate are derived to fully understand this inherent mixed mode dynamic fracture problem. Numerical simulations are made to show the effects of the material mismatch, the thickness of the interfacial layer, the crack position, and the crack speed upon the dynamic fracture behavior. A critical state for the electromechanical loading applied to the medium is identified, which determines whether the traditional impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric model considering the effect of dielectric medium crack filling.