Mode-I stress intensity factor derivation by a suitable Green's function

A suitable Green's function is developed for the infinite elastic solid, containing internal penny-shaped crack and loaded by a singular co-axial tensile and radial ring-shaped source acting outside or on crack faces. The corresponding boundary integral equation (BIE) is solved by the BEM for the calculation of the mode-I stress intensity factor of cracked axisymmetric finite bodies under tension. The proposed technique has three advantages: (a) it does not require discretization of the crack surface, (b) it does not require multiregion modeling and (c) it reduces the 3-D discretization of the solid to 1-D, resulting in substantially reduced effort. Numerical results are derived for the case of a cylindrical bar with a central penny-shaped crack located in a plane normal to its axis, loaded by tensile force. Comparison with results of other methods are included indicating excellent agreement.     

Boundary integral equations in elastodynamics of interface cracks

The paper concerns the validation of a method for solving elastodynamics problems for cracked solids. The proposed method is based on the application of boundary integral equations. The problem of an interface penny-shaped crack between two dissimilar elastic half-spaces under harmonic loading is considered as an example.     

The effect of internal pressure on a penny-shaped crack at the interface of two bonded dissimilar elastic half-spaces

The problem considered is that of determining the displacement field in the vicinity of a penny-shaped crack situated at the interface of two half-spaces of different elastic materials bonded together along their common plane boundary. The deformation in the two half-spaces is a result of the application of a symmetrically-distributed pressure to the faces of the crack. The representation of the displacement field in the form of Hankel transforms leads to a set of simultaneous dual integral equations for two unknown functions. These equations are transformed by the use of Abel operators to a similar set involving both the Fourier cosine and the Fourier sine transforms of each of these two unknown functions.These equations are transformed in turn to a singular integral equation for a complex-valued function in terms of whose real and imaginary parts it is possible to completely specify the stress and displacement fields. An explicit formula is obtained for the crack energy in the case in which the applied pressure is constant; it is also indicated how simple integral expressions may he obtained for the components of stress along the interface in this case.     

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.     

Torsional impact response of a penny-shaped interface crack in a layered composite

The torsional impact response of a penny-shaped interface crack in a layered composite is considered in this study. The geometry of the composite consists of two bonded dissimilar elastic layers which are sandwiched between two half-spaces made of a different material. 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 result is expressed in terms of a Fredholm integral equation of the second kind. A numerical Laplace inversion routine is used to recover the time dependence of the solution. The dynamic stress intensity factor is determined and its dependence on time, the material properties and the geometry parameters is discussed.     

Modelling Crack Closure for an Interface Crack Under Harmonic Loading

The paper is devoted to a linear crack located between two dissimilar elastic half-spaces under normally incident harmonic tension-compression loading. The system of boundary integral equations for displacements and tractions is derived from the dynamic Somigliana identity. The dynamic stress intensity factors (the opening and the transverse shear modes) are computed as functions of the loading frequency taking the contact interaction of the opposite crack faces into account. The results are compared with those obtained neglecting the crack closure.     

The influence of layer thickness on the stress intensity factor of a penny-shaped crack in a sandwiched viscoelastic bimaterial

In this paper we consider a penny-shaped crack embedded in the central layer of a composite viscoelastic material. In order to study the possibility of fracture within the central layer, we study the response of the crack, namely the stress intensity factor (SIF), to a remote applied mode. Mode I deformation is considered and we investigate how the SIF varies with the thickness of the interior layer by modelling the crack with a continuous distribution function (representing special crack opening displacements) and hence reducing the problem to a singular integral equation. The integral equation becomes insufficient when the layer is infinitesimal; for this region a Wiener–Hopf technique is employed and a formula for the SIF is derived using matched asymptotic expansions.     

Dynamic propagation of a finite crack in a micropolar elastic solid

Summary The dynamic propagation of a finite crack under mode-I loading in a micropolar elastic solid is investigated. By using an integral transform method, a pair of two-dimensional singular integral equations governing stress and couple stress is formulated in terms of displacement transverse to the crack, macro and micro rotations, and microinertia. These equations are solved numerically, and solutions for dynamic stress intensity and couple stress intensity factors are obtained by utilizing the values of the strengths of the square root singularities in macrorotation and the gradient of microrotation at the crack tips.     

Antiplane crack analysis of a functionally graded material by a BIEM

Elastostatic analysis of an antiplane crack in a functionally graded material (FGM) is performed by using a hypersingular boundary integral equation method (BIEM). An exponential law is applied to describe the spatial variation of the shear modulus of the FGM. A Galerkin method is applied for the numerical solution of the hypersingular traction BIE. Both unidirectional and bidirectional material gradations are investigated. Stress intensity factors for an infinite and linear elastic FGM containing a finite crack subjected to an antiplane crack-face loading are presented and discussed. The influences of the material gradients and the crack orientation on the stress intensity factors are analyzed.     

Axisymmetric dynamic response of a penny-shaped crack to a pair of transient concentrated forces

The three-dimensional axisymmetric elastodynamic response of a penny-shaped crack embedded in an infinite elastic solid subjected to a pair of transient concentrated forces is investigated. The forces are applied on the symmetry axis perpendicular and symmetric to the crack surfaces, including the special case when the forces act precisely on the crack surfaces. A time-domain boundary integral equation method is applied for computing the crack-opening displacement and subsequently the time dependence of the dynamic stress intensity factors. Numerical calculations are carried out for various geometry parameters and the results are discussed. It is found that the location of the applied concentrated forces inducing the highest dynamic stress intensity factors differs from that producing the highest static values.     

Boundary-integral equation analysis of twisted internally cracked axisymmetric bimaterial elastic solids

 Green's function is obtained for the infinite bimaterial elastic solid, containing an internal circular interface crack, loaded by a unit tangential co-axial circular source. An axisymmetric direct boundary integral equation (BIE) is used for the analysis of a finite bimaterial axisymmetric body containing an internal circular interface crack and a finite homogeneous cracked cylinder, both under torsional loading. Using the proposed technique, no discretization of the crack surface is necessary. Numerical results for both examples as obtained by the proposed method are presented and discussed. Received: 29 October 2001 / Accepted: 29 May 2002     

Application of Boundary Integral Equations to Elastodynamics of an Interface Crack

We considers application of boundary integral equations to the problem of an interface crack between two elastic half-spaces with different mechanical properties under dynamic loading. The derived system of equations allows evaluation of the displacements at the crack faces, and the traction and the displacements at the interface.     

The indentation of a precompressed penny-shaped crack

The paper examines the axisymmetric problem related to the indentation of the plane surface of a penny-shaped crack by a smooth rigid disc inclusion. The crack is also subjected to a far-field compressive stress field which induces closure over a part of the crack. The paper presents the Hankel integral transform development of the governing mixed boundary value problem and its reduction to a single Fredholm integral equation of the second kind and an appropriate consistency condition which considers the stress state at the boundary of the crack closure zone. A numerical solution of this integral equation is used to develop results for the axial stiffness of the inclusion and for the stress intensity factors at the tip of the penny-shaped crack.     

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.     

On integral equation approaches to the mechanics of fibre-crack interaction

The paper examines the problem of a penny-shaped crack which is formed by the development of a crack in both the fibre and the matrix of a composite consisting of an isolated elastic fibre located in an elastic matrix of infinite extent. The composite region is subjected to a uniform strain field in the direction of the fibre. The paper presents two integral-equation based approaches for the analysis of the problem. The first approach considers the formulation of the complete integral equations governing the associated elasticity problem for a two material region. The second approach considers the boundary integral equation formulation of the problem. Both methods entail the numerical solution of the governing integral equations. The solutions to these integral equations are used to evaluate the stress intensity factor at the boundary of the penny-shaped crack.     

Time-domain boundary element analysis of elastic wave interaction with a crack

The mathematical formulation of the problem of transient wave interaction with a crack in a homogeneous, isotropic, linearly elastic solid has been reduced to the solution of an integral equation over the insonified crack face. The integral equation relates the unknown crack-opening displacement, which depends on time and position, to the incident wave field. The integral equation has been solved numerically by a time-stepping method in conjunction with a boundary element discretization of the crack surface. For normal incidence of a longitudinal step-stress wave on a penny-shaped crack, results as functions of time have been obtained for the crack-opening displacement, the elastodynamic Mode-I stress intensity factor and the scattered far-field.     

Stress intensity factors for penny and half-penny shaped cracks subjected to a stress gradient

Stress intensity factors at any point on the crack front of penny and half-penny shaped cracks subjected to stress gradients are presented. The SIF's which are exact for a penny shaped crack are based on the well known solution for a point load acting normally to such a crack. The line load solution which is derived from this is different in form to those given by previous workers and is more readily integrated to give SIF's for stress gradient loading. This is demonstrated by the derivation of a general equation for the SIF at any point on a penny-shaped crack due to polynomial stress gradients. These results are extended to produce a similarly general, albeit approximate, equation for the SIF at any point on the circumference of a half-penny crack due to polynomial loading. The usefulness of the approach developed here is further indicated by the derivation of an approximate SIF for exponential stress gradients over a half-penny crack.     

A fiber-reinforced matrix containing a penny-shaped crack under mode III loading condition

One of the fundamental problems related to the fracture of composite materials, that is, a penny-shaped crack in a fiber-reinforced matrix is solved under the Mode III loading condition, where the fibers are perpendicular to the crack plane and located along the crack border. An elastic fiber model is developed to the above torsional problem, yielding a Fredholm-type integral equation of the second kind for a set of fibers distributed symmetrically on a circle concentric with the crack. The integral equation is numerically evaluated, and the stress intensity factors are presented with the parameter of the fiber to matrix Young's modulus ratio for various geometries.     

Pulse shape effects on the dynamic stress intensity factor

The interaction of a transient stress pulse with a penny-shaped crack embedded in an infinite elastic solid is investigated. The front of the incident stress pulse is assumed to be planar and parallel to the crack surfaces. A time-domain boundary integral equation method is applied for computing the time history of the crack opening displacement, from which the time dependence of the dynamic stress intensity factor is subsequently calculated. Numerical calculations are carried out for several stress pulses of different shape and time dependence, to explore the effects of the shape, duration, rise and descent time of a transient stress pulse, or the period and the mean stress of a cyclic stress pulse on the dynamic stress intensity factor. Implications regarding crack surface penetrations or crack surface interactions caused by certain stress pulses are also discussed.     

An iterative BEM for the dynamic analysis of interface crack contact problems

The multiparametric algorithm for solving a fracture dynamics problem is presented in the paper. The linear crack between two dissimilar elastic isotropic half-spaces under normal tension-compression load is considered. The system of boundary integral equations for displacements and tractions is derived from the dynamic Somigliana identity. The crack closure is taken into account by imposing the constraints to the normal components of contact force and displacement discontinuity vectors. The convergence of the algorithm is investigated in detail. The study of the effect of frequency of the loading on the stress intensity factors for opening and transverse shear modes is carried out.