Green’s function for KI determination of axisymmetric elastic solids containing external circular crack

A new Green’s function is derived to determine the mode-I stress intensity factor for axisymmetric solids containing external circular crack. The formulated boundary integral equation is applied to a finite cylindrical bar with an external crack, and the obtained solution is compared with existing published results, indicating good agreement. The proposed method compared with the finite element method or the conventional application of the boundary element method provides the following main advantages: (a) it does not require discretization of the crack surface, (b) it does not require multi-region modeling and (c) it reduces the 3-D discretization of the solid to 1-D resulting in substantially reduced effort.     

3-D dynamic interaction between a penny-shaped crack and a thin interlayer joining two elastic half-spaces

Three-dimensional (3-D) elastodynamic interaction between a penny-shaped crack and a thin elastic interlayer joining two elastic half-spaces is investigated by an improved boundary integral equation method or boundary element method. The penny-shaped crack is embedded in one of the half-spaces, perpendicular to the interlayer and subjected to a time-harmonic tensile loading on its surfaces. Effective “spring-like” boundary conditions are applied to approximate the effects of the thin layer in the mathematical model. Integral representations for the displacement and the stress components are derived by using modified Green’s functions, which satisfy the “spring-like” boundary conditions identically. Then, application of the dynamic loading condition on the crack-surfaces results in a boundary integral equation (BIE) for the crack-opening-displacement over the crack-surfaces only. A solution procedure is developed for solving the BIE numerically. Numerical results for the mode-I dynamic stress intensity factor (SIF) are presented and discussed to show the variations of the mode-I dynamic SIF with the angular coordinate of the crack-front points, the dimensionless wave number, the material mismatch and the crack-layer distance.     

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.     

Boundary element analysis of three-dimensional crack problems in two joined transversely isotropic solids

The present paper presents a boundary element analysis of penny-shaped crack problems in two joined transversely isotropic solids. The boundary element analysis is carried out by incorporating the fundamental singular solution for a concentrated point load in a transversely isotropic bi-material solid of infinite space into the conventional displacement boundary integral equations. The conventional multi-region method is used to analyze the crack problems. The traction-singular elements are employed to capture the singularity around the crack front. The values of the stress intensity factors are obtained by using crack opening displacements. The numerical scheme results are verified with the closed-form solutions available in the literature for a penny-shaped crack parallel to the plane of the isotropy of a homogeneous and transversely isotropic solid of infinite extent. The new problem of a penny-shaped crack perpendicular to the interface of a transversely isotropic bi-material solid is then examined in detail. The crack surfaces are subject to the three normal tractions and the uniform shear traction. The associated stress intensity factor values are obtained and analyzed. The present results can be used for the prediction of the stability of composite structures and the hydraulic fracturing in deep rock strata and reservoir engineering.     

Note on the growth of a penny-shaped crack in a general uniform applied stress field

Energetic arguments are used to discuss the growth of a penny-shaped crack situated within an infinite solid which is subject to tensile and shear stresses that are respectively normal and parallel to the crack plane. The most favourable growth mode is that for which the circular periphery becomes an ellipse, such that there is no growth perpendicular to the direction of application of the. shear stress; the appropriate growth condition is derived and compared with that obtained by assuming the circular crack to expand uniformly.     

Axisymmetric crack with a slightly non-flat surface

A three-dimensional axisymmetric crack with a slightly non-flat surface in an isotropic linear solid under axisymmetric loading is analyzed. The problem is formulated by using the Hankel transform and a perturbation solution is obtained, which is accurate to the first order of the parameter representing the non-flatness. The stress intensity factor for the problem is evaluated. In particular, the stress intensity factor at the onset of axisymmetric kinking from a penny-shaped crack is obtained. It is also shown that the two-dimensional Cotterell-Rice theory for the effect of tensile stress acting parallel to the crack surface on the stability of crack path is valid for the axisymmetric crack.     

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     

The application of post-yield fracture mechanics to penny-shaped and semi-circular cracks

Bilby, Cottrell and Swinden model solutions for a penny-shaped crack subject to an axisymmetric stress are given. Approximate solutions for a semi-circular crack in a half-plane are constructed and applied to predicting the fracture stress of a crack subject to uniform tensile loading. The analytically determined fracture stresses are in good agreement with published experimental results.     

Stresses from arbitrary loads on a penny-shaped crack

An implementation of a solution to the problem of a penny-shaped crack in an infinite elastic solid with arbitrary normal and shear loads is described, and is used to generate the stresses corresponding to some simple crack loads. The program described is fast and stable, and is shown to give accurate results even if the crack loads are not of the desired polynomial form. Stress intensity factors are obtained directly from combinations of load constants.     

Axisymmetric crack terminating at the interface of transversely isotropic dissimilar media

In this paper, the axisymmetric elasticity problem of an infinitely long transversely isotropic solid cylinder imbedded in a transversely isotropic medium is considered. The cylinder contains an annular or a penny shaped crack subjected to uniform pressure on its surfaces. It is assumed that the cylinder is perfectly bonded to the medium. A singular integral equation of the first kind (whose unknown is the derivative of crack surface displacement) is derived by using Fourier and Hankel transforms. By performing an asymptotic analysis of the Fredholm kernel, the generalized Cauchy kernel associated with the case of `crack terminating at the interface' is derived. The stress singularity associated with this case is obtained. The singular integral equation is solved numerically for sample cases. Stress intensity factors are given for various crack geometries (internal annular and penny-shaped cracks, annular cracks and penny-shaped cracks terminating at the interface) for sample material pairs.     

Axisymmetric finite cylinder with one end clamped and the other under uniform tension containing a penny-shaped crack

This study considers the axisymmetric analysis of a finite cylinder containing a penny-shaped transverse crack. Material of the cylinder is assumed to be linearly elastic and isotropic. One end of the cylinder is bonded to a fixed support while the other end is subjected to uniform axial tension. Solution is obtained by superposing the solutions for an infinite cylinder loaded at infinity and an infinite cylinder containing four cracks and a rigid inclusion loaded along the cracks and the inclusion. When the radius of the inclusion approaches the radius of the cylinder, its mid-plane becomes fixed and when the radius of the distant cracks approach the radius of the cylinder, the ends become cut and subject to uniform tensile loads. General expressions for the perturbation problem are obtained by solving Navier equations with Fourier and Hankel transforms. Formulation of the problem is reduced to a system of five singular integral equations. By using Gauss-Lobatto and Gauss-Jacobi integration formulas, these five singular integral equations are converted to a system of linear algebraic equations which is solved numerically. Stress distributions along the rigid support, stress intensity factors at the edges of the rigid support and the crack are calculated.     

Shape prediction and stability analysis of Mode-I planar cracks

This paper presents a numerical technique for simulating stable growth of Mode-I cracks in two and three dimensions, using energy release rate and its derivatives. The crack growth model used in the numerical simulation is based on the concept of maximizing potential energy of the system released as cracks evolve. Therefore, a series of quadratic programming (QP) problems with linear constraints and bounds are solved to simulate stable growth of Mode-I planar cracks. The derivative of energy release rate provides a stability condition for crack growth in structures and can be regarded as a discretized influence function that represents the strength of the interaction among crack extensions at different crack tips in 2-D and different locations along a crack front in 3-D. The energy release rate and its derivative are accurately calculated by the analytical virtual crack extension method [Engng. Fract. Mech. 59 (1998) 521; 68 (2001) 925] in a single analysis. Numerical examples are presented to demonstrate the capabilities of the proposed approach. Examples include a central crack subjected to wedge forces in a 2-D finite plate, a system of interacting thermally induced parallel cracks in a two-dimensional semi-infinite plane and a 3-D penny-shaped crack embedded in a large cylinder, pressurized in a central circular region.     

Uncoupled numerical method for fracture analysis

An uncoupled numerical method for the analysis of dynamic crack propagation is proposed. The approach consists of two main steps. Firstly, the internal stresses in the intact, unfractured, elastic body are calculated with the use of the finite element method. Firstly, the internal stresses in the intact, unfractured, elastic body are calculated with the use of the finite element method. In this calculation it is assumed that no cracks are present and that fracture does not occur. Secondly, a theoretical crack is initiated and possible crack paths are derived from the elastic stress data. The stress-intensity factors for the planar fracture modes I and II, for the anti-plane mode III, and for the bending modes 1 and 2 are calculated from the well-known, linearized expressions for arbitrary, slightly curved cracks in thin plate-like and shell-like structures. The direction and speed of crack propagation are determined from a dynamic fracture criterion based on the energy release rate. Several applications of the uncoupled numerical method are presented, concerning standard fracture specimens loaded by tensile forces and bending moments, a single-edge notched beam loaded by shear forces, and a three-dimensional cylindrical tube loaded by torsional moments. Good agreement with both experimental and numerical results from the literature has been obtained. The major advantages of the uncoupled approach are its ease-of-use and the limited computational effort. This revised version was published online in August 2006 with corrections to the Cover Date.     

Stress intensity factors for functionally graded solid cylinders

This paper presents the mode I stress intensity factors for functionally graded solid cylinders with an embedded penny-shaped crack or an external circumferential crack. The solid cylinders are assumed under remote uniform tension. The multiple isoparametric finite element method is used. Various types of functionally graded materials and different gradient compositions for each type are investigated. The results show that the material property distribution has a quite considerable influence on the stress intensity factors. The influence for embedded cracks is quite different from that for external cracks.     

Diffraction of high frequency torsion waves by a penny-shaped crack

The diffraction of high frequency torsion waves by a penny-shaped crack situated in an infinite isotropic elastic solid is considered. Asymptotic expressions for the dynamic stress intensity factors are derived for a variety of incident excitations, and the results predict an oscillatory behaviour of these factors at high frequencies.     

On solutions of crack surface opening displacement of a penny-shaped crack in an elastic cylinder subject to tensile loading

A simple analytical expression for the surface displacement of a penny-shaped crack in an elastic cylinder subject to remote tensile loading is proposed based on a modified shear-lag model. The results are then compared with the dilute solution [1] and those of finite element calculation. It is found that the present work gives much better result than the dilute model.     

Three-dimensional dynamic stress analysis on a penny-shaped crack interacting with a suddenly transformed spherical inclusion

In this paper, the interaction between a penny-shaped crack and a near-by suddenly transformed spherical inclusion in 3-dimensional solid is investigated to assess the dynamic effect of the transformation. To simplify the solution procedure, the current problem is divided into two sub-problems by using the superposition principle. A time domain boundary integral equation method (BIEM) is adopted to evaluate the stress and displacement fields. 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. The impact effect of the spherical inclusion when it experiences a pure dilatational eigenstrain on the penny-shaped crack is studied. The relationship between the relative location of the inclusion and its impact effect on the time history of the Mode I crack intensity factor is discussed in detail.     

Axisymmetric problems for an externally cracked elastic solid. I. Effect of a penny-shaped crack

The present paper examines the problem of a penny-shaped flaw which is located in the plane of an external crack in an isotropic elastic solid. The penny-shaped flaw is subjected to uniform internal pressure. The paper develops power series representations for the stress intensity factors at the boundary of the penny-shaped flaw and at the perimeter of the externally cracked region. These series representations are in terms of a non-dimensional parameter which is the ratio of the radius of the penny-shaped flaw to the radius of the externally cracked region.     

Three-dimensional asymptotic stress field in the vicinity of the circumference of a bimaterial penny-shaped interfacial discontinuity

An eigenfunction expansion method is presented to obtain three-dimensional asymptotic stress fields in the vicinity of the circumference of a bimaterial penny-shaped interfacial discontinuity, e.g., crack, anticrack (infinitely rigid lamella), etc., located at the center, edge or corner, and subjected to the far-field torsion (mode III), extension/bending (mode I), and sliding shear/twisting (mode II) loadings. Five different discontinuity-surface boundary conditions are considered: (1) bimaterial penny-shaped interface anticrack or perfectly bonded thin rigid inclusion, (2) bimaterial penny-shaped interfacial jammed contact, (3) bimaterial penny-shaped interface crack, (4) bimaterial penny-shaped interface crack with partial axisymmetric frictionless slip, and (5) bimaterial penny-shaped interface thin rigid inclusion alongside penny-shaped crack. Solutions to these cases except (3) are hitherto unavailable in the literature. Closed-form expressions for stress intensity factors subjected to various far-field loadings are also presented. Numerical results presented include the effect of the ratio of the shear moduli of the layer materials, and also Poisson’s ratios on the computed lowest real parts of eigenvalues for the case (5). Interesting and physically meaningful conclusions are also presented, especially with regard to cases (1) and (2).     

Lack-of-contact conditions for a penny-shaped crack under a polynomial normal loading

Summary The classical problem of a penny-shaped crack inside an infinite three-dimensional isotropic elastic medium and under a polynomial normal loading (with axial symmetry) on both crack faces is reconsidered. By using elementary results from computational quantifier elimination techniques in computer algebra and applied logic, such as cylindrical algebraic decomposition and Sturm (or Sturm-Habicht) sequences, it is possible to satisfy the funcdamental inequality constraint about the positivity of the crack opening displacement inside the whole crack. This constraint assures us about the lack of contact of the crack faces, due to the loading of the crack, and the derived quantifier-free formula constitutes the related necessary and sufficient condition involving the loading parameters, that is the coefficients of the loading polynomial. Several such low-degree polynomial loadings are considered in detail (with the help of elementary and well-known solutin techniques for the present penny-shaped crack problem) as an application of the approach. Further possibilities for generalizations are also discussed in brief.