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.     

A non-hypersingular time-domain BIEM for 3-D transient elastodynamic crack analysis

A three-dimensional (3-D) time-domain boundary integral equation method (BIEM) is presented for transient elastodynamic crack analysis. A non-hypersingular traction BIE formulation is used with the crack opening displacements and their derivatives as unknown quantities. A collocation method in conjunction with a time-stepping scheme is developed to solve the non-hypersingular time-domain BIEs. To simplify the analysis and to describe the proper behaviour of the unknown quantities at the crack front, a constant spatial shape function is applied for elements away from the crack front, while a spatial ‘square-root’ crack-tip shape function is adopted for elements near the crack front. A linear temporal shape function is used in the time-stepping scheme. Numerical calculations, have been carried out for penny-shaped and square cracks. Results for the elastodynamic stress intensity factors are presented as functions of the temporal and the spatial variables. For the test examples considered, good agreement between the present results and those of other authors is obtained.     

3D elastodynamic crack analysis by a non-hypersingular BIEM

A boundary integral equation method (BIEM) is presented for 3D elastodynamic crack analysis. The method is based on a non-hypersingular BIE formulation, where the unknown quantities are the crack opening dispacements and their derivatives. 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 front to describe the proper behavior of the unknown quantities at the crack front. A collocation method is applied to convert the non-hypersingular BIEs to a system of linear algebraic equations which are solved numerically. For several geometrical configurations, numerical results are presented for both the elastodynamic stress intensity factors and the scattering cross section. They are in good agreement with those obtained by other authors.     

On the transient elastic field for an expanding spherical inclusion in 3-D solid

Summary. The three-dimensional transient elastic field of an infinite isotropic elastic medium is investigated when a phase transformation is nucleated from a point and proceeds through the crystal dynamically. The phase transformation keeps the spherical shape and expands at a speed of arbitrary time profile. This process is modeled by an expanding spherical inclusion with a spatially uniform eigenstrain. The objective of this paper is to present a general method to determine the transient displacement field for points either covered or not covered by the transformation area. This method can be applied to investigate the nucleation and expanding mechanism of phase transformation. Using a Green's function approach, an explicit procedure is presented to evaluate the 3-D displacement field when the expanding history of the spherical inclusion is given. As numerical examples, the explicit formulations are given for the transient elastic fields, when the spherical inclusion expands at a constant or an exponent damping speed with a pure dilatational eigenstrain or pure shear eigenstrain. It is found that the elastic field inside the expanding inclusion remains constant with respect to time, which is consistent with the well-known Eshelby solution for a static inclusion case.     

An integral equation approach to the inclusion-crack interactions in three-dimensional infinite elastic domain

 In this paper, an integral equation method to the inclusion-crack interaction problem in three-dimensional elastic medium is presented. The method is implemented following the idea that displacement integral equation is used at the source points situated in the inclusions, whereas stress integral equation is applied to source points along crack surfaces. The displacement and stress integral equations only contain unknowns in displacement (in inclusions) and displacement discontinuity (along cracks). The hypersingular integrals appearing in stress integral equation are analytically transferred to line integrals (for plane cracks) which are at most weakly singular. Finite elements are adopted to discretize the inclusions into isoparametric quadratic 10-node tetrahedral or 20-node hexahedral elements and the crack surfaces are decomposed into discontinuous quadratic quadrilateral elements. Special crack tip elements are used to simulate the variation of displacements near the crack front. The stress intensity factors along the crack front are calculated. Numerical results are compared with other available methods. Received: 28 January 2002 / Accepted: 4 June 2002 The work described in this paper was partially supported by a grant from the Research Grant Council of the Hong Kong Special Administration Region, China (Project No.: HKU 7101/99 E).     

Nonstationary torsion of a tapered shaft weakened by a spherical crack

We obtain an approximate effective solution of the problem of nonstationary stress concentration near a spherical crack located inside a tapered shaft whose face is subjected to the action of impact tangential stresses. To solve the problem, we propose to use an approach based on the discretization of the problem in time with the help of a difference scheme. By the method of integral transformations, the problem is reduced to an integral equation for the unknown jump of displacements on the crack. This equation is solved by the method of orthogonal polynomials. The relation for the evaluation of the stress intensity factors is obtained. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 1, pp. 49–55, January–February, 2008.     

Analysis of dynamic interaction between an inclusion and a nearby moving crack by BEM

In this paper, the dynamic interaction between an inclusion and a nearby moving crack embedded in an elastic medium is studied by the boundary element method (BEM). To deal with this problem, the multi-region technique and two kinds of time-domain boundary integral equations (BIEs) are introduced. The system is divided into two parts along the interface between the inclusion and the matrix medium. Each part is linear, elastic, homogeneous and isotropic. The non-hypersingular traction boundary integral equation is applied on the crack surfaces; while the traditional displacement boundary integral equation is used on the interface and external boundaries. In the numerical solution procedure, square root shape functions are adopted as to describe the proper asymptotic behavior in the vicinity of the crack-tips. The crack growth is modeled by adding new elements of constant length to the moving crack tip, which is controlled by the fracture criterion based on the maximum circumferential stress. In each time step, the direction and the speed of the crack advance are evaluated. The numerical results of the crack growth path, speed, dynamic stress intensity factors (DSIFs) and dynamic interface tractions for various material combinations and geometries are presented. The effect of the inclusion on the moving crack is discussed.     

Stress singularity for a crack with an arbitrarily curved front

To investigate the possibility of a point of a crack front propagating in a direction not in the normal plane of the crack front, the three-dimensional form of the crack front stress field is obtained. To simplify comparisons of the states of stress at various points lying on a spherical surface centered at a point of the crack front, a local spherical coordinate system is used. It is found that the crack propagation will be from each point of the crack front in a direction lying in the normal plane.The results are used in conjunction with the strain energy density fracture criterion for the problem of an elliptical crack. The plane of the flat elliptical crack makes an arbitrry angle with the field of uniform applied tensile stress. Crack growth directions for various positions along the crack front are determined, and loads required for fracture for various angles are obtained.     

Viscoelastic crack analysis using symplectic analytical singular element combining with precise time-domain algorithm

In this study, the crack problem in linear viscoelastic material is investigated numerically. The time dependent two-dimensional (2D) viscoelastic crack problem is treated by the precise time-domain expanding algorithm (PTDEA), such that the original problem is transformed into a series of quasi-elastic crack problems. The relationships among these quasi-elastic problems are expressed in terms of the time-domain expanding coefficients of displacement and stress in an improved recursive manner. Then a symplectic analytical singular element (SASE) which has been demonstrated to be effective and efficient for 2D elastic fracture problem is applied to solve the quasi-elastic crack problems obtained above. The SASE is constructed by using the symplectic eigen solutions with higher order expanding terms. An improved convergence criterion employing both displacement and stress for PTDEA is proposed. Taking advantage of the SASE, the stress intensity factors, crack opening and sliding displacements (COD and CSD) and strain energy release rate of the studied problem can be solved directly without any post-processing. Numerical examples show that the results of the present method can be solved accurately and effectively.     

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.     

Variations of stress intensity factor of a semi-elliptical surface crack subjected to mixed mode loading

Maximum stress intensity factors of a surface crack usually appear at the deepest point of the crack, or a certain point along crack front near the free surface depending on the aspect ratio of the crack. However, generally it has been difficult to obtain smooth distributions of stress intensity factors along the crack front accurately due to the effect of corner point singularity. It is known that the stress singularity at a corner point where the front of 3 D cracks intersect free surface is depend on Poisson's ratio and different from the one of ordinary crack. In this paper, a singular integral equation method is applied to calculate the stress intensity factor along crack front of a 3-D semi-elliptical surface crack in a semi-infinite body under mixed mode loading. The body force method is used to formulate the problem as a system of singular integral equations with singularities of the form r ⁻³ using the stress field induced by a force doublet in a semi-infinite body as fundamental solution. In the numerical calculation, unknown body force densities are approximated by using fundamental density functions and polynomials. The results show that the present method yields smooth variations of mixed modes stress intensity factors along the crack front accurately. Distributions of stress intensity factors are indicated in tables and figures with varying the elliptical shape and Poisson's ratio.     

A simple and less-costly meshless local Petrov-Galerkin (MLPG) method for the dynamic fracture problem

A simple and less-costly MLPG method using the Heaviside step function as the test function in each sub-domain avoids the need for both a domain integral, except inertial force and body force integral in the attendant symmetric weak form, and a singular integral for analysis of elasto-dynamic deformations near a crack tip. The Newmark family of the methods is applied into the time integration scheme. A numerical example, namely, a rectangular plate with a central crack with plate edges parallel to the crack axis loaded in tension is solved by this method. The results show that the stresses near the crack tip agree well with those obtained from another MLPG method using the weight function of the moving least square approximation as a test function of the weighted residual method. Time histories of dynamic stress intensity factors (DSIF) for mode-I are determined form the computed stress fields.     

A MODEL FOR THE BEHAVIOUR OF MATERIALS WITH CRACKS UNDER HYDROGEN EMBRITTLEMENT CONDITIONS

Abstract— We consider the slow growth of normal tension cracks as quasi-brittle behaviour under hydrogen embrittlement conditions. Experiments show that the cracking resistance of a material in such cases is not a constant of the material, but is characterized by some function that relates the rate of crack growth to the stress intensity factor. We propose a numerical method for the calculation of opening mode crack growth when the kinetics are controlled by the gas diffusion into the material. The problems under consideration model the fracture phenomena inherent to structures (e.g. pressure vessels, pipelines) that operate in an aggressive medium and in particular a hydrogen environment.
In such problems it is necessary to calculate the pressure variation inside a crack as a result of gas diffusion and crack growth under the action of this pressure. Hence it is necessary to solve problems of diffusion theory and elasticity theory for a cracked medium together with some additional conditions that provide the link between these two fundamental problems.
We study the case of an infinite medium containing a crack which occupies a plane domain of arbitrary shape. To avoid difficulties related to the three-dimensionality of the problems, we reduce them to two-dimensional integro-differential equations for the crack domain. The integro-differential equation of the elasticity problem of the crack is solved on the basis of the Boundary Element Method (BEM). The crack kinetics are calculated using a scheme previously introduced by one of the authors and then the BEM is used to solve the integral equation for the diffusion-into-the-crack problem similar to the analogous problem of filtration of the fluid into a crack.     

Three-dimensional crack problem analysis using boundary element method with finite-part integrals

A set of hypersingular integral equations of a three-dimensional finite elastic solid with an embedded planar crack subjected to arbitrary loads is derived. Then a new numerical method for these equations is proposed by using the boundary element method combined with the finite-part integral method. According to the analytical theory of the hypersingular integral equations of planar crack problems, the square root models of the displacement discontinuities in elements near the crack front are applied, and thus the stress intensity factors can be directly calculated from these. Finally, the stress intensity factor solutions to several typical planar crack problems in a finite body are evaluated. This revised version was published online in August 2006 with corrections to the Cover Date.     

Fracture problems of rubbers: J-integral estimation based upon η factors and an investigation on the strain energy density distribution as a local criterion

The plane elasticity problem studied is of a circular inclusion having a circular arc-crack along the interface and a crack of arbitrary shape in an infinite matrix of different material subjected to uniform stresses at infinity. The solution of the problem is given using Muskhelishvili's complex variable method with sectionally holomorphic functions. First, the solution to the (auxiliary) problem of a dislocation (or force) applied at a point in the matrix with the circular inclusion partially bonded is derived fully in its general form by solving the appropriate Rieman-Hilbert problem. It is subsequently used as the Green's function for the initial problem by introducing an unknown density function associated with a distribution of dislocations along the crack in the matrix. The initial problem is then reduced to a singular integral equation (SIE) over the crack in the matrix only. The SIE is solved numerically by appropriate quadratures and the stress intensity factors reported for the arc-cut and a straight crack in the matrix for a range of values of the geometrical parameters.     

A simple and efficient XFEM approach for 3-D cracks simulations

In this work, a simple and efficient XFEM approach has been presented to solve 3-D crack problems in linear elastic materials. In XFEM, displacement approximation is enriched by additional functions using the concept of partition of unity. In the proposed approach, a crack front is divided into a number of piecewise curve segments to avoid an iterative solution. A nearest point on the crack front from an arbitrary (Gauss) point is obtained for each crack segment. In crack front elements, the level set functions are approximated by higher order shape functions which assure the accurate modeling of the crack front. The values of stress intensity factors are obtained from XFEM solution by domain based interaction integral approach. Many benchmark crack problems are solved by the proposed XFEM approach. A convergence study has been conducted for few test problems. The results obtained by proposed XFEM approach are compared with the analytical/reference solutions.     

The stress intensity factor Green's function for a crack interacting with a circular inclusion

The Green's function is constructed for the stress intensity factor due to the unit dipole force applied to the crack surface in the presence of a circular inclusion in front of the crack tip. An explicit functional form of the Green's function is proposed in terms of dipole force location, Young's modulus ratio and the inclusion size and position with respect to the crack tip. This is achieved through a combination of the dimensional analysis and parametric studies by means of the finite element method. The purpose of this paper is to provide the basis for further studies of a crack interaction with an array of microdefects and/or inclusions.     

Numerical life estimation after fatigue failure of a complex component

Failure of a large ethylene‐reciprocating compressor was found to be due to fatigue growth of cracks in the crosshead of one of the cylinders, initiated at material defects near stress raisers. Total fatigue crack growth time was required in order to identify the cause of the failure. The applied stress field near the initiation sites and along fatigue path was estimated using FEM. The stresses were found to vary steeply and become partly compressive along a large part of the fatigue crack path. A weight function based on numerical method was developed, which was able to predict exactly the shape of the crack front during propagation. Fatigue crack initiation was traced to a disassembly 6 months before final failure. This failure was found to be jointly the result of non‐conformities in manufacture and maintenance.     

Variation of mixed modes stress intensity factors of an inclined semi-elliptical surface crack

In this paper, a singular integral equation method is applied to calculate the stress intensity factor along crack front of a 3D inclined semi-elliptical surface crack in a semi-infinite body under tension. The stress field induced by displacement discontinuities in a semi-infinite body is used as the fundamental solution. Then, the problem is formulated as a system of integral equations with singularities of the form r ^–3. In the numerical calculation, the unknown body force doublets are approximated by the product of fundamental density functions and polynomials. The results show that the present method yields smooth variations of mixed modes stress intensity factors along the crack front accurately for various geometrical conditions. The effects of inclination angle, elliptical shape, and Poisson's ratio are considered in the analysis. Crack mouth opening displacements are shown in figures to predict the crack depth and inclination angle. When the inclination angle is 60 degree, the mode I stress intensity factor F _I has negative value in the limited region near free surface. Therefore, the actual crack surface seems to contact each other near the surface.     

The interacting stress field around a composite crack with a misfitting inclusion

The problem of a composite crack interacting with a circular misfitting inclusion in an infinite elastic medium is investigated. The close-formed solutions of the stress fields in the inclusion and the matrix are obtained by using the dislocation model of cracks and the point force method as well as the complex function technique. The stress intensity factors at two tips of the crack are calculated from the stress components.