Three-dimensional Crack Problems Research Articles

Some recent fatigue and fracture mechanics research in BIAM

The paper presents a brief review of some of the major research activities on fatigue and fracture mechanics in recent years at the Beijing Institute of Aeronautical Materials. Attention is mainly given to the studies on weight function methods for analyses of two- and three-dimensional crack problems, fatigue crack growth under variable amplitude loading, small crack effects and a fracture-mechanics-based total fatigue life prediction method.

An asymptotic procedure and numerical study for the analysis of an elastic body with a thin sub-surface crack

A class of three-dimensional crack problems is considered, of which a prototype example is provided by a half-space containing a long internal crack, located in a plane perpendicular to the boundary. By means of an asymptotic procedure, the original three-dimensional problem is split up into a sequence of two-dimensional formulations. Results of its numerical implementation are in good agreement with results of more computer-intensive finite-element calculations.

A study on three-dimensional crack tip fields

In this paper several kinds of crack problems are numerically analyzed by the three-dimensional finite element method. They are (1) CT specimens with different thicknesses; (2) stable crack growth in CT specimens: (3) surface cracked plate subjected to bending: (4) pipe with surface crack subjected to bending: (5) CT and surface cracked specimens made of welded plate. The crack tip singular fields are compared with HRR solutions and the Q-factor is evaluated. The effects of the J-integral and Q-factor on the crack growth behavious are discussed with the comparison of the experimental results. It is concluded that the J-integral does not uniquely control the stable crack growth behaviors in many three-dimensional crack problems. It is also found that both J-integral and Q-factor play important roles in the stable crack growth especially in the welded plate.

Error estimation of numerical stress intensity factors for semi-elliptical cracks

Error estimates of numerical results and empirical formulae for stress intensity factors are usually based on author's experience and on comparison of some test results with closed analytical solutions (e.g., circular and elliptical cracks in an infinite solid). Such evaluation of K~-data in three-dimensional crack problems may become insufficient, as it was demonstrated for some part-through crack configurations [1]. Another way to check the quality of K~-results [2,3] is based on the evaluation of average, or integral, errors through the calculation of energy integrals associated with a cracked body. This procedure has been developed for the error estimation of some known stress intensity factor equations for part-elliptical cracks in plates [1,3]. It can also be a useful tool for the verification of numerical results on K~-factor provided the latter are derived for a wide range of configuration parameters of a cracked body and different locations of points on the crack front. The present paper deals with the evaluation of accuracy of boundary element data on K~-distdbution for circumferential and axial part-elliptical cracks in a hollow cylinder. Average errors are estimated at different kinds of polynomial loading applied to crack faces. It is concluded that the evaluation procedure can be applied for an approximate error analysis of local K~-estimates at some characteristic points of a crack front.

Three-dimensional thermoelastic problems of interface cracks in periodic two-layered composites

This paper outlines the potential function method for solving three-dimensional interface crack problems of a two-layered periodic space, treated within the framework of linear stationary thennoelasticity with microlocal parameters. By constructing the appropriate harmonic functions, the resulting boundary-value problems regarding an interface crack are reduced to corresponding ordinary problems dealing with mechanical loading in homogeneous isothermal elasticity. This result—known as an analogy between the thermal crack problems and their mechanical counterparts—may be exploited to yield closed-form solutions following directly from that obtained in the isotropic case. In connection with the theory of fracture mechanics, the thermal stress singularities as well as the effects of the geometrical and mechanical parameters of the composite structure on them are discussed.

Stress intensity factors for two offset parallel circular cracks: Part I — Infinite elastic solid

Abstract The method (W.D. Collins, Proc. R. Soc. London A274, 507–528 (1963); W.S. Fu and L.M. Keer, Int. J. Eng. Sci. 7, 361–372 (1969) [1,2]) used to solve co-planar penny-shaped cracks is generalized to investigate interaction of arbitrarily located penny-shaped cracks. The solution (M.K. Kassir and G.C. Sih, Three-dimensional Crack Problems (Noordhoff International, 1975) [3]) for the problem of an isolated crack in an infinite solid is applied together with the superposition principle to reduce the problem to a system of Fredholm integral equations of the second kind. These integral equations are then solved iteratively when the cracks are far apart. Some asymptotic solutions for the stress intensity factors are presented and comparisons are made whenever possible. Numerical solutions reveal some interesting phenomena.

The discontinuity displacement method applied to three-dimensional co-planar crack problem for any boundary value condition

The fundamental solution of discontinuity displacement for a three-dimensional problem is put forward by use of the Hankel transformation method and then the accurate solutions of discontinuity displacement in rectangular and triangular areas are obtained. One set of linear algebraic equations is given for arbitrary boundary values according to stress and displacement conditions on the crack surface, and a numerical method for obtaining the stress intensity factor using crack surface discontinuity displacements is established. It can be seen from many calculation results that the degree of accuracy of this method (discontinuity displacement method, DDM) is very high even though the element number is small. Combined with the boundary element method, this numerical method can also solve the finite cracked body problem.

Transient analysis of three-dimensional crack problems by the laplace transformed boundary element method

The Laplace transformed boundary element method is used to analyse the transient responses of a three-dimensional crack. The choice of Laplace inversion parameters and the calculation of dynamic stress intensity factors (SIF) are discussed. In order to verify the computation procedure and choose the discretization format, the transient displacement response of a prismatical beam under various dynamic loadings and the transient SIF response of a thick plate with a through-thickness crack under two different discretization schemes are calculated. Finally, the transient SIF response of a plate with a semi-circular surface crack under several different loading conditions is also computed, and the results are shown to be quite good.

Three-dimensional crack and contact problems with a general geometric configuration

A new method, based on point set theory and properties of orthogonal functions, is developed for determining exact solutions to three-dimensional crack and contact problems with complicated geometric configurations (e.g. a star-convex domain) in an infinite linear elastic medium. The governing equation is a two-dimensional Fredholm integral equation of the first kind. The central idea of this method is the chain extension of an exact solution from a regular subdomain to an irregular entire domain. Examples are given illustrating how this solution procedure can be used to obtain exact closed form solutions for a general Hertz contact problem and various crack problems in an inhomogeneous isotropic medium with an elastic modulus which is a power function of depth.

Curvilinear formulation of the 3-D J integral: Application to delamination cracking of curved laminates

This paper presents a formulation of the J integral for three-dimensional crack problems in local curvilinear coordinates. Infinitesimal deformations, elastic material behaviour and zero body forces are assumed. The development proceeds from the definition of the potential energy to emphasize the equivalence with the energy release rate per unit area of crack extension. By employing a formulation in terms of local curvilinear coordinates, calculations of energy release rates for nonplanar crack problems can be performed without knowledge of the stress field along the crack faces near the crack front. This feature can simplify the evaluation of the J integral for crack problems with nonplanar cracks. As an illustration of its utility, the technique is used to obtain a closed-form expression for the energy release rate for delamination cracking in arbitrarily loaded 2-D curved composite beams.

A new boundary integral equation method of three-dimensional crack analysis

Introducing the mode II and mode III dislocation densities W2(y) and W3(y) of two variables, a new boundary integral equation method is proposed for the problem of a plane crack of arbitrary shape in a three-dimensional infinite elastic body under arbitrary unsymmetric loads. The fundamental stress solutions for three-dimensional crack analysis and the limiting formulas of stress intensity factors are derived. The problem is reduced to solving three two-dimensional singular boundary integral equations. The analytic solution of the axisymmetric problem of a circular crack under the unsymmetric loads is obtained. Some numerical examples of an elliptical crack or a semielliptical crack are given. The present formulations are of basic significance for further analytic or numerical analysis of three-dimensional crack problems.

On the three-dimensional interface crack problems in periodic two-layered composites

Within the framework of the linear thermoelasticity with microlocal parameters the solutions of some three-dimensional interface crack problems in a two-layered periodic space are obtained. Effective results have been achieved through the use of similarity in governing equations of the homogenized model for the laminated body and transversely isotropic elastic solid. Some specific examples concerning interface crack problems have been considered and discussed from the point of view of fracture theory.

Eigenstrain methods in three-dimensional crack problems: An alternative integration procedure

In this paper three-dimensional crack problems are analysed using the eigenstrain method (body force method). The formulation leads to a set of hyper-singular integral equations which is solved numerically using a discretization technique. An analytical expression is derived for the evaluation of the singular integrals, i.e. the two-dimensional singular integrals are reduced to regular integrals along the contour of the domain, thus avoiding the direct calculation of cumbersome singular integrals and minimizing the numerical effort. The analytical expression is independent of the size and shape of the domain, allowing the use of high order elements in numerical solutions. Some well-known problems have been re-evaluated to test the method.

An Analysis of Crack Trapping by Residual Stresses in Brittle Solids

The residual stress distribution in a brittle polycrystalline solid may have a significant influence on its toughness. Grains in a state of residual compression are less likely to be fractured by a growing crack and may trap the crack front or be left behind as bridging particles (Evans et al., 1977). This paper estimates the toughness enhancement due to intergranular residual stresses, using a three-dimensional model. The residual stress is approximated as a doubly sinusoidal distribution acting perpendicular to the plane of an initially straight semi-infinite crack. An incremental perturbation method developed by Bower and Ortiz (1990) for solving three-dimensional crack problems is extended here to cracks loaded by nonuniform remote stresses. It is used to calculate the shape of the semi-infinite crack as it propagates through the doubly sinusoidal residual stress. It is shown that the local regions of compression may trap the crack front and give rise to some transient toughening. In addition, if the residual stress exceeds a critical magnitude, pinning particles may be left in the crack wake. However, for practical values of residual stress and grain size, the predicted toughness enhancement is insignificant. Furthermore, the analysis cannot account for the large bridging zones observed in experiments. It is concluded that the R-curve behavior and bridging particles observed in monolithic ceramics are caused by mechanisms other than residual stresses acting perpendicular to the crack plane.

Boundary element crack closure calculation of three-dimensional stress intensity factors

A general method for boundary element-crack closure integral calculation of three-dimensional stress intensity factors is presented. An equation for the strain energy release rate in terms of products of nodal values of tractions and displacements is obtained. Embedded and surface cracks of modes I, II, and III are analyzed using the proposed method. The multidomain boundary element technique is introduced so that the crack surface geometry is correctly modeled and the unsymmetrical boundary conditions for mode's II and III crack analysis are handled conveniently. Conventional quadrilateral elements are sufficient for this method and the selection of the size of the crack front elements is independent of the crack mode and geometry. For all of the examples demonstrated in this paper, 54 boundary elements are used, and the most suitable ratio of the width of the crack front elements to the crack depth is 1/10 and the calculation error is kept within ±1.5 percent. Compared to existing analytical and finite element solutions the boundary element-crack closure integral method is very efficient and accurate and it can be easily applied to general three-dimensional crack problems.

New Method of Analysis of Three-Dimensional Crack Problems. 1st Report. Basic Theory and Applications to Infinite Body Problems.

New Method of Analysis of Three-Dimensional Crack Problems. 1st Report. Basic Theory and Applications to Infinite Body Problems.

New Method of Analysis of Three-Dimensional Crack Problems. 2nd Report. Application to Semi-Infinite Body Problems with an Arbitrary Surface or Internal Crack under Complex Loading Conditions.

In this paper, a new versatile method for analizing a semi-infinite body with an arbitrary surface or internal crack under complex loading conditions was proposed. New numerical results were obtained in many cases, and compared with the other solutions. As a result, it was found that the present method is very powerful and gives highly accurate results with no extrapolation procedures even in the case of coarse meshes.

The method of caustics in conjunction with stress frozen technique for the investigation of three-dimensional crack problems

The method of caustics in conjunction with stress frozen technique for the investigation of three-dimensional crack problems

The rigid bodies—spring models and their applications to three-dimensional crack problems

The rigid bodies—spring models (RBSM) were applied to the phenomenon of crack growth in three-dimensional problems and demonstrated the effectiveness of the models. The characteristic of RBSM is to give practical answers withing a relatively small number of degrees of freedom in the limit analyses such as strongly non-linear and fracture analyses; the models have already proved useful for the analysis of two-dimensional problems. In this paper, the practical application of the three-dimensional RBSM is presented, and the conventional constitutive law of steels is discussed. Compared with the experimental data, the potential fracture criteria are determined. The stable crack growth in an arbitrarily shaped intitial crack is simulated. The plastic zone of the crack tip demonstrate that Mohr—Coulomb's law may be more suitable than von Mises'. The effectiveness of the cutting criterion for the springs of the RBSM is verified in the application to an arbitrarily shaped crack. In an application to dynamic crack propagation of brittle material, the simulated crack propagation is in resonable agreement with the actual crack propagated in the residual stress field of the welded plate.

Determination of three-dimensional stress intensity factor using virtual crack extension technique based on singular integral equations

A numerical procedure is presented for the virtual crack extension technique based on singular integral equations of the finite-part integral to determine three-dimensional stress intensity factors for arbitrarily shaped crack problems. Basic formulation is derived for the potential energy release rate. Numerical calculations are implemented through obtaining related coefficient matrices and their derivatives with respect to the crack front nodal coordinates. Numerical results are obtained for the penny-shaped crack and the elliptical crack under uniform and linearly varying normal tractions in an infinite body, and show satisfactory agreement with exact solutions. The method presented here can be used to obtain auxiliary solutions for the Schwarz alternating method to solve more general three-dimensional crack problems of the finite body.