Singularity Analysis and Boundary Integral Equation Method for Frictional Crack Problems in Two-Dimensional Elasticity

This study investigates the stress singularities in the neighborhood of the tip of a sliding crack with Coulomb-type frictional contact surfaces, and applies the boundary integral equation method to solve some frictional crack problems in plane elasticity. A universal approach to the determination of the complex order of stress singularity is established analytically by using the series expansion of the complex stress functions. When the cracks are open, or when no friction exists between the upper and lower crack faces, our results agree with those given by Williams. When displacement and traction are prescribed on the upper and lower crack surfaces (or vice versa), our result agrees with those by Muskhelishvili. For the case of a closed crack with frictional contact, the only nonzero stress intensity factor is that for pure shear or sliding mode. By using the boundary integral equation method, we derive analytically that the stress intensity factor due to the interaction of two colinear frictional cracks under far field biaxial compression can be expressed in terms of E(k) and K(k) (the complete elliptic integrals of the first and second kinds), where k=[1-(a/b)2]1/2 with 2a the distance between the two inner crack tips and b- a the length of the cracks. For the case of an infinite periodic colinear crack array under remote biaxial compression, the mode II stress intensity factor is found to be proportional to [2b tan(π a/2b)]1/2 where 2a and 2b are the crack length and period of the crack array. This revised version was published online in July 2006 with corrections to the Cover Date.     

A general Boundary Element Analysis of 2-D Linear Elastic Fracture Mechanics

This paper presents a boundary element method (BEM) analysis of linear elastic fracture mechanics in two-dimensional solids. The most outstanding feature of this new analysis is that it is a single-domain method, and yet it is very accurate, efficient and versatile: Material properties in the medium can be anisotropic as well as isotropic. Problem domain can be finite, infinite or semi-infinite. Cracks can be of multiple, branched, internal or edged type with a straight or curved shape. Loading can be of in-plane or anti-plane, and can be applied along the no-crack boundary or crack surface. Furthermore, the body-force case can also be analyzed. The present BEM analysis is an extension of the work by Pan and Amadei (1996a) and is such that the displacement and traction integral equations are collocated, respectively, on the no-crack boundary and on one side of the crack surface. Since in this formulation the displacement and/or traction are used as unknowns on the no-crack boundary and the relative crack displacement (i.e. displacement discontinuity) as unknown on the crack surface, it possesses the advantages of both the traditional displacement BEM and the displacement discontinuity method (DDM) and yet gets rid of the disadvantages associated with these methods when modeling fracture mechanics problems. Numerical examples of calculation of stress intensity factors (SIFs) for various benchmark problems were conducted and excellent agreement with previously published results was obtained. This revised version was published online in August 2006 with corrections to the Cover Date.     

Boundary element analysis of three‐dimensional cracks in anisotropic solids

This paper presents a boundary element analysis of linear elastic fracture mechanics in three‐dimensional cracks of anisotropic solids. The method is a single‐domain based, thus it can model the solids with multiple interacting cracks or damage. In addition, the method can apply the fracture analysis in both bounded and unbounded anisotropic media and the stress intensity factors (SIFs) can be deduced directly from the boundary element solutions. The present boundary element formulation is based on a pair of boundary integral equations, namely, the displacement and traction boundary integral equations. While the former is collocated exclusively on the uncracked boundary, the latter is discretized only on one side of the crack surface. The displacement and/or traction are used as unknown variables on the uncracked boundary and the relative crack opening displacement (COD) (i.e. displacement discontinuity, or dislocation) is treated as a unknown quantity on the crack surface. This formulation possesses the advantages of both the traditional displacement boundary element method (BEM) and the displacement discontinuity (or dislocation) method, and thus eliminates the deficiency associated with the BEMs in modelling fracture behaviour of the solids. Special crack‐front elements are introduced to capture the crack‐tip behaviour. Numerical examples of stress intensity factors (SIFs) calculation are given for transversely isotropic orthotropic and anisotropic solids. For a penny‐shaped or a square‐shaped crack located in the plane of isotropy, the SIFs obtained with the present formulation are in very good agreement with existing closed‐form solutions and numerical results. For the crack not aligned with the plane of isotropy or in an anisotropic solid under remote pure tension, mixed mode fracture behavior occurs due to the material anisotropy and SIFs strongly depend on material anisotropy. Copyright © 2000 John Wiley & Sons, Ltd.     

Simulation of Crack Propagation in Functionally Graded Materials Under Mixed-Mode and Non-Proportional Loading

Automatic simulation of crack propagation in homogeneous and functionally graded materials is performed by means of a remeshing algorithm in conjunction with the finite element method. The crack propagation is performed under mixed-mode and non-proportional loading. Each step of crack growth simulation consists of calculation of mixed-mode stress intensity factors by means of a novel formulation of the interaction integral method, determination of crack growth direction based on a specific fracture criterion, and local automatic remeshing along the crack path. The present approach requires a user-defined crack increment at the beginning of the simulation. Crack trajectories obtained by the present numerical simulation are compared with available experimental results.     

An Approximate Method to Measure Crack Opening Displacement of Fast Propagating Crack in Araldite B

Abstract: The method of crack opening displacement (COD) has been used to obtain the energy release rate of fast propagating cracks just before and just after crack bifurcation. In this method, COD is measured on the microscopic photographs of the cracks. But, in the case of cracks in Araldite B, the corner made of a crack surface and a specimen surface is chipped out of the specimen; then, it is often difficult to measure CODs from the photographs of the cracks. The present study proposes a method to approximately measure the COD in the region where COD is difficult to be measured directly on the photograph. The accuracy of the approximate method is good enough that the measured CODs can give the energy release rate of the crack at bifurcation with the accuracy of about 7%.     

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.     

Elastic stress intensity factors and crack opening displacements for circumferential through-walled cracked elbows

This paper provides tabulated solutions of elastic stress intensity factors and crack opening displacements for circumferential through-wall cracked elbows under internal pressure and under in-plane bending, based on extensive three-dimensional elastic finite element analyses covering a wide range of crack lengths and elbow/pipe geometries. The effect of crack length and elbow/pipe geometry on the results is discussed, with particular emphasis on the crack closure behaviour under in-plane bending.     

Domain-Decomposition Singular Boundary Method for Stress Analysis in Multi-Layered Elastic Materials

This paper applies an improved singular boundary method (SBM) in conjunction with domain decomposition technique to stress analysis of layered elastic materials. For problems under consideration, the interface continuity conditions are approximated in the same manner as the boundary conditions. The multi-layered coating system is decomposed into multiple subdomains in terms of each layer, in which the solution is approximated separately by the SBM representation. The singular boundary method is a recent meshless boundary collocation method, in which the origin intensity factor plays a key role for its accuracy and efficiency. This study also introduces new strong-form regularization formulas to accurately evaluate the origin intensity factors for elasticity problem. Consequently, we dramatically improve the accuracy and convergence of SBM solution of the elastostatics problems. The proposed domain-decomposition SBM is tested on two benchmark problems. Based on numerical results, we discuss merits of the present SBM scheme over the other boundary discretization methods, such as the method of fundamental solution (MFS) and the boundary element method (BEM).     

Dual boundary element method for three-dimensional thermoelastic crack problems

This paper describes the formulation and numerical implementation of the three-dimensional dual boundary element method (DBEM) for the thermoelastic analysis of mixed-mode crack problems in linear elastic fracture mechanics. The DBEM incorporates two pairs of independent boundary integral equations; namely the temperature and displacement, and the flux and traction equations. In this technique, one pair is applied on one of the crack faces and the other pair on the opposite one. On non-crack boundaries, the temperature and displacement equations are applied. This revised version was published online in July 2006 with corrections to the Cover Date.     

Improved numerical method for the Traction Boundary Integral Equation by application of Stokes' theorem

This paper concerns the direct numerical evaluation of singular integrals arising in Boundary Integral Equations for displacement (BIE) and displacement gradients (BIDE), and the formulation of a Traction Boundary Integral Equation (TBIE) for solving general elastostatic crack problems. Subject to certain continuity conditions concerning displacements and tractions at the source point, singular integrals in the BIE and the BIDE corresponding to coefficients of displacement and displacement gradients at the source point are shown to be of a form that allows application of Stokes' theorem. All the singular integrals in 3-D BIE and BIDE are reduced to non-singular line integrals, and those in 2-D BIE and BIDE are evaluated in closed form. Remaining terms involve regular integrals, and no references to Cauchy or Hadamard principal values are required. Continuous isoparametric interpolations used on continuous elements local to the source point are modified to include unique displacement gradients at the source point which are compatible with all local tractions. The resulting numerical BIDE is valid for source points located arbitrarily on the boundary, including corners, and a procedure is given for constructing a TBIE from the BIDE. Some example solutions obtained using the present numerical method for the TBIE in 2-D and 3-D are presented. © British Crown Copyright 1997/DERA.     

Crack analysis in unidirectionally and bidirectionally functionally graded materials

Mixed-mode crack analysis in unidirectionally and bidirectionally functionally graded materials is performed by using a boundary integral equation method. To make the analysis tractable, the Young's modulus of the functionally graded materials is assumed to be exponentially dependent on spatial variables, while the Poisson's ratio is assumed to be constant. The corresponding boundary value problem is formulated as a set of hypersingular traction boundary integral equations, which are solved numerically by using a Galerkin method. The present method is especially suited for straight cracks in infinite FGMs. Numerical results for the elastostatic stress intensity factors are presented and discussed. Special attention of the analysis is devoted to investigate the effects of the material gradients and the crack orientation on the elastostatic stress intensity factors.     

Singular Boundary Method for Heat Conduction in Layered Materials

In this paper, we investigate the application of the singular boundary method (SBM) to two-dimensional problems of steady-state heat conduction in isotropic bimaterials. A domain decomposition technique is employed where the bimaterial is decomposed into two subdomains, and in each subdomain, the solution is approximated separately by an SBM-type expansion. The proposed method is tested and compared on several benchmark test problems, and its relative merits over the other boundary discretization methods, such as the method of fundamental solution (MFS) and the boundary element method (BEM), are also discussed.     

Effect of Reference Loading and Crack Configuration on the Stress Intensity Factors In Weight Function Method

In calculating the stress intensity factors (SIFs) using the weight function method, the accuracy of the result depends upon proper selection of the set of reference loadings. The objective of the present work is to find a unique set of reference loadings to calculate the SIF at the tip of a crack of any configuration. Additionally, the universality of these reference loadings with respect to calculating the SIF for various crack configurations is examined. Two sets of crack configurations were considered, with and without a pre-existing crack. In each set, a horizontal crack and a slanted crack were analyzed.     

A Method for Measuring Current Values of the Crack-Tip Opening Displacement under Cyclic Loading

The analysis of the methods used for the evaluation of forces required for the opening of the tip of a fatigue crack enables us to choose and substantiate the approach to the evaluation of current values of the crack-tip opening displacement. On the basis of this approach, we propose an original procedure for measuring current values of the crack-tip opening displacement for a propagating fatigue crack at a constant distance behind its tip.     

An Analysis of a Quarter-Infinite Solid with a Corner Crack of Arbitrary Shape

In this paper, a versatile body force method for a quarter-infinite solid with a corner crack of arbitrary shape is proposed under two types of pressure: constant and linear. New numerical results are obtained for different corner crack cases. Fatigue crack growth from a corner crack has been analysed successively with the present method. Moreover, the stress intensity factor of a corner crack is proposed in a simple form for an arbitrary shape.     

A weight function method for mixed modes hole‐edge cracks

A weight function approach is proposed to calculate the stress intensity factor and crack opening displacement for cracks emanating from a circular hole in an infinite sheet subjected to mixed modes load. The weight function for a pure mode II hole‐edge crack is given in this paper. The stress intensity factors for a mixed modes hole‐edge crack are obtained by using the present mode II weight function and existing mode I Green (weight) function for a hole‐edge crack. Without complex derivation, the weight functions for a single hole‐edge crack and a centre crack in infinite sheets are used to study 2 unequal‐length hole‐edge cracks. The stress intensity factor and crack opening displacement obtained from the present weight function method are compared well with available results from literature and finite element analysis. Compared with the alternative methods, the present weight function approach is simple, accurate, efficient, and versatile in calculating the stress intensity factor and crack opening displacement.     

A direct method for calculating the stress intensity factor in BEM

The proposed algorithm employs singular crack tip elements in which the stress intensity factor appears as a degree of freedom. The additional degrees of freedom are compensated by constraint conditions which originate from imposing continuity across elements and a contour integration formula. The two benchmark problems indicate the proposed algorithm can accurately predict the stress intensity factor and the distribution of the primary and secondary variables in fracture problems.     

Weight Function Method for Stress Intensity Factor of Internal Surface Semi-elliptical Crack in Elliptical Holes

The hatches for inspecting are usually designed with elliptical holes in airplane structures, so computation of the stress intensity factor of three dimensional crack at elliptical holes is pivotal for damage tolerance analysis of these structures. In this paper, weight function is derived for a two dimensional through cracks at elliptical holes by applying a compounding method. Stress intensity factor formulas for an internal surface semi-elliptical crack in elliptical holes are obtained wing the three dimensional weight function method. Stress intensity factors for an internal surface semi-elliptical crack in elliptical holes under remote tension are computed. At the same time, research on how radius of curvature for elliptical holes affect stress intensity factors was conducted. Stress intensity factors decrease when radius of curvature increases. Some results and conclusions which are of practical value are given.     

Application of fracture mechanics for weld integrity assessment

The structural integrity assessment of a weld joint by conventional techniques is inadequate, because of unavoidable defects in the weld composite. The stress situation in a component having a defect is quite different from that of a homogeneous material. The significance of fracture mechanics to deal with such integrity assessments is brought out. A brief review on the basic formulations in the application of fracture mechanics is followed by established guidelines for evaluating the integrity of engineering components containing crack-like defects.     

An analytical method for mixed-mode propagation of pressurized fractures in remotely compressed rocks

An analytical method for mixed-mode (mode I and mode II) propagation of pressurized fractures in remotely compressed rocks is presented in this paper. Stress intensity factors for such fractured rocks subjected to two-dimensional stress system are formulated approximately. A sequential crack tip propagation algorithm is developed in conjunction with the maximum tensile stress criterion for crack extension. For updating stress intensity factors during crack tip propagation, a dynamic fictitious fracture plane is used. Based on the displacement correlation technique, which is usually used in boundary element/finite element analyses, for computing stress intensity factors in terms of nodal displacements, further simplification in the estimation of crack opening and sliding displacements is suggested. The proposed method is verified comparing results (stress intensity factors, propagation paths and crack opening and sliding displacements) with that obtained from a boundary element based program and available in literatures. Results are found in good agreements for all the verification examples, while the proposed method requires a trivial computing time.