Multiple Crack Problems Research Articles

Fredholm integral equation for the multiple circular arc crack problem in plane elasticity

An elementary solution for the multiple circular arc problem is obtained in this paper. The elementary solution is defined as a particular case of the single circular arc crack problem, in which remote stresses are equal to zero, and two pairs of concentrated forces are applied at a prescribed point of crack face. By using the principle of superposition, Fredholm integral equation for the multiple circular arc problem in plane elasticity is obtained. The suggested approach is illustrated by several numerical examples. If a smaller arc crack is surrounded by a larger arc crack, the stress intensity factors for the former become rather small. The phenomenon of shielding is illustrated by examples.

Non-singular terms for multiple cracks in anisotropic elastic solids

The non-singular and bounded terms for the stresses near the crack tip are investigated. This paper deals with the multiple crack problem for an infinite plate. The original problem is decomposed into three elementary subproblems: (1) the problem for remote uniform stress filed without cracks; (2) the single crack problem with traction applied along the crack face and (3) the problem for the influence of the other cracks. Several examples for collinear and non-collinear cracks are discussed and the results are shown.

FRACTURE AND DAMAGE ANALYSIS OF A CRACKED BODY BY A NEW BOUNDARY ELEMENT MODEL

Based on the concept of discontinuity displacement, an analytical solution for cracked thin plates has been derived in which displacements and stresses in a solid can be expressed by the linear distributed discontinuity displacements on the whole boundary. By way of the potential variational principle and the analytical solution newly developed, a boundary element model for 2D multiple crack problems has been presented and applied to fracture and damage analysis of thin plates with many cracks. Two numerical examples are considered to illustrate applications of the proposed element model.

FRACTURE AND DAMAGE ANALYSIS OF A CRACKED BODY BY A NEW BOUNDARY ELEMENT MODEL

Based on the concept of discontinuity displacement, an analytical solution for cracked thin plates has been derived in which displacements and stresses in a solid can be expressed by the linear distributed discontinuity displacements on the whole boundary. By way of the potential variational principle and the analytical solution newly developed, a boundary element model for 2D multiple crack problems has been presented and applied to fracture and damage analysis of thin plates with many cracks. Two numerical examples are considered to illustrate applications of the proposed element model.© 1997 John Wiley & Sons, Ltd.

Complex variable approach to the BEM for multiple crack problems

A boundary element method for straight multiple center and edge crack problems is developed in this paper. The method is constructed upon the systematic use of the elastic singularity solutions in complex variables. The crack opening is represented by the continuous distribution of dislocation dipoles and the effect of the non-crack boundary by the continuous distributions of point forces and dislocation dipoles. The crack-tip singularity is embedded into the interpolation using orthogonal polynomials (i.e. Chebyshev and Jacobi) and their associated singular weight functions. The proposed analytical integration procedure of the Cauchy-type integrals defined over the crack eliminates the need for the quadrature formulae for numerical integration, streamlines, and enhances the accuracy of the traditional singular integral equation method for crack problems. The stress intensity factors for the fifteen problems analyzed in this paper have been accurate enough to substitute those given in stress intensity factor handbooks. Since non-crack boundary of arbitrary shape can be introduced at will the method is expected to give accurate stress intensity factors for complex real life problems.

Analysis of Multiple Cracks Close to Bimaterial Interface

In this paper we present a methodology and a rigorous solution formulation for multiple crack problems in nonhomogeneous plates using the dislocation complex potential technique. The fundamental perturbation problem is investigated by establishing a system of singular integral equations, whose solution gives mixed-mode stress intensity factors [(SIFs), k] and strain energy release rate [(SERR), G]. The numerical integration of the singular integral equations is carried out by using a Lobatto type collection technique. Finite-element analysis (FEA) of the representative problem is provided for accuracy comparison. Finally, studies for two cracks located on the opposite side of the interface are presented.

Computational modeling of elastic and plastic multiple cracks by the fundamental solutions

The fundamental solutions of elasticity are used to establish a numerical method for elastic and plastic multiple crack problems in two dimensions. The continuous distributions of the point forces, dislocations, and the plastic sources are used systematically to model the crack, non-crack boundary, and the plastic deformation. Use of these singularities are guided strictly by the physical interpretation of the problem. We adopt Muskhelishvili's complex variable formalism that facilitate the analytical evaluation of the integrals representing the continuous distributions of the singularities. The resulting numerical method is concise and accurate enough to be used for elastic and plastic multiple crack problems.

Interacting multiple cracks with complicated crack surface conditions

In solving multiple cracks problems for the strong crack interactions, crack surfaces that are not always open even under remote tension loading, some crack surfaces may contact or even partially contact. This kind of complicated multiple cracks problem has not been treated before

A numerical green's function approach for boundary elements applied to fracture mechanics

AbstractThe most accurate boundary element formulation to deal with fracture mechanics problems is obtained with the implementation of the associated Green's function acting as the fundamental solution. Consequently, the range of applications of this formulation is dependent on the availability of the appropriate Green's function for actual crack geometry. Analytical Green's functions have been presented for a few single crack configurations in 2‐D applications and require complex variable theory. This work extends the applicability of the formulation through the introduction of efficient numerical means of computing the Green's function components for single or multiple crack problems, of general geometry, including the implementation to 3‐D problems as a future development. Also, the approach uses real variables only and well‐established boundary integral equations.

A survey of new integral equations in plane elasticity crack problem

The formulation of the integral equations in plane elasticity crack problem is presented in this review. Emphasis is laid on some proposals for formulating the Fredholm integral equation (FIE) and regularizing the existing singular integral equation (SIE) in the crack problem. Two kinds of regularization are suggested for the multiple crack problems. The first kind of regularization is obtained by the use of an appropriate substitution of the unknown functions in SIE while the second one is reached by operating some integral operator on SIE. Alternatively, the FIE will be obtained directly if one chooses the traction applied along the crack as the unknown function in the equation. For the curve crack problem, the resultant force function rather than the traction applied on the crack face is chosen as the right hand term in the resulting integral equation. Therefore, a weaker singular integral equation with logarithm kernel is obtainable. Some modified complex potentials (MCP) similar to the Green's function for solving the Laplace's equation are introduced. The MCP will provide possibilities for solving some relevant problems. It is proved that the proposed FIE for the crack problem is easier to compute. Miscellaneous problems, say, the thermally insulated crack problem, are also discussed.

Hypersingular integral equation approach for the multiple crack problem in an infinite plate

In this paper, a hypersingular integral equation for the multiple crack problem in an infinite plate is formulated. The unknown functions involved in the equation are the crack opening displacements (CODs) while the right hand terms are the tractions applied on the crack faces. Some particular hypersingular integrals are quadratured in a closed form. After the CODs are approximated by a weight function multiplied by a polynomial, the hypersingular integrals in the equation can be evaluated in a closed form, and the regular integrals can be integrated numerically. Numerical examples with the calculated stress intensity factors (SIFs) at the crack tips are given.

Stress intensity factors for radial cracks in thick walled cylinders—III. Asymmetrical cracks

For many years, a great amount of work has been done on the problems of symmetrical multiple radial cracks in the thick walled cylinder. But, up to now, nothing has been done on asymmetrical multiple radial cracks in thick walled cylinders. This paper deals with two and three asymmetrical radial cracks, and the stress intensity factors (SIF) due to two kinds of internal pressures and full autofrettage process has been calculated using finite element method and the influence of crack configurations on SIF is discussed.

T stress in multiple crack problem for an infinite plate

The non-singular and bounded term in the expansion form of stress near the crack tip is called the T stress. The T stress in multiple crack problems for an infinite plate is analyzed. The original problem is decomposed into a uniform stress field and many single crack problems. The T stress at the crack tip for the multiple crack problem consists of three parts: (a) the influence from the uniform stress field; (b) the influence from the considered crack; and (c) the influence from the remaining cracks. Numerical examples are given to demonstrate the T stress at the crack tips.

Moving template analysis of crack growth—I. Procedure development

Based on a moving template procedure, this two part series will develop a method to follow the crack tip physics in a self-adaptive manner which provides a uniformly accurate prediction of crack growth. For multiple crack environments, this is achieved by attaching a moving template to each crack tip. The templates are each individually oriented to follow the associated growth orientation and rate. In this part, the essentials of the procedure are derived for application to fatigue crack environments. Overall the scheme derived possesses several hierarchical levels, i.e. the global model, the interpolatively tied moving template and a multilevel element death option to simulate the crack wake. To speed up computation, the hierarchical polytree scheme is used to reorganize the global stiffness inversion process. In addition to developing the various features of the scheme, the accuracy of predictions for various crack lengths is also benchmarked. Part II [Y. H. Guo and J. Padovan, Engng Fracture Mech. 48, 427–444 (1994)] extends the scheme to multiple crack problems. Extensive benchmarking is also presented to verify the scheme.

Moving template analysis of crack growth—II. Multiple crack extension and benchmarking

In Part I of this series [J. Padovan and Y. H. Guo, Engng Fracture Mech. 48,405–425 (1994)], a moving template scheme was developed which can model the propagation of long cracks possessing complex topologies and mixed mode behavior. This part extends the procedure to the propagation of multiple fatigue cracks. The methodology is benchmarked over a wide variety of single and multiple crack problems involving complex mixed mode behavior. This includes such problem features as compound loading, multiple cracks, crack wake induced shading of the unfolding stress distribution, different growth rates for the various cracks as defined by the stress redistribution, as well as mixed mode behavior. Additionally, formal expressions are derived which illustrate the advantages of employing the HPT scheme (hierarchical substructuring) to streamline the computational effort. These point to significant cost effective improvements.

New Fredholm integral equation for multiple crack problem in plane elasticity and antiplane elasticity

A singular integral equation for the multiple crack problem of plane elasticity is formulated in this paper. In the formulation we choose the crack opening displacement (COD) as unknown function and the resultant force as the right hand term of the equation. After using Vekua's regularization procedure or making inversion of the Cauchy singular integral in the equation, a new Fredholm integral equation is obtainable. The obtained Fredholm integral equation is compact in form and easy for computation. After solving the equation, the CODs of the cracks and the stress intensity factors (SIFs) at the crack tips can be derived immediately. Similar formulation for the multiple crack problem of antiplane elasticity is also presented. Finally, numerical examples are given to demonstrate the use of the proposed integral equation approach.

An alternative fredholm integral equation approach for multiple crack problem and multiple rigid line problem in plane elasticity

An alternative Fredholm integral equation approach for the multiple crack problem and the multiple rigid line problem in plane elasticity is suggested in this paper. After using some operators on both sides of the singular integral equation for the relevant problem, an alternative Fredholm integral equation is obtainable. The solutions of the suggested integral equations satisfy the single-valuedness condition of displacement (for the multiple crack problem) and the zero resultant force condition (for the multiple rigid line problem), respectively. The obtained integral equation is compact in form and easy for computation. A technique for evaluating the rotation in the multiple rigid line problem is also proposed. Numerical examples are given to demonstrate the efficiency of the proposed approach.

A general method for multiple crack problems in a finite plate

The multiple crack problem in a finite plate is calculated by a new method. The basic stress functions of the solution consist of two parts. One is the Fredholm integral equation solution for the crack problem in an infinite plate. The other is that of weighted residual method in a common plane problem. Some examples are shown that when the cracks are very short, the results compare very favorably with the existing results for an infinite plate. Furthermore, the method can be used for arbitrary multiple crack problems in a finite plate.

Multiple crack interaction and its effect on stress intensity factor

This paper presents a versatile and numerically accurate and efficient method to study multiple crack problems. The solution technique is based on a singular integral method which uses distributions of edge dislocations to represent the cracks in the mathematical model. Plane strain multiple crack problems are considered to investigate the effect on the stress intensity factor due to the elastic interaction between multiple cracks. The results obtained show that interaction between microcracks could produce either an enhancement or a shielding effect on the stress intensity factor depending on the positions and orientations of the microcracks.

Multiple crack problem for an infinite strip

An elementary solution of the multiple crack problem for an infinite strip is derived in this paper. The proposed elementary solution is a particular solution of the infinite strip containing a single crack. The solution corresponds to the following boundary loading conditions: 1. (1) the edges of the strip are traction free 2. (2) two pairs of the concentrated forces and some distributed tractions are applied on both the borders of crack. Using the obtained elementary solution and taking some undetermined density of the elementary solution along each crack, a system of Fredholm integral equations can always be obtained. Three numerical examples are given.