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

Summary 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.     

Solution for Eshelby's elastic inclusions in a finite plate using boundary integral equation method

This paper provides a solution for Eshelby's elastic inclusions in a finite plate based on the complex variable boundary integral equation (CVBIE) method. In the formulation, an inclusion with Eshelby's eigenstrains is embedded in an elliptic plate, and the exterior boundary is applied by some static loading. Two BIEs are suggested in the present study. One of BIEs is used for the finite inclusion region, and the other is used for region bounded by interface and the exterior boundary. After the discretization of BIEs, a numerical solution is suggested. In the solution, an inverse matrix technique is suggested which can eliminate one unknown vector in advance. Three numerical examples under different generalized loadings are provided. Interaction between the prescribed eigenstrains and the static loading along the exterior boundary is studied in detail.     

Solution of the problem of a crack in a finite plane region using an indirect boundary-integral method

In a previous paper [1], an indirect boundary-integral approach was developed for the treatment of a finite, plane, linear-elastic region weakened by a hole of arbitary shape. It was suggested there that this method would yield excellent results on and near the hole boundary.

In this presentation, the method is applied to the problem of a finite, plane, linear-elastic region containing a sharp crack. Numerical results are obtained for a simple geometry and the crack-opening-displacement and crack tip stresses are compared to the known solution. It is shown that the boundary-integral solution is quite accurate.     

Solutions of multiple crack problems of a circular plate or an infinite plate containing a circular hole by using Fredholm integral equation approach

Presented is an elementary solution, which is a particular solution of the circular plate containing one crack. The solution consists of two parts and satisfies the following conditions: (i) the first part corresponds to a pair of normal and tangential concentrated forces acting at a prescribed point on both edges of a single crack; (ii) the second part corresponds to some distributed tractions along both edges of the crack; (iii) the obtained elementary solution, i.e. the sum of the first and second parts, satisfies a traction free condition on the circular boundary. Using this elementary solution and taking some undetermined density of the elementary solution along each crack, a system of Fredholm integral equations of multiple crack problems can always be obtained. The multiple crack problems of an infinite plate containing a circular hole can be solved in a similar way. Several numerical examples are given in this paper.     

A general method for multiple crack problems in a finite plate

A novel method for the multiple crack problems in a finite plate is proposed in this paper. 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, and the other is that of the weighted residual method for general plane problems. The combined stress functions are used in the analysis and the boundary conditions on the crack surfaces and the boundary are considered. After the coefficients of the functions have been determined, the stress intensity factors (SIF) at the crack tips can be calculated. Some numerical examples are given and it was observed that when the cracks are very short, the results compare very favorably with the existing results for an infinite plate. Furthermore, the influence of the boundary can be considered. This method can be used for arbitrary multiple crack problems in a finite plate.     

A boundary integral equation method for an inverse problem related to crack detection

This paper discusses an application of a boundary integral equation method (BIEM) to an inverse problem of determining the shape and the location of cracks by boundary measurements. Suppose that a given body contains an interior crack, the shape and the location of which are unknown. On the exterior boundary of this body one carries out measurements which are interpreted mathematically as prescribing Dirichlet data and measuring the corresponding Neumann data, or vice versa, for a field governed by Laplace's equation. The inverse problem considered here attempts to determine the geometry of the crack from these experimental data. We propose to solve this problem by minimizing the error of a certain boundary integral equation (BIE). The process of this minimization, however, is shown to require solutions of certain are proposed. Several 2D and 3D numerical examples are given in order to test the performance of the present method.     

Boundary integral equation method for two dissimilar elastic inclusions in an infinite plate

This paper provides a numerical solution for an infinite plate containing two dissimilar elastic inclusions, which is based on complex variable boundary integral equation (CVBIE). The original problem is decomposed into two problems. One is an interior boundary value problem (BVP) for two elastic inclusions, while other is an exterior BVP for the matrix with notches. After performing discretization for the coupled boundary integral equations (BIEs), a system of algebraic equations is formulated. The inverse matrix technique is suggested to solve the relevant algebraic equations, which can avoid using the assembling of some matrices. Several numerical examples are carried out to prove the efficiency of suggested method and the hoop stress along the interface boundary is evaluated.     

Anti-plane shear problem for an edge crack in a finite orthotropic plate

The problem of an edge crack in a finite orthotropic plate under anti-plane shear is considered. The boundary collocation method is used to calculate the mode III stress intensity factor (SIF). For the case in which the material is isotropic, the present results agree very well with those obtained by using the integral equation method. Furthermore, the method can be extended readily for general cases with arbitrary geometrical and boundary loading conditions and material properties.     

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.     

Eigenfunction expansion variational method for the solution of a cusp crack problem in a finite plate

Summary. In this paper, the EEF (eigenfunction expansion form) for the cusp crack in a finite plate is obtained, and the EEVM (eigenfunction expansion variational method) is used to solve the cusp crack problem in a finite plate. Each term in the EEF satisfies the governing equation of elasticity and the traction free condition along the cusp crack. As a result of using EEVM, the final solution for complex potentials is obtainable. It is found that the slenderness of the cusp crack has a significant influence to the SIF (stress intensity factor) at the crack tip. Particular attention is paid to a compression loading applied in the direction of the cusp crack axis. This can make an explanation for the rupture of rock with cusp crack under compression. Finally, numerical examples with the calculated results are presented.     

Solution of two-dimensional elastic crack problems using a localized finite element method

Two-dimensional linear elastic fracture mechanics analysis of the opening-mode crack problem is carried out, in order to use a localized finite element method. The stress distribution near the crack-tip is stated in the form of eigensolutions obtained by a classical separation variables technique.     

Solution of periodic group crack problems by using the Fredholm integral equation approach

Summary Periodic group cracks composed of infinitely many groups numbered from j = -∞,...-2,-1,0,1,2,...to j = ∞ placed periodically in an infinite plate are studied. The same loading condition and the same geometry are assumed for cracks in all groups. The Fredholm integral equation is formulated for the cracks of the central group (or the 0-th group) collecting the influences from the infinite neighboring groups. The influences from many neighboring groups on the central group are evaluated exactly, and those from remote groups approximately summed up into one term. The stress intensity factors can be directly evaluated from the solution of the Fredholm integral equation. Numerical examples show that the suggested technique provides very accurate results. Finally, several numerical examples are presented, and the interaction between the groups is addressed.     

An effective numerical method for an interfacial crack in a finite bi-material plate

In this paper an effective numerical method is presented for analyzing the stress intensity factors associated with the stress field near a partially debonded interface in a finite bi-material plate. The strees functions are assumed such that they can represent the stress singularity at the crack tips, satisfying not only the equilibrium equations in the domain, but also the stress and displacement conditions on the crack surfaces and across the interface. Therefore, only the boundary conditions of the plate need be considered, and they can be satisfied approximately by the Boundary Collocation Method. Numerical examples demonstrated that the proposed method gives satisfactory results and has many advantages compared to other methods.     

A variational boundary integral equation method for an elastodynamic antiplane crack

This paper investigates the transient wave scattering by a crack by means of the Boundary Integral Equation Method (BIEM). The author has developed a new formulation to solve the BIE for the Crack Opening Displacement (COD). The resolution is done directly in the time domain. The solution is represented by means of a retarded double layer potential, and the resulting BIE, with the COD as unknown, has a hypersingular kernel. The corresponding difficulty is overcome by using a variational method. We present the application of this method to an antiplane crack, describe the approximate problem and finally give some numerical results.