T-stress solutions for two-dimensional crack problems in anisotropic elasticity using the boundary element method

The importance of a two‐parameter approach in the fracture mechanics analysis of many cracked components is increasingly being recognized in engineering industry. In addition to the stress intensity factor, the T stress is the second parameter considered in fracture assessments. In this paper, the path‐independent mutual M‐integral method to evaluate the T stress is extended to treat plane, generally anisotropic cracked bodies. It is implemented into the boundary element method for two‐dimensional elasticity. Examples are presented to demonstrate the veracity of the formulations developed and its applicability. The numerical solutions obtained show that material anisotropy can have a significant effect on the T stress for a given cracked geometry.     

Boundary integral equation formulation for Interface cracks in anisotropic materials

We present a boundary integral formulation for anisotropic interface crack problems based on an exact Green's function. The fundamental displacement and traction solutions needed for the boundary integral equations are obtained from the Green's function. The traction-free boundary conditions on the crack faces are satisfied exactly with the Green's function so no discretization of the crack surfaces is necessary. The analytic forms of the interface crack displacement and stress fields are contained in the exact Green's function thereby offering advantage over modeling strategies for the crack. The Green's function contains both the inverse square root and oscillatory singularities associated with the elastic, anisotropic interface crack problem. The integral equations for a boundary element analysis are presented and an example problem given for interface cracking in a copper-nickel bimaterial.     

Unilateral buckling of thin elastic plates by the boundary integral equation method

The unilateral buckling of thin elastic plates, according to Kirchhoff's theory, is studied by using a boundary integral method. A representation for the second member of the equation is given. In the matrix formulatiea, boundary unknowns are eliminated; therefore, the unilateral buckling problem reduces to compute the eigenvalues and the eigenvectors of a matrix depending on the contact zone with the rigid foundation. An iterative process allows this zone and the buckling load to be computed. The capacities of the proposed method are illustrated by four examples.     

A new boundary integral equation method for cracked 2-D anisotropic bodies

By using integration by parts to the traditional boundary integral formulation, a traction boundary integral equation for cracked 2-D anisotropic bodies is derived. The new traction integral equation involves only singularity of order 1/r and no hypersingular term appears. The dislocation densities on the crack surface are introduced and the relations between stress intensity factors and dislocation densities near the crack tip are induced to calculate the stress intensity factors. The boundary element method based on the new equation is established and the singular interpolation functions are introduced to model the singularity of the dislocation density (in the order of ) for crack tip elements. The proposed method can be directly used for the 2-D anisotropic body containing cracks of arbitrary geometric shapes. Several numerical examples demonstrate the validity and accuracy of BEM based on the new boundary integral equation.     

A meshless local boundary integral equation method for two-dimensional steady elliptic problems

A novel meshless local boundary integral equation (LBIE) method is proposed for the numerical solution of two-dimensional steady elliptic problems, such as heat conduction, electrostatics or linear elasticity. The domain is discretized by a distribution of boundary and internal nodes. From this nodal points’ cloud a “background” mesh is created by a triangulation algorithm. A local form of the singular boundary integral equation of the conventional boundary elements method is adopted. Its local form is derived by considering a local domain of each node, comprising by the union of neighboring “background” triangles. Therefore, the boundary shape of this local domain is a polygonal closed line. A combination of interpolation schemes is taken into account. Interpolation of boundary unknown field variables is accomplished through boundary elements’ shape functions. On the other hand, the Radial Basis Point Interpolation Functions method is employed for interpolating the unknown interior fields. Essential boundary conditions are imposed directly due to the Kronecker delta-function property of the boundary elements’ interpolation functions. After the numerical evaluation of all boundary integrals, a banded stiffness matrix is constructed, as in the finite elements method. Several potential and elastostatic benchmark problems in two dimensions are solved numerically. The proposed meshless LBIE method is also compared with other numerical methods, in order to demonstrate its efficiency, accuracy and convergence.     

A boundary integral equation method in the frequency domain for cracks under transient loading

This paper concerns the fracture mechanics problem for elastic cracked materials under transient dynamic loading. The problem is solved by use of the boundary integral equations in the frequency domain, and the components of the solution are presented by the Fourier exponential series. The dynamic stress intensity factors are computed for different stress pulses and compared with those obtained for the case of the Heaviside loading.     

Material constant sensitivity boundary integral equation for anisotropic solids

In this paper, the material constant sensitivity boundary integral equation is presented, and its numerical solution proposed, based on boundary element techniques. The formulation deals with plane problems with general rectilinear anisotropy. Expressions for the computation of sensitivities for displacements, tractions, strains and stresses are derived, both for boundary and interior points. The sensitivities can be computed with respect to the bulk material properties or to the properties of part of the domain (inclusions, coatings, etc.). To assess the accuracy of the proposed approach, the computed results are compared to analytical ones derived from exact solutions obtained by complex potential theory, when possible, or finite difference derivatives otherwise. Copyright © 2005 John Wiley & Sons, Ltd.     

A boundary integral equation method for the two-dimensional diffusion equation subject to a non-local condition

A boundary integral equation method is proposed for the numerical solution of the two-dimensional diffusion equation subject to a non-local condition. The non-local condition is in the form of a double integral giving the specification of mass in a region which is a subset of the solution domain. A specific test problem is solved using the method.     

A new boundary integral equation method for analysis of cracked linear elastic bodies

Abstract

A novel integral equation method is developed in this paper for the analysis of two‐dimensional general anisotropic elastic bodies with cracks. In contrast to the conventional boundary integral methods based on reciprocal work theorem, the present method is derived from Stroh's formalism for anisotropic elasticity in conjunction with Cauchy's integral formula. The proposed boundary integral equations contain boundary displacement gradients and tractions on the non‐crack boundary and the dislocations on the crack lines. In cases where only the crack faces are subjected to tractions, the integrals on the non‐crack boundary are non‐singular. The boundary integral equations can be solved using Gaussian‐type integration formulas directly without dividing the boundary into discrete elements. Numerical examples of stress intensity factors are given to illustrate the effectiveness and accuracy of the present method.     

A hypersingular boundary integral equation for antiplane crack problems for a class of inhomogeneous anisotropic elastic materials

An antiplane multiple crack problem is considered for a class of inhomogeneous anisotropic elastic materials. The problem is reduced to a boundary integral equation involving hypersingular integrals. The boundary integral equation may be solved numerically using standard procedures. Some crack problems for a particular inhomogeneous material is considered in detail and the stress intensity factors are obtained in order to assess the effect of the anisotropy and inhomogeneity on the stress field near the crack tips.     

The boundary integral equation method for the solution of problems involving elastic slabs

Three boundary integral equations for the solution of an important class of elastic slab problems are considered. Some numerical examples are examined in order to illustrate the application of the integral equations to particular boundary-value problems.     

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.     

Boundary integral equation method for conductive cracks in two and three-dimensional transversely isotropic piezoelectric media

Using the fundamental solutions and the Somigliana identity of piezoelectric medium, the boundary integral equations are obtained for a conductive planar crack of arbitrary shape in three-dimensional transversely isotropic piezoelectric medium. The singular behaviors near the crack edge are studied by boundary integral equation approach, and the intensity factors are derived in terms of the displacement discontinuity and the electric displacement boundary value sum near the crack edge on crack faces. The boundary integral equations for two dimensional crack problems are deduced as a special case of infinite strip planar crack. Based on the analogy of the obtained boundary integral equations and those for cracks in conventional isotropic elastic material and for contact problem of half-space under the action of a rigid punch, an analysis method is proposed. As an example, the solution to conductive Griffith crack is derived.     

A boundary integral equation method for radiation and scattering of elastic waves in three dimensions

A vector boundary integral equation (BIE) formulation and numerical solution procedure is presented for problems of three-dimensional elastic wave radiation and scattering from arbitrarily shaped obstacles. The formulation is explicitly in terms of surface traction and displacement, rather than wave potentials, and the BIE on which numerical work is based is written in a form entirely free of Cauchy principal value integrals. Indeed, the subsequent computational process, based on quadratic isoparametric boundary elements, renders all integrals free of singularities, so that ordinary Gaussian quadrature may be used. Numerical examples include scattering from spherical surfaces and radiation from a cube.     

Stress intensity factors for semi-elliptical surface cracks in pressurised cylinders using the boundary integral equation method

Three dimensional linear elastic fracture mechanics analyses of the problem of an internally pressurised thick-walled cylinder with a semi-elliptical surface crack are carried out using the Boundary Integral Equation method. The cases treated are for ratios of external to internal cylinder radii of 2 and 3, with maximum crack depths ranging from 20% to 80% of the wall thickness. For a typical crack, the predicted value of stress intensity factor decreases slightly along the crack front moving away from the point of deepest penetration, reaching a minimum before increasing rapidly as the free internal surface of the cylinder is approached. The results presented will be of considerable use in the prediction of residual or safe life of, for example, chemical reactor tubes known to be cracked to a certain depth.

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Résumé Une analyse tri-dimensionnelle élastique et linéaire en mécanique de rupture du problème d'un cylindre à paroi épaisse soumis à pression interne et comportant une fissure de surface semi-elliptique a été effectuée en utilisant la méthode d'équation intégrale aux limites. Les cas envisagés dans l'étude sont relatifs à des rapports des rayons de cylindre externe et interne de 2 et de 3 avec des profondeurs maximum de fissure s'étalant de 20% à 80% de l'épaisseur de paroi. Dans le cas d'une fissure typique, la valeur prédite pour le facteur d'intensité des contraintes décroit légèrement le long du front de la fissure lorsque l'on se meut du point de pénétration le plus profond, et passe par un minimum pour ensuite s'accroître rapidement lorsqu'on approche de la surface libre interne du cylindre. Les résultats présentés s'avèreront d'une aide considérable pour prédire la vie résiduelle ou la durée de service fiable, par exemple, de tubes de réacteur chimique dont on sait qu'ils sont fissurés sur une certaine profondeur.

     

Regularized boundary integral equations for two-dimensional crack problems in multi-field media

A systematic procedure is followed to develop a set of regularized boundary integral equations for modeling cracks in 2D linear multi-field media. The class of media treated is quite general and includes, as special cases, anisotropic elasticity, piezoelectricity and magnetoelectroelasticity. Of particular interest is the development of a pair of weakly-singular, weak-form integral equations for ‘generalized displacement’ and ‘generalized stress’; these serve as the basis for a weakly-singular symmetric Galerkin boundary element method.     

Stress intensity factor K and the elastic T-stress for corner cracks

The stress intensity factor K and the elastic T-stress for corner cracks have been determined using domain integral and interaction integral techniques. Both quarter-circular and tunnelled corner cracks have been considered. The results show that the stress intensity factor K maintains a minimum value at the mid-plane where the T-stress reaches its maximum, though negative, value in all cases. For quarter-circular corner cracks, the K solution agrees very well with Pickard's (1986) solution. Rapid loss of crack-front constraint near the free surfaces seems to be more evident as the crack grows deeper, although variation of the T-stress at the mid-plane remains small. Both K and T solutions are very sensitive to the crack front shape and crack tunnelling can substantially modify the K and T solutions. Values of the stress intensity factor K are raised along the crack front due to crack tunnelling, particularly for deep cracks. On the other hand, the difference in the T-stress near the free surfaces and at the mid-plane increases significantly with the increase of crack tunnelling. These results seem to be able to explain the well-observed experimental phenomena, such as the discrepancies of fatigue crack growth rate between CN (corner notch) and CT (compact tension) test pieces, and crack tunnelling in CN specimens under predominantly sustained load.     

Kinked cracks in bonded half-planes modeled by an integral equation method

Based on the integral equation for resultant forces along a crack, a boundary integral equation method for the solution of kinked cracks in bonded half-planes is presented. The equation only contains a weak logarithmic singularity and is valid for every point along the crack lines. Numerical results are presented to illustrate the efficiency and reliability of the method.     

Simply supported plates by the boundary integral equation method

Plates governed by Kirchhoff's equation have been analysed by the boundary integral equation method using the fundamental solution of the biharmonic equation. In the case of supported plates, the boundary conditions permit the uncoupling of the field equation into two harmonic equations that originate, due to the nature of the fundamental solution, easier integration kernels and a simpler system of equations. The calculation of bending and twisting moments and transverse shear force can be formed, combining derivatives of the integral equation which defines the expression of the deflection on any point of the plate. The uncoupling of the biharmonic equation into two Poisson's equations involves the discretization of the domain of the studied problems. Nevertheless, the unknown quantity of the problem does not appear in the domain integrations for which a refined discretization is unnecessary. In the paper, however, a numerical alternative is considered to express the domain integral by means of boundary integrals. In this way, we need only discretize the boundary of the plate, making it necessary to solve a supplementary system of equations in order to calculate the coefficients of the approximation carried out.