A numerical Green's function and dual reciprocity BEM method to solve elastodynamic crack problems

A computational model based on the numerical Green's function (NGF) and the dual reciprocity boundary element method (DR-BEM) is presented for the study of elastodynamic fracture mechanics problems. The numerical Green's function, corresponding to an embedded crack within the infinite medium, is introduced into a boundary element formulation, as the fundamental solution, to calculate the unknown external boundary displacements and tractions and in post-processing determine the crack opening displacements (COD). The domain inertial integral present in the elastodynamic equation is transformed into a boundary integral one by the use of the dual reciprocity technique. The dynamic stress intensity factors (SIF), computed through crack opening displacement values, are obtained for several numerical examples, indicating a good agreement with existing solutions.     

Upper bounds for the stress intensity factors along the boundaries of interacting coplanar cracks in three-dimensional elasticity

An elementary method for obtaining upper bounds for the stress intensity factors along the boundaries of interacting coplanar cracks inside an infinite isotropic elastic medium is presented. This method is based on the singular integral equation of the aforementioned elasticity problem and on the solutions of this equation for each particular crack problem, assumed known. The method is applied to the simple problem of interaction of two circular cracks, as well as to the similar problem of two cracks having the shape of a straight strip. The present results constitute a generalization of the corresponding method for crack problems in two-dimensional elasticity and can easily be further generalized to apply to more complicated crack problems in three-dimensional elasticity.     

Three-dimensional BEM for piezoelectric fracture analysis

A boundary element approach with quadratic isoparametric elements, quarter-point elements and singular quarter-point elements for three-dimensional crack problems in piezoelectric solids under mechanical and electrical loading conditions, is presented in this paper for the first time. The procedure is based on Deeg's fundamental solution for anisotropic piezoelectric materials, and the classical extended displacement boundary integral equation. Stress and electric displacement intensity factors are directly evaluated as system unknowns, and also as functions of the computed nodal displacements and electric potentials at crack faces. Special attention is paid to the fundamental solution evaluation. Several three-dimensional crack problems in transversely isotropic bodies under mechanical and electrical loading conditions are analysed. Numerical solutions computed for prismatic cracked 3D plate problems with a plane strain behaviour are in very good agreement with their corresponding 2D BE solutions. Results for a penny shape crack in a piezoelectric cylinder are presented for the first time. The proposed approach is shown to be a simple, robust and useful tool for stress and electric displacement intensity factors evaluation in piezoelectric media.     

Displacement discontinuity formulation for modeling cracks in orthotropic shear deformable plates

A displacement discontinuity formulation is presented for modeling cracks in orthotropic Reisnner plates. Fundamental solutions for displacement discontinuity are derived for the first time using a Fourier transform method. Boundary integral equations are presented in terms of discontinuity rotations on the crack surfaces for opening mode problems. As the fundamental solutions have singularity of O (1/r ²), Chebyshev polynomials of the second kind are used to evaluate the integral equations. By solving for coefficients of the Chebyshev polynomials, the stress intensity factors at the crack tips are obtained directly. Comparisons are made with solutions using the finite element method to demonstrate that the displacement discontinuity method is an efficient and accurate method for solving crack problems in orthotropic Reissner plates.     

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.     

On the concentrated loading of certain elastic half-space problems and related external crack problems. A new approach

Using a new integral equation connecting different types of fundamental solutions, a variety of external crack problems are solved when the crack is situated in an infinite space and occupies the region exterior to a circle, with the loading on the crack faces being of a concentrated nature. As a consequence of a particular integral representation we are also able to discuss certain “punch” problems for an elastic half-space when the punch is rigid and frictionless, with a circular cross section, but with an arbitrary base profile     

SOLVING TRANSIENT DYNAMIC CRACK PROBLEMS BY THE HYPERSINGULAR BOUNDARY ELEMENT METHOD

Abstract— The subject of hypersingular boundary integral equations is a rapidly developing topic due to the advantages which this kind of formulation offers compared to the standard boundary integral method. The hypersingular formulation is particularly well suited for fracture mechanics problems, where there are important gradients of the stress field and singularities. This formulation for time domain antiplane problems has been recently addressed by the authors and in the present paper, the formulation for time domain plane problems is presented and applied for the first time. A mixed Boundary Element approach based on the standard integral equation and the hypersingular integral equation is developed. The mixed formulation allows for a very simple discretization of the problem, where no subregion is needed. Conforming quadratic elements are used for the crack and the external boundaries. The hypersingular integral equation is used for collocation points within the crack elements, while the standard integral representation is used for the external boundaries. Several examples with different crack geometries are studied to illustrate the possibilities of the method. The Stress Intensity Factor (S.I.F.) is very accurately computed from the crack tip opening displacements along the crack tip element. The results show that the proposed approach for S.I.F. evaluation is simple and produces accurate solutions.     

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.     

Effect of curvature at the crack tip on the stress intensity factor for curved cracks

In this paper, the numerical solution of the hypersingular integral equation using the body force method in curved crack problems is presented. In the body force method, the stress fields induced by two kinds of standard set of force doublets are used as fundamental solutions. Then, the problem is formulated as a system of integral equations with the singularity of the form r ^–2. In the numerical calculation, two kinds of unknown functions are approximated by the products of the fundamental density functions and power series. The calculation shows that the present method gives rapidly converging numerical results for curved cracks under various geometrical conditions. In addition, a method of evaluation of the stress intensity factors for arbitrary shaped curved cracks is proposed using the approximate replacement to a simple straight crack.     

Frictional contact problems of kinked cracks modelled by a boundary integral method

A numerical method is presented for the solution of two dimensional crack problems including the effects of crack kinks and frictional contact between crack faces. The metod is based on an integral equation for the resultant forces along a crack. Coulomb friction between contacting crack surfaces is taken into account. The numerical implementation is demonstrated by considerations of surface and sub-surface piece-wise straight line cracks in a half-plane. Numerical results are presented to illustrate the efficiency and the reliability of the presented method.     

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

Introducing the mode II and mode III dislocation densities W ₂(y) and W ₃(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.     

A direct traction boundary integral equation method for three-dimension crack problems in infinite and finite domains

This paper presents a direct traction boundary integral equation method (TBIEM) for three-dimensional crack problems. The TBIEM is based on the traction boundary integral equation (TBIE). The TBIE is collocated on both the external boundary and one of the crack surfaces. The displacements and tractions are used as unknowns on the external boundary and the relative crack opening displacements (CODs) are introduced as unknowns on the crack surface. In our implementation, all the surfaces of the considered structure are discretized into discontinuous elements to satisfy the continuity requirement for the existence of finite-part integrals, and special crack-front elements are constructed to capture the crack-tip behavior. To calculate the finite-part integrals, an adaptive singular integral technique is proposed. The stress intensity factors (SIFs) are computed through a modified COD extrapolation method. Numerical examples of SIFs computation are presented to demonstrate the accuracy and efficiency of our method.     

Fracture analysis of magnetoelectroelastic solids by using path independent integrals

A solution scheme based on the fundamental solution for a generalized edge dislocation in an infinite magnetoelectroelastic solid is presented to analyze problems involving single, multiple and slowly growing impermeable cracks. The fundamental solution for a generalized dislocation is obtained by extending the complex potential function formulation used for anisotropic elasticity. The solution for a continuously distributed dislocation is derived by integrating the solution for an edge dislocation. The problem of a system of cracks subjected to remote mechanical, electric and magnetic loading is formulated in terms of set of singular integral equations by applying the principle of superposition and the solution for a continuously distributed dislocation. The singular integral equation system is solved by using a numerical integration technique based on Chebyshev polynomials. The J_i and M-integrals for single crack and multi-cracks problems are derived and their dependence on the coordinate system is investigated. Selected numerical results for the M-integral, total energy release rate and mechanical energy release rate are presented for single, double and multiple crack problems. The case of a slowly growing crack interacting with a stationary crack is also considered. It is found that M-integral presents a reliable and physically acceptable measure for assessment of fracture behaviour and damage of magnetoelectroelastic materials.     

Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media Part II: Numerical analysis

In this paper, elliptical cracks and rectangular cracks embedded in a three-dimensional infinite transversely isotropic piezoelectric solid are analyzed under combined mechanical tension and electric fields. The hypersingular integral equation method is used to solve the mentioned problems. The unknown function in the hypersingular integral equations is approximated with a product of the fundamental density function and polynomials. The hypersingular integrals can be numerically evaluated by using a method of Taylor series expansion. Therefore, the hypersingular integral equations for the crack problems can be solved immediately. Finally, numerical examples of the stress and electric displacement intensity factors as well as the energy release rates for these crack configurations are presented. The numerical results demonstrate the present approach to be very efficient.     

Domain element local integral equation method for potential problems in anisotropic and functionally graded materials

An efficient numerical method is proposed for 2-d potential problems in anisotropic media with continuously variable material coefficients. The method is based on the local integral relationships (integral form of balance equation and/or integral equations utilizing fundamental solutions) and consistent approximation of field variable using standard domain-type elements. The accuracy and convergence of the proposed method is tested by several examples and compared with benchmark analytical solutions.     

Numerical solutions of hypersingular integral equation for curved cracks in circular regions

In this paper, numerical solutions of a hypersingular integral equation for curved cracks in circular regions are presented. The boundary of the circular regions is assumed to be traction free or fixed. The suggested complex potential is composed of two parts, the principle part and the complementary part. The principle part can model the property of a curved crack in an infinite plate. For the case of the traction free boundary, the complementary part can compensate the traction on the circular boundary caused by the principle part. Physically, the proposed idea is similar to the image method in electrostatics. By using the crack opening displacement (COD) as the unknown function and traction as right hand term in the equation, a hypersingular integral equation for the curved crack problems in the circular regions is obtained. The equation is solved by using the curve length coordinate method. In order to prove that the suggested method can be used to solve more complicated cases of the curved cracks, several numerical examples are given.     

A general integral equation formulatin for homogeneous orthotropic potential problems

In this paper an integral equation formulation is proposed for the analysis of orthotropic potential problems. The two primary integral equations of the method are derived from the original governing differential equation firstly by rewriting it in a slightly different form and then applying the direct boundary element method formulation. The solution procedure is based on the use of the fundamental solutions for the isotropic potential case and special attention is given to the differentiation of a singular integral which yields an additional term as well as to the evaluation of the resulting Cauchy principal value integral. A simple discretization for the boundary and its interior domain is adopted in order to express the primary integral equations of the method in matrix form. Three examples are presented, the results of which illustrate the satisfactory accuracy of the method. The main feature of the proposed formulation is its generality, which makes possible its direct extension to solve such as heat conduction or subsurface flow in anisotropic media and, foremost, to orthotropic and anisotropic elasticity or elastoplasticity.     

Dislocations and internal loading in a semi-infinite elastic medium with surface stresses

This paper presents analytical solutions for shear and opening dislocations in an elastic half-plane with surface stresses by using the Gurtin–Murdoch continuum theory of elastic material surfaces. The fundamental solutions corresponding to buried vertical and horizontal loads are also presented. Fourier integral transforms are used in the analysis. It is found that a characteristic length parameter that depends on the surface and bulk elastic moduli exists for this class of problems, and it represents the influence of surface stresses on the bulk elastic field. Selected numerical results are presented to demonstrate the influence of surface stresses on the bulk stress field. The fundamental solutions presented in this study can be used to develop boundary integral equation and other methods to analyze complicated fracture and boundary-value problems associated with nano-scale structures and soft elastic solids.     

Numerical solutions of the singular integral equations in the crack analysis using the body force method

In this paper, numerical solutions of the singular integral equations of the body force method in the crack problems are discussed. The stress fields induced by two kinds of displacement discontinuity are used as fundamental solutions. Then, the problem is formulated as a hypersingular integral equation with the singularity of the form r ². In the numerical calculation, two kinds of unknown functions are approximated by the products of the fundamental density function and the Chebyshev polynomials. As examples, the stress intensity factors of the oblique edge crack, kinked crack, branched crack and zig-zag crack are analyzed. The calculation shows that the present method gives accurate results even for the extremely oblique edge crack and kinked crack with extremely short bend which has been difficult to analyze by the previous method using the approximation by the products of the fundamental density function and the stepped functions etc.     

Transient response of a piezoelectric material with a semi-infinite mode-III crack under impact loads

The problem of a semi-infinite impermeable mode-III crack in a piezoelectric material is considered under the action of impact loads. For the case when a pair of concentrated anti-plane impact loads and electric displacements are exerted symmetrically on the upper and lower surfaces of the crack, the asymptotic electroelastic field ahead of the crack tip is determined in explicit form. The dynamic intensity factors of electroelastic field and dynamic mechanical strain energy release rate are obtained. The obtained results can be taken as fundamental solutions, from which general results may directly be evaluated by integration. The method adopted is to reduce the mixed initial-boundary value problem, by using the Laplace and Fourier transforms, into two simultaneous dual integral equations. One may be converted into an Abel's integral equation and the other into a singular integral equation with Cauchy kernel, and the solutions of both equations can be determined in closed-form, respectively. For some particular cases, the present results reduce to the previous results.