A hypersingular Green's function generation for fracture mechanics problems (original) (raw)
Related papers
A numerical green's function approach for boundary elements applied to fracture mechanics
International Journal for Numerical Methods in Engineering, 1995
The 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.
Green's function: a numerical generation for fracture mechanics problems via boundary elements
Computer Methods in Applied Mechanics and Engineering, 2000
The paper discusses the application of the hyper-singular boundary integral equation to obtain Green's function solution to general geometry fracture mechanics problems, such as curved multifracture crack simulation, static and harmonic (extended to transient dynamic through inverse numerical transforms), in 2D and 3D. The numerical Green's function (NGF) can be implemented into a boundary element computer program, as the fundamental solution, to produce an accurate and ecient boundary element procedure. The complete formulation is presented in a uni®ed manner, generalizing previous problem oriented procedures proposed by the authors. The results to some typical linear fracture mechanics problems are presented.
Hypersingular integrals in boundary element fracture analysis
International Journal for Numerical Methods in Engineering, 1990
A new general purpose boundary element method for domains with cracks has been recently developed. This technique avoids the use of a multi-domain decomposition by including an additional integral equation expressing the boundary condition on the crack. The principal requirement of this technique is the analytic determination of certain hypersingular integrals of the Green's function which arise from this equation. In order to establish the applicability of this method for fracture, these integrals are evaluated herein for the Kelvin solution of the three-dimensional Navier equations of linear elasticity. Numerical results for fracture problems using the single-domain boundary element analysis are also presented.
2007
Abstract The Green's functions for the line force and dislocation that satisfy the traction free boundary condition on the surfaces of arbitrary multiple straight cracks in the isotropic solids are obtained. We develop the hybrid Green's functions combining the analytical and numerical Green's function methods. The Green's function is split into the singular and the image terms. The crack opening displacement, represented as the continuous distribution of dislocation dipoles over each crack, serves as the source of the image term.
Numerical Green's functions for some electroelastic crack problems
Engineering Analysis With Boundary Elements, 2009
A plane electroelastic problem involving planar cracks in a piezoelectric body is considered. The deformation of the body is assumed to be independent of time and one of the Cartesian coordinates. The cracks are traction free and are electrically either permeable or impermeable. Numerical Green's functions which satisfy the boundary conditions on the cracks are derived using the hypersingular integral approach and applied to obtain a boundary integral solution for the electroelastic crack problem considered here. As the conditions on the cracks are built into the Green's functions, the boundary integral solution does not contain integrals over the cracks. It is used to derive a boundary element procedure for computing the crack tip stress and electrical displacement intensity factors.
A numerical Green's function for multiple cracks in anisotropic bodies
Journal of Engineering Mathematics, 2004
The numerical construction of a Green's function for multiple interacting planar cracks in an anisotropic elastic space is considered. The numerical Green's function can be used to obtain a special boundary integral method for an important class of two-dimensional elastostatic problems involving planar cracks in an anisotropic body.
European Journal of Computational Mechanics
In the traditional boundary element methods, the numerical modelling of cracks is usually carried out by means of a hypersingular fundamental solution, which involves a 1=r 2 kernel for two-dimensional problems. A more natural procedure should make use of fundamental solutions that represent the square root singularity of the gradient field around the crack tip (a Green's function). Such a representation has been already accomplished in a variationally based framework that also addresses a convenient means of evaluating results at internal points. This paper proposes a procedure for the numerical simulation of two-dimensional problems with a fundamental solution that can be in part or for the whole structure based on generalised Westergaard stress functions. Problems of general topology can be modelled, such as in the case of unbounded and multiply-connected domains. The formulation is naturally applicable to notches and generally curved cracks. It also provides an easy means of evaluating stress intensity factors, when particularly applied to fracture mechanics. The main features of the theory are briefly presented in the paper, together with several validating examples and some convergence assessments.
Hypersingular quarter-point boundary elements for crack problems
International Journal for Numerical Methods in Engineering, 1995
The present paper deals with the study and effective implementation for Stress Intensity Factor computation of a mixed boundary element approach based on the standard displacement integral equation and the hypersingular traction integral equation. Expressions for the evaluation of the hypersingular integrals along general curved quadratic line elements are presented. The integration is carried out by transformation of the hypersingular integrals into regular integrals, which are evaluated by standard quadratures, and simple singular integrals, which are integrated analytically. The generality of the method allows for the modelling of curved cracks and the use of straight line quarter-point elements. The Stress Intensity Factors can be computed very accurately from the Crack Opening Displacement at collocation points extremely close to the crack tip. Several examples with different crack geometries are analyzed. The computed results show that the proposed approach for Stress Intensity Factors evaluation is simple, produces very accurate solutions and has little dependence on the size of the elements near the crack tip.
Non-hypersingular boundary integral equations for 3-D non-planar crack dynamics
Computational Mechanics, 2000
We derive a non-hypersingular boundary integral equation, in a fully explicit form, for the time-domain analysis of the dynamics of a 3-D non-planar crack, located in an in®nite homogeneous isotropic medium. The hypersingularities, existent in the more straightforward expression, are removed by way of a technique of regularization based on integration by parts. The variables are denoted in terms of a local Cartesian coordinate system, one of the axes of which is always held locally perpendicular to the potentially curved surface of the crack. Also given, in a fully explicit form, are the expressions for the off-fault stress and displacement ®eld, as well as the special form of the equations for the case in which the fault is planar.
Nonsingular BEM for fracture modeling
Computers & Structures, 1998
ÐA new three-dimensional formulation for the fracture mechanics problem based on a nonsingular form of the hypersingular Somigliana stress identity (SSI) is presented. The resulting formulation permits the use of low-order numerical integration algorithms and allows the crack opening displacement variable to be modeled in a piecewise-C 1,a manner which is well suited to standard boundary element method (BEM) algorithms. The method is not limited to particular crack shapes and is suciently general that it holds excellent prospects for providing a general class of numerical Green's functions for this important class of problems.