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.     

An integral equation approach to eddy-current calculations

An integral-equation approach has been used to solve eddy current problems. The conducting material is represented by a network of current-carrying line elements. Consequently, Maxwell's field equations can be replaced by Kirchhoff's circuit rules. The loop equations for voltages, supplemented by the node equations for the currents, comprise a set of linear equations that can be solved repeatedly to give the time development of the eddy currents. Currents, magnetic fields, and power are calculated at each step. For a two-dimensional geometry, either thin plates or infinite cylinders can be calculated. Rectangular and circular cross sections have been calculated with good agreement to analytical expressions. Thin curved shells have also been calculated.     

Axisymmetric formulation for boundary integral equation methods in scalar potential problems

Axisymmetric geometries often appear in electromagnetic device studies. The authors present an original formulation for Boundary Integral Equation methods in scalar potential problems. This technique requires only 2D boundary in the r-z plane and evaluation of the equations only on those boundaries.     

Complex variable formulations for usual and hypersingular integral equations for potential problems — with applications to corners and cracks

This paper first presents an unified discussion of real and complex boundary integral equations (BIEs) for two-dimensional potential problems. Relationships between real and complex formulations, for both usual and hypersingular BIEs, are discussed. Potential problems in bounded as well as in unbounded domains are of concern in this work. Quantities of particular interest are derivatives of the primary field that exhibit discontinuities across corners, as well as stress intensity factors at the tips of mode III cracks. The latter problem in an application of a recent generalization of the well-known Plemelj-Sokhotsky formulae. Numerical implementations and results for interior problems in bounded domains, as well as for crack problems in unbounded domains, are presented and discussed.C. Y. Hui is supported by the Material Science Center at Cornell University, which is funded by the National Science Foundation (DMR-MRL program). S Mukherjee acknowledges partial support from NSF grant number ECS-9321508 to Cornell University.     

On singular integral equations for kinked cracks

Chebyshev polynomial techniques for solution of singular integral equations leading to square root singularities at the ends of the interval of integration are studied. It is shown that the results are less accurate when a singularity, albeit a weak one, appears between the interval ends. Typical examples are problems involving kinked cracks. Some attempts to improve the accuracy are discussed.

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Résumé On considère l'application de la technique polynominale de Chebyshev pour solutionner les intégrales singulières conduisant à des singularités d'ordre % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaamaalyaabaGaaG% ymaaqaaiaaikdaaaaaaa!38EB!\[{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\] aux extrémités de l'intervale d'intégration. On montre que les résultats sont moins précis lorsqu'une singularité, même faible, apparait entre les extrémités de l'intervale. Les problèmes comportant des fissures tortueuses constituent des exemples types d'application. On discute de diverses tentatives pour améliorer la précision des résultats.

     

The volume integral method of eddy-current modeling: Verification

The volume integral method of eddy-current modeling represents a flaw in metal as a set of electric dipoles located within volume elements or cells defining the flaw volume. Given this dipole distribution, impedance changes may be computed. The electric field of the dipole distribution is determined by an integral equation relating, by means of the electric field Green's tensor, the electric field due to the source to the total electric field in the flaw. The integral equation is solved by assuming that the total electric field is constant in each volume element, resulting in a matrix equation. The method has been programmed for use on a microcomputer. The method and computer program are verified using the analytical solution for a small spherical flaw and three sets of measured impedance data, measured by air-core coils along profiles overlying both surface-breaking and buried simulated flaws of known dimensions. Operating frequencies ranged between 900 and 4000 Hz. Generally agreement is good at lower frequencies ( 1000 Hz). At higher frequencies ( 4000 Hz), the agreement is not as good. This is thought to be due to the inability of the constant electric field approximation to model the steep electric field gradients present in the host metal at high frequency. The results are also sensitive to the method of computation of the electric field due to the source. Some improvements can and should be made to the method.     

Solution of integral equation for almost circular cracks

The article deals with the two-dimensional integral equation of the first kind to which spatial problems of the theory of cracks can be reduced. The domain of integration in such an equation differs little from a circle. The article suggests a method of solving this equation based on the use of the Fréchet derivative of some nonlinear operator and the variational formula. Examples of the application of this method to problems of the theory of cracks are presented. In particular an inversion formula of the theory of cracks is obtained when the boundary contour of the crack (situated in an infinite elastic body) is almost circular.Translated from Problemy Prochnosti, No. 4, pp. 50–56, April, 1993.     

Numerical models of volumetric insulating cracks in eddy-current testing with experimental validation

This paper concerns fast electromagnetic modeling of volumetric cracks in conductive materials under eddy-current inspection. The underlying numerical method is described. The model is tested on cracks in aluminum structures employed in aeronautical manufacture. The computational results obtained with the method display satisfactory agreement with the respective experimental and numerical results obtained by representing cracks as nonconductive surfaces.     

From a boundary integral formulation to a vortex method for viscous flows

For the numerical solution of flow problems past a solid body it is worth to consider boundary integral techniques for their inherent capability to manage efficiently the far-field boundary conditions as well as the approximation of the solid body contour. However, for the analysis of large Reynolds number flows, of major interest in the applications, several computational difficulties appear when using the integral representation for the velocity or for the vorticity field in its classical form with interpolating functions (BEM). In particular, the evaluation of the volume integrals is a serious drawback while the steepness of their kernel introduces artificial diffusion in the calculation. To satisfy the opposite requirements of the advective and of the diffusive part of the Navier-Stokes equations, we adopt an operator splitting scheme according to the Chorin-Marsden product formula (Chorin et al. 1978), together with a proper vorticity generation scheme at the solid boundary. A solution procedure based on the approximation of the vorticity field by a finite number of point vortices (PVM) follows as a natural evolution of the boundary integral formulation.The numerical results given by the two methods for the merging of two like-signed vortices in free space reveal the excessive numerical diffusion of BEM. The better accuracy of PVM is also established through the evaluation of some first integrals of motion. Several results are also reported for flows in presence of solid boundaries where the vorticity generation is crucial. In this case accurate solutions are only obtained with PVM, while BEM is even less satisfactory than in free space. Finally, the proposed vortex-like method (PVM) is tested on the classical problem of the wake behind a cylinder, in comparison with other well established techniques.This work was partly supported by the Italian Ministry for Scientific Research through a MURST grant and by C.N.R. through Progetto Finalizzato Trasporti II.     

J integral applications for short fatigue cracks at notches

Elastic and elastic-plastic fracture mechanics solutions are modified to predict the behaviour of short cracks. An effective crack length, l ₀ is introduced into the solutions for both the linear elastic stress intensity factor and the J integral. Crack growth results for short cracks, in both elastic and plastic strain fields of unnotched specimens, when interpreted in terms of the modified solutions, show excellent agreement with elastic long crack data. The modified J integral solutions are extended to plastically strained notches, and the solutions obtained are tested in the correlation of data for growth of sort cracks near notches of varying severity with data for long crack under elastic loading. Although constant stress amplitude tests of these notches gave crack growth rate versus crack length curves which varied from monotonically increasing for blunt notches, to an initial decrease followed by an increase of sharp notches, all the data fell within the long crack data when correlated by the J integral solutions. Conversely, these solutions can be used to predict elastic and inelastic short crack growth curves for notches of various severities.

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Résumé On a modifié les solutions de mécanique de rupture élastique et élastoplastique afin de prédire le comportement de fissures courtes. On a introduit une longueur effective de fissure l ₀ dans les solutions donuant le facteur d'intensité de contrainte linéaire élastique et l'intégrale J. Les résultats de croissance de fissure dans le cas de fissures courtes dans des éprouvettes non entaillées soumises à des champs de déformation élastique ou plastique, font état d'un excellent accord avec les données afférant à des fissures longues en condition élastique, lorsqu'ils sont interprétés sous forme de solutions modifiées. Les solutions des intégrales J sont extrapolées aux cas des entailles sollicitées dans le domaine plastique, et les solutions obtenues sont éprouvées dans une corrélation des données de croissance de fissures courtes au voisinage d'une entaille de sévérités diverses, avec les données de croissance de fissures longues sous mise en charge élastique.Les essais à amplitude de contrainte constante sur ces entailles conduisent à une vitesse de croissance qui, en fonction de la longuer de fissure, varie d'un accroissement régulier dans le cas d'entailles arrondies, à une diminution suivie d'un accroissement, dans le cas d'entailles aiguës. Ce nonobstant, toutes les données se sont révélées similaires aux données pour de longues fissures, lorsque l'on établit la corrélation des solutions des intégrales J.Complémentairement, ces solutions peuvent être utilisées pour prédire les courbes de croissance des fissures courtes élastique et inélastique, dans le cas d'entailles de sévérités différentes.

     

Discrete constitutive equations in A-/spl chi/ geometric eddy-current formulation

Using a geometric formulation for eddy currents, we present a geometric approach to constructing approximations of the discrete magnetic and Ohm's constitutive matrices. In the case of Ohm's matrix, we also show how to make it symmetric. We compared the impact on the solution of the proposed Ohm's matrices, and an iterative technique to obtain a consistent right-hand-side term in the final system is described.     

Boundary integral equation formulation for the generalized micro-polar thermoviscoelasticity

A formulation of the boundary integral equation method for generalized linear micro-polar thermoviscoelasticity is given. Fundamental solutions, in Laplace transform domain, of the corresponding differential equations are obtained. The initial, mixed boundary value problem is considered as an example illustrating the BIE formulation. The results are applicable to the generalized thermoelasticity theories: Lord-Shulman with one relaxation time, Green-Lindsay with two relaxation times, Green-Naghdi theories, and Chandrasekharaiah and Tzou with dual-phase lag, as well as to the dynamic coupled theory. The cases of generalized linear micro-polar thermoviscoelasticity of Kelvin-Voigt model, generalized linear thermoviscoelasticity and generalized thermoelasticity can be obtained from the given results.     

A symmetric boundary integral formulation for cohesive interface problems

An incremental symmetric boundary integral formulation for the problem of many domains connected by non-linear cohesive interfaces is here presented. The problem of domains with traction-free cracks and/or rigid connections are particular instances of the proposed cohesive formulation. The numerical approximation of the considered problem is achieved by the symmetric Galerkin boundary element method.     

A geometric integral formulation for eddy‐currents

In the computational electromagnetism community it is known how the differential formulation of an eddy‐current problem, can be translated into a finite dimensional system of equations involving circulations and fluxes, by means of the so‐called Discrete Geometric Approach. This is done by exploiting the geometric structure behind Maxwell's equations. In this paper, we will show how the same Discrete Geometric Approach can be profitably used also to discretize an eddy‐current problem formulated in an integral way. We rely on a purely geometric definition of a novel set of face vector basis functions that we use to construct the discrete counterparts—matrices—of both the Ohm's constitutive relation and of the integral relation between the magnetic vector potential and the eddy‐current density vector. The symmetry and positive‐definiteness of such matrices will be demonstrated and their geometric structure will be apparent. Copyright © 2010 John Wiley & Sons, Ltd.     

An integral formulation applied to the diffusion and Boussinesq equations

A new integral method is proposed here to solve the diffusion equation (confined flow) and the Boussinesq equation (unconfined flow) in a two-dimensional porous medium. The method, based on Green's theorem, derives its integral representation from the portion of the original differential equation with the highest space derivatives so that the resulting kernel of the integral representation is not time dependent. Compared to an earlier integral formulation, namely the direct Green function, based on the same theorem, the kernel is simpler so that the present theory provides a more efficient numerical model without compromising accuracy. An iterative scheme is employed along with the theory to achieve solutions to the non-linear Boussinesq equation. Concepts used in the finite difference and finite element methods enable simplification of the temporal derivative. The method is tested with success on a number of numerical examples from groundwater flow.     

An alternative integral equation approach for curved and kinked cracks

An integral-equation representation of cracks is presented which differs from the well-known dislocation-layer representation in that the equations are written in terms of the displacement-discontinuity across the crack surfaces rather than derivatives of the displacement-discontinuity. The advantages of such a representation is that, unlike the dislocation layer, the displacement discontinuity is not singular at crack tips and kinks. The method is demonstrated for two-dimensional infinite domains.

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Résumé On propose une représentation de fissures par une équation intégrale qui diffère de la représentation bien connue des couches de dislocation en ce que les équations constitutives sont écrites en termes de discontinuité de déplacement au travers de la section de la fissure, plutôt que dérivées de ces discontinuités de déplacement. L'avantage de cette représentation est que, à la différence de la couche de dislocation, le discontinuité de déplacement n'est pas singulière à l'extrémité ou aux branchements de la fissure. La méthode est soumise à démonstration pour des domaines infinis à deux dimensions.

     

Scalar potential model of eddy-current interactions with three-dimensional flaws

Magnetic scalar potential theory is applied to a model of eddy-current detection of a surface-breaking flaw in a conductor. A general boundary integral equation for the potential is derived first in a form suitable for numerical solution. The problem is then specialized to a flaw in a perfectly conducting half-space. A solution algorithm based on the boundary element method is outlined and demonstrated by application to a three-dimensional rectangular slot. Methods for accounting for the effects of nonvanishing skin depth are discussed.     

Magnetic vector potential based model for eddy-current loss calculation in round-wire planar windings

An analytical development of the magnetic vector potential is used to investigate the losses in round-wire planar windings. The current distribution in wires is affected by the skin effect and the field created by adjacent wires (proximity effect). This field depends on the current distribution in conductors, resulting in a closed-form problem. In this paper, we obtain the vector potential outside a conductor to estimate the effect of induced currents in the field shape over the neighboring conductors. We use the results to calculate the losses in planar windings such as those in domestic induction heaters. We obtain an equivalent resistance representing the losses in windings and compare it with measurements. This solution provides an accurate analytical approach to modeling the losses in close-packed windings.     

Eigenstrain formulation of boundary integral equations for modeling particle-reinforced composites

A novel computational model is presented using the eigenstrain formulation of the boundary integral equations for modeling the particle-reinforced composites. The model and the solution procedure are both resulted intimately from the concepts of the equivalent inclusion of Eshelby with eigenstrains to be determined in an iterative way for each inhomogeneity embedded in the matrix. The eigenstrains of inhomogeneity are determined with the aid of the Eshelby tensors, which can be readily obtained beforehand through either analytical or numerical means. The solution scale of the inhomogeneity problem with the present model is greatly reduced since the unknowns appear only on the boundary of the solution domain. The overall elastic properties are solved using the newly developed boundary point method for particle-reinforced inhomogeneous materials over a representative volume element with the present model. The effects of a variety of factors related to inhomogeneities on the overall properties of composites as well as on the convergence behaviors of the algorithm are studied numerically including the properties and shapes and orientations and distributions and the total number of particles, showing the validity and the effectiveness of the proposed computational model.     

Direct second kind boundary integral formulation for Stokes flow problems

A direct boundary element method is formulated for the Stokes flow problem based on an integral equation representation for the components of traction. For problems in which the components of velocity are prescribed on the boundary of the domain, this new formulation results in a hypersingular Fredholm integral equation of the second kind. A method of regularization to evaluate the hypersingular integral is discussed. For certain problems involving flows about particles, the integral equation representation for the tractions is not unique because of the existence of rigid body eigenmodes. A method to constrain out these rigid body modes is also discussed. Several example problems are considered in which this new formulation is compared to more traditional boundary element formulations.