Solution Of Crack Problems Research Articles

Strain gradient elasticity theory for antiplane shear cracks. Part II: Monotonic displacements

Solutions of crack problems for a strain gradient elastic theory whose crack opening displacements are monotonic rather than oscillatory in profile are examined in this (portion of a two-part) paper. A new analytical solution having a simple form is also obtained through the introduction of parabolic cylindrical coordinates and expansion of the imposed displacement along the crack surfaces in terms of orthogonal polynomials. Uniqueness of solution is also discussed, as well as finite-difference approximations of the governing equations for use in numerical solutions of boundary value problems.

Exact Representations on Artificial Interfaces and Applications in Mechanics

In various areas of applied mechanics, there are instances where it is necessary or beneficial to represent the behavior of a mechanical system on an artificial boundary, or artificial interface, which is introduced into the system. Examples include, among others, the computational treatment of mechanical problems in infinite media, the solution of crack problems in fracture mechanics, the dynamical analysis of a mechanical system attached to a number of smaller subsystems, iterative domain decomposition methods, and the mathematical formulation of inverse problems. The representation of the solution on the interface may be approximate or exact. This article is concerned with exact representations. It explains the benefit in using such representations, compares them to approximate representations in various respects, surveys work that has been done in this field, and highlights applications in applied mechanics. It is the author’s opinion that despite the fact that approximate interface representations are more popular than exact ones, the latter have definite advantages in many situations. References cited in this review article number 163.

“Solution of crack problems — the distributed dislocation technique” by D.A. Hills, P.A. Kelly, D.N. Dai and A M. Korsunsky

<b>“Solution of crack problems — the distributed dislocation technique”</b> by D.A. Hills, P.A. Kelly, D.N. Dai and A M. Korsunsky

A new thermoelectroelastic solution for piezoelectric materials with various openings

A new solution is obtained for thermoelectroelastic analysis of an insulated hole of various shapes embedded in an infinite piezoelectric plate. Based on the exact electric boundary conditions on the hole boundnary, Lekhnitskii's formulation and conformal mapping, the solution for elastic and electric fields has been obtained in closed form in terms of complex potential. The solution has a simple unified form for various holes such as ellipse, circle, triangle and square openings. As an application of the solution, the hoop stress and electric displacement (SED) and the solution for crack problems are discussed. Using the above results, the SED intensity factor and strain energy release rate can be obtained analytically. One numerical example is considered to illustrate the application of the proposed formulation and compared with those obtained from impermeable model.

An exact solution of crack problems in piezoelectric materials

An assumption that the normal component of the electric displacement on crack faces is thought of as being zero is widely used in analyzing the fracture mechanics of piezoelectric materials. However, it is shown from the available experiments that the above assumption will lead to erroneous results. In this paper, the two-dimensional problem of a piezoelectric material with a crack is studied based on the exact electric boundary condition on the crack faces. Stroh formalism is used to obtain the closed-form solutions when the material is subjected to uniform loads at infinity. It is shown from these solutions that: (i) the stress intensity factor is the same as that of isotropic material, while the intensity factor of the electric displacement depends on both material properties and the mechanical loads, but not on the electric load. (ii) the energy release rate in a piezoelectric material is larger than that in a pure elastic-anisotropic material, i. e., it is always positive, and independent of the electric loads. (iii) the field solutions in a piezoelectric material are not related to the dielectric constant of air or vacuum inside the crack.

Some applications of singular fields in the solution of crack problems

This paper reviews some recent developments in superposition methods for calculating linear elastic stress intensity factors and eigenvalues for cracks and notches, presents some new results for pairs of edge cracks and provides new insights into the nature of the errors in these processes. The procedure requires a numerical solution to the full cracked problem and a second solution on the same mesh using the known form of the singularity in an infinite region. This is equivalent to the well-known Subtraction of Singularity (SST) method. The advantages of this procedure over conventional SST are: (1) no modifications need to be made to a standard computer program; (2) multiple crack tips may be analysed without the difficulty of unknown rigid body displacements at the crack tips; (3) solutions with different boundary conditions on the same mesh may be obtained simply in one step by re-using one singular field solution; The singular crack tip field may also be studied independently leading to estimates of the eigenvalues and some insight into mesh-induced errors. The additional computational cost of a two-step procedure is minimal since the solution matrix from step one may be re-used with a new right-hand side. Numerical experiments using the boundary element method demonstrate the accuracy and simplicity of the superposition approach for notches, simple cracks, mixed-mode cracks, two edge cracks of different lengths and eigenvalues under various boundary conditions. © 1998 John Wiley & Sons, Ltd.

Solution of Crack Problems: The Distributed Dislocation Technique

Preface. 1. Introduction to Fracture Mechanics. 2. Distributed Dislocation Fundamentals. 3. Further Topics in Plane Crack Problems. 4. Interface Cracks. 5. Solution of Axi-Symmetric Crack Problems. 6. Three-Dimensional Cracks: An Introduction. 7. Three-Dimensional Cracks: Further Concepts. 8. Concluding Remarks. A: Dislocation Influence Functions. B: Numerical Solution of SIEs with Cauchy Kernel. C: Plane and Ring Dipole Influence Functions. D: Contour Integral and Kernel Function. References. Index.

The Somigliana ring dislocation revisited

Solutions for Somigliana ring dislocations in an elastic half space and in two perfectly bonded dissimilar half spaces, with the dislocation ring parallel to the interface, are obtained from the Papkovich potential solutions by using Aderogba's Theorem. Interaction between a dislocation and a free surface or a perfectly bonded interface is considered. Implications of the results for the solution of crack problems are discussed.

The Somigliana ring dislocation revisited

Solutions for Somigliana ring dislocations in an elastic half space and in two perfectly bonded dissimilar half spaces, with the dislocation ring parallel to the interface, are obtained from the Papkovich potential solutions by using Aderogba's Theorem. Interaction between a dislocation and a free surface or a perfectly bonded interface is considered. Implications of the results for the solution of crack problems are discussed.

The Solution of Crack Problems by Using Distributed Strain Nuclei

Crack problems may be solved by first establishing the stress field in the crack's absence and distributing strain nuclei along the line of the crack in order to render its faces traction-free. The relationship between the different possible forms of nucleus and the kinds of singular integral equation to which they lead are explored. The merits of each are then highlighted.

Fundamental eigenstrain solutions for axisymmetric crack problems

In this paper the fundamental eigenstrain solutions are derived for axisymmetric crack problems. The solutions are found in terms of Papkovich-Neuber potentials, which in turn are expressed using one function from the family of Lipschitz-Hankel integrals. In order to achieve the most concise form, two methods are used in the analysis: integration method for the axial opening eigenstrain ring and direct solution method for the radial opening eigenstrain ring and the ring of shear. The behaviour of the elastic stress fields in the vicinity of each type of eigenstrain ring is analysed. It is shown that the relevant component of stress exhibits a second order of singularity as the point of observation approaches the eigenstrain ring. It is also demonstrated that the ring curvature a −1 serves as the measure of the deviation of the stress field from the appropriate plane strain solution. Implications of the results for the solution of crack problems are discussed.

The stress field due to a dislocation in layered media

We consider the problem of an elastic layer perfectly bonded to an elastically dissimilar substrate, when an edge dislocation, of arbitrary orientation, is present either in the layer or in the substrate. The solution given includes the complete stress field, which is needed for the solution of crack problems, together with the Peach-Koehler force.

A superposition method for the solution of crack problems

AbstractWe discuss a two‐step superposition method for calculating linear elastic stress intensity factors. The procedure requires the solution to the full cracked problem and the solution to a problem on the same mesh assuming the singularity due to a crack tip in an infinite region. We show that this is equivalent to the well known subtraction of singularity method if the two solutions are characterized by crack tip stress. The advantages of our procedure are that no modifications need to be made to a standard computer program and that once one singular solution is available on a given cracked mesh, solutions with different boundary conditions on the same mesh may be obtained in one step without including any singular crack effects. The mesh required to represent the singular crack tip field may also be studied independently of the complete problem. The additional computational cost of a two‐step procedure is minimal since the solution matrix from step one may be reused with a new right‐hand side. Numerical experiments using the boundary element method demonstrate the high accuracy and simplicity of the superposition approach.

The solution of plane crack problems by dislocation dipole procedures

The solution of crack problems by employing dislocation loops (or dipoles) as strain nuclei is described. The relationship between this technique and dislocation density methods is outlined, the advantages of the former being illustrated by practical examples. The procedure also has the merit of being capable of extension to three-dimensional problems.

Weight Functions for Stress-Intensity Factors for Cracks Emanating From the Base of Edge Notches

Over the past two decades, numerous analytical solutions of crack problems have appeared in the open literature. A great number of these solutions are concerned with idealized crack geometries. Only a few of them involving the interaction of cracks with neighboring boundaries have been solved satisfactorily. One such problem of great practical interest is that of cracks emanating from the base of edge notches. A Green function for the problem of the edge crack was first given by Hartranft and Sih using the alternating method. The present analysis extends these results to apply to the case of edge cracks at notches.

Locating a crack of arbitrary but known shape by the method of path-independent integrals

The solution of a crack problem of an arbitrary, but known, shape inside an infinite plane isotropic elastic medium can be achieved in general by the method of complex singular integral equations and their numerical solution by using the Gauss- or the Lobatto-Chebyshev methods. In a few special cases, like straight or circular-arc-shaped cracks, this solution is available in closed form. In this paper we will not contribute to the above methods of solution of crack problems, but we will propose a method for the determination of the exact position of such a crack inside a closed contour in the elastic medium by gathering and using information along this contour only (by experimental techniques) and applying the method of complex path-independent integrals for the location of the crack. This paper constitutes a nontrivial generalization of relevant previous results by the author and it is of quite general applicability in fracture mechanics for nondestructive testing. Numerical results for the particular case of a straight crack are displayed for the illustration of the efficiency of the method. The generalization of the present results to the determination of additional geometric and loading parameters of the crack is also suggested very briefly and related numerical results, concerning the length of the crack and the pressure distribution on it, in the aforementioned numerical application are also presented.

On the elastic fields of interacting defense and main hole systems

In this study, a general solution describing the interaction between a main hole and an arbitrarily located defense hole, under uniaxial and biaxial loading conditions, was developed. The analysis was based upon the complex potentials of Muskhelishvili, an appropriate superposition procedure and the Laurent series expansion method adopted earlier by Isida [[1] Methods of Analysis and Solutions of Crack Problems (edited by G. C. Sih), pp. 56–130. Noordholf, Leydon (1973)] for the internal crack problem. In particular, the variation in the tangential stress and stress concentration factor (SCF) at the main hole were obtained in a general asymptotic form and explicit closed form expressions pertaining to the fourth order solution, using an appropriate perturbation technique, were developed. In addition to the explicit closed form expressions, higher order solutions were computed and used to provide a detailed description of the effect of the applied boundary loads and defense hole configurations upon SCF at main and defense holes. The present solution, which can be obtained to any desired degree of accuracy, is verified by comparison with existing ones and the results show excellent agreement. The work provides a useful quantitative design tool and a valuable insight into the effect of defense holes upon the stress distribution at a main hole in engineering structures.

Interface tilt walls, shear walls and cracks

The problem of the tilt (and the shear) boundary near or at an interface separating two elastic half spaces of differing elastic constants is related to the problem of the mode I and the mode II interface crack. The existing solutions for crack problems are based on the solution of the stress-displacement field of a discrete edge dislocation near an interface boundary that is obtained using discrete image dislocations, image line forces, image line pressure sources, etc. acting at the cores of the discrete image dislocations. "Peculiarities" arise in interface crack and interface tilt boundary solutions whose reality might be questioned because of use of unusually strong image stress singularities in generating these solutions. (The stress singularities are so strong that the elastic displacements produced by them, at the image point, are infinitely greater than the distance from the singularity to the interface.) The analysis of this paper is based on a solution of the edge dislocation which is obtained without use of image stress singularities other than that of image dislocations. The need of an image stress singularities is removed through the introduction of image infinitesimal dislocation distributions at the interface. It is concluded that the "peculiarities" of the solutions of interface type crack problems are real and are not artifacts unintentionally created from some (hitherto unrecognized) improper use of strong image stress singularities.

Construction of singular integral equations for interacting straight cracks by using reduce

The method of singular integral equations has been applied to the solution of crack problems in plane and antiplane elasticity many times during the last 20 years. Here we revisit the case of an arbitrary number of interacting straight cracks in plane elasticity and we illustrate the possibility of constructing (algebraically) the corresponding system of singular integral equations by using computer algebra software. We present a procedure (computer program) using REDUCE as well as several examples of application of the present approach, extensions and generalizations of which follow rather trivially.

強磁場下の構造材料の力学的挙動と電磁破壊力学

Design and development of superconducting structures require basic research on electromagnetic fracture mechanics. In the theory of brittle fracture in a strong magnetic field, usually we consider linear electromagnetic elastic solutions of crack problems. Examples including analysis of electromagnetic elastic crack problems are reviewed to show the effect of magnetic field on the mechanical behavior of cracked materials. Many electrically conducting materials are used in magnetic and electric current fields. First, the electromagnetic elastic problems of conducting cracked materials are analized to show the effect of electromagnetic force. The current flow is disturbed by the presence of the crack and the electromagnetic body force is caused by the interaction between the magnetic field and the disturbed current. If the electrically conducting material is used in the strong magnetic field, we must consider the effect of induced current. Secondly, scattering of time harmonic waves by a crack in an electrically conducting material under a magnetic field is analized to show this effect. If a ferromagnetic material is used in a magnetic field, we must consider the effect of induced magnetization. Lastly, the fracture and deformation of a soft ferromagnetic material in a magnetic field is investigated to show this effect.