Elastic Crack Problems Research Articles

The mixed mode crack problem in an inhomogeneous orthotropic medium

The mixed mode crack problem in plane elasticity for a graded and oriented material is considered. The material property grading is intentional, whereas the property orientation or orthotropy is usually the consequence of material processing. It is assumed that the crack is located in a plane perpendicular to the direction of property grading and the principal axes of orthotropy are parallel and perpendicular to the crack. The four independent engineering constants E11, E22, G12, and ν12 are replaced by a stiffness parameter, E = √E11 E22, a stiffness ratio, δ = (E11/E22)1/4, a Poisson's ratio, ν = √ν12 ν21, and a shear parameter κ0 = (E/2G12) - ν. The corresponding mixed boundary value problem is reduced to a system of integral equations which is solved for various loading conditions and material parameters. The results presented consist of the strain energy release rate, the stress intensity factors and the crack opening displacements. It is found that generally the stress intensity factors increase with increasing material inhomogeneity parameter and shear parameter and with decreasing stiffness ratio.

Stress Intensity Factors and Displacements in Elastic Contact and Crack Problems

This article is designed to be a step in creation of the “contact and crack calculus.” This term is introduced to denote a mathematical description of the infinitesimal variations in the elastic fields due to infinitesimal variation of domain of contact or crack, respectively. Rice has created foundations of this apparatus for cracks. Here, similar results were obtained for normal contact problems. It is also shown that the relationships between the stress intensity factors and displacements in tangential crack problems can be made a working apparatus in both directions.

Singularity Analysis and Boundary Integral Equation Method for Frictional Crack Problems in Two-Dimensional Elasticity

This study investigates the stress singularities in the neighborhood of the tip of a sliding crack with Coulomb-type frictional contact surfaces, and applies the boundary integral equation method to solve some frictional crack problems in plane elasticity. A universal approach to the determination of the complex order of stress singularity is established analytically by using the series expansion of the complex stress functions. When the cracks are open, or when no friction exists between the upper and lower crack faces, our results agree with those given by Williams. When displacement and traction are prescribed on the upper and lower crack surfaces (or vice versa), our result agrees with those by Muskhelishvili. For the case of a closed crack with frictional contact, the only nonzero stress intensity factor is that for pure shear or sliding mode. By using the boundary integral equation method, we derive analytically that the stress intensity factor due to the interaction of two colinear frictional cracks under far field biaxial compression can be expressed in terms of E(k) and K(k) (the complete elliptic integrals of the first and second kinds), where k=[1-(a/b)2]1/2 with 2a the distance between the two inner crack tips and b- a the length of the cracks. For the case of an infinite periodic colinear crack array under remote biaxial compression, the mode II stress intensity factor is found to be proportional to [2b tan(π a/2b)]1/2 where 2a and 2b are the crack length and period of the crack array.

Fredholm integral equation for the multiple circular arc crack problem in plane elasticity

An elementary solution for the multiple circular arc problem is obtained in this paper. The elementary solution is defined as a particular case of the single circular arc crack problem, in which remote stresses are equal to zero, and two pairs of concentrated forces are applied at a prescribed point of crack face. By using the principle of superposition, Fredholm integral equation for the multiple circular arc problem in plane elasticity is obtained. The suggested approach is illustrated by several numerical examples. If a smaller arc crack is surrounded by a larger arc crack, the stress intensity factors for the former become rather small. The phenomenon of shielding is illustrated by examples.

Computational modeling of elastic and plastic multiple cracks by the fundamental solutions

The fundamental solutions of elasticity are used to establish a numerical method for elastic and plastic multiple crack problems in two dimensions. The continuous distributions of the point forces, dislocations, and the plastic sources are used systematically to model the crack, non-crack boundary, and the plastic deformation. Use of these singularities are guided strictly by the physical interpretation of the problem. We adopt Muskhelishvili's complex variable formalism that facilitate the analytical evaluation of the integrals representing the continuous distributions of the singularities. The resulting numerical method is concise and accurate enough to be used for elastic and plastic multiple crack problems.

On the Definition of Crack Problems in Elasticity

On the Definition of Crack Problems in Elasticity

Two and Three-Dimentsional Crack Problems in Gradient Elasticity

Two and Three-Dimentsional Crack Problems in Gradient Elasticity

New integration scheme for the branch crack problem

The branch crack problem in plane elasticity is modeled by a reasonable distribution of the dislocation. A new integration scheme is proposed in the following manner. A point dislocation is placed at the branch point and the distributed dislocations are assumed along all the branches. Thus, the singular integral equation and the constraint equation can be formulated for the branch crack problem. A semi-open quadrature rule is used, which can ensure that the number of unknowns is equal to the number of equations. The results of the numerical solution directly relate to the stress intensity factors at the branch tip. Finally, several numerical examples are given.

Weight function for an internal crack in plane elasticity

In this paper a fundamental field which is composed of the singular displacement field and the complementary regular field is suggested. By the use of the Betti's reciprocal theorem to the fundamental field and the physical field, the weight function for an internal crack problem in plane elasticity is formulated. The formulation proposed is compact in form and easy to understand.

Weight function for an internal crack in plane elasticity

In this paper a fundamental field which is composed of the singular displacement field and the complementary regular field is suggested. By the use of the Betti's reciprocal theorem to the fundamental field and the physical field, the weight function for an internal crack problem in plane elasticity is formulated. The formulation proposed is compact in form and easy to understand.

Double crack problem in nonlocal elasticity

In the present work, the problem of two collinear cracks in an isotropic, homogeneous elastic medium which is subjected to uniform anti-plane shear loading at infinity is investigated in the context of nonlocal theory of elasticity. Governing equations of the problem are obtained by employing the field equations of the nonlocal elasticity. By use of the boundary conditions, the solution of the problem is reduced first to the investigation of the triple integral equations and then to a singular Fredholm integral equation. Numerical calculations are carried out for different values of the chosen parameter. Then stress analyses are given for single and double cracks.

Hypersingular integral equation for curved crack problem in antiplane elasticity

Hypersingular integral equation for curved crack problem in antiplane elasticity

A space-time variational formulation for the boundary integral equation in a 2D elastic crack problem

— This paper investigates the transient elastic wave scattering by a crack in R by means of Boundary Integral Equation Method. The analysis of the Laplace-Fourier transform (in time) of the integral operator allows to obtain existence, uniqueness and continuity dependence of the solution with respect to the data, in a Sobolev functional framework. A regularisation of the hypersingular BIE is applied in order to remove the hypersingularity and to write the associated time-space variational formulation on a tractable form. A Galerkin-type approximation is then performed to solve this variational formulation and we finally present some numerical results. Resume. — Nous nous interessons a la resolution par une methode d'equations integrales d'un probleme de diffraction d'ondes elastiques transitoires par une fissure. L'analyse de la transformee de F ourier-Laplace (en temps) de Voperateur integral permet d'obtenir des resultats d'existence, d'unicite et de continuite de la solution par rapport aux donnees dans des espaces fonctionnels de type Sobolev. Le noyau de V equation integrale etant hypersingulier, on ne dispose pas directement d'une expression calculable de la forme bilineaire associee. Cette difficulte est surmontee en appliquant une methode de regularisation qui fournit finalement la formulation variationnelle espace-temps du probleme. Cette formulation est enfin approchee a l'aide d'un schema de Galerkin et nous presentons quelques resultats numeriques.

New Fredholm integral equation for multiple crack problem in plane elasticity and antiplane elasticity

A singular integral equation for the multiple crack problem of plane elasticity is formulated in this paper. In the formulation we choose the crack opening displacement (COD) as unknown function and the resultant force as the right hand term of the equation. After using Vekua's regularization procedure or making inversion of the Cauchy singular integral in the equation, a new Fredholm integral equation is obtainable. The obtained Fredholm integral equation is compact in form and easy for computation. After solving the equation, the CODs of the cracks and the stress intensity factors (SIFs) at the crack tips can be derived immediately. Similar formulation for the multiple crack problem of antiplane elasticity is also presented. Finally, numerical examples are given to demonstrate the use of the proposed integral equation approach.

Continuation of some solutions in potential theory and their applications to elastic contact and crack problems

The following mixed boundary value problem is considered: arbitrary tangential displacements are prescribed inside a circle, while the tangential and normal stresses outside the circle are zero. In this case, a direct and simple formula is derived for the tangential displacements outside the circle in terms of prescribed displacements inside, thus making the tangential displacement known all over the boundary. The original problem is no longer mixed, and the complete solution becomes readily available. Another solution for the case, when arbitrary tangential displacements are prescribed outside a circle, is derived in a similar manner. The reciprocal theorem is used to derive the continuation formulae for the tangential stresses inside and outside a circle. Application of these results to contact and crack problems is demonstrated.

Derivation of the singular integral equations for curvilinear cracks with computer algebra software

The classical Cauchy-type complex singular integral equation for the general curvilinear crack problem in plane isotropic elasticity is rederived by using an appropriate computer procedure, based on the MATHEMATICA computer algebra system. Both cases of one crack (or a system of cracks) and of a periodic array of cracks are considered in detail and the MATHEMATICA procedure is displayed. The approach can also be generalized to additional arrays of cracks (like a doubly-periodic or a star-shaped array of curvilinear cracks). The present approach permits the direct error-free construction of sufficiently complicated complex singular integral equations of direct use in fracture mechanics in a simple and powerful way, which rid us of complicated manual, probably error-prone, computations. The approach is based on the knowledge stored in the MATHEMATICA kernel and on additional knowledge (like the Plemelj formulae) provided to the procedure itself. The subsequent use of a numerical algorithm for the solution of the derived complex singular integral equation is also possible.

A SIMPLE PATH-INDEPENDENT INTEGRAL FOR CALCULATING MIXED-MODE STRESS INTENSITY FACTORS

A path-independent integral is introduced for calculating stress intensity factors. The derivation of the integral is based on the application of the known Bueckner's fundamental field solution for a crack in an infinite body and on the reciprocal theorem. The method was applied to two-dimensional linear elastic mixed-mode crack problems. The key advantage of the present path-independent integral is that the stress intensity factor components for any irregular cracked geometry under any kind of loading can be easily obtained by a contour integral around the crack tip. The method is simple to implement and only the far field displacements and tractions along the contour must be known. The required stress analysis can be made by using any analytical or numerical method, e.g. the finite element method, without special consideration of the modelling of crack tip singularity. The application of this integral is also independent of the crack type, that is, there is no difference between an edge crack and an embedded crack, provided that the crack tip asymptotic behaviour exists.

An alternative fredholm integral equation approach for multiple crack problem and multiple rigid line problem in plane elasticity

An alternative Fredholm integral equation approach for the multiple crack problem and the multiple rigid line problem in plane elasticity is suggested in this paper. After using some operators on both sides of the singular integral equation for the relevant problem, an alternative Fredholm integral equation is obtainable. The solutions of the suggested integral equations satisfy the single-valuedness condition of displacement (for the multiple crack problem) and the zero resultant force condition (for the multiple rigid line problem), respectively. The obtained integral equation is compact in form and easy for computation. A technique for evaluating the rotation in the multiple rigid line problem is also proposed. Numerical examples are given to demonstrate the efficiency of the proposed approach.

The dual boundary element method: Effective implementation for crack problems

AbstractThe present paper is concerned with the effective numerical implementation of the two‐dimensional dual boundary element method, for linear elastic crack problems. The dual equations of the method are the displacement and the traction boundary integral equations. When the displacement equation is applied on one of the crack surfaces and the traction equation on the other, general mixed‐mode crack problems can be solved with a single‐region formulation. Both crack surfaces are discretized with discontinuous quadratic boundary elements; this strategy not only automatically satisfies the necessary conditions for the existence of the finite‐part integrals, which occur naturally, but also circumvents the problem of collocation at crack tips, crack kinks and crack‐edge corners. Examples of geometries with edge, and embedded crack are analysed with the present method. Highly accurate results are obtained, when the stress intensity factor is evaluated with the J‐integral technique. The accuracy and efficiency of the implementation described herein make this formulation ideal for the study of crack growth problems under mixed‐mode conditions.

Dual boundary element method for three-dimensional fracture mechanics analysis

In this paper, an effective numerical implementation of the three-dimensional dual boundary element method, for linear elastic crack problems, is presented. Displacement and traction integral equations which constitute the dual boundary formulation are used independently on crack surfaces to overcome the numerical difficulties associated with co-planar crack surfaces in boundary element analysis. The crack surfaces are modelled with discontinous quadrilateral quadratic elements. The use of discontinous elements allow for accurate integration of finite part integrals. The accuracy of the proposed method is demonstrated by solving a number of problems including edge and embedded cracks.