Multiple crack problems of antiplane elasticity in an infinite body by using fredholm integral equation approach

Two elementary solutions are present to solve the proposed problem. The first (second) elementary solution is defined as a solution that one pair of longitudinal concentrated forces is applied at a prescribed point of both edges of a single crack in infinite isotropic medium, with same magnitude and opposite direction (with same magnitude and same direction). Using the two elementary solutions and the principle of superposition, we found the proposed problem can be easily converted into a system of Fredholm integral equations. The system is solved numerically and SIF values at the crack tips can be easily calculated. Several numerical examples are given.     

Singular integral equation method for multiple Zener–Stroh crack problems in antiplane elasticity

In this paper, the multiple Zener–Stroh crack problems in anti-plane elasticity are studied. The crack faces are assumed to be traction free, and dislocation distributions on the cracks are chosen as the unknown functions in the solution. The singular integral equations for the problem are obtained. The constraint equations are also derived from the condition of the accumulation of dislocation on the cracks. After solving the integral equations, the stress intensity factors at crack tips can be evaluated immediately. Numerical examples are given. It is found that interactions between the Zener–Stroh cracks are quite different from those for the Griffith cracks, in qualitative and quantitative aspects.     

Multiple Zener-Stroh crack problem in an infinite plate

Summary. In this paper, the multiple Zener-Stroh crack problem is studied. We choose the distributed dislocations as unknown functions in the integral equation. The crack faces are assumed to be traction free. The applied generalized loading for cracks is the initial displacement jump (abbreviated as IDJ), which in turn is the increment of displacement when a moving point goes around the crack in a closed loop. A system of singular integral equations is obtained. After solving the integral equations, the stress intensity factors at crack tips can be evaluated immediately. Numerical examples are given. It is found that interactions between Zener-Stroh cracks are quite different from those for the Griffith cracks, in the qualitative and quantitative aspects.AcknowledgementThe research project is supported by National Natural Foundation of China.     

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.     

Solutions of multiple crack problems of a circular region with free or fixed boundary condition in antiplane elasticity

Four cases of multiple crack problems in antiplane elasticity, circular region or an infinite region exterior to a circle with free or fixed boundary condition, are considered in this paper. For all these cases, the presented elementary solution is a particular solution of the circular region containing one crack. The solution consists of two parts and satisfies the following conditions: (a) The first part corresponds to a pair of longitudinal forces acting at a prescribed point on both edges of a single crack; (b) The second part corresponds to some distributed forces along both edges of the crack; (c) The elementary solution, i.e. the sum of the first and second parts, satisfies the above mentioned boundary value conditions along the circular boundary. As we have done in [1–3], the system of Fredholm integral equations for the undetermined densities of the elementary solutions can be easily estabished and the stress intensity factors at crack tips can also be evaluated.     

General case of multiple crack problems in an infinite plate

General case of multiple crack problems in an infinite plate is a case that the tractions applied on two edges of each crack are arbitrary, generally, are not in equilibrium. Two elementary solutions are present to solve the proposed problem. The first (second) elementary solution is defined as a solution that two pairs of normal and tangential concentrated forces are applied at a point of both edges of a single crack in an infinite isotropic elastic medium, with same magnitude and opposite direction (with same magnitude and same direction). Using the two elementary solutions and the principle of superposition, we found the proposed problem can be converted into a system of Fredholm integral equations. Finally, the system is solved numerically and SIF values at the crack tips can be easily calculated. In order to explain our study, one numerical example is given in this paper.     

A wedge crack in an anisotropic material under antiplane shear

A crack emanating from the apex of an infinite wedge in an anisotropic material under antiplane shear is investigated. An isotropic wedge crack subjected to concentrated forces is first solved by using the conformal mapping technique. The solution of an anisotropic wedge crack is obtained from that of the transformed isotropic wedge crack based on a linear transformation method. Expressions for the stress intensity factor for the anisotropic wedge crack with both concentrated and distributed loads are derived. The stress intensity factors are numerically calculated for generally orthotropic wedge cracks with various crack and wedge angles as well as anisotropic parameters.     

A new singular integral equation for the classical crack problem in plane and antiplane elasticity

A new singular integral equation (with a kernel with a logarithmic singularity) is proposed for the crack problem inside an elastic medium under plane or antiplane conditions. In this equation the integral is considered in the sense of a finite-part integral of Hadamard because the unknown function presents singularities of order ?3/2 at the crack tips. The Galerkin and the collocation methods are proposed for the numerical solution of this equation and the determination of the values of the stress intensity factors at the crack tips and numerical results are presented. Finally, the advantages of this equation are also considered.     

Fixed rigid line problem in antiplane elasticity

In this paper the properties of eigenfunction expansion form (EEF) in the fixed rigid line problem in antiplane elasticity are discussed in detail. After using Betti's reciprocal theorem for a body containing a fixed rigid line, several new path-independent integrals are obtained. Finally, many solutions for this problem are proposed in a closed form.     

Property of eigenvalues and eigenfunctions for an interface V-notch in antiplane elasticity

The so-called pseudo-orthogonal property of the eigenfunction expansion form is proved to be valid for the case of an antiplane interface V-notch and the corresponding path-independent integral is derived. The relation between the path-independent integral and the stress intensity factor of the notch is found. The influence of loads on the related integral is also presented.     

Rigid body rotation at the tip of a slant crack in an infinite plate

Using Kolosov-Muskhelishvili relations of stresses the rigid body rotation is obtained in the form of complex potentials. The rotation at a point near the tip of a slant crack is expressed in terms of stress intensity factors and the coordinates (r, ) of the point. The relation of rigid body rotation near the crack tip are used to describe some features of mode I and mode II crack tip plastic zone. The rotation field surrounding the tip of a slant crack in infinite plate is obtained and its properties are discussed.

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Résumé En utilisant les relations de Kolosov-Muskhelishvili relatives aux contraintes, on obtient la rotation d'un corps rigide sous forme de potentiels complexes. La rotation en un point près de l'extrémité d'une fissure inclinée est exprimée en fonction du facteur d'intensité d'entaille et des coordonnées (r-) do point. On utilise les relations de rotation d'un corps rigide au voisinage de l'extrémité d'une fissure pour décrire certaines caractéristiques de la zone plastique à l'extrémité d'une fissure de mode I et de mode II. Le champ rotationnel autour de l'extrémité d'une fissure inclinée dans une plaque infinie est obtenue et ses propriétés sont discutées.

     

A hybrid-element approach to crack problems in plane elasticity

The hybrid-element concept and the complex variable technique have been adopted for constructing a special super-element to be used jointly with conventional finite elements for the analysis of elastic stress intensity factors for plane cracks. The use of the complex variable technique permits the proper consideration of the stress intensity at the crack tip, and it also leads to very efficient programming. The use of such a super-element in the finite element solution has been shown to be highly accurate when only a very coarse element mesh is used near the crack.     

Some triple equations with an application to crack problems in the theory of elasticity

In this paper the authors investigate some triple integral equations involving inverse Mellin type transforms. The use of these equations is then illustrated by their application in the solution of certain crack problems in the theory of elasticity.