Crack Problems In Plane Elasticity Research Articles

Stability and convergence analysis of singular integral equations for unequal arms branch crack problems in plane elasticity

In this paper, an unequal arms branch crack problem in a plane elasticity is treated. Using distribution dislocation function and complex variable potential method, the problem is formulated into a singular integral equation. The appropriate integration scheme, in which a point dislocation is set at the origin and the distribution dislocation, is applied through all arms of the branch crack to solve the obtained singular integral equations numerically. Stability, convergence, the order of convergence, and the error term of the solution are analyzed. Some numerical examples are examined to describe the behavior of stress intensity factors at the endpoints of each branch crack.

Fracture analysis for a cusp-type crack problem

This paper provides an exact solution for a cusp-type crack problem in plane elasticity. The solution of the studied problem depends on the conformal mapping technique and complex potentials. The stress intensity factors at the cusp-type crack tip can be evaluated accordingly. Some particular characters for the SIF (stress intensity factor) solution have been found. Particularly, if a compression loading is applied along the direction of the cusp-type crack, the relevant SIF will take a positive value. This phenomenon is useful in the rupture of the rock with the cusp-type crack.

Stress intensity factor for an elastic half plane weakened by multiple curved cracks

• The multiple curved cracks problem in an elastic half plane is formulated by using the complex potential methods. • The integral equations are solved numerically using quadrature rule for the stress intensity factor. • The closer the cracks to the boundary the higher the value of SIF. Modified complex potential with free traction boundary condition is used to formulate the curved crack problem in a half plane elasticity into a singular integral equation. The singular integral equation is solved numerically for the unknown distribution dislocation function. Numerical examples exhibit the stress intensity factor increases as the cracks getting close to each other, and close to the boundary of the half plane.

Solution of stress intensity factors of multiple cracks in plane elasticity with eigen COD formulation of boundary integral equation

The concept of eigen crack opening displacement (COD) can be defined as the COD of a crack in infinite plate under the tractions acting on the crack surface. By introducing this concept, the eigen COD formulation of boundary integral equation is proposed in this paper, together with the solution procedures for multiple crack problems in plane elasticity. With the proposed approach, the multiple crack problems can be solved with the conventional displacement discontinuity boundary integral equations in an iterative fashion with a small size of system matrix as that in the numerical Green’s function (NGF) approach but without the trouble to determine the complementary solutions since the standard boundary element discretization on the crack surface is no longer required with the proposed approach. Some numerical examples computing the stress intensity factors are presented and compared with those in literature to show the accuracy and the effectiveness of the proposed approach.

T-stress in the Zener–Stroh arc crack problem in plane elasticity

A Zener–Stroh curved crack is defined such that the crack undergoes an initial displacement discontinuity. A singular integral equation is suggested to solve the Zener–Stroh curved crack problem. General formulation for evaluating the stress intensity factors and the T-stresses at the crack tips of a Zener–Stroh curved crack is carried out. For the Zener–Stroh arc crack, T-stresses at the crack tips can be evaluated in a closed form.

An Iterative Algorithm of Hypersingular Integral Equations for Crack Problems

Crack problems are reducible to singular integral equations with strongly singular kernels by means of the body force method. In the ordinary method, the integral equations are reduced to a system of linear algebraic equations. In this paper, an iterative method for the numerical solution of the hypersingular integral equations of the body force method is proposed. This method is based on the Gauss- Chebyshev numerical integration rule and is very simple to program. The solution is achieved without solving the system of linear algebraic equations. The proposed method is applied to some plane elasticity crack problems and is seen to give convergent results.

Collinear Zener–Stroh crack problem in plane elasticity

This paper investigates the collinear Zener–Stroh crack problem in plane elasticity. Two cracks in series are chosen as an example in analysis. Different to the Griffith crack problem, an initial displacement discontinuity exists in the Zener–Stroh crack problem, which in turn is the increment of displacement when a moving point goes around the crack in a closed loop. From the traction free condition along the cracks, the dislocation distribution function as well as the complex potential with undetermined coefficients is suggested. The involved undetermined coefficients can be evaluated from the condition of the assumed initial displacement discontinuity. Finally, a closed form solution for the problem is obtained and the calculated stress intensity factors at crack tips are presented. A problem for two collinear Zener–Stroh cracks with different lengths is also studied.

Integral Equation Methods for Multiple Crack Problems and Related Topics

The content of this review consists of recent developments covering an advanced treatment of multiple crack problems in plane elasticity. Several elementary solutions are highlighted, which are the fundamentals for the formulation of the integral equations. The elementary solutions include those initiated by point sources or by a distributed traction along the crack face. Two kinds of singular integral equations, three kinds of Fredholm integral equations, and one kind of hypersingular integral equation are suggested for the multiple crack problems in plane elasticity. Regularization procedures are also investigated. For the solution of the integral equations, the relevant quadrature rules are addressed. A variety of methods for solving the multiple crack problems is introduced. Applications for the solution of the multiple crack problems are also addressed. The concept of the modified complex potential (MCP) is emphasized, which will extend the solution range, for example, from the multiple crack problem in an infinite plate to that in a circular plate. Many multiple crack problems are addressed. Those problems include: (i) multiple semi-infinite crack problem, (ii) multiple crack problem with a general loading, (iii) multiple crack problem for the bonded half-planes, (iv) multiple crack problem for a finite region, (v) multiple crack problem for a circular region, (vi) multiple crack problem in antiplane elasticity, (vii) T-stress in the multiple crack problem, and (viii) periodic crack problem and many others. This review article cites 187 references.

Numerical analysis of the stress intensity factor for two kinds of mixed-mode crack specimens

By means of a boundary element method (BEM), this paper is concerned with two kinds of mixed-mode crack specimens, i.e. the three-point bending and shear specimen with an offset edge crack and the ring specimen with a pair of radial cracks under diametral compression. The boundary element method consists of the constant displacement discontinuity element preented by Crouch and starfield and the crack-tip diplacement discontinuity element proposed by the author. Test examples are included to show that the numerical approach is very efficient and accurate for the stress intensity factor calculation of plane elasticity crack problems in both infinite and finite domains. Specifically, the stress intensity factors (SIFs) for these two kinds of mixed-mode crack specimens are given.

An effective boundary element method for analysis of crack problems in a plane elastic plate

A simple and effective boundary element method for stress intensity factor calculation for crack problems in a plane elastic plate is presented. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity elements proposed by YAN Xiangqiao. In the boundary element implementation the left or the right crack-tip displacement discontinuity element was placed locally at the corresponding left or right each crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Test examples (i. e., a center crack in an infinite plate under tension, a circular hole and a crack in an infinite plate under tension) are included to illustrate that the numerical approach is very simple and accurate for stress intensity factor calculation of plane elasticity crack problems. In addition, specifically, the stress intensity factors of branching cracks emanating from a square hole in a rectangular plate under biaxial loads were analysed. These numerical results indicate the present numerical approach is very effective for calculating stress intensity factors of complex cracks in a 2-D finite body, and are used to reveal the effect of the biaxial loads and the cracked body geometry on stress intensity factors.

Numerical analysis of a few complex crack problems with a boundary element method

In this paper, numerical analysis of a few complex crack problems is made by using a boundary element method which consists of the non-singular displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity elements due to the author. In the boundary element implementation the left or the right crack-tip displacement discontinuity element is placed locally at the corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. The numerical results of a few complex crack problems, such as “a singly branched crack in an infinite plate”, “interaction between a circular hole and a crack in an infinite plate”, “a pair of branched cracks emanating from an elliptical hole in a rectangular plate under biaxial uniform tensions” and “fatigue growth analysis of a branched crack emanating form the corner of a square hole”, indicate that the numerical approach is very efficient and accurate for analyzing complex plane elasticity crack problems.

Stress intensity factors for cracks emanating from a triangular or square hole in an infinite plate by boundary elements

This paper concerns stress intensity factors of cracks emanating from a triangular or square hole in an infinite plate subjected to internal pressure calculated by means of a boundary element method, which consists of constant displacement discontinuity element presented by Crouch and Starfied and crack tip displacement discontinuity elements proposed by the author. In the boundary element implementation the left or the right crack tip displacement discontinuity element is placed locally at corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Numerical examples are included to show that the method is very efficient and accurate for calculating stress intensity factors of plane elasticity crack problems. Specifically, the numerical results of stress intensity factors of cracks emanating from a triangular or square hole in an infinite plate subjected to internal pressure are given.

Solution of Zener–Stroh arc crack problem in plane elasticity

In this paper, solution of the Zener–Stroh arc crack in plane elasticity is present. The problem is reduced to a solution of singular integral equation. After using some formulae and equations in complex variable function a closed form solution is obtained.

A special crack tip displacement discontinuity element

Based on the analytical solution to the problem of a constant discontinuity in displacement over a finite line segment in the x, y plane of an infinite elastic solid and the note of the crack tip element by Crouch, in the present paper, the special crack tip displacement discontinuity element is developed. Further the analytical formulas for the stress intensity factors of crack problems in general plane elasticity are given. In the boundary element implementation the special crack tip displacement discontinuity element is placed locally at each crack tip on top of the non-singular constant displacement discontinuity elements that cover the entire crack surface. Numerical results show that the displacement discontinuity modeling technique of a crack presented in this paper is very effective.

Singular integral equation method for the solution of multiple curved crack problems

A method based on complex potentials for distributions of dislocations along curved cracks is used to solve multiple curved crack problems in plane elasticity. The method allows evaluation of the interaction between curved cracks. A crack problem is reduced to a system of singular integral equations and the crack curve length is taken as the coordinate in the associated integral equations. A crack is then mapped on the real axis in an interval (− a, a), where 2 a is the length of crack and the original singular integral equations are transformed accordingly. The method allows cracks with a general curvature and is not restricted to slightly curved crack configurations. The resulting singular integral equation system is solved through Gauss quadrature. A few numerical examples of problems with two cracks are given and crack interaction, i.e., the shielding effect of a curved crack surrounding by another, is also studied.

THE INDIRECT BOUNDARY INTEGRAL METHOD FOR CURVED CRACKS IN PLANE ELASTICITY

For curved crack problems in plane elasticity, subjected to the traction conditions on the crack faces, we present a system of boundary integral equations. The procedure is based on the indirect boundary integral method in terms of real variables. For efficient mathematical analysis, we decompose the singular kernel into the Cauchy singular part and the regular one. As a result, solvability of the presented system is proved and availability of the present approach is shown by the numerical example of a circular arc crack.

Closed form solutions of T-stress in plane elasticity crack problems

The T-stress is considered as an important parameter in linear elastic fracture mechanics. In this paper, several closed form solutions of T-stress in plane elasticity crack problems in an infinite plate are investigated using the complex potential theory. In the line crack case, if the applied loading is the remote stress or the concentrated forces, the T-stress can be derived from the basic field. Here, the basic field is defined as the field caused by the applied loading in the infinite plate without the crack. For the circular arc crack, the T-stress can be abstracted from a known solution. For the cusp crack problems, the T-stress can be separated from the obtained stress solution for which the conformal mapping technique is used.

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.

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.