Cracks emanating from a square hole in rectangular plate in tension

A numerical analysis of cracks emanating from a square hole in a rectangular plate in tension is performed using a hybrid displacement discontinuity method (a boundary element method). Detailed solutions of the stress intensity factors (SIFs) of the plane elastic crack problem are given, which can reveal the effect of geometric parameters of the cracked body on the SIFs. By comparing the calculated SIFs of the plane elastic crack problem with those of the centre crack in a rectangular plate in tension, in addition, an amplifying effect of the square hole on the SIFs is found. The numerical results reported here also prove that the boundary element method is simple, yet accurate, for calculating the SIFs of complex crack problems in finite plate.     

Approximate weight function technique using single-layer potential for a cracked circular disc

An efficient weight function technique using the indirect boundary integral method was presented for cracked circular discs. The crack opening displacement field was presented by a single layer whose kernel was a modified form of the fundamental solution in elastostatics. The application of a single-layer potential to the weight function method leads to a unique closed-form SIF (stress intensity factor) solution. The solution can be applied to a cracked circular discs with or without an internal hole or opening. For these crack geometries over a wide range of crack ratios, the SIF solution can be applied without any modification.

The calculation procedure of SIFs for the various cracked circular discs using only one analytical solution is very simple and straightforward. The information necessary in the analysis includes only two or three reference load cases. In most cases the SIF solution using two reference SIFs gives reasonably accurate results while the SIF solution with three reference load cases may be used to improve the solution accuracy of the crack configurations, with an internal opening or hole, compared with the solutions of the available literature.     

Evaluation of T‐stresses in multiple crack problems of finite plate

This paper provides a solution for T‐stresses for multiple cracks in a finite plate. The results for stress intensity factors (SIFs) are also presented. The case of two cracks in a rectangular plate is taken as an example. In the problem, the crack faces are applied by some loadings, and tractions are free along edges of a rectangular plate. The whole stress field is considered as a superposition of three particular stress fields. The first and second stress fields are initiated by loadings on the first and second crack faces in an infinite plate. The third field is chosen in a polynomial form of complex potentials. After discretization, the loadings on two cracks and the undetermined coefficients in the complex potentials become the unknowns. The relevant algebraic equations are formulated. The solution of algebraic equations will lead to the results of SIFs and T‐stresses at the crack tips. Several numerical examples are presented, which were not reported previously.     

Thermal stress singularities at tips of a crack in a semi-infinite medium under uniform heat flow

In the framework of plane thermoelastic problems is discussed the thermal stress field near the tips of an arbitrarily inclined crack in an isotropic semi-infinite medium with the thermally insulated edge surface under uniform heat flow. The crack is replaced by continuous distributions of quasi-Volterra dislocations corresponding to line heat sources and edge dislocations, and we obtain a set of simultaneous singular integral equations for dislocation density functions, whose solution is given in the forms of series in terms of Tchebycheff polynomials of the first kind. By means of this method, the thermal stress singularities at the crack tips are estimated exactly and the stress intensity factors can be readily evaluated. Numerical results are given for the particular case where the surface of the inclined crack is maintained at constant temperature and the heat supplied across the surface of the crack vanishes as a whole. The effects of the distance from the crack tip to the edge surface of the semi-infinite medium and the angle of inclination of the crack on the stress intensity factors and the initial direction of crack extension are shown graphically.     

Stress intensity solutions for the interaction between a hole edge crack and a line crack

The principle of superposition is used to solve the problem and the original problem is converted into two particular hole edge crack problems. The remote stresses are applied at infinity in the first problem. Meantime, a dislocation distribution is assumed outside the hole contour in the second problem. Singular integral equation is proposed for the solution of the second problem, in which the right hand side of the integral equation is obtained from the solution of the first problem. The first problem as well as the elementary solution of the second problem are solved by means of the rational mapping approach. Finally, numerical examples with the calculated results of stress intensity factors are presented.     

The extended finite element method in thermoelastic fracture mechanics

The extended finite element method (XFEM) is applied to the simulation of thermally stressed, cracked solids. Both thermal and mechanical fields are enriched in the XFEM way in order to represent discontinuous temperature, heat flux, displacement, and traction across the crack surface, as well as singular heat flux and stress at the crack front. Consequently, the cracked thermomechanical problem may be solved on a mesh that is independent of the crack. Either adiabatic or isothermal condition is considered on the crack surface. In the second case, the temperature field is enriched such that it is continuous across the crack but with a discontinuous derivative and the temperature is enforced to the prescribed value by a penalty method. The stress intensity factors are extracted from the XFEM solution by an interaction integral in domain form with no crack face integration. The method is illustrated on several numerical examples (including a curvilinear crack, a propagating crack, and a three‐dimensional crack) and is compared with existing solutions. Copyright © 2007 John Wiley & Sons, Ltd.     

Rectangular tensile sheet with single edge crack or edge half-circular-hole crack

By using the displacement discontinuity method with crack-tip elements (a boundary element method) proposed recently by the author, this note presents the stress intensity factors (SIFs) of a rectangular tensile plate with single edge crack. Further this note studies the SIFs of crack emanating from an edge half-circular hole. By comparing the calculated SIFs of the single edge half-circular-hole crack with those of the single edge crack, a shielding effect of the half-circular hole on the SIFs of the single edge crack is discussed. It is found that the boundary element method is simple, yet accurate for calculating the SIFs of complex crack problems in finite plate.     

Cracks emanating from circular hole or square hole in rectangular plate in tension

A numerical analysis of cracks emanating from a circular hole (Fig. 1) or a square hole (Fig. 2) in rectangular plate in tension is performed by means of the displacement discontinuity method with crack-tip elements (a boundary element method) presented recently by the author. Detail solutions of the stress intensity factors (SIFs) of the two plane elastic crack problems are given, which can reveal the effect of geometric parameters of the cracked bodies on the SIFs. By comparing the SIFs of the two crack problems with those of the center crack in rectangular plate in tension (Fig. 3), in addition, an effect of the circular hole or the square hole on the SIFs of the center crack is discussed in detail. The numerical results reported here also illustrate that the boundary element method is simple, yet accurate for calculating the SIFs of complex crack problems in finite plate.     

Rectangular Tensile Sheet with Double Edge Notch Cracks

This paper deals with the rectangular tensile sheet with symmetric double edge notch cracks. Such a crack problem is called an edge notch crack problem for short. By using a hybrid displacement discontinuity method (a boundary element method), two edge notch models are analyzed in detail. By changing the geometrical forms and parameters of the edge notch, and by comparing the stress intensity factors (SIFs) of the edge notch crack problem with those of the double edge cracked plate tension specimen (DECT), which is a model frequently used in fracture mechanics, the effect of the geometrical forms and parameters of the edge notch on the SIFs of the DECT specimen, is revealed. Some geometric characterestic parameters are introduced here, which are used to formulate the notch length and the branch crack length, which are to be determined in mechanical machining of the DECT specimen So we can say that the geometric characterestic parameters and the formulae used to determine the notch length and the branch crack length presented in this paper perhaps have some guidance role for mechanical machining of the DECT specimen.     

Stress intensity factors for surface fatigue crack in a round bar under cyclic axial loading

In this paper, the surface fatigue crack growth shape for an initial straight-fronted edge crack in an elastic bar of circular cross-section is determined through experiments under pure fatigue axial loading. Three different initial notch depths are discussed. The relations of the aspect ratio (b/c) and relative crack depth (b/D) are obtained, and it is shown that there is a great difference in the growth of cracks with different initial front shapes and crack depths. Further, using the three-dimensional finite element method, the stress intensity factors (SIFs) are determined under remote uniform tension loading. Since the relationship of b/c and b/D changes during the fatigue crack growth, the SIFs are determined for different surface crack configurations.     

Stress intensity factors for cracks emanating from a circular hole in an infinite quasi‐orthotropic plane

An infinite quasi‐orthotropic plane with a cracked circular hole under tensile loading at infinity is studied analytically. To this end, complex variable theory of Muskhelishvili is used. In addition, to obtain analytical functions, a new conformal mapping is proposed and expanded to series expressions. Stress intensity factors (SIFs) for two unequal cracks emanating from a circular hole are obtained. To validate the analytical SIFs in a quasi‐orthotropic plane, the results are compared with FEM and the results of isotropic plane. The SIFs for small cracks in a quasi‐orthotropic and an isotropic plane are different, because of difference between stress concentrations in points which cracks emanate from the hole. However, the results of quasi‐orthotropic plane converge to isotropic plane for the large cracks. Therefore, the SIFs of the large cracks in a quasi‐orthotropic plane can be replaced by the results of the center crack with equivalent length in an isotropic plane.     

Interaction between crack and arbitrarily shaped hole with stress and displacement boundaries

The interaction between a crack and an arbitrarily shaped hole under stress and displacement boundaries in an infinite plane subjected to a remote uniform load is studied. The Green's functions of a point dislocation for the problems are derived and are then used to analyze the interaction problem. The superposition principle is employed to reduce the original problem to two subsidiary problems. The complex stress functions of each problem are composed of two parts, in which the second parts are always holomorphic. Using analytical continuation in conjunction with rational mapping function, the stress functions are obtained in closed form. The interaction of a hole or an inclusion with a crack is solved using dislocations to model the crack and solving a system of singular integral equations. Stress intensity factors for crack tips and stress concentration factors for inclusion corner are determined and plotted for various cases. The affecting ranges of hole and inclusion are investigated.     

Variable thermal singularity boundary elements in the study of neighbouring singularities

Two new boundary elements are presented for the simulation of variable order thermal singularities in two dimensions. The first can model the variable order temperature derivative and heat flux singularities at one end of the element. The second element can simulate the temperature derivative and heat flux singularities at both the ends of the element simultaneously. These elements are useful for studying the interaction of variable order thermal neighbouring singularities. They are employed here for the computation of stress intensity factors in the crack–crack interaction problems under thermal load. To improve the accuracy of such computations a modified crack closure integral based method is adapted. Examples of mode I and mode II centre crack, two collinear neighbouring cracks, kinked crack in a plate, and tee joint with a kinked edge crack under thermal or thermal and mechanical loading are studied to illustrate the usefulness of these elements in the study of neighbouring thermal singularities. Copyright © 2001 John Wiley & Sons, Ltd.     

General solution for arc crack problem in thermoelastic medium

A problem of a circular arc-shaped crack in an infinite plate under the thermal loading is solved by using the complex variable function and the integral equation method. General solution for arbitrary heat flux along the crack face is obtained. For some particular cases, for example, the constant heat flux case and remote heat flux case, a closed form solution is obtained. The solution technique is effective in derivation and compact in form.     

An integral representation of the stress intensity factors for three-dimensional static problems

For finding suitable expressions for the stress intensity factors (SIFs) under a general three-dimensional condition, the first stress invariant and the displacement tangent to a crack edge are analyzed. By using Green's theorem, the SIFs are expressed by integrals for the most general situations. K _I and K _II are expressed by integrals of the first stress invariant and its partial derivative. K _III is expressed by an integral of the displacement tangent to the crack edge and its partial derivative. The integrals include a surface integral on a smooth surface of arbitrary shape, and a line integral along part of the surface's boundaries. The expressions are valid for an arbitrarily shaped elastic medium with stationary cracks of arbitrary shape. The expressions provide a new approach for the determination of the SIFs.     

Closure of circular arc cracks under general loading: effects on stress intensity factors

The two-dimensional circular arc crack solution of Muskhelishvili (Some basic problems of the mathematical theory of elasticity, P. Noordhoff Ltd, Groningen, Holland, 1953) has been used widely to study curved crack behavior in an infinite, homogeneous and isotropic elastic material. However, for certain orientations and magnitudes of the remotely applied loads, portions of the crack will close. Since the analytical solution is incorrect once the crack walls come into contact, the displacement discontinuity method is combined with a complementarity algorithm to solve this problem. This study uses stress intensity factors (SIFs) and displacement discontinuities along the crack to define when the analytical solution is not applicable and to better understand the mechanism that causes partial closure under various loading conditions, including uniaxial tension and pure shear. Closure is mainly due to material from the concave side of the crack moving toward the outer crack surface. Solutions that allow interpenetration of the crack tips yield non-zero mode I SIFs, while crack tip closure under proper contact boundary conditions produce mode I SIFs that are identically zero. Partial closure of a circular arc crack will alter both mode I and II SIFs at the crack tips, regardless of the positioning or length of the closed section along the crack. Friction on the crack surfaces in contact changes the total length and positioning of closure, as well as generally decreases the magnitude of opening along the portions of the crack that are not closed.     

Mode-III stress intensity factors for two circular inclusions subject to a remote uniform shear load

ABSTRACT

An analytical solution to the antiplane elasticity problem associated with two circular inclusions interacting with a line crack is provided in this article. A series solution for the stress field is derived in an elegant form by using complex variable theory in conjunction with the alternation method. Based on the superposition method, a singular integral equation (SIE) is established from the traction-free condition along the crack surface. After solving the SIE, the mode-III stress intensity factors (SIFs) can be obtained to quantify the singular behavior of the stress field ahead of the crack tips. Numerical results of the SIFs, when a crack is embedded either in the inclusion or in the matrix, are discussed in detail and displayed in graphic form.     

An infinitely large plate weakened by a circular-arc crack subjected to partially distributed loads

On the basis of the complex-variable approach for the first boundary condition problems, a mapping function is proposed to transform the contour surface of a circular arc crack into a unit circle. By this mapping, direct stress integration along the contour surface can be performed for the case when uniform tractions are applied on part of the crack edge. General complex stress functions are obtained by evaluating the Cauchy integral for the governing boundary equation. After the obtained stress functions are differentiated with respect to a reference angle in the mapped plane, the general complex stress functions for the circular-arc crack problem, when concentrated loads are applied on the crack surface, can be obtained. The importance of this solution lies in its general applicability to crack problems with arbitrary loading.     

Limiting state of a semiinfinite plate with edge crack in bending with tension

We solve the problem of stressed state and limiting equilibrium of a semiinfinite plate with edge crack under the action of symmetric bending with tension. In the two-dimensional case, the possibility of crack closure is taken into account by using a model of contact along a line. An algorithm for the numerical solution of the mixed problem is proposed. The diagram of limiting equilibrium of the cracked plate shows that the range of safe states enlarges if the effect of crack closure is taken into account.Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 40, No. 2, pp. 73–77, March–April, 2004.     

Stress intensity factors for a surface-breaking crack in a half-plane subject to contact loading

Modes I and II stress intensity factors are derived for a crack breaking the surface of a half-plane which is subject to various forms of contact loading. The method used is that of replacing the crack by a continuous distribution of edge dislocations and assume the crack to be traction-free over its entire length. A traction free crack is achieved by cancelling the tractions along the crack site that would be present if the half-plane was uncracked. The stress distribution for an elastic uncracked half-plane subject to an indenter of arbitrary profile in the presence of friction is derived in terms of a single Muskhelishvili complex stress function from which the stresses and displacements in either the half-plane or indenter can be determined. The problem of a cracked half-plane reduces to the numerical solution of a singular integral equation for the determination of the dislocation density distribution from which the modes I and II stress intensity factors can be obtained. Although the method of representing a crack by a continuous distribution of edge dislocations is now a well established procedure, the application of this method to fracture mechanics problems involving contact loading is relatively new. This paper demonstrates that the method of distributed dislocations is well suited to surface-breaking cracks subject to contact loading and presents new stress intensity factor results for a variety of loading and crack configurations.