The penny-shaped crack problem in micropolar elasticity

In the linear theory of micropolar elasticity, the problem of a penny-shaped crack in a transverse field of constant uniaxial tension is studied. By means of Hankel transforms and dual integral equations the problem is reduced to a regular Fredholm integral equation of the second kind and is then solved numerically. The singular fields arising at the crack-tip are studied in detail and the results are compared with those of the couple stress theory. Classical results are derived as limiting case.The stress environment at the periphery of the crack is found to depend on, apart from Poisson's ratio and a material length-parameter, another parameter which characterises the coupling of the microstructure with the displacement field. This parameter does not occur in the analogous problem in couple stress theory.     

A new boundary integral equation for notch problem of antiplane elasticity

In this paper the notch problem of antiplane elasticity is discussed and a new boundary integral equation is formulated. In the problem, the distributed dislocation density is taken to be the unknown function. Unlike the usual choice, the resultant force function is taken as the right hand term of the integral equation; therefore, a new boundary integral equation for the notch problem of antiplane elasticity with a weaker singular kernel (logarithmic) is obtained. After introducing a particular fundamental solution of antiplane elasticity, the notch problem for the half-plane is discussed and the relevant boundary integral equation is formulated. The integral equations derived are compact in form and convenient for computation. Numerical examples demonstrated that high accuracy can be achieved by using the new boundary equation.     

Upper bounds for the stress intensity factors along the boundaries of interacting coplanar cracks in three-dimensional elasticity

An elementary method for obtaining upper bounds for the stress intensity factors along the boundaries of interacting coplanar cracks inside an infinite isotropic elastic medium is presented. This method is based on the singular integral equation of the aforementioned elasticity problem and on the solutions of this equation for each particular crack problem, assumed known. The method is applied to the simple problem of interaction of two circular cracks, as well as to the similar problem of two cracks having the shape of a straight strip. The present results constitute a generalization of the corresponding method for crack problems in two-dimensional elasticity and can easily be further generalized to apply to more complicated crack problems in three-dimensional elasticity.     

Axisymmetric crack terminating at the interface of transversely isotropic dissimilar media

In this paper, the axisymmetric elasticity problem of an infinitely long transversely isotropic solid cylinder imbedded in a transversely isotropic medium is considered. The cylinder contains an annular or a penny shaped crack subjected to uniform pressure on its surfaces. It is assumed that the cylinder is perfectly bonded to the medium. A singular integral equation of the first kind (whose unknown is the derivative of crack surface displacement) is derived by using Fourier and Hankel transforms. By performing an asymptotic analysis of the Fredholm kernel, the generalized Cauchy kernel associated with the case of `crack terminating at the interface' is derived. The stress singularity associated with this case is obtained. The singular integral equation is solved numerically for sample cases. Stress intensity factors are given for various crack geometries (internal annular and penny-shaped cracks, annular cracks and penny-shaped cracks terminating at the interface) for sample material pairs.     

Multiple crack problems of antiplane elasticity in an infinite body

Twe elementary solutions are presented for case of a pair of normal or tangential concentrated unit forces acting at a point of both edges of a single crack in an infinite plane isotropic elastic medium. Using these two elementary solutions and the principle of superposition, we found that the multiple crack problems can be easily converted into a system of Fredholm integral equations. Finally, the system obtained is solved numerically and the values of the stress intensity factors at the crack tips can be easily calculated. Two numerical examples are given in this paper. A system of Fredholm integral equations is complex form is also presented. We found that the system of Fredholm integral equations can be easily reduced from the system of singular integral equations given by Panasyuk[1]     

Various integral equations for a single crack problem of elastic half-plane

Two kinds of the complex potentials used for the crack problem of the elastic half-plane are suggested. First one is based on the distribution of dislocation along a curve, and second one is based on the distribution of crack opening displacement along a curve. Depending on the use of the complex potentials and the right hand term in the integral equation, two types of the singular integral equation for a single crack problem of elastic half-plane are derived. Regularization of the suggested singular integral equations gives three types of the Fredholm integral equation for the relevant problem. The weaker singular integral equation and the hypersingular integral equation are also introduced. Seven types of the integral equation are finally obtainable. The relation between the kernels of the various integral equations is also discussed.     

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.     

Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity

Summary A modification of the form of the singular integral equation for the problem of a plane crack of arbitrary shape in a three-dimensional isotropic elastic medium is proposed. This modification consists in the incorporation of the Laplace operator into the integrand. The integral must now be interpreted as a finite-part integral. The new singular integral equation is equivalent to the original one, but simpler in form. Moreovet, its form suggests a new approach for its numerical solution, based on quadrature rules for one-dimensional finite part integrals with a singularity of order two. A very simple application to the problem of a penny-shaped crack under constant pressure is also made. Moreover, the case of straight crack problems in plane isotropic elasticity is also considered in detail and the corresponding results for this special case are also derived.With 2 Figures     

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.     

A natural approach to the introduction of finite-part integrals into crack problems of three-dimensional elasticity

The classical singular integral equation for the problem of a plane crack inside an infinite isotropic elastic medium and under an arbitrary normal pressure distribution was recently modified and written without the use of the Laplace operator Δ or the derivatives of the unknown function, but with the use of a finite-part integral. In this paper, a second complete derivation of the same equation is made (not based on previous forms of this equation) by using a limiting procedure, which makes it clear why the finite-part integral results in this equation. It is believed that the present results will be used in future for the introduction of finite-part integrals into a lot of crack problems in the theory of three-dimensional elasticity.     

Regularization of hypersingular boundary integral equations: a new approach for axisymmetric elasticity

A hypersingular boundary integral equation (HBIE) formulation, for axisymmetric linear elasticity, has been recently presented by de Lacerda and Wrobel [Int. J. Numer. Meth. Engng 52 (2001) 1337]. The strongly singular and hypersingular equations in this formulation are regularized by de Lacerda and Wrobel by employing the singularity subtraction technique. The present paper revisits the same problem. The axisymmetric HBIE formulation for linear elasticity is interpreted here in a ‘finite part’ sense and is then regularized by employing a ‘complete exclusion zone’. The resulting regularized equations are shown to be simpler than those by de Lacerda and Wrobel.     

Analytical study and numerical experiments for degenerate scale problems in the boundary element method for two‐dimensional elasticity

For a plane elasticity problem, the boundary integral equation approach has been shown to yield a non‐unique solution when geometry size is equal to a degenerate scale. In this paper, the degenerate scale problem in the boundary element method (BEM) is analytically studied using the method of stress function. For the elliptic domain problem, the numerical difficulty of the degenerate scale can be solved by using the hypersingular formulation instead of using the singular formulation in the dual BEM. A simple example is shown to demonstrate the failure using the singular integral equations of dual BEM. It is found that the degenerate scale also depends on the Poisson's ratio. By employing the hypersingular formulation in the dual BEM, no degenerate scale occurs since a zero eigenvalue is not embedded in the influence matrix for any case. Copyright © 2002 John Wiley & Sons, Ltd.     

Three iterative methods for the numerical determination of stress intensity factors

Three iterative methods for the numerical determination of stress intensity factors at crack tips (by using the method of singular integral equations with Cauchy-type kernels) are proposed. These methods are based on the Neumann iterative method for the solution of Fredholm integral equations of the second kind. Two of these methods are essentially used for the solution of the system of linear algebraic equations to which the singular integral equation is reduced when the direct Lobatto-Chebyshev method is used for its approximate solution, whereas the third method is a generalization of the first two and is related directly to the singular integral equation to be solved. The proposed methods are useful for the determination of stress intensity factors at crack tips. Some numerical results obtained in a crack problem show the effectiveness of all three methods.     

The elasticity problem for a thick-walled cylinder containing a circumferential crack

The elasticity problem for a long hollow circular cylinder containing an axisymmetric circumferential crack subjected to general nonaxisymmetric external loads is considered. The problem is formulated in terms of a system of singular integral equations with the Fourier coefficients of the derivative of the crack surface displacement as density functions. The stress intensity factors and the crack opening displacement are calculated for a cylinder under uniform tension, bending by end couples, and self-equilibrating residual stresses.     

A weakly singular integral formulation for displacement prescribed problems of elasticity

Summary. This paper introduces a new integral formulation for displacement prescribed problems in linear elasticity. The formulation uses a weakly singular kernel and extends to the case of linear elasticity the integral formulation introduced by Mikhlin [1] to solve Dirichlet problems for Laplaces equation in multiply connected domains. A detailed proof of the proposed formulation is given for displacement prescribed problems in two-dimensional multiply-connected domains. The proposed method can also be readily extended to solve three-dimensional problems.     

Distributed dislocation approach for cracks in couple-stress elasticity: shear modes

The distributed dislocation technique proved to be in the past an effective approach in studying crack problems within classical elasticity. The present work aims at extending this technique in studying crack problems within couple-stress elasticity, i.e. within a theory accounting for effects of microstructure. As a first step, the technique is introduced to study finite-length cracks under remotely applied shear loadings (mode II and mode III cases). The mode II and mode III cracks are modeled by a continuous distribution of glide and screw dislocations, respectively, that create both standard stresses and couple stresses in the body. In particular, it is shown that the mode II case is governed by a singular integral equation with a more complicated kernel than that in classical elasticity. The numerical solution of this equation shows that a cracked material governed by couple-stress elasticity behaves in a more rigid way (having increased stiffness) as compared to a material governed by classical elasticity. Also, the stress level at the crack-tip region is appreciably higher than the one predicted by classical elasticity. Finally, in the mode III case the corresponding governing integral equation is hypersingular with a cubic singularity. A new mechanical quadrature is introduced here for the numerical solution of this equation. The results in the mode III case for the crack-face displacement and the near-tip stress show significant departure from the predictions of classical fracture mechanics.     

Bonded Cracked Half Planes with an Interface and an Interesting Crack

The solution of the plane elasticity problem of two bonded isotropic linearly elastic half-planes of different elastic properties having a crack L along the interface as well as a crack S in one of the half planes which intersects the interface crack, is given by using the Muskhelishvili's complex variable method with sectionally holomorphic functions. The initial problem is reduced to a Hilbert problem, the solution of which in the case of a dislocation existing in either half-planes constitutes the Green's functions of the problem. Finally, a singular integral equation is derived for the problem only along the crack S. The singular integral equation is solved numerically and results are presented for the stress intensity factors. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dual boundary integral equation formulation in antiplane elasticity using complex variable

This paper investigates the dual boundary integral equation formulation in antiplane elasticity using complex variable. Four kinds of boundary integral equation (BIE) are studied, and they are the first complex variable BIE for the interior region, the second complex variable BIE for the interior region, the first complex variable BIE for the exterior region, and the second complex variable BIE for the exterior region. The first BIE for the interior region is derived from the Somigliana identity, or the Betti’s reciprocal theorem in elasticity. A displacement versus traction operator is suggested. After using this operator, the second BIE for the interior region is derived. Similar derivations are performed for the first and second BIEs for the exterior region. In the case of the exterior boundary, two degenerate boundary cases are studied. One is the curved crack case, and other is the case of a deformable line. All kernels in the suggested BIEs are expressed in terms of complex variable.     

A generalized crack problem in plane elasticity

The plane isotropic elasticity problem of a simple curvilinear crack with non-coincident edges (contrary to the idealization usually made) is considered. The maximum opening between the edges of the crack may be as great as 0.2 of the crack length. For the solution of this problem, the model of replacing the real crack by a continuous distribution of poles (concentrated forces and edge dislocations) along a single are lying between the real crack edges is introduced. The problem is reduced to an almost singular integral equation and an approximate method for its numerical solution is proposed. An application to the case of a symmetric crack in an infinite plane medium under uniform loading at infinity is also made.     

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.