Kinked cracks in anisotropic elastic materials

A mixed mode crack problem in functionally graded materials is formulated to a system of Cauchy singular Fredholm integral equations, then the system is solved by the singular integral equation method (SIEM). This specific crack problem has already been solved by N. Konda and F. Erdogan (Konda & Erdogan 1994). However, many mathematical details have been left out. In this paper we provide a detailed derivation, both analytical and numerical, on the formulation as well as the solution to the system of singular Fredholm integral equations. The research results include crack displacement profiles and stress intensity factors for both mode I and mode II, and the outcomes are consistent with the paper by Konda & Erdogan (Konda & Erdogan 1994).

A half-space containing a surface-breaking crack of uniform depth is subjected to three-dimensional dynamic loading. The elastodynamic stress-analysis problem has been decomposed into two problems, which are symmetric and antisymmetric, respectively, relative to the plane of the crack. The formulation of each problem has been reduced to a system of singular integral equations of the first kind. The symmetric problem is governed by a single integral equation for the opening-mode dislocation density. A pair of coupled integral equations for the two sliding-mode dislocation densities govern the antisymmetric problem. The systems of integral equations are solved numerically. The stress-intensity factors are obtained directly from the dislocation densities. The formulation is valid for arbitrary 3-D loading of the half-space. As an example, an applied stress field corresponding to an incident Rayleigh surface wave has been considered. The dependence of the stress-intensity factors on the frequency, and on the angle of incidence, is displayed in a set of figures.

Fracture analysis of a cracked orthotropic strip bonded to a magneto-electro-elastic layer

An essential part of describing the damage state and predicting the damage growth in a multicracked plate is the accurate calculation of stress intensity factors (SIF). Here, a methodology and rigorous solution formulation for SIF of a multicracked plate, with fully interacting cracks, subjected to a far-field arbitrary stress state is presented. The fundamental perturbation problem is derived, and the steps needed to formulate the system of singular integral equations whose solution gives rise to the evaluation of the SIF are identified. This analytical derivation and numerical solution are obtained by using intelligent application of symbolic computations and automatic FORTRAN generation capabilities in form of symbolic/FORTRAN package, named SYMFRAC, that is capable of providing accurate SIF at each crack tip. The accuracy of the results has been validated for the two parallel interacting crack problem. Limits and sensitivity of the results for the problem of a horizontal notch interacting with ten microcracks have been analyzed.

The material-toughening mechanism based on the crack-path deflection is studied. This investigation is based on a model which consists of a macrocrack (semi-infinite crack), with a curvilinear segment at the crack tip, situated in a brittle solid. The effect of material toughening is evaluated by comparison of the remote stress field parameters, such as the stress intensity factors (controlled by a loading on a macroscale), to effective values of these parameters acting in the vicinity of a crack tip (microscale). The effects of the curvilinear crack path are separated into three groups: crack-tip direction, crack-tip geometry pattern-shielding, and crack-path length change. These effects are analyzed by investigation of selected curvilinear crack patterns such as a macrocrack with simple crack-tip kink in the form of a circular arc and a macrocrack with a segment at the crack tip in the form of a sinusoidal wave. In conjunction with this investigation, a numerical procedure has been developed for the analysis of curvilinear cracks (or a system of cracks) in a two-dimensional linear elastic solid. The formulation is based on the solution of a system of singular integral equations. This numerical scheme was applied to the cases of finite and semi-infinite cracks.

An analytical solution to the fracture analysis of a functionally graded (FG) orthotropic substrate with FG piezoelectric coating weakened by a screw dislocation is carried out. The material properties are assumed to vary exponentially through the thickness of the layers. The problem is solved under various types of anti-plane shear and in-plane electric loadings. At first, by considering a single-screw dislocation at crack location, an analytical solution is developed. Next, by using the Fourier transform, the problem is reduced to a system of singular integral equations with Cauchy-type singularities. Then, by computing the dislocation densities, both the stress intensity factors and the stress fields at the crack tips under different electromechanical loadings are determined. In this investigation, various examples are solved to show the applicability of the proposed solution by studying the effects of the cracks configurations, material properties, and non-homogeneity parameter on the stress intensity factors.

In the present study, an alternative strategy for elasto-dynamic crack problems using boundary element method is discussed. The Laplace transformation is applied to time variation in the formulation. Since the numerical solutions are obtained in the Laplace-transformed domain, the numerical-inverse Laplace transformation is adopted to obtain time histories of the solutions. In the present method, the distributions of the displacement and traction in the vicinity of the crack tip are expressed as the superposition of singular terms and regular ones. The singular terms are the analytical solutions including stress intensity factors of Modes I and II, while the regular terms are approximated by the discrete polynomial functions in general BEM formulations. Additionally, the ligament side is discretized by singular-type crack elements in order to increase the accuracy of the traction near the crack tip. It is shown that the dynamic stress intensity factors (DSIFs) of Modes I and II are determined directly without supplementary procedures such as extrapolation or J-integral scheme. The validity of the present method is confirmed through the analysis of mixed mode dynamic problems of crack.

The interface crack problem for a piezoelectric semi-infinite strip under concentrated electromechanical loading

Cracks around railway wheel flats exposed to rolling contact loads and residual stresses

This paper investigates the thermoelectromechanical interaction among multi parallel axisymmetric cracks in a piezoelectric material under a uniform heat flow far away from the crack region. The crack faces are supposed to be insulated thermally and electrically. By using the Hankel transform, the thermal and electromechanical problems are reduced to systems of singular integral equations, respectively. The singular integral equations are solved numerically by using the Gauss-Jacobi integration formula. Numerical calculations are carried out, and detailed results are presented to illustrate the influence of the thermoelectromechanical interaction on the stress and electric displacement intensity factors. From numerical results, we can find that the normalized stress and electric displacement intensity factors are under the great influence of the crack spacing and the number of cracks. Especially, the influence of the thermoelectromechanical interaction on the fracture behavior is complicated when the crack spacing becomes small.

In this paper, the fracture problem of functionally graded material (FGM) was studied, and the shear modulus was assumed to be an exponential function. The influences of inhomogeneous parameter, crack size and crack angle on the stress intensity factors have been analyzed by the finite element method. The results indicated that the stress intensity factors of mode I decreased with the increasing of the crack angle, the stress intensity factors of mode II increased with the increasing of the crack angle, and the crack stress intensity factor of mode I and mode II decreased with the increasing of the inhomogeneous parameters at crack tips, which was of certain directive significance for the FGM design and manufacture in the actual engineering.

The processes of reflection of three-dimensional electromagnetic waves by locally irregular media interfaces are investigated. The boundary value problem for the system of Maxwell equations in an infinite half-space is reduced to the solution of two systems of singular integral equations. To construct a numerical algorithm for solving these systems of singular integral equations, the approximation and collocation method is used. Special attention is focused on calculation of the kernels of these equations. Solutions of systems of singular integral equations are used to calculated the far-field pattern for the scattered field in the far zone.

This paper investigates the theimoelectromechanical interaction among multi parallel cracks in a piezoelectric material under a uniform heat flow and a uniform mechanical load far away from the crack region The crack faces are supposed to be insulated thermally and electrically. By using the Fourier transform, the thermal and electromechanical problems are reduced to systems of singular integral equations, respectively. The singular integral equations are solved numerically by using the Gauss-Jacobi integration formula. Numerical calculations are carried out, and detailed results are presented to illustrate the influence of the thermoelectromechanical interaction on the stress and electric displacement intensity factors. The temperature-stress distributions are also presented. From numerical results, we can find that the normalized stress and electric displacement intensity factors are under the great influence of the crack spacing and the number of cracks. Especially, the influence of the thermoelectromechanical interaction on the fracture behavior is complicated when the crack spacing becomes small.

Underlying fracture trends and triggering on Mode-II crack branching and kinking for quasi-brittle solids

The problem for bonded half-planes containing a crack at an arbitrary angle to the graded interfacial zone