Half-plane Crack Research Articles

Influence of FGM coating on the dynamic fracture behavior of multiple cracks in a homogeneous half-plane under in-plane loading

Influence of FGM coating on the dynamic fracture behavior of multiple cracks in a homogeneous half-plane under in-plane loading

Fundamental solutions for penny-shaped and half-plane cracks in one-dimensional hexagonal quasicrystals: Shear mode

The present article is to investigate the penny-shaped and half-plane crack problems in an infinite space of one-dimensional hexagonal quasicrystals. The cracks are subjected to a pair of anti-symmetrical point shear loads exerted on the crack surfaces. By means of the potential theory method, the governing integral equation is developed and the complete fundamental solutions of the phonon and phason fields are obtained. Moreover, the important quantities on the crack plane, including the crack slip displacement and the stress intensity factor, are derived in terms of elementary functions. The fundamental solutions presented in the article are important to boundary element analysis and may serve as benchmarks for numerical research.

Indirect boundary element method for modelling 2‐D poroelastic wave diffraction by cavities and cracks in half space

AbstractAn indirect boundary element method (IBEM) is developed to solve two‐dimensional (2‐D) wave diffraction by cavities and cracks in a fluid‐saturated poroelastic half space. The dynamic Green's functions of forces and fluid source in a poroelastic full‐space, rather than their half‐space counterparts, are utilized, because of their closed‐form expressions. The diffracted wave field is constructed by distributing forces and sources in fictitious densities over the boundaries, based on the single‐layer potential theory. The densities are determined by matching the approximate solution with the boundary conditions. The nonuniqueness of integral equation caused by the degenerated geometry of a crack is treated by a zoned approach. The numerical accuracy and stability of the IBEM are verified by comparing with available results in the literature. Problems of wave scattering by cavity and flat crack in saturated half plane are solved. The displacement amplification effect on the free surface is investigated. The numerical solutions show that the response is strongly dependent on factors such as wave frequency, incident angle, the length and depth of the crack, and material properties.

Dynamics of an Elastic Punch on an Elastic Half-Plane with Crack Formation

The dynamics of buildings and structures in an earthquake zone is studied relying on a model boundary value problem describing the dynamics of an elastic punch on an elastic half-plane under diffraction and refraction of waves generated by stress release on a crack. The problem is solved by applying an explicit difference scheme constructed using the method of bicharacteristics combined with dimensional splitting. The problems of wave diffraction and refraction by an elastic inclusion in the release of vertical and shear stresses on a horizontal crack in an elastic half-plane are considered, which model the dynamics of seismic processes in the formation of mode I and II cracks. Computer experiments are performed for media with various elastic parameters and various punch–crack distances. Their results are presented in the form of oscillograms on the ground surface and the punch, as well as diffraction displacement velocity fields and stress tensor invariants in the medium and the punch.

A screw dislocation interacting with a semi-infinite interface crack and two semi-infinite cracks perpendicular to the bimaterial interface

Using conformal mapping and the image method, we derive an analytic solution to the problem of an isotropic elastic bimaterial involving a screw dislocation interacting with a semi-infinite interface crack, a semi-infinite crack in the upper half-plane perpendicular to the interface, and another semi-infinite crack in the lower half-plane also perpendicular to the interface. Closed-form expressions of the image force acting on the screw dislocation and the mode III stress intensity factors at the three crack tips are derived.

Half Circle Position for Arc Cracks in Half Plane

In this paper, the problem of arc cracks that lie in the boundary of half circle in an elastic half plane is investigated. The complex potential variables with free traction boundary condition is used to formulate the problem into a singular integral equation. The singular integral equation is solved numerically for the unknown distribution dislocation function with the help of curve length coordinate method. The numerical results have shown that our results are in good agreement with the previous works. Stress intensity factors for different cracks position are presented.

Numerical solutions for cracks in an elastic half-plane

The behavior of the stress intensity factor at the tips of cracks subjected to uniaxial tension $$\sigma ^{\infty }_{x}=p$$ with traction-free boundary condition in half-plane elasticity is investigated. The problem is formulated into singular integral equations with the distribution dislocation function as unknown. In the formulation, we make used of a modified complex potential. Based on the appropriate quadrature formulas together with a suitable choice of collocation points, the singular integral equations are reduced to a system of linear equations for the unknown coefficients. Numerical examples show that the values of the stress intensity factor are influenced by the distance from the cracks to the boundary of the half-plane and the configuration of the cracks.

The linear steady state analysis of multiple moving cracks in a piezoelectric half-plane under in-plane electro-elastic loading

In this work, the steady state problem of multiple Yoffe-type cracks propagating in a piezoelectric half-plane within the framework of linear electroelasticity under in-plane electro-elastic loading is studied. At first, the closed-form solution of the moving electric and Volterra type climb and glide edge dislocations are derived using the complex Fourier transforms to achieve the integral equations of a piezoelectric half-plane with several moving cracks. Then, the integral equations with Cauchy-kind singularity are solved numerically to determine the mixed mode stress intensity factors and the electric displacement intensity factors in a piezoelectric medium. Finally, the effects of the loading conditions, crack moving speed, cracks lengths, cracks interactions and geometrical parameters on the field intensity factors are considered.

A Problem of Fatigue Fracture Mechanics of a Two-Layer Material with Edge Cracks

A problem for an elastic half-plane consisting of two tightly coupled homogeneous isotropic elastic media and containing edge cracks is considered. Its lateral surface is subjected to a given external load. The growth of a fatigue crack in the two-layer half-plane is investigated. For some materials, the number of cycles to failure is calculated. The stress intensity factor is determined, and a numerical analysis is performed.

Response of an orthotropic half-plane subjected to transient anti-plane loading with multiple edge cracks

This study deals with the problem of multiple edge cracks in an elastic orthotropic half-plane under transient loading. The dislocation solution is utilized to derive integral equations for multiple interacting edge cracks in an orthotropic half-plane. These equations are solved numerically thereby obtaining the dislocation density function on the crack faces and stress intensity factors at crack tips. Numerical results are obtained to illustrate the variation of the dynamic stress intensity factors as a function of crack length and material properties.

A semi-numerical model for near-critical angle scattering.

Numerous phenomena in the fields of physics and mathematics as seemingly different as seismology, ultrasonics, crystallography, photonics, relativistic quantum mechanics, and analytical number theory are described by integrals with oscillating integrands that contain three coalescing criticalities, a branch point, stationary phase point, and pole as well as accumulation points at which the speed of integrand oscillation is infinite. Evaluating such integrals is a challenge addressed in this paper. A fast and efficient numerical scheme based on the regularized composite Simpson's rule is proposed, and its efficacy is demonstrated by revisiting the scattering of an elastic plane wave by a stress-free half-plane crack embedded in an isotropic and homogeneous solid. In this canonical problem, the head wave, edge diffracted wave, and reflected (or compensating) wave each can be viewed as a respective contribution of a branch point, stationary phase point, and pole. The proposed scheme allows for a description of the non-classical diffraction effects near the "critical" rays (rays that separate regions irradiated by the head waves from their respective shadow zones). The effects include the spikes present in diffraction coefficients at the critical angles in the far field as well as related interference ripples in the near field.

Cracking in orthotropic half-plane with a functionally graded coating under anti-plane loading

This investigation evaluates, by the dislocation method, the dynamic stress intensity factors of cracked orthotropic half-plane and functionally graded material coating of a coating-substrate material due to the action of anti-plane traction on the crack surfaces. First, by using the complex Fourier transform, the dislocation problem can be solved and the stress fields are obtained with Cauchy singularity at the location of dislocation. The dislocation solution is utilized to derive integral equations for multiple interacting cracks in the orthotropic half-plane with functionally graded orthotropic coating. Several examples are solved and dynamic stress intensity factors are obtained.

Dynamic fracture analysis in an orthotropic half-plane with FGM coating containing several moving cracks

This paper considers several finite moving cracks in an orthotropic half-plane with non-homogenous coating subjected to anti-plane deformation. The distributed dislocation technique is used to carry out stress analysis in the medium containing moving cracks under anti-plane loading. By utilizing the complex Fourier transformation technique, the stress fields are obtained for an orthotropic half-plane containing a screw dislocation. The Galilean transformation is employed to express the wave equations in terms of coordinates that are attached to the moving crack. Stress fields are derived in the medium under anti-plane point forces. These solutions are Cauchy singular at the location of dislocations and the point of application of forces. The solution is employed to derive integral equations for a half-plane weakened by moving cracks. Numerical results are provided to show the effects of material properties, the crack length and the speed of the crack propagating upon the stress intensity factor.

Development of a distributed dislocation dipole technique for the analysis of multiple straight, kinked and branched cracks in an elastic half-plane

A distributed dislocation dipole technique for the analysis of multiple straight, kinked and branched cracks in an elastic half plane has been developed. The dipole density distribution is represented with a weighted Jacobi polynomial expansion where the weight function captures the asymptotic behaviour at each end of the crack. To allow for opening and sliding at crack kinking and branching the dipole density representation contains conditional extra terms which fulfills the asymptotic behaviour at each endpoint. Several test cases involving straight, kinked and branched cracks have been analysed, and the results suggest that the accuracy of the method is within 1% provided that Jacobi polynomial expansions up to at least the sixth order are used. Adopting even higher order Jacobi polynomials yields improved accuracy. The method is compared to a simplified procedure suggested in the literature where stress singularities associated with corners at kinking or branching are neglected in the representation for the dipole density distribution. The comparison suggests that both procedures work, but that the current procedure is superior, in as much as the same accuracy is reached using substantially lower order polynomial expansions.

Evaluation of the T-stress for multiple cracks in an elastic half-plane using singular integral equation and Green’s function method

This paper studies the T-stress problem for multiple cracks in an elastic half-plane using the singular integral equation and Green’s function method. In the relevant perturbation field, the boundary of the elastic half-plane is traction free. In the solution, the complex potentials are decomposed into two parts, the principal part and the complementary part. The principal part of the complex potentials is derived from dislocation distributions along the crack faces. The role of the complementary part is to eliminate the traction along the boundary of half-plane caused by the principal part. Finally, a singular integral equation is formulated in which the dislocation distributions are unknown function. The explicit formulae for evaluating the stress intensity factors and the T-stresses at the crack tips are provided. Several numerical examples are presented.

Влияние тонкой накладки на коэффициент интенсивности напряжений в задачах о поперечной трещине в полуплоскости и полосе

Влияние тонкой накладки на коэффициент интенсивности напряжений в задачах о поперечной трещине в полуплоскости и полосе

Thermo-Elastic Analysis of a Cracked Half-Plane Under a Thermal Shock Impact Using the Hyperbolic Heat Conduction Theory

The transient thermal stresses around a crack in a thermo-elastic half-plane are obtained under a thermal shock using the hyperbolic heat conduction theory. Fourier, Laplace transforms and singular integral equations are applied to solve the temperature and thermal stress fields consecutively. The integral equations are solved numerically and the asymptotic fields around the crack tip are obtained. Numerical results show that the hyperbolic heat conduction have significant influence on the dynamic temperature and stress field. It is suggested that to design materials and structures against fracture under thermal loading, the hyperbolic model is more appropriate than the Fourier heat conduction model.

Thermostressed state of a piezoceramic body with a plane crack in a symmetric heat flow from its surfaces

A problem of thermoelectroelasticity for a piezoelectric material with a plane crack of arbitrary shape in a symmetric heat flow from the crack surfaces is solved. The crack is in a plane perpendicular to the polarization axis. We establish a correspondence between the stress intensity factor (SIF) and electric displacement intensity factor (EDIF) for a piezoelectric material with a plane crack and the SIF for a thermoelastic isotropic material with a crack of the same shape under the same thermal load. This correspondence allows finding the SIF and EDIF for an electroelastic material from the solution of the thermoelastic problem. The SIF and EDIF for a penny-shape crack (for different cases of thermal loading) and a half-plane crack are found as examples

The G2 constant displacement discontinuity method – Part II: Solution of half-plane crack problems

In the previous Part I, the G2 constant displacement discontinuity element was presented that is dedicated for the fast (only one collocation point per element), stable and accurate numerical solution of modes I, II and III cracks of arbitrary shape in an infinite plane isotropic elastic body. Herein, another G2 constant displacement discontinuity element is constructed for the case of cracks in the half-plane. It is successfully validated against existing semi-analytical and numerical solutions of crack problems in the half-plane.

Evaluation of the Lazarus–Leblond constants in the asymptotic model of the interfacial wavy crack

The paper addresses the problem of a semi-infinite plane crack along the interface between two isotropic half-spaces. Two methods of solution have been considered in the past: Lazarus and Leblond [1998a. Three-dimensional crack-face weight functions for the semi-infinite interface crack-I: variation of the stress intensity factors due to some small perturbation of the crack front. J. Mech. Phys. Solids 46, 489–511, 1998b. Three-dimensional crack-face weight functions for the semi-infinite interface crack-II: integrodifferential equations on the weight functions and resolution J. Mech. Phys. Solids 46, 513–536] applied the “special” method by Bueckner [1987. Weight functions and fundamental fields for the penny-shaped and the half-plane crack in three space. Int. J. Solids Struct. 23, 57–93] and found the expression of the variation of the stress intensity factors for a wavy crack without solving the complete elasticity problem; their solution is expressed in terms of the physical variables, and it involves five constants whose analytical representation was unknown; on the other hand, the “general” solution to the problem has been recently addressed by Bercial-Velez et al. [2005. High-order asymptotics and perturbation problems for 3D interfacial cracks. J. Mech. Phys. Solids 53, 1128–1162], using a Wiener-Hopf analysis and singular asymptotics near the crack front. The main goal of the present paper is to complete the solution to the problem by providing the connection between the two methods. This is done by constructing an integral representation for Lazarus–Leblond's weight functions and by deriving the closed form representations of Lazarus–Leblond's constants.