Elastodynamic Stress Intensity Factors Research Articles

Scattering of SH waves by an arbitrarily orientated closed crack

The 2D problem of a time-harmonic plane shear horizontal (SH) wave scattered by a finite closed crack in an isotropic material is presented in the paper. The crack is arbitrarily orientated with regard to the incident wave. A spring model based on the assumption that the traction components on the crack surfaces are linearly related to the crack opening displacement (COD) is used to model the closed crack. The problem is formulated in a set of boundary integral equations which contains the CODs as unknowns. Numerical examples are presented for the CODs, elastodynamic stress intensity factors, and the scattered displacement field for various parameters, such as spring stiffness, crack sizes and crack orientations. The results show that both the crack closure and orientation have significant effects on the scattered displacement field for the closed crack.

Finite-Element Calculation of Elastodynamic Stress Intensity Factors along a 3D Nonplanar Curved Crack

A numerical procedure is presented for calculation of the mixed-mode elastodynamic stress intensity factors for a stationary three-dimensional curved crack with nonplanar surfaces. The computational scheme is developed by modifying the concept of the elastodynamic Jk integrals, which are shown to be surface independent in a modified sense. As the near-front region needs to be included in the calculation, special treatment is thus addressed to modeling the elastodynamic singular behavior. Neither extra auxiliary solutions nor singular finite elements are required for the computation

A hypersingular time‐domain BEM for 2D dynamic crack analysis in anisotropic solids

AbstractA hypersingular time‐domain boundary element method (BEM) for transient elastodynamic crack analysis in two‐dimensional (2D), homogeneous, anisotropic, and linear elastic solids is presented in this paper. Stationary cracks in both infinite and finite anisotropic solids under impact loading are investigated. On the external boundary of the cracked solid the classical displacement boundary integral equations (BIEs) are used, while the hypersingular traction BIEs are applied to the crack‐faces. The temporal discretization is performed by a collocation method, while a Galerkin method is implemented for the spatial discretization. Both temporal and spatial integrations are carried out analytically. Special analytical techniques are developed to directly compute strongly singular and hypersingular integrals. Only the line integrals over an unit circle arising in the elastodynamic fundamental solutions need to be computed numerically by standard Gaussian quadrature. An explicit time‐stepping scheme is obtained to compute the unknown boundary data including the crack‐opening‐displacements (CODs). Special crack‐tip elements are adopted to ensure a direct and an accurate computation of the elastodynamic stress intensity factors from the CODs. Several numerical examples are given to show the accuracy and the efficiency of the present hypersingular time‐domain BEM. Copyright © 2008 John Wiley & Sons, Ltd.

A time-domain collocation-Galerkin BEM for transient dynamic crack analysis in anisotropic solids

This paper presents a time-domain boundary element method (BEM) for transient elastodynamic crack analysis in homogeneous and linear elastic solids of general anisotropy. A finite crack subjected to a transient loading is investigated. Two-dimensional (2D) generalized plane-strain or plane-stress condition is considered. The initial-boundary value problem is described by a set of hypersingular time-dependent traction boundary integral equations (BIEs), in which the crack-opening displacements (CODs) are unknown quantities. The hypersingular time-domain BIEs are first regularized to weakly singular ones by using spatial Galerkin method, which transfers the derivatives of the fundamental solutions to the unknown CODs and the weight functions. To solve the time-domain BIEs numerically, a time-stepping scheme is developed. The scheme applies the collocation method for temporal discretization of the time-domain BIEs. As spatial shape-functions, two different functions are implemented. For elements away from crack-tips, linear spatial shape-function is used, while for elements near the crack-tips a special ‘crack-tip shape-function’ is applied to describe the local ‘square-root’ behavior of the CODs at the crack-tips properly. Special attention of the analysis is devoted to the numerical computation of the transient elastodynamic stress intensity factors for cracks in general anisotropic and linear elastic solids. Numerical examples are presented to verify the accuracy of the present time-domain BEM.

3-D Transient Dynamic Crack Analysis by a Novel Time-Domain BEM

A novel non-hypersingular time-domain traction BEM is presented for three-dimensional (3D) transient elastodynamic crack analysis. The initialboundary value problem is formulated as a set of nonhypersingular time-domain traction boundary integral equations (BIEs). To solve the time-domain traction BIEs, a time-stepping scheme based on the convolution quadrature formula of Lubich (1988a,b; 1994) for temporal discretization and a collocation method for spatial discretization is adopted. Numerical examples are given for an unbounded solid with a penny-shaped crack under a tensile and shear impact loading. A comparison of the present time-domain BEM with the conventional one shows that the novel time-domain method is much more stable and less sensitive to the choice of the used timesteps. keyword: 3-D Time-domain boundary element method, Non-hypersingular boundary integral equations, Transient elastodynamic crack analysis, Elastodynamic stress intensity factors.

Elastodynamic stress intensity factors for a semi-infinite crack due to three-dimensional concentrated shear loadings on the crack faces

The dynamic stress intensity factors for a semi-infinite crack in an otherwise unbounded elastic body is investigated. The crack is subjected to a pair of suddenly-applied shear point loads on its faces at a distance l away from the crack tip. This problem is treated as the superposition of two problems. The first problem considers the disturbance by a concentrated shear force acting on the surface of an elastic half space, while the second problem discusses a half space with its surface subjected to the negative of the tangential surface displacements induced by the first problem in the front of the crack edge. A fundamental problem is proposed and solved by means of integral transforms together with the application of the Wiener-Hopf technique and Cagniard-de Hoop method. Exact expressions are then derived for the mode II and III dynamic stress intensity factors by taking integration over the fundamental solution. Some features of the solutions are discussed.

Elastodynamic stress intensity factor due to three-dimensional concentrated transient loading on the faces of a crack

The dynamic stress intensity factor for a semi-infinite crack in an otherwise unbounded elastic body is analyzed. The crack is subjected to a pair of suddenly applied point loads on its faces at a distancel away from the crack tip. The solution of the problem is obtained by superposition of the solutions of two simpler problems. The first of these problems is Lamb's problem, while the second problem considers a half space with its surface subjected to the negative of the normal displacement induced by Lamb's problem in the rangex>0. The latter is sovled by means of integral transforms together with the application of Weiner-Hopf technique and Cagniard-de Hoop method. An exact expression is derived for the mode I stress intensity factor as a function of time for any point along the crack edge. Some features of the solution are discussed.

Stress intensity factor histories of a steadily propagating crack under 3-D combined mode traction

Elastodynamic stress intensity factor histories of an unbounded solid containing a semi-infinite plane crack that propagates at a constant velocity under 3-D time-independent combined mode loading are considered. The fundamental solution, which is the response of point loading, is obtained. Then, stress intensity factor histories of a general loading system are written out in terms of superposition integrals. The methods used here are the Laplace transform methods in conjunction with the Wiener-Hopf technique.

Elastodynamic analysis of nonplanar cracks in a half-space

The interaction of time harmonic antiplane shear waves with nonplanar cracks embedded in an elastic half-space is studied. Based on the qualitatively similar features of crack and dislocation, with the aid of image method, the problem can be formulated in terms of a system of singular integral equations for the density functions and phase lags of vibrating screw dislocations. The integral equations, with the dominant singular part of Hadamard's type, can be solved by Galerkin's numerical scheme. Resonance vibrations of the layer between the cracks and the free surface are observed, which substantially give rise to high elevation of local stresses. The calculations show that near-field stresses due to scattering by a single crack and two cracks are quite different. The interaction between two cracks is discussed in detail. Furthermore, by assuming one of the crack tips to be nearly in contact with the free surface, the problem can be regarded as the diffraction of elastic waves by edge cracks. Numerical results are presented for the elastodynamic stress intensity factors as a function of the wave number, the incident angle, and the relative position of the cracks and the free surface.

Effect of an inclusion on the interaction of elastic waves with a crack

The problem of the scattering of time-harmonic elastic waves by a configuration of a crack and an adjacent inclusion embedded in a 2-D infinite elastic medium is investigated by the traction boundary integral equation (TBIE0 method. The result of the formulation is a system of six coupled singular integral equations for the unknown displacement jumps across the crack as well as for the displacement and traction components on the interface boundary between the inclusion and elastic medium. A simple regularization method is used to regularize the hypersingular arisen in teh TBIE. Numerical results for elastodynamic stress intensity factors have been obtained for several geometrical configurations, where the influence of the incident angle of plane longitudinal waves, the size and location of the inclusion, and the hardness of the inclusion on the stress intensity factors is fully discussed.

Elastodynamic stress-intensity factors for a semi-infinite crack under 3-D combined mode loading

The dynamic stress intensity factor histories for a half plane crack in an otherwise unbounded elastic body are analyzed. The crack is subjected to a traction distribution consisting of two pairs of suddenly-applied shear point loads, at a distance L away from the crack tip. The exact expression for the combined mode stress intensity factors as the function of time and position along the crack edge is obtained. The method of solution is based on the direct application of integral transforms together with the Wiener-Hopf technique and the Cagniard-de Hoop method, which were previously believed to be inappropriate. Some features of solutions are discussed and the results are displayed in several figures.

Exact elastodynamic analysis of some fracture specimen models involving strip geometries

A procedure for analysing a class of transient elastodynamic crack problems is presented. These problems model certain experimental situations which can be used to infer fracture toughness values for materials under stress-wave loadings. In particular, the present analysis provides exact expressions for the elastodynamic stress intensity factor at the tip of a long external crack in a striplike body whose lateral (upper and lower) boundaries are parallel to the crack line. Plane stress/strain conditions are assumed to prevail. In this class of problems, the crack may be situated asymmetrically with respect to the mid-strip line, and various dynamic loadings are considered, including crack face tractions and lateral face displacements. The loadings considered will have an arbitrary time dependence, but will be spatially uniform. The problem analysis is based on integral transforms and an asymptotic usage of the Wiener-Hopf technique. Two useful cases are considered in detail as examples; the symmetrically cracked strip with traction-free lateral boundaries under sudden crack face pressure, and the symmetrically cracked strip with suddenly displaced shear-free lateral boundaries.

A non‐hypersingular time‐domain BIEM for 3‐D transient elastodynamic crack analysis

AbstractA three‐dimensional (3‐D) time‐domain boundary integral equation method (BIEM) is presented for transient elastodynamic crack analysis. A non‐hypersingular traction BIE formulation is used with the crack opening displacements and their derivatives as unknown quantities. A collocation method in conjunction with a time‐stepping scheme is developed to solve the non‐hypersingular time‐domain BIEs. To simplify the analysis and to describe the proper behaviour of the unknown quantities at the crack front, a constant spatial shape function is applied for elements away from the crack front, while a spatial ‘square‐root’ crack‐tip shape function is adopted for elements near the crack front. A linear temporal shape function is used in the time‐stepping scheme. Numerical calculations, have been carried out for penny‐shaped and square cracks. Results for the elastodynamic stress intensity factors are presented as functions of the temporal and the spatial variables. For the test examples considered, good agreement between the present results and those of other authors is obtained.

Interaction of antiplane cracks with elastic waves in transversely isotropic materials

The interaction of plane time-harmonic SH-waves with micro-cracks in transversely isotropic materials is investigated. Elastic wave scattering by a single micro-crack is first analyzed. The scattered displacement is expressed as a Fourier integral containing the crack opening displacement. By using this representation formula and by invoking the traction-free boundary condition on the faces of the crack, a boundary integral equation for the unknown crack opening displacement is obtained. Expanding the crack opening displacement into a series of Chebyshev polynomials and adopting a Galerkin method, the boundary integral equation is converted into an infinite system of inear algebraic equations for the expansion coefficients which is solved numerically. Numerical results are presented for the elastodynamic stress intensity factors, the scattered far-field and the scattering cross section of a single crack. Then, propagation of plane time-harmonic SH-waves in a transversely isotropicmaterial permeated by a random and dilute distribution of micro-cracks is investigated. The effects of the micro-crack density on the attenuation coefficient and the phase velocity are analyzed by appealing to a simple energy consideration and by using Kramers-Kronig relations.

Shear and torsional impact of cracked viscoelastic bodies—A Numerical integral equation/transform approach

The transient elastodynamic stress intensity factor was determined for a cracked linearly viscoelastic body under impact loading. Two separate geometries along with associated loading conditions were considered. In the first case, the body is in the form of an infinite strip containing a central finite-length crack and is subjected to anti-plane shear tractions. Various strip heights are considered including the possibility of a body of nearly infinite extent. In the second case, the body is of infinite extent containing a finite-length penny-shaped crack and is subjected to radial shear and torsional (twisting) tractions. The analytical parts of the solutions are either given by a previous analysis of H. G. Georgiadis or are obtained from results by G. C. Sih through the use of the correspondence principle. The numerical procedure consists of solving integral equations and then inverting the Laplace transformed solution by the Dubner-Abate-Crump technique. Numerical results were given for the standard linear solid by considering several combinations of material constants.

Elastodynamic Analysis of a Periodic Array of Mode III Cracks in Transversely Isotropic Solids

Time-harmonic elastodynamic analysis is presented for a periodic array of collinear mode III cracks in an infinite transversely isotropic solid. The scattering problem by a single antiplane crack is first formulated, and the scattered displacement field is expressed as Fourier integrals containing the crack opening displacement. By using this representation formula and by considering the periodicity conditions in the crack spacing, a boundary integral equation is obtained for the crack opening displacement of a reference crack. The boundary integral equation is solved numerically by expanding the crack opening displacement into a series of Chebyshev polynomials. Numerical results are given to show the effects of the crack spacing, the wave frequency, the angle of incidence, and the anisotropy parameter on the elastodynamic stress intensity factors.

Transient stress intensity factors for a cracked plane strip under anti-plane point forces

The transient elastodynamic stress intensity factors of a semi-infinite crack in an elastic infinite strip of finite width are analyzed. The crack is subjected to a pair of suddenly applied anti-plane concentrated point loadings on its faces at a distance l away from the crack-tip. The crucial steps in the analysis are the direct application of integral transforms together with the Wiener-Hopf technique. The functions of exponential type, which are introduced by the fixed characteristic lengths in geometry and in loading, must be split explicitly into the form of product and sum, respectively, of regular functions in the Wiener-Hopf equation. Instead of performing the quotient splitting directly, however, the corresponding term is expanded first in a series. The solution is then constructed as a series accordingly. Each term in the solution series can be interpreted as the contribution of waves that have reflected at the strip surfaces different times. Exact expressions are obtained for the resulting mode-III stress intensity factors as functions of time. For illustration, the first three terms in the series for the stress intensity factor history are then computed. The results are exact for the time interval from initial loading until the first wave scattered at the crack tip is reflected three times at the strip surface and returns to the crack tip. Numerical results show that the maximum of the transient stress intensity factors of symmetric strips are larger than the ones for asymmetric strips. Moreover the smaller strip height is, the larger is the dynamic overshot on the stress intensity factors for the cases of the symmetric strips.

3D elastodynamic crack analysis by a non-hypersingular BIEM

A boundary integral equation method (BIEM) is presented for 3D elastodynamic crack analysis. The method is based on a non-hypersingular BIE formulation, where the unknown quantities are the crack opening dispacements and their derivatives. The numerical scheme applied here uses a constant shape function for elements away from the crack front, and a “square-root” crack-tip shape function for elements near the crack front to describe the proper behavior of the unknown quantities at the crack front. A collocation method is applied to convert the non-hypersingular BIEs to a system of linear algebraic equations which are solved numerically. For several geometrical configurations, numerical results are presented for both the elastodynamic stress intensity factors and the scattering cross section. They are in good agreement with those obtained by other authors.

Elastodynamic response of an interface crack in a layered composite under anti-plane shear impact load

The elastodynamic response of a four-layered composite with an interface crack under anti-plane shear impact load is considered in this paper. Laplace and Fourier transforms are applied to reduce this mixed boundary value problem to a Cauchy-type singular integral equation of the first kind in Laplace transform plane, which is solved numerically. A Laplace inversion technique developed by Miller and Guy is then used to get the solution in the physical plane. Finally, the elastodynamic stress intensity factors are obtained as function of time, geometrical parameters and material properties.

Elastodynamic analysis of periodic antiplane cracks near and parallel to an interface

Time-harmonic elastodynamic analysis is presented for periodic antiplane cracks near and parallel to an interface of two joined half-planes of different material properties. The scattering problem by a single crack near and parallel to an interface is first formulated, and the scattered displacements are expressed as Fourier integrals. By using this representation formula and by considering the periodicity condition in the crack spacing, a boundary integral equation is obtained for the crack opening displacement of a reference crack. The boundary integral equation is solved numerically by the use of a Galerkin method. Elastodynamic stress intensity factors are computed for several values of geometrical and material parameters.