Crack-face Loading Research Articles

Weight functions for an array of collinear cracks in an infinite anisotropic elastic plate

Abstract Weight functions are provided for calculating the stress intensity factors for collinear cracks in an infinite anisotropic elastic plate. The number of cracks as well as the length of the cracks are arbitrary. The weight functions are for crack-face loading, which may be non-self-equilibrium. It is shown that for self-equilibrium loading, the weight functions are independent of elastic constants. For non-self-equilibrium loading, however, an additional constant term that is material-dependent appears.

Generalized T-stress solutions for penny-shaped cracks in transversely isotropic piezoelectric media

A novel set of analytical and semi-analytical solutions is obtained for the generalized T-stress for both electrically impermeable and permeable penny-shaped cracks embedded in a transversely isotropic, linear piezoelectric, infinite medium under uniform far-field loading and general mechanical and electrical crack-face loading. The generalized stress field of the crack for two basic loading cases, viz. uniformly distributed self-equilibrated crack-face loading and a pair of crack-face self-equilibrated unit normal point forces and unit point electrical charges, are employed together with the asymptotic near-front expansion to establish fundamental solutions for the generalized T-stress. Such fundamental solutions are explicit and involve only elementary functions, and are then utilized to form an explicit integral formula for computing the generalized T-stress of a penny-shaped crack under general crack-face loading. The exact integration of the integral formula is also established for specific loading conditions, whereas its evaluation via an efficient quadrature for general loading conditions is also addressed. A selected set of numerical results is presented to verify the proposed quadrature and significant and useful findings are also reported.

An antiplane shear crack moving in one-dimensional hexagonal quasicrystals

An investigation is presented of an antiplane shear strip crack subjected to representative general non-constant crack-face loading conditions which is moving uniformly in one-dimensional hexagonal quasicrystals, using an extended technique of continuous dislocation layers. Readily-calculable explicit closed-form representations are determined and discussed for the components of the phonon and phason stress fields created throughout the media. Illustrative numerical results are presented graphically. As a special case, the corresponding expressions are deduced explicitly for a non-uniformly loaded stationary crack.

Weight function for an edge-cracked rectangular plate

In-situ edge-cracked rectangular plates (width H, length L) comprised of first-year sea ice on McMurdo Sound were tested during recent field trips to Antarctica, and expressions applicable for a wide range of crack lengths are sought for the stress-intensity-factor (SIF), and crack-opening-displacement (COD), given that the edge-crack is subjected to arbitrary crack-face loading. A weight function able to provide the required accurate wide-ranging expressions for an edge-cracked rectangular plate (ECRP) subject to arbitrary crack-face loading is developed in this paper, given H/L=0.25,0.5,1.0,1.5,2.0, and 4.0. The accuracy of the ECRP weight function is assessed by making comparisons with the edge-cracked strip subjected to pure bending (using H/L=4.0), using the essentially identical SIF and CMOD results independently obtained by Kaya and Erdogan (1987) and Bakker (1995). Comparisons are also made with the SIF values obtained by Fett (1999) for the ECRP subjected to pure bending (for H/L=1.0,1.5, and 2.0). The double-cantilever-beam (DCB) study in Foote and Buchwald (1985) is used to assess the accuracy of the ECRP SIF for the case of concentrated loads at the crack mouth, for H/L=1/4 and H/L=1/2. For the same loading and H/L=4, the study by Kaya and Erdogan (1980) of an edge-cracked infinite strip is used to assess accuracy. For crack-face concentrated loading acting at X/L=yi, given H/L=4, the accuracy of the ECRP SIF weight function is assessed by examining associated data reported by Kaya and Erdogan (1978).

Comparative study of time and frequency domain BEM approaches in frictional contact problem for antiplane crack under harmonic loading

Abstract Two different boundary element methods (BEM) for crack analysis in two dimensional (2-D) antiplane, homogeneous, isotropic and linear elastic solids by considering frictional contact of the crack edges are presented. Hypersingular boundary integral equations (BIE) in time-domain (TD) and frequency domain (FD), with corresponding elastodynamic fundamental solutions are applied for this purpose. For evaluation of the hypersingular integrals involved in BIEs a special regularization process that converts the hypersingular integrals to regular integrals is applied. Simple regular formulas for their calculation are presented. For the problems solution while considering frictional contact of the crack edges a special iterative algorithm of Udzava's type is elaborated and used. Numerical results for crack opening, frictional contact forces and dynamic stress intensity factors (SIFs) are presented and discussed for a finite III-mode crack in an infinite domain subjected to a harmonic crack-face loading and considering crack edges frictional contact interaction using the TD and FD approaches.

T-Stresses for Edge Cracks and Vanishing Ligaments

An edge-cracked half-plane 0 < x < A and a half-plane x > 0 with a semi-infinite crack x > a perpendicular to the edge are examined in this paper. Uniform crack-face loading is thoroughly examined, with a thorough exposition of the Koiter Wiener–Hopf approach (Koiter, 1956, “On the Flexural Rigidity of a Beam Weakened by Transverse Saw Cuts,” Proc. Royal Neth. Acad. of Sciences, B59, pp. 354–374); an analytical expression for the corresponding T-stress is obtained. For the additional cases of (i) nonuniform edge-crack crack-face loading σ(x/A)k (ℜ(k)>-1), (ii) concentrated loading at the edge-crack crack mouth, the Wiener–Hopf solutions and analytical T-stress expressions are provided, and tables of T-stress results for σ(x/A)k and σ(1-x/A)k are presented. A Green's function for the edge-crack T-stress is developed. The differing developments made by Koiter (1956, “On the Flexural Rigidity of a Beam Weakened by Transverse Saw Cuts,” Proc. Royal Neth. Acad. of Sciences, B59, pp. 354–374, Wigglesworth (1957, “Stress Distribution in a Notched Plate,” Mathematika, 4, pp. 76–96), and Stallybrass (1970, “A Crack Perpendicular to an Elastic Half-Plane,” Int. J. Eng. Sci., 8, pp. 351–362) for the case of an edge-cracked half-plane are enhanced by deducing a quantitative relationship between the three different Wiener–Hopf type factorizations. An analytical universal T-stress expression for edge-cracks is derived. Finally, the case of a vanishing uncracked ligament in a half-plane is examined, and the associated Wiener–Hopf solution and analytical T-stress expression are again provided. Several limiting cases are examined.

A cohesive edge crack

The nucleation and growth of a cohesive edge crack is studied. This topic is germane to the initiation and early stage of crack growth during unnotched beam testing, the growth of short edge cracks in finite test pieces, and the formation of tension cracks of geological origin. This paper focuses on an edge crack in a semi-infinite plane, under a uniform far-field tensile stress acting parallel to the plane boundary. Expressions for the Mode I stress-intensity-factor and crack-opening-displacement for an edge crack subjected to arbitrary crack face loading are determined via the weight function method. All of the constants needed to define the weight function and associated integrals are themselves explicit functions of just two constants: f r and ψ. Two types of softening behavior in the cohesive zone are examined: rectangular softening, and linear softening. In each case the process zone size, energy-release-rate, crack-opening displacement and load-ratio are examined. The different test behavior exhibited under load-control versus fixed-grip displacement control is explored. The test control conditions alter the fracture behavior significantly. For a linear softening cohesive edge crack, it is found out that under fixed-grip control (load-control), the process zone size decreases (increases) steadily with increasing traction-free crack length, approaching the semi-infinite crack asymptote from above (below). The differences between load-control versus fixed-grip control decrease rapidly with increasing traction-free crack length.

Asymptotic fields ahead of mixed mode frictional cohesive cracks

AbstractThis paper discusses the asymptotic fields ahead of mixed mode frictional cohesive cracks in quasi‐brittle materials. These fields have been derived after reformatting the cohesive‐law in a special but universal polynomial containing fractional or integer powers. This special form ensures that the radial and angular variations of the asymptotic fields are separable as in the Williams expansions for a traction‐free crack. The coefficients of the expansions however depend nonlinearly on the softening law and the boundary conditions. As expected, the asymptotic field of a frictional cohesive crack reduces to that of a frictionless cohesive crack with normal cohesive separation when the friction coefficient becomes zero. It is also shown that the asymptotic field of a frictionless cohesive crack can be used to a non‐cohesive crack opened by crack face loading, after including the singular terms corresponding to the eigenvalue 1/2. Furthermore, the coefficients are given explicitly for two special softening laws that are commonly used in practice. The leading terms of the true displacement asymptotic field are also derived explicitly. These are especially useful as the enrichment functions at the tip of a mixed mode cohesive crack for accurate simulation of crack growth in quasi‐brittle materials using the extended finite element (XFEM).

Reconsidering the boundary conditions for a dynamic, transient mode I crack problem

A careful examination of a dynamic mode I crack problem leads to the conclusion that the commonly used boundary conditions do not always hold in the case of an applied crack face loading, so that a modification is required to satisfy the equations. In particular, a transient compressive stress wave travels along the crack faces, moving outward from the loading region on the crack face. This does not occur in the quasistatic or steady state problems, and is a special feature of the transient dynamic problem that is important during the time interval immediately following the application of crack face loading. We demonstrate why the usual boundary conditions lead to a prediction of crack face interpenetration, and then examine how to modify the boundary condition for a semi-infinite crack with a cohesive zone. Numerical simulations illustrate the resulting approach.

Moving antiplane shear crack in transversely isotropic magnetoelectroelastic media

An antiplane shear strip crack moving uniformly in transversely isotopic magnetoelectroelastic media when subjected to representative non-constant crack-face loading conditions is studied. Readily calculable explicit closed-form representations are determined and discussed for the components of the stress, electric and magnetic fields created throughout the material. Representative numerical data are presented. Alternative boundary conditions for which corresponding analyses can be derived analogously are listed.

The T-stress solutions for through-wall circumferential cracks in cylinders subjected to general loading conditions

The elastic T-stress is a parameter used to define the level of constraint at a crack tip. It is important to provide T-stress solutions for practical geometries to apply the constraint-based fracture mechanics methodology. In the present work, T-stress solutions are provided for circumferential through-wall cracks in thin-walled cylinders. First, cylinders with a circumferential through-wall crack were analyzed using the finite element method. Three cylinder geometries were considered; defined by the mean radius of the cylinder ( R) to wall thickness ( t) ratios: R/ t = 5, 10, and 20. The T-stress was obtained at eight crack lengths ( θ/π = 0.0625, 0.1250, 0.1875, 0.2500, 0.3125, 0.3750, 0.4375, and 0.5000, θ is the crack half angle). Both crack face loading and remote loading conditions were considered including constant, linear, parabolic and cubic crack face pressures and remote tension and bending. The results for constant and linear crack face pressure were used to derive weight functions for T-stress for the corresponding cracked geometries. The weight functions were validated against several linear and non-linear stress distributions. The derived weight functions are suitable for T-stress calculations for circumferential cracks in cylinders under complex stress fields.

A comparative study of three BEM for transient dynamic crack analysis of 2-D anisotropic solids

Three different boundary element methods (BEM) for transient dynamic crack analysis in two-dimensional (2-D), homogeneous, anisotropic and linear elastic solids are presented. Hypersingular traction boundary integral equations (BIEs) in frequency- domain, Laplace-domain and time-domain with the corresponding elastodynamic fundamental solutions are applied for this purpose. In the frequency-domain and the Laplace-domain BEM, numerical solutions are first obtained in the transformed domain for discrete frequency or Laplace-transform parameters. Time-dependent results are subsequently obtained by means of the inverse Fourier-transform and the inverse Laplace-transform algorithm of Stehfest. In the time-domain BEM, the quadrature formula of Lubich is adopted to approximate the arising convolution integrals in the time-domain BIEs. Hypersingular integrals involved in the traction BIEs are computed through a regularization process that converts the hypersingular integrals to regular integrals, which can be computed numerically, and singular integrals which can be integrated analytically. Numerical results for the dynamic stress intensity factors are presented and discussed for a finite crack in an infinite domain subjected to an impact crack-face loading.

2D transient dynamic crack analysis in piezoelectric solids by BEM

This paper presents a two-dimensional (2D) transient dynamic crack analysis in homogeneous and linear piezoelectric solids. A hypersingular traction boundary integral equation formulation is used for this purpose. A finite crack in an infinite piezoelectric solid subjected to an impact crack-face loading is investigated, which could be either an impact mechanical loading, or an impact electrical loading or a combination of both. In the time-domain boundary element method (BEM), the Lubich’s quadrature formula is adopted for approximating the arising convolution integrals. Hypersingular integrals arising in the traction boundary integral equations (BIEs) are computed through a regularization technique, which converts the hypersingular integrals into regular integrals plus strongly singular and another hypersingular integrals. The regular integrals are computed numerically, while the strongly singular and the new hypersingular integrals are integrated analytically. Numerical results for the dynamic intensity factors will be presented and discussed.

Weight Functions for T-Stress for Edge Cracks in Thick-Walled Cylinders

This paper presents T-stress solutions for an internal edge crack in thick-walled cylinders. Elastic fracture mechanics analysis using the boundary element method (BEM) is performed to determine the T-stress solutions for a wide range of radius ratios and relative crack lengths. The loading cases considered in the BEM analysis for the cracked cylinder are crack-face pressures with polynomial stress distributions acting on the crack face. T-stress results for the uniform and linearly varying crack-face pressure cases are subsequently used as the reference solutions to derive weight functions for T-stress. Boundary element results of T-stress for other stress distributions, namely, other nonlinear crack face loading, internal pressure, and steady-state thermal loading, are used to validate the derived T-stress weight functions. Excellent agreement between the results from the weight function predictions and those directly computed is shown to be obtained. The weight functions derived are suitable for obtaining T-stress solutions for thick-walled cylinders with an internal edge crack under any complex stress fields.

The Coupled Thermoelastic Transversely Isotropic Bimaterial: Interface Crack Extension

A semi-infinite crack grows at a constant subcritical speed along the interface of rigidly bonded, dissimilar transversely isotropic, coupled thermoelastic half-spaces. Shear and normal loads that move on the crack faces drive the process. A dynamic steady state of plane strain is considered. Robust asymptotic full-field solutions for the related problem of translating interface disturbances are first obtained. These lead to coupled singular integral equations for the crack problem that are solved analytically. Expressions for the crack opening components and discontinuity in temperature between crack faces, the traction and temperature change ahead of the crack, and debonding energy rate are presented. These show that the critical crack speed is the minimum of the two Rayleigh speeds and, if it exists, the Stoneley speed. The case of zinc bonded to a thermally inert rigid solid is examined, and calculations for interface temperature change and debonding energy rate given. Apart from any fracture criterion, these parameters show sensitivity to crack speed and to the extent which compressive crack face loading dominates shear loading. Indeed, interface temperature change may decrease in magnitude with crack speed when shear loading dominates.

Antiplane crack analysis of a functionally graded material by a BIEM

Elastostatic analysis of an antiplane crack in a functionally graded material (FGM) is performed by using a hypersingular boundary integral equation method (BIEM). An exponential law is applied to describe the spatial variation of the shear modulus of the FGM. A Galerkin method is applied for the numerical solution of the hypersingular traction BIE. Both unidirectional and bidirectional material gradations are investigated. Stress intensity factors for an infinite and linear elastic FGM containing a finite crack subjected to an antiplane crack-face loading are presented and discussed. The influences of the material gradients and the crack orientation on the stress intensity factors are analyzed.

Effects of material gradients on transient dynamic mode-III stress intensity factors in a FGM

This paper presents a transient dynamic crack analysis for a functionally graded material (FGM) by using a hypersingular time-domain boundary integral equation method. The spatial variations of the material parameters of the FGM are described by an exponential law. A numerical solution procedure is developed for solving the hypersingular time-domain traction BIE. To avoid the use of time-dependent Green’s functions which are not available for general FGM, a convolution quadrature formula is adopted for approximating the temporal convolution, while a Galerkin method is applied for the spatial discretization of the hypersingular time-domain traction BIE. Numerical results for the transient dynamic stress intensity factors for a finite crack in an infinite and linear elastic FGM subjected to an impact anti-plane crack-face loading are presented and discussed. The effects of the material gradients of the FGM on the transient dynamic stress intensity factors and their dynamic overshoot over the corresponding static stress intensity factors are analyzed.

Intersonic crack growth under time-dependent loading

The problem investigated in this paper is a mode II crack extending at a uniform intersonic speed in an otherwise unbounded elastic solid subjected to time dependent crack-face tractions. The fundamental solution for this problem is obtained analytically, which is then used to construct the general solution for an intersonic crack subjected to arbitrary time-dependent loading. For time-independent loading, this solution reduces to Huang and Gao’s [J. Appl. Mech 68 (2001) 169] fundamental solution. We have also studied two crack-face loadings that are of interest for engineering applications.

Transient dynamic analysis of a cracked functionally graded material by a BIEM

A hypersingular time-domain traction boundary integral equation method (BIEM) is presented for transient dynamic crack analysis in a functionally graded material (FGM). A finite crack in an infinite and linear elastic FGM subjected to an impact antiplane crack-face loading is investigated. The spatial variation of the materials constants is described by an exponential law. To solve the hypersingular time-domain traction BIE, a numerical solution procedure is developed. The numerical solution procedure uses a convolution quadrature formula for approximating the temporal convolution and a Galerkin method for the spatial discretization of the hypersingular time-domain traction BIE. Numerical examples are presented to show the effects of the materials gradients on the dynamic stress intensity factors.

Alternative Fracture Criterion in Terms of the Energy-Based Theory of Strength

Within the framework of the energy-based strength criterion, the transition from the stress space to the stress-intensity-factor space is performed in a such way so that the fracture criterion obtained contains all characteristics of the material crack resistance \(K_{ic} \left( {i = I,\,II,\,III} \right)\), as well as dimensions \(r_{i*} \left( {\theta } \right)\) and \(r_* \left( {\theta } \right)\) of the local fracture zone at the crack-tip vicinity under different modes of the crack-face loading. We propose an alternative approximation of the fracture criterion, which is exact in terms of the model under consideration. The theoretical results are compared with some known experiments. An example of the calculation of a structural element with a macrocrack is given.