Crack-inclusion interaction for mode I crack analyzed by Eshelby equivalent inclusion method

Several simple formulas have been developed to predict the variations of stress intensity factors for mode I crack induced by the stiffness and geometry of the near crack-tip inclusion. The derivation of the fundamental formula is based on the transformation toughening theory. The unconstrained mismatch strains between matrix and inclusion, which induce the variation of the near crack-tip field, are estimated based on the Eshelby equivalent inclusion approach. As validated by numerical examples, the developed formulas have satisfactory accuracy for a wide range of the modulus ratio between inclusion and matrix as long as the inclusion is located in the K ₀-controlled field.     

Estimations of stress intensity factor for a short crack in an elastic–plastic notch

This note presents a simple method for estimating the stress intensity factor (SIF) for a short crack emanating from an elastic–plastic notch.     

Transient thermal stress intensity factors for an internal longitudinal semi-elliptical crack in a thick-walled cylinder

In this paper, the stress intensity factors are derived for an internal semi-elliptical crack in a thick-walled cylinder subjected to transient thermal stresses. First, the problem of transient thermal stresses in a thick-walled cylinder is solved analytically. Thermal and mechanical boundary conditions are assumed to act on the inner and outer surfaces of the cylinder. The quasi-static solution of the thermoelasticity problem is derived analytically using the finite Hankel transform and then, the stress intensity factors are extracted for the deepest point and the surface points of the semi-elliptical crack using the weight function method. The results show to be in accordance with those cited in the literature in the special case of steady-state problem. Using the closed-form relations extracted for the transient thermal stress intensity factors, some conclusive results are drawn.     

Dynamic stress intensity factor (Mode I) of a penny-shaped crack in an infinite poroelastic solid

This paper considers the transient stress intensity factor (Mode I) of a penny-shaped crack in an infinite poroelastic solid. The crack surfaces are impermeable. By virtue of the integral transform methods, the poroelastodynamic mixed boundary value problems is formulated as a set of dual integral equations, which, in turn, are reduced to a Fredholm integral equation of the second kind in the Laplace transform domain. Time domain solutions are obtained by inverting Laplace domain solutions using a numerical scheme. A parametric study is presented to illustrate the influence of poroelastic material parameters on the transient stress intensity. The results obtained reveal that the dynamic stress intensity factor of poroelastic medium is smaller than that of elastic medium and the poroelastic medium with a small value of the potential of diffusivity shows higher value of the dynamic stress intensity factor.     

Crack tip stress intensity factors for a crack emanating from a sharp notch

Accurate calibrations are provided for the crack tip stress intensity factor for a crack of finite length emanating from the symmetric tip of a sharp notch, of arbitrary angle, in terms of the generalised stress intensity quantifying remote loading of the notch. The solution is applied to example problems and shown to be accurate for cases where the crack is much shorter then the notch depth.     

On the dynamic interaction between a penny-shaped crack and an expanding spherical inclusion in 3-D solid

Three-dimensional stress investigation on the interaction between a penny-shaped crack and an expanding spherical inclusion in an infinite 3-D medium is studied in this paper. The spherical transformation area (the inclusion) expands in a self-similar way. By using the superposition principle, the original physical problem is decomposed into two sub-problems. The transient elastic filed of the medium with an expanding spherical inclusion is derived with the dynamic Green's function. A time domain boundary integral equation method (BIEM) is then adopted to solve the current problem. 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 tip to describe the proper behavior of the unknown quantities near the crack front. A collocation method as well as a time stepping scheme is applied to solve the BIEs. Numerical examples for the Mode I stress intensity factor are presented to assess the dynamic effect of the expanding inclusion.     

Stress intensity factor for a Griffith crack interacting with a coated inclusion

Stress investigation for the interaction problem between a coated circular inclusion and a near-by line crack has been carried out. The crack and the coated inclusion (a coated fiber) are embedded in an infinitely extended isotropic matrix, with the crack being along the radial direction of the inclusion. Two loading conditions, namely, the tensile and shear loading ones are considered. During the solution procedure, the crack is treated as a continuous distribution of edge dislocations. By using the solution of an edge dislocation near a coated fiber as the Green's function, the problem is formulated into a set of singular integral equations which are solved by Erdogan and Gupta (1972) method. The expressions for the stress intensity factors of the crack are then obtained in terms of the asymptotic values of the dislocation density functions evaluated from the integral equations. Several numerical examples are given for various material and geometric parameters. The solutions obtained from the integral equations have been checked and confirmed by the finite element analysis results.     

Computation of mode I stress intensity factors for three-dimensional bodies using displacements at an arbitrary location

An approach is developed to determine stress intensity factors using the displacements due to a crack increment in a three-dimensional body. By computing displacements for both the actual loading and a virtual force some distance from the crack, stress intensity factors may be obtained by the finite element method without using special crack elements. Numerical results obtained using coarse meshes for several crack configurations are found to be in good agreement with those available in the literature. This revised version was published online in July 2006 with corrections to the Cover Date.     

Mode II stress intensity factors for central slant cracks with frictional surfaces in uniaxially compressed plates

The effect of crack surface friction on mode II stress intensity factor (SIF) of a central slant crack in a plate uniformly loaded in uniaxial compression is quantified. A previously developed two-dimensional finite element analysis was utilised after its modification to accommodate the friction between the crack surfaces. The plane strain state was assumed. A new numerical technique was devised to avoid the iteration procedures, which had to be employed due to the existence of frictional forces.

The crack inclination angle varied between zero and 75° measured from the horizontal direction. The coefficient of friction of the crack surfaces changed from zero to 1. In case of relatively sliding crack surfaces, mode II SIF existed. As is well known, the resulting mode II SIF decreased with increasing the coefficient of friction of the crack surfaces. Further, mode II SIF increased with increasing crack line inclination angle and then decreased after reaching a maximum value. The angle corresponding to that maximum SIF increased as the coefficient of friction of the crack surfaces increased.     

Computation of stress intensity factors (KI, KII) and T-stress for cracks reinforced by composite patching

To increase the operational life of defected structures, a repairing method using composite patches has been used to reinforce cracked components. Due to various advantages of composite materials, this method has received much attention from researchers and engineers. Considerable investigations have been performed to highlight the effect of bonded composite patches on the fracture parameters such as stress intensity factors (SIF) and J-integral. However the effect of composite patches on the T-stress, the constant stress term acting parallel to the crack, has not been investigated in the past. In this paper, the finite element method is carried out to analyze the effect of bonded composite patches for repairing cracks in pure mode I and also mixed mode I/II conditions, by computing the stress intensity factors and the T-stress, as functions of the crack length, the crack inclination angle and the type of composite material. In pure mode I condition, the finite element analysis is carried out for three different specimens: centre crack, double edge crack and single edge crack specimens. For mixed mode I/II condition the analysis is conducted on an inclined central crack of various slant angles. For both pure mode I and mixed mode I/II, the numerical results show that composite patching has considerable effect on the T-stress.     

The interaction of mode I crack with multi-inclusions in a three-phase model

An approximately close form solution has been developed for mode I crack interacting with multi-inclusions in composite materials. The crack-tip stress intensity factor is evaluated in a three-phase model, which combines the present knowledge that the inclusions only in the immediate neighborhood of the crack-tip have strong effect on the stress intensity factor and that the far inclusions have an overall effects which can be estimated by effective properties of the composites. As validated by numerical examples, the solution has good accuracy for a wide range of the modulus ratios between the inclusion and matrix material.     

The effect of an elliptical inclusion on a crack

The interaction between an elliptical inclusion and a crack is analyzed by body force method. The investigated stress field is simulated by superposing the fundamental solutions for a point force applied at a point in an infinite plate containing an elliptical inclusion. Based on numerical results, effects of the inclusion shape on the crack tip stress intensity factor are discussed. It is found that for small cracks emanating from a stress-higher point on the inclusion interface the stress intensity factors are mainly determined by the stresses, occurring at the crack starting point before the crack initiation, and the inclusion root radius, besides the crack length. However, for the cracks occurring in a stress-lower region around the inclusion, it is difficult to characterize the effect of the inclusion geometry on the stress intensity factors of small cracks by the inclusion root radius alone. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stress intensity factor for an embedded elliptical crack under arbitrary normal loading

In this paper we introduce the boundary value problem of three-dimensional classical elasticity for an infinite body containing an elliptical crack. Using the method of simultaneous dual integral equations, the problem is transformed to the system of linear algebraic equations. Stress intensity factor is obtained in the form of the Fourier series expansion. Several solutions for specific cases of applied polynomial stress fields are derived and compared with existing results. Eligibility of the method for more complicated stress fields is demonstrated on the example of partially loaded elliptical crack.     

Mode I stress intensity factor estimate for a half-penny shaped crack in a trilayer structure by reduction to a bilayer model

The stress intensity factor (SIF) of a half-penny shaped crack normal to the interface in the top layer of a three-layer bonded structure is obtained by the finite element method for a wide range of parameters. To obtain a simple estimate of the SIF, the method of reduction of an idealized cracked trilayer domain to that of a corresponding bilayer domain has been introduced based on the notion of an equivalent homogeneous material substitution for the two bottom layers. The results obtained are utilized in estimating the SIF of a small crack at the interface in a trilayer structure subjected to an indentation load based on the stress calculations in a corresponding uncracked structure. The simplification method may be useful in predicting brittle failure initiating from interfacial flaws in layered structural components with complex geometries that would normally require extensive computational modeling.     

Dynamic stress intensity factors around three cracks in an infinite elastic plane subjected to time-harmonic stress waves

The time-harmonic problem for an infinite elastic plane weakened by three parallel cracks has been solved. In this problem, two cracks are situated symmetrically on either side of a central crack and incident stresses impinge perpendicular to the cracks. Using the Fourier transform technique, the boundary conditions are reduced to four simultaneous integral equations. To solve the equations, the differences of displacements inside the cracks are expanded in a series. The unknown coefficients in the series are solved by the Schmidt method. The dynamic stress intensity factors are calculated numerically for several crack configurations. This revised version was published online in August 2006 with corrections to the Cover Date.     

Computation of stress intensity factors for a 2-D body from displacements at an arbitrary location

An approach to stress intensity factor computation is presented which allows accurate estimation using a simple finite element program and a coarse mesh of elements. The stress intensity factor is obtained from what we term the Crack-Displacement (C-D) Factor. This involves the rate of change of displacements with crack length at the same location remote from the crack due to two loading conditions. One loading condition is merely that applied to the cracked body. The other loading condition is a virtual line force applied at the location at which the displacement rates are computed. The accuracy of the procedure is demonstrated for uniform tensile stresses applied to a center-cracked panel and an edge-cracked strip.     

On the Mode I stress intensity factor for an external circular crack with fibre bridging

This paper presents a theoretical analysis of an external matrix crack located in a unidirectional fibre-reinforced elastic solid modelled as a transversely isotropic material. The presence of matrix cracking with fibre continuity introduces bridging action that has an influence on the stress intensity factors at the crack tip of the external crack. This paper presents a model for the bridged crack, where the fibre ligaments induce a constant displacement-dependent traction constraint over the external crack. This gives rise to a Fredholm integral equation of the second kind, which can be solved in an approximate fashion. We examine the specific problem where the bridged external circular crack is loaded by a doublet of concentrated forces. Numerical results are presented to illustrate the influence of the fibre–matrix modular ratio and the location of the loading on the bridged-crack opening mode stress intensity factor.     

Thermal stress intensity factors of an infinite orthotropic layer with a crack

Stress intensity factors are determined for a crack in an infinite orthotropic layer. The crack is situated parallel to the plane surfaces of the layer. Stresses are solved for two kinds of the boundary conditions with respect to temperature field. In the first problem, the upper surface of the layer is heated to maintain a constant temperature T ₀, while the lower surface is cooled to maintain a constant temperature –T ₀. In the other problem, uniform heat flows perpendicular to the crack. The surfaces of the crack are assumed to be insulated. The boundary conditions are reduced to dual integral equations using the Fourier transform technique. To satisfy the boundary conditions outside the crack, the difference in temperature at the crack surfaces and differences in displacements are expanded in a series of functions that vanish outside the crack. The unknown coefficients in each series are evaluated using the Schmidt method. Stress intensity factors are then calculated numerically for a steel layer that behaves as an isotropic material and for a tyrannohex layer that behaves as an orthotropic material.     

Stress intensity factors for an inclined edge crack in a semiplane

Mode I and II Stress Intensity Factors under uniform general biaxial loadings were derived for an inclined edge crack in a semiplane. By interpolating Finite Element results in the angular range [0°÷80°], analytical expressions were obtained for both K_I and K_II with an accuracy better than 1%. Influence coefficients were defined in the crack reference frame thus highlighting the coupling effects between Modes I and II due to the loss of symmetry when the crack is not normal to the surface.     

Stress intensity factors of an inclined elliptical crack near a bimaterial interface

In this paper the stress intensity factors are discussed for an inclined elliptical crack near a bimaterial interface. The solution utilizes the body force method and requires Green’s functions for perfectly bonded semi-infinite bodies. The formulation leads to a system of hypersingular integral equation whose unknowns are three modes of crack opening displacements. In the numerical calculation, unknown body force densities are approximated by using fundamental density functions and polynomials. The results show that the present method yields smooth variations of stress intensity factors along the crack front accurately. Distributions of stress intensity factors are presented in tables and figures with varying the shape of crack, distance from the interface, and elastic modulus ratio. It is found that the inclined crack can be evaluated by the models of vertical and parallel cracks within the error of 24% even for the cracks very close to the interface.