Corner Point Singularities Research Articles

Peaked Stokes waves as solutions of Babenko’s equation

Babenko’s equation describes traveling water waves in holomorphic coordinates. It has been used in the past to obtain properties of Stokes waves with smooth profiles analytically and numerically. We show in the deep-water limit that properties of Stokes waves with peaked profiles can also be recovered from the same Babenko’s equation. In order to develop the local analysis of singularities, we rewrite Babenko’s equation as a fixed-point problem near the maximal elevation level. As a by-product, our results rule out a corner point singularity in the holomorphic coordinates, which has been obtained in a local version of Babenko’s equation.

SIF formula based on exact SIF distribution for semi‐elliptical surface cracks subjected to mode I, II, III uniform loading

AbstractThe stress intensity factor (SIF) distribution of a semi‐elliptical surface crack is often used for evaluating the fatigue strength of structures. Virtually exact distributions of SIFs can be provided along the crack front by solving the hypersingular integral equation of the body force method. In this paper, to create a very accurate SIF variation formula, the elliptical crack SIF solutions , , are used and the SIF ratios , , are mainly focused. Paying attention to the corner point singularity, by applying the least squares method to the ratio , in the whole range of parametric angle , and formulas can be proposed. Instead, formula can be proposed by applying the method of least squares to the ratio of in the range but applying directly to in the range . In this way, all of formulas proposed in this paper provide the SIFs with better than 1.0% accuracy.

Second-order diffusion limit for the phonon transport equation: asymptotics and numerics

We investigate the numerical implementation of the limiting equation for the phonon transport equation in the small Knudsen number regime. The main contribution is that we derive the limiting equation that achieves the second order convergence, and provide a numerical recipe for computing the Robin coefficients. These coefficients are obtained by solving an auxiliary half-space equation. Numerically the half-space equation is solved by a spectral method that relies on the even-odd decomposition to eliminate corner-point singularity. Numerical evidences will be presented to justify the second order asymptotic convergence rate.

Modes I-II-III stress intensity factors of a semi-elliptical surface crack at a round bar under torsion loading by FEM and DBEM

The corner point singularity of surface cracks by finite element method (FEM) has become a numerical concern decades ago. The literature showed that the stress intensity factors (SIFs) at the corner points were often excluded. Further, most SIFs were reported for larger ratios of the crack depth over cylinder diameter. This paper presents the SIFs (Modes I, II and III) of a semi-elliptical surface crack at a solid round bar under torsion. The tetrahedral and hexahedral elements were used in the finite element modelling. The effects of the loading mode and the crack aspect ratio on the corner point singularity were discussed. The tetrahedral meshing was generally observed to be more suitable for modelling relatively small surface cracks, particularly in respect to the corner point singularity. For all loading modes, the SIFs away from the corner points of using the tetrahedral meshing were found to have fairly good agreement with those by dual boundary element method (DBEM).

Numerical fracture mechanics as a practical failure investigatory tool: The outlook of cracked round bars

Many failure investigations related to the failed round bars/shafts for various applications have been reported in literature. The cracks were often initiated at the regions with a high stress concentration. However, numerical fracture mechanics was frequently not used in the failure investigations. Nowadays advancement of computational development on the interactive numerical techniques allows the analysts to more accessibly figure out the fracture phenomena when performing engineering analyses. In addition, the computational cost is no longer the issue of performing numerical fracture mechanics analyses. The stress intensity factors (SIFs) of a surface planar flaw at a stepped round bar are of practical interest, however, the SIF solutions along the flaw front were not yet documented in literature. Here, the practical abilities of the numerical fracture mechanics on the fracture analyses are demonstrated by evaluating a planar surface flaw at the shoulder fillet through numerical simulations by the finite element method (FEM) and the dual boundary element method (DBEM). The use of tetrahedral meshing of FEM is presented, and the concern of corner point singularity on a surface planar flaw is highlighted. The stress contours around the front of planar flaws that are hardly acknowledged in literature are also presented. The SIF behaviors of the semi-elliptical planar flaws are discussed. The SIF data for relatively small crack sizes are valuable to be used for judging a condition of crack size whether or not it would grow to a critical size.

Tetrahedral meshing for a slanted semi-elliptical surface crack at a solid cylinder

This technical note demonstrates the use of tetrahedral meshing for modelling a slanted semi-elliptical surface crack at a solid cylinder by finite element method (FEM). For a relatively small ratio of the crack depth over cylinder diameter, Modes I and II (in-plane) stress intensity factors (SIFs) at the corner points by FEM were observed to be in good agreements with those by the dual boundary element method (DBEM). Effect of the crack aspect ratio, crack size and loading mode on the corner point singularity was noted. The tearing loading mode was found to promote a greater corner point singularity.

3D effects on Fracture Mechanics: corner point singularities

Plane elasticity has allowed the development of the current knowledge on fracture mechanics since the dawn of this discipline. Linear elastic fracture mechanics (LEFM) has taken advantage of the possibility to obtain analytical closed-form solutions. Indeed, three dimensional effects, related basically to the Poisson’s ratio and the plate thickness, have been mostly neglected for a long time. The proof of the existence of corner point singularities, i.e. stress singularity at the intersection of crack front with a free surface, has contributed to explain fracture phenomena, such as the bowed crack fronts and the behavior of stress intensity factors along the thickness of a cracked/notched body. Although fatigue crack growth is Mode I dominated and corner point singularities become insignificant, for the estimation of brittle failure of cracked/notched they may represent a substantial source of error, both in terms of magnitude of the stress field and failure location. In this paper we provide before a brief overview on 3D corner point singularities. By exploiting of 3D finite element (FE) modeling, to understand the theoretical basis of this phenomenon, numerical experiments on sharply-V-notched linear elastic plates under mode 1 loading are performed. A quantitative and qualitative description of the causes and effects of 3D vertex singularities under mode 1 in relation with Poisson’s ratio is given, including negative values of this latter material parameter, never before investigated. Besides, by means of numerical simulations of the same notched plates subjected to mode 3 loading, observations on the possible connection between 3D corner point singularities and the out-of-plane shear stress distribution near the model surface are discussed. Finally, future perspectives on this research field are addressed.

Cracked components under anti-plane loading: recent outcomes and future developments

The existence of three-dimensional effects at cracks has been known for many years, but understanding has been limited, and for some situations still is. Understanding improved when the existence of corner point singularities and their implications became known. Increasingly powerful computers made it possible to investigate three-dimensional effects numerically in detail. Despite increased understanding, three-dimensional effects are sometimes ignored in situations where they may be important. The purpose of the present investigation is to study by means of accurate 3D finite element (FE) models a coupled fracture mode generated by anti-plane loading of a straight through-the-thickness crack in linear elastic plates. An extended version of the present work has recently been published in the literature. The results obtained from the highly accurate finite element analyses have improved understanding of the behaviour of through cracked components under anti-plane loading. The influence of plate bending is increasingly important as the thickness decreases. It appears that a new field parameter, probably a singularity, is needed to describe the stresses at the free surfaces. Discussion on whether KIII tends to zero or infinity as a corner point is approached is futile because KIII is meaningless at a corner point. The intensity of the local stress and strain state through the thickness of the cracked components has been evaluated by using the strain energy density (SED) averaged over a control volume embracing the crack tip. The SED has been considered as a parameter able to control fracture in some previous contributions and can easily take into account also coupled three-dimensional effects. Calculation of the SED shows that the position of the maximum SED is independent of plate thickness. Both for thin plates and for thick ones the maximum SED is close to the lateral surface, where the maximum intensity of the coupled mode II takes place.

Corner point singularities under in-plane and out-of-plane loading: a review of recent results

Corner point singularities under in-plane and out-of-plane loading: a review of recent results

State of the art of corner point singularities under in-plane and out-of-plane loading

The linear elastic analysis of homogeneous, isotropic cracked bodies started in the 1900s. The existence of three dimensional corner point effects in the vicinity of a corner point where a crack front intersects a free surface was investigated in the late 1970s. An approximate solution by Bažant and Estenssoro explained some features of corner point effects but there were various paradoxes and inconsistencies. Results derived from finite element models showed that the analysis is incomplete. The stress field in the vicinity of a corner point appears to be the sum of two singularities.

Coupled fracture modes under anti-plane loading

The linear elastic analysis of homogeneous, isotropic cracked bodies is a Twentieth Century development. It was recognised that the crack tip stress field is a singularity, but it was not until the introduction of the essentially two dimensional stress intensity factor concept in 1957 that widespread application to practical engineering problems became possible. The existence of three dimensional corner point effects in the vicinity of a corner point where a crack front intersects a free surface was investigated in the late 1970s: it was found that modes II and III cannot exist in isolation. The existence of one of these modes always induces the other. An approximate solution for corner point singularities by Bažant and Estenssoro explained some features of corner point effects but there were various paradoxes and inconsistencies. In an attempt to explain these a study was carried out on the coupled in-plane fracture mode induced by a nominal anti-plane (mode III) loading applied to plates and discs weakened by a straight crack. The results derived from a large bulk of finite element models showed clearly that Bažant and Estenssoro’s analysis is incomplete. Some of the results of the study are summarised, together with some recent results for a disc under in-plane shear loading. On the basis of these results, and a mathematical argument, the results suggest that the stress field in the vicinity of a corner point is the sum of two singularities: one due to stress intensity factors and the other due to an as yet undetermined corner point singularity.

The linear elastic analysis of cracked bodies, crack paths and some practical crack path examples

The linear elastic analysis of cracked bodies is a Twentieth Century development, with the first papers appearing in 1907. The stress intensity factor concept, introduced in 1957, developed rapidly in the 1960s, with widespread application to practical problems. Finite element analysis had a significant influence on the development of linear elastic fracture mechanics. Corner point singularities were investigated in the late 1970s, but a new field parameter is still needed to describe the stresses at surfaces. There are geometric constraints on permissible fatigue crack paths. Some case studies of fatigue crack paths in metallic materials are included.

Crack paths and the linear elastic analysis of cracked bodies

The linear elastic analysis of cracked bodies is a Twentieth Century development, with the first papers appearing in 1907, but it was not until the introduction of the stress intensity factor concept in 1957 that widespread application to practical engineering problems became possible. Linear elastic fracture mechanics (LEFM) developed rapidly in the 1960s, with application to brittle fracture and fatigue crack growth. The first application of finite elements to the calculation of stress intensity factors for two dimensional cases was in 1969. Finite element analysis had a significant influence on the development of LEFM. Corner point singularities were investigated in the late 1970s. It was soon found that the existence of corner point effects made interpretation of calculated stress intensity factors difficult and their validity questionable. In 1998 it was shown that the assumption that crack growth is in mode I leads to geometric constraints on permissible fatigue crack paths. Current open questions are. The need for a new field parameter, probably a singularity, to describe the stresses at surfaces. How best to allow for the influence of corner point singularities in three dimensional numerical predictions of fatigue crack paths. Adequate description of fatigue crack path stability.

Three-dimensional effects on cracked discs and plates under nominal Mode III loading

The existence of three-dimensional effects at cracks has been known for many years, but understanding has been limited, and for some situations still is. Understanding improved when the existence of corner point singularities and their implications became known. Increasingly powerful computers made it possible to investigate three-dimensional effects numerically in detail. Despite increased understanding, threedimensional effects are sometimes ignored in situations where they may be important. The purpose of the present contribution is to review the study carried out by the same authors in some recent investigations, in which a coupled fracture mode generated by anti-plane loading of a straight through-the-thickness crack in linear elastic discs and plates has been analysed by means of accurate 3D finite element (FE) models. The results obtained from the highly accurate finite element analyses have improved understanding of the behaviour of through cracked components under anti-plane loading. The influence of plate bending is increasingly important as the thickness decreases. It appears that a new field parameter, probably a singularity, is needed to describe the stresses at the free surfaces. Discussion on whether KIII tends to zero or infinity as a corner point is approached is futile because KIII is meaningless at a corner point. The intensity of the local stress and strain state through the thickness of the cracked components has been evaluated by using the strain energy density (SED) averaged over a control volume embracing the crack tip. The SED has been considered as a parameter able to control fracture in some previous contributions and can easily take into account also coupled three-dimensional effects. Calculation of the SED shows that the position of the maximum SED in the discs case is a function of the thickness. In the plates case instead the position of the maximum SED is independent of plate thickness, contrary to disc results.

Multiscale investigation of stress-corrosion crack propagation mechanisms in oxide glasses

Abstract Fracture propagation involves the coupling of many length scales ranging from the sample loading geometry to the molecular level. In brittle materials, the length scales of the damage process zone are reduced to a submicrometric scale and the coupling with the macroscopic scale is expected to be the domain of linear elastic fracture mechanics. However, although 2D elastic analyses are generally adequate to describe the sample deformation at macroscopic scales, local investigations of failure mechanisms at the sample-free surface require the use of 3D mechanical tools due to the crack front local curvature and to the corner point singularities at the intersection between the crack front and the external surfaces of the sample. We present here a thorough multiscale investigation of the slow crack growth of a sharp crack in oxide glasses in the stress-corrosion regime, combining experimental and numerical analyses of the displacement fields from the millimeter scale to the nanoscale range. The principal aim of the study is identifying the length and time scales of the mechanisms of damage and interaction between water and glass, which have been the subject of an extensive debate in the last decades.

Three-dimensional effects on cracked components under anti-plane loading

The existence of three-dimensional effects at cracks has been known for many years, but understanding has been limited, and for some situations still is. Understanding improved when the existence of corner point singularities and their implications became known. Increasingly powerful computers made it possible to investigate three-dimensional effects numerically in detail. Despite increased understanding, threedimensional effects are sometimes ignored in situations where they may be important. The purpose of the present investigation is to study by means of accurate 3D finite element (FE) models a coupled fracture mode generated by anti-plane loading of a straight through-the-thickness crack in linear elastic plates. An extended version of the present work has recently been published in the literature. The results obtained from the highly accurate finite element analyses have improved understanding of the behaviour of through cracked components under anti-plane loading. The influence of plate bending is increasingly important as the thickness decreases. It appears that a new field parameter, probably a singularity, is needed to describe the stresses at the free surfaces. Discussion on whether KIII tends to zero or infinity as a corner point is approached is futile because KIII is meaningless at a corner point. The intensity of the local stress and strain state through the thickness of the cracked components has been evaluated by using the strain energy density (SED) averaged over a control volume embracing the crack tip. The SED has been considered as a parameter able to control fracture in some previous contributions and can easily take into account also coupled three-dimensional effects. Calculation of the SED shows that the position of the maximum SED is independent of plate thickness. Both for thin plates and for thick ones the maximum SED is close to the lateral surface, where the maximum intensity of the coupled mode II takes place.

The linear elastic analysis of cracked bodies and crack paths

The linear elastic analysis of cracked bodies is a Twentieth Century development, with the first papers appearing in 1907, but it was not until the introduction of the stress intensity factor concept in 1957 that widespread application to practical engineering problems became possible. Linear elastic fracture mechanics (LEFM) developed rapidly in the 1960s, with application to brittle fracture and fatigue crack growth. The first application of finite elements to the calculation of stress intensity factors for two dimensional cases was in 1969. Finite element analysis had a significant influence on the development of LEFM. Corner point singularities were investigated in the late 1970s. It was soon found that the existence of corner point effects made interpretation of calculated stress intensity factors difficult and their validity questionable. In 1998 it was shown that the assumption that crack growth is in mode I leads to geometric constraints on permissible fatigue crack paths. Current open questions are. The need for a new field parameter, probably a singularity, to describe the stresses at surfaces. The use of strain energy density is promising. Adequate description of fatigue crack path stability. How best to allow for the influence of corner point singularities in three dimensional numerical predictions of fatigue crack paths. Development of an adequate plane strain fracture toughness testing standard. Some background information is given in appendices.

Coupled fracture mode of a cracked disc under anti-plane loading

The existence of three-dimensional effects at cracks has been known for many years, but understanding has been limited, and for some situations still is. Understanding improved when the existence of corner point singularities and their implications became known. Increasingly powerful computers made it possible to investigate three-dimensional effects numerically in detail. Despite increased understanding, three-dimensional effects are sometimes ignored in situations where they may be important. The purpose of the present investigation is to study by means of accurate 3D finite element (FE) models a coupled fracture mode generated by anti-plane loading of a straight through-the-thickness crack in linear elastic discs. The results obtained from the highly accurate finite element analyses have improved understanding of the behaviour of through cracked discs under anti-plane loading. The influence of plate bending is increasingly important as disc thickness decreases. Bažant and Estenssoro’s analysis works well for the symmetric mode (mode I), but it is incomplete for the asymmetric mode (a combination of modes II and III). It appears that a new field parameter, probably a singularity, is needed to describe the stresses at the disc surfaces. Discussion on whether KIII tends to zero or infinity as a corner point is approached is futile because KIII is meaningless at a corner point. Calculation of the strain energy density (SED) in a control volume at the crack tip shows that the position of the maximum SED is a function of disc thickness.

Coupled fracture mode of a cracked disc under anti-plane loading

The existence of three-dimensional effects at cracks has been known for many years, but understanding has been limited, and for some situations still is. Understanding improved when the existence of corner point singularities and their implications became known. Despite increased understanding, three-dimensional effects are sometimes ignored in situations where they may be important. The purpose of the present investigation is to study by means of accurate 3D finite element (FE) models a coupled fracture mode generated by anti-plane loading of a straight through-the-thickness crack in linear elastic discs. The results obtained from the highly accurate finite element analyses have improved understanding of the behaviour of through cracked discs under anti-plane loading. The influence of plate bending is increasingly important as disc thickness decreases. It appears that a new field parameter, probably a singularity, is needed to describe the stresses at the disc surfaces. Calculation of the strain energy density (SED) in a control volume at the crack tip shows that the position of the maximum SED is a function of disc thickness.

A 50‐year retrospective review of three‐dimensional effects at cracks and sharp notches

ABSTRACTThis review is a brief survey of three‐dimensional effects at cracks and sharp notches. The overall aim is to review developments over the past 50 years leading up to the current state of the art. The review is restricted to linear elastic, homogeneous, isotropic materials, with any yielding confined to a small region at a crack or notch tip. It is also restricted to static loading and to constant amplitude fatigue loading. An enormous amount of theoretical and experimental information relevant to three‐dimensional effects has been published in the past five decades, so the review is, of necessity, highly selective. Theoretical topics covered are linear elastic fracture mechanics, including Volterra distorsioni, stress intensity factors, corner point singularities, crack front line tension, displacement analysis of cracks and notches, and through thickness distributions of stresses and stress intensity factors. Crack path topics covered are fatigue crack path constraints, determination of fatigue crack paths, oscillating crack fronts in thin sheets and the transition to slant crack propagation in thin sheets. Plane strain fracture toughness testing, including standards, is covered. Overall, it can be concluded that the existence of three‐dimensional effects at cracks and sharp notches has been known for many years, but understanding has been limited, and for some situations still is. Understanding improved when the existence of corner point singularities and their implications became known. Increasingly powerful computers made it possible to investigate three‐dimensional effects numerically in detail. Despite increased understanding, three‐dimensional effects are sometimes ignored in situations where they may be important.