Closed-form solution for the size of plastic zone in an edge-cracked strip

This paper is concerned with the problem of plastic zone at the tip of an edge crack in an isotropic elastoplastic strip under anti-plane deformations. By means of complex potential and Dugdale model, the stress intensity factor and the size of plastic zone are obtained in closed-form. Furthermore, the analytic solutions for an edge crack at the free boundary of a half-space and a semi-infinite crack heading towards a free surface are determined as the limiting cases of the strip geometries.     

A finite element method for calculating stress intensity factors and its application to composites

A concept which allows for the development of efficient finite element techniques in the analysis of plane elastic structures containing cracks is discussed. It consists of combining a specially defined finite element in the region surrounding each crack tip with conventional CST elements describing the remaining portion of the geometry considered. For the special element a pair of displacement functions is chosen which adequately represents the singular character of the elastic solution at the crack tip. The application of this concept is illustrated through a specific numerical method developed by W. K. Wilson for the calculation of mode I stress intensity factors.

Wilson's method was coded and used to analyze an infinitely long strip under tension with a line crack perpendicular to its axis of symmetry. Circular inclusions of different material properties were assumed to be present near the tips of the crack and their effect on the mode I stress intensity factor was investigated.

It was found that more flexible inclusions increase the intensity factor while more rigid inclusions decrease it. These results are quite similar to those obtained by analytical methods in an analogous problem involving an infinite sheet, but in the case of a strip, the influence of inclusions on the intensity factor was found to be more pronounced.     

Crack kinking under nonsymmetric loading

This paper considers an asymmetrically kinked, semi-infinite crack in a two-dimensional solid under mixed-mode loading and a stress acting parallel to the main crack, the latter providing the non-singular stress term, T, in the Irwin-Williams expansion of the crack tip field. The aim of the study is twofold: First, to obtain an almost closed-form solution for the stress intensity factors at the tip of the kink with a view to explaining a curious result observed by many authors that under predominantly Mode I loading the first order solution in small kink angle is accurate for considerably large kink angles and, second, to study the effect of the in-plane tensile stress on the stability of crack growth. Where possible, the results are compared with those available in the literature.     

Steady-State Crack Velocity in a Viscoelastic Composite Material

The velocity of a semi-infinite crack slowly propagating in aninfinitely long strip made of a viscoelastic composite material isdetermined according to Christensen's fracture criterion. The edges ofthe strip are subjected to uniform opposite displacements normal to thecrack plane. The crack velocity is obtained from an energy balanceequation involving the energy dissipated in the whole strip; the latteris evaluated using an approximate, but sufficiently accurate, expressionof the stress field in the structure obtained by taking into account thestrong anisotropy of the long fibre composite material of interest. Thisnew version of Christensen's criterion compares favourably with theoriginal one and gives crack velocity predictions very close to thoseprovided by Schapery's criterion.     

The effect of repeating load on the crack growth initiation and crack propagation in delayed failure

The delayed failure test under repeating load was carried out with pre-cracked specimen. The incubation time and the crack propagation rate were correlated with the stress intensity factor K.

The incubation time is decreased by the superposition of repeating load, as the range of stress intensity factor ΔK or the repeating frequency f increase. The reason can be explained by the promotion of corrosion reaction due to, e.g. the destruction of oxide film on the crack tip, which facilitates the invasion of hydrogen atoms into the material.

The crack propagation rate da/dt is decreased by the superposition of repeating load, and there exist two valleys of crack propagation rate minima on the da/dt vs f and da/dt vs ΔK curves. One valley corresponds to the interaction between the cyclic movement of the region with tri-axial tensile stress and the hydrogen atoms diffused from crack tip, which disturbs the concentration of hydrogen atoms. Another seems te correspond to the generation of retained compressive stress which reduces the effective stress intensity at crack tip and supresses the invasion and diffusion of hydrogen atoms.     

On the intersonic crack propagation in an orthotropic medium

Motivaded by recent theoretical studies the elastodynamic response of an orthotropic material with a semi-infinite line crack, which propagates intersonically. is revisited through an approach which differs from those used in previous studies. The near tip stress and displacement fields are obtained for Mode I and Mode II of steady state crack propagation. The strain energy release rate analysis confirms that the Mode I is physically impossible due to the order of stress singularity, which is larger then one half. For Model II the order of stress is less than one half and it is shown that a steady state intersonic propagation is allowed only for a particular crack tip velocity which is a function of the material orthotropy.     

Rotation at the crack tip under uniform heat flow in an infinite medium

A complex analysis of rigid body rotation is presented. The crack-tip rotation for a line crack subjected to steady uniform heat flow is obtained in terms of thermal stress intensity factor in shear mode of the crack, the material and thermal parameters and coordinates of points close to the crack tip. The shear strip configuration is analysed on the basis of rotation and displacement at the end of the shear strip.     

The interaction of two inclined cracks with dynamic stress wave loadings

To gain insight into the phenomenon of the interaction of stress waves with material defects and the linkage of two cracks, the transient response of two semi-infinite inclined cracks subjected to dynamic loading is examined. The solutions are obtained by the linear superposition of fundamental solutions in the Laplace transform domain. The fundamental solution is the exponentially distributed traction on crack faces proposed by Tsai and Ma [1]. The exact closed form solutions of stress intensity factor histories for these two inclined cracks subjected to incident plane waves and diffracted waves are obtained explicitly. These solutions are valid for the time interval from initial loading until the first wave scattered at one crack tip returns to the same crack tip after being diffracted by another crack tip. The result shows that the contribution of diffracted waves to stress intensity factors is much less than the incident waves. The probable crack propagation direction is predicted from the fracture criterion of maximum circumferential tensile stress. The linkage of these two cracks is also investigated in detail.     

Shielding and amplification of a penny-shape crack due to the presence of dislocations

The interaction between a penny-shape crack and a dislocation in crystalline materials is investigated within the framework of dislocation dynamics. The long-range and singular stress field resulting from the crack is determined by modeling the crack as continuous distribution of dislocation loops. This distribution is determined by satisfying the traction boundary condition at the crack face, resulting into a singular integral equation of the first kind that is solved numerically. This crack model is integrated with the dislocation dynamics simulation technique to yield the stress field of the combine system of crack and different types of dislocations situated at different positions in a three dimensional space. The integrated system is then used to investigate the dislocation behavior and its influence on the crack opening displacement and the characteristic of the stress field near the crack tip. It is shown that, depending on the relative position of the dislocation and its character, the dislocation may result in reduction in the stress amplitude at the crack tip and in some cases in closure of the crack tip. These analyses yield shielding and amplification zones near the crack providing an insight of the dislocation influence on the crack. The full dislocation dynamic analysis reveals the nature of the crack dislocation interaction and the manner in which the dislocation morphology changes as it is attracted to the crack surfaces, as well as the changes it causes to the crack profile.     

狭长磁电弹性体中Ⅲ型半无限裂纹的场强度因子

通过引入合适的保角映射, 利用复变函数方法研究了裂纹面上受反平面剪应力和面内磁电载荷共同作用下狭长磁电弹性体中半无限裂纹的断裂行为, 给出了磁电全非渗透型边界条件下裂纹尖端场强度因子和能量释放率的解析解。当狭长体高度趋于无限大时, 可得到无限大磁电弹性体中半无限裂纹的解析解。若不考虑磁场或电场作用, 所得解可退化为已知解。通过数值算例, 分析了裂纹面上受载长度、狭长体高度以及磁、电和机械载荷对能量释放率的影响。     

Some problems involving distributions of screw dislocations in a finite slab

This paper considers two problems involving distributions of screw dislocations in a finite slab. The first problem is concerned with the spread of plasticity from a crack ¦x₁¦c, x₂ = 0 in the finite slab ¦x₂¦ h/2 subject to an externally applied shear stress p₂₃ = σ, and is treated in terms of the theory of continuous distributions of dislocations. The extent to which the dislocations describing the plastic relaxation spread from a crack tip is determined, together with the relative displacement of the crack surfaces at its tips. Thus the criterion for crack extension is calculated, particular attention being given to the case when fracture occurs at low applied stresses. The second problem studied is the special case of the first problem that arises when no plastic relaxation is allowed; the criterion for crack extension is then determined from energetic considerations; the similarity between the two sets of results for low stress fractures is emphasized.     

Rigid body rotation surrounding the mode II crack

Using Westergaard function in conventional and also in modified form the rigid body rotation at the crack tip and near the crack edges of mode II cracks are obtained. An attempt is made to explain the deformed configuration of a mode II crack. The rotation at the end of the plastic shear strip ahead of the crack tip is shown to be independent of the stress intensity factor. The rotation field in the vicinity of a crack tip under combined mode I and mode II conditions is obtained.     

Fatigue crack closure study based on whole-field displacements

This study is concerned with crack tip strain field fluctuations at loads below the point of crack closure in fatigue cycling. Moiré interferometry was used to investigate crack tip fields in compact tension specimens, cracked under constant stress intensity range and fixed R-ratio conditions. An elastic-plastic finite element model of simulated closure was developed to provide a theoretical cross-reference for the moiré studies. The ‘stretched zone’, which is believed to be the most significant source of closure effects, was simulated by inserting a constant thickness strip of elements into the crack before unloading from the maximum load point. Analysis of the crack tip fields in the experimental and theoretical cases was made in terms of crack face opening profiles, compliance changes and elastic stress intensity parameters. The latter were inferred through stress and displacement measurements made along circular and radial paths relative to the crack tip. Closure on the stretched zone was found to generate non-proportional loading in the crack tip field, so that the resulting stress changes were not well characterized by the asymptotic elastic equations. It is concluded firstly, that significant strain fluctuations occur below the point of closure load and that these should not be ignored in crack propagation studies. Secondly, the effective stress intensity range in fatigue cycling is not simply related to the open-crack stress intensity range and the need therefore remains for R-ratio and geometry effects to be treated as variables in crack propagation data collection programmes.     

Crack tip stress singularities and dislocation interactions with anisotropic viscoelastic bimaterial interfaces

The time evolution of the force on a dislocation and the near tip stress field of a semi-infinite crack, lying at an angle towards the welded interface of a bimaterial and just touching it, is studied. The bimaterial consists of viscoelastic media with the most general anisotropy. Applying the Laplace transform to the field equations gives similar relations to those considered in [3], provided that appropriate elastic coefficients are replaced by their Laplace-transformed viscoelastic counterparts. Numerical methods are then used for the inversion of both the Laplace and Mellin transforms and numerical results for near crack tip stress fields are presented along with asymptotic ones for small and large times. Corresponding results are presented for the force on a dislocation. For simple anisotropies a relation between the force on a dislocation and the time dependence of appropriate crack tip stresses is established.     

In-plane bending problem of a double edge cracked plate

In this paper the boundary collocation method is used for evaluating the stress intensity factors (SIF) of a double edge cracked plate under in-plane bending. For the case with a large ratio of the plate height to the width, h/b, the results obtained compare very favorably with existing solutions for an infinite strip. Moreover, this method has been used for different finite plates, and a series of conclusions is provided for application.     

Micro-crack initiation at the tip of a semi-infinite rigid line inhomogeneity in piezoelectric solids

A dislocation emission mechanism for micro-crack initiation at the tip of a semi-infinite rigid line inhomogeneity in a piezoelectric solid is proposed in the present paper. For a rigid line inhomogeneity embedded in a piezoelectric matrix, dislocations of one sign are driven away from the tip due to high stress level, while the stationary dislocations of the opposite sign are left behind near the tip of the inhomogeneity. As a result, a micro-Zener–Stroh crack is initiated ahead of the line inhomogeneity. In the current study, a dislocation pileup mechanism for micro-crack initiation at the inhomogeneity tip is proposed. An interesting result is that the critical stress intensity factors for a line inhomogeneity perpendicular to the poling direction can be related to the fracture toughness of a conventional crack in the same material. Analytical solutions show that the critical plane shear stress intensity factor depends on the plane shear mechanical and displacement loadings, and the critical opening stress and electric displacement intensity factors depend on not only the mechanical and displacement loadings, but the electric field and displacement loadings as well.     

Energy-density concept in fracture mechanics

A theory of fracture mechanics is proposed in which attention is focused on the intensity of the energy field in the crack tip region. This energy field possesses a 1/r-type of singularity for both elastic and plastic materials. The strength or amplitude of this field will be referred to as the “energy-density factor”, S. Unlike the stress-intensity factor k in classical fracture mechanics which is only a measure of the local stress amplitude, the energy-density factor is also direction sensitive. The difference between k and S is analogous to the difference between a scalar and vector quantity. In this sense, the critical value S_cr specifies the direction of crack initiation as well as the fracture toughness of the material.     

Thermal effect of plastic dissipation at the crack tip on the stress intensity factor under cyclic loading

Plastic dissipation at the crack tip under cyclic loading is responsible for the creation of an heterogeneous temperature field around the crack tip. A thermomechanical model is proposed in this paper for the theoretical problem of an infinite plate with a semi-infinite through crack under mode I cyclic loading both in plane stress or in plane strain condition. It is assumed that the heat source is located in the reverse cyclic plastic zone. The proposed analytical solution of the thermo-mechanical problem shows that the crack tip is under compression due to thermal stresses coming from the heterogeneous stress field around the crack tip. The effect of this stress field on the stress intensity factor (its maximum and its range) is calculated analytically for the infinite plate and by finite element analysis. The heat flux within the reverse cyclic plastic zone is the key parameter to quantify the effect of dissipation at the crack tip on the stress intensity factor.     

The elastic contact problem for dissimilar orthotropic semi-infinite and infinite strips

This article examines the stress state which develops when the end of a symmetrically loaded orthotropic semi-infinite strip is supported on an orthotropic infinite strip. Both smooth and bonded contact between the strips are considered. By using Fourier transforms the problem is reduced to a system of singular integral equations. The dependence of the order of the stress singularity on the orthotropic properties of the materials is investigated. Numerical solutions are presented for the contact stresses corresponding to different loadings and material properties.