An inclined surface crack subject to biaxial loading

The elastic–plastic stress fields and mode mixity parameters for semi-elliptical surface cracks on biaxial loaded plates have been investigated using detailed three-dimensional finite element calculations. Different degrees of mode mixity are given by combinations of the far-field stress level, biaxial stress ratio and inclined crack angle. These analyses were performed for different surface flaw geometries to study the combined load biaxiality and mode mixity effects on the crack-front stress fields and the size and shape of the plastic zones. It is clear from considering the local stress distributions along the crack front that the elastic crack tip singularities have been derived for several particular cases of mixed mode biaxial loading. By theoretical analysis, the new formulae have been introduced for both the elastic and plastic mode-mixity parameters, accounting for ratios between the I/II, II/III and III/I modes. Particular attention was paid to the strong variations of the mode-mixity parameters along the semi-elliptical surface crack front. The mixed-mode behavior of the crack growth direction angle along the semi-elliptical crack front for different combinations of biaxial loading and inclination crack angles was also determined. It was done using methods based on the maximum tangential stress and the strain energy density criteria.     

Coulomb traction on a penny-shaped crack in a three dimensional piezoelectric body

The axisymmetric problem of a penny-shaped crack embedded in an infinite three-dimensional (3D) piezoelectric body is considered. A general formulation of Coulomb traction on the crack surfaces can be obtained based on thermodynamical considerations of electromechanical systems. Three-dimensional electroelastic solutions are derived by the classical complex potential theory when Coulomb traction is taken into account and the poling direction of piezoelectric body is perpendicular to the crack surfaces. Numerical results show that the magnitude of Coulomb tractions can be large, especially when a large electric field in connection with a small mechanical load is applied. Unlike the traditional traction-free crack model, Coulomb tractions induced by an applied electric field influence the Mode I stress intensity factor for a penny-shaped crack in 3D piezoelectric body. Moreover, compared to the current model, the traditional traction-free crack model always overestimates the effect of the applied electric load on the field intensity factors and energy release rates, which has consequences for 3D piezoelectric fracture mechanics.     

The failure of a strain-softening material: II. The effect of geometrical and loading parameters

This paper presents theoretical analyses of a variety of models which simulate crack growth in a strain-softening material, with attention being focused on the fully developed softening zone length and the value of the crack tip stress intensity associated with the attainment of such a state. Results from the models show that both these parameters can be very sensitive to both the initial crack configuration and the loading characteristics, and can differ appreciably from the values appropriate to a semi-infinite crack in a remotely loaded infinite solid. The present paper's results underline the view that the analytical results obtained in Part I, and other workers' numerical results for a specific material, are rather special. Part I analyzed the behaviour of a crack in a double cantilever beam specimen, and it was shown that the value of the crack tip stress intensity associated with a fully developed softening zone is essentially independent of the initial crack size and beam height, and is equivalent to the value for a semi-infinite crack in a remotely loaded infinite solid.     

Linear Elastic Solutions for Slotted Plates

Two-dimensional problems of finite-length blunted cracks cut into infinite plates subject to remote tractions are solved using complex variable theory. The slot geometry is composed of two flat surfaces connected by rounded ends. This special geometrical shape was derived by Riabouchinsky in the study of two-dimensional ideal fluid flow around parallel plates. The simpler antiplane slotted plate problem is addressed initially for this geometry. From this exact solution, the equivalent of a Westergaard stress potential is found and applied to the two other principal modes of fracture, which are plane elasticity problems. For a plate subject to uniform radial tension at infinity, an analytical solution is obtained that will reduce to the familiar mode I singular crack solution as the separation between the parallel faces of the slot becomes zero. For finite-width mode I slots, the rounded ends have tensile tractions which terminate at the adjoining flat surfaces of the slot, which remain traction-free. In this respect, the finite-width mode I slot problem resembles a Barenblatt cohesive zone model of a plane crack or a Dugdale plastic strip model of a plane crack, although the tractions will vary in magnitude along the slot ends rather than remaining uniform as in the former type of crack problems. Similarly, in the case of the finite-width mode II slot problem, the rounded ends of the slot have shear tractions, while the flat surfaces remain load-free. A distinguishing feature of the mode II slot solution over the mode I slot problem is that the maximum in-plane shear stress is constant along the rounded ends of the slot. Because of this, those particular regions of the boundary can represent incipient plastic yield based on either the Mises or Tresca yield condition under plane strain loading conditions. In this way, the problem resembles the plastic strip models of Dugdale, Cherepanov, Bilby-Cottrell-Swinden, and others. Notably, the mode III slot problem also has a constant maximum shear stress along the curved portions of the slot, while the entire slot boundary remains traction-free, unlike the mode II slot problem. Consequently, the mode III slot problem represents both a generalization of the standard mode III crack problem geometry, while simultaneously satisfying the boundary conditions of a plastic strip model.     

Stability of an advancing crack to small perturbation of its path

In this work we investigate the stability of a nominally straight two-dimensional quasistatically growing crack to a small perturbation of its path. Formulae for perturbations of stress intensity factors induced by slight deviation of the crack trajectory were developed by Movchan et al. (Int. J. Solids Struct. 35, 3419) Their solution is exploited to derive an equation for the perturbation of the crack path on the assumption that the crack advances in pure “opening” mode (i.e. local K_II=0). Various types of loading conditions are considered, including a cracked body loaded by a pair of point body forces and a crack whose faces are subjected to given tractions acting in the direction normal to the crack boundary. The body is also subjected to a remotely maintained uniaxial stress, aligned with the direction of the unperturbed crack. The loading is assumed to advance as the crack advances, to maintain the critical value of Mode I stress intensity factor. Numerical computations of possible crack paths have been performed, extending results on crack stability obtained by Cotterell and Rice (Int. J. Fract. 16, 155). The results show that in the case of loading by point body forces the stability of the crack path depends on the positions of the points of application of the applied forces and the magnitude of the applied stress acting parallel to the crack. There exists a critical value of this stress such that the crack path is stable for values less than critical and unstable otherwise. It is shown that the crack is always unstable in the case of point force tractions applied normal to the crack faces.     

Penny-shaped cracks

The Somigliana formula is used to reduce an arbitrary elastic crack problem to a system of three integral equations for the components of displacement discontinuity. For the case of a penny shaped crack situated in an infinite isotropic medium with the crack faces subjected to arbitrary tractions, the integral equations are solved explicitly. In particular integral formulae are obtained for the stresses on the plane of the crack beyond the crack-tip, and hence for the stress intensity factors. The special case of uni-directional shear traction on the crack is examined.     

Dynamic crack tip fields and dynamic crack propagation characteristics of anisotropic material

Based on mechanics of anisotropic material, the dynamic crack propagation problem of I/II mixed mode crack in an infinite anisotropic body is investigated. Expressions of dynamic stress intensity factors for modes I and II crack are obtained. Components of dynamic stress and dynamic displacements around the crack tip are derived. The strain energy density theory is used to predict the dynamic crack extension angle. The critical strain energy density is determined by the strength parameters of anisotropic materials. The obtained dynamic crack tip fields are unified and applicable to the analysis of the crack tip fields of anisotropic material, orthotropic material and isotropic material under dynamic or static load. The obtained results show Crack propagation characteristics are represented by the mechanical properties of anisotropic material, i.e., crack propagation velocity M and fiber direction α. In particular, the fiber direction α and the crack propagation velocity M give greater influence on the variations of the stress fields and displacement fields. Fracture angle is found to depend not only on the crack propagation but also on the anisotropic character of the material.     

A novel method in determination of dynamic fracture toughness under mixed mode I/II impact loading

The objective of this paper is to propose a novel methodology for determining dynamic fracture toughness (DFT) of materials under mixed mode I/II impact loading. Previous experimental investigations on mixed mode fracture have been largely limited to qusi-static conditions, due to difficulties in the generation of mixed mode dynamic loading and the precise control of mode mixity at crack tip, in absence of sophisticated experimental techniques. In this study, a hybrid experimental–numerical approach is employed to measure mixed mode DFT of 40Cr high strength steel, with the aid of the split Hopkinson tension bar (SHTB) apparatus and finite element analysis (FEA). A fixture device and a series of tensile specimens with an inclined center crack are designed for the tests to generate the components of mode I and mode II dynamic stress intensity factors (DSIF). Through the change of the crack inclination angle β (=90°, 60°, 45°, and 30°), the K_II/K_I ratio is successfully controlled in the range from 0 to 1.14. A mixed mode I/II dynamic fracture plane, which can also exhibit the information of crack inclination angle and loading rate at the same time, is obtained based on the experimental results. A safety zone is determined in this plane according to the characteristic line. Through observation of the fracture surfaces, different fracture mechanisms are found for pure mode I and mixed mode fractures.     

Crack at the apex of a loaded notch

This paper deals with the plane elastostatic problem of an infinite wedge, subjected to arbitrary surface tractions, and cracked along the wedge angle bisector. The problem is reduced to a single Fredholm integral equation, which is solved numerically for normal loads on the crack faces and various loads on the wedge faces. It is shown that the crack tip intensity factor depends strongly on the wedge angle. An approximation to a half plane with a notch of finite angle, cracked at its apex, is also obtained.     

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

Aprocedure is described for determining dynamic stress intensity factor histories for a half plane crack in an otherwise unbounded elastic body, with the crack faces subjected to tractions that result in variation of the stress intensity factor along the crack edge. The procedure is based on integral transform methods and the properties of analytic functions of a complex variable. The procedure is illustrated for the case of a pair of opposed line loads suddenly applied on the crack faces along a line perpendicular to the crack edge. An exact expression is obtained for the resulting mode I stress intensity factor as a function of time for any point along the crack edge. Some features of the solution, as well as possible extensions of the procedure, are discussed.     

Stress distribution in a two-dimensional infinite anisotropic medium with collinear cracks

The plane problem of an anisotropic material with cracks, whose surfaces are subject to surface tractions of a general kind, is studied. The medium considered if of infinite extent and the cracks are located on a single line. The Fourier transform method is employed to derive the stress and displacement components at an arbitrary point of the medium in terms of the dislocation densities and the stress discontinuities on the crack line.These formulae for stress and displacement components involve the roots of a quartic equation whose coefficients are the material constants. The cases of different roots and pairwise coincident roots are examined separately. An orthotropic medium is an important example for the case of different roots while an isotropic medium is that for the case of pairwise coincident roots. These examples are discussed in detail.As an illustration of the use of these formulae the problem of a single crack in an infinite anisotropic medium is examined in detail.Work supported in part by a grant from Council of Scientific and Industrial Research, New Delhi, India.     

CTS试件中复合型疲劳裂纹扩展

针对复合型循环载荷作用下的金属构件中的裂纹扩展问题进行了实验分析和理论建模. 首先 采用紧凑拉剪试件(CTS)和 Richard研制的复合型载荷加载装置，对承受复合型循环载荷的裂纹进行了实验研究. 实验选择了两种金属材料试件，分别承受3种形式的复合型循环载荷的作用，在裂纹尖端具 有相同的初始应力场强度的条件下考察复合型循环载荷对裂纹扩展规律的影响. 实验结果表明，疲劳裂纹的扩展速率与加载角度有关. 对于同样金属材料的试件，当裂尖处 初始应力场强度相等时，载荷越接近于II型，裂纹增长速率越快. 采用等效应力强度 因子(I型和II型应力强度因子的组合)、裂纹扩展速率及复合强度等参数，以实验数据为 基础，建立了一个疲劳裂纹扩展模型，用来预测裂纹在不同模式疲劳载荷作用下的扩展速率. 为验证其有效性，该模型被应用于钢制试件的数值模拟计算中. 实验结果与模拟计算曲线保 持一致，表明该模型可以用来估算带裂纹金属构件的寿命.     

Opening-Function Simulation of the Three-Dimensional Nonstationary Interaction of Cracks in an Elastic Body

Integral relations between three-dimensional dynamic displacements (stresses) in an infinite elastic body with arbitrarily located plane cracks and discontinuities in the displacements of the opposite crack faces are presented. The influence of opening cracks on each other is considered in the problem on crack faces loaded by pulse forces. This problem is reduced to a system of boundary integral equations of the wave-potential type in a time domain. The dynamic mode I stress intensity factors are determined for two coplanar elliptic cracks under forces in the form of the Heaviside function     

The fracture mechanics of slit-like cracks in anisotropic elastic media

U method of continuously distributed dislocations, the problem of a slit-like crack in an arbitrarily-anisotropic linear elastic medium stressed uniformly at infinity is formulated and solved. The crack faces may be either freely-slipping or loaded by arbitrary equal and opposite tractions. If there is no net dislocation content in the crack, then the tractions and stress concentrations on the plane of the crack are independent of the elastic constants and the anisotropy; the same is true of the elastic stress intensity factors. The crack extension force depends on anisotropy only through the inverse matrix elements K _mg⁻¹, where [K] is the pre-logarithmic energy factor matrix for a single dislocation parallel to the crack front. Numerical results for crack extension forces are presented for three media of cubic symmetry.     

Dynamic crack kinking from a PMMA/Homalite interface

The behavior of a pre-existing, dynamically loaded, interfacial crack kinking away from the interface separating two materials was experimentally investigated under dynamic loading conditions. Dynamic fracture experiments were performed on pre-cracked bimaterial panels of PMMA bonded with Homalite-100 impact loaded using a high-speed gas gun. By varying the location of impact, a large range of mixed mode loading at the crack tip was produced. Information about the stress state surrounding the crack tip was obtained through use of the lateral shearing interferometer of coherent gradient sensing in conjunction with high-speed photography. The high-speed interferogram corresponding, to the time of crack initiation was analyzed in each case to find the preinitiation mode mixity at the crack tip. Measurement of both the local initiation mode mixity and the crack kink angle allows for possible extension of existing quasi-static interface crack kinking criteria, such as maximum opening stress or maximum energy release rate, to the case of dynamic loading. It was found that for bimaterial systems with small material property mismatch, such as the PMMA/Homalite system, the maximum opening stress criterion accurately predicts the relation between crack tip mode mixity and resulting kink angle for small initial crack kinking speeds.     

Thermal shock analyses for a strip containing an infinite row of periodically distributed cracks normal to its edge

The transient thermal stress problem of a semi-infinite plate containing an infinite row of periodically distributed cracks normal to its edge is investigated in this paper. The elastic medium is assumed to be cooled suddenly on the crack-containing edge. By the superposition principle, the formulation leads to a mixed boundary value problem, with the negating tractions arisen from the thermal stresses for a crack-free semi-infinite plate. The resulting singular integral equation is solved numerically. The effects on the stress intensity factors due to the presence of periodically distributed cracks in a semi-infinite plate are illustrated. For both the edge crack and the embedded crack arrays, the stress intensity factors increase, due to the reduction of the shielding effect, as the stacking cracks are more separated. For the case of embedded crack array, one has the further conclusion that the stress intensity factors decline as the crack array shifts from the plate edge.     

On the asymptotic crack-tip stress fields in nonlocal orthotropic elasticity

The crack-tip stress fields in orthotropic bodies are derived within the framework of Eringen’s nonlocal elasticity via the Green’s function method. The modified Bessel function of second kind and order zero is considered as the nonlocal kernel. We demonstrate that if the localisation residuals are neglected, as originally proposed by Eringen, the asymptotic stress tensor and its normal derivative are continuous across the crack. We prove that the stresses attained at the crack tip are finite in nonlocal orthotropic continua for all the three fracture modes (I, II and III). The relative magnitudes of the stress components depend on the material orthotropy. Moreover, non-zero self-balanced tractions exist on the crack edges for both isotropic and orthotropic continua. The special case of a mode I Griffith crack in a nonlocal and orthotropic material is studied, with the inclusion of the T-stress term.     

Penny-shaped and half-plane cracks in a transversely isotropic piezoelectric solid under arbitrary loading

Summary The problem of a penny-shaped crack in a transversely isotropic piezoelectric material loaded by both normal and tangential tractions and by electric charges is analyzed. Closed-form solutions are obtained for the full electroelastic fields as well as for the stress and electric displacement intensity factors. Solutions are also obtained for the (non-trivial) limiting case of a half-plane crack. The results are illustrated on the example of piezoceramics PZT-6B. Received 12 July 1999; accepted for publication 20 July 1999     

Oblique edge crack in an anisotropic material under antiplane shear

An oblique edge crack in an anisotropic material under antiplane shear loadings is investigated. The antiplane problems are formulated based on a linear transformation method. An anisotropic solid containing an edge crack subjected to concentrated forces is first considered. The stress intensity factor for the edge crack with concentrated forces is obtained from the solution of the transformed edge crack in an isotropic material which is solved by using conformal mapping technique and complex function theory. The solution of the edge crack under concentrated loads is used to construct the stress intensity factor for the oblique edge crack in the anisotropic material subjected to antiplane distributed loads. Some numerical computations are carried out to calculate the stress intensity factors for the edge crack in inclined orthotropic materials subjected to point forces as well as distributed tractions.