Closed interface crack with singular spring stiffness model

The singularity of the stress field at the tip of a partially closed interface crack is discussed based on a non-uniform spring stiffness model that includes an inverse function of the radial distance from the crack edge. The stress singularity is found to be governed by the coefficient of the inverse function, henceforth referred to as “stiffness intensity” due to its resemblance to the well-known stress intensity factor. Two distinct non-oscillatory singular stress fields, corresponding to Modes I and II deformations, respectively, are found to coexist. An oscillatory singularity may also appear, but only when the two stiffness intensities are close to each other. This means that the oscillatory singularity exists when the normal and tangential interactions between the upper and lower crack faces are close.     

Numerical analysis of blunting of a crack tip in a ductile material under small-scale yielding and mixed mode loading

The blunting of the tip of a crack in a ductile material is analysed under the conditions of plane strain, small-scale yielding, and mixed mode loading of Modes I and II. The material is assumed to be an elastic-perfectly plastic solid with Poisson's ratio being 1/2. The stress and strain fields for a sharp crack under mixed mode loading are first determined by means of elastic-plastic finite element analysis. It is shown that only one elastic sector exists around the crack tip, in contrast with the possibility of existence of two elastic sectors as discussed by Gao. The results obtained for a sharp crack are used as the boundary conditions for the subsequent numerical analysis of crack tip blunting under mixed mode loading, based on slip line theory. The characteristic shapes of the blunted crack tip are obtained for a wide range of Mode I and Mode II combinations, and found to resemble the tip of Japanese sword. Also the stress field around the blunted crack tip is determined.     

Experimental and numerical investigations on the influence of the loading direction on the fatigue crack growth

During a service loading fatigue cracks can be subjected to a mixed mode loading if, due to the alteration of the loading direction, the basic crack modes (Modes I, II and III) are combined. An alteration of the loading direction, e.g. can occur either occasionally paired with an overload (mixed mode overload) or permanently in terms of a mixed mode block loading as a combination of normal and shear stresses.Within the scope of this paper, experimental investigations on both mixed mode overloads, which are interspersed into a Mode I baseline level loading, and mixed mode block loadings are presented. The experimental investigations show that the retardation effect decreases with an increasing amount of Mode II of the overload. Due to the block loading, the fatigue crack growth rate is retarded as well, and the crack is also deflected. The kinking angle depends on the fraction of shear stresses. Furthermore, a detailed elastic–plastic finite element analysis of the fatigue crack growth after mixed mode overloads is presented in order to understand the mechanism of the load interaction effects. By such numerical simulations, it can be shown that, due to mixed mode overloads, plastic deformations occur, which on the one hand reduce the near-tip closure and on the other hand cause a far-field closure. Also the stress distribution before and after the crack tip changes. A mixed mode overload causes lower closure and the crack tip deformations become asymmetrical, which is a reason for the smaller retardation effect of a mixed mode overload.     

Crack kinking under high pressure in an elastic-plastic material

Directional crack growth criteria in compressed elastic–plastic materials are considered. The conditions at the crack tip are evaluated for a straight stationary crack. Remote load is a combined hydrostatic stress and pure shear, applied via a boundary layer assuming small scale yielding. Strains and deformations are assumed to be small. Different candidates for crack path criteria are examined. Maximum non-negative hoop stress to judge the risk of mode I and maximum shear stress for mode II extension of the crack are examined in some detail. Crack surfaces in contact are assumed to develop Coulumb friction from the very beginning. Hence, a condition of slip occurs throughout the crack faces. The plane in which the crack extends is calculated using a finite element method. Slip-line solutions are derived for comparison with the numerically computed asymptotic field. An excellent agreement between numerical and analytical solutions is found. The agreement is good in the region from the crack tip to around halfway to the elastic–plastic boundary. The relation between friction stress and yield stress is varied. The crack is found to extend in a direction straight ahead in shear mode for sufficiently high compressive pressure. At a limit pressure a kink is formed at a finite angle to the crack plane. For lower pressures the crack extends via a kink forming an angle to the parent crack plane that increases with decreasing pressure.     

奇异薄层单元及其在双材料界面断裂分析中的应用

构造了一种平面应力奇异薄层单元并证明了其具有-1/2阶奇异性。用此单元研究了双材料界面层的刚度对界面裂纹尖端场的影响。研究发现:对于I 型界面断裂, 减小界面法向刚度对K_Ⅰ、K_Ⅱ的影响远大于减小界面切向刚度, 且法向和切向刚度的减小对K_Ⅱ的影响均大于对K_Ⅰ的影响, 降低法向刚度会显著改变裂尖正应力和剪应力的分布, 而降低切向刚度只明显改变剪应力的分布, 对正应力的分布影响不大;对于界面II 型断裂, 则减小切向刚度对K_Ⅰ、K_Ⅱ的影响远大于减小法向刚度, 且切向、法向刚度的减小对K_Ⅰ影响均大于对K_Ⅱ影响, 降低切向刚度会显著改变裂尖正应力和剪应力的分布, 而降低法向刚度只明显改变裂尖正应力的分布, 对剪应力的分布影响不大。随着界面刚度增大, 应力强度因子和裂尖应力分布均趋近无厚度理想界面情况。     

ANALYSIS OF CRACK TIP HYDROGEN DISTRIBUTION UNDER I/II MIXED MODE LOADS

Abstract— The distribution of hydrogen in the vicinity of a crack tip was studied using SIMS (Secondary Ion Mass Spectrometry) under different ratios of I/II mixed mode loads. Modified WOL specimens with kinked slits were employed in the course of the experimental work. Spectrographic measurements show that under I/II mixed mode loading, both in the HIC and in the r ^max_p directions, there are two hydrogen accumulation peaks ahead of the crack tip, corresponding to the location of the maximum hydrostatic stress and maximum equivalent plastic strain, respectively. Based on results obtained over a range of loading conditions from mode I to a high K_II/ K_I, ratio, it is shown that the mode II component has a clear influence on both peaks. The conditions for hydrogen redistribution are discussed in terms of crack tip stress-strain fields.     

On the contact zone of a subinterfacial Zener-Stroh crack

Summary A Zener-Stroh crack is initiated by dislocations pile-up. Due to this displacement loading mechanism, only one of the two crack tips is sharp, and crack propagation is possible along the sharp tip only. When such a crack is initiated near an interface, the crack faces behind the sharp crack tip may contact each other due to material mismatch and loading combination. In the present study, a subinterface Zener-Stroh crack is analyzed with contact zone consideration near the tip. The problem is formulated as a set of nonlinear Cauchy-type singular integral equations which are solved numerically using Erdogan and Gupta's method. The physically pathological features of interpenetration of the crack surfaces and oscillation of the near tip fields are eliminated in the solutions due to the presence of a contact zone near the crack tip. It is found that the normal traction is bounded at the crack tip where a contact zone exists; while the shear traction has square-root singularities at both the crack tips. This result, is totally different to the case of an interface crack where Mode I and Mode II stress intensity factors, are inter-related at the sharp crack tip.     

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.     

The effect of T‐stress on crack‐tip plastic zones under mixed‐mode loading conditions

In this paper, the influence of T‐stress on crack‐tip plastic zones under mixed‐mode I and II loading conditions is examined. The crack‐tip stress field is defined in terms of the mixed‐mode stress intensity factors and the T‐stress using William's series expansion. The crack‐tip stress field is incorporated into the Von Mises yield criteria to develop an expression that determines the crack‐tip plastic zone. Using the resultant expression, the plastic zone is plotted for various combinations of mode II to mode I stress intensity factor ratios and levels of T‐stress. The properties of the plastic zone affected by T‐stress and mixed‐mode phase angle are discussed. The observations obtained on plastic zones variations are important for further fatigue and fracture analyses for defects in engineering structures under mixed‐mode loading conditions.     

VECTOR CTD CRITERION APPLIED TO MIXED MODE FATIGUE CRACK GROWTH

Abstract— This work is aimed at developing a general parameter based on the deformation intensity at a mixed mode crack tip to predict crack growth behaviour, especially in the near threshold region. Being a mechanisms-related parameter, the vector crack tip displacement (CTD) is defined as a vector summation of CTOD and CTSD_c which act, respectively in the directions of mode I and mode II fatigue crack growth. The basic assumption is that both direction and rate of mixed mode fatigue crack growth are governed by the vector ΔCTD, which represents the resultant of the "driving force"at the crack tip. The analytical predictions obtained by using the vector ΔCTD are in good agreement with the reported experimental results of mixed mode I and II fatigue cracks.     

Comparison of elastic interaction of a dislocation and a crack for four bonding conditions of the crack plane

A comparison of elastic interaction of a dislocation and a crack for four bonding conditions of the crack plane was made. Four cases of single crystalline material, sliding grain boundary, perfectly bonded interface, and sliding interface were considered. The stress intensity factors arising from edge and screw dislocations and their image forces for the above four cases were compared. The stress intensity factor at a crack tip along the perfectly bonded interface arising from screw dislocation can be obtained from that in a single crystalline material if the shear modulus in the single crystalline material is replaced by the harmonic mean of both shear moduli in the bimaterial. The stress intensity factor at a crack tip along the sliding interface arising from edge dislocation in the bimaterial can be obtained from that along the sliding grain boundary in the single material if the μ/(1−ν) in the single material is substituted by the harmonic mean of μ/(1− ν) in the bimaterial where μ and ν are the shear modulus and Poisson's ratio, respectively. The solutions of screw dislocation near a crack along the sliding grain boundary and sliding interface are the same as that of screw dislocation and its mirror image. Generally, the effect of edge dislocation for perfectly bonded interface on the crack propagation is more pronounced than that for the sliding interface. The effect of edge dislocation on the crack propagation is mixed mode for the cases of perfectly bonded interface and single crystalline material, but mode I fracture for the cases of sliding interface and sliding grain boundary. All curves of Fx versus distance r from the dislocation at interface to the right-hand crack tip are similar to one another regardless of dislocation source for both sliding interface and perfectly bonded interface. The level of Fx for m=0 is larger than that for m=−1. This revised version was published online in August 2006 with corrections to the Cover Date.     

Fracture mechanics parameters for cracks on a slightly undulating interface

Typical bimaterial interfaces are non-planar due to surface facets or roughness. Crack-tip stress fields of an interface crack must be influenced by non-planarity of the interface. Consequently, interface toughness is affected. In this paper, the crack-tip fields of a finite crack on an elastic/rigid interface with periodic undulation are studied. Particular emphasis is given to the fracture mechanics parameters, such as the stress intensity factors, crack-tip energy release rate, and crack-tip mode mixity. When the amplitude of interface undulation is very small relative to the crack length (which is the case for rough interfaces), asymptotic analysis is used to convert the non-planarity effects into distributed dislocations located on the planar interface. Then, the resulting stress fields near the crack tip are obtained by using the Fourier integral transform method. It is found that the stress fields at the crack tip are strongly influenced by non-planarity of the interface. Generally speaking, non-planarity of the interface tends to shield the crack tip by reducing the crack-tip stress concentration.     

Branch crack development from the flank of a fatigue crack propagating in mode II

The propagation of fatigue cracks in mode II often leads to the development of a branch starting from a crack flank, some distance behind the tip and not to the expected bifurcation at the crack tip. This type of branch is suggested to initiate by decohesion along a secondary slip plane and to grow in mode I due to the tensile component of the mode II stress field. Finite element calculations are performed to evaluate the stress intensity factors for the main crack and the branch as a function of the position of the latter. It is shown that the branch has a substantial shielding effect on the main crack and generates contact forces along its flanks. The simultaneous and competitive growth of the main crack and the branch in fatigue is simulated step by step using kinetic data for mode II and mode I obtained for a maraging steel.     

Threshold and growth mechanism of fatigue cracks under mode II and III loadings

ABSTRACT The fatigue crack growth behaviour of 0.47% carbon steel was studied under mode II and III loadings. Mode II fatigue crack growth tests were carried out using specially designed double cantilever (DC) type specimens in order to measure the mode II threshold stress intensity factor range, ΔK_IIth. The relationship ΔK_IIth > ΔK_Ith caused crack branching from mode II to I after a crack reached the mode II threshold. Torsion fatigue tests on circumferentially cracked specimens were carried out to study the mechanisms of both mode III crack growth and of the formation of the factory‐roof crack surface morphology. A change in microstructure occurred at a crack tip during crack growth in both mode II and mode III shear cracks. It is presumed that the crack growth mechanisms in mode II and in mode III are essentially the same. Detailed fractographic investigation showed that factory‐roofs were formed by crack branching into mode I. Crack branching started from small semi‐elliptical cracks nucleated by shear at the tip of the original circumferential crack.     

Acoustic emission source characterization in concrete under biaxial loading

The results are reported of a mode identification study on cracks produced during mixed mode loading of concrete. The locations of cracks were found to agree very well with the surface crack patterns for a variety of loading paths. The classification of cracks into mode I and mode II were conducted using a simplified moment tensor analysis. The results indicated that, in general, cracking occurs in mode I, even when the loading is pure shear. Mode II deformations generally follow the mode I cracks, as the crack interface is subjected to shear.     

Characterizing delamination of fibre composites by mixed mode cohesive laws

A novel method is used for the determination of mixed mode cohesive laws and bridging laws for the characterisation of crack bridging in composites. The approach is based on an application of the J integral. The obtained cohesive laws were found to possess high peak stress values. Mixed mode cohesive stresses were found to depend on both the normal and tangential crack opening displacements. The bridging laws, which are to be used together with a mode mixity dependent crack tip fracture energy, were found to possess relative low bridging stresses; the peak normal bridging stress was approximately 2 MPa during pure Mode I and the maximum shear stress during pure Mode II was about 10 MPa.     

Delamination in elastic shear deformable circular layers

This work extends the analytical solution of an interface crack in straight layered structures to circular layered structures. A small segment at the vicinity of an interface crack tip in a circular laminated beam is analyzed by a novel shear deformable bi-layered circular beam theory. Two concentrated forces are found existing at the crack tip due to the requirement of the equilibrium condition. Closed-form solution of the total energy release rate of the interface crack is obtained as the half of the product of the concentrated forces and the corresponding displacement gradient discontinuities at the crack tip. Closed-form expressions of the mode I and II components of the energy release rate are also obtained by global and local methods. Numerical verifications are conducted by analyzing the interlaminar delamination of a circular beam with an edged crack and comparing with the baseline results obtained through finite element analysis. Excellent agreements between the present method and finite element analysis on the predictions of total energy release rate and mode partition verify the accuracy and efficiency of the present solution.     

Cracked Elastic Bi-Material Layer Under Compressive Loads

The boundary value problem of an elastic bi-material layer containing a finite length crack under compressive mechanical loadings has been studied. The crack is located on the bi-material interface and the contact between crack surfaces is frictionless. Based on Fourier integral transformation techniques the solution of the formulated problem is reduced to the solution of singular integral equation, then, with Chebyshev`s orthogonal polynomials, to infinite system of linear algebraic equations. The expressions for contact stresses in the elastic compound layer are presented. Based on the analytical solution it is found that in the case of frictionless contact the shear and normal stresses have inverse square root singularities at the crack tips. Numerical solutions have been obtained for a series of examples. The results of these examples are illustrated graphically, exposing some novel qualitative and quantitative knowledge about the stress field in the cracked layer and their dependence on geometric and applied loading parameters. It can be seen from this study that the crack tip stress field has a mixture of mode I and mode II type singularities. The numerical solutions show that an interfacial crack under compressive forces can become open in certain parts of the contacting crack surfaces, depending on the applied forces, material properties and geometry of the layers.     

The anti-plane shear problem of debonding of two solids made of power law hardening materials

Debonding of two different solids made of power law hardening materials is studied for the case of anti-plane shear loading mode by using an interface crack model. The stresses and the stress intensity factor at the interface crack are determined analytically. Using these analytical results, the constitutive equations by Hencky–Ilyushin and the general equation of energy in the neighborhood of the crack tip, the adhesion energy for the loading mode under consideration is found analytically. It can be observed that for the particular case of two linearly elastic materials and a homogeneous linearly elastic material the solution found here is in excellent agreement with the solutions found in the literature.     

Computation of the stress intensity factors for repaired cracks with bonded composite patch in mode I and mixed mode

In this study, the finite element method is used to analyse the behaviour of repaired cracks with bonded composite patches in mode I and mixed mode by computing the stress intensity factors at the crack tip. The effects of the patch size and the adhesive properties on the stress intensity factors variation were highlighted. The plot of the stress intensity factors according to the crack length in mode I, shows that the stress intensity factor exhibits an asymptotic behaviour as the crack length increases. In mixed mode, the obtained results show that the Mode I stress intensity factor is more affected by the presence of the patch than that of mode II.