The practical evaluation of stress intensity factors at semi-infinite crack tips

The solution of crack problems in plane or antiplane elasticity can be reduced to the solution of a singular integral equation along the cracks. In this paper the Radau-Chebyshev method of numerical integration and solution of singular integral equations is modified, through a variable transformation, so as to become applicable to the numerical solution of singular integral equations along semi-infinite intervals, as happens in the case of semi-infinite cracks, and the direct determination of stress intensity factors at the crack tips. This technique presents considerable advantages over the analogous technique based on the Gauss-Hermite numerical integration rule. Finally, the method is applied to the problems of (i) a periodic array of parallel semi-infinite straight cracks in plane elasticity, (ii) a similar array of curvilinear cracks, (iii) a straight semi-infinite crack normal to a bimaterial interface in antiplane elasticity and (iv) a similar crack in plane elasticity; in all four applications appropriate geometry and loading conditions have been assumed. The convergence of the numerical results obtained for the stress intensity factors is seen to be very good.     

On modelling of piece-wise smooth cracks in two-dimensional finite bodies

An integral equation method for the solution to problems of piece-wise smooth cracks in two-dimensional finite bodies is presented. The method is based on an integral equation for the resultant forces along the crack line, coupling to an integral equation for the displacements on the outer boundary. Two kinds of integral kernels have been derived. One is based on the fundamental solutions for an infinite plane and the other is based on the fundamental solutions for a half-plane. The latter can be directly applied to both internal crack problems and surface crack problems. Four numerical examples are presented and compared either to existing results or to the author's FEM calculations.     

Growth characteristics of two interacting cracks

This paper deals with the growth characteristics of two interacting cracks in a mixed mode condition. The problem considered is that of two cracks of different lengths, arbitrarily placed in a plane. A uniform load is applied in the plane at a large distance away from the cracks. The objectives of this study are threefold: (1) Solve the mixed boundary value problem described in the foregoing; (2) Predict the most vulnerable crack tip where the instability would first occur as the load is increased; (3) Determine the directions in which the cracks may grow. The numerical results given are only for the specific case of one crack lying along a radial line which emanates from the tip of the other crack at an arbitrary angle.     

拉—拉疲劳下15MnMoVN多裂纹扩展研究

为分析单裂纹或多裂纹在裂纹面承受疲劳拉伸载荷作用下尖端应力强度因子变化规律和裂纹形貌变化以及疲劳寿命情况,以含不同初始长深比的半椭圆单裂纹或双裂纹的薄片试样为研究对象,对试样在应力比R=0.1的疲劳拉伸载荷下单裂纹或双裂纹情况进行了仿真分析。建立含裂纹试样的有限元模型,仿真分析了裂纹在扩展过程中尖端应力强度因子的分布情况,并将单裂纹扩展结果与双裂纹相互作用影响下的结果进行了对比研究;进行含裂纹试样的疲劳实验,分析了含单裂纹或双裂纹的试样的断裂面的形成原因,并验证仿真结果正确性。结果表明,裂纹面之间的相互作用会逐渐影响裂纹的扩展方向、扩展速率以及在扩展过程中尖端应力强度因子的变化趋势;而且初始形貌为半椭圆形的双裂纹在相互作用影响下会逐渐过渡到半圆形。     

压电复合材料中幂函数型曲线裂纹的反平面问题

通过构造新保角映射, 利用Stroh公式研究了远场受反平面剪应力和面内电载荷共同作用下无限大压电复合材料中幂函数型曲线裂纹的断裂行为。给出了电不可渗透边界条件下裂纹尖端场强度因子和机械应变能释放率的解析解。该解析解在幂函数的幂次为零时, 可退化为已有文献中无限大压电复合材料含直线裂纹的结果, 证明了其合理性。由解析解可知, 裂纹几何形状一定时, 电场分布将不受机械载荷的影响。最后, 通过数值算例讨论了幂函数的幂次、 系数及其在 x₁轴上的投影长度对机械应变能释放率的影响。结果表明, 当压电体仅受 x₂方向载荷作用时, 对于给定幂次与开口的曲线裂纹, 在 x₁轴上的投影长度存在一临界值使其最容易开裂; 而对于给定投影长度与幂次的曲线裂纹, 开口越大裂纹越容易扩展。      

Crack growth from naturally occurring material discontinuities under constant amplitude and operational loads

This paper discusses how cracks that grow from small naturally occurring material discontinuities under operational load spectra behave. The growth of small cracks under a representative maritime aircraft flight load spectrum is discussed first. The results of this study, when taken in conjunction with the authors previous studies into cracks growing under combat aircraft load spectra, illustrate how for cracks that grow from small naturally occurring material discontinuities under operational load spectra crack growth can often be easily and accurately computed. It is also shown that the Hartman–Schijve variant of the NASGRO crack growth equation is able to accurately represent the growth of small cracks in two different rail steels. It is further shown that the growth of both small and long cracks can be described by a family of da/dN versus ΔK curves and that, for 7050-T7451, the experimental procedures commonly used to determine a closure free da/dN versus ΔK curve produce curves that are consistent with those obtained using the Hartman–Schijve equation and allowing for small variations in the term ΔK_thr.     

Transient analysis of multiple parallel cracks under anti-plane dynamic loading

The problem of a homogeneous linear elastic body containing multiple non-collinear cracks under anti-plane dynamic loading is considered in this work. The cracks are simulated by distributions of dislocations and an integral equation relating tractions on the crack planes and the dislocation densities is derived. The integral equation in the Laplace transform domain is solved by the Gaussian–Chebyshev integration quadrature. The dynamic stress intensity factor associated with each crack tip is calculated by a numerical inverse Laplace scheme. Numerical results are given for one crack and two or three parallel cracks under normal incidence of a plane horizontally shear stress wave.     

Numerical simulation of plasticity-induced fatigue crack closure with emphasis on the crack growth scheme: 2D and 3D analyses

In this paper a numerical simulation of plasticity-induced fatigue crack closure is performed using the finite element method. Emphasis is placed on the crack growth scheme usually adopted for modelling fatigue crack growth in crack closure problems. The number of load cycles between node releases usually reported in the literature has been, in general, one or two. The present work shows that increasing the number of load cycles between node releases has a strong effect on the opening stresses, particularly, under plane strain conditions and 3D fatigue cracks, in contrast plane stress shows little variation with increasing number of load cycles. This investigation also suggests that ratchetting may take place close to the crack tip in both plane strain and 3D crack problems. The problem of discontinuous crack closure under plane strain conditions, often reported in the literature, is also addressed.     

The J and M Integrals and Energy of Two Unequal Collinear Cracks

The energetics of two unequal-length collinear cracks for modes- I, II, III is considered. The analysis is performed for remote uniform load normal to the plane of cracks by applying the J and M path-independent integrals and the relationship between them. The material forces for each crack as well as the total crack energy of the system are evaluated. The exact closed form solution to this interaction problem is expressed as a function of cracks’ dimensions and spacing between them. The expression derived for the energy of interaction between a main crack and a nearby microcrack can be used for an estimate of the toughness degradation due to microcracks. Another application of the two-crack energy solution is related to the elastic compliance of cracked solids relevant for the cases when crack faces are in contact along certain areas.     

An integral equation approach to the inclusion-crack interactions in three-dimensional infinite elastic domain

 In this paper, an integral equation method to the inclusion-crack interaction problem in three-dimensional elastic medium is presented. The method is implemented following the idea that displacement integral equation is used at the source points situated in the inclusions, whereas stress integral equation is applied to source points along crack surfaces. The displacement and stress integral equations only contain unknowns in displacement (in inclusions) and displacement discontinuity (along cracks). The hypersingular integrals appearing in stress integral equation are analytically transferred to line integrals (for plane cracks) which are at most weakly singular. Finite elements are adopted to discretize the inclusions into isoparametric quadratic 10-node tetrahedral or 20-node hexahedral elements and the crack surfaces are decomposed into discontinuous quadratic quadrilateral elements. Special crack tip elements are used to simulate the variation of displacements near the crack front. The stress intensity factors along the crack front are calculated. Numerical results are compared with other available methods. Received: 28 January 2002 / Accepted: 4 June 2002 The work described in this paper was partially supported by a grant from the Research Grant Council of the Hong Kong Special Administration Region, China (Project No.: HKU 7101/99 E).     

Upper bounds for the stress intensity factors along the boundaries of interacting coplanar cracks in three-dimensional elasticity

An elementary method for obtaining upper bounds for the stress intensity factors along the boundaries of interacting coplanar cracks inside an infinite isotropic elastic medium is presented. This method is based on the singular integral equation of the aforementioned elasticity problem and on the solutions of this equation for each particular crack problem, assumed known. The method is applied to the simple problem of interaction of two circular cracks, as well as to the similar problem of two cracks having the shape of a straight strip. The present results constitute a generalization of the corresponding method for crack problems in two-dimensional elasticity and can easily be further generalized to apply to more complicated crack problems in three-dimensional elasticity.     

Effect of biaxial loads on elastic-plastic <Emphasis Type="Italic">J</Emphasis> and crack tip constraint for cracked plates: finite element study

Based on detailed two-dimensional (2-D) and three-dimensional (3-D) finite element (FE) analyses, this paper attempts to quantify in-plane and out-of-plane constraint effects on elastic-plastic J and crack tip stresses for a plate with a through-thickness crack and semi-elliptical surface crack under positive biaxial loading. For the plate with a through-thickness crack, plate thickness and relative crack length are systematically varied, whereas for the plate with a semi-elliptical surface crack, the relative crack depth and aspect ratio of the semi-elliptical crack are systematically varied. It is found that the reference stress based approach for uniaxial loading can be applied to estimate J under biaxial loading, provided that the limit load specific to biaxial loading is used, implying that quantification of the biaxiality effect on the limit load is important. Investigation on the effect of biaxiality on the limit load suggests that for relatively thin plates with small cracks, in particular with semi-elliptical surface cracks, the effect of biaxiality on the limit load can be neglected for positive biaxial loading, and thus elastic-plastic J for a biaxially loaded plate could be estimated, assuming that such plate is subject to uniaxial load. Regarding the effect of biaxiality on crack tip stress triaxiality, it is found that such effect is more pronounced for a thicker plate. For plates with semi-elliptical surface cracks, the crack aspect ratio is found to be more important than the relative crack depth, and the effect of biaxiality on crack tip stress triaxiality is found to be more pronounced near the surface points along the crack front.     

Influence from geometrical features on the growth rate of a micro-structurally short fatigue crack

The influence on the crack growth rate on a micro-structurally short edge crack subjected to fatigue loading from changes in crack length, distance to grain boundaries and applied load has been investigated. The crack is assumed to grow in a single shear mechanism due to nucleation, glide and annihilation of dislocations along preferred slip planes in the material. The external geometry is modelled by distributed dislocation dipole elements in a boundary element approach under quasi-static and plane strain conditions. The evolving plasticity is described by individual discrete dislocations along a slip plane emanating from the crack in the crack direction. The crack growth rate is shown to be controlled by the plasticity, which in turn is controlled by geometrical parameters in combination with the external load.     

Determination of the opening load for a growing fatigue crack: evaluation of experimental data reduction techniques and analytical models

In metallic materials, growing cracks will remain closed or partially closed for a portion of the applied cyclic load as a consequence of plastically deformed material left in the wake of a growing crack, surface roughness along the crack surfaces, or corrosion debris. Proper characterization of this crack closure and the subsequent opening load is required for accurate prediction of crack growth. In the laboratory, global load–displacement data are commonly used in conjunction with a data reduction technique to estimate the opening load for a growing crack. Different data reduction techniques will be compared, and the influence of data smoothing will be demonstrated, using AA 7075-T651 specimens tested under constant amplitude cyclic loading with load ratios R = 0.1, 0.2, and 0.3. The ratio of maximum stress intensity factor to plane strain fracture toughness was approximately K _max / K _Ic = 0.5. The measured crack opening loads are also compared with predictions obtained from two different strip-yield models and three-dimensional elastic–plastic finite element analyses. Results show the necessity of using smoothed data, and the poor behaviour of the compliance offset data reduction technique, when analysing high load ratio data. A modification to this technique is proposed which improves crack opening load estimates. Overall, the analytical model predictions compare well with the experimental results; especially those results generated using the modified compliance offset technique.     

Contact model of cracks in orthotropic materials

We consider a plane strain problem for an orthotropic half plane loaded at infinity and containing a crack along its fixed edge. To remove a singularity near the right crack tip, we introduce an artificial contact zone. The problem under consideration is reduced to the mixed Dirichlet-Riemann boundaryvalue problem. We present the exact solution of this problem and deduce formulas for stresses in the contact zone and on the continuation of the crack and for the stress intensity factors. By using both analytic and numerical methods, we prove that the energy-release rate is quasiinvariant in the process of crack propagation relative to the size of the contact zone. On the basis of these results, we propose an algorithm for the evaluation of the paramenters of fracture of composites of finite dimensions with interface cracks. As a special case, we develop a model of interface cracks with actual contact zone and establish the dependences of the length of this zone on the external load and elasticity moduli of the material. Dnepropetrovsk State University, Dnepropetrovsk. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 5, pp. 59–66, September–October, 1999     

TEM investigations of intergranular stress corrosion cracking in austenitic alloys in PWR environmental conditions

Abstract

Analytical transmission microscopy has been used to investigate the initiation of stress corrosion cracking in Inconel 600 subjected to constant load testing under simulated pressured water reactor primary water conditions. The observations revealed that intergranular attack proceeded by the development of a zone of polycrystalline chromia along the boundary plane intersecting either the free surface or a blunted, open crack in contact with the free surface. Ni-rich metal particles were interspersed within the chromia. Conversely, open cracks were filled with nanocrystalline NiO and large compound particles of spinel and NiO, indicating a difference in potential between closed, attacked boundaries and open cracks. Open cracks appeared to have initiated by fracture of the chromia zones, such fracture being strongly dependent on boundary geometry with respect to loading direction. The observations suggest that stress corrosion crack initiation and propagation is dependent on diffusion of oxygen through the porous oxides. Dislocations and stress could enhance diffusion as chromia was observed along slip planes at the arrested tips of blunt cracks.     

Crack propagation in AlZnMgCu 0.5 during static loading

Fatigue cracked bend specimens of the age hardened aluminum alloy AlZnMgCu 0.5 were investigated in interrupted static load tests. The purpose of the work described in the present paper was to get some informations about the behaviour of cracks under static loads. The fractographic observations showed that crack propagation due to static loads can be divided into two stages, namely initial crack propagation by minute amounts during the blunting of the crack tip and crack propagation due to dimple formation. Both stages of crack propagation can be described quantitatively by simple models. In addition, the stress intensity at the onset of the second crack propagation stage and the plane strain crack toughness can be predicted quite accurately.     

Two coplanar Griffith cracks in an infinite elastic layer under torsion

Summary We consider the problem of determining the stress distribution in an infinitely long isotropic homogeneous elastic layer containing two coplanar Griffith cracks which are opened by internal shear stress acting along the lengths of the cracks. The faces of the layer are assumed to be stress free. The cracks are located in the middle plane of the layer parallel to its faces. By using Fourier transforms, we reduce the problem to the solution of a set of triple integral equations with a cosine kernel and a weight function. These equations are solved exactly by using finite Hilbert transform techniques. Finally we derive the closed form expressions for the stress intensity factors and the crack energy. Solutions to the following problems are derived as particular cases: (i) a single crack in an infinite layer under torsion, (ii) two coplanar cracks in an infinite space under torsion, (iii) a single crack in an infinite space under torsion.     

A general method for solving dynamically accelerating multiple co-linear cracks

We present a general method for analyzing dynamically accelerating multiple co-linear cracks that can be applied to the contexts of plane strain or antiplane shear in an elastic material. The difficulty in solving such problems lies in the fact that the space-time regions containing known data evolve as the crack propagates in an a priori unknown manner. Using an analog to a Dirichlet-to-Neumann map, we can find complete knowledge of the stress and displacement along the fracture plane, facilitating the application of fracture criteria that require these values away from the crack tip. The method is demonstrated for a semi-infinite or finite mode III crack as well as for a pair of cracks in elastic material, using a stress intensity factor fracture criterion for simplicity.     

Failure modes and serviceability of high strength self compacting concrete deep beams

The behaviour of deep beams is significantly different from shallow beams. In deep beams, the plane section does not remain plane after deformation. The main purpose of this study is to facilitate the prediction of deep beam failure related to tensile bar and web reinforcement percentage variations. Six high strength self compacting concrete (HSSCC) deep beams were tested until failure. Strains were measured on concrete surface along mid span, tensile bar and compression strut trajectory. The load was incrementally applied and at each load increment new cracks, their widths and propagation were monitored. The results clearly show that, at ultimate limit condition, the strain distribution on concrete surface along mid-span is no longer parabolic. In deep beams several neutral axes were obtained before ultimate failure is reached. As the load increases, the number of neutral axis decreases and at failure load it reduces to one. The failure of deep beams with longitudinal tensile steel reinforcement less than that suggested by ACI codes is flexural and is accompanied by large deflections without any inclined cracks. As the longitudinal tensile steel reinforcement increases, the failure due to crushing of concrete at nodal zones was clearly observed. The first flexural crack at mid-span region was always vertical. It appeared at 25–42% of peak load. The crack length was in the range of 0.24–0.6 times the height of section. As the tensile bar percentage increases number of cracks increases with reduced crack length and crack width. The appearance of first inclined crack in compression strut trajectory is independent of tensile and web bar percentage variations.