Fractal effects of crack propagation on dynamic stress intensity factors and crack velocities

Crack extension paths are often irregular, producing rough fracture surfaces which have a fractal geometry. In this paper, crack tip motion along a fractal crack trace is analysed. A fractal kinking model of the crack extension path is established to describe irregular crack growth. A formula is derived to describe the effects of fractal crack propagation on the dynamic stress intensity factor and on crack velocity. The ratio of the dynamic stress intensity factor to the applied stress intensity factor K(L(D, t), V)/K(L(t), 0), is a function of apparent crack velocity V_o, microstructure parameter d/a (grain size/crack increment step length), fractal dimension D, and fractal kinking angle of crack extension path . For fractal crack propagation, the apparent (or measured) crack velocity V_o, cannot approach the Rayleigh wave speed Cr. Why V_o is significantly lower than Cr in dynamic fracture experiments can be explained by the effects of fractal crack propagation. The dynamic stress intensity factor and apparent crack velocity are strongly affected by the microstructure parameter (d/a), fractal dimension D, and fractal kinking angle of crack extension path . This is in good agreement with experimental findings.     

Interfacial fatigue crack growth in foam core sandwich structures

This paper deals with the experimental measurement of face/core interfacial fatigue crack growth rates in foam core sandwich beams. The so-called ‘cracked sandwich beam’ specimen is used, slightly modified, which is a sandwich beam that has a simulated face/core interface crack. The specimen is precracked so that a more realistic crack front is created prior to fatigue growth measurements. The crack is then propagated along the interface, in the core material, during fatigue loading, as is assumed to occur in a real sandwich structure. The crack growth is stable even under constant amplitude testing. Stress intensity factors are obtained from the FEM which, combined with the experimental data, result in standard da/dN versus ΔK curves for which classical Paris’ law constants can be extracted. The experiments to determine stress intensity factor threshold values are performed using a manual load-shedding technique.     

Prediction of fatigue crack growth rates using crack closure finite element analysis

A three-dimensional finite element fatigue crack closure model of a corner crack and of a through thickness crack has been developed to evaluate the range of effective stress intensity factor from the distribution of the range of stress ahead of the crack tip. The corresponding fatigue crack growth rate was evaluated from a Paris law fit to experimental data from high stress ratio tests. The point of origin for the range of stress distribution was adjusted in accordance with Irwin’s plastic zone correction. Encouraging comparisons of finite element predictions of fatigue crack growth rate incorporating closure effects with experimental measurements were obtained.     

Methods for crack opening load and crack tip shielding determination: a review

In this paper a review of the literature on crack closure/opening load and crack tip shielding effects determination methods is presented. Commonly used ‘subjective’ (visual) and ‘non‐subjective’ approaches have been included. Procedures associated with the determination of an effective crack driving force for both Elber type and that of partial (or incremental) crack closure models have been covered. Comparison among different methods of analyses based on compliance and fatigue crack growth rate measurements is discussed together with their implications and difficulties in fatigue crack growth correlations.     

The competition between the crack kinking away from the interface and crack propagation along the interface in elastic bicrystals

The problem analyzed is of the crack kinking away from the interface between the two different anisotropic materials. The attention is concentrated on the initiation of the crack kinking and the condition that the length of the crack segment that is leaving the interface is small in comparison to the crack segment that remains along the interface. The emphasis is placed to the application of the fracture mechanics concept for the interfacial crack that propagates dynamically between the two orthotropic materials. The simulations and calculations were done by application of the Mathematica ^® programming routine. The stress intensity factors and the energy release rate are obtained for the kinked crack, as functions of the corresponding values for the interfacial crack prior to kinking. The analysis was performed of the influence of anisotropy on the crack kinking versus crack propagating along the interface competition. Due to anisotropy the kinking is easier, i.e., it is easier for the crack to kink away from the interface into the “softer” of the two materials. The oscillatory index for the case of the dynamic crack growth along the interface between the two orthotropic materials increases with crack tip speed v and with increase of the difference in stiffnesses. The practical application of this analysis could be for the interface in the glued joints and protective coatings.     

The influence of stress state on the exponent in the power law equation of fatigue crack growth

Paris et al. were the first to show that the fatigue crack growth rate in metals shows a power law relationship with the stress intensity factor range. It is generally accepted that the exponent is material-dependent. However, it is also true that the empirical Paris equation is dimensionally correct only when the dimensions of the constant in the equation are changed with the power law exponent. In the present paper it will be shown that for 29 identical fatigue crack growth tests on aluminium alloy 7075-T7351 the exponent changes between specimens and crack lengths. Fractography and the crack length measurements show that the exponent is higher and varies at crack lengths where crack growth is dominated by plane strain conditions. The power law exponent decreases and is similar for all specimens after the transition to plane stress conditions at higher crack lengths. The mathematical concept of a pivot point is used to model crack growth with two different exponents using a dimensionally correct equation. It also allows modelling the crack growth variation in all specimens by varying only one parameter, the power law exponent for the plane strain condition.     

Micro- and macro-approach to the fatigue crack growth in progressively drawn pearlitic steels at different R-ratios

This paper deals with the role of microstructure on the fatigue behaviour of pearlitic steels with different degrees of cold drawing. The analysis is focussed on the region II (Paris) of the fatigue behaviour, measuring the constants (C and m) for the different degrees of drawing. From the engineering point of view, the manufacturing process by cold drawing improves the fatigue behaviour of the steels, since the fatigue crack growth rate decreases as the strain hardening level in the material increases. In particular, the coefficient m (slope of the Paris laws) remains almost constant and independent of the drawing degree, whereas the constant C decreases as the drawing degree rises. The paper focuses on the relationship between the pearlitic microstructure of the steels (progressively oriented as a consequence of the manufacturing process by cold drawing) and the macroscopic fatigue behaviour. To this end, a detailed metallographic analysis was performed on the fatigue crack propagation path after cutting and polishing on a plane perpendicular to the crack front (fracto-metallographic analysis). It is seen that the fatigue crack growth path presents certain roughness at the microscopic level, such a roughness being related to the pearlitic colony boundaries more than to the ferrite/cementite lamellae interfaces. Fatigue cracks are transcollonial and exhibit a preference for fracturing pearlitic lamellae, with non-uniform crack opening displacement values, micro-discontinuities, branchings, bifurcations and frequent local deflections that create microstructural roughness. The net fatigue surface increases with cold drawing due to the higher angle of crack deflections. With regard to the influence of the R-ratio, an increase of such a stress ratio produces microcracking with a higher number of branchings for the same stress intensity range.     

The effect of microstructures on fatigue crack growth in Q345 steel welded joint

It is a traditional that the fatigue crack growth behavior is sensitive to microstructure in threshold regime, while it is sensitive to R‐ratio in Paris regime. Fatigue test is carried out for welded joints of a Q345 steel where the compact tension specimens with 3.8 and 12.5 mm thickness are used, and comparisons of fatigue crack growth behavior between base metal and a few different locations in the welded joint are considered in Paris regime. Welding residual stresses are removed by heat treatment to focus the study on the microstructural effect. It is shown that fatigue crack growth rate (FCGR) in the base metal is not sensitive to R‐ratio, but the FCGR increases in the overheated zone, the fusion zone and the weld metal zone with R‐ratio increasing. To the low R‐ratio, FCGR in the three zones is smaller than that in the base metal, but they approximate the same with base metal under the high R‐ratio. The mechanism of fatigue crack growth is analyzed through crack path in microstructures and SEM fractograph. The coarse‐grained ferrite in the base metal is of benefit to relaxation of the average stress at the crack tip, and the fatigue crack growth predicts branching and deflection within above different locations in the welded joint. These tortuous crack paths with crack branching and deflection will promote crack closure as well as crack‐tip stress shielding and then resulted in higher crack growth resistance.     

基于分形理论分析裂缝形态对纤维/混凝土渗透性的影响

分形维数可以表征裂缝形态,能够用来分析混凝土裂缝断面的粗糙程度。裂缝形态对开裂混凝土的渗透性有重要影响,为研究这种影响,利用劈裂试验获得不同宽度的裂缝,使用不同的纤维种类,并设置多种纤维掺量,得到粗糙程度不同的裂缝断面,通过水渗透试验测量不同裂缝宽度时混凝土的渗透系数。采用激光扫描仪扫描裂缝断面并重构3D断面几何形态,采用立方体覆盖法计算断面分形维数。采用分形维数将实测裂缝宽度和有效裂缝宽度联系起来,联立达西定律和泊肃叶定律建立开裂混凝土渗透系数和分形维数的函数关系。结果表明:使用相同的网格划分法,分形维数随着纤维掺量的增加而增大;渗透系数随着纤维掺量的增加而减小;函数关系式中分形维数的指数绝对值和修正系数都随裂缝宽度增加而减小。      

A multifractal analysis of fatigue crack growth and its application to concrete

As is well-known, strength of materials is influenced by the specimen or structure size. In particular, several experimental campaigns have shown a decrease of the material strength under static or fatigue loading with increasing structure size, and some theoretical arguments have been proposed to interpret such a phenomenon. As far as fatigue crack growth is concerned, limited information on size effect is available in the literature, particularly for so-called quasi-brittle materials like concrete. In the present paper, by exploiting concepts of fractal geometry, some definitions of fracture energy and stress intensity factor based on physical dimensions different from the classical ones are discussed. A multifractal size-dependent fatigue crack growth law (expressing crack growth rate against stress intensity factor range) is proposed and used to interpret relevant experimental data related to concrete.     

Application of Paris’ law for estimation of hydrogen-assisted fatigue crack growth

Based on the general form of Paris’ law a new method is proposed for estimating the rate of cracking of metals under Hydrogen-Assisted Fatigue. It is based on relating the fatigue crack growth rate of hydrogen embrittled metal to the fatigue crack growth rate of the metal without hydrogen embrittlement. One-dimensional hydrogen diffusion is assumed. Simulations using the proposed method on steel types X52, X70–80 and X80 are in agreement with published experimental tests results. The new method obviates numerical modelling of crack propagation and much reduces the computational costs.     

Microstructural effects on the short crack behaviour of a stainless steel weld metal during low-cycle fatigue

An experimental study into microstructural effects on short fatigue crack behaviour of 19 stainless steel weld metal smooth specimens during low-cycle fatigue is performed by a so-called ‘effective short fatigue crack criterion’. This material has a mixed microstructure in which it is difficult to distinguish the grains and measure the grain diameter. The columnar grain structure is made up of matrix-rich δ ferrite bands, and the distance between the neighbouring rich δ ferrite bands is an appropriate measurement for characterizing this structure. Particularly, the effective short fatigue cracks (ESFCs) always initiate from the bands of δ ferrite in the matrix in the weakest zone on one of the specimen surface zones which is orientated in accordance with the inner or outer surface of welded pipe from which the specimens were machined. These cracks exhibit characteristics of the microstructural short crack (MSC) and the physically small crack (PSC) stages. The average length of the ESFCs at the transition between MSC and PSC behaviour is ≈40 μm, while the corresponding fatigue life fraction is ≈0.3 at this transition. Different from previous test observations, the growth rate of the dominant effective short fatigue crack in the MSC stage still shows a decrease with fatigue cycling under the present low-cycle fatigue loading levels. A statistical evolution analysis of the growth rates reveals that the short fatigue crack growth is a damage process that gradually evolves from a non-ordered (chaotic) to a perfectly independent stochastic process, and then to an ordered (history-dependent) stochastic state. Correspondingly, the microstructural effects gradually evolve from a weak effect to a strong one in the MSC stage, which maximizes at the transition point. In the PSC stage, the effects gradually evolve from a strong to weak state. This improves our understanding that the short crack behaviour in the PSC stage is mainly related to the loading levels rather than microstructural effects.     

Influence of loading path on fatigue crack growth under multiaxial loading condition

In this study, the specimens made of carbon steel S45 with an initial surface straight edge notch were subjected to combined cyclic axial‐torsion loading at room temperature. The fatigue life, surface crack extension direction and crack length were experimentally investigated. The effects of loading path, stress amplitude ratio and phase angle on the crack growth behaviour were also discussed. The results showed that, under the combination of cyclic axial and torsion loading, the tension stress amplitude had more effect on the initial crack growth path than the latter. The shear stress amplitude contributed mainly to the latter crack extension. The crack extension path was mainly determined by the stress amplitudes and the ratio of the normal stress to the shear stress, and almost independent of the mean stresses. The increase of the tension stress amplitude and shear stress amplitude would both accelerate the crack growth rate.     

Investigation on the fatigue crack behavior of Zr702/TA2/Q345R composite plate with a crack normal to interface

In this paper, the effects of maximum load, load ratio, and average load on fatigue crack propagation of Zr702/TA2/Q345R composite plate with a crack normal to the interface are studied by experiment and finite element method. When crack propagates to the interface from the compliant material side, the crack growth rate decreases to the minimum at first. After crack penetrates through the interface, the fatigue crack growth rate accelerates continuously. When crack propagates to the interface from the stiff material side, the fatigue crack growth rate generally increases with the crack length. Regardless of the direction of crack growth, the increase of load ratio will weaken the difference of crack growth rate near the interface caused by material property mismatch. Finite element results show that elastic modulus mismatch significantly changes the variation of the stress intensity factor amplitude. All results demonstrate that crack growth rate is dependent on the competition of the stress intensity factor amplitude, the fatigue crack growth rate in the corresponding material, and the interface strength.     

Effects of T-stresses on fracture initiation for a closed crack in compression with frictional crack faces

This paper studies crack extension resulting from a closed crack in compression. The crack-tip field of such a crack contains a singular field relative to K _II and non-singular T-stresses T _x and T _y parallel and perpendicular to the crack plane, respectively. Using a modified maximum tensile stress criterion with the singular and non-singular terms, the kinking angle at the onset of crack growth is determined by a two parameter field involving the mode-II stress intensity factors and T-stresses, and at fracture initiation a wing crack may be created at an arbitrary angle from 0° to 90°. A compressive T _y increases the kinking angle and reinforces apparent mode-II fracture toughness, while a compressive T _x decreases the kinking angle and enhances apparent mode-II fracture toughness. The direction and resistance of fracture onset is strongly affected by T-stresses as well as frictional stress. The von Mises effective stress is determined for small-scale yielding near the crack tip. The effective stress contour shape exhibits a marked asymmetrical behavior unless 2T _x  = T _y  ≤ 0 for plane stress state. Coulomb friction between two crack faces generally increases the kinking angle, shrinks the size enclosed by the effective stress contour and enhances apparent fracture toughness. Field evidence and experimental observations of many phenomena involving the growth of closed cracks in compression agree well with theoretical predictions of the present model.     

An incremental formulation for the prediction of two-dimensional fatigue crack growth with curved paths

This paper presents a new incremental formulation for predicting the curved growth paths of two-dimensional fatigue cracks. The displacement and traction boundary integral equations (BIEs) are employed to calculate responses of a linear elastic cracked body. The Paris law and the principle of local symmetry are adopted for defining the growth rate and direction of a fatigue crack, respectively. The three governing equations, i.e. the BIEs, the Paris law and the local symmetry condition, are non-linear with respect to the crack growth path and unknowns on the boundary. Iterative forms of three governing equations are derived to solve problems of the fatigue crack growth by the Newton–Raphson method. The incremental crack path is modelled as a parabola defined by the crack-tip position, and the trapezoidal rule is employed to integrate the Paris law. The validity of the proposed method is demonstrated by two numerical examples of plates with an edge crack. Copyright © 2007 John Wiley & Sons, Ltd.     

Which is the most unfavourable crack orientation?

The most unfavourable orientation of a straight crack of given length is investigated, assuming mode I growth. It is shown that this orientation is not always the one perpendicular to the largest principal stress. If the smallest in-plane principal stress is compressive and its magnitude more than about one third of the largest (tensile) in-plane stress, then some other orientations are more unfavourable. Crack growth then takes place after kinking.     

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.     

On the Hertzian fatigue cone crack propagation in ceramics

The Hertzian cone crack initiation and propagation in ceramics under cyclic fatigue loading with a spherical indenter is studied. Unlike the so-called quasi-static Hertzian cone crack, the fatigue Hertzian cone crack propagation eliminates the dynamic effect on unstable crack propagation. As such, the crack is found to propagate following the path of pure mode I type. We use an elasticity approach, a finite element analysis, and an empirical analysis to investigate the Hertzian cone crack in three stages: crack initiation, crack propagation, and crack kinking. The mechanism of the multiple concentric cone cracks is also explained. The purpose is to understand and predict the behavior of the formation of the Hertzian fatigue cone crack using available modeling tools.Zheng Chen is currently with Space Power Institute, Auburn University.     

An improved model for predicting the crack size and plasticity dependence of fatigue crack propagation

The angled crack problem has been given special attention in the recent years by fracture mechanics investigators due to its close proximity to realistic conditions in engineering structures. In this paper, an investigation of fatigue crack propagation in rectangular steel plates containing an inclined surface crack is presented. The inclined angle of the crack with respect to the axis of loading varied between 0° and 90°. During the fatigue tests, the growth of the fatigue crack was monitored using the AC potential drop technique. A series of modification factors, which allow accurate sizing of such defects, is recommended. Paris power law is normalized and adopted for data analysis. Subsequently, this concept is applied to predict crack growth due to fatigue loads. The results obtained are compared with those obtained using the commonly employed fracture criterion and the experimental data.