Self-similarity and crack growth instability in the correlation between the Paris’ constants

In this note the question about the existence of a correlation between the parameters C and m of the Paris’ law is re-examined. According to dimensional analysis and incomplete self-similarity concepts applied to the linear range of fatigue crack growth, a power-law asymptotic representation relating the parameter C to m and to the governing variables of fatigue is derived. Then, from the observation that the Griffith-Irwin instability must coincide with the Paris’ instability at the onset of rapid crack growth, the exponents entering this correlation are determined. A fair good agreement is found between the proposed theory and extensive experimental data.     

On the application of multiaxial high-cycle fatigue criteria using the theory of critical distances

This paper proposes that the application of multiaxial fatigue criteria in terms of the theory of critical distances requires the use of a distance which may be different from the widely adopted value given by half of the El Haddad’s intrinsic crack length. Three criteria (Modified Wöhler Curve Method, Crossland and Dang Van) are evaluated at the appropriate critical distance and compared with experimental data obtained from specimens containing small and/or sharp notches under proportional loading. The Modified Wöhler Curve Method provided the best estimates. It is also shown that this theory can be extended to the prediction of the loading ratio, R, effect on the threshold stress intensity factor range for propagation of long cracks.     

A unified fractal approach for the interpretation of the anomalous scaling laws in fatigue and comparison with existing models

In this paper, a critical reexamination of the fractal models for the analysis of crack-size effects in fatigue is proposed. The enhanced ability to detect and measure very short cracks has in fact pointed out the so-called anomalous behavior of short cracks with respect to their longer counterparts. The crack-size dependencies of both the fatigue threshold and the Paris’ constant C are only two notable examples of these anomalous scaling laws. In this context, a unified theoretical model seems to be missing and the behavior of short cracks can still be considered as an open problem. A new generalized theory based on fractal geometry is herein proposed, which permits to consistently interpret the short crack-related anomalous scaling laws within a unified theoretical framework. The proposed model is used to interpret relevant experimental data related to the crack-size dependence of the fatigue threshold in metals. As a main result, the model gives an explanation to the experimentally observed variability in the slope of the asymptote of the scaling law for the fatigue threshold in the short crack regime.     

Effect of material uncertainties on fatigue life calculations of aircraft fuselages: A cohesive element model

The present paper deals with the effect of uncertainties on the prediction of fatigue failure of aerospace and mechanical components. Typically the design of such structures has been based on costly experiments or modified versions of Paris’ law which are applicable to very restricted range of conditions. The present formulation employs cohesive zone elements in order to resolve the fractured zone in combination with an extrapolation scheme that makes the analysis over hundred of thousands of cycles computationally efficient. The effect of randomness in the cohesive strength is examined with respect to the total lifetime of the specimen.     

A parametric fracture mechanics study of welded joints with toe cracks and lack of penetration

The existing design rules give quite general guidelines to the fatigue assessment of different types of welded joints. The goal of this investigation was to give designers some tools, which would allow more precise assessment of the effect of dimensional variations on the fatigue strength. Therefore the fatigue behaviour of 12 common types of welded joints has been studied parametrically. Two-dimensional (2D) finite element models of the joint were made and evaluated using plane strain linear elastic fracture mechanics (LEFM) calculations. The as-welded condition was assumed with the result that no crack initiation period was considered and stress ranges were fully effective. A maximum tangential stress criterion with the Paris’ crack growth law was used to predict the growth rate and direction of root and toe cracks under mixed mode K_I-K_II conditions. The effects of weld size and joint dimension ratios on the fatigue strength were systematically studied. In addition to tensile loading, bending and combined tension/bending moment loading in both directions are examined for positive and negative mean stress.     

Multiple random crack propagation using a boundary element formulation

This paper proposes a boundary element method (BEM) model that is used for the analysis of multiple random crack growth by considering linear elastic fracture mechanics problems and structures subjected to fatigue. The formulation presented in this paper is based on the dual boundary element method, in which singular and hyper-singular integral equations are used. This technique avoids singularities of the resulting algebraic system of equations, despite the fact that the collocation points coincide for the two opposite crack faces. In fracture mechanics analyses, the displacement correlation technique is applied to evaluate stress intensity factors. The maximum circumferential stress theory is used to evaluate the propagation angle and the effective stress intensity factor. The fatigue model uses Paris’ law to predict structural life. Examples of simple and multi-fractured structures loaded until rupture are considered. These analyses demonstrate the robustness of the proposed model. In addition, the results indicate that this formulation is accurate and can model localisation and coalescence phenomena.     

An analytical approach for fracture and fatigue in functionally graded materials

The problem of brittle crack propagation and fatigue crack growth in functionally graded materials (FGMs) is addressed. The proposed analytical approach can be used to estimate the variation of the stress-intensity factor as a function of the crack length in FGMs. Furthermore, according to the Paris’ law, the fatigue life and the crack-tip velocity of crack propagation can be predicted in the case of fatigue crack growth. A comparison with numerical results obtained according to the Finite Element method will show the effectiveness of the proposed approach. Detailed examples are provided in the case of three-point bending beam problems with either a FGM interlayer, or a FGM external coating. A comparison is presented between two types of grading in the elastic modulus: a continuous linear variation in the FGM layer and a discrete approximation with a multi-layered beam and a constant Young’s modulus in each layer.     

Fatigue growth of short cracks in Ti-17: Experiments and simulations

The fatigue behaviour of through thickness short cracks was investigated in Ti-17. Experiments were performed on a symmetric four-point bend set-up. An initial through thickness crack was produced by cyclic compressive load on a sharp notch. The notch and part of the crack were removed leaving an approximately 50 μm short crack. The short crack was subjected to fatigue loading in tension. The experiments were conducted in load control with constant force amplitude and mean values. Fatigue growth of the short cracks was monitored with direct current potential drop measurements. Fatigue growth continued at constant R-ratio into the long crack regime. It was found that linear elastic fracture mechanics (LEFM) was applicable if closure-free long crack growth data from constant K_Imax test were used. Then, the standard Paris’ relation provided an upper bound for the growth rates of both short and long crack.The short crack experiments were numerically reproduced in two ways by finite element computations. The first analysis type comprised all three phases of the experimental procedure: precracking, notch removal and fatigue growth. The second analysis type only reproduced the growth of short cracks during fatigue loading in tension. In both cases the material model was elastic-plastic with combined isotropic and kinematic hardening. The agreement between crack tip opening displacement range, cyclic J-integral and cyclic plastic zone at the crack tip with ΔK_I verified that LEFM could be extended to the present short cracks in Ti-17. Also, the crack size limits described in the literature for LEFM with regards to plastic zone size hold for the present short cracks and cyclic softening material.     

A continuous damage approach to the fatigue process

The fatigue process is analyzed theoretically by studying the interaction between a macroscopic crack and continuously distributed microdamage near the crack-tips. A mode I loading case is studied, using a Dugdale crack model. The results show good agreement with Paris' law for fatigue crack propagation with an exponent approximately equal to 4. The model also predicts a finite fatigue lifetime in the sense that the crack growth rate increases without limit after a finite number of load cycles.     

Notch root radii effects in the fatigue of polymers

Wöhler-type rotating bending fatigue tests have been performed on PVC cantilever specimens containing various sharp and blunt notches of known geometries. An attempt has been made to analyse the data obtained using linear elastic fracture mechanics concepts. Results from the sharp-notched specimens show a good correlation on a stress intensity factor basis and a fatigue limit is revealed. For blunt notches a stress intensity factor had to be calculated allowing for small flaws at the notch root, and some measure of correlation of the stress intensity factor at the fatigue limit for the various specimens is obtained.     

Automated simulation of fatigue crack propagation for two-dimensional linear elastic fracture mechanics problems by boundary element method

In this paper, automated simulation of multiple crack fatigue propagation for two-dimensional (2D) linear elastic fracture mechanics (LEFM) problems is developed by using boundary element method (BEM). The boundary element method is the displacement discontinuity method with crack-tip elements proposed by the author. Because of an intrinsic feature of the boundary element method, a general growth problem of multiple cracks can be solved in a single-region formulation. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not necessary. Local discretization on the incremental crack extension is performed easily. Further the new adding elements and the existing elements on the existing boundaries are employed to construct easily the total structural mesh representation. Here, the mixed-mode stress intensity factors are calculated by using the formulas based on the displacement fields around crack tip. The maximum circumferential stress theory is used to predict crack stability and direction of propagation at each step. The well-known Paris’ equation is extended to multiple crack case under mixed-mode loadings. Also, the user does not need to provide a desired crack length increment at the beginning of each simulation. The numerical examples are included to illustrate the validation of the numerical approach for fatigue growth simulation of multiple cracks for 2D LEFM problems.     

Fatigue and post-fatigue tensile behaviour of non-crimp stitched and unstitched carbon/epoxy composites

An experimental study is described in this paper dealing with the tensile–tensile fatigue and the quasi-static post-fatigue tensile behaviour of a structurally stitched multi-ply carbon composite and the unstitched counterpart. The influence of the stitching on the fatigue life and on the residual post-fatigue quasi-static properties in two principal direction is investigated. The fatigue behaviour of both composites is represented by Wöhler-like diagrams. The damage imparted during fatigue is studied by X-ray analyses. The residual mechanical properties of the fatigued composites after different number of cycles are compared in term of stiffness and strength. The post-fatigue quasi-static tensile tests include acoustic emission (AE) registration and full-field surface strain mapping (SM) to investigate the damage onset and development. The main conclusions of the experimental work are: the fatigue life is improved in the direction of the structural stitching and is reduced in the orthogonal direction; for the considered cyclic stress level the post-fatigue reduction of the mechanical properties is limited by the structural stitching.     

On damage accumulations in the cyclic cohesive zone model for XFEM analysis of mixed-mode fatigue crack growth

Predicting mixed-mode fatigue crack propagation is an important and troublesome issue in structure assessment for decades. In the present paper an extended finite element method (XFEM) combined with a new cyclic cohesive zone model (CCZM) is introduced for simulating fatigue crack propagation under mixed-mode loading conditions, which has been implemented in the commercial general purpose software ABAQUS. The algorithm allows introducing a new crack surface at arbitrary locations and directions in a finite element mesh, without re-meshing. The cyclic cohesive zone model is based on the known S–N curves and Goodman diagram for metallic materials and validated by uniaxial tension results. Furthermore, the sensitivity of the model parameter is investigated for mixed-mode fatigue. The virtual crack closure technique has been extended to the cohesive zone model and proposed to calculate the energy release rate for the generalized Paris’ law. Finally, the crack propagation rate and direction under mixed-mode fatigue loading conditions are studied.     

Numerical analysis of the AC losses of 500-m HTS power cable in Super-ACE project

The alternating current (AC) losses of 500-m high-T_C superconducting (HTS) power cable in Super-ACE project are calculated by using an electric-circuit model. The cable core is consisted of a former (copper-stranded wire conductor), HTS conductor (Ag/Bi-2223 tapes, 1 layer), electrical insulation (LN₂-impregnated laminated paper), HTS shield (Ag/Bi-2223 tapes, 1 layer) and protection (copper-braided wire). Numerically calculated AC loss (the total loss) is obtained by a sum of the self-field loss and external-field loss (hysteresis losses) consumed at the HTS conductor and HTS shield, the ohmic loss caused by a resistance of copper of the former and protection, and the eddy-current loss caused by an axial field at the former. The calculations are compared with experimental results measured by a calorimetric method. The calculations are almost equal to the measurements at wide transport-current range. At transporting 1 kA_rms to the cable, the calculation indicates that the total loss of 1.29 W m⁻¹ (the measurement is 1.3 W m⁻¹) is obtained by a sum of 0.89 W m⁻¹ as the self-field loss, 0.32 W m⁻¹ as the external-field loss, 0.06 W m⁻¹ as the ohmic loss and 0.02 W m⁻¹ as the eddy-current loss. On the other hand, Central Research Institute of Electric Power Industry (CRIEPI) has estimated that the AC loss is obtained by a sum of 0.5 W m⁻¹ consumed at the HTS conductor and HTS shield as the hysteresis loss, and 0.8 W m⁻¹ consumed at the former as the eddy-current loss. The author’s result contradicts CRIEPI’s estimation. It is considered that the difference is caused by an overestimation of the axial field at the former in CRIEPI’s loss calculation.     

Variable amplitude loading of spray‐formed hypereutectic aluminium silicon alloy DISPAL® S232 in the VHCF regime

Spray‐formed hypereutectic aluminium silicon alloy DISPAL® S232–T6x is cycled with variable amplitude at ultrasonic frequency up to the very high cycle fatigue (VHCF) regime under fully reversed tension–compression loading. The Powder Metallurgy alloy is tested using a Gaussian cumulative frequency distribution of load cycles, and lifetimes are compared with constant amplitude data. Miner calculation delivers mean damage sums between 0.5 and 0.9 for mean lifetimes between 8 × 10⁷ and 1.6 × 10¹⁰ cycles, respectively. Cracks are initiated at voids, at inclusions or at distributed inhomogeneities (porous areas or oxides) at the surface or in the interior. In situ analysis of vibration properties indicates that cracks are formed and start growing from the beginning of fatigue cycling, even if failure occurs in the very high cycle fatigue regime. Crack initiation stage is negligible. Lifetime prediction calculation is performed using an adapted Paris‐law and considering lifetime as cycles necessary to propagate an initial crack to failure. Measured and predicted mean lifetimes differ by factor 0.4–1.0. Large crack‐initiating defects strongly reduce the fatigue lifetimes, which is successfully covered in the crack propagation model.     

Mechanical and thermal fracture of plastics under cyclic strains

It is shown on the basis of the theory of cyclic strain-induced heating of plastics that two forms of fatigue fracture are possible (thermal and mechanical), this being reflected in the fact that the appropriate Wöhler curves consist of two branches. The conditions under which a transition from one form of fracture to another takes place are analyzed, and the theoretical predictions are proved by experiment.The authors wish to thank Z. S. Shmakova who took part in the experimental work.     

On the life-controlling microstructural fatigue mechanisms in ductile metals and alloys in the gigacycle regime

The high-cycle fatigue (HCF) behaviour of ductile metals and alloys, and the life-controlling microstructural fatigue mechanisms known from HCF are reviewed critically with respect to their possible role in the gigacycle or ultra-high-cycle fatigue (UHCF) regime. Arguments are presented to support the hypothesis that, at the very low amplitudes of the UHCF regime, fatigue crack initiation, resulting from cyclic strain localization, and slow early Stage I fatigue crack propagation are the life-controlling mechanisms and that these processes can essentially be described in terms of the microstructurally irreversible portion of the cumulative cyclic plastic strain. Emphasis is placed on the important role of the so-called slip irreversibility which decreases as the amplitude becomes lower and lower. Finally, the Manson–Coffin law is reformulated for very low amplitudes in terms of microstructurally relevant parameters, and a fatigue life diagram is developed, based on these preceding microstructural considerations. Important features of this diagram are: (i) the plastic strain fatigue limit in the HCF regime which is related to the threshold for cyclic strain localization in persistent slip bands; and (ii) the transition from this plastic strain fatigue limit to a threshold of negligible slip irreversibility at still lower amplitudes in the UHCF regime.     

A note on Eringen’s moment balances

The present study provides a comparison of Eringen’s [Eringen, A.C. (1970). Balance laws of micromorphic mechanics. International Journal of Engineering Science, 8, 819–828] general moment balances for micromorphic continua with Germain’s [Germain, P. (1973). The method of virtual power in continuum mechanics. Part 2: Microstructure. SIAM Journal on Applied Mathematics, 25, 556–575] momentum balances based on virtual work principles, and with those derived in the present paper by a two-scale Fourier analysis of heterogeneous media. It has not been possible to establish a clear-cut correspondence between Eringen’s balances and either of the latter, partly because Eringen’s balances involve a mixture of surface and volume averages over microdomains.There is disagreement between the last two methods, arising from the fact that Germain’s treatment involves spatial gradients not occurring in the elementary two-scale Fourier analysis. A brief discussion is given of the possible extension of the latter to achieve agreement with the former.As a separate matter, a construction of path-moments of density fields serves to establish a source-flux duality in continuum balances, which inter alia establishes a fairly direct connection between Newton’s and Cauchy’s laws and provides an expression for stress suggested by the statistical mechanics of point-particles.     

The fatigue limit of bearing steels – Part I: A pragmatic approach to predict very high cycle fatigue strength

Bearing steels and other high strength steels exhibit complex fatigue behavior in excess of 10⁷ cycles due to their sensitivity to defects like inclusions. Failure occurring in the very high cycle fatigue regime and the lack of an asymptote in the measured S–N data raise the questions as to the existence of fatigue limit and prediction of the fatigue strength of the high strength steel components. A series of two papers are written to discuss on the characteristics of the very high cycle fatigue and their implication for engineering applications. In the present paper (Part I) a deterministic defect model is developed to describe the fatigue crack growth from de-bonded hard inclusions. The model is shown to provide a unified prediction of fatigue behavior in different regimes, i.e. low cycle fatigue regime dictated by the tensile strength, high cycle fatigue regime obeying Basquin’s law and the very high cycle fatigue regime featured by the fish-eye and ODA (optically dark area) surrounding an interior fatigue-initiating inclusion on the fracture surface. The model predictions agree well with experiments. A combination of the deterministic model with a stochastic model that describes the inclusion size distribution allows prediction of fatigue strength and fatigue limit associated with certain reliability of a steel component. It is found that very high cycle fatigue, associated with interior inclusions, is attributed to the very slow crack propagation in vacuum condition, and that an asymptote for fatigue limit observed for mild steels also exists for high strength steels such as bearing steels, but extends beyond the very high cycle fatigue regime normally measured to-date. Monte Carlo simulation shows that such a fatigue limit asymptote becomes clearly visible in excess of 10¹² cycles, which is difficult to measure with today’s testing devices. Furthermore, the effects of steel cleanliness and specimen type and shape are studied by means of Monte Carlo simulations.     

Ermüdungsschwellwert: Materialeigenschaft und Auslegungsparameter

Fatigue Crack Propagation Threshold: Material Property and Design Criteria The fatigue crack propagation threshold has been determined by two experimental methods covering the tension-tension fatigue regime. The two experimental methods are boundary conditions for this fatigue regime. A third experimental method positions the threshold somewhere between K_max and K_min such that because of the ?Closure Parameter K_op”? the threshold is not positioned below 0.5 K_max. The results obtained show that a threshold exists under all fatigue loading conditions and – depending on the subject, material – is either a material property or a material parameter which depends on K_max