Probabilistic high cycle fatigue behaviour of nodular cast iron containing casting defects

Theoretical and experimental investigations were combined to characterize the influence of surface casting defects (shrinkages) on the high cycle fatigue (HCF) reliability. On fracture surfaces of fatigue samples, the defect is located at the surface. The shape used for the calculation is a spherical void with variable radius. Finite-element simulations were then performed to determine stress distribution around defects for different sizes and different loadings. Correlated expressions of the maximum hydrostatic stress and the amplitude of the shear stress were obtained by using the response surface technique. The loading representative point in the HCF criterion was then transformed into a scattering surface, which has been obtained by a random sampling of the defect sizes. The HCF reliability has been computed by using the Monte Carlo simulation method. Tension and torsion fatigue tests were conducted on nodular cast iron with quantification of defect size on the fracture surface. The S – N curves show a large fatigue life scattering; shrinkages are at the origin of the fatal crack leading to the final failure. The comparison of the computed HCF reliability to the experimental results shows a good agreement. The capability of the proposed model to take into account the influence of the range of the defect sizes and the type of its statistical distribution has been demonstrated. It is shown that the stress distribution at the fatigue limit is log-normal, which can be explained by the log-normal defect distribution in the nodular cast iron tested.     

The nature of initiation and propagation S-N curves at and below the fatigue limit

The traditional stress-life method of life prediction relies on an S – N curve of stress versus total life. However, the total life of a sample can be divided into two phases—an initiation phase and a propagation phase that leads to ultimate failure. Although this break-up of total life into two phases has been recognized in theory, there has been no experimental method to generate initiation and propagation S – N curves. In this paper a methodology to generate initiation and propagation S – N curves is presented. Acoustic emission technology is used to detect the transition from the initiation phase to the propagation phase. The phenomenon of fatigue limits is also explored and it is shown that the fatigue limit of the traditional S – N curve corresponds to the fatigue limit of the initiation phase and that initiated cracks continue to propagate at stress levels below the initiation endurance limit. It is also shown that no damage is accrued at stress levels below the fatigue limit. A method to extend the propagation life curve below the initiation endurance limit is also presented. The proposed two-phase S – N curve will greatly extend the life-predicting capability of the stress-life method and can explain some of the contradictions observed in experiments.     

Determination of high cycle fatigue properties of a wide range of steel sheet grades from self-heating measurements

The development of a self-heating method, based on self-heating measurements is proposed in order to predict S–N–P curves (i.e., amplitude stress-number of cycles to failure-probability of failure). Two dissipative phenomena can be observed on self-heating curves for the 16 steel grades of interest, the first for low amplitudes of cyclic loading and the second for higher amplitudes. In order to predict the fatigue properties accurately, a two scale probabilistic model, with two dissipative mechanisms (to account for the two dissipative phenomena) is proposed. Finally, the prediction made using the proposed approach is validated by its comparison with traditional fatigue tests, thus demonstrating time-saving advantages in the determination of steel grade fatigue properties.     

Fatigue life prediction based on equivalent stresses for laser beam welded 6156 Al‐alloy joints under variable amplitude loading

In the present work, a simple fatigue life prediction approach is proposed using fracture mechanics for laser beam welded Al‐alloy joints under variable amplitude loading. In the proposed approach, variable amplitude loading sequence is transformed into an equivalent constant amplitude loading using the root mean square model. The crack growth driving force K^* is chosen to describe the fatigue crack growth rate. The influences of residual stress and its relaxation on fatigue life are taken into account in the proposed approach. The fatigue lives are also predicted using the traditional approach based on the S‐N curves and the rainflow counting method. The predicted results show that the proposed approach is better than the traditional approach.     

Fitting fatigue data with a bi‐conditional model

The formulation of a probability‐stress‐life (P‐S‐N) curve is a necessary step beyond the basic S‐N relation when dealing with reliability. This paper presents a model, relevant to materials that exhibits a fatigue limit, which considers the number of cycles to failure and the occurrence of the failure itself as statistically independent events, described with different distributions and/or different degree of scatter. Combining these two as a parallel system leads to the proposed model. In the case where the S‐N relation is a Basquin's law, the formulations of the probability density function, cumulative distribution function, quantiles, parameter and quantile confidence interval are presented in a procedure that allows practically any testing strategy. The result is a flexible model combined with the tools that deliver a wide range of information needed in the design phase. Finally, an extension to include static strength and applicability to fatigue crack growth and defects‐based fatigue approach are presented.     

Mean stress and ratcheting corrections in fatigue life prediction of metals

Ratcheting occurs easily because of the presence of mean stress during the stress‐control fatigue of engineering components. For ductility exhaustion dominated fatigue failure, a new fatigue life prediction model is developed by introducing the mean ratcheting strain rate to incorporate the effects of ratcheting and mean stress on fatigue life. The prediction accuracy of the proposed model was compared with that of the generalised damage parameter, Xia–Kujawski–Ellyin, Walker and Goswami models. Specifically, the model predictions and tested lives were compared using nine sets of experimental data from the literature. In the statistical analysis of these five models, the proposed model provides the highest accuracy and robust life predictions with the lowest model prediction errors.     

A method for assessing critical plane‐based multiaxial fatigue damage models

Fatigue failure is a complex phenomenon. Therefore, development of a fatigue damage model that considers all associated complexities resulting from the application of different cyclic loading types, geometries, materials, and environmental conditions is a challenging task. Nevertheless, fatigue damage models such as critical plane‐based models are popular because of their capability to estimate life mostly within ±2 and ±3 factors of life for smooth specimens. In this study, a method is proposed for assessing the fatigue life estimation capability of different critical plane‐based models. In this method, a subroutine was developed and used to search for best estimated life regardless of critical plane assumption. Therefore, different fatigue damage models were evaluated at all possible planes to search for the best life. Smith‐Watson‐Topper (normal strain‐based), Fatemi‐Socie (shear strain‐based), and Jahed‐Varvani (total strain energy density‐based) models are compared by using the proposed assessment method. The assessment is done on smooth specimen level by using the experimental multiaxial fatigue data of 3 alloys, namely, AZ31B and AZ61A extruded magnesium alloys and S460N structural steel alloy. Using the proposed assessment method, it was found that the examined models may not be able to reproduce the experimental lives even if they were evaluated at all physical planes.     

Optimization research on S‐N curve of ring welding structure based on structural stress method

In engineering, △F‐N curves are usually used to predict the fatigue life of ring welding, which is time‐consuming, laborious, and not universal. To improve the above inadequacies, an S‐N curve for evaluating the fatigue life of the ring welded specimen is proposed. The fatigue life of ring welded specimens with different materials, plate thicknesses, and hole diameters is obtained by tensile and shear fatigue tests. Shell elements, CBar beam elements (a kind of beam element in Nastran that can simulate bending), and rigid elements are used to establish the finite element model of ring welding. The stress of the ring welding structure under tensile shear load is calculated according to the structural stress method. The stress range △ σs of the welding core is taken as the longitudinal coordinate and the experimental life N as the horizontal coordinate, using two‐parameter log‐log model and the least square method of the fatigue data for linear fitting to obtain the S‐N curve equation of fatigue life evaluation. Most of the data are located within five times of the life span, which proves that the predicted life is close to the actual life of the test, and it can provide a certain reference for design and life prediction of the ring welding structures.     

A fatigue damage accumulation model for reliability analysis of engine components under combined cycle loadings

Rotor components of an aircraft engine in service are usually subjected to combined high and low cycle fatigue (CCF) loadings. In this work, combining with the load spectrum of CCF, a modified damage accumulation model for CCF life prediction of turbine blades is first put forward to take into account the effects of load consequence and load interaction caused by high‐cycle fatigue (HCF) loads and low‐cycle fatigue (LCF) loads under CCF loading conditions. The predicted results demonstrate that the proposed model presents a higher prediction accuracy than Miner, Manson‐Halford model does. Moreover, to evaluate the fatigue reliability of rotor components, reliability model with the failure mode of CCF is proposed on the basis of the stress‐strength interference method when considering the strength degeneration, and its results show that the reliability model with CCF is more suitable for aero‐engine components than that with the failure mode of single fatigue.     

S–N curve characteristics and subsurface crack initiation behaviour in ultra-long life fatigue of a high carbon-chromium bearing steel

The S – N curve obtained from cantilever-type rotary bending fatigue tests using hour-glass-shaped specimens of high carbon-chromium bearing steel clearly distinguished the fracture modes into two groups each having a different crack origin. One was governed by crystal slip on the specimen surface, which occurred in the region of short fatigue life and a high stress amplitude level. The other was governed by a non-metallic inclusion at a subsurface level which occurred in the region of long fatigue life and low stress amplitude. The inclusion developed a fish-eye fracture mode that was distributed over a wide range of stress amplitude not only below the fatigue limit defined as the threshold for fracture due to the surface slip mode but also above the fatigue limit. This remarkable shape of the S – N curve was different from the step-wise one reported in previous literature and is characterized as a duplex S – N curve composed of two different S – N curves corresponding to the respective fracture modes. From detailed observations of the fracture surface and the fatigue crack origin, the mechanisms for the internal fracture mode and the characteristics of the S – N curve are discussed.     

Experimental investigation on bending fatigue failure of corrugated paperboard

A series of 3‐point bending fatigue tests were conducted to investigate the bending fatigue behaviour of flute type B and C corrugated paperboard samples under cyclic loading. The S‐N curve was obtained. The fatigue failure of corrugated paperboard may be described by both Basquin‐type and exponential‐type S‐N curves; however, the exponential‐type S‐N curve is more appropriate. The stiffness is gradually degraded with almost same energy dissipation in most stress cycles, but it decreases abruptly with the enlarged energy dissipation when the testing cycle is very close to the ultimate cycles of fatigue failure. The corrugated board deforms constantly under the action of cyclic loading, and no visible crack appears. The fatigue failure modes and mechanisms are same for the corrugated boards with B‐flute and C‐flute; however, the S‐N curve of corrugated board is closely related to the flute structure. The results obtained in this paper may be applied to the dynamic design and accelerated vibration test of stacked corrugated boxes.     

Effect of microstructure of simulated heat‐affected zone on low‐ to high‐cycle fatigue properties of low‐carbon steels

To clarify the effect of microstructural changes on the fatigue property of the weld heat‐affected zone (HAZ), low‐ to high‐cycle fatigue tests were conducted on 16 types of simulated HAZ specimens that had been prepared using thermal processes. The results showed the fatigue S‐N curves of the HAZ to be widely scattered as a function of strength level. These fatigue data were divided into two groups: coarse grain (CG) and fine grain (FG) HAZ, when strain amplitude was used to represent S‐N curves. The fatigue data for the CGHAZ group showed a relatively short fatigue life. Based on surface observations, the initiated fatigue crack size of CGHAZ was larger than that of FGHAZ as a function of microstructural unit size. Hence, fatigue crack growth life, which is almost the same as total fatigue life of CGHAZ, decreased.     

Experimental characterisation of a CuAg alloy for thermo‐mechanical applications. Part 2: Design strain‐life curves estimated via statistical analysis

Strain‐life fatigue data on copper alloys, especially type CuAg, are seldom available in the literature. This work fills this gap by estimating the strain‐life curves of a CuAg alloy used for thermo‐mechanical applications, from isothermal low‐cycle fatigue tests at 3 temperatures (room temperature, 250°C, 300°C). Regression analysis is used to estimate the median fatigue curves at 50% survival probability. The comparison of median curves with the Universal Slopes Equation model, calibrated on monotonic tensile properties, shows a fairly good agreement. Design strain‐life curves with a lower failure probability and given confidence are estimated by several approximate statistical methods (“Equivalent Prediction Interval,” univariate tolerance interval, Owen's tolerance interval for regression). When higher survival probabilities are considered, the results show a marked decrease in the allowable design strain at a prescribed fatigue life. The suggested procedure thus improves the durability analysis of components loaded thermo‐mechanically.     

Criteria of multiaxial random fatigue based on stress,strain and energy parameters of damage in the critical plane

In this paper generalized criteria of multiaxial random fatigue based on stress, strain and strain energy density parameters in the critical plane have been discussed. The proposed criteria reduce multiaxial state of stress to the equivalent uniaxial tension–compression or alternating bending. Relations between the coefficients occurring in the considered criteria have been derived. Thus, it is possible to take into account fatigue properties of materials under simple loading states during determination of the multiaxial fatigue life. Presented models have successfully correlated fatigue lives of cast iron GGG40 and steel 18G2A specimens under constant amplitude in‐phase and out‐of‐phase loadings including different frequencies.     

Probabilistic anisomorphic constant fatigue life diagram approach for prediction of P–S–N curves for woven carbon/epoxy laminates at any stress ratio

A general engineering methodology to construct a family of anisomorphic constant fatigue life (CFL) diagrams with probability of failure as the parameter that allows efficiently predicting P–S–N curves at any stress ratios is developed and validated for a plain weave fabric carbon/epoxy laminate. Constant amplitude fatigue tests are first performed to obtain statistical samples of fatigue life at different stress levels and stress ratios, respectively. Static tensile and compressive strength data are also collected. The Kolmogorov–Smirnov and Anderson–Darling goodness-of-fit tests suggest that both two-parameter lognormal and Weibull distributions are acceptable as the distributions for the static strength and fatigue life data, respectively, at the significance level of 5%. Then, we attempt to develop a methodology for efficient construction of the anisomorphic CFL diagrams for different constant values of probability of failure. It requires the P–S–N curves for any percentile points of the distribution for the critical stress ratio. To come up with this requirement, a probabilistic scaling law is formulated. It takes account of the probability-of-failure dependence of the critical stress ratio and the stress-ratio dependence of the P–S–N curve for the critical stress ratio. Finally, the anisomorphic CFL diagrams for different constant values of probability of failure are predicted using the proposed methodology, and they are shown to be in good agreement with the experimental results. It is also demonstrated that the P–S–N curves can efficiently and accurately be predicted for the woven CFRP laminate at any stress ratios using the proposed probabilistic anisomorphic CFL diagram approach.     

Modelling curved S–N curves

The fatigue limit distribution is estimated using fatigue data and under the assumption that the fatigue limit is random. The stress levels for the broken and unbroken specimens are used. For the broken specimen the number of cycles to failure is also used. By combining the finite life and fatigue limit distribution it is possible to get the probability of not surviving a certain life. This probability is used to estimate a curved S–N curve by using the method of likelihood. The whole S–N curve is estimated at the same time. These curves show the predictive life given a certain stress level. The life and the quantile of the fatigue limit distribution are also predicted by using profile predictive likelihood. In this way the scatter around the S–N curve as well as the uncertainty of the S–N curve are taken into account.     

There is no infinite fatigue life in metallic materials

Generally, the shape of the S – N curve beyond 107 cycles is unknown except in some statistical approaches, and this is also true for the fatigue limit. In the case of a statistical approach, the standard deviation applied to the average fatigue limit is certainly not the best way to reduce the risk of rupture in fatigue. Only the exploration of the life range between 106 and 10¹⁰ cycles will create a safer basis for modelling.
Today, some piezoelectric fatigue machines are very reliable, capable of producing 10¹⁰ cycles in less than 1 week. We based our research on accelerated fatigue tests which were performed at 20  kHz in the gigacyclic fatigue regime in order to study several typical alloys from the aeronautical and space industries.     

Thermography in high cycle fatigue short‐term evaluation procedures applied to a medium carbon steel

This paper focuses on the fatigue life calculation for an unalloyed medium carbon steel SAE1045 (German DIN‐standard: C45E), by applying an energy dissipation‐based approach quantified through thermographic measurements. The purpose of this approach is to establish an intrinsic dissipation model and to predict characteristics derived from the cyclic deformation behavior of stress‐controlled fatigue tests, eg, the fatigue limit and the S‐N data by using simplified (zero‐dimensional, 0D) thermodynamic equations. In order to investigate the possibilities for a rapid evaluation while simultaneously reducing the experimental effort, one load increase test (LIT) and two constant amplitude tests (CATs) were carried out. The S‐N data evaluated on such a basis is competitive to conventionally determined S‐N data as will be shown.     

A stress-based method to predict lifetime under multiaxial fatigue loadings

This paper extends to low/medium‐cycle fatigue a stress‐based method recently proposed by the same authors for high‐cycle multiaxial fatigue assessments. By considering the plane of maximum shear stress amplitude coincident with the microcrack initiation plane, the method requires the calculation both of the maximum shear stress amplitude and the maximum normal stress relative to the same plane. Multiaxial fatigue life estimates are made by means of bi‐parametric modified Wöhler curves, which take into account the mean stress effect, the influence of the out‐of‐phase angle and the presence of notches by using a generalization to multiaxial fatigue of the fatigue strength reduction factor K_f. Approximately 700 experimental data taken from the literature are used to demonstrate that the method is a useful tool to summarize fatigue strength data of both smooth and notched components, subjected to either in‐phase or out‐of‐phase loads. Finally, a simple practical rule for the calculation of the multiaxial fatigue strength reduction factor is proposed.     

On the application of the stochastic approach in predicting fatigue reliability using Miner's damage rule

The present paper investigates the application of the stochastic approach when the commonly adopted Miner's linear damage rule is implemented, both in its traditional and modified forms to include the presence of a random stress threshold (random fatigue limit), below which the rate of damage accumulation is reduced. Main steps are provided to obtain the simulated distribution of the accumulated damage under variable amplitude loading. When the stochastic approach is applied in the presence of a random fatigue limit, an additional correlation structure, which takes into account the fatigue limit value, must be introduced in the analysis. If the number of cycles to failure under constant amplitude loading is Weibull (Log‐Normal) distributed, then the corresponding accumulated damage is Fréchet (Log‐Normal) distributed. The effects of the correlation structure on reliability prediction under variable amplitude loading are also investigated. To this aim, several experimental datasets are taken from the literature, covering various metallic materials and variable amplitude block sequences. The results show that the choice of the damage accumulation model is a key factor to value the improvement in the accuracy of reliability predictions introduced by the stochastic approach. Comparison of the predicted number of cycles to failure with experimental data shows that larger errors are non‐conservative, regardless of the adopted correlation structure. When the analysis is limited to reliability levels above 80%, for these large non‐conservative errors, it is the quantile approach to be closer to actual experimental data, thus limiting the overestimation of component's life. For the experimental datasets considered in the paper, adoption of a stochastic approach would improve the accuracy of Miner's predictions in 10% of cases.