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.     

Microstructure based fatigue life predictions for thick plate 7050-T7451 airframe alloys

The purpose of this work was to test a microstructure based fatigue life prediction Monte-Carlo model for potential use in quantifying fatigue quality and reliability of metallic structural alloys. The model used was of the crack growth type with the sizes of crack initiating pores, local crack geometry and crack tip texture as random variables. The model was verified using data for 7050-T7451 plate alloy fatigue tested in smooth sample configuration at σ_max of 241 and 276 MPa and R=0.1. The mechanical testing was supplemented with characterizations of the size distributions of the fatigue performance limiting bulk porosity and measurements of the actual sizes of the fatigue crack initiating pores. Predicted fatigue lives were in good agreement with the experimental results and the model identified the size distribution of the crack initiating pores as the dominant variable controlling fatigue performance. The distributions of those pores were predicted from the bulk pore size data using the statistics of extremes. The developed approach proved effective in incorporating microstructural information in modeling fatigue and could be used in ranking fatigue quality and reliability of materials based on microstructural data.     

A method to develop a unified fatigue life prediction model for filled natural rubbers under uniaxial loads

Rubber components are widely used in many fields because of their superior elastic properties. Fatigue failures, commonly encountered in rubber components, however, remain a critical issue. In this study, the effect of strain ratio R on the fatigue life of filled natural rubbers used in automotive mounts is investigated experimentally and numerically. A uniaxial tension/compression fatigue experiment was conducted on dumb‐bell cylindrical rubber specimens subject to loads representing different R ratios. The experimental fatigue data are used to formulate two preliminary fatigue models based on peak strain and strain amplitude as the damage parameters. The deficiencies of these two models in predicting fatigue life over a wide range of R ratios are discussed, and an alternative life prediction model is proposed. The proposed model incorporates the effect of R ratio using an equivalent strain amplitude. It is shown that the proposed model could effectively predict fatigue life over a wide range of R ratios with an improved accuracy.     

Stress‐Strength Reliability Analysis with Extreme Values based on q‐Exponential Distribution

When dealing with practical problems of stress–strength reliability, one can work with fatigue life data and make use of the well‐known relation between stress and cycles until failure. For some materials, this kind of data can involve extremely large values. In this context, this paper discusses the problem of estimating the reliability index R = P(Y < X) for stress–strength reliability, where stress Y and strength X are independent q‐exponential random variables. This choice is based on the q‐exponential distribution's capability to model data with extremely large values. We develop the maximum likelihood estimator for the index R and analyze its behavior by means of simulated experiments. Moreover, confidence intervals are developed based on parametric and nonparametric bootstrap. The proposed approach is applied to two case studies involving experimental data: The first one is related to the analysis of high‐cycle fatigue of ductile cast iron, whereas the second one evaluates the specimen size effects on gigacycle fatigue properties of high‐strength steel. The adequacy of the q‐exponential distribution for both case studies and the point and interval estimates based on maximum likelihood estimator of the index R are provided. A comparison between the q‐exponential and both Weibull and exponential distributions shows that the q‐exponential distribution presents better results for fitting both stress and strength experimental data as well as for the estimated R index. Copyright © 2016 John Wiley & Sons, Ltd.     

Models for Variable-Stress Accelerated Life Testing Experiments Based on Wener Processes and the Inverse Gaussian Distribution

Variable-stress accelerated life testing trials are experiments in which each of the units in a random sample of units of a product is run under increasingly severe conditions to get information quickly on its life distribution. We consider a fatigue failure model in which accumulated decay is governed by a continuous Gaussian process W(y) whose distribution changes at certain stress change points to < t _l < < … <t _k , Continuously increasing stress is also considered. Failure occurs the first time W(y) crosses a critical boundary ω. The distribution of time to failure for the models can be represented in terms of time-transformed inverse Gaussian distribution functions, and the parameters in models for experiments with censored data can be estimated using maximum likelihood methods. A common approach to the modeling of failure times for experimental units subject to increased stress at certain stress change points is to assume that the failure times follow a distribution that consists of segments of Weibull distributions with the same shape parameter. Our Wiener-process approach gives an alternative flexible class of time-transformed inverse Gaussian models in which time to failure is modeled in terms of accumulated decay reaching a critical level and in which parametric functions are used to express how higher stresses accelerate the rate of decay and the time to failure. Key parameters such as mean life under normal stress, quantiles of the normal stress distribution, and decay rate under normal and accelerated stress appear naturally in the model. A variety of possible parameterizations of the decay rate leads to flexible modeling. Model fit can be checked by percentage-percentage plots.     

The effect of interference-fit on fretting fatigue crack initiation and ΔK of a single pinned plate in 7075 Al-alloy

The effect of interference-fit on fretting fatigue crack initiation and ΔK was studied numerically for available experimental results in a single pinned plate in Al-alloy 7075-T6. The role of interference ratio was investigated alongside friction coefficient through finite element. Cyclic stress distributions in the plate ligament and fretting stresses on the contact interface were evaluated using 3-D elastic–plastic finite element models. Additionally a 3-D elastic finite element model was utilized to discuss ΔK of cracks emanating from interference fitted holes. Results demonstrate that fretting was the main reason for crack nucleation, and furthermore, the location was precisely predicted and fatigue life enhancement was explained.     

Estimations of parameters in Pareto reliability model in the presence of masked data

Estimations of parameters included in the individual distributions of the life times of system components in a series system are considered in this paper based on masked system life test data. We consider a series system of two independent components each has a Pareto distributed lifetime. The maximum likelihood and Bayes estimators for the parameters and the values of the reliability of the system's components at a specific time are obtained. Symmetrical triangular prior distributions are assumed for the unknown parameters to be estimated in obtaining the Bayes estimators of these parameters. Large simulation studies are done in order: (i) explain how one can utilize the theoretical results obtained; (ii) compare the maximum likelihood and Bayes estimates obtained of the underlying parameters; and (iii) study the influence of the masking level and the sample size on the accuracy of the estimates obtained.     

复合材料疲劳寿命预测

在疲劳载荷作用下,复合材料的弹性模量会随着载荷循环数的增加而不断下降,而材料中的内部损伤则不断增大。为此,本文提出复合材料的疲劳模量和累积应变的概念,并由此定义出三种预测复合材料疲劳寿命的疲劳损伤模型。文中应用这三种模型对单应力水平和多应力水平下的玻璃纤维增强环氧树脂复合材料的疲劳寿命进行了估算,并同实验结果进行了比较。     

A model for life prediction in low-cycle fatigue with hold time

In high-temperature and low-cycle fatigue, creep damage reduces fatigue life. In this investigation, a model for life prediction in low-cycle fatigue with hold time at tensile peak strain is suggested in the temperature region of 0.57T _m. This model is based on previously reported theories for creep cavitatation and we predict the creep-fatigue life. It is proposed that the fatigue life may be predicted in terms of plastic strain range, test temperature, hold time and other parameters. An equation for life prediction is given and checked using other investigators' experimental results with various hold times. The predicted creep-fatigue lives are in good agreement with those observed experimentally for 304 stainless steel, 316 stainless steel, CrMoV steel and 13CrMo44 steel.     

Derzeitiger Stand der Lebensdauervorhersage für Bauteile

The State of the Art in the Fatigue Live Evaluation of Components For fatigue life estimations of components many compromises must be entered in the utilization of available fatigue design data. The parameters of the chosen design data which do not fit the component to be evaluated are in most cases taken into consideration by very global factors. These factors and the inaccuracies of Miner's Rule provide only a rough estimate of the fatigue life of a component. When the component is allready available, these uncertainties can be reduced by performing a fatigue test with service like load-time histories and a relative Miner's Calculation. For the same reason the total knowledge about the service fatigue behaviour of similar components should be taken into account in the life estimation.     

Life Testing in Variably Scaled Environments

Typically, accelerated life-testing models postulate a specific functional relationship between the stress level at which an experiment is performed and the parameters of the assumed family of lifetime distributions. These models, and the statistical analyses that accompany them, are often criticized on the basis of the dubious validity of the assumed functional relationship and of the uncertainty involved in the extrapolation of experimental results to low stress levels at which little or no data have been obtained. This study focuses on an exponential factorial model for accelerated life tests that postulates that the lifetime distributions of different component types tested under varying environmental conditions are linked via environmental or component-related scale changes. Necessary and sufficient conditions are given for the identifiability of model parameters. For both censored and complete data, the derivation and properties of maximum likelihood estimates of these parameters are discussed in detail. Under the conditions that guarantee identifiability, the existence and the uniqueness of the maximum likelihood estimators are demonstrated, and their computation and large-sample behavior are discussed. In the final section, the model is fitted to published data from an accelerated life-testing experiment.     

Application of a material fatigue damage model in 4PB tests

The fatigue property of an asphalt mix is an important issue in pavement design. This property is often determined with the aid of a four-point bending (4PB) test in controlled deflection mode. The fatigue property is related to the decrease in the calculated complex stiffness modulus, however, due to the non- homogenous stress and strain field in the beam, the measured response does not represent the stiffness modulus of the material but a weighted stiffness value. For a correct interpretation, a fatigue damage material model like the Asphalt Concrete Pavement-Fatigue model is needed. After integration, the calculated and measured responses are compared. By varying the model parameters, an excellent comparison between the two responses is obtained up to a certain number of cycles. This number of cycles is denoted as the fatigue life N _PH . The accumulated dissipated energy at the surface of the beam in the midsection can be expressed as a constant times the fatigue life N _PH to the power z and also as a constant times the product of the fatigue life N _PH and the initial dissipated energy in the first cycle. Using these two findings, a Wöhler curve was established similar to the one directly based on the strain amplitudes and fatigue life data.     

On fatigue life variability in bone cement

Fatigue life of Simplex P bone cement was tested at three different stress amplitudes by using specimens produced by two different mixers. Fatigue life data showed high variability in all instances. Statistical analysis showed that fatigue life was not affected by the type of mixer. Analysis of fracture surfaces showed that fatigue life variability could be attributed to the presence of defects, such as bubbles and mixing defects. Both Weibull and Tiryakio?lu distributions provided excellent fits to the fatigue life data. Moreover, the Gumbel parameters for fatigue initiator size data estimated in the Tiryakio?lu distribution agreed closely with fractographic measurements.     

Reliability approximation using finite Weibull mixture distributions

The shape of measured or design life distributions of systems can vary considerably, and therefore frequently cannot be approximated by simple distribution functions. The scope of the paper is to prove that the reliability of an arbitrary system can be approximated well by a finite Weibull mixture with positive component weights only, without knowing the structure of the system, on condition that the unknown parameters of the mixture can be estimated. To support the main idea, five examples are presented. In order to estimate the unknown component parameters and the component weights of the Weibull mixture, some of the already existing methods are applied and the EM algorithm for the m-fold Weibull mixture is derived. The fitted distributions obtained by different methods are compared to the empirical ones by calculating the AIC and δ_C values. It can be concluded that the suggested Weibull mixture with an arbitrary but finite number of components is suitable for lifetime data approximation. For parameter estimation the combination of the alternative and EM algorithm is suggested.     

Experiment and modeling of uniaxial tension fatigue performances for filled natural rubbers

A tension fatigue model of filled natural rubbers is investigated to study the contributions of two key factors, namely, the damage parameter and the specimen geometry used in the fatigue experiment. The uniaxial tension fatigue experiments were carried out for three filled natural rubber specimens with different geometry: a dumbbell simple tension specimen (STS), a dumbbell cylindrical specimen (DCS), and a hollow cylindrical specimen (HCS). The commonly used damage parameters for fatigue life prediction are discussed. The fatigue life prediction models are formulated using the measured tension fatigue life of the STS together with different damage parameters. The effectiveness of the models is established in terms of a correlation coefficient characterizing the error between the measured and predicted fatigue lives. It is concluded that all the damage parameters considered in the study can effectively estimate the tension fatigue life with correlation coefficients exceeding 0.9. The fatigue life model formulated for the STS was also found to be appropriate for predicting the fatigue life of specimens with different geometry (DCS and HCS) suggesting that the relationship between the tension fatigue life and the damage parameters is independent of the specimen geometry. One may thus conduct tension fatigue tests with STS alone in order to model the tension fatigue life of rubbers with alternate geometry.     

A high-cycle fatigue life model for variable amplitude multiaxial loading

A high‐cycle fatigue life model for structures subjected to variable amplitude multiaxial loading is presented in this paper. It treats any kind of repeated blocks of variable amplitude multiaxial loading without using a cycle counting method. This model based on a mesoscopic approach is characterized by the following features: (i) the choice of a damage factor related to the accumulated mesoscopic plastic strain per stabilised cycle; (ii) the use of a mesoscopic mechanical behaviour taking into account the fatigue mechanisms such as plasticity and void growth. This behaviour is a von Mises elastoplastic model with linear kinematic hardening and hydrostatic stress dependent yield stress. The fatigue life model has six parameters identified with one SN curve and two fatigue limits. In‐phase and out‐of‐phase experimental tests from the literature are simulated. The predicted fatigue lives are compared to experimental ones.     

An engineering method for reliability analyses of mechanical structures for long fatigue lives

Reliability analyses of mechanical structures designed for long fatigue lives require: some information on the probability distribution of material fatigue strength at long fatigue lives. In order to address this need, three-parameter P-S-N curves are adopted to represent the results of fatigue tests based on the conventional method. The parameters of the P-S-N curves are estimated by using the least squares fitting method and maximizing the correlation coefficient. The three-parameter P-S-N curves obtained are then used to estimate the fatigue strengths for different survival probabilities at an arbitrarily long fatigue life. These fatigue strengths are used to define the probability distribution of fatigue strength at the long fatigue life when a normal. distribution is used to approximate the probability distribution of fatigue strength. The method is illustrated by analyzing the results of fatigue tests of steel #45 (Chinese steel) notched specimens subject to axial loads with a stress concentration factor k_t = 2·0. Then, the method is applied to the fatigue reliability analysis of the runner of a hydraulic turbine. It is shown here that the reliability at long lifetimes can be easily estimated by the proposed engineering method and the conventional method may give a non-conservative design at long fatigue lives due to the assumption of bi-linear P-S-N curves.     

Generalized renewal process for repairable systems based on finite Weibull mixture

Repairable systems can be brought to one of possible states following a repair. These states are: ‘as good as new’, ‘as bad as old’ and ‘better than old but worse than new’. The probabilistic models traditionally used to estimate the expected number of failures account for the first two states, but they do not properly apply to the last one, which is more realistic in practice. In this paper, a probabilistic model that is applicable to all of the three after-repair states, called generalized renewal process (GRP), is applied. Simplistically, GRP addresses the repair assumption by introducing the concept of virtual age into the stochastic point processes to enable them to represent the full spectrum of repair assumptions. The shape of measured or design life distributions of systems can vary considerably, and therefore frequently cannot be approximated by simple distribution functions. The scope of the paper is to prove that a finite Weibull mixture, with positive component weights only, can be used as underlying distribution of the time to first failure (TTFF) of the GRP model, on condition that the unknown parameters can be estimated. To support the main idea, three examples are presented. In order to estimate the unknown parameters of the GRP model with m-fold Weibull mixture, the EM algorithm is applied. The GRP model with m mixture components distributions is compared to the standard GRP model based on two-parameter Weibull distribution by calculating the expected number of failures. It can be concluded that the suggested GRP model with Weibull mixture with an arbitrary but finite number of components is suitable for predicting failures based on the past performance of the system.     

A genetic algorithm for unified approach-based predictive modeling of fatigue crack growth

This paper presents a framework to derive models of fatigue crack growth in real-life applications based on the unified approach.The unified approach enunciates that two parameters-namely, the stress intensity amplitude ΔK and the peak stress intensity K_max-drive fatigue crack growth. It captures and explicates the various fatigue phenomena coherently. However, its application for damage prediction is still in its infancy. Mathematical models that are consistent with the approach and the various observed characteristics under various environments are imperative for fatigue damage life prediction. These models will reduce cumbersome experimentation that is usually needed for the fatigue crack growth analysis. The framework presented in this paper consists of using the unified approach to design the structure of a model that relates fatigue crack growth with the specified microstructure, applied stress and environmental conditions. The fatigue growth model is derived by parametrizing, using a genetic algorithm, these structural relationships from the known experimental data. This model can quantitatively estimate crack growth rate under the given combination of microstructure, applied stress and environmental conditions. The initial research on modeling fatigue crack growth dynamics in Al-5052 under vacuum and air has revealed that the models resulting from the framework can capture the actual crack growth pattern to within 12% accuracy, and that an automatic rendering of ΔK^* vs. trajectories is possible for a given material and environmental conditions.     

Theory for Optimal Test Plans for the Random Fatigue-Limit Model

This article presents methodology for planning life tests based on the random fatigue-limit model (RFLM). The RFLM describes the relationship between fatigue life and the applied load in the presence of fatigue limits. A standardization of the RFLM is presented, and corresponding expressions for the elements of the Fisher information matrix are given. Different test plan criteria that express various objectives that the practitioner may have in conducting experiments under the RFLM are discussed. These criteria include estimation of life quantiles, estimation of stress/strain levels that yield a specific life quantile, estimation of fatigue-limit quantiles, and D and D_s optimality. Equivalence theorems used to verify optimality of test plans are also presented. The methods are sufficiently general so they can be applied to criteria based on any function of the model parameters. They are demonstrated using a nickel-base superalloy experiment.