Quadratic damage rule in random loading case

The simple mathematical structure of the quadratic damage rule (QDR) enables mathematical expression of the failure condition in the case of random loading. Random loading after modified rainflow analysis is considered as a random marked point process. Moments of the damage distribution can be expressed by use of ensemble averages. A condition for validity of the first damage moment solution is formulated. The QDR damage first moment solution involves failure predictions for both stationary and non‐stationary loadings. In these cases, final fracture conditions suitable for practical use are derived. QDR predicts fatigue lives that are always shorter than the linear damage rule (LDR) in the case of stationary loading with short interval of statistical dependence of random variables. QDR appears to be a suitable additional engineering tool for random loading fatigue life predictions considering loading sequence and loading level interactions.     

Cumulative fatigue damage of stress below the fatigue limit in weldment steel under block loading

To investigate the cumulative fatigue damage below the fatigue limit of multipass weldment martensitic stainless steel, and to clarify the effect of cycle ratios and high‐stress level in the statement, fatigue tests were conducted under constant and combined high‐ and low‐stress amplitude relative to stress above and below the fatigue limit. The outcomes indicate that neither modified Miner's nor Haibach's approach provided accurate evaluation under repeated two‐step amplitude loading. Moreover, effect of cycle ratios has been determined. Additionally, the cumulative fatigue damage saturated model is established and validated. Cumulative fatigue damage contributed by low‐stress below the fatigue limit in high stress of 700 MPa is higher than that with 650 MPa at identical conditions (fatigue limit 575 MPa). Thus, high stress affects fatigue damage behaviour below the fatigue limit. A new predicted approach has been proposed based on Corten‐Dolan law, whose accuracy and applicability have been proven.     

On the interaction of normal and shear stresses in multiaxial fatigue damage

One of the important issues in assessing multiaxial fatigue damage is interactions between different components of stress such as normal and shear stresses. The present study investigated this interaction effect on the fatigue behavior of materials with shear failure mode when subjected to multiaxial loading conditions. A method is introduced to model this interaction based on the idea that two types of influence are caused by the normal stress acting on the critical plane orientation. These two types of influence are affecting roughness induced closure, as well as fluctuating normal stress which affects the growth of small cracks in mode II. Shear‐based critical plane damage models which use normal stress as a secondary input, such as FS damage model, could then use the summation of these terms. In order to investigate the effect of the method, constant amplitude load paths with different levels of interaction between the normal and shear stresses, as well as variable amplitude tests with histories both taken from service loading conditions and generated using random numbers were designed for an experimental program. The proposed method was observed to result in improved fatigue life estimations where significant interactions between normal and shear stresses exist.     

Cumulative fatigue damage and life prediction of elastomeric components

Elastomeric components are widely used in many applications due to their good damping and energy absorption characteristics. The type of loading normally encountered by these components in service is variable amplitude cyclic loading. Therefore, fatigue failure is a major consideration in their design. In this work capabilities of Rainflow cycle counting procedure, maximum principal strain as a damage criterion, and Miner's linear cumulative damage rule are evaluated with both specimen and component tests. An automotive cradle mount is used as an illustrative component. Comparison of predicted and experimental fatigue lives in both specimen and cradle mount variable amplitude load tests indicate satisfactory predictions in both cases.     

An experimental approach to estimate damage and remaining life of metals under uniaxial fatigue loading

An experimental procedure to estimate damage evolution and remaining fatigue life of metals associated with fatigue loading is presented. Experimental phase involves uniaxial tension–compression fatigue tests performed with solid API 5L X52 and tubular carbon steel 1018 specimens subjected to both constant and variable amplitude loading. A correlation between the so-called damage parameter and the thermal response of a material at different damage levels is proposed. Results demonstrate that the correlation can estimate damage evolution with reasonable accuracy in both constant and variable amplitude fatigue processes. It is shown that under the conditions tested the evolution of damage parameter with respect to the normalized fatigue life is independent of the load amplitude, load ratio, loading sequence, material properties, and specimen geometry. The proposed correlation and the relationship between the damage parameter and the normalized fatigue life are employed to develop a non-destructive method to predict the remaining fatigue life of metallic specimens with prior fatigue damage. The method is applied to both constant and variable amplitude loading and the predicted results are found to be in good agreement with those obtained from the experiments.     

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.     

Multiaxial fatigue damage parameter and life prediction for medium-carbon steel based on the critical plane approach

The tension–torsion fatigue characteristics were investigated under proportional and non-proportional loading in this paper. The fatigue cracks on the surface of multiaxial fatigue specimens were observed and analyzed by a scan electron microscope. On the basis of the investigation on the Kindil–Brown–Miller and Fatemi–Socie’s critical plane approaches, a shear strain based multiaxial fatigue damage parameter was proposed by von Mises criterion based on combining the maximum shear strain and the normal strain excursion between adjacent turning points of the maximum shear strain on the critical plane. The proposed multiaxial fatigue damage parameter does not include the weight constants. According to the proposed multiaxial fatigue damage parameter, the multiaxial fatigue life prediction model was established with the Coffin–Manson equation, which is used to predict the multiaxial fatigue life of medium-carbon steel. The results showed that the proposed multiaxial fatigue damage parameter could be used under either multiaxial proportional or non-proportional loading.     

Multiaxial notch fatigue life prediction based on the dominated loading control mode under variable amplitude loading

A new calculation approach is suggested to the fatigue life evaluation of notched specimens under multiaxial variable amplitude loading. Within this suggested approach, if the computed uniaxial fatigue damage by the pure torsional loading path is larger than that by the axial tension–compression loading path, a shear strain‐based multiaxial fatigue damage parameter is assigned to calculate multiaxial fatigue damage; otherwise, an axial strain‐based multiaxial fatigue damage parameter is assigned to calculate multiaxial fatigue damage. Furthermore, the presented method employs shear strain‐based and axial strain‐based multiaxial fatigue damage parameters in substitution of equivalent strain amplitude to consider the influence of nonproportional additional hardening. The experimental data of GH4169 superalloy and 7050‐T7451 aluminium alloy notched components are used to illustrate the presented multiaxial fatigue lifetime estimation approach for notched components, and the results reveal that estimations are accurate.     

A review of multiaxial fatigue criteria for random variable amplitude loads

Nowadays, the estimation of fatigue life under multiaxial random loading is still an extremely complex task. In this paper, a comprehensive review of the multiaxial random fatigue criteria available in the literature is presented. Such a review is mainly devoted to stress‐based criteria for the evaluation of fatigue life in high‐cycle regime. Time and frequency domain approaches are examined. The focus of this paper is related to uniform stress/strain distribution, but also the effect of stress/strain gradient is tangentially addressed. More than 200 references are cited.     

A hybrid frequency–time domain method for predicting multiaxial fatigue life of 7075‐T6 aluminium alloy under random loading

A hybrid frequency–time domain method for predicting multiaxial fatigue life under random loading is developed on the basis of combination of the frequency domain and time domain analysis. The critical damage point of the structure is determined by the frequency domain equivalent stress method. Then, the fatigue life prediction is made in time domain by generating random load‐time histories from the power spectral density of the critical point. The method is validated with the random vibration fatigue test of 7075‐T6 aluminium alloy. It has been shown that the results of fatigue life calculated by hybrid method are well correlated with the experiment.     

Evaluation of fatigue life for titanium alloy TC4 under variable amplitude multiaxial loading

Fatigue tests under variable amplitude multiaxial loading were conducted on titanium alloy TC4 tubular specimens. A method to estimate the fatigue life under variable amplitude multiaxial loading has been proposed. Multiaxial fatigue parameter based on Wu–Hu–Song approach and rainflow cycle counting and Miner–Palmgren rule were applied in this method. The capability of fatigue life prediction for the proposed method was checked against the test data of TC4 alloy under variable amplitude multiaxial loading. The prediction results are all within a factor of two scatter band of the test results.     

Research on fatigue reliability prediction model and structural improvement of welded drive axle housing based on master S-N Curve method

Fatigue reliability prediction of welded structures is mainly based on nominal stress or hot spot stress method, but there are some problems such as grid sensitivity and joint geometry dependence. The Master S-N curve method can solve these problems well, but the corresponding reliability model needs to be studied. In this paper, the fatigue reliability model of welded structures based on the Master S-N curve method is studied. Considering the randomness of life and the correlation of failure, a reliability model is proposed, which reduces the computational burden by establishing a median damage-random threshold rule. Taking the welded drive axle housing as an object, the system reliability is analyzed under the bench test condition, and verified by the experimental data. After the verification, this method is used to predict the reliability of the axle housing under variable amplitude loading collected in the test field, and the results are verified by Monte Carlo (MC) method. When the P-S-N curves are parallel, the model is accurate, which is the characteristic of the Master S-N curve method. This method only needs to input the median damage value of the weak part, which is easy to be applied. This method can speed up the reliability prediction cycle of welded structures, which is beneficial to product innovation and optimal design. Finally, an improved design scheme is proposed for the weak parts of welding, and the effects of welding leg width, welding depth, and closed weld on fatigue life are revealed.     

Online multiaxial fatigue damage evaluation method by real‐time cycle counting and energy‐based critical plane criterion

To realize online multiaxial fatigue damage assessment for the mechanical components in service, an online multiaxial cycle counting method is proposed coupled with the segment processing technique and Wang‐Brow's relative equivalent strain concept. Meanwhile, considering all the stress and strain components, which contribute to the fatigue damage on the critical plane, a multiaxial fatigue damage model without any weight coefficients is also proposed in an equivalent form of shear strain energy. Then, an online fatigue damage evaluation method for multiaxial random loading is developed by combining with the proposed damage model and online cycle counting method. The experimental results showed that the proposed online cycle counting method can be successfully applied to the calculation of multiaxial fatigue damage under random loading. Moreover, the proposed online multiaxial fatigue damage evaluation method can provide satisfactory predictions.     

A fatigue driving energy approach to high‐cycle fatigue life estimation under variable amplitude loading

In this paper, a concept of fatigue driving energy is formulated to describe the process of fatigue failure. The parameter is taken as a combination of the fatigue driving stress and strain energy density. By assessing the change of this parameter, a new non‐linear damage model is proposed for residual life estimation within high‐cycle fatigue regime under variable amplitude loading. In order to consider the effects of loading histories on damage accumulation under such condition, the load interaction effects are incorporated into the new model, and a modified version is thus developed. Life predictions by these two models and Miner rule are compared using experimental data from literature. The results show that the proposed model gives lower deviations than the Miner rule, while the modified model shows better prediction performances than the others. Moreover, the proposed model and its modifications are ease of implementation with the use of S–N curve.     

Fatigue strength of small-notched specimens under variable amplitude loading within the fatigue limit diagram

Although the fatigue limit diagram is defined in principle for constant stress amplitude, it is often considered that fatigue failure would not occur, even in varying loading, if applied stresses were kept within the fatigue limit diagram. However, it was shown in the case of small‐notched specimens that fatigue failure occurred in some special cases of variable amplitude loading, even when all stress amplitudes were kept within the fatigue limit diagram. The cause of this phenomenon was examined using two‐step stress and repeated two‐step stress patterns in which the first step stress was chosen to be equal to the fatigue limit with zero mean stress and a mean stress was superposed on the second step stress. A non‐propagating crack was formed by the first step stress. This crack functioned as a pre‐crack for the second step stress with high mean stress. Consequently, fatigue failure occurred even when all stress amplitudes were kept within the fatigue limit diagram. It was an unexpected fracture caused by the interference effect of a non‐propagating crack and a mean stress change.     

Cumulative damage model based on equivalent fatigue under multiaxial thermomechanical random loading

A new creep–fatigue damage cumulative model is proposed under multiaxial thermomechanical random loading, in which the damage at high temperature can be divided into the pure fatigue damage and the equivalent fatigue damage from creep. During the damage accumulation process, the elementary percentage of the equivalent fatigue damage increment is proportional to that of the creep damage increment, and the creep damage is converted to the equivalent fatigue damage. Moreover, combined with a multiaxial cyclic counting method, a life prediction method is developed based on the proposed creep–fatigue damage cumulative model. In the developed life prediction method, the effects of nonproportional hardening on the fatigue and creep damages are considered, and the influence of mean stress on damage is also taken into account. The thermomechanical fatigue experimental data for thin‐walled tubular specimen of superalloy GH4169 under multiaxial constant amplitude and variable amplitude loadings were used to verify the proposed model. The results showed that the proposed method can obtain satisfactory life prediction results.     

On the connection between Palmgren‐Miner rule and crack propagation laws

The classical Palmgren‐Miner (PM) rule, despite clearly approximation, is commonly applied for the case of variable amplitude loading, and to date, there is no simple alternative. In the literature, previous authors have commented that the PM hypothesis is based on an exponential fatigue crack growth law, ie, when da/dN is proportional to the crack size a, the case that includes also Paris law for m=2, in particular. This is because they applied it by updating the damage estimate during the crack growth. It is here shown that applying PM to the “initial” and nominal (Stress vs Number of cycles) curve of a cracked structure results exactly in the integration of the simple Paris power law equation and more in general to any crack law in the form da/dN=HΔσ^haⁿ. This leads to an interesting new interpretation of PM rule. Indeed, the fact that PM rule is often considered to be quite inaccurate pertains more to the general case when propagation cannot be simplified to this form (like when there are distinct initiation and propagation phases), rather than in long crack propagation. Indeed, results from well‐known round‐robin experiments under spectrum loading confirm that even using modified Paris laws for crack propagation, the results of the “noninteraction” models, neglecting retardation and other crack closure or plasticity effects due to overloads, are quite satisfactory, and these correspond indeed very closely to applying PM, at least when geometrical factors can be neglected. The use of generalized exponential crack growth, even in the context of spectrum loading, seems to imply the PM rule applies. Therefore, this seems closely related to the so‐called lead crack fatigue lifing framework. The connection means however that the same sort of accuracy is expected from PM rule and from assuming exponential crack growth for the entire lifetime.