Variable Amplitude Loading Research Articles

Time-based subcycle formulation for fatigue crack growth under arbitrary random variable loadings

A time-based subcycle fatigue crack growth (FCG) formulation and validation are proposed to calculate the fatigue crack growth under general random variable amplitude loadings. The intrinsic difficulties of the classical cycle-based formulation for general random variable loadings are discussed first. Several typical spectrums that are not appropriate for cycle-based FCG are illustrated, such as the “Christmas tree” spectrums. A time-based subcycle formulation is then proposed to address this difficulty. The proposed model includes three major component: (1) a time-based crack growth kinetics function at the subcycle (time) scale; (2) an efficient crack tip opening displacement (CTOD) estimation method; (3) a crack tip plasticity zone tracking algorithms for crack opening level determination of a growing crack. Detailed derivation and calculation procedures are given. Following this, several numerical examples are illustrated for the proposed model under different loading spectrums for the crack growth and CTOD calculation. Randomly generated loading spectrums are used to illustrate the capability of the proposed method under arbitrary loadings. Next, in-house testing for “Christmas tree spectrum” and literature data on several representative variable loading spectrums are used for model validation. Finally, some conclusions and future work are drawn based on the proposed study.

On the applicability of Miner’s rule for multiaxial fatigue life calculations under non-proportional load histories

Fatigue design routines and computer codes must use some damage accumulation rule to deal with variable amplitude loadings (VAL), usually Palmgren-Miner (or Miner’s) linear rule for lack of a clearly better option. Nevertheless, fatigue lives are intrinsically sensitive to the order of VAL events, which may e.g. induce residual stresses and thus change the critical point stress state, much affecting its subsequent residual life. On the other hand, in general, non-linear damage accumulation rules are not robust, resulting in better predictions than Miner’s rule only for some specific load orders, requiring a case-by-case analysis. Therefore, Miner’s linear damage rule still is the usual choice in practical fatigue calculations and assessments, giving reasonable predictions at least when properly combined with approaches that sequentially consider plasticity-induced effects, following the critical point stress/strain history in a cycle-by-cycle basis. In this work, Miner’s rule is evaluated for non-proportional tension-torsion loadings on annealed tubular 316L stainless steel specimens. Normal-shear strain histories following either cross, diamond, circular or square paths are applied, and their fatigue lives are measured. Then, more complex VAL paths consisting of combinations of these individual path shapes are applied in other specimens, whose associated fatigue lives are predicted based on Miner’s rule.

A model to quantify fatigue crack growth by cyclic damage accumulation calculated by strip-yield procedures

Elber's hypothesis that DeltaKeff can be assumed as the driving force for fatigue crack growth (FCG) is the basis for strip-yield models widely used to predict fatigue lives under variable amplitude loads, although it does not explain all load sequence effects observed in practice. To verify if these models are indeed intrinsically better, the mechanics of a typical strip-yield model is used to predict FCG rates based both on Elber's ideas and on the alternative view that FCG is instead due to damage accumulation induced by the cyclic strain history ahead of the crack tip, which does not need or use DeltaKeff ideas. The main purpose here is to predict FCG using the cyclic strains induced by the plastic displacements calculated by strip-yield procedures, assuming there are strain limits associated both the with the FCG threshold and with the material toughness. Despite based on conflicting principles, both models can reproduce quite well FCG data, a somewhat surprising result that deserves to be carefully analyzed.

Gradient approach for the evaluation of the fatigue limit of welded structures under complex loading

Welded ‘T-junctions’ are tested at different load ratio for constant and variable amplitude loading. Fatigue results are analyzed through the type of fatigue mechanisms depending on the loading type. A gradient approach (WSG: Welded Stress Gradient) is used to evaluate the fatigue limit and the comparison with experimental results shows a relative good agreement. Nonlinear cumulative damage theory is used to take into account the variable amplitude loading.

Fatigue behaviour of additive manufactured Ti6Al4V, with as-built surfaces, exposed to variable amplitude loading

Additive Manufacturing (AM) allows for great design freedom compared to conventional manufacturing. This is very attractive for the aerospace industry in which AM could contribute to lightweight designs and thereby reduce fuel consumption, increase payload and extend flight range. The fatigue behaviour for rough as-built AM surfaces has previously been characterized with constant amplitude testing but in aerospace applications, most parts are exposed to variable amplitude loading. The fatigue behaviour for variable amplitude is not always consistent with the behaviour for constant amplitude due to effects of overloads and local plastic deformations. Therefore, variable amplitude loading behaviour of laser sintered and electron beam melted Ti6Al4V, with rough as-built surfaces have been investigated in this study using the Short-FALSTAFF (Fighter Aircraft Loading STAndard For Fatigue) load sequence. The predicted and the experimental fatigue life was overall consistent even though most experimental results exceeded the predicted life, especially for the laser sintered material. These findings show that conventional cumulative damage fatigue life predictions give reliable predictions for AM materials with rough as-built surfaces for the type of tension dominated load sequence used.

A plain linear rule for fatigue analysis under natural loading considering the sequence effect

Fatigue under variable amplitude loading is currently assessed with the Palmgren-Miner rule in structural standards, ignoring the order of loading, which would require non-linear or mixed rules, especially for the random loading sequences applied to certain structures. Therefore, the goal is to develop a practical and simple correction factor ensuring the linear summation of damage is conservative, so as to take the sequence effect into account in random loading from natural sources. The theoretical consistency of this approach is demonstrated and a case study is developed to test the feasibility of the new rule and its simplicity.

Fatigue life prediction based on equivalent initial flaw size for Al-Li alloy 2297 under spectrum loading

Fatigue life prediction methods based on the Equivalent Initial Flaw Size (EIFS) concept under variable amplitude (VA) spectrum loading conditions were studied in this paper. Fatigue tests of Al-Li alloy 2297 under constant amplitude (CA) and VA loading cases were carried out. Based on the comparison of predictions and test results, different ways to determine the EIFSs and whether the load-interaction effects should be considered in VA fatigue life prediction were discussed to recommend their applicable cases. A good agreement is observed between EIFS model predictions and experimental data, especially in case of high maximum stresses in VA loading.

Multiaxial fatigue: From materials testing to life prediction

Fatigue damage has special relevance on the life span of mechanical components and structures, as it takes responsibility for a large majority of the registered structural failures in service. The continuous need for the weight reduction and improved design optimization of components and a growing need for greater lifespans of equipment, forced the understanding of the fatigue behavior of materials either under multiaxial loading cycles or under increased number of loading cycles. Fatigue tests were always performed under uniaxial fatigue loads. However, it is generally recognized that multiaxial stresses occur in many full-scale structures, being rare the occurrence of pure uniaxial stress states. In the following, a wide range of multiaxial fatigue failure criteria have been developed that may be applied to different metallic materials. These failure criteria may be classified as strain based for high strain loading conditions or stress based for higher lifespans in the elastic loading regime. These criteria are nowadays used in design optimization also contributing to the weight reduction and improved lifespans of modern equipment. Present manuscript is a review of the main specimens and equipment where research can be carried out for performing multiaxial fatigue tests under bi-axial stress states. Being a personal view, it will be based on the author experience on the performance of multiaxial fatigue tests under axial/torsion loading using mechanical actuated or servo-hydraulic fatigue testing machines or biaxial fatigue tests using electromagnetic actuators in cruciform specimens. Further research is needed and is being carried out concerning multiaxial very high cycle fatigue in order to achieve better improvements in weight reduction of components and structures and also in complex variable amplitude loading with adequate damage accumulation models.

Fatigue life prediction method for AISI 316 stainless steel under variable-amplitude loading considering low-amplitude loading below the endurance limit in the ultrahigh cycle regime

To investigate the effect of cycle ratio of combined high- and low-amplitude loading corresponding to loading above and below the fatigue limit on fatigue life (cumulative fatigue damage), fatigue tests were conducted under repeated two-step amplitude loading condition with varying cycle ratios (the number of high amplitude cycles was kept at 10 cycles, the number of low amplitude loading cycle was set at four different values 103, 104, 5×104, and 10/105 cycles in one block) using AISI 316, a type of commercial austenitic stainless steel. The test results indicate that low-amplitude loading following high-amplitude loading contributed to the fatigue damage and affected fatigue life. Furthermore, both the EN (European) and KHK (The high pressure gas safety institute of Japan) standards, which describe predictive approaches for variable-amplitude loading conditions that consider stress levels below the fatigue limit in an S–N curve based on the oblique hyperbola model, gave inaccurate evaluation of fatigue life because of the effect of cycles of low-amplitude loading below the fatigue limit in one block. Fatigue damage at saturation in one block contributed by low-amplitude was proved. A reasonable predictive approach that considering the number of cycles of low-amplitude loading in one block using linear cumulative fatigue damage rule has been established based on repeated two-step amplitude loading test results, and its precision was confirmed through random-amplitude loading tests.

Fatigue failure of 350WT steel under large-strain seismic loading at room and subfreezing temperatures

Due to its high ductility, weldability and toughness at low temperature, CSA G40.21-350WT steel in Canada is primarily used in bridge construction, ship building, and seismic energy dissipation systems. This article presents uniaxial tensile tests and constant- and variable-amplitude cyclic testing performed on 350WT steel at room and subfreezing temperatures. The variable-amplitude tests include common step-loading patterns as well as tests under strain signals obtained from the brace response in building structures subjected to three different types of earthquakes. The ductility of 350WT steel from monotonic tensile tests is essentially same at room and low temperatures (−40°C). The cyclic test results revealed that cold temperatures as low as −35°C did not have adverse effects on the low cycle fatigue life of 350WT steel. The benchmark constant-amplitude tests were employed to predict fatigue life under different large-strain variable-amplitude loading patterns both for room and subfreezing temperature conditions. In addition to the common strain-life approach, the adequacy of two well-established energy-life models for predicting fatigue life under variable-amplitude loading was evaluated. The strain-life approach generally performed better than the energy-life methods, particularly for step-loading histories. Comparison between predictions and laboratory observations showed that the fatigue failure life under large strain seismic loading can be accurately estimated, especially at room temperature.

Fatigue modeling for a thermoplastic polymer under mean strain and variable amplitude loadings

The applicability of several fatigue damage models for polyether ether ketone (PEEK) under mean strain and variable amplitude multi-block loadings is evaluated. The models utilized in this study are assessed against experimental data for PEEK under various cyclic loading conditions, including (1) constant amplitude loading with non-zero mean strains, (2) two-block loading with zero and non-zero mean strains, and (3) three- and four-block loading with zero mean strain. Several fatigue models, including a strain-based, a strain-stress-based, and an energy-based, are employed to correct for the effect of mean strain and stress on fatigue behavior of PEEK. Among these models, the energy-based approach, which considers the deformation response of the material throughout its entire life, better correlates the PEEK experimental data in the presence of mean strain. For specimens under block loading, three different damage accumulation models are employed to evaluate their applicability for PEEK. These include the Linear Damage Rule, as well as the non-linear Damage Curve Approach and Hashin-Rotem model. Additionally, a Direct Cumulative Damage (DCD) approach using an energy-based parameter is also employed to account for the load history and sequence effect on PEEK fatigue behavior. The DCD method is found to provide acceptable fatigue life predictions for PEEK under multi-block loading with various strain ratios and frequencies.

A simple finding on variable amplitude (Gassner) fatigue SN curves obtained using Miner’s rule for unnotched or notched specimen

In this note, starting from the SN curve under Constant Amplitude (CA) for the fatigue life of the uncracked (plain) specimen, we obtain that Gassner curves for Variable Amplitude (VA) loading using the simple Palmgren-Miner’s law are simply shifted CA curves. Further, using the Critical Distance Method in a very clean and powerful form proposed by Susmel and Taylor for VA loading, we find similar result for notched specimen, the spectrum loading results in the same multiplicative term for notched, cracked and unnotched specimen. Hence, the present proposal can be considered as a simple empirical unified approach for rapid assessment of the notch effect under random loading, which simplifies the recent proposal by Susmel and Taylor. To their extensive validations, we add some specific comparison with experimental data from the Literature on our further findings.

Prediction of fatigue crack growth retardation using a cyclic cohesive zone model

Cohesive zone modeling of fatigue crack growth retardation in aerospace titanium alloy Ti–6Al–4V subjected to a single overload during constant amplitude is presented in this work. The cyclic softening behavior of the bulk material is simulated according to the Ohno–Wang’s cyclic plasticity theory. The fracture process zone is represented by an irreversible cohesive law which governs the material separation of fatigue crack. The material degradation mechanism is described by the gradual reduction of the unloading cohesive stiffness after each loading cycle. The fatigue crack growth behaviors are examined using the proposed cohesive model under both constant and variable amplitude loadings. The computational results are verified according to the experimental data, which confirm that the present model can be applied to predict the transient retardation in fatigue crack growth rate of the Ti–6Al–4V alloy accurately.

Fatigue crack closure: a review of the physical phenomena.

Plasticity‐induced, roughness‐induced and oxide‐induced crack closures are reviewed. Special attention is devoted to the physical origin, the consequences for the experimental determination and the prediction of the effective crack driving force for fatigue crack propagation. Plasticity‐induced crack closure under plane stress and plane strain conditions require, in principle, a different explanation; however, both types are predictable. This is even the case in the transition region from the plane strain to the plane stress state and all types of loading conditions including constant and variable amplitude loading, the short crack case or the transition from small‐scale to large‐scale yielding. In contrast, the prediction of roughness‐induced and oxide‐induced closures is not as straightforward.

A one-parameter nonlinear fatigue damage accumulation model

A nonlinear model for fatigue damage accumulation under variable amplitude loading is presented. The known assumption that the isodamage curves are converging at the knee point of the S-N curve of the material, has been adopted. The proposed model does only require one parameter, in addition to the S-N curve of the material, in order to calculate the remaining fatigue life. A single value for the parameter has been found to give satisfying agreement with experimental data for four arbitrary selected steels, indicating a general trend.

Experimental results and fatigue life evaluation of magnesium laserbeam‐welded joints under proportional and non‐proportional multiaxial fatigue loading with variable amplitudes

Fatigue life of magnesium laserbeam‐welds (AZ31 and AZ61 alloys) was assessed experimentally under variable amplitude loadings. The specimens were subjected to load‐controlled cyclic loadings. The tests were carried out using a Gauss‐distributed amplitude sequence of length LS = 5 · 104 cycles and loading ratio R = –1 under pure axial, pure torsion as well as in‐phase and out‐of‐phase combined loadings. The notch stresses were obtained from a linear‐elastic FE‐model using the reference radius approach with rref = 0.05 mm. The stress‐based hypotheses were applied: Effective equivalent stress hypothesis (EESH), shear stress intensity hypothesis (SIH), Findley, and modified Gough‐Pollard. A non‐proportionality factor is introduced and steps required for computing are presented in order to improve fatigue life assessment under non‐proportional loadings.

Probabilistic S-N curves for constant and variable amplitude

Welded details in steel bridges are subjected to fatigue lives up to 200·106 cycles due to traffic loads. While the majority of these stress ranges are below the Constant Amplitude Fatigue Limit (CAFL), some few high stress cycles can trigger the start of fatigue damage and lead the remaining load spectra to become damaging. The behaviour under spectra loading is thus of major importance for the fatigue design of steel bridges. This paper focuses on the fatigue behaviour of welded joints under variable amplitude loads. Fatigue tests have been conducted under constant and variable amplitudes on a typical bridge detail. Experimental crack growth curves were obtained using the Alternative Current Potential Drop method (ACPD), which showed the detrimental effect of stress ranges below the conventional CAFL. A two-step model with initiation-propagation was established to estimate the experimental fatigue lives, using a local strain approach for the initiation life and fracture mechanics for crack propagation. The model was implemented in a probabilistic Monte Carlo framework to include variability on the main parameters and establish S-N curves for Constant and Variable amplitude. The results of the simulations show that load spectra shape can be correlated with the S-N curves, namely the 2nd slope value below the CAFL.

A new method for fatigue life evaluation under out-of-phase variable amplitude loadings and its application to thin-walled magnesium welds

Depending on material ductility out-of-phase loads can result in fatigue life increase or fatigue life reduction compared to in-phase loading. Especially the latter case is crucial and must be taken into account as fatigue life evaluation is performed.Area covered by rotating shear stress vectors in section planes is used in order to provide a numerical out-of-phase measure of the load. Fatigue life evaluation for the variable amplitudes loadings is performed using the Palmgren-Miner linear damage accumulation, whereas the total damage of every cycle is split up into the stress amplitude component and the out-of-phase component. In order to compute the two components a modification of the Rainflow counting method, which keeps track of the time intervals, where the cycles occur, must be used. The method takes into account different slopes of the pure axial and the pure torsional Wöhler curves using a Wöhler curve interpolation.Fatigue life tests under constant and variable amplitude (Gauß-distributed spectrum with LS=5·104 cycles) loadings were performed on the tube-tube thin-walled welded specimens made of magnesium (AZ31 and AZ61) alloys. The tests included pure axial, pure torsional and combined in-phase and out-of-phase loadings with the load ratio R,R‾=-1. Magnesium welds show a fatigue life reduction under out-of-phase loads.FE-models of the specimens according to the notch stress approach were built in order to obtain local linear elastic stresses in the weld.The fatigue life assessment under multiaxial variable amplitude in-phase and out-of-phase loading with the developed method, showed a good agreement with the experimental results.

Probabilistic Modeling of Damage Accumulation for Fatigue Reliability Analysis

For high-speed trains, reliability prediction of railway axles plays an important role in preventing accidents due to fatigue. Using a nonlinear damage accumulation concept, a probabilistic S-N curve and one-to-one probability density function transformation, a probabilistic approach for modeling damage accumulation is developed to analyze the fatigue reliability of railway axle steels. Through characterizing the damage accumulation process as a distribution of degradation path, the proposed method captures a nonlinear damage accumulation phenomenon under variable amplitude loadings. The fatigue reliability is then analyzed and demonstrated through probabilistic modeling of cumulative damage, and an agreement between the predicted results and the experimental measurements under different loading conditions is obtained.

Testing of Ultra-Low Cycle Fatigue at Complex Loading Scenarios

This article describes systematic investigations on ULCF under multiaxial stress states and variable loading sequences as they occur under earthquake loading. The tests are performed on circular hollow sections (CHS) of structural steel S355 which are welded on baseplates. The multiaxial stress states are applied by bending respectively twisting the CHS, so that true strains up to 100 % per cycle and stress triaxialities between 0 and 0.6 occur. With this specimen and test setup arbitrary stress states in this range could be tested as well as predefined load sequences along different paths. In total 120 tests have been subjected to constant and variable amplitude loading. Different fracture modes and origins of fracture initiation have been observed. These experiments, their evaluation and documentation are presented within the article.