Simulation des Mikrorisswachstums unter Schwingbeanspruchung. Teil 2: Simulationsergebnisse – Einfluss des Spannungszustandes und Reihenfolgeeffekte

Simulation of fatigue micro crack growth. Part 2: Results of simulation – influence of stress state and sequence effects In part one the modelling of micro crack growth due to alternating loading has been presented. Simulation results for tension/compression, torsion and proportional multiaxial loading, the scatter of the simulated lifetimes to a macroscopic crack length of 500 μm as well as the influence of the density of crack seeds and the grain size have been presented. In part two the influence of the stress state under proportional and non‐proportional loading is examined. Additionally the sequence effect of High‐Low and Low‐High as well as consecutive load sequences will be discussed. The comparison of the simulation results to experimental results shows that the influences of multiaxial loading and sequences can be simulated qualitatively correctly. The simplifications of the modelling have been to be considered. If the length of the maximum crack is interpreted as a measure of the damage, it can be concluded that the damage accumulation is non linear and non continuous. The main proportion of lifetime from the crack seed to the macro crack is contributed in the phase in which the length of the maximum micro crack comes close to the size of the grain size.     

Fatigue assessment of laserbeam‐welded aluminium joints under multiaxial loading

This paper presents the results and evaluation of the multiaxial fatigue behaviour of laserbeam‐welded overlapped tubular joints made from the artificially hardened aluminium alloy AlSi1MgMn T6 (EN AW 6082 T6) under multiaxial loadings with constant and variable amplitudes. Several fatigue test series under pure axial and pure torsional loadings as well as combined axial and torsional proportional and non‐proportional loadings have been carried out in the range of 2·10⁴ to 2·10⁷ cycles. The assessment of the investigated thin‐walled joints is based on a local notch stress concept. In this concept the fatigue critical area of the weld root is substituted by a fictitious notch radius r_ref = 0.05 mm. The equivalent stresses in the notch, considering especially the fatigue life reducing influence of non‐proportional loading in comparison to proportional loading, were calculated by a recently developed hypothesis, which is called the Stress Space Curve Hypothesis (SSCH). This hypothesis is based on the time evolution of the stress state during one load cycle. In addition, the fatigue strength evaluation of multiaxial spectrum loading was carried out using a modified Gough‐Pollard algorithm.     

Multiaxial low‐cycle fatigue life evaluation under different non‐proportional loading paths

This paper presents analytical and experimental investigations for fatigue lives of structures under uniaxial, torsional, multiaxial proportional, and non‐proportional loading conditions. It is known that the rotation of principal stress/strain axes and material additional hardening due to non‐proportionality of cycle loading are the 2 main causes resulting in shorter fatigue lives compared with those under proportional loading. This paper treats these 2 causes as independent factors influencing multiaxial fatigue damage and proposes a new non‐proportional influencing parameter to consider their combined effects on the fatigue lives of structures. A critical plane model for multiaxial fatigue lives prediction is also proposed by using the proposed non‐proportional influencing factor to modify the Fatemi‐Socie model. The comparison between experiment results and theoretical evaluation shows that the proposed model can effectively predict the fatigue life due to multiaxial non‐proportional loading.     

CONSIDERATION OF MULTIAXIALITY IN FATIGUE LIFE PREDICTION USING THE CLOSURE CONCEPT

Abstract— Based on fracture mechanics a model has been developed for predicting the fatigue lifetime to initiate a crack of technical size (2 a ∼ .5mm) in engineering components under multiaxial-proportional loading. Using material data determined for uniaxial loading, the model describes and evaluates the elasto-plastic multiaxial notch stresses and strains using the effective range of the J -integral, Δ J _eff, as a crack tip parameter. Mean stresses, load sequence effects and various modes of crack propagation due to variable amplitude loading with individual multiaxiality ratios can be explicitly considered. The prediction accuracy of the model is demonstrated using experimental fatigue life data determined with unnotched specimens of FeE460 and A15083 for proportional fully-reversed constant and variable amplitude loading with various multiaxiality ratios.     

Kurzrisswachstum bei mehrachsig nichtproportionaler Beanspruchung

Short fatigue crack growth under multiaxial nonproportional loading Initiation and short fatigue crack growth have been investigated under nonproportional cyclic loading. A critical plane approach based on fracture mechanics is used for modelling the fatigue process. A Paris‐type crack growth law, formulated using the effective cyclic J‐integral as crack driving force parameter, is integrated to give crack growth curves. Crack opening stresses and strains are calculated with approximation equations. Jiang's plasticity model is used to predict the stress‐strain path. The good agreement between model and real damage evolution is shown comparing experimentally determined crack growth curves, crack orientations, and life curves.     

Prediction methods of fatigue critical point for notched components under multiaxial fatigue loading

Two methods based on local stress responses are proposed to locate fatigue critical point of metallic notched components under non‐proportional loading. The points on the notch edge maintain a state of uniaxial stress even when the far‐field fatigue loading is multiaxial. The point bearing the maximum stress amplitude is recognized as fatigue critical point under the condition of non‐mean stress; otherwise, the Goodman's empirical formula is adopted to amend mean stress effect prior to the determination of fatigue critical point. Furthermore, the uniaxial stress state can be treated as a special multiaxial stress state. The Susmel's fatigue damage parameter is employed to evaluate the fatigue damage of these points on the notch edge. Multiaxial fatigue tests on thin‐walled round tube notched specimens made of GH4169 nickel‐base alloy and 2297 aluminium‐lithium alloy are carried out to verify the two methods. The prediction results show that both the stress amplitude method and the Susmel's parameter method can accurately locate the fatigue critical point of metallic notched components under multiaxial fatigue loading.     

Low‐cycle fatigue life prediction of various metallic materials under multiaxial loading

This paper proposed a simple life prediction model for assessing fatigue lives of metallic materials subjected to multiaxial low‐cycle fatigue (LCF) loading. This proposed model consists of the maximum shear strain range, the normal strain range and the maximum normal stress on the maximum shear strain range plane. Additional cyclic hardening developed during non‐proportional loading is included in the normal stress and strain terms. A computer‐based procedure for multiaxial fatigue life prediction incorporating critical plane damage parameters is presented as well. The accuracy and reliability of the proposed model are systematically checked by using about 300 test data through testing nine kinds of material under both zero and non‐zero mean stress multiaxial loading paths.     

Fatigue limit evaluation of notches,small cracks and defects: an engineering approach

In this paper, the average stress method for the fatigue limit evaluation of stress raising geometrical features is revised and extended. In particular, an analytical close‐form approach was used and the linear elastic stress equations were modified by taking into account the effect of nominal stress on the local stress distribution. Hence, the average tangential stress was correctly evaluated over a distance of 2a₀, where a₀ was El Haddad's short crack constant, for long and small notches as well as for crack‐like notches. When this model is applied to a wide range of geometrical features subjected to mode I fatigue loading, the classical shape of the curves of the Kitagawa–Takahashi diagram was obtained for changes in crack‐like notch size. Similarly, notch sensitivity was estimated by reducing the notch tip radius. The accuracy of the proposed method in predicting fatigue limits was then checked by using experimental data taken from the literature and generated on testing specimens weakened by rounded and sharp notches as well as by small artificial defects.     

Short-crack-growth-based fatigue assessment of notched components under multiaxial variable amplitude loading

A short crack model originally proposed for multiaxial constant amplitude loading is extended and applied to multiaxial variable amplitude loading. Load sequences have a significant influence on variable amplitude life; they are taken into account using algorithms originally proposed only for uniaxial loading. The estimated fatigue lives of unnotched tubular specimens and notched shafts under different in- and out-of-phase multiaxial constant and variable amplitude load histories are compared with the experimental results. The comparison reveals that the proposed short crack approach enables sufficiently accurate estimation. Moreover, the estimated critical planes, i.e., the planes of maximum crack growth rate or minimum life, are in good agreement with the experimental observations.     

Corrosion fatigue behaviour of 304 stainless steel under proportional and non‐proportional multiaxial loading condition

Corrosion fatigue and electrochemical tests under proportional loading and non‐proportional loading were conducted on 304 stainless steel in 0.63 mol L^?1 NaCl solution at room temperature. Two biaxial loading paths were applied to study the effect of proportional loading and non‐proportional loading on corrosion fatigue behaviour. Surface and fractographic observations of multiaxial corrosion fatigue specimens were carried out by scanning electron microscopy. It was shown that proportional loading had a more significant effect on the occurrence of local corrosion compared with non‐proportional loading because the continuous rotation of the principal stress plane under non‐proportional loading inhibits the pit formation.     

Multiaxial notch fatigue life prediction based on pseudo stress correction and finite element analysis under variable amplitude loading

A new computational methodology is proposed for fatigue life prediction of notched components subjected to variable amplitude multiaxial loading. In the proposed methodology, an estimation method of non‐proportionality factor (F) proposed by authors in the case of constant amplitude multiaxial loading is extended and applied to variable amplitude multiaxial loading by using Wang‐Brown's reversal counting approach. The pseudo stress correction method integrated with linear elastic finite element analysis is utilized to calculate the local elastic‐plastic stress and strain responses at the notch root. For whole local strain history, the plane with weight‐averaged maximum shear strain range is defined as the critical plane in this study. Based on the defined critical plane, a multiaxial fatigue damage model combined with Miner's linear cumulative damage law is used to predict fatigue life. The experimentally obtained fatigue data for 7050‐T7451 aluminium alloy notched shaft specimens under constant and variable amplitude multiaxial loadings are used to verify the proposed methodology and equivalent strain‐based methodology. The results show that the proposed methodology is superior to equivalent strain‐based methodology.     

Biaxial fatigue for proportional and non‐proportional loading paths

In order to study the use of a local approach to predict crack‐initiation life on notches in mechanical components under multiaxial fatigue conditions, the study of the local cyclic elasto‐plastic behaviour and the selection of an appropriate multiaxial fatigue model are essential steps in fatigue‐life prediction. The evolution of stress–strain fields from the initial state to the stabilized state depends on the material type, loading amplitude and loading paths. A series of biaxial tension–compression tests with static or cyclic torsion were carried out on a biaxial servo‐hydraulic testing machine. Specimens were made of an alloy steel 42CrMo4 quenched and tempered. The shear stress relaxations of the cyclic tension–compression with a steady torsion angle were observed for various loading levels. Finite element analyses were used to simulate the cyclic behaviour and good agreement was found. Based on the local stabilized cyclic elastic–plastic stress–strain responses, the strain‐based multiaxial fatigue damage parameters were applied and correlated with the experimentally obtained lives. As a comparison, a stress‐invariant‐based approach with the minimum circumscribed ellipse (MCE) approach for evaluating the effective shear stress amplitude was also applied for fatigue life prediction. The comparison showed that both the equivalent strain range and the stress‐invariant parameter with non‐proportional factors correlated well with the experimental results obtained in this study.     

Multiaxial fatigue of rubber: Part I: equivalence criteria and theoretical aspects

This paper investigates commonly used approaches for fatigue crack nucleation analysis in rubber, including maximum principal strain (or stretch), strain energy density and octahedral shear strain criteria. The ability of these traditional equivalence criteria, as well as a recent equivalence criterion (the cracking energy density) to predict multiaxial fatigue behaviour is explored. Theoretical considerations are also introduced relating to the applicability of various fatigue life analysis approaches. These include the scalar nature of traditional equivalence criteria, robustness of the criteria investigated for a wide range of multiaxial loadings, effects of crack closure and applications to non‐proportional multiaxial loadings. It is shown that the notion of a stress or strain amplitude tensor used for the analysis of multiaxial loading in metals is not appropriate in the analysis of rubber due to nonlinearity associated with finite strains and near incompressibility. Taken together, these considerations illustrate that traditional criteria are not sufficiently consistent or complete to permit confident analysis of arbitrary multiaxial loading histories, and that an analysis approach specific to the failure plane is needed. Of the three traditional criteria, maximum principal strain is shown to match most closely to the cracking energy density criterion, in terms of a failure locus in principal stretch space.     

Fatigue analysis of railway wheel using a multiaxial strain‐based critical‐plane index

A fatigue damage model to assess the development of subsurface fatigue cracks in railway wheels is presented in this paper. A 3‐dimensional finite element model (FEM) is constructed to simulate repeated cycles of contact loading between a railway wheel and a rail. The computational approach includes a hard‐contact over‐closure relationship and an elastoplastic material model with isotropic and kinematic hardening. Results from the simulation are used in a multiaxial critical‐plane fatigue damage analysis. The employed strain‐based critical‐plane fatigue damage approach is based on Fatemi‐Socie fatigue index that takes into account the non‐proportional and out‐of‐phase nature of the multiaxial state of stress occurs when a railway wheel rolls on a rail. It predicts fatigue‐induced micro‐crack nucleation at a depth of about 3.7 mm beneath the wheel tread, as well as the crack plane growth orientation which indicates the possible failure pattern. Additionally, the influence of various factors such as contribution of normal stresses, higher wheel load, and material model have been investigated.     

On the estimation of the material fatigue properties required to perform the multiaxial fatigue assessment

To accurately perform the fatigue assessment of engineering components subjected to in‐service multiaxial fatigue loading, the adopted design criterion must properly be calibrated, the used information usually being the fatigue strength under both pure uniaxial and pure torsional fatigue loading. Because of the complex fatigue response of metallic materials to multiaxial loading paths, the only reliable way to generate the necessary pieces of calibration information is by running appropriate experiments. Unfortunately, because of a lack of both time and resources, very often, structural engineers are requested to perform the multiaxial fatigue assessment by guessing the necessary fatigue properties. In this complex scenario, initially, the available empirical rules suitable for estimating fatigue strength under both pure axial and pure torsional fatigue loading are reviewed in detail. Subsequently, several experimental results taken from the literature and generated by testing metallic materials under a variety of proportional and non‐proportional multiaxial loading paths are used to investigate the way such empirical rules affect the accuracy in estimating fatigue strength, the damage extent being evaluated according to the modified Wöhler curve method. Such a systematic validation exercise allowed us to prove that under proportional loading (with both zero and non‐zero mean stresses), an adequate margin of safety can be reached even when the necessary calibration information is directly estimated from the material ultimate tensile strength. On the contrary, in the presence of non‐proportional loading, the use of the empirical rules reviewed in the present paper can result, under particular circumstances, in a non‐conservative fatigue design.     

Verformungsverhalten und rechnerische Abschätzung der Ermüdungslebensdauer metallischer Werkstoffe unter mehrachsig nichtproportionaler Beanspruchung

Deformation behaviour and numerical fatigue lifetime prediction of metallic materials under multiaxial nonproportional loading The development and evaluation of a model for lifetime prediction under multiaxial nonproportional loading is the aim of the current research project. It is assumed that the technical crack initiation life is consumed by short crack growth. This phenomenon is described using a fracture mechanics based approach. Herein, the effective cyclic J‐integral is used as crack tip parameter. Crack opening levels and J‐integral values are calculated applying approximation formulas. A plasticity model that is based on the Jiang model [Jia93] and extended to describe nonproportional hardening is used to predict the deformation behaviour. Experimental investigations on tubular and notched specimens with a wide range of different loading spectra serve for the verification of the model and for the identification of damage mechanisms.     

PSEUDO STRESS AND PSEUDO STRAIN BASED APPROACHES TO MULTIAXIAL NOTCH ANALYSIS

Abstract The analysis of notch stresses and strains is one of the key parts of fatigue life prediction of components and structures, In this paper two related approaches are introduced, covering the whole field from uniaxial to multiaxial non-proportional loading. The pseudo stress at the notch root computed by theory of elasticity is introduced as the governing variable for elastic-plastic notch analysis. Although the pseudo stress is just a specially defined nominal stress, it eases notch analysis in comparison to using arbitrarily definable nominal stresses, especially for non-proportional multiaxial loading. Additionally a pseudo strain based approach is introduced and compared to the stress approach. Both proportional and non-proportional loading are discussed, and compared with the approaches of other workers.     

A non-local theory applied to high cycle multiaxial fatigue

ABSTRACT The stress gradient effect on the fatigue limit is an important factor which has to be taken into account for an efficient transfer of fatigue data from laboratory tests to design of industrial components. A short review of some multiaxial high cycle fatigue criteria considering this effect is presented. On the basis of the two local mesoscopic approaches of Papadopoulos, two new non‐local high cycle multiaxial fatigue criteria are developed. These proposals are based on the concept of volume influencing fatigue crack initiation. Their predictions are compared with experimental multiaxial fatigue data on four materials (a mild steel, two high strength steels and a spheroidal graphite cast iron). The accuracy of the two local Papadopoulos criteria and of the non‐local proposals are compared and discussed, together with the physical interpretation of the threshold defining the volume influencing fatigue crack initiation.     

EFFECT OF NON-PROPORTIONAL OVERLOADING ON FATIGUE LIFE

Abstract— The effect of non-proportional overloading on both low cycle and high cycle fatigue life has been studied. Low cycle multiaxial fatigue tests were performed on EN 15R (a low alloy steel) using sequential loading blocks which comprised uniaxial "ordinary" cycles and torsion "overload" cycles, and vice versa. In high cycle fatigue, the behaviour of mode I crack growth in a medium carbon steel subjected to mixed (I and II) mode overloading was examined.
Under tension-torsion sequential overloading, crack growth behaviour shows an earlier transition from Stage I to Stage II with a pronounced reduction in accumulated fatigue life. Tensile overloading on torsion cycles was found to be more damaging compared to torsion overloading on repeated tensile cycles. The crack-load interaction in sequential overloading and its influence on crack growth and fatigue life is discussed. In low strain fatigue, Stage II crack growth retardation closely relates to the overload plastic zone size, crack tip blunting and crack surface shielding. Mixed mode overloading is shown to have a significant effect only if the mode I component of overloading is large enough to keep the crack open. Under both low cycle and high cycle fatigue conditions non-proportional overloading is shown to be more damaging than proportional overloading.     

Multiaxial fatigue life modelling using hybrid approach of critical plane and genetic algorithm

This paper presents a new hybrid approach for multiaxial fatigue life estimation, based on continuum damage mechanics theory and a genetic algorithm with critical plane model formulation. The hybrid model employs a genetic algorithm based setup for calibration with standard proportional and non‐proportional profiles to predict fatigue life for complex loading profiles. The model is evaluated using experimental fatigue life data for SS304 steel. Calibration using simplified profiles is in agreement with the requirement for cost‐effective experimental fatigue life testing. In‐phase and out‐of‐phase loads are used for calibration, and fatigue life is predicted for more complicated profiles. The results show good agreement between the estimated and experimental fatigue life, and calibration through simple loading histories to predict fatigue life for complex histories appears to be an effective solution using the proposed model. A brief comparison is presented with fatigue life estimation performance of the proposed model with models available in commercial codes. Proposed model found to be more consistent in fatigue life prediction against various loading conditions.