Hao Wu

A multiaxial incremental fatigue damage formulation using nested damage surfaces

Multiaxial fatigue damage calculations under non-proportional variable amplitude loadings still remains a quite challenging task in practical applications, in part because most fatigue models require cycle identification and counting to single out individual load events before quantifying the damage induced by them. Moreover, to account for the non-proportionality of the load path of each event, semi-empirical methods are required to calculate path-equivalent ranges, e.g. using a convex enclosure or the MOI (Moment Of Inertia) method. In this work, a novel Incremental Fatigue Damage methodology is introduced to continuously account for the accumulation of multiaxial fatigue damage under service loads, without requiring rainflow counters or path-equivalent range estimators. The proposed approach is not based on questionable Continuum Damage Mechanics concepts or on the integration of elastoplastic work. Instead, fatigue damage itself is continuously integrated, based on damage parameters adopted by traditional fatigue models well tested in engineering practice. A framework of nested damage surfaces is introduced, allowing the calculation of fatigue damage even for general 6D multiaxial load histories. The proposed approach is validated by non-proportional tensiontorsion experiments on tubular 316L stainless steel specimens.

On the applicability of multi-surface, two-surface and non-linear kinematic hardening models in multiaxial fatigue

In this work, a comparison between NLK and Mroz-Garud’s multi-surface formulations is presented. A unified common notation is introduced to describe the involved equations, showing that the Mroz-Garud model can be regarded as a particular case of the NLK formulation. It is also shown that the classic two-surface model, which is an unconventional simplified plasticity model based on the translation of only two surfaces, can also be represented using this formulation. Such common notation allows a direct quantitative comparison among multi-surface, two-surface, and NLK hardening models.

Shortcuts in multiple dimensions: the multiaxial racetrack filter

Filtering techniques have been proposed for multiaxial load histories, usually aiming to filter out non-reversals, i.e. sampling points that do not constitute a reversal in any of its stress or strain components. However, the path between two reversals is needed to evaluate the equivalent stress or strain associated with each event. Filtering out too many points in such path would almost certainly result in lower equivalent stresses or strains than expected. To avoid such issues, it is important to consider how a measured multiaxial loading path deviates from its course using some metric, such as the von Mises stress or strain. In this work, a multiaxial version of the racetrack filter is proposed, which is able to perform efficient filtering even for 6D nonproportional histories. In the Multiaxial Racetrack algorithm, the stress or strain history is represented in a 6D space, only requiring from the user a desired scalar filtering amplitude r. For uniaxial histories, the proposed algorithm exactly reproduces the classic racetrack filter. The efficiency of the proposed Multiaxial Racetrack filter is qualitatively verified from a tension-torsion history example, showing the reduction in the number of data points for larger filter amplitudes r. The procedure can efficiently filter out non-damaging events but preserving the overall multiaxial path shape and multiaxial reversion points, which usually do not coincide with the reversion points of individual stress or strain components.

Issues With Multiaxial Fatigue Assessment in the ASME Boiler and Pressure Vessel Code

This work analyzes the applicability of the ASME Boiler and Pressure Vessel Code procedures to calculate fatigue crack initiation under multiaxial stresses and/or strains, in particular when caused by non-proportional loads that lead the principal directions at the critical point to vary with time, e.g. due to outof-phase bending and torsion loads induced by independent sources. Classic uniaxial fatigue damage models are usually inappropriate for analyzing multiaxial loads, since they can generate highly inaccurate predictions. Moreover, it is shown that the ASME procedures can lead to non-conservative results for non-proportional load histories. INTRODUCTION Service loads can be applied on one or on several points of a structural component. They can be generated by only one or by multiple sources, coherent or not. In general, such loads cause bending, torsion, normal, and/or shear efforts which, when combined, can (and usually do) generate multiaxial strains and stresses at the critical point(s) of the component. Multiaxial fatigue deals with the initiation and/or the propagation of fatigue cracks under such general conditions. Multiaxial fatigue load histories can be proportional or non-proportional (NP). They are proportional when the principal axes of the stresses and strains induced by them, and thus their maximum-shear planes, remain fixed during their entire duration. On the other hand, NP loads induce principal stress/strain directions that change in time (1). Consider for instance a tension-torsion problem where a shaft is loaded by a normal stress x(t) in the x direction combined with a shear stress xy(t), where t stands for time. In this case, the principal stresses 1 and 2 and the angle 1 between 1 and the x axis are given by 2 x x 2 1,2 xy 2 2              and xy 1 1 x 2 (t) 1 (t) tan 2 (t)            (1) When the shear and normal stresses are proportional, the ratio xy(t)/x(t) and the angle 1(t) remain fixed for all time t, thus this simple multiaxial loading history is proportional. If xy(t)/x(t) and hence 1(t) vary with time, then the loading is non-proportional. The relative degree of non-proportionality of any multiaxial load history is quantified by its so-called nonproportionality factor 0  FNP  1, with FNP  0 standing for a proportional load history and FNP  1 for a fully NP history. If all stress and strain components are periodic and have the same frequency, they can also be classified as in-phase or out-of-phase. Figure 1 shows in-phase as well as 90 and 180 out-of-phase tension-torsion load histories. Both the in-phase and the 180 out-of-phase loadings have a constant xy(t)/x(t) ratio, so they are proportional histories with FNP  0. The 90 out-of-phase tension-torsion loading, on the other hand, always results in a NP history, with the FNP value depending on the ratio between the shear and normal amplitudes. It is usually accepted (1) that, in tension-torsion histories, the maximum value FNP  1 is achieved for sinusoidal 90 out-of-phase loadings with equal amplitudes for x and xy3, when the von Mises stress 2 2 x xy 3    remains constant along the load path.

A two-step multiaxial racetrack filter algorithm for non-proportional load histories

The recently proposed multiaxial racetrack filter (MRF) is able to deal with general non-proportional multiaxial load histories. While only requiring a single user-defined scalar filter amplitude, the MRF is able to synchronously eliminate non-damaging events from any noisy multiaxial load history without changing the overall shape of its original path, a necessary condition to avoid introducing errors in fatigue damage assessments. The MRF procedures are optimized here by the introduction of a pre-processing “partitioning” step on the load history data, which selects candidates for the reversal points in a robust partitioning process, highly increasing the filter efficiency and decreasing its computational time. The improved MRF is evaluated through the fatigue analyses of over-sampled tension-torsion data measured in 316L stainless steel tubular specimens under non-proportional load paths.

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.

Incorporation of Mean/Maximum Stress Effects in the Multiaxial Racetrack Filter

This work extends the Multiaxial Racetrack Filter (MRF) to incorporate mean or maximum stress effects, adopting a filter amplitude that depends on the current stress level along the stress or strain path. In this way, a small stress or strain amplitude event can be filtered out if associated with a non-damaging low mean or peak stress level, while another event with the very same amplitude can be preserved if happening under a more damaging high mean or peak stress level. The variable value of the filter amplitude must be calculated in real time, thus it cannot depend on the peak or mean stresses along a load event, because it would require cycle identification and as so information about future events. Instead, mean/maximum stress effects are modeled in the filter as a function of the current (instantaneous) hydrostatic or normal stress along the multiaxial load path, respectively for invariantbased and critical-plane models. The MRF efficiency is evaluated from tension-torsion experiments in 316L stainless steel tubular specimens under non-proportional (NP) load paths, showing it can robustly filter out nondamaging events even under multiaxial NP variable amplitude loading histories. KEYWORDS. Multiaxial racetrack filter; Mean/peak stress effects; Nondamaging events; Multiaxial loads.

Application of the Moment Of Inertia method to the Critical-Plane Approach

The Moment-Of-Inertia (MOI) method has been proposed by the authors to solve some of the shortcomings of convex-enclosure methods, when they are used to calculate path-equivalent ranges and mean components of complex non-proportional (NP) multiaxial load histories. In the proposed 2D version for use with critical-plane models, the MOI method considers the non-proportionality of the projected shear-shear history on each candidate plane through the shape of the load path, providing good results even for challenging non-convex paths. The MOI-calculated path-equivalent shear stress (or strain) ranges from each counted load event can then be used in any shear-based critical-plane multiaxial fatigue damage model, such as Findley’s or Fatemi-Socie’s. An efficient computer code with the shear-shear version of the MOI algorithm is also provided in this work. KEYWORDS. Multiaxial fatigue; Non-proportional loadings; Equivalent ranges; Critical-Plane Approach.