Issue 41

M. A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 41 (2017) 1-7; DOI: 10.3221/IGF-ESIS.41.01 2 Consequently, a most important practical issue in multiaxial fatigue analyses is how to reduce a large amount of redundant multiaxial data to a manageable size to decrease their intrinsically high computational cost, while maintaining all the essential features of the load history that contribute to plasticity memory effects. Needless to say, this is an unavoidable step to not underestimate fatigue damage, an inadmissible feature in practical applications. Uniaxial amplitude filters can be directly implemented in the cycle counting algorithm, usually based on the rainflow method [1-4]. But the original rainflow procedure can only be started after the entire load history is known, increasing even more the computational cost as well as computer memory requirements, which can be quite significant for very long histories. Computational cost can be dramatically reduced with ‘‘real-time” rainflow algorithms, such as the pioneer Martin–Topper–Sinclair’s 1971 method [5], which essentially reproduces in real time the uniaxial rainflow algorithm as the load events are provided or measured. The original racetrack filter, proposed by Fuchs et al. in 1973 [6], can sequentially filter small amplitudes, but it is limited to uniaxial histories. Simplistic amplitude filters based on the uniaxial racetrack are not recommended in multiaxial analyses, because the path between two load reversals is needed to evaluate the path-equivalent stress or strain amplitude associated with each rainflow count, e.g. using a convex-enclosure method [7, 8]. Moreover, the reversal points obtained from a multiaxial rainflow algorithm might not occur at the reversal of one of the stress or strain components [9]. In this work, a multiaxial version of the racetrack filter proposed by the authors in [10-13] is reviewed and optimized. While only requiring a single user-defined scalar filter amplitude, it is able to synchronously filter complex loading histories while preserving all key features of the loading path. The filtering process is optimized through the introduction of a pre-processing “partitioning” operation on the load history data, highly increasing its efficiency. Over-sampled experimental data from tension-torsion experiments in 316L stainless steel tubular specimens under NP load paths are used to verify the efficiency and robustness of the proposed method. T HE M ULTIAXIAL R ACETRACK F ILTER (MRF) n the multiaxial racetrack filter (MRF) algorithm originally proposed in [10], the load history path must first be represented in an appropriate stress or strain space. Several spaces were proposed in [11], some of them separating the stresses or strains in their deviatoric and hydrostatic components, to allow for a filtering metric based on Mises equivalent values or even damage parameters. Fig. 1 shows an example of a tension-torsion loading following an elliptical stress path, represented in a normal vs. effective shear stress space  x   xy  3 through a series of 238 over-sampled points (for each periodic cycle), represented as “  ” marks. However, spaces with higher dimensions would be necessary for general 6D multiaxial load histories. Several of these points could be filtered out without compromising the subsequent multiaxial fatigue life calculations. This amplitude-filtering process is a most desirable step in practical applications, to eliminate unavoidable measurement noise and redundant over-sampled data, as well as small amplitudes that do not cause fatigue damage [12]. But it is important to avoid filtering out important counting points from multiaxial rainflow algorithms, or significant history paths that can affect the calculation of a path-equivalent stress or strain, since all stress or strain components contribute altogether for the reversals that can be eliminated. The MRF requires a user-defined scalar filter amplitude r, which is graphically represented in Fig. 1 as the radius of the small dashed circle centered at point 1. In this example, r  80MPa was chosen as the amplitude, which in the  x   xy  3 space has a clear physical meaning: r is the Mises distance between two stress states, due to the adopted  3 scaling factor used in the shear component. The MRF algorithm, thoroughly described in [10-13], was able to reduce the 238 over- sampled measurements to only 8, guaranteeing that no filtered-out data lies beyond r  80MPa of the resulting polygonal path 1-2-3-4-5-6-7-8-1. This filtering process results in a dramatic decrease of the computational time needed for further multiaxial fatigue life calculations, especially considering that the original 238 points were from a single elliptical path. More refined outputs can be achieved simply by reducing the user-defined value of the (combined) amplitude filter r. For instance, in the above example, r  5MPa would better describe the elliptical shape of the original stress path, but it would need 31 points instead of 8 to represent it; still, 31 is much better than the original 238 points per cycle. The filter amplitude r can also be varied as a function of the instantaneous normal stresses, to better consider mean and peak-stress effects, as discussed in [12]. Moreover, the MRF also works with non-periodic load histories, using the same algorithm. I