Issue 33

D. G. Hattingh et alii, Frattura ed Integrità Strutturale, 33 (2015) 382-389; DOI: 10.3221/IGF-ESIS.33.42 385 The generated experimental data were post-processed under the hypothesis of a log-normal distribution of the number of cycles to failure for each stress level with a confidence level equal to 95% [9]. The results of the statistical reanalysis are summarised in Tab. 1 in terms of SN curves. In particular, B R =  nom,a /  nom,a is the ratio between the amplitudes of the axial and torsional nominal stress, R is the nominal load ratio (R=  nom,min /  nom,max =  nom,min /  nom,max ),  is the out-of-phase angle, k is the negative inverse slope,  A and  A are the amplitudes of the axial and torsional endurance limits extrapolated at N A =2  10 6 cycles to failure, and, finally, T  is the scatter ratio of the amplitude of the endurance limit for 90% and 10% probabilities of survival. To conclude, it is worth observing that the fatigue curves summarised in Tab. 1 are determined in terms of nominal stresses referred to the annular section of the parent tube. B R R  N. of data k  A  A T  [°] [MPa] [MPa]  -1 - 9 6.5 33.5 - 1.58  0.1 - 10 4.4 18.6 - 1.82 0 -1 - 11 10.8 - 38.9 1.49 0 0 - 10 9.5 - 32.9 1.52 3 -1 0 8 5.3 26.2 15.1 1.54 1 -1 0 7 5.3 23.1 23.1 2.13 3 0 0 7 4.2 17.2 9.9 1.74 1 0 0 7 3.2 12.8 12.8 2.00 3 -1 90 7 3.7 18.5 10.7 2.25 3 0 90 7 10.4 23.4 13.5 1.38 Table 1 : Summary of the generated experimental results. F UNDAMENTALS OF THE M ODIFIED W ÖHLER C URVE M ETHOD he MWCM is a critical plane approach which predicts the number of cycles to failure via the maximum shear stress amplitude,  a , as well as via the mean value,  n,m , and the amplitude,  n,a , of the stress normal to the critical plane. In this setting, the critical plane is defined as that material plane experiencing the maximum shear stress amplitude,  a [10]. The combined effect of the relevant stress components relative to the critical plane is taken into account by means of stress index  eff which is defined as follows [11]: , , n m n a eff a m        (1) In definition (1), m is the so-called mean stress sensitivity index [10]. This index is a material property ranging in between 0 and 1 whose value has to be determined by running appropriate experiments [11]. Ratio  eff is a stress quantity which is sensitive not only to the presence of superimposed static stresses, but also to the degree of non-proportionality of the applied loading path [10]. The way the MWCM estimates fatigue lifetime under multiaxial fatigue loading is schematically shown via the modified Wöhler diagram sketched in Fig. 5. This log-log diagram plots the shear stress amplitude relative to the critical plane,  a , against the number of cycles to failure, N f . By performing a systematic investigation based on numerous experimental results generated under multiaxial fatigue loading [12-14], it was observed that fatigue damage tends to increase as  eff increases. This results in the fact that the corresponding fatigue curve tends to shift downward in the modified Wöhler diagram with increasing of  eff (Fig. 5). By taking full advantage of the classic log-log schematisation which is commonly adopted to summarise stress based fatigue data, the position and the negative inverse slope of any Modified Wöhler curve can unambiguously be defined via the following linear laws [10, 12-14]: T