Issue34

Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01 3 - Weight V – 1 1 1 max 1min 1 0 k f k af k k k af a for a W for a                      – this weight was developed as a result of combining weights II and III, - Weight VI – 1 1 1 0 m k k af f k k af for a a W for a                           – in this proposal only those positions of main axes are taken for averaging, in which  1 (t) is higher than fatigue limit fraction, while their share is in exponential function dependent on Wöhler curve inclination. However, the selection of proper angles for averaging creates problems, and there are no physical guidelines, which angles should be averaged. The issue of averaging proper angles is discussed in the study [10], where direction cosines were made dependent on Euler angles. Matrix of direction cosines defined in this way is expressed in the following form cos cos cos sin sin cos cos sin sin cos cos sin sin cos cos cos sin sin cos sin cos cos sin sin sin cos sin sin cos                                               . (3) Nevertheless, some transformations are required in order to obtain values of Euler angles. The first step involves calculation of the quantity:   1 2 3 1 arccos 1 2 l m n      , 3 2 1 2sin m n u    , 1 3 2 2sin n l u    , 2 1 3 2sin l m u    . (4) Then, Euler - Rodriguez parameters are used: 1 sin 2 u    , 2 sin 2 u    , 3 sin 2 u    , cos 2    (5) to determine values of angles arctg arctg                    , 3 arcsin sin m          , arctg arctg                    . (6) Euler angles calculated in this way are averaged using the following relations:     1 1 ˆ L k k W k W      ,     1 1 ˆ L k k W k W      ,     1 1 ˆ L k k W k W      . (7) Then, Macha and Będkowski [11] developed variance method to determine critical plane position. In this method, the critical plane is considered to be the plane, for which the variance of equivalent stress reduced by selected criterion reaches maximum. The study [12] contains comparison of lives of steel specimens using variance method with damage accumulation method for criterion of maximum shear stress in the critical plane. According to this criterion, the equivalent stress  eq (t) takes the following form