Issue 9

An. Carpinteri et alii, Frattura ed Integrità Strutturale, 9 (2009) 46 - 54; DOI: 10.3221/IGF-ESIS.09.05 48 with T = period of the loading cycle and   tW = weight function given by:   max ,1 1 )( )(     t H tW (2) where   H is the Heaviside function (   1  xH for 0  x ,   0  xH for 0  x ), and max ,1  is the maximum value (in the loading cycle) of the maximum principal stress 1  . The proposed weight function is such that no averaging procedure is actually required (this makes the implementation of the criterion rather simple), since the averaged principal stress axes coincide with the instantaneous principal directions corresponding to the time instant at which the maximum principal stress 1  achieves its maximum value in the loading cycle. Figure 1 : Principal stress directions 1 , 2 , 3 described through the Euler angles  ,  ,  . Critical plane and final fatigue fracture plane Fatigue crack propagation can be distinguished into two stages [22]: a first stage in which a crack nucleates along a shear slip plane (Stage 1, fatigue crack initiation plane or critical plane) and a second stage in which crack propagates in a plane normal to the direction of the maximum principal stress (Stage 2, final fatigue fracture plane). According to the present criterion, the normal to the estimated final fatigue fracture plane (Stage 2), which is the one observed post mortem at the macro level, is assumed to be coincident with the averaged direction 1ˆ of the maximum principal stress 1  [14]. On the other hand, the critical plane (Stage 1) is the verification material plane, where fatigue strength assessment is to be performed. The orientation of the critical plane has been proposed to be correlated with the averaged directions of the principal stress axes [15] and the empirical expression of the off angle  between the normal w to the critical plane (where w belongs to the averaged principal plane 3ˆ1ˆ ) and the averaged direction 1ˆ of the maximum principal stress 1  is given by:                     2 1, 1, 1 8 3 af af     (3) Equation (3) is valid for hard metals, which are characterised by values of the ratio 1, 1,     af af ranging from 3 1 to 1 . The off angle  is assumed to be equal to 0 for 1 1, 1,    af af   , whereas  is taken to be equal to 4/  for 3 1 1, 1,    af af   . Mean value and amplitude of normal stress and shear stress The critical plane  passing through a given point P in a solid and the attached orthogonal coordinate system Puvw are considered (Fig.2), where the u-axis belongs to the plane formed by the w-axis (normal to the critical plane) and the Z-