PSI - Issue 28

Sabrina Vantadori et al. / Procedia Structural Integrity 28 (2020) 1055–1061

1057

3

Author name / Structural Integrity Procedia 00 (2019) 000–000

1 2 3 , ,    instantaneous principal stresses , 1 af  

fully-reversed shear stress fatigue strength

2. The Carpinteri et al. criterion The main steps of the criterion employed in the case of multiaxial constant amplitude cyclic loadings on metals and metallic alloys are described hereafter (Carpinteri et al. (2017) and Vantadori et al. (2020)). Firstly, the critical plane has to be determined. Let us consider a generic material point P and a fixed XYZ coordinate system with its origin in P (Fig.1). At a given time instant t , the principal stress directions 1, 2 and 3 can be defined through the principal Euler angles , ,    .

Y'

Z

 w 

1

X

P

3 



Y

Y'

2

Fig. 1. PXYZ reference frame and ˆ ˆ ˆ P123 averaged principal stress frame at point P . The normal to the critical plane w is also shown. The averaged principal stress directions 1ˆ , 2ˆ and 3ˆ are deduced by averaging the instantaneous values of the above Euler angles on the cyclic loading period T :

0 ˆ 1 ( ) ( ) T W    

0 ˆ 1 ( ) ( ) T W    

0 ˆ 1 ( ) ( ) T W    

t W t dt

t W t dt

t W t dt

(1)

being   W T the weight function expressed by:

0, 1,

1 t       ( ) ( ) t

 

1,max

( )

W t

 

(2)

1

1,max

  1 t  is the maximum principal stress (   1 t

  t t      ) at a given time instant t , and 1,max  is the   2 3

where

  1 t  during a period T . According to the weight function in Eq.(2), the averaged

maximum value achieved by

  1 t  achieves its maximum value

principal stress axes coincide with the instantaneous ones at the time instant when

during T . The unit vector w normal to the critical plane is obtained from a rotation  of the 1ˆ -direction in the averaged principal plane 1ˆ 3ˆ (Fig.1). Such an off-angle  is given by: