PSI - Issue 79

46 D. Marhabi et al. / Procedia Structural Integrity 79 (2026) 34–52 The definition of the assumed critical plane sees the crack appear, is based on the maximization of a physical quantity. In practice, this involves sweeping the space with many vectors �⃗ to maximize a quantity that call ��⃗ . McDiarmid criterion [13] The chosen design is not directly related to the expression of the fatigue function as is done for the criteria of the critical plane type. The latter determine an indicator of damage and then search among all the possible plans for the one where this quantity is maximum.

3 2

   

   

1 2



2



1

m nnm



(23)





( 





(1

)

)

E

nnm

na

MD

2

R



The constants α and β are determined by two fatigue limits under tension and torsion

  

   .



1



1

  









and





1

1

3

  2  

2

1 2

R 

The criterion imposes two conditions to be respected 1 1 1 2     

1 2

nnm m



and

Dang Van 2 criterion [14] The author asserts that the basic mechanism for crack initiation is the shearing of the most unfavorably oriented crystallographic planes. The local shear stress acting on these planes is important. The influence of hydrostatic pressure P H (t) is predominant in the crack opening phenomenon. Dang Van has developed a microscopic macroscopic approach to calculate the stress in grains to the known stress at the macroscopic scale. ( ) ( ) ( ) max n pr H t P t E t      (24.a) 1 ( ) ( ) ( ) , ( ) ( ) , ( ) ( ) 2 ( ) ax pr Ia IIa IIa IIIa IIIa Ia t S t S t S t S t S t S t M t      (24.b) The two constants α and β of the criterion are expressed: 1 1 1 1 2 3( )                 (24.c)

The damage Indicator E n relating to the normal plane is expressed by : 2 max( ) DV n E E n  At the endurance limit of the material, we have: 2 1 DV E 

Robert criterion ([15], [16], [21], [24]) The author associates the shear stress on face with the normal stress, and not with the octahedral normal stress. However, the average and alternating normal stresses do not play the same role on the fatigue. Robert criterion is expressed by: ( ) ( ) ( ) max n a a m t t E t         (25.a) The material constants are given by the following expressions: they require three fatigue endurance limits.