PSI - Issue 79

D. Marhabi et al. / Procedia Structural Integrity 79 (2026) 34–52

52

2 2 2 2 m Rb m To 

2 2 2 1 m Rb    

  

    

       

r       R

 , we consider the following ratio :

2

  







q

( 2 2 ) m To   

W

Rb To 

2 E E

2

The hypothesis that the crack initiated energy * W

 Rb To is combined triaxial parameters dT a and dT m and non-damaging

* 2 Eq

1 E RdT Rb To dT Rb To a m    ( ( ), (

*

WRb To

elastic energy is such as:

 

), ) 

2

* 2 Eq

*          r R



1

So











1

( R dTa Rb To dT Rb To m   ( ), (

), ) 

2 2 (1 2 ) 2 m Rb    

q

So that the over-energy under rotating bending combined to torsion supplied on a cycle is:   * 2 2 2 * 2 2(1 ) (1 )(1 ) 2 Rb To Rb To Rb W m E r R q                    * 2 2 2 2(1 ) 1 * (1 ) 2 2 2 ( ( ), ( ), ) m Eq Rb W Rb To Rb To E E RdT Rb To dT Rb To a m q                 We propose an over energy introducing Experimental tensile stress. This allows us to write

2 *

2  

1



, Eq Te   (

)

Eq

Uniax

), ) 

( E E R dT

dT

(

),

(

Rb To 

Rb To 

a

m

2

2





To  

2

The experimental test of [10] suggested the endurance limit

(1 ) 

Rb





1

1

So that the over-energy criterion under rotating bending combined to torsion supplied on a cycle is:

    

    

    

    

2

*2

2

2

2 m

2









   

, Eq Te

Rb m To RdT dT

, ) 













where

(

,

2

2.07

a m

2

2

2

2









2*

Rb

To

Rb

Rb









1

1

1

1

, , 2.07) 1 R dTa dTm 

We consider the relationship by triaxiality: (

The following expression of fatigue criterion under combined rotating bending and torsion is :

2

2

* 2 Eq

2 2 2       , mRb Rb mTo To , 2   





, Eq Te

1

) 1 

0 2 



where E

(

2 To

2

2

E





0

, 1   This elliptical function depends on parameters from standard tests initiating the small crack in material and a particular threshold stress. , 1  , 1  , 1 Rb Rb Rb

, , 1 , 1 Rb To     , tension loading (

, ) , m Te Te   to