PSI - Issue 79

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

44

Figure 6: Representation of the empirical functions F and R The representation of the functions (13) for ߚ ൌ 2.07 are analysed under the software. Note that the triaxiality function of the multi-axial load R(dTa, dTm, β ) varies slightly with respect to the uniaxial degree function F(dTa (Rb), β ). The ratio of the previous functions (12b), (13b) is lower than the unity. The hypothesis that the crack initiated energy * W  Rb To is combined parameters varies to dT a and dT m . 1 * * ( ( ), ( ), ) Te Rb To Rb To Rb To W W R dT dT a m      (14) The non-damaging elastic energy (1.f.2) introduced in (14) is such as:

2

*



Te

,

1

Eq

(15)

*



W

Rb To 

), ) 

( R dT

dT

(

),

(

Rb To 

Rb To 

E

a

m

The multiaxial over-energy associated by unit surface of the specimen on normal section is: 1 0 ( ( ), ( ), ) Te Rb T Rb To Rb To R dT dT a m         We propose an over energy to introduce relation (1.f.3) in (16). This allows us to write

(16)

2 *

2  

1

 

, Eq Te   (

)

(17)

Eq

), ) 

Rb To

( E E R dT

dT

(

),

(

Rb To 

Rb To 

a

m

The Multiaxial Fatigue Criterion including Shear Stress The fatigue criterion is defining a parameter function for the stress tensor path and allows determining if the structure is going to fail or not. The fatigue criterion can be represented by the following synthetic shape:



,  

, 1    , To

, E 0 is an indicator damage parameter.

,

,

,...)



(





F

E

,

To

, 1

, Rb m To

Rb

0

, m Rb

Over-Energy Criterion Under Rotating Bending and Torsion For the alternate rotating bending combined to torsion on cylindrical smooth test, the stresses are:   sin ) , ( , .sin Rb and t mTo To r m Rb R r t R            

(18)

The multiaxial energy of the material under rotating bending combined to torsion is: