PSI - Issue 79

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

39

equations near the small half axis and the large half axis of the ellipse (Fig. 1): 2 * 2 * 2 0, ( ) , b For Eq Eq Rb           



R  

(6)

  

2

*       a R

2



*

2











For

,

(

)

Eq

Rb



2

The equations (5), (6) and (1.f) are considered to define the function of variable coefficients (7 a):     * 2 2 * 2 0 B C A Eq Rb Eq Rb Rb Te Rb         

(7.a)

For the simplicity of the writing, we consider 4 

ARb B Rb

2

2 Te

2





) ( 







2[(2

)

(

)]







(7.b)

Rb

( , ) Eq Rb

, 1 

, 1 

Rb

( , ) Eq Te

2

2

2 Te

2

2











)] ( 



) 2 



C

[2[

(2

](

)





Rb



Rb

Rb

( , ) Eq Rb

, 1 

, 1 ( , ) Eq Rb 

( , ) Eq Te

Rb

From the equation (7) according to the over-energy we use the numeric database (8) choice the results of the fatigue tests in 10 6 cycles for the rotating bending realized on 30NCD16 steel by [24].

 



Rb

              



, m Rb

2 4

2 B B A C A    Rb Rb Rb Rb



 

MPa

427

*

2

Te





(8)

Eq



MPa

290

RB

, mTe





MPa

658

, 1 

Rb



MPa

560





, 1 

Te

The analysis of the over-energy allows the evaluation of the stress threshold and requires increased vigilance regarding the limit endurance of the chosen material. 5. ANALYZE OF THE NON-DAMAGING STRESS NEAR SMALL CRACK [2] Our proposal consists to study the polynomial model (7) and predict the threshold stress *  :     * ( ) * 2 2 * 2 Eq B C A Eq Rb Eq Rb Rb        (9) We also need the tensile loading values 427 MPa, 290 MPa respectively of the dynamic and the average stress when the material 30NCD16 steel is testing for initiating small crack and growth in middle-cycle regime (# 10 5 Cycles). The energy coefficients (7b) are calculated by loads at the endurance limit , 1 , 1 , Te Rb     and various stresses values on the experimental fatigue tests realized by [4] in rotating bending , ) ( , Rb mRb   . The threshold stress *  is obtained for the upper dynamic stress mode if , Rb m Rb    The objective of the following polynomial representation is to define the threshold stress σ *. For means of various rotating bending loads with average stress, we deduce the roots of the analytical over-energy (9) drawn by the beam curves