Issue 55

A. Yankin et alii, Frattura ed Integrità Strutturale, 55 (2021) 327-335; DOI: 10.3221/IGF-ESIS.55.25

In the article [37], the authors considered multiaxial fatigue models and compared them based on the results of biaxial non proportional tests with a constant component of the shear and normal loading axes. Among the models, the most accurate is the modified Sines++ model. This model allows to take into account the beneficial effect of the mean compressive axial stresses, the higher mean stress effect in the axial direction compared with the torsion case and different slopes of the S-N curves under tension-compression and torsion. Simplistically, the model can be represented as follows:

2

2

) ( +

)

(

+ +  CI DI

A I

B I

1

(1)

2

a

2

m

1

m

1

a

2

2

2

(

) (

) (

)

(

)

2

2

2

1

=

  −

+ −

 

+ −

 

+

   + +

I

6

(2)

2

a

11

a

22

a

22

a

33

a

11

a

33

a

12

a

23

a

13

a

6

2

2

2

(

) (

) (

)

(

)

2

2

2

1

=

  −

+ −

 

+ −

 

+

   + +

I

6

(3)

2

m

11

m

22

m

22

m

33

m

11

m

33

m

12

m

23

m

13

m

6

   = + + 1 11 22 33 m m m m I

(4)

   = + + 1 11 22 33 a a a a I

(5)

where I 2a and I 2m are the amplitude and the mean value of the second invariant of the stress deviator tensor, I 1m and I 1a are the amplitude and the mean value of the first invariant of the stress tensor. The model parameters A , B , C and D were determined as follows:

1

1

1 '(2 ) bo N

  1 1 3 u

;

;

;

= 1



=

−

=

= −

B

D

A

(6)

C

u

b

bo





 3 '(2 ) N

'(2 ) N

f

eq

f

eq

u

f

eq

   = N N 

  = eq N N N ;

(7)





where N eq is the equivalent fatigue life, N σ and N τ are values of the predicted fatigue life (normal and shear axis), σ u is the ultimate tensile strength, τ u is the shear strength, τ ' f is the shear fatigue strength coefficient, b 0 is the shear fatigue strength exponent, σ ' f is the fatigue strength coefficient, b is the fatigue strength exponent, υ σ and υ τ are load frequencies for normal and shear axis. Discovering the number of cycles to failure using the Sines ++ model is an iterative process. In this regard, a program was developed in Python software: https://github.com/yanicen1/multiaxial-fatigue-modified-model-Sines-. The prediction results are shown in Tab. 2 and Fig. 3. In addition, the standard error was calculated using the formula:

n

2

(

)

(

)



log

N N

pr

ex

=

i

1

=

SE

(8)

−

n

2

where n is a number of tests. According to the results of the study, only 1 experimental point out of 28 (or 3.6%) lies outside the ±2-factor error and not a single point outside the ±3-factor error. The standard error was 0.175 or ±1.5 factors (±50 %). It should be noted that although the model describes the experimental data quite well, it does not describe the change in the number of cycles when changing the values of the phase shift angle between loading modes, i.e. the predicted number of cycles remains constant (see Tab. 2, loading No. 5-8 and 16-23). However, for the 2024 alloy, this dependence is not essential, which allows the

332