Issue 77

A. Sivtseva et alii, Fracture and Structural Integrity, 77 (2026) 138-172; DOI: 10.3221/IGF-ESIS.77.10

n n

lg

1

N

lg

− − E E E E 0

(

)

A

(

)

f

B

*

=

= − − 1 1

= + − 1

= B k

(13)

D

n

A B

,

1

,

,

R

2

) σ

E

B

− −

n n

1 1

(

−

R

1

max

lg

2

f

0

σ

B

U

1

where A, B are the linearly related model parameters; n 1 , n 2 are the values of n corresponding to the boundaries of the second stage of damage accumulation (0 < n 1 < n 2 < 1); k is the proportionality coefficient; R is the cycle ratio. For the residual strength, Stojković N., Folić R., Pasternak H. [5 8] used the model of Shaff J. R., Davidson B. D. [44, 45] and introduced the concept of strength reserve σ res :

) , B n

− )(1 ), B n

σ σ σ σ = − − ( R U U

σ σ σ σ σ = − = − (

(14)

res

R

U

max

max

max

and then, for three-stage dependencies, they proposed using a combination of power functions, called the normalized strength reserve model, which, when transformed, yields the expression:

σ σ σ σ − − R

σ σ − U R

(1 ) , B A n

− )(1 ) , B A n

B A

*

σ

=

= −

σ σ σ σ = + − (

=

= − −

(15)

D

1 (1 ) n

max

σ

, res n

R

U

max

max

σ σ −

U

U

max

max

Thus, the general form of the model and its derivatives is: ( ) ( ) ( ) ( ) ( ) ( − − − − ′ = − − = −  ′′ = − − − + − − 1 1 2 2 2 1 1 , 1 , 1 1 1 1 A A B B B A B B D n D AB n n D AB A B n n B

(16)

)

  

−

A

1

−

B

B

2

n

n

.

 

It should be noted that the model is only applicable when using the relative number of cycles and “normalized damage”. The constraints on the range of values of the function imply that A > 0 and B > 0. In this case, for A = 1, the model reduces to a simple power function, while for B = 1, it is a model of type (Eq. 11). Since A , B > 0, the condition for the first derivative to be positive is satisfied. Depending on the values of parameters A and B , the following options are possible (Tab. 5): ‒ when A < 1 and B ≥ 1, then D ′′ ≥ 0 and the function is applicable to describing two-stage dependencies with accelerated damage accumulation; ‒ when A > 1 and B ≤ 1, then D ′′ ≤ 0 and the function is applicable to describing two-stage dependencies with decelerated damage accumulation; ‒ when A < 1 and B < 1, then the function is suitable for describing three-stage “fast–slow–fast” dependencies; ‒ when A > 1 and B > 1, then the function is suitable for describing three-stage “slow–fast–slow” dependencies. Model of Yang J. N. and Du S. type. The authors [59] proposed the following equation for predicting the residual strength of polymer composites:

σ σ σ σ − − V V U C C U max

(

)

Y

σ σ V V

σ A N B range

(17)

= −

,

max

(

)

R

U

Y

where σ R is the residual strength normalized by β; σ max is the maximum fatigue stress normalized by β; V , A , B , C and Y are the model parameters; σ range is the stress range. This expression can be reduced to the following form: ( ) ( ) ( ) ( ) ( ) σ σ σ − − − − ′ = − = − − = − ′′ = − − + − 1 1 2 2 1 1 1 , 1 , 1 1 1 . C C B B B R U B C B B D An D ABC An n D ABC An n An BC B (18)

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