Issue 77

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

D ( n )

D′ ( n )

No

Model

1

0< α <1,

Beta distribution based model [81]

1

D

D′

α >1, 0< β <1

0

0

1

0

1

n

n

1

A >0, 0 < B ≤ 1

A > 0, B > 1

Model based on the Weibull distribution [81]

A > 0, B > 1

2

D

D′

A >0, 0 < B ≤ 1

0

0

1

0

1

n

n

1

A >0, 0 < B ≤ 1

Model based on the Weibull distribution with a linear section [82]

A > 0, B > 1

A > 0, B > 1

3

D

D′

A >0, 0 < B ≤ 1

0

n 0 n 0 +∆ n

n 0 n 0 +∆ n

0

1

0

1

n

n

Table 9: The typical dependencies D(n) and D′(n) of models based on cumulative distribution functions Model based on the Weibull distribution. Based on the use of the Weibull cumulative distribution function, the authors [81] developed a model:

σ σ

E E

1

1

(

)

(

)

( ln 1 A n = − = − − R 1

)

( ln 1 A n = − = − − R 1

)

(61)

D

D

or

,

B

B

σ

E

U

0

where A and B are parameters which values depend on the loading conditions. Within this model, the damage function and its derivatives have the form:

A

1

1

1

(

)

(

)

−

1

( ln 1 = − −

)

( ln 1 D A n = − −

)

′

D

n

,

,

B

B

−

B

n

1

(62)

A

) n B  1 1 2

1



(

)

−

2

(

)

( ln 1 ′′ = − −

)

D

n

 − − −    1 ln 1 . n

B

(

B

−

1

This model can only be used with a relative number of cycles and only with “normalized damage”, since D → ∞ as n → 1. The conditions for the damage function to be positive and its value to be zero at n = 0 imply that A > 0, B > 0. This same condition is sufficient for the first derivative of the damage function to be positive. Depending on the value of parameter B , the following options are possible (Tab. 9): ‒ when B ≤ 1, then D ′′ ≥ 0 and the function is applicable to describing two-stage dependencies with accelerated damage accumulation;

163