Issue 77

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

C/N f 2 < A ,

B > 0, A > B

C > A/N f 2

+ ln(1 + N f ) – – 1/ A ) + ln(1 + N f ) – – 1/ A

α < 1, β < 1 α < 1, β > 1 α > 1, β < 1

1 < C < 2

0 < B < 1

0 < A < 1,

0 < B < 0.5

0 < A < 0.5,

0 < B < 1 + B < 0 or B ≥ 1 +

B > 1

B > 1

+

+

+

+

+

+

+

+

+

+*

+*

3-stage

C ≤ A/N f 2 C/N f 2 ≥ A

or A ≥ 1, B = 1

0 < B ≤ 0.5

2-stage, D ′′ ≤ 0

A < B

C ≥ 2

A ≥ 1,

0 < B < 1,

A ≥ 0.5,

–

–

+

+

+

+

+

+

+

+*

+*

A = 1, B ≥ 1

Descriptive capability

0 < A ≤ 0.5,

2-stage, D ′′ ≥ 0

B ≤ 1

B ≤ 1

B ≥ 1

0 < A < 1,

–

–

–

B < 0

+

+

+

+

+

+

+

B ≥ 0.5, or

+*

– / +

– / +

– / +

– / +

– / +

– / +

– / +

– / +

+ / +

+ / +

+ / +

normalized damage

Damage /

– / +

+ / –

+ / –

+ / –

– / +

– / +

– / +

– / +

– / +

– / +

– / +

number

/relative

of cycles

Absolute

A ≠ 0, A < 1/ln(1 + N f ) B ≠ 0, AB > 0

B – A ln( C – 1) ≤ 1

– 1)sin( An )sin( Bn A ) +

0 ≤ n 0 < 1 – ∆ n ,

+ cos( An )cos( Bn A ) > 0

A > 0, B > 0, C > 0,

A ln( N f ) + B ( N f – 1) + + C ln( N f ) ≤ 1

– π /2 < B < π /2,

– π /2 < B < π /2,

0 < A < π ,

β > 0, β ≠ 1

0 < A < π ,

( A – B ) < π /2

α > 0, α ≠ 1,

A > 0, B > 0

A > 0, B > 0

A > 0, B > 0

A > 0, B > 0

Constraints for parameters A > 0, C > 1,

( Bn A –1 –

A > 0, B > 0,

0 < ∆ n < 1

parameters 3

2

4

3

2

2

2

2

2

2

4

Number of

Damage function Model name

Table 10: Classification of the phenomenological models of residual mechanical properties

Yuan Y., Wu Z., Xu R.

Liu H., Zhang Z., Jia H., Liu Y., Leng J.

Reiter H., Harris B. Yao W.X.,

Himmel N.

Based on beta distribution

Based on Weibull distribution

Based on Weibull distribution

Ramakrishnan V., Jayaraman N.

Gao J., Zhu P.,

Adam T., Dickson R.F., Jones C.J.,

Whitworth H.A.

Wang C., Zhang J.

with a linear section

Logarithmic models

Trigonometric models

Models based on the cumulative distribution functions

Model class

18

19

20

21

22

23

24

25

26

27

28

No.

*The descriptive capability is proved by parameters variation **The simplified version of the model is analyzed

167