Issue 77

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

(

)

′ D AB =−

′′ D A B =− 2

1 e , An

e , An

e . An

(31)

= −

D B

Here, A and B are model parameters (the conditions A ≠ 0, B > 0, B (1 – e A ) < 1 must be satisfied). These parameters are selected experimentally and depend on the loading conditions. The model can be used with both damage and “normalized damage”, as well as with both absolute and relative values of the number of cycles. The D ′ ( n ) function is positive for A < 0, while the second derivative of the damage function is always negative, and the model is applicable to describing two-stage dependencies with decelerated damage accumulation (Tab. 6). Post N. L., Bausano J., Case S. W. and Lesko J. J. model . The authors [29] proposed using a combination of exponential and linear functions to describe the processes of stiffness degradation in the form:

S

S

(

)

(

)

− − A A Cn D e , Bn

− = − = − + 1 e Bn R A 1

= − + 1

(32)

Cn

,

R

S

E

E

0

0

where A , B , and C are fitting coefficients whose values depend on the loading conditions. Within this model, the damage function and its derivatives will have the form: ( ) − − − ′ ′′ = − + = + =− 2 1 e , e , e . Bn Bn Bn D A Cn D AB C D AB (33) It is noted that this model can be represented as a function of both the relative and absolute number of cycles. For A = 0 or B = 0, the model is linear; for C = 0, the model takes the form (31). The condition D (0) = 0 is satisfied for any values of parameters A, B , and C ; the condition D (1) ≤ 1 is satisfied for A (1 – e – B ) + C ≤ 1; accordingly, both damage and “normalized damage” can be used. The function D ′ ( n ) is positive if AB + C > 0 for B ≥ 0, and ABe – B + C > 0 for B < 0. Depending on the value of parameter A , the following options are possible (Tab. 6): ‒ when A > 0, then D ′′ < 0 and the function is applicable to describing two-stage dependencies with decelerated damage accumulation; ‒ when A < 0, then D ′′ > 0 and the function is applicable for describing two-stage dependencies with accelerated damage accumulation. No D ( n ) D′ ( n ) No D ( n ) D′ ( n ) Kassapouglou C. [66, 67] Philippidis T. P., Passipoularidis V. A. [11]

1

1

A < 0, B > 0, B (1 – e A )<1

A >0, B

A >0, B 0

A < 0, B > 0, B (1 – e A )<1

A >0, B

1

3

D

D

D′

D′

A >0, B 0

0

0

0

1

0

1

0

1

0

1

n

n

n

n

Post N. L., Bausano J., Case S. W., Lesko J. J. [29]

1

A >0

A > 0

2

D

D′

A < 0

A < 0

0

0

1

0

1

n

n

Table 6: The typical dependencies D(n) and D′(n) of exponential models

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