Issue 77

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

(

)

(

)

−

−

A

+ + − 1 B Cn BC n n A B

1

1

An

1

A

−

n

1

′

= − 1

=

D

D

,

.

(26)

(

)

2

B

+

Cn

1

B

+

Cn

1

Like the previous two models, this model can be applied without introducing additional parameters only when using the relative value of the number of cycles as the argument of the function and “normalized damage”. The condition of equality of the initial damage to zero is automatically satisfied for this function when A , B > 0. In the case of С = 0, this model reduces to a simple power law dependence. A sufficient condition for the positivity of the damage accumulation rate is C > 0. It is noted, that, in general, combinations of parameters with C < 0 are possible that do not violate the condition of positivity of the damage function and its derivative. Since the second derivative of the damage function is too cumbersome for analysis, the possibility of describing both two stage and three-stage dependencies was confirmed by selecting parameters A, B and C . The typical form of the dependencies D(n) and D′ (n) is presented in Tab. 5. Wang Z., Song L., Lei J., Xu S., Qui Y. and Shen W. model . The authors [21] proposed using the following function to describe the dependencies of the residual Young’s modulus and residual strength on the relative number of cycles:

A

σ σ − −

σ σ U R

n

*

=

=

(27)

D

,

σ

(

)

A

+ − 1 n B n A

U

max

where A and B are the model parameters ( A ≠ 1 or B ≠ 1 condition is required). Derivatives of the damage function are:

(

)

(

)

( n n A

) ( − − + − 1

) ( A A B n n

)

−

A

2

−

A

2

−

− +

n

n

A

1

2 1

2 1

(

)

−

A

1

−

A

1

−

n

n

1

′ D AB =

′′ D AB =

,

.

(28)

(

)

(

)

2

3

(

)

(

)

A

A

+ − 1 n B n A

+ − 1 n B n A

This model requires using the relative value of the number of cycles and “normalized damage”. The function implies A > 0, the first derivative of the damage function is positive when B > 0. 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, then D ′′ (0) < 0, D ′′ ( n = 1) > 0, therefore, the function is applicable to describing three-stage dependencies. Exponential models of residual mechanical characteristics Adam T., Dickson R. F., Jones C.J., Reiter H., Harris B. and Kassapoglou C. model. The authors [65] proposed to approximate the data on residual strength by an expression of the form:

σ σ

− e , AN

− 1 e , An

σ σ = R U

= − = − 1 R

(29)

D

σ

U

where A is a parameter depending on the loading conditions. After this, in the work [66, 67], Kassapoglou C. proposed a strength degradation model of the type:

f N

) σ σ σ σ     −  max E U E − 

B

−

N

1

(

= − e AN C

σ σ σ = − U E R

σ

σ

+

or

,

(30)

R

E

A

where σ E is the endurance limit stress, A, B, C are model parameters that depend on loading conditions and fatigue resistance characteristics. Given that D ( N = 0) = 0, after transformations, models (29) and (30) can be reduced to the general form:

153