Issue 77

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

σ σ σ σ = − − ( A A A A R U U max

) , n

1985 [54]

1

4

Hashin Z.

A is an empirical parameter

  

   

  

  

A

σ

σ σ

A

= − = −  − − 1 1 1 R 1

D

n

max

σ

A

σ

U

U

σ σ σ = − A A

max B N ( A

−

1),

R

U

1

Sendeckyj G. P.

1991 [55]

A, B are the two dimensionless functions that do not depend on σ

   

   

A A

5

 

  

σ

σ σ

= − = − − −  1 1 ( 1) R B N 1

D

max

σ

σ

U

U

N

σ σ σ σ = − − ( С С С C U U max R

)

,

N

f

σ σ − С C

σ σ С С

= −

N

,

U

max / V C

R

U

    

   

   

   

  

  

σ

B

1

 − −

exp

1 1

U

A, V are the parameters that depends on the applied stress; B, C are the experimentally determined constants

σ

A

Whitworth H. A.

2000 [56]

max

6

1

       

      

С

 

 

С

σ

N

 −  max 1

С U

σ

σ σ

= − = − − 1 1 1 R

D

σ

    

    

   

   

/ V C

  

  

σ

B

1

U

 − − 1 1

exp

U

σ

A

max

Table 4: Power law models of the type of Hahn H. T., Kim R. Y.

1

A < 1 , A ≠ 0, B < 0 or B > 1

B < 0 or B > 1

D

D′

A < 1, A ≠ 0, 0< B < 1

0 < B < 1

0

0

1

0

1

n

n

Figure 5: The typical dependencies D(n) and D′(n) of Hahn H. T., Kim R. Y. type power law model. Tang H. C., Nguyen T., Chuang T., Chin J., Lesko J. and Wu H. F. model. The authors [57] proposed a general model of stiffness degradation, approximating the initial and final sections of the damage dependence on the number of cycles in the form:

= + 1 − E dN D D 1 E dD A A E

= −

D

0 1 , R

.

2

(12)

E

(

)

B

B E

1

2

E

Here, A 1 , A 2 , B 1 , B 2 are model parameters selected experimentally. The authors propose using the first term of the sum to describe the initial section of the stiffness degradation diagram, and the second term for the final section. It is noted that this differential equation cannot be solved explicitly. Therefore, a simplified model consisting of one of the two terms can be considered, assuming A 1 = 0 or A 2 = 0. It can be shown that when A 2 = 0, this model reduces to the simple power law (Eq. 9), and when A 1 = 0, this model reduces to the general form (Eq. 11). Wu F., Yao W.X. and Stojković N., Folić R., Pasternak H. model. Wu F. and Yao W.X. [28] proposed using the following combination of power functions to approximate the residual Young's modulus data:

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