Issue 77

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

Future scopes Based on the review, several promising directions for the further development of phenomenological models of residual mechanical characteristics can also be identified. The first direction is related to the introduction of stochasticity in the distribution of the material’s elastic and strength characteristics into the models. Such approaches have been proposed in works [29, 30, 39, 40, 44, 45, 47, 49–53, 55, 56, 59, 71], with the Weibull distribution law being the most frequently used. The necessity of this line of development is determined by the presence (in some cases significant) of scatter in the mechanical properties of polymer composites. The second direction concerns the application of the considered models to the problem of damage summation under variable loading conditions. This direction has been addressed in papers [28, 38, 78, 79, 80] (Tab. 11). The change in residual strength under multiple damage levels was considered in works [12, 36, 44, 45, 49, 50, 54, 66, 67, 84] (Tab. 12). The relevance of this direction is justified by the fact that fatigue damage accumulation rarely occurs under constant loading conditions. Moreover, changes in the mechanical characteristics themselves can lead to alterations in the stress-strain state within the cycle.

No

Authors

Model Damage accumulation under the variable amplitude fatigue loading

Owen M. J., Howe R. J. 1972 [38]

σ   ∑ =∑  −  2 i i D Bn An

1

1

     

     

− A B 1 i

i

   

   

   

   

i B A

B

Ai

   

   

   

   

−

i

i

1

+ N N

− N N + 1 i

Wu F., Yao W.X. 2010 [28]

2

− , 1

− − i

i

i i

i

1, 2

− −

= − −

− , 1 N N = i i

D n

( ) 1 1

,

1 1

f

i

N

N

i

f

f

−

i

i

1

− (1 )

w A R B

     

    

σ

log

− max max 1 i i

w B A R

w

σ

log

+ N N

  

  

− −

A n

− A B 1 i

sin( sin(

)cos(

) )

− (1 )

w

− , 1

i

i i

=

− , 1 N N = i i

D n

− − i

−

( )

,

,

i

i

1 1

1

Shiri S., Yazdani M., Pourgol-Mohammad M. 2015 [78]

f

i

N

− − − i A A n B 1 1 1 )cos( i i

i

3

−

f

i

1

i

 =

σ σ

+ B B A A A = + ,

B

min

−

−

w

i

i

w

i

i

1

1

max

   

   

   

   

N

N N

A

cos( ) B

A

B

sin

sin

cos(

)

−

i

i

1

Gao J., Zhu P., Yuan Y., Wu Z., Xu R. 2022 [79] Liu H., Zhang Z., Jia H., Liu Y., Leng J. 2020 [80]

−

−

i

i

i

i

1

1

N

f

f

=

=

D n

( D n

( )

,

)

−

i

i

1

4

− − 1 i

i

i

i

1

   

   

   

   

A

A

        i i f N N

   

   

−

i

i

1

N N

A B

)cos A B

sin( )cos

sin(

−

i

1

−

−

i

i

i

i

1

1

f

−

i

1

1

     

     

− A B 1 i

    

    

i

   

   

   

   

i B A

− B A 1 i

   

   

   

   

i

+ N N

− N N + 1 i

π

i

5

− , 1

− − i

i

i i

i

1, 2

− −

= − ( ) 1 sin 1

−

− , 1 N N = i i

D n

,

1 1

i

f

N

N

2

i

f

f

−

i

i

1

Table 11: Damage accumulation in the considered models Model Residual strength models for several stress levels

No

Authors

σ σ σ σ = − − ( U U ( ) R n

= − −

σ

σ σ σ (

) , n

n

)

Broutman L. J., Sahu S. 1972 [36]

+ R n n (

R

U

max 1

)

max 2

1 1

1

2 1 2

1

2

1

= − −

− − ( U

σ

σ σ σ ( U U

σ σ

n

n

)

)

+ R n n (

)

max 1

max 2

2 1 2

1

2

1



      

  

  

−

j

1

A

∑

j

σ σ U R

N

Shaff J. R., Davidson B. D. 1997 [44, 45]

 −

A

   

+ 

j

N N

i

  

  

j

    

∑

2

eff

j

=

i

1

=

σ

σ σ σ ( U U

= − −

  

N

N

N

)

,

j

R

i

eff

j

max

σ σ −

N

j

j

=

i

1

f

U

max

j

j

168