Issue 46

M. Hack et alii, Frattura ed Integrità Strutturale, 46 (2018) 54-61; DOI: 10.3221/IGF-ESIS.46.06

the stiffness in longitudinal and transverse directions, whereas in unidirectional plies, the effect of matrix degradation on longitudinal behaviour can be neglected. However, the coupling between 22 D and 12 D remains mandatory and has been maintained. Finally, the deletion of this coupling imposes the addition of a propagation term in the formulation of 12 D , so that a pure shear load in a ply remains able to lead to its collapse. With these assumptions applied to the formulations taken from [1], the evolution laws for the damage variables become:

  

  

11 ( ) d d 



d





1,11 11    c exp c  

 



2

11

c d  

5,11 11 4,11   c

exp c

2,11

3,11 11 11

 

dN



11

3

   

   

  

   

c

11 ( ) 



  

  

d d

d

5,11

1,11 11    c exp c  

 

  

2

11

c d  

exp

c

2,11

3,11 11 11

11 4,11

dN

3





11

   

   

22 ( ) 



d d

d











3,22 22 22 c d   



f

2

2

22

1 D exp c     c



5,22 22 4,22   c

exp c





1,22

12

22

2,22

dN



f

2





D

1

12

22

3

    

    

   

   

c

22 ( ) 



  

  

d d

d





5,22



3,22 22 22 c d   

  

f

2

2

22



D exp c   



c

exp

c

1

(3)





1,22

12

22

2,22

22 4,22

dN

3



f

2





D

1

12

22

   

   

12 ( ) 



d d

d









2       exp c 1 c d

3,12 12 12 c d   



2

12



5,12 12 4,12   c

exp c





1,12

12

12

2,12

dN





2





d

2 1

12

12

   

   

12 ( ) 



d d

d









2       exp c 1 c d

3,12 12 12 c d   



2

12



5,12 12 4,12   c

exp c





1,12

12

12

2,12

dN





2





d

2 1

12

12

where , i jk c  give the connection to residual strength as the ratio between the effective stress and the (actual) ultimate strength of the material in the ij component, and defined as are the 15 fatigue material coefficients that must be identified, the fatigue failure indices ij

   

   

   

 

ij   ij

ij 

  

  

  

;





max

max

(4)

ij

cycle

ij

ij

cycle

ij

ij 

ij 

0  

0,  

ij 

 

where

,    



0



ij

ij

Accumulated permanent strain In addition to these damage evolution laws, the model takes into account the permanent strain which appears in the ply due to a cyclic shear loading. Some matrix debris formed by the shear stress is accumulated in the opening matrix cracks during tension stress [1], which leads to a non-reversible deformation of the ply. The 9 c parameter drives the fatigue permanent strain accumulation following the formulation:

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