PSI - Issue 18

Alexey N. Fedorenko et al. / Procedia Structural Integrity 18 (2019) 432–442 A.N. Fedorenko, B.N. Fedulov, E. V. Lomakin / Structural Integrity Procedia 00 (2019) 000–000

439

8

  

 

1 3 3 2  1 3 3 2 

3 2

  

 

 

 

2

,



A

A





 

11 1 0        



 

 

11

1111

11

1122

22

  

  

3 2

  

  

 

 

2

(9)

,



A

A





 

     



 

 

22

1122

11

2222

22

22

1 0





   

  

1212 A Q  ,



  

  

1 2

3 2





2      1 0 12

1212 A Q  ,

,





 



12

Q



1 2

 

 

 





where  , prime denotes the derivative with respect to parameter  and only parameter A 1212 depends on shear parameter Q . The coefficients A ijkl must be defined to guarantee positive-definiteness of the potential of Eq. (8). At first step of approximation a linear dependency from  may be supposed. The relation for 12  in Eq. (9) allows a significant reduction. For example, assuming independency of 1212 A on triaxiality and quadratic dependency on shear parameter Q , the equation for shear strain takes form derived by Fedorenko and Fedulov (2018): 2   2   2 1 1111 11 2222 22 1122 11 22 1212 A Q  12 2 ,   A A A              

1 1 2 G  

  

2      2 1 0 12 3

.

(10)



12     

12

Eq. 9 allows a combination with damage model assuming dependency of functions A ijkl from damage parameters in consistency with Eq. (3).

  

 

1 3 3 2 

3 2

  

 









2

1 11    ,

,   

,



A

A





 

11 1 0        



11

1111

1122

2 22

  

  

1 3 3 2 

3 2

  

  









2

(11)

2 11    ,

,   

,



A

A





 

     



22

1122

2222

2 22

22

1 0





   

  

, ,  

1212 A Q



  

  

1 2

3 2





2

2      1 0 12

, ,  

,

1212 A Q





 



12

2

Q



Fig. 7. The stress–strain diagrams for laminate composite material under conditions of tension at the angles 0°, 22.5° and 45° to the direction of the warp of the cloth for longitudinal strain and for transverse strain