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

434

3

shown in Eq. (2):

11 C E E E E E E G G G G G G                            1 11 22 2 22 33 2 33 12 2 12 13 2 13 23 2 23 1 2 12 13 1 2 13 23 1 2 23 , , , , , , , , C C C C C C C C 12

(2)

where index C denotes a current value of elastic constants, corresponding to current level of damage of the material. Eventually, constitutive relations for damaged material can be written as:

1

 

 

    

                  

0

0

0





2 31

2 21

E

E

E



1 11  

22

33

1

 

0

0

0

2 32





2 12

11                             22 33 12 13 23       

11                   22       33 12 13 23

E

E

E



11

2 22

33

1

2 13     2 23

0

0

0



E

E

E



11

22

2 33

(3)

1

0

0

0

0

0

2 12 G



1

0

0

0

0

0 1

2 13 G



    

0

0

0

0

0

2 23 G



A simple failure criterion based on commonly available values from standard unidirectional tests represents a condition for failure envelope:

, X Y

,

X

Y

S

11   

22   





12

C

T C

T

where X c – compression failure stress in fiber direction X t – tension failure stress in fiber direction Y c – compression failure stress in transversal direction Y t – tension failure stress in transversal direction S – in-plane shear failure stress

(4)

Next step is to determine the damage evolution low for parameters 1  and 2  . Let us assume that only parameter 2  is varied to keep a stress vector within the failure envelope. The stress vector response to material loading is shown schematically in Fig. 1. Once after stress el ij  reaches outer side of failure envelope, corresponding elastic constants are multiplied by damage parameter 2  in such a way to return stress vector back on envelope in consistency with Eq.(3), as shown in Fig.1. If it is not possible and value of 11  exceeds corresponding critical values X c or X t , the material is considered failed and 1  is assumed equal to zero 1 ( 0)   .