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

435

4

Fig. 1. Case of transverse tension. A scheme of return

22 t Y   due to the stiffness modification, loading of unidirectional specimen.

A complete formulation of damage parameters modification is presented below:

1       2  

el   

1,

,

0

if X

X

otherwise





11 1 12 22 13 23 33 c t      2 2 2 , , , , 2 2

(5)





,

min

22 2  – solution of equation 22 el

where

t Y   , if

22 el t Y   ,

22 2  – solution of equation 22 el

c Y   , if

22 el c Y   ,

22  

1

2

otherwise; 12 2 12 / el S    if 13 / el S    if 23 / el S    if 13 2 23 2

12 el S   , S   , S   , 13 el 23 el

12 2 1   otherwise; 1   otherwise; 1   otherwise 13 2 23

2

33 2  – solution of equation 33 el 33 2  – solution of equation 33 el

t Y   , if c Y   , if

22 el t Y   , c Y   , 22 el

33 1   otherwise 1   otherwise 2 33

2

and el ij  – stress tensor components, obtained by Eq. (3) before update of damage parameters. For the formulation of Eq. (5) in case of 22 el  reaches failure envelope, an equation   22 22 2 el t Y   

has a cubic

polynomial form and can always be resolved as follows:      2 22 13 33 1 13 22 33 12 22 23 12 1 11 11 2 c E E E Y E               

   22 2   

3



  



   22 2 





2





2 E E E Y  11 11 22 33

2      13 1 E

2   

(6)

1 12       12 1 c Y

E E E Y 



11 33 23 22

11 33 23 1 c

c

 

2 22 11 22 22 2   

0

E E

11 22 E E Y









c

The solution of Eq.(6) shall be a minimum positive real root less than one. If these requirements are not satisfied then 22 2 1.   In order to obtain load drop in corresponding test curve, the dependencies for   2 c Y  ,   2 t Y  and   2 S  can be used. However, to avoid the need of complex experiential data analysis and use the minimum available data, we assume that material has no load drop stage in loading diagrams, as shown in Fig. 2. The assumption of constant stress after arrive to failure envelope is close to Zinoviev et al (1998) model.