PSI - Issue 35

Toros Arda Akşen et al. / Procedia Structural Integrity 35 (2022) 82 – 90 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

85

4

In the equation above, a 1-9 are the parameters related to the material anisotropy. These parameters can be determined from the yield stress and r value directionalities as well as the biaxial data obtained from the uniaxial and biaxial tensile tests. The five parameters a 1-5 can be calculated using Eq. (3).

4. . 1 r +

a r

1

4.

r

1 1 a = ,

,

,

,

(3)

a

= −

5 90

a

=

1 2 a a a a  = − + + + 4 4 ( 1 ) (

)

a

a

= −

0

4

5

4 90

3

5



2

1

r

+

90

b

0

a 6 and a 8 parameters cannot be calculated through an explicit equation and an optimization method is required to calibrate these two parameters. Within the scope of this work, these two parameters were calculated by using the least square optimization method. To optimize these parameters, minimization of the error between the numerical and experiment predictions of r values and yield stress ratios were considered for the interval angles. Required r values and yield stress ratios of 15°, 30°, 60° and 75° were calculated by regarding the arithmetic mean of the experimental data given in the Table 1. Main function to be optimized is given as follows (Şener et al. (2021)) .

2 1 . [ .( i H w r = =  ( ) i 1 1 2

( ) i

( ) i

2

( ) i

( ) i

( ) i

2 ) ]

)

.(

r

w

(4)

−

+



−



,

,exp

2

,

,exp

prediction

prediction









Here, w 1 and w 2 are the weight functions regulates the effect of the r value and yield stress ratio directionalities. 15° and 75° were selected as interval angles in the optimization. Besides, these two anisotropy parameters should satisfy the following equations to ensure the convexity.

0

8 5 9 6 . a a a  

0

6 .

,

(5)

a  

a a

6

1 9

a7 can be calculated by using Eq. (6).

2

2

(6)

(

) ( − − +

)

a

a a

=

7

6 8

4   4

(1 ). r +

45 45

b

Regarding the convexity and the positivity measures, a 9 can be determined by Eq. (7).

4 ( 2 ) .

r

45

4 45

 =

, where,

(7)

4

1 ( 1 ) b  =

B

0

a

B + 

9

1

1

r

+

45

For the convexity requirement, r 45 should satisfy the following inequality.

1 2

3 2

b 

b 

b 

b 

(8)

4 ) 8] ( .(2. + +

4 ) )[(2.

4 ) 8] +

4

[5.(2.

(2.

)

−









1

45   r

−

45

45

45

45

b 

2

4 ) 8] +

[9.(2.



45

Isotropic hardening rule was assumed, and associated flow rule was adopted in this work. Associated flow rule can be expressed as Eq. (9).

df

(9)

.   =

d d

p

d



ij