PSI - Issue 61

Toros Arda Akşen et al. / Procedia Structural Integrity 61 (2024) 268 – 276 Toros Arda Akşen, Bora Şener, Emre Esener, Ümit Kocabıçak, Mehmet Fırat / Structural Integrity Procedia 00 (2019) 000 – 000 3

270

2.1. Fourth-degree complete polynomial yield criterion (HomPol4) A yield function should produce a convex yield locus for the uniqueness of the flow direction for each loading path. Furthermore, the plastic flow should be independent of the hydrostatic pressure. In this study, the HomPol4 criterion was incorporated into the FE approach to depict the anisotropic features of the aluminum sheet, and the equivalent stress based on this criterion is expressed in Eq. (1) (Soare et al. (2008)).

4 a a   + + + + + + + = + 3 3 4 4 2 2 a a 4 ( ) a a a a a

2 2 yy xx

2 y

(1)



1

2

3

4

1

6

7

8

9

eqv

xx

yy

xx

yy xx

yy

xy

xx

xx yy

y

x

y

Where, the yield function is given in Eq. (2).

0 (2) σ 0 refers to as the yield stress. The parameters a 1-9 should be adjusted. The four parameters a 2-5 are adjusted based upon the Eq. (3), while the a 1 is 1. eqv   = − f

4

r

(3)

a

0

=−

2

1

r

+

0

4 1

a r

(4)

a

5 90

=−

4

r

+

90

1

(5)

a

=

5

4 90



1  = − + + + 2 4 4 b 1 ( ) ( a a a a

(6)

)

a

3

5

An optimization step is required so as to adjust the a 6 and a 8 parameters. For this optimization process, the anisotropic features (directionalities of Lankford coefficients and the yield stresses) of distinct intermediate angles are necessary. These features can be pairs of 15°-75°, 25°-75° or 30°-60°, separately. In the current study, the parameters corresponding to pairs of 15°-75° were picked out and calculated by the arithmetic mean rule (Ak ş en et al. (2023)). The objective function is given in Eq. (7).

pr

pr

( ) ( )    

( ) ( ) r r  

n

H w = 

2 − + 1)

2 1) )

(7)

1 ( (

(

w

−

m

m

2

exp

exp

1

m

=

m

m

Besides, the parameters a 6 and a 8 should also obey the following inequalities.

0

6 a a a  

0

6 1 9 6 a aa   ;

(8)

8

5 9

Finally, the parameters a 7 and a 9 can be found using Eq. (9).

2 ( )

4

r

4 1 ( ) 0 

45

2

2



;

(9)

a

45

=

+

(

) ( − − +

)

a

a a

=

9

7

6

8

1

r

4

4 b

+



(1 ) +

45 45 r  

b

45

The correlation between the incremental plastic strain and the yield potential is constructed regarding the associated flow rule expressed in Eq. (10).

df

(10)

p d d  =



ij

d



ij