PSI - Issue 2_B

So-Dam Lee et al. / Procedia Structural Integrity 2 (2016) 847–854 So-Dam LEE et al. / Structural Integrity Procedia 00 (2016) 000–000

851

5

3. Result

3.1. FE result of C* and J(0)

FE results of C * for four different L r values using Eq. (2) is shown in Table 1. Since steady state C * values are not supposed to be dependent on plastic constitutive equations, calculated FE C * showed within 1% difference for all assumed cases. In Table 2, FE results of J values when t =0 are summarized and shows dependency of FE J (0) values on plastic constitutive equations. Due to difference in plastic strain, J (0) values are the largest for Eq. (3) and the smallest for Eq. (4) for a given load compare to those of Eq. (2). Differences tends to increase as L r and m increase.

Table 1. FE C * values [MPa ∙ mm/h] m = n

Equation L r =0.3

L r =0.5

L r =0.8

L r =1.0

5

0.05 1.06 17.84

67.78

(2)

10

0.025 6.83 1201.99 13983.60

Table 2. FE J (0) values [ MPa∙mm ]. m Equation L r =0.3

L r =0.5

L r =0.8

L r =1.0

(2) (3) (4) (2) (3) (4)

2.00

6.03

20.23

42.43

5

1.96 (-2.00%) 1.97 (-1.70%)

5.54 (-8.15%) 5.59 (-7.29%)

15.25 (-24.61%) 28.51 (-32.81%) 16.93 (-16.31%) 36.91 (-13.02%)

1.97

5.67

17.91

42.30

10

1.96 (-0.38%) 1.97 (-0.22%)

5.56 (-1.86%) 5.59 (-1.44%)

15.61 (-12.84%) 31.57 (-25.37%)

16.52 (-7.76%)

37.92 (-10.34%)

3.2. Influence of plastic material properties on C(t) values

n

1

+

(

)

*

1

+

τ

C t

AC

( )

(7)

,

1 = −

=

φ

( ) 0

n

*

1

+

BJ

(

)

C

1

τ + −

φ

( ) * 0

J

t

(8)

; red t

τ =

=

t

C

red

In this section, results of FE C ( t ) values and redistribution time are given in normalized form. For the reference of normalization, FE C ( t ) values of Eq. (2) are used. Similarly, time is normalized by t red of Eq. (2) for consistency. The relaxation behavior of C ( t )/ C * suggested by Ainsworth and co-workers (Joch and Ainsworth, 1992) can be given by Eq. (7). Constants A and B are defined in Eq. (1) and (2). In Fig. 3, FE C ( t ) values with normalized time t = t / t red are compared with predictions of Eq. (7) using FE J (0) and C * for assumed cases and both generally show good agreements. The redistribution time t red can be calculated with FE J(0) and C* values in Eq. (8). In Fig. 4, normalized FE C ( t ) values using Eq. (3) are depicted and shows that calculated FE C ( t ) values of Eq. (3) are almost similar with those of Eq. (2) at long times. However, it shows differences at short times when t / t red <0.2. Figure 4c shows the normalized results at smaller scale and it shows that differences are limited to very small normalized times. In Fig. 5, normalized FE C ( t ) values using Eq. (4) are depicted and overall tendencies resembles those in Fig. 4 with smaller differences.