PSI - Issue 2_B

Tuncay YALÇINKAYA et al. / Procedia Structural Integrity 2 (2016) 1716–1723 Tuncay Yalc¸inkaya and Alan Cocks / Structural Integrity Procedia 00 (2016) 000–000

1719

4

Physics based traction seperation law

Physics based traction seperation law

300

f 0 =0.01 f 0 =0.04 f 0 =0.1 f 0 =0.2

h 0 =0.01 h 0 =0.1 h 0 =0.5 h 0 =1

250

250

200

200

150

150

T n

T n

100

100

50

50

0

0

δ n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

δ n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 3. Dependence of mode-I traction-separation law on initial height of voids and volume fraction with σ y = 100MPa and h 0 = 0 . 01 µ m (top) and f 0 = 0 . 01 (bottom) using the Minkowski inequality.

Here f = a 2 / l 2 is the area fraction of pores in the plane of the cavitated zone. In order to retain analytical ex pressions for the constitutive response of the cohesive, the integral in (4) can be estimated by using the Minkowski inequality �� ( f + g ) k dx � 1 k > �� f k dx � 1 k + �� g k dx � 1 k or Jensen’s inequality � ( f + g ) k dx < � f k dx + � g k dx (see e.g. Hardy et al. (1996)). Applying the Minkowski inequality and using (4) gives

1 2

1 √

1 f

) 2 �

T n ≈ σ y � (1 − f )

2 + (

(5)

ln

.

3

Applying Jensen’s inequality would retain the formal nature of the upper bound calculation (see Yalcinkaya and Cocks (2015)), but the Minkowski inequality generally provides a better estimate of the full integral and results in a continuous single equation for the yield function under multiaxial loading (see section 2.2) rather than the discontinu ous function when using Jensen’s inequality (see Yalcinkaya and Cocks (2015)). However, here we have indicated that formally the result is an approximation and no longer an upper bound, because the Minkowski inequality is the oppo site sense to that of the upper bound. Practically (5) provides a yield function, which can be calibrated by comparison with experimental behavior. Assume that f = f 0 when h = h 0 and consider the following relation again: d f / (1 − f ) = dh / h and integration gives ln([1 − f 0 ] / [1 − f ]) = ln( h / h 0 ) or simply h = h 0 (1 − f 0 ) / (1 − f ), and if δ n = h − h 0 this gives δ n = h 0 ( f − f 0 ) / (1 − f ) and f = ( δ n + h 0 f 0 ) / ( δ n + h 0 ). Then using (5) we get

1 2

( δ n + h 0 ) � 2

2   

( δ n + h 0 ) ( δ n + h 0 f 0 ) ��

� h 0 (1 − f 0 )

1 √

T n = σ y   

+ �

ln �

(6)

3

which is an alternative representation of the mode-I traction separation law and its dependence on the initial volume fraction of pores and height of the pore. The mode–I traction–separation relation is illustrated in Fig. (3) for di ff erent initial height of voids and volume fraction values and in Fig. (4) traction versus normalized separation / initial height ratio is presented.