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

1720

5

Physics based traction seperation law

300

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

250

200

150

T n

100

50

0

0

2

4

6

8 10 12 14 16 18 20

n /h 0

δ

Fig. 4. Traction versus separatoin / initial height ratio for mode-I loading with σ y = 100MPa using the Minkowski inequality.

2.2. Mixed-mode loading

Now consider combined normal and tangential loading. Note that under shear loading pores become more crack like and elongate in the direction of shear Fleck and Hutchinson (1986). The previous analysis shows that as a result of ˙ δ n pore volume increases and therefore f increases d f = d δ n / h (1 − f ). Under shear loading the shape of the pores change (see Fig. (5a)). We consider a new shape where cylinders have modified radius and height (see Fig. (5b). Considering s hear deformation does not change the volume we have π a 2 h = π ( a + d δ t ) 2 ( h − dh ) = π a 2 h + 2 π ahd δ t − π a 2 dh resulting in dh / h = 2( d δ t / a ). Writing f as f = a 2 / l 2 gives d f = (2 a / l )( da / l ) = 2 f ( d δ t / l ). Then we have the following relations for combined normal and shear deformation.

d δ n h

d δ t l

(1 − f ) + 2 f

d f =

(7)

h a

d δ t

dh = d δ n − 2

Fig. 5. Geometry change under shear loading.

For mixed-mode loading the upper bound becomes

˙ δ 2

˙ δ 2 t 3

1

r 2 l 2 .

1 3 v 2 +

n 1 +

T n ˙ δ n + T t ˙ δ t ≤ σ y

dv

v =

where

(8)

f