PSI - Issue 21

Tuncay Yalçinkaya et al. / Procedia Structural Integrity 21 (2019) 52–60

55

4

Yalc¸inkaya et al. / Structural Integrity Procedia 00 (2019) 000–000

T n for Jensen’s inequality. And, substituting to (8) gives, T n = σ y [ ( h 0 (1 − f 0 ) ( δ n + h 0 ) ) 2 + ( 1 √ 3 ln (

2 ]

1 2

( δ n + h 0 ) ( δ n + h 0 f 0 ) ))

(10)

T n for Minkowski inequality. Figure 2 shows the variation of T n with δ n for Jensen’s and Minkowski inequality respectively.

Physics based traction seperation law - Jensens f 0 =0.01 f 0 =0.04

Physics based traction seperation law - Jensens h 0 =0.01

400

400

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

300

300

f 0 =0.1 f 0 =0.2

200

200

T n

T n

100

100

0

0

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

n

n

Physics based traction seperation law - Minkowski f 0 =0.01 f 0 =0.04

Physics based traction seperation law - Minkowski h 0 =0.01

300

300

250

250

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

f 0 =0.1 f 0 =0.2

200

200

150

150

T n

T n

100

100

50

50

0

0

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

n

n

Fig. 2. Dependence of Mode-I traction-separation law on initial volume fraction and height of voids for σ y = 100 MPa and h 0 = 0 . 1 µ m (left) and f 0 = 0 . 01 (right), by using Jensen’s inequality (top) and Minkowaski inequality (bottom).

2.2. Traction-separation relation for mixed mode loading

Fig. 3. Geometry change under shear loading.

Shear loading elongates the pores in the direction of shear and causes them to be more like a crack (Fleck and Hutchinson (1986))(see Fig. 3(left)). Assuming constant volume for shear deformation we can write π (2 a ) 2 h = π (2 a + d δ t ) 2 ( h − dh ) = π 4 a 2 h + 4 π ahd δ t − 4 π a 2 dh , or dh / h = 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 ). For pure mode-II e ff ective strain rate can be written as, ˙ ε e = √ ˙ γ 2 / 3 where ˙ γ = ˙ δ t / h . Upper bound theorem for mode-II loading is, π l 2 T t ˙ δ t ≤ ∫ l a ˙ ε e σ y 2 π rhdr (11)