PSI - Issue 21

Tuncay Yalçinkaya et al. / Procedia Structural Integrity 21 (2019) 52–60 Yalc¸inkaya et al. / Structural Integrity Procedia 00 (2019) 000–000

56

5

Substituting and applying the change of variables given in (4), T t ≤ σ y ∫ 1 f √ ( 1 3 ) d ν or T t ≤ σ y √ 3 Noting that f = f 0 when δ t = 0 and taking integral of d f for pure mode-II case we get, T t = σ y √ 3 (1 − ( √ f 0 + δ t l ) 2 ) (1 − f )

(12)

(13)

Traction separation law for uniform shear

Traction separation law for uniform shear

60

60

f 0 =0.01 f 0 =0.04

l=1 m l=2 m l=3 m l=4 m

50

50

f 0 =0.1 f 0 =0.2

40

40

30

30

20 T t [MPa]

20 T t [MPa]

10

10

0 0.5 1 1.5 2 2.5 3 3.5 4 t [ m] 0

0

0

0.2

0.4

0.6

0.8

1

t [ m]

Fig. 4. Mode-II traction-separation equation for σ y = 100 MPa and l = 1 µ m with changing f 0 (left), and f 0 = 0 . 01 (right) with changing l (right).

Remember for mode-I loading, dh = d δ n and d f = d δ n / h (1 − f ). Superposing, for combined normal and shear deformation we get,

(1 − f ) + 2 √ f

d δ t l

d δ n h

d f =

h a

d δ t

(14)

dh = d δ n −

Upper bound theorem for mixed-mode gives, T n ˙ δ n + T t ˙ δ t ≤ σ y ∫ 1 f √ ˙ δ 2 n ( 1 + 1 3 ν 2 ) + ˙ δ 2 t 3 Applying Minkowski inequality results in the yield function, g = [ T 2 n (1 − f ) 2 + ( 1 √ 3 ln 1 f ) 2 + 3 T 2 t (1 − f ) 2 ] And, applying Jensen inequality gives for a specified T t / T n ,

r 2 l 2

d ν where

(15)

ν =

1 2

(16)

− σ y = σ − σ y

σ y [ ( (1 − f ) +

¯˙ δ √ 3 ]

1 f ) + (1 − f )

1 √

3 ln

˙ δ t ˙ δ n

T n ≤

(17)

where ˙ δ =

( 1 + T t T n

¯˙ δ )

R.H.S is minimized by ¯˙ δ = 0 or ¯˙ δ = ∞ which gives,

1 √ 3

1 f

]

(18)

T n ≤ σ y [(1 − f ) +

ln

σ y √ 3

T t ≤ (19) respectively. Notice that letting T t = 0 in (16) would give the same result for T n as previous section (8), and (18) is already the same as before (5). And, by letting T n = 0 in (16), we get the same form as (19) for pure mode-II loading. (1 − f )