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

1721

6

Fig. 6. Yield function representation of normal and tangential traction through Minkowski inequality.

Applying Minkowski inequality gives eventually the following quadratic yield function representation (see also Fig. 6)

1 2

    

    

3 T 2 t (1 − f ) 2

T 2 n (1 − f ) 2 + �

g =

− σ y = ¯ σ − σ y

(9)

2 +

1

1

f �

ln

√

3

Both Minkowski inequality and the Jensen’s inequality (see Yalcinkaya and Cocks (2015)) give the same functional form of the yield condition under pure mode II loading. The yield function of (9) can be combined with suitable hardening laws for the yield strength σ y to provide an incremental model for variable mixed-mode loading. Noting that,

∂ ¯ σ ∂ T t

∂ ¯ σ ∂ T n

˙ δ n = λ

˙ δ t = λ

and

(10)

and using

T n

3 T t (1 − f ) 2 ¯ σ

∂ ¯ σ ∂ T n

∂ ¯ σ ∂ T t

(11)

and

=

=

2   ¯ σ

1

1 √

  (1 − f ) 2 + �

f �

ln

3

The equivalent separation rate is written as follows

1 2

˙ δ 2

2    +

2

t (1 − f )

1

1 √

λ = ˙ δ e =    ˙ δ 2

n   

  

f �

(1 − f ) 2 + �

(12)

ln

3

3