PSI - Issue 7

Stanislav Žák et al. / Procedia Structural Integrity 7 (2017) 254 – 261 Stanislav Žák et al. / Structural Integrity Procedia 00 (201 7 ) 000 – 000

258

5

Angle α is obviously a function of position along the precrack front. However, the maximal kink angle α m can be evaluated from the angle β and precrack characteristic dimensions: ( ) 1 m m tan tan 2 d a β α − ⋅ = ⋅ ∆       . (2)

Fig. 3. Simplified model of tortuous crack geometry

Let us now assume the stress tensor T σ components around the crack front (Anderson (1995)) for the remote mode II loading (smooth long crack, plane strain conditions, Poisson’s ratio ν ) and normalized to external loading and overall crack geometry:

  

  

2   α    

3             cos 2 2 α α ⋅

sin

2 cos +

,

σ

= −

⋅

xx

3                   cos cos 2 2 2 α α α ⋅ ⋅

sin

,

σ

=

yy

α       

(3)

2 sin , 2 ν

σ

= − ⋅ ⋅

zz

  

2   α    

3             sin 2 2 α α ⋅

cos

1 sin

,

xy σ σ σ =

⋅ −

 

0.

= =

xz

yz

To obtain the tensor T σ

* associated with the kinked and twisted facets of the tortuous crack front, the transformation

of the tensor T σ to the new coordinate system, related to elements of the inclined facet, leads to:

T σ ∗ = ⋅ ⋅ T L T L , σ

(4)

where L is matrix of direction cosines for double rotation of tensor coordinate system:

( ) ( ) α α

( ) α

cos sin

sin

0 0

−

1 0 0 cos 0 sin

0

( ) α

( ) ( ) β β

( ) β

x z L L L = ⋅

sin

cos

.

(5)

=

−

⋅

( ) β

0

0 1

cos

From equations (3) - (5) the normalized stress intensity factors can be derived as components of transformed tensor T σ * :

∗

T

I k K k K k K II II III / / /

,

≈

2,2

σ

∗

T

(6)

≈

,

II

1,2

σ

∗

T

.

≈

II

2,3

σ