PSI - Issue 13

Liviu Marșavina et al. / Procedia Structural Integrity 13 (2018) 1867 – 1872 Author name / Structural Integrity Procedia 00 (2018) 000–000

1869

3

with  the shear modulus,  the Poisson ratio, r the distance from the crack tip to the corresponding node and u i , v i , w i are the nodal displacements of point i , measured on x, y, z axis. For the case of plain stress  is replaced by   Most accurate results could be obtained using quarter point crack tip elements, Fig.2. b.

y

y

a

b

c

b

b d

r

x

x

a

a

Crack

Crack

e

Fig. 2. (a) triangular elements; (b) quarter point crack tip elements.

The SIF's expressions according with Shih et al. (1976) and Tracey (1977) are:

















 2r 1 4 v v v v e c b d 

          2r 1 4 u u u u e c b d

       

(2)

K

, K

I

II

The use of displacement correlation technique is simply and allow the separation of different mode of fracture. However, to obtain accurate results a highly refine mesh at the crack tip is necessary and care in selecting the nodes from crack singularity zone. Another alternative is to compute an apparent stress intensity factor K AP for a series of points approaching the crack tip, to interpolate the results and then extrapolate at the crack tip for r  0. The processing of data is based on the displacement field near crack tip for a mixed mode load:

2 r         4 + K 2 r 4 - K II II

r

2 



2 



 

   

 

I

(2 +3) sin 

2 +cos 3 2 +sin 3

4 v = K 4 u = K

(2 - 1)cos 

2 - cos 3

2  

(3)

r

2 



2 



 

 

I

(2 - 3)cos 

(2 +1) sin 

2 - sin 3

2  

with (r,  ) the polar coordinates,  the shear modulus,  the Poisson ratio and   for plane stress, respectively  for plane strain. In order to practically apply this technique the displacements on a certain direction  , along a radial line for different distances r (Fig 3.a) are selected and the apparent SIF's are calculated:

 2 2 



    

    

  ( 2 1)cos

 2 sin 3 2 cos 3



v u

  

   

K

r 4 2

 

I

  ( 2 1) sin



(4)

 2



    

    

( 2 3 ) sin

2 sin 3

  



v u

  

   

K

r 4 2

 

II

 2



  ( 2 3 )cos

2 cos 3



Then an interpolation is performed and the value of SIF's is obtained by extrapolation at the crack tip (r  0), Fig.3.b.