PSI - Issue 13

Jan Klusák et al. / Procedia Structural Integrity 13 (2018) 1261–1266 Jan Klusák & Ond ř ej Krepl/ Structural Integrity Procedia 00 (2018) 000–000

1264

4

 

 

1

 

 

2

2

2





(6)

  

   



m  

1  

 

r

k

k

2

1 4

r m 

m rrm m

m rrm m



8

m

Let’s substitute for each stress component its stress series form of n terms and integrate the equation over specific distance d . By doing that, the final form of the formula for the mean value of the SEDF is: 2 1 1 1 1 1 ( ) 8 n n k l m k l klm m k l k l k H d U                  (7) in which the  k 1,  l 1 are the ratios between GSIFs defined:

H H

H H

, k

l

 

 

k

l

1

1

1

1

and the augmented shape functions in the following form:

2

2

2



( ) 

( )( 

( ) 

( ))( 

1) 4 ( ) for r km f    

 



k l 

f

f

k

f

f

k

2

1) (

km rrkm m 



km

rrkm

m

  

 

U

( )  

( ) lm rrkm f 

( ))( 

( ) 

( ) 



 



f

( ) km rrlm f

f

k

f

f

2(   

1) 2(

klm









m

km

lm

 

1) 8 ( ) f 

( ) 

 

k l 

f

( ) rrkm rrlm f

k

f

( ))(

for

m  The crack initiation angle  0 is found as a minimum of mean value of the SEDF. Mathematically, the function minimum can be found when: 2 2 0, 0 m m          (8) By substitution of Eq. (7) into Eq. (8) we obtain equation by which we find the crack initiation angle  0 : 1 1 1 1 ( ) 0 n n k l klm k l k l k l k U d                   (9) We assume that the crack initiation mechanism in the case of sharp and bi-material notches is identical to the mechanism of crack propagation in homogeneous media. The employment of equation (7) for mean value of the SEDF together with equation (4) for the critical value of the SEDF leads to formula for determination of critical value of GSIF: r km r lm   Note that all the critical values H 1 c, 1 , H 1 c, 2 and H 1c , interface should be evaluated for calculated corresponding angles of crack initiation  0,1 ,  0,1 and  0,interface respectively. Once the critical fracture toughness values are known, in order to assess stability, the generalized stability condition as stated in equation (2) is used. 4. Numerical example Three point bending specimen with bi-material notch is modeled with geometry as shown the Fig. 2. The geometric parameters are L = 76.2 mm, h = 17.8 mm and thickness b = 12.7 mm. The notch opening angle 2α = 120° and the notch depth to specimen height ratio is a/h = 0.2. The specimen is loaded with F = 100 N. The material part 1 consists of epoxy and the material part 2 of aluminum with material properties given in the Table 1. The problem is examined by the use of multi-parameter SEDF criterion with specific distance chosen to d = 1 mm. The global minimum occurs in epoxy (see Fig. 3), whereas solution by two singular terms gives crack initiation angle of 186.8° and solution by 1c, H K  m Ic, 1 1 1 1 k l k     2 ( )  m m n n k l        k l k l   klm k d U  (10)