PSI - Issue 3

F. Berto et al. / Procedia Structural Integrity 3 (2017) 126–134 F. Berto et al. / Structu al Integrity Procedia 00 (2017) 000–000

128

3

      

1 2 2 1 2 2     

     

2 K I

2

I

K

2 C K D K

2

W





 

1

1

2

2 

 2 1



 2 1



, a FE

1

2

a

a

2

1   E R

2  



1 

2 

R

a

a

(3)

2 K I

2

I

K

2 C K D K

2

W





 

1

1

2

2 

 2 1



 2 1



, b FE

1

2

b

b

2

1   E R 

2  



1 

2 

R

b

b

where ܧ is the Young’s modulus of the material while I 1 and I 2 are the integrals of the angular stress functions (Lazzarin and Zambardi, 2001), which depend on the notch opening angle, 2 α = 2 π - 2 γ , and the Poisson's ratio ν . This method cannot be applied to a crack subjected to mixed mode loading, since an indeterminate system of equations would be obtained. Solving the system of equations, the values of the NSIFs can be determined: Fig. 2. Control volumes in the Lazzarin et al. approach (a) and in the Treifi et al. approach (b)

, a b FE DW D W D C D C   a b b a

, b a FE

K



(4)

1

2

, a FE W C K a

D 

1

(5)

K



2

a

2.2. Treifi et al. approach The second method has been proposed by Treifi et al. (Treifi and Oyadiji, 2013) and it is based on the evaluation of the averaged SED on two different control volumes (semi-circular sectors with a central angle equal to γ ) centred at the notch tip and characterized by a radius R (Fig. 2b). Known the SED values ( a W and b W ) by means of a FE analysis, and defined the control radius ( R ), it is possible to obtain a system of two equations in two unknowns ( K 1 and K 2 ):

      

2

I

I

I

     

     

2

2

1

K

K

K K

1,

2,

12,

s

s

s

2 MK NK QK K    2

W







1

2 

1 2  

 2 1



 2 1



, a FE

1

2

1 2

2

2

(

)    

1 2  

E

R

1  

2  



1 

2 

R

R

1

2

(6)

2

I

I

I

2

2

1

K

K

K K

1,

2,

12,

s

s

s

2 MK NK QK K    2

W







1

2 

1 2  

 2 1



 2 1



, b FE

1

2

1 2

2

2

(

)    

1 2  

E

R

1  

2  



1 

2 

R

R

1

2

where I 1, s , I 2, s and I 12, s are the integrals of the angular stress functions (Treifi and Oyadiji, 2013), which depend on the notch opening angle, 2 α , the angle defined by the semi-circular sector, γ , and the Poisson's ratio ν . In this case also the contribution of the mixed term ( K 1 . K 2 ሻ is present because the integration for the strain energy evaluation is not performed on a control volume symmetrical with respect to the notch bisector line (Fig. 2b). Due to the presence