PSI - Issue 3

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

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F. Berto et al. / Structural Integrity Procedia 00 (2017) 000–000

of the mixed term it is possible to decouple the contributions of the loading modes, obtaining a solution of the system even in the crack case. Solving the system of equations, the values of the NSIFs can be determined:





2

4

, a FE MN W W 



  



2

, b FE

W W W W  



(7)

, a FE

, b FE

, a FE

, b FE

2

Q

K



1

4

M

, 2 a FE b FE W W QK  ,

K



(8)

2

1

Fig. 3. Control volumes in the new approach (a) and in the modified version of the new approach (b)

2.3. New approach based on the deviatoric SED In the present contribution a new method is proposed. It is based on the evaluation of the total and deviatoric SED averaged in a single control volume (circular sector) centred at the notch tip and characterized by a radius R (Fig. 3a). Two independent equations can be obtained: one linked to the total SED and the other to the deviatoric one. In this way it is possible to obtain a solution of the system even in the case of a crack. As previously, knowing the SED values ( W and dev W ), by means of a FE analysis, and defining the control radius ( R ), it is possible to obtain a system of two equations in two unknowns ( K 1 and K 2 ):

      

1 2 2   

  

2 K I

2

I

K

2 SK TK

2

W





 

1

1

2

2 

 2 1



 2 1



1

2

FE

2

1   E R

2  



1 

2 

R

(9)

I

I

 

  

2

2

1

K

K





1,

2,

dev

dev

2 S K T K 

2

W







1

2 

 2 1



 2 1



, dev FE

1

2

dev

dev

3 2 E 

2

1  

2  



1 

2 

R

R

Where I 1, dev and I 2, dev are the integrals of the angular stress functions related to the deviatoric strain energy density (Lazzarin and Zambardi, 2001), which depend on the notch opening angle, 2 α , and the Poisson's ratio ν . Solving the system of equations, the values of the NSIFs can be determined:

TW T W

, dev FE dev FE TS T S  

K



(10)

1

dev

dev

2

, dev FE dev W S K 

1

(11)

K



2

T

dev