PSI - Issue 23

Nikitin I.S. et al. / Procedia Structural Integrity 23 (2019) 119–124 Author name / Structural Integrity Procedia 00 (2019) 00 – 000

121 3

integration over  and  are determined by relations 2 1 0    and   2 0    . From the condition 2 1 0    it is obtained that the integration limits over  have the form:

2 max[0, ] cos Y ( ) 4 ) / (2 ) Y A B A B A A       The range of permissible values for  in the plane , A B has the form shown in Fig. 1, and represents the outer part of the shaded domain. For 2 B > 1, the integration limit 2 0 Y   and should be replaced by 0. The integration contour Г over  is a part of the circle 2 2 2 3 12 ( ) A S B S    located in the allowed (unshaded) domain in the plane , A B (Fig. 1). 2    , 2 Y   2 2 2 2 2 2 2 2 (

Fig. 1. The range of permissible values for  and the integration contour over  .

The condition   2 0   

can be written for a non-degenerate case 12 0 S  and 12 0 S  in the following form:

  2 



12 2 (  

 

S A A A 

B

S

)cos

2 sin / 

0









2

2

0

12

where 0 13 12 12 13 12 2( ) A S S S S S   is an additional combined parameter characterizing the process of loading a material particle together with independent derivatives 12 S и 13 S . This condition gives additional restrictions on the limits of integration over  . If 12 0 S  and 0 2 A  , then there are no acceptable values  for 2 2 0 Y Y   , 2 2 2 0 max[0, ] cos Y Y     for 2 2 2 0 Y Y Y     , 2 2 2 max[0, ] cos Y Y      for 2 2 0 Y Y   , where 2 2 0 0 / / ( ) Y B A A A   , 0 0 A A   . If 12 0 S  and 0 2 A  , then 2 2 2 max[0, ] cos Y Y      . If 12 0 S  and 0 2 A  , then 2 2 2 max[0, ] cos Y Y      for 2 2 0 Y Y   , 2 2 2 0 cos Y Y     for 2 2 2 0 Y Y Y     , there are no acceptable values  for 2 2 0 Y Y   . If 12 0 S  and 0 2 A  there are no acceptable values  . The integration ranges for degenerate cases 12 0 S  , 12 0 S  and 12 0 S  , 12 0 S  are calculated much simpler and are not written out here (Nikitin (2009)) .