PSI - Issue 3

Nataly Vaysfeld et al. / Procedia Structural Integrity 3 (2017) 526–544 Author name / Structural Integrity Procedia 00 (2017) 000–000

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Thus, all values constituent in the formulas for the displacements (15) and (16) are found, from where it is not complicated to get the stress values inside the cylinders. 5. Finding the stress intensity factors (SIF) SIF values are extremely interesting when solving the problems for solids with cracks. For the proposed problem such SIF values it is important to calculate on the internal and external contours of a crack .     0 lim 2 ,        j j a j z j III r a K a r r d and     0 lim 2 , , 1, .         j j b j z j III r b K r b r d j N With regard to all the executed earlier changeof the variables, integration by parts and truncation of summands having the finite limits when 0   j r a and 0   j r b , one gets     1 1 1 2 1 0 1 1 1 1 lim 1 exp ln 2 2                                            j a j j j III j j d K d d R     1 1 1 2 2 1 0 0 1 1 1 1 lim 1 exp ln 2 2 1                                                 n j n j j j j n T d d d R     1 1 1 2 2 1 0 0 1 1 1 1 1 lim 1 exp ln 2 2 2 1                                              n j n j j j j j n T d R   1 2 1 1 1 exp ln . 2 1                            n j T d d d (24)     1 1 1 2 1 0 1 1 1 1 lim 1 exp ln 2 2                                              j b j j j III j j j d K d d R     1 1 1 2 2 1 0 0 1 1 1 1 lim 1 exp ln 2 2 1                                                   n j n j j j j j n T d d d R     1 1 1 2 2 1 0 0 1 1 1 1 1 lim 1 exp ln 2 2 2 1                                                n j n j j j j j j n T d R   1 2 1 1 1 exp ln . 2 1                            n j T d d d For the next investigation one needs to evaluate the limit values of the integrals     1 0 2 1 ln 1 1      n T t L x dt x t t and     1 1 2 1 ln 1 . 1      n T t d L x dt dx x t t (25)when 1 0    x and 1 0   x . As it shown at Appendix B, for all n the integrals   0 L x are bounded and when 1 0    x and when 1 0   x too. For the integrals   1 L x the next asymptotic formulas were found

   

1 2

2



 1 , 



0

   x

n



2

 

when

1 0    x

1 L x



  1 1  





n

n

1

n



1 2

2 1





        x  1 2  n

j

j

,

1

n

n





  3

2

 

j

0

j

2 ! j

