PSI - Issue 33

4

Author name / Structural Integrity Procedia 00 (2019) 000–000

A. Sapora et al. / Procedia Structural Integrity 33 (2021) 456–464

459

r d dr 

r r    

(3)

0





Upon substitution of Eqs. (2a, b) into equilibrium Eq. (3), a fourth-order ordinary differential equation for r u is obtained (Chen et al. 2018). Its general solution is easily derived as

B

r  

r  

( ) r u r A r

C I     

1             D K c c

(4)

1

r

where c c l        ,  is the Poisson ratio, , , , A B C D are integration constants, whereas I 1 and K 1 are the first-order modified Bessel functions of first and second kind, respectively. In order to determine the integration constants in Eq. (6), four boundary conditions (two classical boundary conditions and two extra boundary conditions) are required: r r R p     , 0 r r    , 2 2 / 0 r R d u dr   , and 2 2 / 0 r d u dr   . Then, the integration constants are calculated as follows: 2 / (1 ) g

     

g         

1 c K R l 

2

2

R L    pc

0 A  ,

,

0 C  ,

(5)

1  

B pR 

D



   

2

RL



R

1 R R L RK l K l l                   0 R g g g

whereas

and K 0 and K 1 are the zero and first-order modified Bessel functions of second

kind. Inserting Eqs. (4) and (5) into Eq. (2b) yields:

      

g         

2 l K R l 1 g

2

2           g R l  

   

g   r       l

g   r       l

2

1

1

R

(6)

1   p

2

p

K

K







   

1

0



2

RL

r

L r

l

R

g

At the boundary of the hole ( r R  ), the circumferential stress attains its maximum value

2

p

(7)

r R      p

1 K R l                   0 K R l g g g R l

1

  

Equation (7) thus provides the stress concentration factor   ( r=R ) / p , which will be used in Section 4 in order to calculate the dimensionless failure pressure.