PSI - Issue 50

Kirill E. Kazakov et al. / Procedia Structural Integrity 50 (2023) 131–136 Kirill E. Kazakov / Structural Integrity Procedia 00 (2022) 000 – 000

133

3

2 dr d u r

1 K r r K r ( ) ( )  

 K r K r

  

  

  

  

( ) 2

( )

( )

( ) ~ ( ) 1 ~ ( )      r r

r u r

dr du r

(5)

0,







( )

r

and ~ ( ) ( ) r r 

in the plane stress case and

where ( ) ( )/[1  and ~() ()/[1 ()] r r r      K r E r  

 

() ()[1 ()]/[1 () 2 ()] 2 r r r K r E r       

2 r

( )]

in the plane strain case. Radial displacements of the inner and outer surfaces of the cylinder are known and do not depend on the angular coordinate:

(6)

( ) out sh in u r R r u u r r r    , ( ) in in

,

out R r u c

  

out

where sh R is the outer radius of the shaft,

c R is the inner radius of the coupling, in r and out r are the inner and outer

radii of the cylinder in the undeformed state. The problem is to find the displacement u ( r ) from equations (5), (6) and further determine the strains ( ) r r  , ( ) r   and stresses ( ) r r  , ( ) r   from equations (2) – (4). It should be noted that, for arbitrary functions E ( r ) and ν ( r ), the obtained differential equation (5) does not have an analytical solution. 3. Construction of an analytical solution for inhomogeneity of a special kind In the works Horgan and Chan (1999), Tutuncu and Ozturk (2001), Xiang et al. (2006), Shi et al. (2007), several special cases were considered where the construction of an analytical solution is possible. Analytical solutions were obtained for the cases where the Poisson's ratio is constant, and the dependence of the Young's modulus on the radial coordinate is described by linear, exponential, and power functions. Let us consider another case where the construction of an analytical solution is possible. Suppose that the Poisson's ratio is constant, and the Young's modulus varies logarithmically, that is

r

,

(7)

( ) r

0 ,  

( ) E r E 

ln



0

r

0

0  , 0 E , and 0 r are known parameters. In this case, differential equation (5) takes the form

where

ln( / ) ~ r r 

  

  

  

1      r

2 dr d u r 2

( ) 1

ln( / ) 1 r r

( )

( )

dr du r

2 r u r

1  

0,



(8)

0

0

~

~

where for the plane strain case. We reduce the resulting equation (8) to Kummer ’s differential equation (see, for example, Zwillinger (1997)): 0   for the plane stress case and  /(1 ) 0 0     

(9)

) ( )     xy x k xyx kyx ( ) ( 

( ) 0, 

1

2

where k 1 and k 2 are some constants. To do this, we will make the replacement 0, ( ) ( ), ln( / ), ( ) 2 1 0    xk rr ur fxfx k

(10)

where x is a new variable, k is an unknown constant ( k > 0 because x must increase with increasing r ), f 1 ( x ) and f 2 ( x ) are new unknown functions. Then equation (8) will take the form