PSI - Issue 71

Ritesh Kumar et al. / Procedia Structural Integrity 71 (2025) 364–371

367

1



*



=

r r E E    −

   +

θ and are two appropriate Poisson ratio. , , and all four material properties are related as r r r E E      = r 

(15)

(16)

The strain-displacement relations for the axisymmetric deformation are considered as, , r r r du u dr r    = = Strain compatibility relation between radial and tangential strain is given by ( ) r d r dr    =

(17)

(18) It is noticed that Poisson’s ratio varies slightly in most of the engineering applications so it can be taken as constant, = = , Eq.14 and 15 can be rewritten as: * r 1 r r r E E        = − + (19) * θ 1 r E E          = − + (20) Multiplying Eq. (20) by r and differentiating concerning r gives * θ 1 r r r r r E E          = − + (21) ( ) ( ) * θ r r r d d d d r r dr dr E dr E dr              = − +         (22) Simplifying the above Eq. (22) to get

 

  

  

  

'

' rE E E E  −

2

rE

d

d





2

1 + − 

r

r

+









2

dr

E dr

r





(23)

  

  

'

rE

(

)

2 3     − ' 2 4 r r r

2 *' − − − r

*  

*

'

3 =− + − 

E



θ

θ

r

r

r



E



For the accuracy, computational cost, and ease of the implementation of the solution of the equation collocation method has been used that can be collaborated with other numerical techniques to improve the efficiency and accuracy. The above differential eq. (23) standard equation of second order, which can be solved by numerous techniques. The above equation can be written as, 2

2 d M N P Q dr dr    + + = d

(24)

where,

2 M r =

'  =  −    ' 1 rE E   

N

r

  

  

rE E E E  −

P



=



(25)

r



  

  

'

rE

(

)

2 3 r     − ' 2 4 2 *' r r

*  

*

'

3 =− + − 

Q

r

E

− − −



θ

θ

r

r

r



E



Free-Free boundary condition, For free-free BCs, the displacement is unrestricted at the inner and outer radius of the disk and therefore the radial stresses