PSI - Issue 41

Christos F. Markides et al. / Procedia Structural Integrity 41 (2022) 351–360 Christos F. Markides et al. / Structural Integrity Procedia 00 (2019) 000 – 000

356

6

    

         

   

       

  z Z z Z

  z Z z Z

  z Z z Z



n



  z

j

j

log

(

) log log

(  

)

iP

 F iP

F iP







m

m

j

j

j

j

2 (1 )   

1

j



m

j

j

         

  

Z

Z

  

  

1

Z

Z

n

2 (   z

j

j

)

(  

)

iP

 F iP

F iP







m

m

m

j

j

j

j

2 z Z 

2

2 z Z 

2

2 (1 ) 1 2 (1 )      

 z Z z Z 

1

j



m

m

j

j

    

   

   

2 R Z z R Z z 2   j

2 R Z z R Z z 2   j

2 R Z z

 

 n

(

) log

(  

) log

log

 F iP

F iP

iP





m

j

j

j

j

m

2

m Z z

R

1

j



j

j

2         m iP R

   

2 R Z Z z Z   m m

2 R Z Z R Z z 2  

1

1

1

 z Z

(7)









m

m

m m









2

2

2 R Z z z 

2 R Z z z 

2 (1 )   

2 R Z z 

m

m



m

m

   

   

   

2

2 R Z Z 

 z Z

2 R Z Z R Z z z R Z z 2 1  z Z    j j

1

 n

j

j

j

j

2

( ) F iP R 















j

j

2

2

2 R Z z z R Z z 2 





1

j



j

j

j

j

  

   

  

2 R Z Z z Z  

2 R Z Z R Z z 2   j

 z Z

1

1

j

j

j

j

j

2

(  

) F iP R







 











j

j

2

2

2 R Z z z 

2 R Z z z 

2 R Z z 

  

j

j

j

j

2 R iP  

    

  

   

Z

Z

  

  

 Z

 Z

2 (1 )    

n



j

j

2 R F iP  2 ( 

)

(  

)

F iP







m

m

  

m

j

j

j

j

2 R Z z R Z z 2

4 R Z z 

2 2

4

2 2

z

R

Z z



1

j



j

m

m

j

 

  

1

1

    In turn, stresses and displacements are calculated everywhere on the region occupied by the FBD, in terms of φ ( z ) and ψ ( z ), through the well-known formulae (prime denotes the derivative):   2 4 ( ), 2 2 ( ) ( ) e , 2 ( ) ( ) ( ) ( ) i r r r z i z z z u iv z z z z                                (8) It is to be mentioned here that the solution exhibits singularities at the points z = Z j . In this context, stresses and dis placements at these points on the flat edge of the FBD should only be calculated for j z Z  (e.g., for 0.99 j z Z  ). In addition, it is to be clearly emphasized that as the forces P j , F j are essentially applied to infinitesimal holes (containing the points Z j ), the stresses depend on the material the FBD is made of (obviously diversifying from plane strain to generalized plane stress conditions). 3. Numerical analysis For comparison and validation purposes, the present problem was also studied numerically by means of the Finite Element Method with the aid of the commercially available software ANSYS. The geometry of the flattened disc (in three dimensions) and its material properties were exactly the ones presented in sub-section 2.1. As far as it concerns the material properties of the stamp those of steel were adopted (namely, a modulus of elasticity E =210 GPa and a Poisson‟s ratio ν =0.3, were assigned to the stamps). The interfaces between the stamps and the disc were properly taken into account and a coefficient of friction equal to 0.7 was adopted. The lower nodes of the lower stamp were rigidly clamped while a vertical downwards displacement, equal to 5 mm (corresponding to the overall load adopted in the analytic solution) was applied to the upper nodes of the upper stamp, as it seen in Fig.8. The convergence of the model was studied by considering the equivalent stress developed at the point K (on the flat edge of the central cross section of the FBD, see Fig.8). Based on the results of this analysis, a numerical model consisting of approximately 68,000 elements was chosen. The mesh adopted is shown in Fig.8, while the convergence analysis is shown in Fig.9.  1 2 (   ) F iP Z F iP Z (   ) 2 (1 )            n m m m j j j j j j j iP Z Z z