PSI - Issue 28

Erkan Oterkus et al. / Procedia Structural Integrity 28 (2020) 411–417 Author name / Structural Integrity Procedia 00 (2019) 000–000

413

3

introduced and the cylindrical domain is mapped into a 2-Dimensional domain. In the 2-Dimensional domain y -axis corresponds to tangential direction. Since the geometry is a cylinder, there should be a continuity of interactions between material points located at the left and right edges of the 2-Dimensional domain (blue regions). Each material point has three degrees of freedom; u ,  and w which correspond to displacement components in axial, tangential and radial directions. To obtain the equations of motion of peridynamic shell membrane formulation, Euler-Lagrange equations are utilized as

i i d L L dt u u      d L L dt      i d L L dt w w      i i i

(1a)

0

 

(1b)

0

 

 

(1c)

0

 

where the Lagrangian, L , is defined as L T U  

(2)

In Eq. (2), the total kinetic energy, T and total potential energy, U of the system can be expressed as

1 2

M 





2    2    w

2

T

V u

(3a)





i

i

i

i

1

i



2

     

     

1

  

  













2   sin

cos

sin      

c u u 

w w 



j

i

ij

j

i

ij

i

j

ij

ij

i 

1 2

2

R

M





1 1 j  i  

u V V b u b b w V      w

U

(3b)



j i

i i

i

i

i

i

i

2

ij 

where M is the total number of points in the solution domain, i  is the number of material points inside the horizon of the material point i , R is the radius of the cylinder, c is the bond constant, V is the volume of the material point, ij  and ij  are the length and orientation of the bond between material points i and j , respectively. In Eq. (3b), u i b , i b  and w i b represent body load components in axial, tangential and radial directions, respectively. By substituting Eqs. (3a,b) in Eqs. (2) and (1a-c) yields the equations of motion of the peridynamic shell membrane formulation as       2 2 1 cos sin cos sin cos j i ij j i ij ij i j ij ij ij c u u w w                   

    

    

2

R







i   u 

u  

0

(4a)

V b



j

i

ij 

1

j



1

     

     

  

  













2 sin       

3   sin

sin cos

c u u 

w w 



j

i

ij

ij

j

i

ij

i

j

ij

ij

2

R



i   

0

  

(4b)

V b



j

i

ij 

1

j

