PSI - Issue 28

5

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

415

1

     

     

  

  













2

2 sin cos   

cos

sin cos        

c u u 

w w 

2  

j

i

j

i

i

j

2

R

0 0  

(10a)

i  u h 

d d   





1

     

     

  

  













2 sin        

3   sin

sin cos

c u u 

w w 

2  

j

i

j

i

i

j

2

R

 

0 0  

(10b)

h

d d   



i



2  

1

1

  

  

  

     

  













0 0  

(10c)

2   sin

2

cos

sin       

sin

w h c u u    

w w 

d d    

i

j

i

j

i

i

j

2

2

R

R

After substituting Eqs. (8a-c) in Eqs. (10a-c) and performing integrations result in

3  

2

4

2

  

  

(11a)

2

i  u c h 

u

u

w











, i xx

,

, i x

, i x

i





2

16

3

3

3

R

R

R

3  

4

2 3

2

2

c h

 

  

 

(11b)

u

w



, i xx   

,  i



, i x

,

i

i







2

2

16 3 

R

R

R

3  

1

1

1

  

  

(11c)

w c h  

3 w u R i



i 

 

, R R  2

, i x

i

2

8

Please note that the equations of motion obtained from peridynamics given in Eqs. (11a-c) have the same form as the classical equations of motion given in Eqs. (5a-c). By equating Eqs. (11a-c) and (5a-c) yields the following relationships

9 E h  

(12a)

c



3

and

1 3

(12b)

 

3. Numerical results In order to validate the current peridynamic shell membrane formulation, an isotropic cylindrical shell having 1m edge length, 0.1592 m radius and 0.01 m thickness is considered. The elastic modulus and Poisson’s ratio are specified as 200 GPa and 1/3, respectively. As shown in Fig. 1, the cylindrical shell geometry is mapped into a 2-Dimensional domain for numerical solution. For spatial discretization, a discretization size of 0.01m   is utilized. The horizon size size is chosen as 3    . The steady-state solution is obtained by using adaptive dynamic relaxation scheme presented in Kilic and Madenci (2010). A uni-axial loading condition in the x -direction is applied as a body load with an amount of 9 3 5 10 N/m  along a region of 0.04m at the top and bottom edges. Since the 2-Dimensional domain is used for the numerical solution of the cylindrical shell, an additional condition is enforced so that the material points at the left and right edges of the solution domain inside a region with a thickness of horizon size interact with each other.