PSI - Issue 28

Bingquan Wang et al. / Procedia Structural Integrity 28 (2020) 482–490 Author name / Structural Integrity Procedia 00 (2019) 000–000

485

4

sin

(7d)

y u v    



Hence, the stretch s between two material points becomes     cos sin u u v v s         

(8) The peridynamic equation of motion with small displacement assumption can be written in longitudinal direction as







u u  

v v 

cos

sin





 

 

  

  ,

   



(9)

u t x 

cos

C

dV





 



V

where   C  is the bond constant. Eq. (9) can be rewritten in cylindrical coordinates as









u u  

v v 

cos

sin

2  





 

  

  

  ,

  C h 

0 0  

(10)

u t x 

cos

d d    







where h is the thickness of the geometry. While considering the waves propagating in positive x -direction and the transverse displacement of each material point is zero, i.e.     , , 0 v t v t    x x , the equation of motion in the longitudinal direction can be simplified as

2  

  ,

  C h u u      



(11)

u t x 

cos cos

d d     





0 0

Inserting the plane wave solution     , i k t u t Ue     x n x

(12) , where n is the unit vector describing the direction of the wave propagation, into the equation of motion in the longitudinal direction leads to ( x  n e )





2  





  C h

0 0 2         0 0 2 0 0

i k x x     

2    

2

1

cos

e

d d   



L

0 0 2  





 i k



  C h 

cos  

2

1

cos

e

d d   





(13)

 

  C h 









2

1 cos cos k  

i sin cos cos k   

d d   





 





 

  C h 





2

1 cos cos cos k     

d d   







Please note that 

 sin cos k   is an odd function and its integration leads to zero. Performing the integration in Eq.

(13) yields





2BesselJ 1,

k

  C h          2

  



(14)

L

k

