PSI - Issue 2_B

Aylin Ahadi et al. / Procedia Structural Integrity 2 (2016) 1343–1350 Ahadi, Hansson and Melin./ Structural Integrity Procedia 00 (2016) 000–000

1346

4

k k d L L dt    

0

        u u 

(2)

where the Lagrangian function is given by L T U   , with T and U being the total kinetic and the total potential energies respectively:

 

1 2 i i i i V  u u   





i      b u i i

T



U W V

V

,

(3)

i

i

i

i

i



1 1 ( ( , ,..., )

2 2 j W w  

with and ij w being a scalar micropotential, which is the energy in a single bond. The factor 1/2 is due to the fact that each endpoint of the bond “owns” only half of the energy in the bond. The equation of motion of a material point k reads 1 2 u u u 1 2 ( , ,..., )) V  u u u i ij ji j w  

j 1   

kj w w 1   

  

jk

, k b

k k  u

V

kj w w 



 





(4)



j

jk

u u

2

k   

k

  with radius δ through bonds.

Fig. 2. Kinematics of the PD theory. Each point x in the body interacts directly with the points in the sphere

The bond force f is introduced as

/ k w    f u and kj kj

/ k w    f u and the equation of motion for a jk jk

material point at x is



(5)

( ( , ) ( , ), t t   f u x u x x x x b x ) ( ) dv  ( , ) t   

( ) ( , ) t x u x  



 