PSI - Issue 28

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

414

4

1

1



  





     

  













1 j w c u u          i j

(4c)

2   sin

2

w  

cos

sin      

sin

0

w w 

V b



ij 



i

ij

j

i

ij

i

j

ij

ij

j

i

2

2

R

R

The peridynamic equations of motion given in Eqs. (4a-c) can be verified by comparing them against classical equations of motion for the shell membrane for the special case of the horizon size approaches zero, i.e. 0   . They can be written as       , , , , 2 2 1 1 2 2 2 1 u i i xx i x i x i i u E u w u b R R R                        (5a)

 1  





 

2

2

E

 1  

  

 

(5b)



u

, w b    i i





,  i







  

, i x

, i xx

i





2

2

R

R

R

2   

2 1

1

1





(5c)

w

w E  

, i x w u b R    i



i 

 





, R R  2

i

i

2

2   

1

where  is density, E is elastic modulus and  is Poisson’s ratio. In Eqs. (5a-c), “dot” symbol represents derivative with respect to time whereas “comma” symbol represents derivative with respect to space. For the horizon size converging to zero, the displacement components of the material point j can be expressed in terms of the displacement components of the material point i by using Taylor’s expansion up to the second order as             2 2 2 , , , , , 2 1 1 1 1 2! 2! j i i x j i i j i i xx j i i x j i j i i j i u u u x x u R u x x u R x x u R R R R                      (6a)             2 2 2 , , , , , 2 1 1 1 1 2! 2! j i i x j i i j i i xx j i i x j i j i i j i x x R x x R x x R R R R                             (6b)             2 2 2 , , , , , 2 1 1 1 1 2! 2! j i i x j i i j i i xx j i i x j i j i i j i w w w x x w R w x x w R x x w R R R R                      (6c) By using the relationships cos j i ij ij x x     (7a)             2 2 2 , , , , , 2 1 1 1 1 2! 2! j i i x j i i j i i xx j i i x j i j i i j i x x R x x R x x R R R R                             (7b) Eqs. (6a-c) can be rewritten as

1

1

1

1

(8a)

2

2

2 sin cos   

2   2 sin

cos  

sin  

cos

u u u  

u

u

u

u







ij 





, i x ij

,

, i xx ij

, i x ij 

,

j

i

ij

i

ij

ij

ij

ij

i

ij

ij





2

2!

2!

R

R

R

1

1

1

1

(8b)

2

2

2 , i x ij        sin cos ij ij ij

2    2 sin

cos               sin

cos



, i x ij

,

, i xx ij

,

j

i

ij

i

ij

ij

i

ij

ij





2

2!

2!

R

R

R

1

1

1

1

(8c)

2

2

2 sin cos   

2   2 sin

cos  

sin  

cos

w w w  

w

w

w

w







ij 





, i x ij

,

, i xx ij

, i x ij 

,

j

i

ij

i

ij

ij

ij

ij

i

ij

ij





2

2!

2!

R

R

R

The volume of the material point j for an incremental volume can be expressed as

(9)

j V h d d    

Then, the equations of motion of the peridynamic shell membrane formulation given in Eqs. (4a-c) can be written in integral form as