PSI - Issue 28

3

Zhenghao Yang et al. / Procedia Structural Integrity 28 (2020) 464–471 Author name / Structural Integrity Procedia 00 (2019) 000–000

466





(3a)

xx E Ez

 

 

xx

x



G w x           

(3b)



xz

where E and G are elastic and shear moduli, respectively. The strain energy of a material point, W , can obtained by using the stress and strain expressions given in Eqs. (2a-c) and (3a,b) as

    

    

2

2

x         

1 2

1 2

w





 





2

(4)

W

Ez

G   



xx xx      xz xz







x   

where  is the shear correction factor. The average strain energy density can be calculated by integrating the strain energy density expression given in Eq. (4) throughout the cross-sectional area and dividing by the cross-sectional area, A , as

   

    

2

2

    

1

1

w

   



 

A 

(5)

W WdA EI  

GA







2

A

A

x

x   



  

where I is the moment of inertia. 3. Peridynamic Timoshenko Beam Formulation The peridynamic equation of motion for a Timoshenko beam can be obtained by using Euler-Lagrange equation and can be written for a particular material point k as

( ) k  q q  d L L dt   

0

(6)

 

( ) k

with the Lagrangian, L T U  

(7a)

and degrees of freedom

( ) k k w         ( )

q

(7b)

( ) k

The total kinetic and potential energy of the beam can be calculated, respectively, as

1 2

( ) k A       I

 



2 w V 

2 k

(8a)

T



( ) k  

( )

( ) k

k





( ) ( ) k k    b q ( ) k 

(8b)

U W V 

V

( ) k

( ) k

k

k

where ( ) k  is the density, ( ) k V is the volume and the body load vector, ( ) k b , is defined as