PSI - Issue 28

Wei Lu et al. / Procedia Structural Integrity 28 (2020) 1559–1571 Author name / Structural Integrity Procedia 00 (2019) 000–000

1563

5

/ 2 dS S   . *  is calculated as the same value as the surface dilatation. In this way, it can be regarded as the equivalence of the surface dilatation. By making an analogy to the equation of the classical energy density, the PD strain energy density expression is proposed as

2

2

 

u

r u       r

r u       r







  

  

(9)

*

*

W

e e

r    

 

  

 

PD

2

2

2

r

where , , ,      are the parameters that need to be determined. The elongation e can be written as

1



(10)

e



ij i j   

i j  

, 1,2

The representative volume element in this model is like a ring. As shown in Fig. 3, assuming the length of the bond equal to l , the infinitesimal volume d V of material point  x can be expressed as





(11)

2   

cos

dV

r l

ldld  



Fig. 3. A single bond in the axisymmetric model

Utilising Eqs. (10), (11) and by defining 1 PD energy equation given in Eq. (9) can be obtained as cos l    , 2

sin l    , and  as the horizon size, the second term of the

        2 4 2     

      2 2 12 1 2   

  

   

4

2

4



l



2  





  





11

1

22

2

0 0  

(12)

2

e e

r ldld  

 

2

2

2

l

    2     11 22 1 2

2

2

 

The weighted volume can be calculated as