PSI - Issue 41

Efstathios E. Theotokoglou et al. / Procedia Structural Integrity 41 (2022) 361–371 Efstathios E.Theotokoglou/ Structural Integrity Procedia 00 (2022) 000–000

363

3

2. Basic Theorem For a nonhomogeneous material the Hooke’s law can be written as (Sadd, (2020)), (Burlayenko, et al., (2017)).   ij ijkl m kl C x    (1) where ijkl C is the elastic moduli which is function of the coordinates m x and the strain tensor is given by (Sadd, (2020)) (Burlayenko, et al., (2017)).   , , kl k l l k u u   

1 2

(2)

The equilibrium equation with zero body forces can be written as (Sadd, (2020)).

 





  

0  

0

, k l ijkl C u u



(3)





, ij j

, l k

x



j

3. Finite element formulation For the problem we considered plane isoparametric element with 8 nodes. The displacements field in terms of the natural coordinates (ξ,η) is given by (Cook, et al., (2001)).

(4) (5)

2

2

2

2

1 2 u a a a a a a a a                                  2 a a a 2 2 2 9 10 a a a a 11 12 13 14 15 16 a 3 4 5 6 7 8

The displacements can be written as a function of the nodal displacements and the shape functions (Cook, et al., (2001)).

 

u             

    

N u

    N d

i i



(6)

i i N 

where [ N ] is the shape functions matrix and { d } is the nodal displacements of an element Coordinates can be written as (Cook, et al., (2001)).

 

x y            

    

i i N x N y i i

(7)

In the case of the plane isoparametric elements Jacobian matrix is given by (Cook, et al., (2001)).

,     , i i i i N x N x  



   

, N y N y   , i i

, ,

y, y,

x x

  

 

  J

i







 

  

(8)





i