PSI - Issue 6

Pavel A. Akimov et al. / Procedia Structural Integrity 6 (2017) 182–189 Author name / Structural Integrity Procedia 00 (2017) 000 – 000

186

5

k k u u (superscript “ ( ) k ” hereinafter corresponds to the

Basic nodal unknowns are displacement components

( ) 1 ,

( ) 2

fe k    ). Thus for node ( , , ) k i j we have the following unknowns: k

number of considered subdomain i.e.

. Bilinear approximation of unknowns is used within finite element (conventional plane rectangular 4

( , , ) 1 k i j u u ,

( , , ) 2 k i j

node finite element of two-dimensional problem of elasticity theory (Fig. 2)).

Fig. 2. Arbitrary finite element with local coordinate system.

Computing of partial derivatives of displacements, deformations and stresses within the finite element, nodal stresses and nodal deformations with allowance for averaging is described in [1]. As known, FEM is reduced to the solution of systems of k N N 1 2, 2 linear algebraic equations:

(15)

k k k K U R  ,

where k U is global vector of nodal unknowns (subscript “ ( ) k ” corresponds to the number of subdomain fe k k    ),

[ ( U u 

u

( ... u

( ... u

u

( ... u

...

) ( T

)

)

) ( ) ( T T

)

)

,1) k N T ( ,

k N

( ,

,2)

( ,1,1) k

( ,2,1) k

T

( ,1,2) k

( ,2,2) k

T

T

1

1

(16)

k

n

n

n

n

n

n

k N

( ,1,

)

( ... u

u

( ... u

)

) ] ; k N N T T n k ) , ( , 1 2.

) k N T ( ,2,

k

2,

k

2.

n

n

;

(17)

u

k i j u u ( , , ) 1

T ] , 1, 2, ..., ,  N i

j

N

[

1, 2, ...,





k i j ( , , )

k i j ( , , ) 2

n

k

1

2,

k K is global stiffness matrix of order Let’s consider arbitrary subdomain

k N N 1 2, 2 ; k R is global right-side vector of order

k N N 1 2, 2 (global load vector).

dc k  . Discrete-continual approximation model is used for two-dimensional problems. It presupposes mesh approximation for non-basic dimension of extended domain (along 1 x ) while in the basic dimension problem remains continual. Subdomain 1  is divided into discrete-continual finite elements

 1 1 1   N i

} dc k i x x x x x x x x         . , { ( , ) : 1 2 2, 1 k 2 2, 1, 1 i 1 1, , b b k i

(18)

,

k 





dc

, dc k i

Lame constants for finite element are defined by formulas:

  

1,

;



   

, dc k i

;



      k i k i , , 

(19)





, where

k

0,

;



k i ,

k i ,

, dc k i

, k i

k

dc k i ,  .

k i ,  is the characteristic function of element