Issue56

A. Mohamed Ben Ali et alii, Frattura ed Integrità Strutturale, 56 (2021) 229-239; DOI: 10.3221/IGF-ESIS.56.19

1 2

1 4

1 4



    



    



   

   1 1

   1 1

   1 1

N

N

N

,

,

1

2

3

(2)

1 4

1 2



 



   



    2 1 1

   1 1



N

N

,

4

5

The stress field in any point is written as:          M

(3)

where   M is the matrix of interpolation functions for stresses and   τ is the vector of nodal stresses.

Figure 1: Mixed finite element RMQ-7.

In the configuration of Fig. 1, the shape functions used to approximate

11 σ are given by:

1 4

1 4

 



 







8

9

   M 1 2 ξ 1 2 η

   M 1 2 ξ 1 2 η

,

11

11

(4)

1 4

1

 



 







10 11

11 11

   M 1 2 ξ 1 2 η

   M 1 2 ξ 1 2 η

,

4 The shape functions used to calculate 12 σ and 22 σ are given as follows:

1 6 1 3

1 6

 



 







6

7

   M 1 2 ξ 1 2 η

   M 1 2 ξ 1 2 η

,

i 2

i 2

(5)

1

 



 







8

9

   M 1 2 ξ 1 η

   M 1 2 ξ 1 η , i 1, 2 

,

i 2

i 2

4 It should be noted that nodes 8, 9, 10 and 11 are inside the element, and which are eliminated by the static condensation technique [18]. The nodal approximation of the displacement and stress fields is expressed by:                                 τ σ M 0 q ε 0 B (6)   

231