Issue 73

N. Laouche et alii, Fracture and Structural Integrity, 73 (2025) 88-107; DOI: 10.3221/IGF-ESIS.73.07

        N 1 1 j

  cos

x

(27)

j

The relation between and ത , and ഥ is defined using rule DQ [8]:   b s z b s z u Qu w Qw , , , , ,

(28)

where

1

0

0

0

0

  

         

         

  1

  1 1,2

  1 1,3

  1

  1

A A A

A A



N

N

1,1

1,

1

1, 0

0

0

1

0



Q

(29)







 



0

0

0

0

1

 

  1

  1

  1

  1

  1

N N A A A ,1 ,2

 N N A A , 1

N

N N ,

,3

Utilizing the DQFEM, by substituting Eqns. (21-29) into Eqns. (8, 12, 14) and employing Lagrange’s principle, we derive the subsequent elementaries matrixes. The element mass matrix:

                  e e e e e e e e e



   M J Q CQ M J Q CA Q M J Q CA Q M JQCG JQA CA Q M M JQCG JQA CA Q M J Q CQ M JQG JQA CA Q 1 11 (1) 2 12 (1) 3 13 (1) (1) 1 5 22 (1) (1) 1 4 23 8 24 (1) (1)                                     T T T T T T T T b T T s T T T

   

  e

(30)

         

 

 

s

1

6

33

   M J Q CQ M J Q CQ 8 34    T T

   

7

44

The element strain matrix:

95