Issue 57

S. Derouiche et alii, Frattura ed Integrità Strutturale, 57 (2021) 359-372; DOI: 10.3221/IGF-ESIS.57.26

Note that subscripts 1, 2, and 3 denote the components corresponding to the tension in the two main axes and the in-plane shear, respectively. The stresses for the coordinate system (x,y) and the stresses for the coordinate system (x’,y’) are related by the transformation matrix [T] such as:    { } [ ]{ '} T (24) where the transformation matrix [T] is given:

   

    

cos ² sin ²  

sin ² cos ²  

 



2 cos sin 2 cos sin

(25)

 

 

[ ] T

2

2

 cos sin cos sin cos  

     sin

It is now possible to have the relation between the two compliance matrices of the two coordinate systems, such as:

         1 ' S T S T

(26) Moreover, the stress-strain relation of the same body in a coordinate system oriented at an angle φ is considered as anisotropic and defined as:          ' ' ' S (27) where

         



    21 1

1



  12

 E E

 2

G

       

     

     

  

  

  

1

  ' S S S S S S S  11 12

13

 

 2 

1

12

 

(28)

12

22

23



 E E    2  1

G

12

S S S

13

32

33

 2

1

1







G G G

12

12

12

here E i ’, ν i ’ and G’ 12 are the Young’s Modulus, Poisson’s coefficient, and shear modulus respectfully for the orientated axes plane, and η i ’ is the coefficient of mutual influence of the first type [41][42] (i = 1; 2). These modules and coefficients for the new axes are well determined by reversing every element of the new stiffness matrix as follow:

E

1

1     S 11

1

E

(29)

E

E E

4 

2

2

4 

  

1   (



1

cos

2 )sin cos

sin

12

G

2

E

1

2     S 22

2

E

(30)

E

E E

4 

2

2

4 

2   (

  



2

cos

2 )sin cos

sin

21

G

1

G

1

12     S 33

12

G

(31)

G E

G

2

2

4    4

2   2



12 (1 2 ) 

12   

12

sin cos

2(

1)sin cos

E

1

2

367