Issue 29

M.L. De Bellis et alii, Frattura ed Integrità Strutturale, 29 (2014) 37-48; DOI: 10.3221/IGF-ESIS.29.05

(2)

= E LU

where the compatibility operator is defined as

 

                

0 0



1  X

              



0

0

X



2





0

2  X X   X X



1

= L

.

(3)

2





2   

1



0

0

X



1



0

0

X



2

According to the strain driven approach, the macroscopic strain components, evaluated at , X are used as input variables for the microscopic level. The kinematic map, expressed in function of the vector , E is imposed on the UC, properly defining a BVP. At the micro-level a repetitive rectangular UC is selected, whose size is 1 2 2 2 a a  and its centre is located at the macroscopic point , X characterized by the displacement field 1 2 ={ , } T u u u , defined at each point   1 2 = , T x x x of the UC domain  . The displacement field, resulting in the UC after solving the BVP, can be represented as the superposition of the assigned field    u X, x and a perturbation field    : u X, x        =   u X, x u X, x u X, x (5) The strain vector at the microscopic level is derived by applying the kinematic operator defined for the 2D Cauchy problem and, in expanded form, it results as:

    

   



1            2  12 

0

,1

 





(6)

= ,

with

and

= 0

ε lu

ε

l x

,2

  

,2

,1

 indicating the partial derivative with respect to . i

x According to Eq. (5), the strain can be written as:

with , i

     ε X x ε X x ε X x  (7) The third order polynomial map, proposed in [5 ,6] and modified in [7], is used. Different material symmetries can be considered ranging between the isotropic and the orthotropic case. In the considered orthotropic case, the kinematic map can be written in compact form as:       , =  u X x A x E X (8) with  , = , ,  

1 2

1 2

    

     









2

2 1 12 1  x

2

3







1 1 2 1 2 s b x x c x  3



2 

3 

x

x

1 1 2 x x

x

0

1

2

2

 

 

(9)

A x

1 1 1 2 2









2

2

2

3







3 



2 1 2 2 1 b x x c x  3 

x x

x

x

2 1 2 x x

s

0

2

1 1

12 2

39