Issue 29

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

Procedure B: 3 Step homogenization This procedure has been proposed by the authors in [11] as an extension to the 2D micropolar computational homogenization of the homogenization of second gradient continua via the asymptotic approach [12, 15]. Here, the basic idea is recalled and the main steps are addressed. It is assumed that the total perturbation displacement vector    u X, x can be expressed as the sum of three fields evaluated in sequence. Initially, only the first order terms of the kinematic map, multiplying the components 1 E , 2 E and 12  in (8), are activated; subsequently, the effects of the quadratic terms related to 1 K and 2 K are considered and, finally, the third order term associated with  is taken into account. Therefore, the vector results as:          1 2 3 , , , , .    u X x r X x r X x r X x    (13) When only the linear terms of the kinematic map are considered, the case of the first order homogenization is recovered. Here, it is assumed that   1 r X, x  is evaluated as the product of unknown functions times the independent components of the first gradient of the kinematic map, as:       1 1 1 = r X, x Λ x γ X, x  (14) where             1 1 1 1 1 1 2 12 1 2 3 = , = T              γ X, x Λ x x x x (15)   1 i  x , =1, 2, 3, i being evaluated by applying the components 1 E , 2 E and 12  on the UC undergoing PBCs. In this case   1

r X, x  results as periodic functions. When the presence of the curvatures 1

2 K is also considered, with = 0  , the perturbation field can be expressed as:

K and

  ,  u X x Λ x γ X x r X x   , =    1   2 1 ,



(16)

 2 r X, x  is the only unknown field. Following the same procedure as for the first term   1

r X, x  , it is assumed

where now 

2 γ X, x , i.e. the nonvanishing components of the

that  2 r X, x  is expressed as the product of unknown functions and   

 1 γ X, x . Then, the unknown field 

 2 r X, x  is represented in the form:

first gradient of 

 , = , r X x Λ x γ X x      2 2 2



(17)

with

  =



T

2



 

       x x x     2 2 2 2 3 1 

  x

 









2

2 4









,

=

(18)

γ X, x

Λ x

1,1 1,2

2,1 2,2

  2 i  x , = 1,.., 4, i being periodic functions evaluated in the UC when 1 0 K  and 2 0 K  . Finally, when the component  is also taken into account, it results that:              1 2 3 1 2 , = , , , .   u X x Λ x γ X x Λ x γ X x r X x 

(19)

The field   3 r X, x  is written as the product of unknown functions times the relevant and nonvanishing components of the first gradient of   2 γ X, x , and it results:       3 3 3 = r X, x Λ x γ X, x  (20) with

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