Issue 49

M. Hamdi et alii, Frattura ed Integrità Strutturale, 49 (2019) 321-330; DOI: 10.3221/IGF-ESIS.49.32

where i u denotes displacements in the original body, i u denotes displacements in homogenized body, i  denotes difference between i u and i u , commonly referred to as the fluctuating functions in micromechanics, and x and y denote global and the local coordinates, respectively. The strains in the original body can then be obtained as:

ij 



ij 





( , ) x y

( , ) x y

(2)

( , ) i j

i  with respect to the local coordinates, and  is a small parameter

( , ) i j  denotes the symmetric gradient of

where

with / y x   . Note that higher order terms have been neglected using the variational asymptotic method [17]. It is beneficial to define the kinematic variables in homogenized body in terms of those in the original body as

i i u u 

ij 

 

and

(3)

ij

where the angle brackets denotes average over the SG. Eq. (3) leads to the following constraints on fluctuating functions:

i  

( , ) i j  

0

0.

and

(4)

The principle of minimum information loss seeks to minimize the difference between the strain energy of original body and that of homogenized body, i.e.

1 2

1 2

* ijkl ij kl  

 

   

 

ij 







 







C

C

(5)

ijkl

( , ) i j

kl

( , ) k l

To minimize  , consider a homogenized body whose *

i  can be obtained by solving

ij  are held fixed.

ijkl C and

variational statement:

1 2

1 2

 

   

 

min

min

C

C







ij  









(6)

.(4)

( , ) i j

( , ) k l

ijkl

kl

Eq

ijkl

ij

kl

i 

i 



Other constraints such as periodic or partially periodic requirements can also be incorporated. Eqn. (6) can be either analytically or numerically solved. For simple structures such as binary composites and periodically layered composites (Tab. 1), analytical solutions are achievable. However, for more general cases, one has to use finite element code (e.g., SwiftComp) for problem solving. Once homogenization is completed, one can dehomogenize the composite. Specifically, introduce the following matrix notations to the global stress and strain column matrices,  and  :   11 22 33 23 13 12 T         (7)

and





T





11 22 33   

23 13 12 2 2 2   

(8)

The global constitutive relations can then be expressed as

, D   

(9)

where D denotes the 6 × 6 effective stiffness matrix rather of the composite.  and  can be partitioned as:

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