Crack Paths 2009

ε

parameter

()uxε

as an asymptotic series of a small

Expanding the structure displacement

Substitute Eq.2 into a virtual work equation

(2) 0 1 2 2 ( ) ( , ) ( , ) ( , ) u x u x y u x y u x y ε ε ε = + + " +

∫

∫

t f vd tvd p Γ Ω + Γ + ∫ ∫ ε l l l l S ε

vdS

ε

ε

(3)

ijkl

E

k l j u v x x ∂ ∂ Ω = i d

Ω

Ω

l l

ε

ε

where and are the real e virtual displacements, respectively. f is the body force applied on the open subset Ω with a smooth boundary on ε u v Γ comprising (where

d Γ

(the traction boundary), and are the tractions. t p

displacements are prescribed) and t Γ

Then, we have：

v d t v d Ω +

∫

∫

∫

∫

E

0 l j u v x x ∂ ∂ Ω = ∂ ∂ l d

f

Γ +

p

ijkl

(4)

Ω

0 H k

Ω

l

ε

0

l

Γ

0 l l

S

l

l

ε

ε

t

ε

0 v d S

ε

where

HijklE is the effective elastic tensor that is defined by Eq.5

kl

1 Y ∫

∂

χ

E

ijkl E E

dY

(5)

iHjkl

ijmn

Y

m

=

−

y

∂

n

where is the periodic solution of the following homogenization equation: klmχ

0,

∫

∫

kl

ν ∀ ∈V

∂

v

∂ ∂ χ

v

i

m i

E dY E =

dY

=

(6)

ijkl

ijmn

Y

i

Y

n

j

Y

∂

y

∂ ∂

y y

where is the period. Eqs.5 and 6 present the macroscopic and microscopic Y

homogenization problems, respectively. By solving the microscopic ones, the

macroscopic material parameters can be calculated, whereas the microscopic boundary

conditions can be identified from the macroscopic ones.

FIXEDP O I N TI T E R A T I OANN DFINITEE L E M E NA NTA L Y S I S

The solution of the effective mechanical parameters of composites with periodic

microstructures

can be used to solve the nonlinear functional equation

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