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ANALYTICAL METHODS FOR TEXTILE COMPOSITES

5-12

σ σ

ij ij

V

V

x dV=

∫

1

( )

(5.6)

the average stress theorem states that

σ

ij j i

S

j

V

i

V

x TdS x FdV= +













∫ ∫

1

(5.7)

where T

i

is a surface traction and F

i

a body force. If

T

i

=

σ

ij

0

n

j

F

i

= 0

(5.8)

then

σ

ij

=

σ

ij

0

(5.9)

Average Virtual Work Theorems

If the homogeneous displacements

u

i

(S) =

ε

ij

0

x

j

(5.10)

are applied to the boundary, S, then the average virtual work, J, can be shown to be

J =

σ

ij

ε

ij

0

V

, (5.11)

with summing implied over repeated indices. Alternatively, if on the bounding surface, S,

T

i

=

σ

ij

0

n

j

(5.12)

then

J =

σ

ij

0

ε

ij

V

(5.13)

Effective Elastic Moduli

Assume that the displacement distribution within the heterogeneous body subject to the

homogeneous boundary conditions of Eq. (5.10) is given by

u

i

x =

ε

kl

0

u

i

(kl)

(x)

. (5.14)