NASA 4750 Manual

A SERVICE OF

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PREDICTION OF ELASTIC CONSTANTS AND THERMAL EXPANSION

5-11

dominated by nominally straight in-plane tows. In a triaxial braid, for example,

λ

i

in the

direction of the axial tows may be effectively zero if the axial tows contain a sufficiently

high fraction of all fibers.

5.1.6 Fundamentals of Heterogeneous Elastica

The concept of homogeneous effective elastic properties existing over

macroscopic length scales is based on some fundamental theorems regarding stresses and

strains in heterogeneous materials. They are summarized here in the notation of Ref.

[5.13].

Average Strain Theorem

Consider a heterogeneous body of volume V consisting of an aggregate of different

phases. If the displacements on its external surface, S, are

u

i

(S) = u

i

0

(5.1)

and displacement continuity is enforced on the internal boundary, S

12

, between phases in

such a way that

u

i

(1)

= u

i

(2)

(on S

12

) (5.2)

then the average strain

,

ε

ij

, in the body is

( )

ε

ij i j j i

S

V

u n u n dS= +

∫

1

2

0 0

$ $

(5.3)

where

n

is the unit normal to S. Further, if

u

i

(S) =

ε

ij

0

x

j

(5.4)

then

ε

ij

=

ε

ij

0

(5.5)

Average Stress Theorem

For an average stress, defined as