Network Theory
The theory of networks of inextensible fibers was formulated by Rivlin [1] as a theory of materials like fish-nets, which have small resistance to distortion but high resistance to stretching. The theory is a continuum theory, in which individual cords or
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e •.
CONTINUUM THEORY OF THE MECHANICS OF FIBRE-REINFO RCED COMPOSITES
EDITED BY
A.J.M. SPENCER THE UNIVERSITY OF NOTTINGHAM
SPRINGER-VERLAG WIEN GMBH
This work is subjtd to copyright. All righ ts are reserved, whether the whole or part of the material is concemed specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks.
© 1984 by Springer-Verlag Wien Originally published by Springer-Verlag Wien-New York in 1984
ISBN 978-3-211-81842-8 DOI 10.1007/978-3-7091-4336-0
ISBN 978-3-7091-4336-0 (eBook)
PREFACE
This booK is made up of tne notes for tne lectures 6ze .-"ive au6zors gave at the International Centre for 1981.
Mecizanica~
Sciences in Udilie in Jdy
The aim of the Scizool was to present an account of rec
(a+b)
/cos~
For this we introduce unit vectors c and d, where ,
d
~(a-b) /sin~
b
:
, (46)
a = : cos 4J + ~
sin~
,
cos~
-
~ sin~
.
On substituting for a and b from (46) into (44), we obtain an expression for W of the same form as (35), with a and b replaced by c and d respectively, and with the coefficients becoming functions of cos
2~.
Hence this case also corresponds to orthotropic symmetry, and if c and d are chosen to lie in the x 1 and x 2 coordinate directions, the constitutive equation assumes the form (37) .
There are again nine independent elastic
coefficients, which are functions of
cos2~.
The constitutive equation in
terms of c and d is obtained by substituting for a and b from (46) into (45). The expression is obviously complicated.
An alternative and rather
simpler procedure is to observe from the beginning that W can be expressed
Constitutive Theory
15
as a function of e, c, d and
cos2~,
with c.d
= 0.
Hence it is possible
to proceed as in the case of orthogonal fibres, with a and b replaced by c and d, and the coefficients in the expression for W regarded as functions of
cos2~.
However, with the alternative procedure it is less
easy to proceed to the case of fibre inextensibility. If the material is incompressible then tr e = 0
A,
involving stress
-p~
y3
andy~
is added.
directions, then a.e.a
and the terms
are omitted from (44) and (45), and a reaction If the material is inextensible in both fibre
=0
and b.e.b
= O,
and (45) is replaced by
(47) The number of independent elastic constants is reduced to five, and the last two terms represent the reaction stress.
In terms of the vectors c
and d, (47) is
a
Az tr :
+ 211: + { h ~ cos 2~ tr : + 2y 2
(:.:.:
cos 2
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