Fracture Mechanics of Fiber-Reinforced Materials
In plane stress of materials composed of two orthogonal families of strong fibers, the analysis based on the assumption of fiber inextensibility is so simple that it becomes possible to predict the course of an advancing crack or tear, and to determine ho
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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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