Page 200 - Plastics Engineering

Mechanical Behaviour of Composites 183

and the shear strain yxy is related to the shear stress, txy, and the shear modulus,
GI29 by
(3.19)

The convention normally used is that direct stresses and strains have one
suffix to indicate the direction of the stress or strain. Shear stresses and strains
have two suffices. The first suffix indicates the direction of the normal to the
plane on which the stress acts and the second suffix indicates the direction of
the stress (or strain). Poisson’s Ratio has two suffices. Thus, u12 is the negative
ratio of the strain in the 2-direction to the strain in the 1-direction for a stress
applied in the 1-direction (u12 = -E~/EI for an applied al). u12 is sometimes
referred to as the major Poisson’s Ratio and u21 is the minor Poisson’s Ratio.
In an isotropic material where 191 = 191, then the suffices are not needed and
normally are not used.
In the analysis of composites it is convenient to use matrix notation because
this simplifies the computations very considerably. Thus we may write the
above equations as (see Appendix E)

{;:y} = 1
0 0 -
Gl2
or in abbreviated form
(3.20)

where [SI is called the compliance matrix.
In order to describe completely the state of triaxial (as opposed to biaxial)
stress in an anisotropic material, the compliance matrix will have 36 terms. The
reader is referred to the more advanced composites texts listed in the Bibliog-
raphy if these more complex states of stress are of interest. It is conventional to
be consistent and use the terminology of the more general analysis even when
one is considering the simpler plane stress situation. Hence, the compliance
matrix [SI has the terms

where