Page 87 - Mechanics Analysis Composite Materials

72 Mechanics and analysis of composite materials

Because of periodic properties of tangent function entering Eq. (3.38), it has k + 1
different roots corresponding to intersection points of the curves z = tan ke and
z = -tan 8/2. For the case k = 4 considered below as an example, these points are
shown in Fig. 3.18. Further transformation allows us to reduce Eq. (3.38) to

2k+ 1
sin-e=o,
2
from which it follows that

27ci
e, =- (i=O,1,2,...,k) . (3.39)
2k+ 1
The first root, 60 = 0, corresponds to A = 0 and Fn = const, Le., to the ply without a
crack in the central fiber. So, Eq. (3.39) specifies k roots (i = 1,2,3,.. .,k) for the
ply under study, and solution in Eqs. (3.29) and (3.37) can be generalized as
k
F,,(js)= Ci[sinn4 - cos n8, tan(k + l)8;]e-’.<? , (3.40)
1= I
where, in accordance with Eq. (3.36)

A; = 2psin- Qi (3.41)
2
and Oj are determined with Eq. (3.39).
Using Eq. (3.38) we can transform Eq. (3.40) to the following final form

k
F,(x)= CjS,,(8i)e-’.ii , (3.42)
1= I
where

(3.43)

Applying Eqs. (3.25) and (3.26) we can find shear and normal stresses, i.e.,

Ik
~(x)= --xAiC;S,(O;)e-;+x (n = 1,2,3,...,k) , (3.44)
a ;=I

(3.45)