Page 259 - Mechanics Analysis Composite Materials

244 Mechanics and analysis of composite materials

k
CCOS24 = 0
i= 1
As can be checked by direct substitutions, for k= I this equation is satisfied if
= 45" and for k = 2 if dl = 0 and 42= 90". Naturally, such one- and two-
layered materials cannot be isotropic even in one plane. So consider the case k 23,
for which the solution has the form

R
qhi=(i-l)-l i= 1,2,3,...,k. (5.55)
k
Using the sums that are valid for angles specified by Eqn. (5.55) i.e.,

k k k
Csin2cbi=Ccos2(bi=-,
i= I i= I 2
Csin4qhi = CCOS~~~
k
k
3k
=-,
i=l i= I 8
k k
sin2dicos2+i =- ,
i=l 8
and calculating stiffness coefficients in Eqs. (5.52) and (4.72) we get

These stiffnesses provide constitutive equations in the form of Eqs. (5.53) and
satisfy conditions (5.54) which can be written as

if

(5.56)

Possible solutions (5.55) providing quasi-isotropic properties of the laminates with
different number of layers are listed in Table 5.1 for k<6.