Page 89 - Mechanics Analysis Composite Materials

74 Mechanics and analysis of composiie materials

k Oi aG,
C~~sin--uo(O) (3.53)
=
i= I 2 2pan1
This set contains k+ 1 equations and includes k unknown coefficients D;and
displacement ~“(0).
The foregoing set of equations allows us to obtain the exact analytical solution for
any number of fibers, k.To find this solution, perform some transformations. First,
multiply Eq. (3.52) by sin[(2n - l)0.,/2] and sum up all the equations from n = 2 to
n = k. Adding Eq. (3.53) for n = 1 multiplied by sin(O,,/2) we obtain

2n - 1 2n - 1 aGm 0.S
22oisin- 0; sin - =-ug (0)sin -
n=l ;=I 2 2 harn 2
Now change the sequence of summation, i.e.,

2n - 1 2n- 1 aGm 0,s
2
20;2sin2 8; sin - = -u0(0) sin- . (3.54)
2wnl
2
i=l
n=l
Using the following known series
k sin 2k0
cos(2n - I )e =-
2sinO ’
n= I
we get in several steps

A 2n - 1 2n - 1
Oisin -
R~,~ sin2 0.5
=
I?= I 2
=-E 2n - 1 (ei - OF) -cos2 (oi+e,,) 1
lk
2n - 1
2
[cos,
1
n=
- O.,) - sink(&+ 6,)
- e,,) sin f (oi+ €5)
-

Using Eq. (3.38) we can conclude that R;., = 0 for i # s. For the case i = s, we
have