Page 126 - Mechanics Analysis Composite Materials

Chapter 3. Mechanics of a unidireclional ply Ill

and derive the following equations for o(x)and t(x)

Em
1
~(x)= -V[sin & (x- c) - sin A,J] + -z’(x)a , (3.1 18)
a 2
a2 2
-t”(x) - -z(x) = -VI.,,( 1 +d)[cosA,(x - c) + cos AIJX] . (3.119)
6Em Gm
We need a periodic solution of Eq. (3.119) and find it in the following form

z(.) = C[COS n,(x - c) +cos A,,x] . (3.120)

Substituting into Eq. (3.119) and taking into account that I, = n/l, we get

(3.121)

Now, using Eqs. (3.119, (3.1 I8), and (3.120) we can write the final expressions for
the stresses acting in the matrix

ZX.” = C[COSi,(x - c) + cos A&],

(3.122)

where C is specified with Eqs. (3.121). The corresponding strain energies of the
typical element in Fig. 3.61 are

Substituting Eqs. (3.122) and integrating we arrive at

adl,
ws =-CC’(l +COSI,C),
2Gm

In conjunction with these results, Eqs.(3.109), (3.11 lb(3.113) and (3.121) allow us
to determine GIwhich acquires the following final form