Page 276 - Mechanics Analysis Composite Materials

Chapter 5. Mechanics of laminates 26I
conditions. The corresponding boundary problem will be discussed further in this
section.
The third equation in Eqs. (5.73) formally yields o, = 0. However, this result is
not correct because the equation corresponds to the plane laminate and is not valid
for the cylinder. In cylindrical coordinates, the corresponding equation has the
following form (see, e.g., Vasiliev, 1993):

a
- [(l fx)crz] = -[(I +-)-+--- at,
z
at,,
az R ax ay R
Taking z,~ = 0 and t, = 0, substituting ov from Eqs. (5.84), and integrating we
obtain

(5.86)

where, A,,,,, (mn= 21, 22) are the step-wise functions of z, i.e.,

A,,,,, =A::: for 0 d z < hi,
A,,,,, =A::; for hi <z < h = hl +h2 ,

and C is the constant of integration. Because no pressure is applied to the inner
surface of the cylinder, a,(z = 0) = 0 and C= 0. Substitution of stiffness
coefficients, Eqs. (5.80), (5.81) and strains, Eqs. (5.79) into Eq. (5.86) yields

(5.87)
P z-hl
= CJ!')(Z= hi) +0.07-.- .
Rh R+z
On the outer surface of the cylinder, z = h and ot2)= 0 which is natural because this
surface is free of loads. Distribution of cr, over the laminate thickness is shown in

2 4 6
Fig. 5.23. Distribution of the normalized radial stress s2 = tr:Rh/P over the laminate thickness.