Page 123 - Mechanics Analysis Composite Materials

108 Mechanics and analysis of composite materials
represented by a sine function as

UI (x) = V sin &x, ~(x)= V sin &(x - c) , (3.108)
where V is an unknown amplitude value, the same for all the fibers, A,, = n/ln,I,, is
a half of a fiber wavelength (see Fig. 3.60), and c = (a +d)cot CI is a phase shift.
Taking c = 0 we can describe the shear mode of buckling (Fig. 3.59(a)), while c = I,
corresponds to extension mode (Fig. 3.59(b)). To find the critical value of stress 01,
we use the Timoshenko energy method (Timoshenko and Gere, 1961) yielding the
following buckling condition

A=W. (3.109)
Here, A is the work of external forces and W is the strain energy accumulated in the
material while the fibers undergo buckling. Work A and energy Ware calculated for
a typical ply element consisting of two halves of fibers 1 and 2 and matrix between
them (see Fig. 3.61). The work, A, can be calculated as

A =a,(a+d)d.d (3.110)

with displacement 6 following from Fig. 3.62, Le.,

Fig. 3.61. A typical ply element.

Fig. 3.62. Deformation of a fiber.