Page 358 - Mechanics Analysis Composite Materials

Chapter I. Environmental, special loading, and manufacturing effects 343

a2u, =o
ax, ax2

The solution for this equation can be readily found and presented as

u2=4I(xl)+42(x2) =41(Z+Ct)+42(Z-Ct) I
where 41and 42 are some arbitrary functions. Using Eq. (7.61) we get

0, = E, Fl (x+ct) +f2 (x - ct)] ]
where

Applying boundary and initial conditions, Eqs. (7.59) and (7.60), we arrive at the
following final result:

0, = E& + ct) -f(x - ct)] , (7.62)

in which the form of function f is governed by the shape of the acting pulse. As
can be seen, the stress wave is composed of two components having the opposite
signs and moving in the opposite directions with one and the same speed c which is
the speed of sound in the material. The first term in Eq. (7.62) corresponds to the
acting pulse that propagates to the free surface z = h (see Fig. 7.35 demonstrating
the propagation of the rectangular pulse), while the second term corresponds to the
pulse reflected from the free surface z = h. It is important that for the compressive
direct pulse (which is usually the case), the reflected pulse is tensile and can cause
material delamination since the strength of laminated composites under tension
across the layers is very low.

Fig. 7.35. Propagation of direct and reflected pulses through the layer thickness.