Page 349 - Mechanics Analysis Composite Materials

334 Mechanics and analysis of composite materials

Strength criteria discussed in Chapter 6 can be generalized for the case of long-
term loading if we change the static ultimate stresses entering these criteria for the
corresponding long-term strength characteristics.

7.3.3. Cyclic loading

Consider the behavior of composite materials under the action of loads
periodically changing in time. For qualitative analysis, consider first the material
that can be simulated with the simple mechanical model shown in Fig. 7.16.
Applying stress acting according to the following law:

a(t) = a0sin ut , (7.53)

where 00 is the amplitude of stress and w is the frequency we can solve Eq. (7.41)
that describes the model under study for strain &(t).The result is

~(t)= Q sin(ot + 0) , (7.54)

where

(7.55)

As follows from these equations, viscoelastic material is characterized with a phase
shift of strain with respect to stress. Eliminating time variable from Eqs. (7.53) and
(7.54) we arrive at the following relationship between stress and strain

(-$+($-2coso-= a& sin2 e .
a0Eo

This is the equation of an ellipse shown in Fig. 7.22(a). The absolute value of the
area A, inside this ellipse (its sign depends on the direction of integration along the
contour) determines the energy dissipation per one cycle of vibration, i.e.
AW = JAl = nao&olsin8). (7.56)

Folowing Zinoviev and Ermakov (1994) we can introduce the dissipation factor as
the ratio of energy loss in a loading cycle, A W, to the amplitude value of the elastic
potential energy in a cycle, W, as

$=- AW
w’

where, in acccordance with Fig. 7.22(b), W = (1/2)00~.Transforming Eq. (7.56)
with the aid of Eqs. (7.55) we arrive at