Page 342 - Mechanics Analysis Composite Materials

Chapter 7. Environmental, special loading, and manufacturing effects 327
Importance of the Laplace transformation for the hereditary theory is associated
with the existence of the so-called convolution theorem according to which
l*

Using this theorem and applying Laplace transformation to Eq. (7.33) we get

1
E*(p) = $*@) + C*(p)O*(p)].

This result can be presented in the form analogous to Hooke's law, i.e.
&(p) = E"(p)E*(p) . (7.43)

where

Applying Laplace transformation to Eq. (7.32) we arrive at Eq. (7.43) in which

E* = E[1 - R*(p)] . (7.44)

Matching Eqs. (7.43) and (7.44) we can link together Laplace transforms of the
creep compliance and the relaxation modulus, i.e.

1 = 1 -R*(p) .
I+C'(p)

With due regard to Eq. (7.43) we can formulate the elastic-viscoelastic analogy
or the correspondence principle, according to which the solution of the linear
viscoelasticity problem can be obtained in terms of the corresponding Laplace
transforms from the solution of the linear elasticity problem if E is replaced with E*
and all the stresses, strains, displacements, and external loads are replaced with
their Laplace transforms.
For an orthotropic material in the plane stress state, e.g., for a unidirectional
composite ply or layer referred to the principal material axes, Eqs. (4.55) and (7.31)
can be generalized as