Page 263 - Bird R.B. Transport phenomena

§8.4 Elasticity and the Linear Viscoelastic Models 247

pression in Eq. 8.4-9 is sometimes more convenient for solving linear viscoelastic prob-
lems than are the differential equations in Eqs. 8.4-5 and 6.
The Maxwell, Jeffreys, and generalized Maxwell models are all examples of linear
viscoelastic models, and their use is restricted to motions with very small displacement
gradients. Polymeric liquids have many internal degrees of freedom and therefore many
relaxation times are needed to describe their linear response. For this reason, the general-
ized Maxwell model has been widely used for interpreting experimental data on linear
viscoelasticity. By fitting Eq. 8.4-9 to experimental data one can determine the relaxation
function G(t — t'). One can then relate the shapes of the relaxation functions to the mole-
cular structure of the polymer. In this way a sort of "mechanical spectroscopy" is devel-
oped, which can be used to investigate structure via linear viscoelastic measurements
(such as the complex viscosity).
Models describing flows with very small displacement gradients might seem to
have only limited interest to engineers. However, an important reason for studying them
is that some background in linear viscoelasticity helps us in the study of nonlinear vis-
coelasticity, where flows with large displacement gradients are discussed.

EXAMPLE 8.4-1 Obtain an expression for the components of the complex viscosity by using the generalized
Maxwell model. The system is described in Fig. 8.2-2.
Small-Amplitude
Oscillatory Motion SOLUTION
We use the yx-component of Eq. 8.4-9, and for this problem the yx-component of the rate-of-
strain tensor is

dv x . n
= -т- = У cos cot (8.4-10)
where со is the angular frequency. When this is substituted into Eq. 8.4-9, with the relaxation
modulus (in braces) expressed as G(t - t'), we get

r yx = - J G(t - t')y° cos cot'dt'

= -y° \ G(s) cos co(t-s)ds
Jo
= —y°\ I G(s) cos cos ds cos cot - y°\ G(s) sin cos ds sin a>f (8.4-11)

in which s = t - t'. When this equation is compared with Eq. 8.2-4, we obtain

г]'{со) = I G(s) cos cos ds (8.4-12)
J о
rf(co) = I G(s) sin cos ds (8.4-13)
J Q
for the components of the complex viscosity rf = r\ — \r\'. When the generalized Maxwell ex-
pression for the relaxation modulus is introduced and the integrals are evaluated, we find that

(8.4-14)
*=i l + u^) 2
= у - (8.4-15)

If the empiricisms in Eqs. 8.4-7 and 8 are used, it can be shown that both 77' and 77" decrease as
l/w 1(1/a ) at very high frequencies (see Fig. 8.2-4).