Page 261 - Bird R.B. Transport phenomena

§8.4 Elasticity and the Linear Viscoelastic Models 245

allow for the inclusion of time derivatives or time integrals, but still require a linear rela-
tion between т and 7. This leads to linear viscoelastic models.
We start by writing Newton's expression for the stress tensor for an incompressible
viscous liquid along with Hooke's analogous expression for the stress tensor for an in-
compressible elastic solid: 1
+
Newton: т = -/x(Vv + (Vv) ) = -fxy (8.4-1)
f
Hooke: т = -G(Vu + (Vu)) = -Gy (8.4-2)
In the second of these expressions G is the elastic modulus, and u is the "displacement
vector," which gives the distance and direction that a point in the solid has moved from
its initial position as a result of the applied stresses. The quantity 7 is called the "infini-
tesimal strain tensor." The rate-of-strain tensor and the infinitesimal strain tensor are re-
lated by 7 = ду/dt. The Hookean solid has a perfect memory; when imposed stresses are
removed, the solid returns to its initial configuration. Hooke's law is valid only for very
small displacement gradients, Vu. Now we want to combine the ideas embodied in Eqs.
8.4-1 and 2 to describe viscoelastic fluids.

The Maxwell Model
The simplest equation for describing a fluid that is both viscous and elastic is the follow-
ing Maxwell model'}

T + \ j T=-r y (8.4-3)
1o
l t
Here A! is a time constant (the relaxation time) and r] is the zero shear rate viscosity.
Q
When the stress tensor changes imperceptibly with time, then Eq. 8.4-3 has the form of
Eq. 8.4-1 for a Newtonian liquid. When there are very rapid changes in the stress ten-
sor with time, then the first term on the left side of Eq. 8.4-3 can be omitted, and when
the equation is integrated with respect to time, we get an equation of the form of Eq.
8.4-2 for the Hookean solid. In that sense, Eq. 8.4-3 incorporates both viscosity and
elasticity.
A simple experiment that illustrates the behavior of a viscoelastic liquid involves
"silly putty." This material flows easily when squeezed slowly between the palms of the
hands, and this indicates that it is a viscous fluid. However, when it is rolled into a ball,
the ball will bounce when dropped onto a hard surface. During the impact the stresses
change rapidly, and the material behaves as an elastic solid.

The Jeffreys Model
The Maxwell model of Eq. 8.4-3 is a linear relation between the stresses and the velocity
gradients, involving a time derivative of the stresses. One could also include a time de-
rivative of the velocity gradients and still have a linear relation:

1
R. Hooke, Lectures de Potentia Restitutiva (1678).
2 This relation was proposed by J. C. Maxwell, Phil. Trans. Roy. Soc, A157,49-88 (1867), to
investigate the possibility that gases might be viscoelastic.