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1 Formulas of Rheology
One of the most important steps in processing polymers is melting the resin, which is
initially in the solid state, and forcing the melt through a die of a given shape. During this
operation, the melt, whose structure plays a key role in determining the quality of the
product to be manufactured, undergoes different flow and deformation processes.
The plastics engineer has therefore to deal with the melt rheology, which describes the flow
behavior and deformation of the melt. The theory of elasticity and hydromechanics can
be considered the frontier field of rheology, because the former describes the behavior of
ideal elastic solids, whereas the latter is concerned with the behavior of ideal viscous fluids.
Ideal elastic solids deform according to Hooke's Law and ideal viscous fluids obey the
laws of Newtonian flow. The latter are also denoted as Newtonian fluids. Plastic melts
exhibit both viscous and elastic properties.
Thus, the design of machines and dies for polymer processing requires quantitative
description of the properties related to polymer melt flow. Starting from the relationships
for Hookean solids, formulas describing viscous shear flow of the melt are treated first,
as far as they are of practical use in designing polymer machinery. This is followed by a
summary of expressions for steady and time-dependent viscoelastic behavior of melts.
1.1 Ideal Solids
The behavior of a polymer subjected to shear or tension can be described by comparing
its reaction to external force with that of an ideal elastic solid under load. To characterize
ideal solids, first of all it is necessary to define certain quantities as follows [I]:
The axial force Fn in Figure 1.1 causes an elongation A/ of the sample of diameter dQ and
length I0 that is fixed at one end. Following equations apply for this case:
Engineering strain:
Hencky strain:
Tensile stress:
(Li)
(1.2)
(1.3)