Consider an incompressible polymer melt, with coarse grains as a concentrated suspension of n nonlinear elastic dumbbell molecules per unit volume. In the flow, the total-viscoelastic, extra-stress tensor can be written as:

(12–1)

where and are the extra-stress contributions from the amorphous and semicrystalline phases, respectively. A purely Newtonian stress component can be added, which is given by

(12–2)

where is the shear viscosity, and is the rate of deformation tensor.

The extra-stress tensor must obey the momentum equation, which, for a steady-state flow in the absence of volume forces, reduces to

or

(12–3)

where p stands for the pressure. Constitutive equations are required for calculating the stress contributions from both amorphous and semicrystalline phases. Individual macromolecules may undergo a conversion process from amorphous to semicrystalline phases in this flow. This conversion is governed by a degree of transformation (x) that may depend on the current stress state in the flow.

The temperature T plays a significant role in the crystallization mechanism, and the associated energy equation incorporates contributions from crystallization (such as enthalpy). These four distinct components are discussed in the following sections.