Chapter 1

Continuum Mechanics, Rheology and Heat Transfer Overview

The aim of this chapter is to provide the readers with the basics of continuum mechanics, rheology and heat transfer, which will be of fundamental importance throughout the remainder of this book. We will keep this presentation as concise as possible by avoiding unnecessarily detailed mathematical manipulations, and referring interested readers to other pertinent references in the literature.

1.1. Continuum mechanics

1.1.1. Strain

Let us consider the deformation of a continuous medium defined by the displacement vector field of components: U(x,y,z), V(x,y,z), W(x,y,z). The corresponding strain (provided that it is small) can be described by a symmetric tensor as follows [SAL 88, AGA 14]:

[1.1]

Hereafter, a symmetric tensor will be considered as a square matrix, involving six independent terms:

1.1.2. Strain rate

We will now consider the velocity field u(x, y, z), v(x, y, z), w(x, y, z) which is associated with the aforementioned strain. Just like the strain tensor, the strain rate tensor can be defined by:

[1.2]

Unlike the strain tensor, defined by relation [1.1] for small deformations, the strain rate tensor is defined in a general manner. It is, therefore, well adapted to the description of fluid flows, for which the deformations are always very large. Just like , the terms for the strain-rate tensor have a particular meaning:

EXAMPLE 1.1.– Let us consider two elementary flows, which we will come across often in the remainder of this book. The first flow is a planar shear flow between two plates (see Figure 1.1(a)). The bottom plate is immobile, whereas the top plate is mobile with a velocity V. The velocity field is, a priori, in the following form: u = u(y), v = 0 and w = 0. The strain rate tensor is reduced to:

[1.3]

The second flow occurs between two immobile plates under a pressure drop (Figure 1.1(b)). This is called a Poiseuille flow. The velocity field has the same form as above, i.e. u = u(y), v = 0 and w = 0.

Figure 1.1. Flow between parallel plates: a) simple shear flow and b) Poiseuille flow

Consequently, the strain rate tensor will also have the same expression. All flows which give rise to this type of tensor (a single symmetric non-zero term) are called simple shear flows. In the corresponding Cartesian coordinate system, Ox is the shear direction, Oxy is the shear plane and du/dy is the shear rate.

1.1.3. Stress

Let us consider a small surface ds upon which a force dF is exerted (Figure 1.2). By definition, the stress vector at point O of this surface is the limit dF / ds as ds tends toward zero. This vector is projected:

Figure 1.2. Stress applied on a surface

The stress vector cannot define a general state of stress, since it is associated with the orientation of the surface on which it is exerted. It is the stress tensor which will define this general state. As with the tensors mentioned above, it is a symmetric tensor written as:

[1.4]

As with the previous tensors, there are six terms involved:

From the stress tensor, the stress vector at any normal point can be calculated using the following equation:

[1.5]

For any stress state, the hydrostatic pressure can be defined by the following equation:

[1.6]

where tr is the trace of the stress tensor, i.e. the sum of the diagonal terms: tr = σxx + σyy + σzz.

This allows the stress tensor to be broken down into the sum of the hydrostatic pressure and the traceless tensor, called the deviator:

[1.7]

where is the identity tensor. It is the deviator that allows us, in section 1.2.2, to define the constitutive laws.

1.1.4. General equations in continuum mechanics

From the above concepts, we can now establish the basic equations that will be used to solve a flow problem.

1.1.4.1. Continuity equation

The continuity equation expresses the conservation of mass in a flow. It is generally written as:

[1.8]

where ρ is the density, is the velocity vector and “div” is the divergence operator.

In the case of an incompressible medium, as in the case of molten polymers, this equation is simplified to:

[1.9]

i.e. in Cartesian coordinates:

Using the definition of the trace of a tensor, this can also be written as:

[1.10]

1.1.4.2. Equilibrium equation

The equilibrium equation expresses the equilibrium of forces being exerted upon an element of material. If F represents the forces of mass (e.g. gravity) and ργ the forces of inertia, the equilibrium equation is written as:

[1.11]

The divergence of a tensor is a vector whose components are the divergence of each row of the tensor.

[1.12]

In the case of an incompressible medium, the unknowns of the problem are the three components of the velocity vector and the six components of the stress tensor. The continuity equation [1.8] and the equilibrium equation [1.11] give us four equations. The five missing equations will be supplied by the constitutive equation, i.e. a relation between the stress tensor (or its deviator) and the strain rate tensor. The definition of these laws is the purpose of rheology, which is presented in the next section.

1.2. Rheology

The molten polymers discussed in this book are naturally viscoelastic. In fact, in the majority of flows encountered in extrusion, it is the viscous characteristic that will predominate and we will limit ourselves solely to the discussion of this behavior. We refer readers who are interested in viscoelasticity to more specialized sources [FER 70, BIR 77, DEA 90].

1.2.1. Newtonian behavior

Let us consider the flow between parallel plates presented in Figure 1.1(a). Viscosity is, by definition, the ratio of the stress exerted upon the top plate to the shear rate:

[1.13]

This viscosity is constant and does not depend on pressure or temperature. The overall equation of Newtonian behavior can be written in terms of the stress and strain rate tensors:

[1.14]

By applying this ratio to the case of shearing between plates, we obtain a shear stress equal to η and three identical normal stresses:

[1.15]

The latter is characteristic of inelastic behavior.

The average viscosity of a molten polymer is approximately 102 to 104 Pa.s; this is very significant compared to the viscosity of conventional fluids such as water (10–3 Pa.s) and oil (10–2to 1 Pa.s).

This has many important consequences for flow, especially since, in the majority of cases, the mass and inertia forces are negligible in comparison to viscous forces. Therefore, in extrusion, even at high rates, the flow will always be laminar, well below turbulent regimes. Mixing operations will certainly become much more difficult.

By introducing the equation for Newtonian behavior [1.14] into the equilibrium equation [1.11], and after deeming the body forces and inertia forces to be negligible, we obtain the Stokes equations:

[1.16]

linking the pressure gradient to the Laplacian of the velocity field. For simple shear flows between plates, which have already been mentioned, this is reduced to:

[1.17]

From this, we deduce that the pressure gradient is constant. For the shear flow (Figure 1.1(a)), this means that the pressure between the two planes is uniform. From this, we can deduce that the velocity profile is linear and independent of viscosity:

[1.18]

For the Poiseuille flow (Figure 1.1(b)), by considering the velocities to be zero on the upper and lower planes, we obtain:

[1.19]

where L is the flow length. The velocity profile is parabolic. The average velocity is obtained from equation [1.20]. It is proportional to the pressure gradient and inversely proportional to the...