Polymer Extrusion

Pierre G. Lafleur, Bruno Vergnes, Pierre G. Lafleur, Bruno Vergnes

• Publisher: Wiley-ISTE
• Year: 2014
• Print ISBN: 9781848216501
• eBook ISBN: 9781118827178
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Mechanical Engineering
• Index: Technology & Engineering

Extrusion is by far the most important and the oldest processing and shaping method for thermoplastic polymers. This process concerns almost all synthetic polymers, as well as elastomers or food materials. Single-screw extrusion is mainly used nowadays to manufacture finished goods or semi-finished products. More than 90 million tons of thermoplastics are therefore processed every year.
Twin-screw extrusion may be divided into two systems: contra-rotating systems used within the context of PVC extrusion, for the manufacture of pipes or profiles; and co-rotating systems experiencing nowadays a very significant development, because of their significant adaptability and flexibility, which enables the manufacture of specific materials (polymer alloys, thermoplastic elastomers, filled polymers, nanocomposites). Extrusion is carried out by passing molten polymer through a tool called die that will give the product its final shape (films and sheets, rolled products, and electric cables). Thanks to the design of dies, we obtain at the output a product with controlled dimensions, uniform speeds and homogeneous temperatures. The book will discuss the same production types, but only in the case of coextrusion flows, i.e. multilayer stratified products. First of all, we will present in this book the physics of the mechanisms at stake, then propose more or less complex models in order to describe these mechanisms and then go forward in the interpretation of results and the control of condition flows.

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:

– the diagonal terms (ε_xx,ε_yy ,ε_zz) correspond to uniaxial deformations of traction or compression;

– the symmetric terms (ε_xy = ε_yx,ε_yz = ε_zy,ε_xz =ε_zx) correspond to shear deformations.

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:

– the diagonal terms are elongation rates, often referred to, below, as

;

– the symmetric terms are shear rates, often referred to, below, as

.

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:

– on the normal

to the surface at point O : σ_n =

.

is the normal stress, traction or compression;

– on the tangent plane: σ_t is the shear stress.

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:

– the diagonal terms, corresponding to normal stresses (along axes x, y and z), of traction or compression;

– the symmetric terms, corresponding to the shear stresses (in the planes xy, xz and yz).

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

= σ_xx + σ_yy + σ_zz.

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...

Table of contents

• Cover
• Table of Contents
• Title Page
• Copyright
• Introduction
• Chapter 1: Continuum Mechanics, Rheology and Heat Transfer Overview
• Chapter 2: Calculation Methods
• Chapter 3: Single-Screw Extrusion
• Chapter 4: Co-rotating Twin-Screw Extrusion
• Chapter 5: Profile Extrusion
• Chapter 6: Production of Films and Sheets
• Chapter 7: Wire Coating and Cable Insulation
• Index