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Materials Science on CD-ROM User Guide

Deformation of Polymers

Version 2.1

Boban Tanovic, MATTER
David Dunning, University of North London

Assumed Pre-knowledge

It is assumed that the user is familiar with the terms elasticity, stress, strain, modulus, viscosity and is able to manipulate basic first order differential equations.

Module structure

This module comprises four sections:

• Introduction
• Mechanical Models of Viscoelastic Behaviour
• Dynamic Properties
• Design methods

Introduction

Stress-strain curves are often used to define several mechanical properties of polymers. This sections starts by defining modulus, tensile strength, elongation at break and yield strength on a typical stress-strain graph. It follows by discussing the difference between elastic, linear viscoelastic and non-linear viscoelastic materials. To reinforce these concepts, a stress-strain graph is plotted for each of the above materials at two values of elapsed time.

A characteristic feature of polymers is the way in which their response to an applied stress or strain depends on the rate, temperature or time period of loading. These effects are shown in turn.

The last two pages in this section concentrate in more detail on the effects of temperature on polymer materials, including descriptions of the four regions of viscoelasticity.

After studying this section, the user should be able to:

• Describe how the behaviour of polymers deviate from that of linear-elastic materials such as metals and ceramics;
• State that time and temperature are critical parameters in the deformation of polymers.
• Explain the importance of time and temperature, in the deformation of polymers.

Mechanical models of viscoelastic behaviour

This section shows how viscoelastic behaviour of polymers can be simulated using elastic springs and viscous dashpots. It starts by discussing the individual properties of elastic springs and viscous dashpots under different loading conditions. The first model to be discussed is the Maxwell model, which consists of a spring and dashpot in series as shown here.

The user can study the effect of different loads and viscosity on the stress-strain curve (creep and recovery) and stress-time curves (relaxation). A tensile test on a Maxwell model is also discussed.

The full derivation for differential equations is not always given, but all the important derivation conditions are clearly stated and the final solution is presented for each of the models discussed. Apart from the Maxwell model the following models are also studied: Kelvin (Voigt) Model, Zener and 4-element model.

This section contains several simple exercises and finishes with a revision questions exercise. It refreshes most of the important points made earlier in the module.

After studying this section, the user should be able to:

• Describe the deformation behaviour of an ideal elastic spring and ideal viscous dashpot under static applied stress.
• Describe the deformation behaviour of an ideal elastic spring and ideal viscous dashpot under static applied strain.
• Manipulate the constitututive equations for an ideal elastic spring and for an ideal viscous dashpot.
• Describe the Maxwell, Kelvin, Zener and 4-element models of viscoelastic behaviour.
• Draw the general shape of the stress-time and strain-time curves for the Maxwell and Kelvin models under a given applied strain and stress respectively.
• Describe how the shapes of these curves are altered by changing the values of the parameters.
• Describe how the addition of extra elements would improve the modelling of real polymers.

Dynamic properties

This section starts by comparing the response of an elastic, viscous and viscoelastic material to cyclic loading. Real and imaginary components of the stress are also defined. An animation is created to show how strain and stress vectors rotate. The angular phase difference between these two vectors is denoted d (delta). By dividing each component of the stress by the strain, two components of a complex modulus are formed:

• In-phase component, known as Storage modulus (E'); and
• Out-of-phase component, known as Loss Modulus (E'').

The ratio of E'/E'' = tan d is a frequently used value, proportional to the ratio of energy lost to energy stored in one cycle (loss factor).

On the last two pages, the time-temperature superposition principle is studied. An animation is created so the user can see the process of shifting the isotherms on the time scale and by doing so, a master curve is produced. This is the basis of time temperature superposition, giving results over a wider time range than is available experimentally.

After studying this section, the user should be able to:

• Define dynamic testing within the context of polymers.
• Distinguish the response of elastic, viscous, and viscoelastic elements to cyclic loading.
• Define the complex modulus of a polymer.
• Define phase angle.
• Explain the relationship between complex modulus and phase angle.
• Predict the effects of temperature and test rate on the behaviour of a polymer.
• Describe the interchangeability of time and temperature in their effect on the mechanical behaviour of polymers.
• Describe the construction of viscoelastic master curves for a polymer.

Design Methods

This short section contains an exercise where the user is asked to create an isometric graph using the given creep curves for rubber-toughened polypropylene at 20⁰C. The last part of this section explains the Boltzmann Superposition principle.

After studying this section, the user should be able to:

• To describe how long-term deformation behaviour of polymers is presented for the purposes of product design.
• Use the Boltzmann Superposition Principle to determine the strain in a sample subjected to a complex loading and unloading cycles.

Bibliography

The student is referred to the following resources in this module: