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Materials Science on CD-ROM User Guide

Rubber Elasticity

Version 2.1

Boban Tanovic, MATTER
David Dunning, University of North London
Philip Withers, University of Cambridge

Assumed Pre-knowledge

Before starting this module, it is assumed that the user is familiar with

  • Simple concepts of mechanical behaviour such as stress, strain, modulus.
  • Basic thermodynamic concepts such as internal energy, work, entropy and the first and second laws.
  • The general long chain structure of carbon based polymers.

Module Structure

This module is presented in one section in a linear fashion. On completion, the student should be able to:

  • Explain the deformation behaviour of rubbery materials in terms of the molecular mechanisms that provide restoring forces.
  • Distinguish entropic and internal energy contributions to stress in a deformed rubber.
  • Calculate relative entropic and internal energy contributions from stress-temperature data.
  • Appreciate that the force (modulus) in an ideal rubber is linear with absolute Temperature.
  • Relate the shear modulus of a rubber sample to the number of molecular chains per unit volume.

Tensile test

The module starts by introducing the standard tensile test plot for natural rubber and emphasising that samples deform uniformly and most deformation is recoverable on release of stress. An alternative measure of deformation is the extension ratio:

l = 1 + e

(1)

Where e is the sample strain.

The freely jointed chain

The freely jointed chain consists of a chain of equal links jointed without the restriction that the valence angles should remain constant. A simple mechanical model consisting of four atoms is used to illustrate how the position of the last atom in a chain depends on all other values of bond angles of previous atoms, each of which can take values between 0 and 2p. This is an idealised freely jointed model.

On the simulation, the student can observe the statistics for the root-mean-square end-to-end distance. Having seen a big enough sample, the results can be compared to the theory, which states that the root-mean-square end-to-end distance is proportional to the square root of the number of links in the chain:

(2)

Where a is the length of chain link and n is the number of links in the molecule.

The Gaussian model

The Gaussian chain (or model) assumes that end to end separation of a macromolecule follows Gaussian statistics. It gives the probability that for a sample chain, one end lies in the volume element dV at r from the other end.

This is the density distribution of end points and shows that if one end of the chains is taken at the origin, the probability is highest of finding the other end in a unit volume near the origin. (The most probable end-to-end distance is not zero!)

This probability decreases continuously with increasing distance from the origin. On the other hand, the probability of finding a chain end within a volume of a spherical shell between distances r and r+dr from the origin has a maximum. This point is emphasised in the exercise where the user is asked to find the correct relationship between the two.

Thermodynamics of rubber elasticity

When a stress is applied to a sample of cross-linked rubber, equilibrium is established fairly rapidly. Once at equilibrium, the properties of the rubber can be described by thermodynamics. Starting with the first law of thermodynamics, we derive the equation for tensile (retractive) force at constant volume and temperature:

(3)

Where A0 is the initial cross-section; N is the number of chains in the network per unit volume and k is Boltzmann’s constant.

The plot of f against T shows that this force should increase in proportion to the absolute temperature in contrast to all other materials.

By using plots of f versus T, users are asked to calculate the relative magnitudes of internal energy and entropic contributions to f.

Bibliography

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

Treloar L.R.G., The Physics of Rubber Elasticity, Clarendon Press, Oxford, 1975

Aklonis J.J., Introduction to Polymer Viscoelasticity, Wiley , 1995

McCrum, N.G., Buckley, C.P., Bucknall,C.B., Principles of Polymer Engineering,

Oxford University Press, 1988

Ward, J.M., Mechanical Properties of Solid Polymers, Wiley , 1979

Rosen, S.L., Fundamental Principles of Polymeric Materials, Wiley, 1993

Gedde, U.W., Polymer Physics, Chapman & Hall, 1995

 

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Copyright of The University of Liverpool 2000