Materials Science on CDROM User Guide
Rubber Elasticity
Version 2.1
Boban Tanovic, MATTER
David Dunning, University of North London
Philip Withers, University of Cambridge
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 stresstemperature
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:
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 rootmeansquare
endtoend distance. Having seen a big enough sample, the results can be compared to the
theory, which states that the rootmeansquare endtoend 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 endtoend 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 crosslinked 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 A_{0} is the initial crosssection; 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
