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Unformatted text preview: E P2 IA Mats. & Min. Sci. - E: Mech. Behaviour of Solids - Practicals Guide EPG13 Rubber Elasticity
1. Background There are two parts to this practical, which can be carried out in either order. Both involve
experimental study of the equations of rubber elasticity. In the first part, the relationship between
the size of a spherical balloon and the pressure within it is explored using a simple manometer
arrangement. The results allow an estimate to be made of the cross-link density within the rubber
of the balloon. In the second part, a long cylindrical balloon made of a similar rubber is subjected
to uniaxial tension and is then heated. This allows both the cross-link density to be measured and
the observed dependence of stiffness on temperature to be compared with theory. 2. Experimental Procedures 2.1 Inflation of a Spherical Balloon Firstly, the balloon is completely inflated, using a bicycle pump. This must be done while the
bung is placed in the end of the U-tube, in order to ensure that the manometer fluid (coloured
water) is not ejected by the pressure pulses associated with operation of the pump. Once the
balloon is fully inflated, the tap is closed, so that air cannot escape from the balloon, and the bung
is removed in order to allow the trapped air out and the internal pressures to equalise. Fig.1 Photograph of the apparatus for studying the pressure-size relationship in a spherical
balloon. The balloon radius is estimated, using calipers to measure the diameter in three orthogonal
directions. The height difference in the manometer is also measured. A small amount of air is then
released from the balloon, by opening the tap for a short period. Further measurements of size and
pressure are then made. This operation is repeated several times, until the pressure falls to
atmospheric and the balloon has reached its uninflated size.
TW C - Len t 2007 E P2 EPG14 Plot the data in the form of pressure, P, against expansion ratio, λ (=R / R0). This plot should be
compared with the form predicted using classical rubber elasticity theory – see Appendix A.
Comment on any discrepancies you observe. Use these results to estimate the number of chain
segments per unit volume in the rubber, nseg. In order to do this, the pressure can be related to the
height difference, h, using this equation
P= h!g (1) where ρ is the density of the liquid and g is the acceleration due to gravity. A convenient method
of evaluating nseg is to use eqn.(A.11) to predict the extension ratio at which the peak in pressure,
Pmax, is expected to occur and then to substitute this value of λ, and the experimentally-obtained
value of Pmax, into this equation, together with measured values of the membrane thickness and
radius of the uninflated balloon, and hence to solve for nseg.
2.2 Heating of Rubber while under Uniaxial Tension Fig.2 Apparatus for heating of a rubber strip under tension The relationship between (nominal) stress and extension ratio during uniaxial loading (see
EH10) may be written 1'
! = nseg kT & " # 2 )
% (2) Use this equation to estimate the number of chain segments per unit volume in the rubber, nseg, by
first measuring the initial (unstressed) length and sectional area of the rubber strip (cylindrical
balloon) and then measuring the lengths with several different loads supported (within the perspex
cylinder). Plot the nominal stress, σ (=F/A0) against extension ratio, λ (=L/L0). Calculate nseg from
several pairs of values of σ and λ, and take the average. Compare this with your value from the
balloon inflation experiment.
Hang a suitable weight on the strip and then switch on the heater, making sure that the free flow
of air is unimpeded, both at the top end (air intake of the hair-dryer) and at the bottom, where the
end of the perspex tube must not be too close to the surface of the bench or the support stand. It
TW C - Len t 2007 E P2 EPG15 follows from eqn.(2) that the “modulus” of rubber (nsegkT) is predicted to rise linearly with the
(absolute) temperature. The weight should rise, as a consequence of the rubber becoming stiffer.
Measure the extension of the strip before heating and then after a new steady state has become
established with the heater switched on. This thermal equilibration should take a couple of minutes
or so. Note the temperature reading at this point.
Check whether the ratio of the temperatures before and after switching on the heater is equal to
the inverse ratio of the corresponding (λ – λ-2) expressions. Outline any relevant effects not
included in this analysis and decide whether they might account for any observed discrepancies. Appendix A – Analysis of Biaxial Tension Case
Consider a square membrane, with initial unstretched sides of length L0. This is now put under
equal biaxial tension, so that the two in-plane extension ratios are equal !1 = !2 = !
" !3 = 1
(since !1! 2 ! 3 = 1)
!2 (A.1) substituting this into the equation relating entropy change to extension ratios:
# nseg k &
!S = " %
( * )1 + )2 + )3 " 3,
$2' (A.2) # nseg k &
!S = " %
' (A.3) leads to
, 2 ) + ) 4 " 3/
. so the work done per unit volume on stretching is
# nseg kT &
!G = " %
, 2 ) + ) 4 " 3/
. (A.4) and hence the work done on the square membrane is given by
" nseg kT %
+ 2 ( + ( 4 ! 3 . L0 t 0
- (A.5) where t0 is the original thickness of the membrane.
Now, consider incrementally extending the membrane in the two in-plane directions, so λ is
increased by δλ. The work done may be written as !W = 2 F ( L0 !" ) (Α.6) where F is the force applied (in both in-plane directions). It follows that F= 1 !W 1
2 !" L0 (Α.7)
TW C - Len t 2007 E P2 EPG16 and hence 4& L
F = nseg kT % 4 ! " 5 ( t 0 0
!' 4 (Α.8) The force per unit length is thus given by F
= nseg kT % 1 " 6 ( t 0
!' (Α.9) Applying this to a spherical balloon, the total force acting across a central section is given by
P π R2, where P is the internal pressure: this must be equal to the force per unit length in the skin of
the balloon, multiplied by its circumference 1&
nseg kT % 1 ! 6 ( t 0 2) R = P) R 2
"' (Α.10) Noting that λ is equal to R / R0, this can be rearranged to give an expression for the pressure in the
balloon as a function of its size
P = 2 nseg kT ! "1 " ! "7 % 0 (
$ R0 ' ( ) (Α.11) TW C - Len t 2007 ...
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This note was uploaded on 07/20/2011 for the course EMA 6165 taught by Professor Brennan during the Spring '08 term at University of Florida.
- Spring '08