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The elasticity and related properties of rubbers
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1973 Rep. Prog. Phys. 36 755
(http://iopscience.iop.org/0034-4885/36/7/001)
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The elasticity and related properties of rubbers
L R G TRELOAR
Department of Polymer and Fibre Science, University of Manchester
Institute of Science and Technology, PO Box no 88, Sackville Street,
Manchester M60 lQD, U K
Abstract
I n Section 1 the relation of rubbers to other classes of polymers and the
molecular basis of rubber elasticity are briefly examined. I n Section 2 the methods
used in the quantitative development of the statistical-thermodynamic theory of a
molecular network are outlined, and the main conclusions, in the form of stressstrain relations, etc, are presented and compared with experimental data. Section 3
examines the photoelastic properties of rubbers from both theoretical and experimental standpoints and discusses in detail the evidence derived from photoelastic
studies on the statistical segment length in the molecular chain and its relation to
intramolecular energy barriers. Section 4 is concerned with the thermodynamic
analysis of stress-temperature data for rubber and other polymers, with particular
reference to the internal energy and entropy changes during extension under constant pressure or constant volume conditions. Methods of deriving these quantities
are compared and the results related to the modified molecular network theory due
to Flory. Section 5 deals with the phenomena of swelling in liquids and considers
both the effect of swelling on the mechanical properties and the effect of different
types of stress or strain on the swelling equilibrium. Sections 1-5 are concerned
mainly with the statistical theory and its applications, T h e final section examines
in considerable detail the formulation of more general theories of large elastic
deformations on a purely empirical or phenomenological basis, so as to overcome
some of the limitations of the statistical theory. T h e logical basis of these formulations is presented and the conclusions are discussed in relation to the available
experimental evidence. Some common pitfalls to be avoided are emphasized.
This review was completed in August 1972.
Rep. Prog. Phys. 1973 36 755-826
756
L R G Treloar
Contents
1. Historical survey .
1.1. Early developments
.
.
.
1.2. Meyer’s kinetic theory .
1.3. Statistical development .
1.4. The glassy state .
1.5, Crystallization
.
2. Network theory
.
2.1. Entropy of single chain .
.
2.2. Definition of the network
2.3. Basic assumptions.
.
2.4. Calculation of network entropy
.
2.5. Particular stress-strain relations
2.6. General stress-strain relations,
2.7. Experimental verification
.
2.8. Limited extensibility of chains : non-gaussian statistics
2.9. T h e equivalent random link .
.
3. Photoelasticity
.
3.1. Basic concepts
3.2. Network theory .
3.3. Significance of theoretical conclusions
.
.
3.4. Experimental examination
3.5. Effect of swelling on the stress-optical coefficient
3.6. T h e equivalent random link .
3.7. Temperature coefficient of optical anisotropy .
,
,
.
.
.
.
.
.
.
.
,
.
,
.
.
.
.
4. Thermodynamics of rubber elasticity
4.1. Elementary theory
.
.
4.2. Later thermodynamic analyses
4.3. Experimental determination of f,/f .
4.4. Thermoelastic data for torsion
4.5. Conclusion .
5. Swelling and mechanical properties .
5.1. Mechanism of swelling .
5.2. Stress-strain relations for swollen rubber .
5.3. Dependence of swelling equilibrium on strain
.
5.4. Torsion of cylinder
.
.
.
.
.
,
.
.
.
.
.
,
.
,
.
6. Phenomenological theories of rubber elasticity .
.
6.1. Rivlin’s formulation of large-deformation theory
6.2. General stress-strain relations.
.
6.3, Particular stress-strain relations
6.4. Stress components: normal stresses in shear
.
6.5. Torsion of cylinder
.
6.6. Experimental determination of form of stored energy function
.
.
.
.
.
.
,
.
.
.
.
.
Page
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804
805
808
757
The elasticity and related properties of rubbers
6.7.
6.8.
6.9.
6.10.
Significance of deviations from statistical theory
Alternative forms of representation .
Compressible rubbers .
Compressibility of ‘solid’ rubbers .
6.11. Ogden’s theory .
References
.
.
.
.
.
.
810
813
819
821
822
,
824
,
758
L R G Treloar
1. Historical survey
T h e recent rapid growth of the polymer industry has been accompanied by a
corresponding expansion in scientific developments in the field of rubbers, plastics
and fibres. Historically these developments have been mainly of a chemical nature,
and have been inspired by the urge to discover new types of materials having
commercially attractive possibilities. More recently, however, the emphasis has
shifted to the physical and engineering aspects of these materials, with a view to
obtaining a rational basis for their more effective utilization in a variety of industrial
applications. I n this development the knowledge gained from the study of the
physical properties of rubbers, and in particular the relation of these properties to
their underlying molecular structure, has provided an essential basis for the interpretation of the properties of a wide variety of more complex materials.
T h e primary purpose of this report is to give an account of the development of
the physics of rubber elasticity, starting from the kinetic or statistical theory, which
provides the essential physical background to the subject, and leading on to the
more recent development of methods of representation of the elastic properties of
rubbers in the most general mathematical form, that is, to the mechanics of rubber
elasticity. At the same time a number of closely related phenomena encountered
in rubbers, particularly their photoelastic properties, swelling in solvents, and
thermoelastic or thermodynamic behaviour, will be examined.
1.1. Eavly developments
For more than a hundred years the only type of rubber of any industrial
importance was natural rubber, derived almost exclusively from the tree Hevea
braxiliensis. T h e rubber latex is a milky liquid consisting of a suspension of rubber
particles of diameter 0.1 to 1.0 pm in a watery medium. From this latex the rubber
is readily precipitated by adding acetic acid or other reagents to form the familiar
raw rubber (cr&peor smoked sheet) of commerce. Faraday, who analysed both the
rubber and non-rubber constitutents of latex, established as long ago as 1526
(Faraday 1526) that the rubber component was a pure hydrocarbon, having a
constitution corresponding exactly to the empirical formula (C,H,),. Raw rubber
as an industrial material suffered from certain disadvantages, the chief of which
were its tendency to become sticky and to lose its elasticity in use, particular in
vi-arm climates, and an opposite tendency, now known to be due to crystallization,
to become hard and inflexible at temperatures around freezing point or below.
Both these undesirable features were effectively suppressed by the discovery by
Goodyear in 1839 of the process known as vulcanixation-a process which rescued
the industry from the very serious difficulties it was experiencing at the time, and
which has remained virtually unchanged until the present day as the basis of rubber
technology. This process consists essentially in a chemical reaction with sulphur ;
at the time of its discovery, and for some hundred years thereafter, the reason for
the changes brought about by this reaction remained a mystery. I t is now known
to be due to a cross-linking of the long-chain molecules, leadipg to the production
of a coherent three-dimensional network which is more perfectly elastic, and at the
same time suppressing the tendency to crystallization.
The elasticity and related properties of rubbers
759
Rubber is peculiar not only in its mechanical properties but also in its thermodynamic or thermoelastic properties. One of the earliest contributions to the subject
of the physics of rubber was that of Gough (1805) who showed that stretched rubber
contracted on heating and extended on cooling. He also demonstrated that heat
was evolved on stretching and absorbed on retraction. These effects were confirmed some fifty years later by Joule (1859) who was able to work with the more
perfectly reversible vulcanixed rubber. T h e subject attracted the attention of Kelvin
(1857) who showed that these two effects were thermodynamically related, that is,
that a material in which the retractive force increases on heating will exhibit a
reversible evolution of heat on extension.
Early attempts to interpret the peculiar mechanical and thermoelastic properties
of rubber, however, could not overcome the difficulty of explaining its very high
extensibility in terms of classical concepts of the structure of matter. This high
extensibility, coupled with values of Young’s modulus of the order 1.0 Nmm-2
(about 10-5 of those for normal solids) defied any explanation in terms of cohesional
forces between molecules as normally understood. T h e structure of the rubber
molecule, so far as was known, could hardly have been simpler, and provided no
clue to the unique properties of this material compared with other hydrocarbons
(paraffins, benzene, etc), which were mostly simple liquids.
T h e solution to the problem was more or less coincident with the emergence of
the concept of a high polymer, that is, of a material composed of molecules of
extremely high molecular weight (in the range 10 000-1000 000), built up by the
successive addition of similar units in the form of a single chain. Prior to this, in the
decade 1920-30, there had been violent controversy on the question of the true
molecular weight of typical colloidal materials, such as gelatin, starch, rubber, as
well as of fibrous materials such as cotton, silk, wool, muscle fibre (myosin), tendons
(collagen), etc (Flory 1953). T h e acceptance of the concept of a high polymer constituted a revolution of thought not only in relation to the chemistry of these
materials, but equally in relation to the interpretation of their physical properties.
1.2. Meyer’s kinetic theory
It was Meyer (Meyer et a1 1932, Meyer and Ferri 1935) who first clearly
appreciated the connection between the chain-like structure of the polymer molecule
and the long-standing problem of rubber elasticity. H e realized that such a
(a)
(6)
Figure 1. (a) Rotation about bonds in long-chain molecule. (b) Resultant statistical form.
molecule was not to be regarded as a rigid structure, like a stiff rod, but rather as a
flexible chain of more or less freely rotating links. H e saw that as a result of internal
rotations of individual links such a chain will take up a randomly kinked form, in
which the distance between its ends is governed by purely statistical considerations
(figure 1). If forcibly extended and released, the molecule will quickly revert to its
760
L R G Treloar
normal crumpled form as a result of the random thermal rotations and vibrations
of the individual atoms. T h e molecule therefore possesses in itself the property of
long-range elasticity.
With this complete break from classical modes of thought Meyer at a single
stroke was able to explain both the elastic and the thermodynamic or thermoelastic
properties of rubber. By introducing the assumption that internal bond rotations
are essentially unrestrained, that is, that all conformations of the molecule have the
same internal energy, he showed that the spontaneous retraction of the extended
chain is due simply to the higher probability of the contracted state. I n thermodynamic terms this implies that the elasticity is associated with a reduction of
configurational entropy on extension. T h e elasticity of rubber is thus not static, as
is that of an ordinary crystalline solid, but kinetic, like the volume elasticity of a gas,
which is similarly a function of configurational entropy.
1.2.1. Thermodynamic consequences of kinetic theory. T h e Gough-Joule effects
follow immediately from the initial assumptions of the kinetic theory. If dU, d W
and d Q represent the changes in internal energy, mechanical work and heat,
respectively, corresponding to an increase dl in length, the first law of thermodynamics states that
d U = dQ+dW.
(1.1)
Introducing the assumption that for the rubber, as for the single chain, the internal
energy (at constant temperature) is independent of the extension, that is, that
d U = 0, it follows that
d Q = -dW.
( 1-2)
Since for an increase in length d W i s necessarily positive, equation (1.2) implies that
dQ is negative, that is, that heat is given out on extension, the heat evolved being
exactly equal to the work done on the rubber by the external force.
T h e second effect-the increase of the retractive force with increase of temperature-is considered in detail in § 4. I t is sufficient for the present to note that since
the elasticity is basically due to the thermal agitation of the atoms in the molecular
chain, the retractive force (at constant stretched length) should be proportional to
this thermal energy, that is, to the absolute temperature, just as the pressure exerted
by an ideal gas (at constant volume) is proportional to absolute temperature. Meyer
and Ferri (1935) showed experimentally that for vulcanized rubber at suitably high
extensions this proportionality was approximately realized (cf figure 16).
1.3. Statistical development
T h e original conception of the kinetic theory was quickly followed up by
detailed mathematical treatments, first of the chain molecule considered as an isolated entity, and later of an assembly of chains corresponding to the bulk rubber.
We shall concern ourselves here only with the first of these problems-the statistics
of the single chain. This development was due mainly to Guth and Mark (1934)
and to Kuhn (1934), who considered an idealized structure corresponding to a
randomly jointed chain of n equal links, each of length 1. If we imagine one end of
such a chain to be fixed at the origin of a Cartesian coordinate system, the probability that the other end shall be situated within a volume element d x d y d x was
shown to be defined by the gaussian probability function
dx dy dx
p(x, y, 2) dx dy dz = (b3/.rr3’z) exp { - b2(x2+y2+ 2)))
(1.3)
The elasticity and related properties of rubbers
761
where
b2 = 3/2n12.
(1.3a)
This function may be resolved into three independent component probabilities of
the type
p ( x) dx = ( b / d 2 )exp ( - b2x2)dx
(1.4)
whose form is reproduced in figure 2(a), implying that the probability of a given
x component of chain end-to-end distance is independent of its y and x components.
r
(b)
Figure 2. Distribution of (a) end-to-end distance Y, and (b) x component of
molecule.
Y,
for long-chain
T h e probability of a given distance r between the ends of the chain, regardless of
direction, is given by a different function P(Y),namely
P ( r ) dr = (4b3/n1/2)
r2exp ( - b2r 2 )dr
(1.5)
whose form is shown in figure 2(b). It is important to note that whereas the component probabilities are a maximum at the origin, the function P ( r ) has a maximum
at a value of Y equal to l / b . T h e root mean square end-to-end distance for the free
chain, obtained from equation (1.5)) is
(p)"2= (3/2b2)'/2= lnlh
and is proportional to the square root of the number of links in the chain.
( 14
762
L R G Treloar
T h e gaussian statistical treatment, as presented above, involves an approximation which is equivalent to the assumption that the number of links is large and that
r<nl
(1.7)
that is, that the end-to-end distance is very much less than the fully extended length
of the chain. T h e more accurate (non-gaussian’ theory, in which this approximation is avoided, will be considered in 5 2.
1.3.1. Application to real molecules. T h e treatment of the molecule in terms of an
idealized randomly jointed chain of equal links is adopted for mathematical convenience only. I n an actual molecule the successive bonds are connected at a
definite angle-the valence angle. T h e simplest of such structures is the paraffintype chain, in which all the bond lengths and bond angles are identical; for such a
structure the RMS length, assuming free rotation about bonds, is given by
where 0 is the supplement to the valence angle. A comparable calculation may be
made for a chain of any structure (Wall 1943). For the chain of natural rubber for
example, in which the repeating unit is cis-isoprene, having the structure
HC=C
/
\
‘CH,
CH;”
Wall, using the accepted values of bond lengths and valence angles, derived the
length of 2.01n1’2 A, where n is the total number nf bonds in the chain.
RMS
1.4. The glassy state
K’ot all polymers are rubberlike, and even in rubber itself the rubbery properties are exhibited only within a certain range of temperature. I t follows, therefore,
that the presence of long-chain molecules, though necessary, is not in itself a
sufficient condition for the appearance of this phenomenon.
One further necessary condition is that the molecular segments in the chain
shall have sufficient thermal energy to free themselves from the fields of force of
their immediate neighbours, so that the chains may be able to undergo the random
changes in conformation on which the phenomenon of rubberlike elasticity depends.
It is only those polymers in which the intermolecular forces are sufficiently weak
which satisfy this condition. I n rubbers, the intermolecular forces are similar to
those in a liquid. On lowering the temperature, however, a point must eventually
be reached at which this condition is no longer satisfied. At this point the thermal
motion becomes limited to an oscillation of atoms about fixed mean positions, as in
a normal solid. I n this state the rubber is hard and brittle, like a glass. For natural
rubber the transformation to the glassy state occurs at about 203 K ( - 70 “C).
T h e rubbers are distinguished from the glassy polymers primarily by the
strength of their intermolecular forces. I n a typical glassy polymer such as polymethyl methacrylate (Perspex), the presence of polar groups attached to the main
chain increases the intermolecular forces, so raising the glass transition temperature
The elasticity and related properties of rubbers
763
to + 100 "C. It is only when heated above this temperature that such a polymer
begins to develop rubberlike properties.
1.5. Crystallization
T h e failure to exhibit rubberlike properties may arise alternatively from
crystallization. Many synthetic polymers, for example, nylon, polyethylene, polypropylene, as well as natural polymers such as wool, silk and cellulose, exist in a
partially crystalline state. Rubber itself may be rendered crystalline either by
prolonged exposure to low temperatures (0 "C or below), or by stretching at room
temperature. I n the latter case the crystallization is first evident at an extension of
200-250~0,and increases in amount with increasing extension (Goppel 1949). When
crystallized in the unstrained state rubber has properties similar to those of an
undrawn crystalline polymer such as polyethylene (polythene). Such polymers,
though not rubberlike, retain a degree of flexibility, due to the fact that they are
only partially crystalline, the remainder of the structure being in the amorphous
state. T h e maximum degree of crystallinity depends on the type of polymer and
on such factors as molecular weight, chain branching, cross-linking, etc, and ranges
from about 30% (unvulcanized rubber) to about 90% (high-density polythene).
(For a discussion of the subject of crystallization and crystalline polymers the
reader is referred to Mandelkern (1964), Geil (1963) and Keller (1968).)
2. Network theory
2.1. Entropy of single chain
T h e application of the statistical theory to the problem of a network of longchain molecules corresponding to a vulcanized or cross-linked rubber starts from the
concept of the entropy of the single chain. Extension of the chain is associated with
a reduction of entropy, and vice versa. Our first requirement, therefore, is to obtain
a quantitative expression for the entropy of the single chain as a function of the distance between its ends. This is derived from the gaussian probability function
(1.3) by the application of the Boltzmann relation between entropy S and probability P, that is,
S =klnP
(2.1)
where K is Boltzmann's constant. Application to equation (1.3) thus gives, for the
chain entropy s,
s = c - kb'(x2 + y 2+ 9 )= c - kb2r2
(2.2)
in which c is a constant which includes the arbitrary volume element dx dy dz, and
r is the distance between the ends.
T h e interpretation of the result (2.2) is that if the two ends of a chain are held
in fixed positions (to within a small volume element of arbitrary size), the difference
of entropy between any two states is proportional only to the corresponding difference
of r2. I n particular, it is to be noted that the entropy (like the probability) is a
maximum when the two ends of the chain are coincident ( r = 0).
2.2. Dejinition of the network
T h e network may be defined either in terms of the number of cross-linkages
introduced between the originally independent molecules, or alternatively in terms
764
L R G Treloar
of the number of ‘chains’ or network segments, a ‘chain’ in this sense being defined
as the segment of a molecule between successive points of cross-linkage. For the
simplest type of cross-linkage, in which four chains (as here defined) radiate from
each junction point, the number of chains is theoretically equal to twice the number
of cross-linkages.
I n the approximate form of the network theory-the so-called gaussian network
theory-a knowledge of the number of chains per unit volume, denoted by N , and
of the mean square end-to-end distance for the assembly of chains in the unstrained
state (r?), is sufficient to determine the properties.
2.3. Basic assumptions
A variety of theoretical models has been devised for the calculation of the elastic
properties of the network (Flory and Rehner 1943a,b; James and Guth 1943, Wall
1942, Treloar 1943). Though differing in degree of mathematical refinement, all
these models embody the same basic physical concepts and lead to substantially
similar conclusions. T h e essence of the problem is to relate the change in molecular
dimensions or chain vector lengths resulting from a deformation to the macroscopic
strain, and hence, by introducing the relation between entropy and vector length
for a single chain, to calculate the entropy of the network, first in the unstrained
state and then in the strained state, the difference being the entropy of deformation.
From this it is possible, by standard thermodynamic procedures, to derive the work
of deformation and hence the forces required to produce any specified state of strain.
2.4. Calculation of network entropy
Following the idea originally introduced by Kuhn (1936) whose theory was
subsequently modified by the writer (Treloar 1943) to bring it into line with later
developments, most authors have based their argument on the assumption that the
x, y and z components of the chain vector length Y change on deformation in the
same ratio as the corresponding dimensions of the bulk rubber (afine deformation
assumption). Although originally introduced as an assumption, this property of
the network is in fact deduced mathematically in the more rigorous treatment of the
gaussian network by James and Guth (1943).
A second assumption normally introduced is that the material deforms without
change of volume. (The limitations of this assumption are dealt with in $4.) On
the basis of this assumption the most general type of pure strain (ie a strain not
involving rotation of the principal axes) may be defined by three principal extension
ratios (or principal semi-axes of the strain ellipsoid) (figure 3) A,, A,, A,, such that
h,h,A,
=
1.
Taking the principal axes of strain to be parallel to the x, y and z axes, the components of length for the individual chain change on deformation from (x, y , z ) to
(X,x, A,y, A,x). T h e change of entropy on deformation for the individual chain is
therefore, from (2.2),
AS = - kb2((AI2 - 1)x2+ (A: - 1)y 2+ (A,,
- 1)2’).
(2.4)
This expression to to be summed over the assembly of N chains contained in unit
volume of the network. Since in the unstrained state the molecules are randomly
The elasticity and related properties of rubbers
765
oriented we have 2 = = z"i = +G, where 3 is the mean square chain vector
length in the unstrained state. Putting this equal to for a corresponding set of
chains in the free or un-cross-linked state (3/2b2), as given by equation (1.6)) we
obtain on summation of (2.4) the entropy of deformation A S for the network in the
form
A S = - hNk(A1' + ',A + - 3 ) .
(2-5)
2
T o obtain the work of deformation we introduce the basic assumption of the
kinetic theory, namely that all states of deformation have the same internal energy
(61
(U1
Figure 3. Pure homogeneous strain: ( a ) unstrained state; ( h ) strained state.
(AU = 0). T h e change in Helmholtz free energy, which for an isothermal reversible
process is equal to the work (U/) done by the applied forces, thus becomes
A A = AU- TAS = - T A S .
(2.6)
Hence from (2.5) the work of deformation per unit volume is
W = iNkT(A1'
+ A,* +
- 3).
(2.7)
2.5. Particular stress-strain relations
Equation (2.7))which represents the work of deformation for the most general
type of strain, implies that the elastic properties of the network are independent
of its detailed structure. T o appreciate its significance it will be convenient to
consider initially its application to certain simple types of strain.
2.5.1. Simple extension. This may be defined in terms of a single extension ratio A,
with equal contractions in the transverse dimensions. T h e condition for constancy
of volume (equation (2.3)) requires that
A,
=
A
A,
=
A, = A-li2.
(2.8)
Insertion in equation (2.7) yields the work of deformation
W = $NkT(A2+2/A-3).
(2.9)
T o obtain the force we consider the specimen in the unstrained state to be in the
form of a cube of unit edge length. If F is the force per unit unstrained crosssectional area, we have then
(2.10)
L R G Treloar
766
2.5.2. Uniaxial compression. Simple extension corresponds to the case h > 1. T h e
same formula applies, however, when h < 1. This represents a uniaxial compression
( F negative). The properties in either extension or compression are thus represented
by a single continuous function (cf figure 5).
2.5.3. Simple shear. For a (large) shear strain in the ( x , y ) plane the principal axes
of strain are defined by the extension ratios
A, = h
A, = 1 j X
A,
=
1
(2.11)
and the corresponding shear strain ( y ) is (Love 1927)
y = X - lih.
(2.12)
The work of deformation (2.7) thus becomes
W = & N k T ( X 2- 2 + l/h2) = $ N k T y 2 .
(2.13)
The only work performed is that done by the shear stress t,,. Hence
t,, = d Wjdy = Ark T y .
(2.14)
2.6. General stress-strain relations
Equations (2.10) and (2.14) are examples of particular types of strain. For the
most general type of deformation the principal stresses t,, t , and t, (figure 3) may
similarly be derived from the stored energy function (2.7). These are represented by
the equation
ti = N k T X i 2 + p i = 1,2,3
(2.15)
in which the ti are true stresses (referred to the strained dimensions) and -9 represents an arbitrary hydrostatic pressure. This arbitrary pressure, which is a direct
consequence of the assumption of constancy of volume or volume incompressibility,
implies that the stresses are to this extent indeterminate. T h e p may be eliminated,
however, by considering the dtflerences of any two of the principal stresses, for
example,
t , - t, = N k T ( h I 2- A,)
etc.
(2.15a)
I n a practical problem one of the principal stresses is usually known, and the
remaining stresses are thus determinable. I n a simple extension, for example, the
lateral stresses ( t , and t3) are zero and hence by introducing (2.8) into ( 2 . 1 5 ~ )we
obtain
t, = N k T ( h 2 - ljh).
(2.16)
Since the cross-sectional area is reduced on extension in the ratio l / h , t , = hF,
where F is referred to the unstrained area. Equations (2.10) and (2.16) are thus
equivalent.
It is interesting to note that the principal stress differences are proportional to
the squares of the corresponding extension ratios. (For small strains these reduce to
differences of the first powers, in accordance with classical theory.)
2.6.1. The shear modulus. I t is seen from equation (2.14) that the quantity N k T ,
which occurs in the stored energy function (2.7) and in the various stress-strain
relations, is equivalent to the shear modulus, subsequently designated G. This
quantity may be expressed alternatively in terms of the mean ‘molecular weight’
The elasticity and related properties of Tubbers
767
M, of the chains (ie segments of molecules between successive points of crosslinkage), thus
G = NkT = pRT/M,
(2.17)
where p is the density of the rubber and R the gas constant per mole.
2.6.2. SigniJicance of stress-strain relations. T h e above stress-strain relations are of
the greatest theoretical interest. They define the properties of an ‘ideal rubber’
which may be regarded as somewhat analogous to the ‘ideal gas’ postulated in the
kinetic theory of gases. Attention is drawn particuarly to the following properties.
(i) A rubber should obey Hooke’s law in simple shear, but not in any other type of
strain (eg simple extension or uniaxial compression).
(ii) T h e stress-strain relations for any type of strain involve only a single physical
parameter or elastic constant. This is NkT, which appears in the stored energy
function (2.7) and is seen from equation (2.14) to be equivalent to the shear modulus.
(iii) T h e form of the stress-strain relations is the same for all rubbers, subject only
to a scale factor (or modulus) which is determined by the number of chains per unit
volume, or degree of cross-linking.
2.7. Experimental veriJication
2.7.1. Stress-strain relations. T h e experimental examination of the above conclusions has been concerned with two main aspects, ( a ) the form of the stress-strain
relations and ( b ) the absolute value of the modulus.
x
Figure 4. Force-extension
curve for uniaxial extension : curve A, experimental; curve R,
theoretical.
T h e form of the experimental stress-strain relations is illustrated by selected
examples taken from the author’s work (Treloar 1944). Figure4 represents a
768
L R G Treloar
typical force-extension curve for simple extension, together with the theoretical
relation (2.10) adjusted to give agreement at small strains. It is seen that beyond
about 50% extension (A = 1.5) the experimental curve falls somewhat below the
theoretical, but that as the highest extension is approached it rises very rapidly.
These deviations are discussed later. Figure 5 shows the same data plotted together
with data for uniaxial compression on the same rubber. T h e compression data show
3
1.2
1.4
1.6
1.8
2
h
Figure 5 . Uniaxial extension and compression (derived from equi-biaxial extension) : broken
curve, experimental; full curve, theoretical.
very close agreement with the theory over the whole range. T h e ‘compression’ data
were derived indirectly from experiments on a uniform two-dimensional extension
produced by the inflation of a sheet, as in a balloon. (The geometry of this type of
strain is identical to that of uniaxial compression. In this way the difficulty of
producing a high degree of compression is avoided.)
Data for simple shear were also obtained indirectly, by making use of the
equivalent ‘pure shear’, in which the principal axes of the strain ellipsoid do not
rotate. This type of strain is approximated by the extension of a wide sheet. From
the tensile force the principal stress t , is obtained directly as a function of A,. T h e
equivalent shear stress is readily obtained in terms of the corresponding shear strain
y . T h e result obtained for the same rubber as that used for the extension and compression experiments shows agreement with the theory up to a shear strain of about
1.0 (figure 6), after which the curve deviates in a manner similar to the simple
extension curve. T h e value of the modulus (0.39 MN m-2) was the same for shear
as for extension and compression.
Bearing in mind the magnitude of the strains involved, the degree of agreement
between theory and experiment, and in particular the agreement in the numerical
value of the modulus for the various types of strain, must be considered satisfactory,
at least as a first approximation. This is further borne out by studies of the more
The elasticity and related properties of rubbers
769
general pure homogeneous strain, considered in $ 6 . T h e deviations, however, are
by no means negligible. These appear most clearly in the curve for simple extension. T h e final upturn in the curve is related to the finite extensibility of the chains,
not considered in the elementary statistical theory, but capable of treatment by a
more exact ' non-gaussian' analysis, discussed later in this section. T h e deviation
in the region from about h = 1.5 to h = 4 has not yet been satisfactorily explained in
molecular terms; it may however be represented mathematically with considerable
accuracy by the two-constant Mooney formula
F = 2(h-1/X2)(C~+C2/h)
(2.18)
which reduces to the form (2.10), with 2C, = N k T , when C, is put equal to zero.
Shear s t r a i n
Figure 6. Stress-strain relation for simple shear: curve A, experimental; curve B, theoretical.
According to this equation, a plot of F/2(X- l / h 2 )against l / h should yield a straight
line, from which the values of C, and C, are readily obtained. Examples of this type
of plot are shown in figure 23. A discussion of the significance of this formula is
deferred till later, when the form of the general stress-strain relations will be more
critically considered (8 6).
2.7.2. Absolute value of modulus. It is clearly important to establish whether the
calculated numerical value of the shear modulus ( N k T ) is in satisfactory agreement
with experiment. For this it is necessary to obtain a direct chemical estimate of the
number of chains per unit volume, that is, of the number of cross-linkages introduced during the cross-linking reaction. Experiments on these lines have encountered considerable difficulties on three main grounds, namely:
(i) the difficulty of dealing with the deviations from the theoretical form of forceextension curve;
(ii) the presence of network imperfections which result in wasted or ineffective
cross-linkages ; and
770
L R G Treloar
(iii) the difficulty of finding a cross-linking agent which yields a determinable
number of cross-linkages.
T o take the first point, it is obvious that deviations from the theoretical form of
curve will result in a variation of modulus (NJZT) with strain. T h e difficulty is
usually evaded by working with rubbers which have been swollen in a solvent;
these show almost perfect agreement with the theoretical form (2.10) (see § 5 ) .
With regard to the second point, in the treatment of the network it is assumed
that every chain is connected at each end to another chain, and that each cross-link
is the terminal point of four chains. Under these conditions the number of chains
is exactly twice the number of junction points, and every chain contributes effectively
to the network elasticity. I n reality, since the original molecules are of finite length,
there will be two ‘loose ends’ for each original molecule which are attached to the
network at one end only and therefore do not contribute to the stress. Flory (1944)
has modified the theory to allow for such ineffective chains, and arrived at the
following more precise formula for the shear modulus:
(2.17a)
in which M is the molecular weight of the primary molecules before cross-linking.
Another type of imperfection is the ‘closed loop’ formed by the cross-linking
of two points on the same chain. No quantitative correction is available for this, or
indeed for any other type of network defect.
T h e most serious difficulty, however, in estimating the number of cross-linkages
present is of a chemical nature. Gee (1947) showed that sulphur vulcanization was
not satisfactory, the sulphur reacting to form polysulphide linkages and other
structures which did not permit of quantitative definition, in addition to the
desired monosulphide cross-linkages. Experiments by Flory et a1 (1949) using
bis-azo dicarboxylates, which gave quantitative cross-linking, yielded agreement
between calculated and observed moduli to within about 25%. T h e later very
careful experiments by Moore and Watson (1956) made use of organic peroxides as
cross-linking agents ; these act as catalysts to produce direct C-C linkages between
chains, without the interposition of a ‘foreign’ chemical group. They worked with
swollen rubbers and applied the loose-end correction (2.17a). Their results
(figure 7) yielded a linear dependence of modulus on degree of cross-linking in
approximate agreement with the theory, but with a definite intercept on the vertical
axis, indicating a finite value of modulus at zero cross-linking. This they attributed
to the presence of a small fraction of physical entanglements between chains which
would have the same effect as genuine chemical cross-links. With this reservation,
these experiments confirm that the statistical theory is substantially correct in predicting the numerical value of the modulus with a fair degree of precision. Even
without this reservation, the degree of agreement over the practical range of crosslinking is within about 25%, which for a calculation which involves no parameter
derived from experiment can be regarded as reasonably satisfactory.
2.8 Limited extensibility of chains :non-gaussian statistics
T h e foregoing account of the network theory is based entirely on the gaussian
statistical formulae (1.3), (1.4) and (1.5) used to describe the properties of the
randomly jointed chain. I n these formulae the probability remains finite for all
The elasticity and related properties of rubbers
77 1
values of end-to-end distance Y, however great. I n reality, the probability must fall
to zero at Y = nl, corresponding to the maximum extensibility of the chain. T h e
gaussian formulae therefore become increasingly inaccurate as the chain extension
approaches this limiting value.
5
Figure 7 . Measured value of modulus plotted against value of NkT calculated from number of
cross-links. From Moore and Watson (1956).
By a more accurate statistical treatment, which avoids the simplifying approximations involved in the gaussian theory following from the condition (1.7), it is
possible to derive a relation between the force f on a single chain and the distance Y
between its ends which is valid over the whole range of extension (James and Guth
1943, Flory 1953, Kuhn and Kuhn 1946). This relation is
kT
f=1)$(I-..
(2.19)
in which 9-l is the inverse Langevin function. T h e form of this function (figure 8)
is such that f+00 as rln1-t 1; its series expansion is
=
3($)
1539 (2)
r 7 + ... .
+ J9 (a)
Y 3 +T7J(;;1)3+-isj
297 r
(2.19a)
T h e gaussian approximation, which corresponds to the first term only in this
series, is equivalent to a linear force-extension relation for the single chain; this
becomes inadequate for fractional chain extensions exceeding about 0.3.
T h e simplicity of the analysis of the gaussian network rests on this linear
relationship for the single chain. When this is abandoned the assumptions of the
simple theory, and in particular the assumption of an ‘affine’ deformation of chains,
L R G Treloar
772
no longer apply. Nevertheless, James and Guth (1943) considered it plausible to
retain this assumption as a first approximation, and in this way derived the nongaussian force-extension relation for the network
f=
h~vkTnl/z(g-1(hn-l/~)- A-%
X-VZ
(2.20)
)>
in which, as before, h is the extension ratio. This expression contains two parameters, of which the first, N (number of chains per unit volume), determines the
vertical scale or modulus in the small-strain region, and the second, n (number of
random links per chain), determines the limiting extensibility of the network, this
g-1(
r lnl
Figure 8. Force on single chain plotted against relative length Y / d ,according to equation (2.19)
(full curve) compared with the gaussian derivation (broken curve).
being given by X = n1/2. T h e form of this expression is shown in figure 9, the
parameters iV and n being adjusted to give the best fit to the experimental data
(reproduced from figure 4). Comparison with figure 4 shows that this admittedly
rather crude treatment nevertheless provides a much more realistic representation
of the properties of rubber over the whole range of extension than does the gaussian
theory.
&lorerefined treatments have proceeded either ( a ) by incorporating in numerical
form the non-gaussian function for the single chain, as given by equation (2.19) or
some comparable relation, in which case a general analytic expression for the
force-extension curve is not obtainable, or ( b ) by introducing the series expansion
( 2 . 1 9 ~ )and retaining only a limited number of terms. Applying the first method
the writer (Treloar 1946), using a four-chain model of the network, examined the
effect of relaxing the affine deformation assumption. This produced a somewhat
higher extensibility but did not greatly alter the form of the curve (figure 10).
Using method ( b ) , Wang and Guth (1952) included only the first two terms of the
non-gaussian distribution function, while the writer (Treloar 1954) retained five
terms in the expansion ( 2 . 1 9 ~ )and compared the resulting force-extension curve
with that obtained by graphical integration of the complete function.
T h e elasticity and related properties of rubbers
773
A
Figure 9. Non-gaussian force-extension relation (equation (2.20) ) for network, fitted to
experimental data, with NkT = 0.273 MNm-*, n = 75, compared with the gaussian
form (broken curve).
A
Figure 10. Non-gaussian force-extension curves derived from different theoretical models,
with n = 25 : curve A, equation (2.20); curve B, four-chain model, affine deformation;
curve C, four-chain model, non-affine deformation, T h e gaussian form is indicated by
the broken curve.
774
L R G Treloar
A modified form of non-gaussian theory which takes into account chain stiffness
has been developed by Smith (1971).
A note of reservation should be added in connection with the application of the
non-gaussian theory to the interpretation of the form of the complete forceextension curve of natural rubber. This is concerned with the possible effects of
crystallization, which sets in at about the same region of strain as the observed
upturn in the force-extension curve. Any such crystallization would be expected
to have a stiffening effect and hence lead to a progressive increase in stress with
increasing extension. This difficulty has been emphasized by Flory (1947) and his
associates. However, the careful work of Smith et a1 (1964), in which a variety of
techniques (swelling, thermoelastic measurements, x ray diffraction, etc) were used
to reveal the precise onset of crystallization, has shown quite definitely that the
initial upturn is a genuine non-gaussian effect. At higher extensions complications
associated with crystallization were observed. I n the early stages of crystallization
the accompanying preferential alignment of the chains tended to reduce the stress
below that corresponding to the amorphous network. It was only at a later stage
(ie at higher extensions) that an increase of stress attributable to crystallization
became observable.
2.9. The equivalent random link
I n fitting the non-gaussian curve to the experimental data in figure 9 the parameters N , the number of chains per unit volume, and n, the number of random
links per chain, were treated as independently adjustable parameters. I n fact these
two parameters are related, since for a given value of chain length or chain molecular
weight Mc (determined by N through equation (2.17)), the number of equivalent
random links is automatically determined.
T h e question of the number of effective randomly jointed links in any real
molecular chain is discussed more fully in 5 3 below. For the present it is sufficient
to note that from the values of N and n required to fit the curve in figure 9 one
obtains the result that one random link is equivalent to 1.63 isoprene units (Treloar
1956).
T h e overall fit in figure 9 is rather poor. If the theoretical curve is adjusted to
give a good fit at high extensions (as in this figure) the agreement at low strains is
poor, and vice versa. I n an attempt to overcome this difficulty Mullins (1959)
introduced a Mooney C, term (equation (2.17) ) into the non-gaussian theory, using
for this purpose the series approximation of Treloar (1954). T h e value of n (the
number of equivalent random links in the chain) was then found from the point at
which the experimental curve began to deviate from the Mooney line. (A deviation
amounting to 2.5% of the C, term was arbitrarily chosen for this purpose.) I n this
way Mullins obtained the somewhat lower figure of 1.1 i 0.15 isoprene units per
random link.
A rather serious criticism of this conclusion has been made by Morris (1964),
who repeated Mullins’s experiments on a series of natural and synthetic poly(cisisoprene) rubbers. He showed the Mullins analysis to be in error in not allowing
for the effect of the non-gaussian terms on the slope of the substantially linear
portion of the Mooney plot. T h e difficulty can only be avoided by choosing a value
of n (together with appropriate values of C, and C,) to give the best fit to the whole
curve. T h e results, shown in figure 11, indicate a very satisfactory agreement with
775
T h e elasticity and related properties of rubbers
this theory. T h e values of n, however, are found to correspond to a figure of 4.3
isoprene units per random link, which is very different from Mullins’s figure of 1.1.
That this difference is not due to experimental uncertainties is shown by the fact
that the application of the Mullins analysis to Morris’s data leads to the same result
as that obtained by Mullins himself.
1
I
I
O.do.2
I
I
I
06
04
I
0.8
1
I/h
Figure 11. ‘ Mooney ’ plot of force-extension data for natural rubber vulcanizates compared
with calculations from non-gaussian theory with added C,term: curve A, M , = 5478;
curve B, M c = 5505; curve C, M c = 5644: open symbols, experimental; full symbols,
calculated. From Morris (1964).
It would appear that Morris’s treatment is the more logical. There is, however,
the further difficulty that Mullins’s values of n were found also to fit very well into
the non-gaussian theory for rubbers swollen with solvents, which relates the
first appearance of the non-gaussian deviation, for any given value of n, to the
degree of swelling. T h e photoelastic data discussed in the following section also
seem to suggest that Morris’s estimate is excessively high. These inconsistencies are
no doubt partly due to the lack of theoretical foundation for the Mooney equation,
and will require further examination before they can be properly assessed.
3. Photoelasticity
3.1. Basic concepts
An unstrained rubber is isotropic in its optical, as well as in its mechanical
properties, that is, it is characterized by a single value of refractive index, O n
straining, however, it becomes birefringent, as can be readily demonstrated by
stretching a film of vulcanized rubber between crossed polaroid plates. On release
of the stress it reverts to the optically isotropic state.
This photoelustic effect is associated with the preferential orientation of the longchain molecules (or molecular segments) in the macroscopic extension of the
material, and is a consequence of the anisotropic polarizability of the molecular
chain itself. By a comparatively simple extension of the statistical theory as already
developed for the treatment of the mechanical properties of a cross-linked rubber it
has been found possible to take account of these anisotropic polarizability effects,
and thus to predict the laws governing the photoelastic behaviour of rubbers.
L R G Treloar
776
T h e basic theory of photoelasticity in rubbers was developed by Kuhn and
Grun (1942). They postulated a randomly jointed chain of identical links, each of
which was characterized by polarizabilities al and a2 for electric fields respectively
parallel and transverse to the length of the link. I t is evident that the chain as a
whole will have a related anisotropy, which will be a function of the distance r
between its ends. Kuhn and Grun's analysis leads to the following expression for the
mean anisotropy of polarizability y1 - y2 for the chain
where n is the number of links, Z the length of the link, and y1 and y z are the mean
polarizabilities for directions of electric field respectively parallel and perpendicular
r lnl
Figure 12. Relative optical anisotropy y 1 - y 2 of single chain as a function of its fractional
extension (equation (3.1) ).
to the chain vector length Y. This expression is valid for all values of r from Y = 0
(chain ends coincident) to r = nl (limiting chain extension); its form is shown in
figure 12.
By expansion of the inverse Langevin function 9-l,equation ( 3 . 1 ) may be
represented in the equivalent series form
y1-y2 = n ( a 1 - a 2 )
3 r 2
[()
J
J
36 r 4 1 0 8 ~ 6
+-175
- +=- - +... .
( n l ) 815 (d)
(3'1a)
For small values of rlnl the first term only of this series provides an adequate
approximation, that is,
(3.lb)
This approximation, which corresponds to the gaussian approximation introduced
in the treatment of the chain statistics, is normally adopted in dealing with the
photoelastic properties of the network.
3.2. Network theory
I n this treatment the same assumptions are made as in the treatment of the
mechanical properties of the gaussian network (§2). Knowing the components of
T h e elasticity and related properties of rubbers
777
polarizability for an individual chain as a function of its vector length r (obtainable
from (3.lb)) it is possible by summation over all chains to derive the components
of polarizability for the whole network in any particular direction. For a simple
extension in the ratio A, the anisotropy of polarizability for the network is found to
be (Kuhn and Grun 1942)
1
pl-p 2 -N("l-"2)
-5
i 3
h2--
where p1 and p2 are polarizabilities per unit volume respectively parallel and
perpendicular to the direction of the extension and N (as before) is the number of
chains per unit volume.
T o convert this result to an actual birefringence, it is necessary to introduce the
Lorentz-Lorenz relation between refractive index n and polarizability P per unit
volume, that is,
n2-1
4-p.
7T
-=
(3-3)
n2+2
3
T h e birefringence so obtained is given by
where no is the mean refractive index (nl+2n2)/3.
3.2.1. Stress-optical coeficient. For an extension ratio X the stress t , per unit crosssectional area measured in the strained state, as given by the statistical theory
(equation (2.16)), is
t = NkT(A2- l / A ) .
(3.5)
T h e ratio of birefringence to stress, or stress-optical coefficient (C), is thus
T h e above treatment has been generalized (Treloar 1947) to the case of a pure
homogeneous strain, corresponding to principal extension ratios A,, A, and A,.
For light propagated in the direction of the A, axis the corresponding birefringence is
n, - n2 = C ( t , - t2)
(3.7)
where t,-tt, is given by equation (2.15a).
3.2.2. Non-gaussian theory. Forms of non-gaussian photoelastic theory, which take
into account higher-order terms in equation ( 3 . l a ) for the anisotropy of the chain,
have been derived, for example, by Treloar (1954) and by Smith and Puett (1966).
Since no essentially new physical principles are involved these will not be discussed
in detail.
3.3. Signijcance of theoretical conclusions
T h e above results are of considerable theoretical interest. Attention is drawn
in particular to the following specific points.
( a ) Equations (3.4), (3.5) and (3.6) imply that although both the birefringence and
the stress are nonlinear functions of the strain, the birefringence is directly proportional to the stress. This means that Brewster's law (previously established
778
L R G Treloar
experimentally for small strains in glassy materials) should apply also to large
strains in rubberlike materials.
( b ) Whilst both the stress and the birefringence depend on the degree of crosslinking (represented by N in equations (3.4) and (3.5)), the ratio of birefringence
to stress (ie the stress-optical coefficient) is independent of the degree of cross-linking,
but depends essentially only on the anisotropy of chain polarizability, as represented
by the parameter a1- c y 2 . T h e stress-optical coefficient should therefore have a
characteristic value for any given polymer, regardless of its state of cross-linking.
(c) From equations (3.7) and ( 2 . 1 5 ~ it
) is seen that in a pure homogeneous strain of
the most general type the birefringence should be proportional to the difference
of the squares of the corresponding extension ratios.
3.4. Experimental examination
3.4.1. Dependence of birefringence on stress. T h e foregoing conclusions have been
extensively studied experimentally, and have been found to give a satisfactory
account, at least approximately, of the actual behaviour of rubbers, though significant differences have also been observed. Figure 13 shows plots of birefringence
Stress on actual section
(MN m-')
Figure 13. Relation between birefringence n l - n 2 and stress for natural rubber in simple
extension.
against stress for natural rubber in simple extension (Treloar 1947). At 75 "C the
expected linear relationship is approximately satisfied over the whole range of
applied stress. At 25 "C, however, deviations of an irreversible character occur at
stresses exceeding about 2.0 MN m-2, corresponding to an extension of about
200%. This is the point at which x ray and other evidence indicates the onset of
strain induced crystallization (Goppel 1949). This tendency to crystallization is
effectively suppressed at the higher temperature.
The elasticity and related properties of rubbers
779
T h e more general case of a pure homogeneous strain, produced by applying
unequal stresses in two perpendicular directions to a sheet of rubber, has also been
studied (Treloar 1948). A plot of birefringence against the difference of principal
stresses showed satisfactory agreement with the theory (figure 14). I n these experiments the maximum extension in either direction (about 200%) was not sufficient
to induce a significant amount of crystallization.
0
t , - t, I M N m+)
Figure 14. Relation between birefringence n, - n2 and difference of principal stresses tl - t 2 for
dry and swollen rubber in pure homogeneous strain.
Table 1. Dependence of stress-optical coefficient on degree of cross-linking
for natural rubber?
Cure
Chain molecular weight
M,
Stress-optical coefficient
C
m2N-l)
Peroxide
2150
4210
5270
7400
0.183
0.195
0.198
0.200
Radiation
7334
13360
15290
0.195
0.201
0.209
From Saunders (1956).
3.4.2. Stress-optical coeficient. Early work by Thibodeau and McPherson (1934)
had shown that the value of the stress-optical coefficient varied considerably
according to the degree of vulcanization in sulphur-vulcanized natural rubber.
This could be attributed to the modification of the chain structure and hence of
the polarizability of the chain by chemical combination with sulphur. This difficulty
was avoided by Saunders (1956), who used both peroxide and radiation curing,
which produce cross-linking without chemical modification of the chain structure.
His results, which are given in table 1, show that over a sevenfold range of
the variation of
cross-linking (as measured by the chain molecular weight iV,)
780
L R G Treloar
stress-optical coefficient was very slight, thus confirming the theoretical prediction.
Similar results were obtained with gutta-percha, the trans isomer of polyisoprene.
For some other polmers, particularly polyethylene, the picture is less clear.
At room temperature polythene is crystalline, and the theory is clearly inapplicable.
Suitable experiments may, however, be carried out on cross-linked polythenes at
temperatures above the crystal melting point (about 110 "C), where it behaves like
a rubber. Under these conditions it has been reported (Saunders 1956,1957) that the
stress-optical coefficient decreased progressively with increase in the degree of
cross-linking. It was suggested that this effect was due to the inadequacy of the
gaussian statistical theory, but although some success was achieved in accounting
quantitatively for the variations in terms of the non-gaussian theory of photoelasticity, later work (Saunders et a1 1968) has thrown doubt on the validity of this
interpretation. Gent and Vickroy (1967), on the other hand, using polyethylenes
cross-linked in the amorphous state, obtained no significant variation of stressoptical coefficient with degree of cross-linking, and attributed Saunders's results to
the fact that his polymers had been cross-linked in the crystalline (ie non-random)
state. However, the more recent work of Saunders et al (1968), in which both
methods of cross-linking were employed, seems to invalidate this claim.
A comparable dependence of stress-optical coefficient on Mc was obtained also
by Mills and Saunders (1968) for silicone rubbers. Since these materials are noncrystalline, the suggested explanation of Gent and Vickroy would not in this case
apply.
3.5. EfSect of swelling on the stress-optical coe8cient
T h e Kuhn-Griin theory assumes that the directional polarizabilities of the chain
element, as represented by a1 and a2,are independent of the general conformation
of the chain and of the local environment of the element. With these assumptions,
the presence of a diluent in the form of a swelling liquid should have no effect on
the value of a1-a2.
An extension of the original theory to take account of swelling (Treloar 1947)
shows the value of the stress-optical coefficient to be unchanged, except to the
extent that the mean refractive index no (equation (3.6)) may be affected, and early
work, using natural rubber swollen in toluene (Treloar 1947), appeared to support
this conclusion (cf also figure 14). More recent work by Gent (1969) on both cisand trans-polyisoprenes suggests, however, that this result may have been fortuitous.
Using a series of nonpolar swelling liquids of widely different refractive indices and
molecular 'shape', he found variations in a l - a 2 ranging, in the case of poly(ciscm3, compared with 48.0 for the
isoprene), for example, from 38.5 to 65-5x
unswollen polymer, T h e values of a1-a2 were found to be closely related to the
geometrical asymmetry, measured by the axial ratio, of the molecule of the swelling
liquid, but not to its refractive index, suggesting that the phenomenon is due to
local ordering of solvent and polymer molecules rather than to an induced polarization or internal field effect. Comparable results have also been obtained for polyethylene (Gent and Vickroy 1967).
3.6. The 'equivalent random link'
I n the statistical theory the polymer chain is represented in terms of an idealized
model of randomly jointed links which are optically anistropic. T h e actual molecule
The elasticity and related properties of rubbers
781
consists of a chain-like backbone in which the bonds are connected at specific
valence angles, together with a number of atoms or chemical groups attached to the
chain backbone. I n rubbery polymers rotation about bonds in the chain backbone
may be assumed, but this rotation will usually be restricted by steric hindrances
between neighbouring atoms or side-groups.
T h e gaussian statistical formula (1.3) may be assumed to apply to a chain of
sufficient length, whatever its geometrical structure, but the constant b will be
determined by the geometry of the chain and the hindrances to rotation about
bonds. Thus for a molecule of any given chain length R, there will be an associated
in the gaussian
value of b, and hence also a specific value of mean square length
region (equation (1.6)). It is therefore possible to postulate an equivalent randomly
jointed chain containing n, links of length 1, which will have the same chain length
and the same mean square length as the actual molecule. For such a ‘statistically
equivalent’ chain the values of n, and 1, are determined by the relations
2
n, 1, = R,
n, 1,”= r i
giving
T h e link of length 1, in the randomly jointed chain, as defined by (3.9), may be
called the ‘equivalent random link’.
T h e length of a monomer unit in the chain (measured along the chain axis) being
known, it is possible to represent the length of the equivalent random link in terms
of the number of monomer units attached end-to-end which would occupy the same
length. This quantity will be denoted by p.
On the basis of the Kuhn-Grun theory the observed stress-optical coefficient
enables the optical anisotropy (a1- a 2 )of the equivalent random link to be derived
directly (equation (3.6)). If it were possible to obtain an independent estimate of
the anisotropy of polarizability for the monomer unit we would thus be able to
calculate the actual value of p corresponding to any given chain structure.
Calculations of this kind have been carried out for a number of polymers, and
are based on the tensor summation of the directional polarizabilities of individual
bonds. Unfortunately, quoted values of directional bond polarizabilities are meagre
and of uncertain accuracy; the most reliable figures are probably those of Denbigh
(1940), which are based on optical measurements on molecules in the vapour state.
(For discussion see Saunders et a1 1968.) Table 2 gives data for the anisotropy of
polarizability for the monomer units of several polymers calculated in this way
(Morgan and Treloar 1972). I n this table PI and Pt are the longitudinal and (mean)
transverse polarizabilities, referred to the chain axial direction. Comparison with
the experimentally derived values of a1- cy2 for the random link yields the number
of monomers per random link, q. An alternative representation in terms of the
number of rotatable links (C-C bonds) in the monomer unit is given in the fifth
column; this is more directly relevant as an indication of the relative ‘stiffnesses’
of different types of chain.
Comparison of the values for cis- and for trans-polyisoprene in the above table
shows the latter to have about twice the equivalent link length, and hence twice the
L R G Treloar
782
mean square length (equations (3.9)), for the same total chain length. T h e high
value of p for polyethylene implies a strong preference for an extended rather than
for a highly kinked conformation, and is in harmony with the great facility with
which this polymer crystallizes.
Table 2. Anisotropy of monomer unit, Pi- P t (calculated) and anisotropy of
random link, u1 - u2 (from stress-optical coefficient)
81 - Pt
Polymer
Natural rubber
(C,H,), (cis)
Gutta percha?
(C5H8), (trans)
Polybutadiene
(C,H,), (96% cis)
Polyethylene?
(CH,),
cm3)
a1-az
cm3)
Number of
monomers
Per
random
link, q
Number of
chain C-C
bonds per
random
link
R d{ln
27.1,
47.0,
1.73
5.19
- 270
28.3,
96.2,
3.39
10.17
+ 40
29.2,
77.7,
2.66
7.94
+ 85
7.2,
133.4
18.5
18.5
- %)I
d(liT)
(cal mol-l)
+ 1090:
?Extrapolated to 20 “C.
LIorgan and Treloar (1972) wrongly quoted the value 1150 cal mol-l, which referred to the energy
difference between t ~ a mand gauche configuraticns.
3.6.1. Comparison with equivalent Yandom link from non-gaussian theory. I t is
interesting to compare the value of equivalent random link derived from the stressoptical coefficient with that obtained from the form of the force-extension curve
by the application of the non-gaussian network theory, referred to in $2. Data are
available only for natural rubber, for which the estimates of q are 1.1 (Mullins 1959),
1.63 (Treloar 1956) and 4.3 (Morris 1964). T h e photoelastic value ( q = 1.73) thus
seems to support the lower values rather than the higher figure obtained by the
more sophisticated treatment of Morris. T h e discrepancy in the latter case may be
due to the lack of any sound theoretical basis, in terms of molecular concepts, for
the Mooney equation.
3.7. Temperature coejicient of optical anisotropy
I n general, the value of the stress-optical coefficient, and hence also of the optical
anisotropy of the random link deduced from it through equation (3.6), shows a
marked dependence on temperature. Data for representative types of polythene,
in the form of a plot of Ig(al-az) against 1/T, are shown in figure 15. T h IS
‘ temperature dependence implies that the statistical conformation of the chain is also
temperature dependent. Saunders et al (1968) have discussed the temperature
dependence of a l - a z for polythene in terms of the steric hindrances or energy
barriers to bond rotation within the chain, and have arrived at values of the energy
difference between trans and gauche configurations of 1150 or 1350 cal mol-l, on the
basis of the theoretical treatments of the statistics of the chain due to Sack (1956)
and to Nagai (1964) respectively. (The trans form corresponds to the planar configuration of successive C-C bonds and the gauche to a bond rotation of 120” out
of this plane.) Since the trans or planar form has the lowest energy, the statistical
length of the chain (P”>”z
therefore decreases with increase in temperature.
783
The elasticity and related properties of rubbers
Data for the temperature coefficients of a1-a2 for a number of polymers, as
given by Morgan and Treloar (1972), are shown in the last column of table 2.
While it has not yet proved possible to analyse the data for molecules of more
complex structure than the polyethylene molecule, it is interesting to see that for
the rubbery polymers the temperature coefficients of a1- cyz are very much smaller
than in the case of polythene. This undoubtedly reflects their greater flexibility,
that is, lower steric hindrances to internal bond rotations, which is of course to be
expected from their physical properties. It is noteworthy also that in natural
I
I
I
I
1.9
2.0
2.1
2.2
I Ir
I
2.3
f
2.4
2 5 x IO3
( KII
Figure 15. Temperature dependence of anisotropy of random link for various polythenes.
0 ‘Hostalen’ high density; 0,‘Hifax’ high density; A ‘DYNK’ low density; x Gent
and Vickroy, low density. The lines correspond to an energy difference between trans
and gauche configurations of 1150 cal mol-1 on Sack‘s theory. From Saunders et a1
(1968).
rubber the temperature coefficient has the opposite sign to that for polythene,
indicating an increase in the statistical length of the chain with increase of temperature. T h e same general conclusions are obtained from thermoelastic studies
($4), though the energy difference between trans and gauche configurations for
polythene from thermoelastic data ( 500 cal mol-l) is significantly lower.
For a more general treatment of the statistical properties of real chains, taking
into account steric hindrances, the reader is referred to the works of Volkenstein
(1963) and Flory (1969).
N
4. Thermodynamics of rubber elasticity
4.1. Elementary theory
I n this section we return to the consideration of the thermodynamics of rubber
elasticity. I n $ 1 reference was made to the importance of the thermodynamic
explanation of the thermoelastic phenomena discovered by Gough and Joule in
establishing the basic concepts of hleyer’s kinetic theory. T h e subsequent success
of the statistical theory in predicting the mechanical, optical and other properties of
cross-linked rubbers has amply justified these original concepts, but has at the same
time tended to shift the centre of interest away from the more academic thermodynamic arguments. Nevertheless, this subject has continued to receive close
attention, and its development has led to a fuller understanding of the mechanical
properties of rubbers and of the significance of some of the more detailed developments of the statistical theory.
784
L R G Treloar
I n the elementary thermodynamic treatment the force f required to maintain a
constant length l in a sample of rubber at an absolute temperature T is represented
as the sum of an internal energy and an entropy component, thus
where A, U and S are respectively the Helmholtz free energy, internal energy and
entropy. T h e terms (SU/i:al), and (aS/aZ), may be evaluated by studying the
variation of force with temperature at constant stretched length and applying the
easily derived thermodynamic relations (Meyer and Ferri 1935)
For the case of an extension without change of internal energy, as postulated by
the kinetic theory, (3Lr/ijal),= 0. We have then
J
f = c ~
where c is a constant, that is, the force is proportional to the absolute temperature.
Early experimental studies (Meyer and Ferri 1935, Anthony et al 1942), which
have been amply confirmed in later works, showed that this conclusion holds
approximately for natural rubber at fairly high extensions. At low extensions,
however, the tension rises less rapidly with increase in temperature, and for
extensions of less than about 10% it actually decreases (figure 16). These anomalies
are reflected in the thermodynamic analysis by the appearance of a significant internal
energy contribution to the force (figure 17). For strains less than about 10% the
entropy of extension, instead of being negative, is in fact positive.
This ' thermoelastic inversion ' arises from the small but thermodynamically very
significant changes of volume which accompany a change of temperature or the
application of a stress. It is easily seen, for example, that an increase of temperature,
by increasing the volume, and hence the unstrained length of the sample, will in effect
reduce the strain at constant length. This can lead to a reduction of stress, even
though the modulus ( N k T in equation (2.10)) increases. Writing (for the case of
small strains) h - 1 = E , we have from equation (2.10)) approximately
and also
f
=
E = € -0 -
3NkTE
(4.4)
+%)
(4.5)
where is the strain at an arbitrary reference temperature T, and /3 is the volume
expansivity. On substituting (4.4) into (4.5)) the strain at which the temperature
coefficient of the stress (i:af/ZT),changes sign is found to be
= g(2T-7;).
The elasticity and related properties of rubbers
ob
lo
io
30
40
50
*
785
do'
Temperature (OC)
Figure 16. Dependence of tensile force at constant length on temperature for natural rubber
at elongation ratios shown : 0 temperature increasing; 0 temperature decreasing.
From Shen et a1 (1967).
8
x
Figure 17. Internal energy (sU/al), and entropy - T(i?S/al),contributions to the force f in
stretched rubber. From Gee (1946a).
786
L R G Treloar
Insertion of typical values, for example, p = 6.6 x
T, = 293 K, T = 343 K,
thus gives (qJinv= 0.088, or 8*8%,which is sufficiently close to the observed value.
T h e early experimental work of Gee (1946a) led to the conclusion that if the
extension ratio h is measured with respect to the unstrained length at the actual
temperature T , the force at constant extension ratio is proportional to the absolute
temperature. From this it follows that, experimentally,
(%Ip,*
f
=T
(4.7)
the subscript p indicating constancy of the surrounding (atmospheric) pressure.
T h e result (4.7) is in harmony with the statistical theory, as represented by
equation (2.10).
I n a more precise thermodynamic analysis (Elliott and Lippmann 1945, Gee
1946a) a distinction is drawn between experiments at constant pressure and at
constant volume. This analysis yields the following expression for the energetic
contribution to the stress at constant pressure
in which H (ie U+$ V )is practically equivalent to U, and /3 is the volume expansivity
of the unstrained rubber. Further, by assuming among other things that the
strained rubber is isotropically compressible, Gee obtained the result
Taken in conjunction with the experimental result (4.7) this leads to the conclusion
(4.10)
4.1.1. Sign$cance of volume changes. T h e result (4.10) states that in an extension
carried out at constant volume (by suitable adjustment of the surrounding pressure)
the internal energy change should be zero. The implication is that the changes of
internal energy which are actually observed in an extension at constant pressure
arise from the accompanying change of volume. Following up this conclusion, and
assuming the classical relation between internal energy and volume for a pure
dilatation to be applicable, Gee derived an expression for the total change of volume
in an extension from an unstrained length lG to a length 1, at constant pressure,
namely,
(4.11)
where K is the volume compressibility. Direct measurements of volume changes
during extension (Gee et al 1950), though somewhat inaccurate, were found to be
consistent with this equation.
4.2. Later thermodynamic analyses
Gee's analysis had carried the subject to the limit of what was possible on the
basis of general thermodynamic arguments coupled with certain simplifying
The elasticity and related properties of rubbers
787
assumptions of a physical character; any further advance could be achieved only
through the introduction of a more specific physical model of the structure. An
appropriate model for this purpose is the gaussian network theory, suitably modified
and amended to take into account volume changes and other associated effects
which the elementary form of this theory, as presented in $2, ignores. T h e modified
theory, which is due primarily to Flory and his associates (Flory et al 1960, Flory
1961), brings out two new considerations, which, though developed within the
framework of a single formula, are essentially distinct. These are presented below.
4.2.1. Anisotropic compressibility. Both Gee and Elliott and Lippman assumed the
rubber sample, acted on by a tensile forcef, to be isotropically compressible under a
superimposed hydrostatic pressure, that is, that the changes in linear dimensions
are the same in all directions. Thus aV/V = 3W1, or
(4.12)
T h e more rigorous theory of Flory, on the other hand, leads to an anisotropic
compressibility in the strained state, the form of which is represented by the relation
1
=
m
(4.13)
where a: is the extension ratio (defined later). This formula, which, as Flory notes,
is substantially the same as that previously derived on an essentially similar basis
by Khasanovich (1959), reduces to (4.12) as the extension (a-1) tends to zero,
representing the fact that for sufficiently small strains the rubber may be considered
to be isotropic.
4.2.2. Intramolecular internal energy. Gee’s provisional conclusion, namely that
the changes in internal energy on extension arise solely from the accompanying
changes of volume, implies that any such internal energy changes are to be associated
with the forces between the polymer molecules (Van der Waals forces), these forces
being of the same kind as those which determine the volume of an ordinary lowmolecular-weight liquid. Flory, however, recognized the need for the inclusion
of an internal energy term associated with the conformation of the polymer molecule
itself, and arising from energy barriers to rotation about bonds within the single
chain. This leads to the possibility of a finite internal energy contribution to the
stress, even under constant volume conditions.
I n Flory’s derivation the force-extension relation (2.10) given by the elementary
gaussian network theory is replaced by the formula
f =+--vkT
J
1
(4.14)
li Y O
in which v is the total number of chains in the network (the chain being defined as
before as the molecular segment between successive cross-linkages), Zi is the length
of the undistorted specimen corresponding to the volume V in the strained state
-(li = V1’3), and 01 = l/li, where 1 is the strained length. T h e additional factor ~ f i r :
is the ratio of the mean square length $ of the network chains in the undistorted
state of the network (at volume V ) to the mean square length of an identical set
of free chains.
3
31
L R G Treloar
788
e/z
will be apparent on recalling that in the
T h e significance of the factor
elementary theory it was assumed that = 2, that is, that the chains in the unstrained state of the network had the same mean square length as a corresponding
set of free chains (cf $2). Even if this assumption should happen to be valid at one
particular temperature (and this cannot be demonstrated), it would still not apply
at any other, since is determined by the volume of the network, which depends
on temperature, while is, of course, an independent statistical property of the free
or unrestricted chain, which will in general also be temperature dependent.
Differentiation of equation (4.14) under conditions corresponding to constant
volume and to constant pressure, respectively, leads to the following expressions
for the corresponding changes in internal energy (or heat content),
2
(4.15)
(4.16)
from which we obtain
(4.17)
This equation enables the stress-temperature coefficient at constant volume to be
derived from experimental data on the stress-temperature coefficient under the
normal constant pressure conditions.
It is convenient to represent the quantity (2UjaZ),,, the internal energy contribution to the force at constant volume, by the symbol fe. T h e fractional contribution of the internal energy to the force, at constant volume, on the basis of
equation (4.15) is therefore
(4.18)
Flory shows further that experiments at constant pressure and constant extension
ratio are not equivalent to experiments at constant volume and constant length, as
proposed by Gee (equation (4.9) ) ; the correct relation between the stress-temperature coefficients under these respective conditions is?
(4.19)
4.3. Experimental determination of f e i f
Despite the experimental difficulties, Allen et aZ (1963) have succeeded in
directly measuring the temperature coefficient of tension at constant volume. A
rubber cylinder bonded to metal end-plates was contained in a mercury-filled
dilatometer to which a hydrostatic pressure could be applied. T h e stress was
determined from transducer measurements of the deflection of a stiff spring. I n
t I n equations (4.16), (4.17), (4.19) and elsewhere no distinction is made between expansion
coefficients at constant 1, constant 01 or in the unstrained state. For justification see eg Price
et al (1969a).
The elasticity and related properties of rubbers
789
addition to measuring (af/a T)v,t directly, they also measured (af/aT),,,, (af/ap),,
and the thermal pressure coefficient (ap/aT),, and were thus able to obtain an
independent derivation of ( a f / a T ) , ,by the use of the exact relation
(4.20)
I n this way an independent check on the directly measured values of (af/aT),, was
obtained, which substantiated the reliability of the method. T h e resultant values
of feiffor natural rubber lay in the range 0.10 to 0.20, there being no significant
dependence either on temperature or on strain. I n a later more precise investigation
Allen et aZ(l971) reported the value fer= 0.12 2 0.02.
Table 3. Values of relative internal energy contribution (fe/f)to the force for
rubbery polymers
Polymer
Reference
I
Allen et a1 (1963)
Allen et a1 (1971)
Roe and Krigbaum (1962)
Natural rubber
I
\
Butyl rubber
Silicone rubber
Polyethylene
{
Ciferri (1961)
Shen (1969)
Boyce‘and Treloar (1970)
Allen et a1 (1968)
Ciferri et a1 (1961)
Price et a1 (1969b)
Ciferri et a1 (1961)
f,if
Method
0.2
0.12 k 0.02
0.25 to 0.11
depending
on strain
0.18
0.15 k 0.3
0.126 k 0,016
- 0.08
- 0.03 k 0.02
0.25 k 0.01
- 0.42 k 0.05
Constant V
Constant V
Constant p
Constant p
Constant p
Torsion
Constant V
Constant p
Constant V
Constant p
fer
I t is important to note that these derivations of
depend only on direct
measurements or on general thermodynamic relations and are therefore not open to
argument on theoretical grounds. Flory’s theory, on the other hand, is based on a
structural model which may or may not correspond precisely to reality. However,
a comparison of the directly measured value of fe/f with the corresponding value
derived from stress-temperature data at constant pressure through the use of
equation (4.17) provides an important check on the applicability of Flory’s theory.
From their constant pressure measurements Allen et a1 (1971) obtained in this way
a figure for f e / f of 0.18 f 0.03, which differs from the above directly measured value
by an amount which is barely outside the spread of their data, and may thus be taken
as a satisfactory confirmation of the theory. T h e authors, however, consider this
result to be to a certain extent fortuitous, since the force-extension curves deviate
significantly from the gaussian form required by the theory (cf 0 2).
T h e value of feris, of course, dependent on the type of rubber employed.
Data for a number of other rubbery polymers are listed in table 3, together with
values for natural rubber obtained by different authors. For a more extensive
summary of the available data reference may be made to the review by Krigbaum
and Roe (1965). For most of the materials examined f,ifis found to be positive, but
there are a few, notably butyl rubber (polyisobutylene) and polyethylene, for which
negative values are obtained. According to equation (4.18) a positive value of
corresponds to a positive temperature coefficient of G, which implies that the more
fer
790
L R G %eloar
contracted form of the chain is energetically favoured, that is, that the intramolecular forces are, on the average, attractive. Conversely a negative value of fe/f
implies that the intramolecular forces are repulsive, This can be understood in the
case of polyethylene, for example, in terms of the steric hindrances between the
hydrogen atoms attached to neighbouring carbon atoms in the chain, as a result of
which the extended (planar) form has a lower energy than the 'gauche' form corresponding to a bond rotation of 120" out of this plane (Flory 1953 p416). T h e value
for polythene given in table 3 is consistent with an energy difference between
these two states of about 500 calmol-1 (Ciferri et al 1961).
T h e effect of temperature on the statistical length of the chain may also be
estimated from the intrinsic viscosity of dilute solutions of the polymer. A comparison of values of d l n z / d T obtained in this way with thermoelastic estimates of
the same quantity shows good agreement in some cases, but considerable discrepancies in others (Krigbaum and Roe 1965).
It should be noted also that these deductions concerning the temperature
dependence of
are in qualitative agreement with the conclusions drawn from
photoelastic studies ( 5 3).
h
Figure 18. Apparent dependence of feif on strain: A Roe and Krigbaum (1962); 0 Smith
et a2 (1964); 13 Ciferri (1961); 0 Shen et al (1967). From Shen et al (1967).
4.3.1. Strain dependence of f J f . A number of authors have reported a marked
dependence of the ratio f e / f on the amount of applied strain. Typical sets of data
are shown in figure 18, taken from the work of Shen et aZ(1967). At high strains
( A > 3.5) the observed reduction in f e / f in the case of natural rubber may reasonably
be atrributed to crystallization (Smith et aZ 1964), but in the region of low strains
(A < 1-5)the apparent strain dependence is not to be expected on theoretical grounds,
The elasticity and related properties of rubbers
791
and is more likely to be associated with the extreme sensitivity of the calculated
value of ferto small errors in the measurement of the unstrained length of the
specimen under conditions involving variations of temperature. From equation
(4.17) it can be seen that the derivation of ferfrom constant pressure experiments
involves the addition of the term /3/(a3- l), which tends to infinity as a3- 1 tends
to zero, with the result that small errors in the measurement of unstrained length,
and hence of cy, will lead to disproportionate errors in ferat small strains.
3
x -1/x2
Figure 19. Plots off against h - l / h 2 for natural rubber pre-swollen with 33.6% hexadecane:
upper, 60" C ; lower, 10" C. From Shen (1969).
I n a careful study of this problem Shen (1969) demonstrated that the apparent
dependence of feron strain could be eliminated by considering, in effect, the
temperature dependence of the shear modulus, obtained by plotting f against
A- 1/A2 at each temperature (figure 19). Since these plots were linear (to within the
experimental error) it follows that
is independent of strain. A variation of the
same principle involved measurements of the linear expansion coefficient under
constant stress (al/aT),, for various values of A, the results being interpreted in
I n all cases feif was found to be independent not only of the strain,
terms of
but also of the presence of a swelling liquid incorporated into the rubber before
cross-linking, thus confirming thatfeifis related purely to the intramolecular energy,
and is not affected by the local environment or intermolecular forces.
fer
fer.
4.4. Thermoelastic data for torsion
Since shear and torsion, on classical elasticity theory, are constant volume
deformations, it might be thought that constant pressure and constant volume conditions would be equivalent for these types of strains, and hence that the difficulties
encountered in deriving constant-volume data in the case of simple extension would
792
L R G Treloar
not arise. Flory et a1 (1960), however, have pointed out that this deduction is not
valid, but have not presented any detailed analysis of the problem. This has since
been carried out by the writer (Treloar 1969b), who considered the temperature
dependence of the torsional couple M in a cylinder of rubber subjected to combined
torsion about the axis and extension in the axial direction.
I n this analysis the basic theory of Flory was used, together with the equations
for the radial stress distribution for an incompressible cylinder in torsion, as
obtained by Rivlin (1949a). T h e general thermodynamic equations for torsion are
similar to those for simple extension, except that the variables f and I are replaced
by M (couple) and (angle of torsion). I n place of equations (4.15) and (4.16) the
following relations are derived for the respective internal energy contributions to
the couple at constant volume (MeV)and at constant pressure (Mep),
(4.21)
T h e difference between these two coefficients is therefore
Mep- MeV= - MTP.
(4.23)
These conclusions are of considerable interest. Equation (4.21) illustrates
Flory’s general conclusion that the temperature dependence of the stress, at constant volume, is directly related to the temperature coefficient of 3, whatever the
type of strain. Equations (4.22) and (4.23) do indeed confirm Flory’s statement in
showing that in the case of torsion constant volume and constant pressure conditions are not equivalent. T h e detailed analysis shows that this is due to the
existence of volume changes of second order (ie proportional to the square of the
torsion) which the classical theory ignores. Nevertheless, there are important
quantitative differences between extension and torsion. Whereas (as has already
been noted) in the case of extension the difference between the constant pressure
and constant volume coefficients (equation (4.17) ) tends to infinity at small strains,
thus making the measurements extremely sensitive to small variations of unstrained
length (which are almost unavoidable), in torsion the corresponding difference is
independent of strain (equation (4.23)). Likewise, the relative slope of the stresstemperature plot at constant pressure, (1/M)(aM/aT),,$, is independent of strain
(ie there is no inversion effect at small strains). I n practice, therefore, torsion
provides a much more accurate basis for deriving the internal energy contribution
to the stress at constant volume from experiments at constant pressure than does
simple extension.
These theoretical expectations are fully confirmed by stress-temperature data
for natural rubber in torsion obtained by Boyce and Treloar (1970). As seen from
figure 20, the slopes of the couple against temperature plots are always positive,
and analysis of the data shows the relative slope, and hence the relative internal
energy contribution to the couple at constant volume, M,,/M, to be independent of
torsional strain. T h e mean value of this ratio, namely 0.126 t 0.016, is in excellent
agreement with the correspondingfJf of 0.12 c 0.02 obtained by Allen et aZ(l971)
from direct measurements at constant volume.
The elasticity and related properties of rubbers
793
4.5. Conclusion
From the foregoing discussion it appears that the model of a gaussian network,
as treated by Flory, provides a substantially accurate basis for the interpretation of
the stress-temperature relations for rubber, and hence for the derivation of the
on temperature. Despite this success, however,
dependence of chain dimensions
there remain some discrepancies between the predictions of this model and the
actual behaviour of rubbers, the most serious of which is the still unexplained
(2)
J
20
I
1
40
Temperature
60
('CO
Figure 20. Stress-temperature relations for natural rubber in torsion for various values of
#ao (# is the twist in radians per unit length, a, is the unstrained radius) : x temperature
increasing; 0 temperature decreasing; - - - repeated after completion of set. From
Boyce and Treloar (1970).
deviation in the form of the force-extension curve, which is examined in more
detail in the following section. Possibly connected with this is the disagreement
between the measured values of the changes of volume which accompany the
extension (Allen et a1 1968, 1971, Christensen and Hoeve 1970) and the predicted
values, the discussion of which cannot be included here. It is important to remember
that on account of such deviations the thermodynamic relations derived on the
basis of the gaussian network model do not have the universal validity of the more
general thermodynamic relations previously employed. Attempts to improve the
L R G Treloar
794
model by the grafting on of a more realistic form of stress-strain relation such as
the Mooney equation have been made, notably by Roe (1966), but in view of the
uncertainties surrounding all such modifications, discussed in $6, it is difficult at
present to assess their significance.
5. Swelling and mechanical properties
5.1. Mechanism of swelling
T h e capacity for swelling in liquids is a characteristic property of high polymers,
and is exemplified by such water-absorbing materials as wood and gelatin, the
hygroscopic fibres (cotton, silk, wool, etc) and the rubbers, which are generally
capable of absorbing large quantities of organic liquids. T h e phenomenon is akin
to solution and is governed by similar thermodynamic relations. We shall not be
primarily concerned with the process of swelling in itself, but rather with the interrelations between swelling and mechanical properties; however, in order to understand these effects some acquaintance with the basic mechanism of the process is
desirable.
For a cross-linked rubber in contact with a low-molecular-weight liquid a
definite equilibrium degree of swelling is established. T h e condition for equilibrium is that the free energy change for further absorption of liquid shall be zero,
that is,
2G _
-0
an,
in which n, is the number of moles of liquid in the swollen polymer, and G is the
Gibbs free energy, defined by
G
=
U- TS+pV
=
A+pV
(54
where A is the Helmholtz free energy. T h e quantity aG/an,, termed the molar
free energy of dilution, is the sum of two terms, one related to the free energy of
mixing G, of the liquid with the rubber molecules in the un-cross-linked state,
and the other G,, related to the elastic expansion of the cross-linked network. T h u s
aG aG,
_
--
aG,
+-.
an, an,
(5.3)
an,
T h e first of these, 8G,/an,, has been derived by Flory (1942) and independently
by Huggins (1942) from a statistical-thermodynamic model involving the calculation of the configurational entropy for a specified number of idealized polymer
molecules together with a number of liquid molecules arranged on lattice sites.
Their formula may be written
'Gm- - RT{ln(1-v2)+v2+xv,2}
(5.4)
an,
in which v 2 is the volume fraction of polymer in the mixture and R the gas constant
per mole. This expression contains a semi-empirical term xv22which is introduced
independently of the entropy calculation to represent the energetic interaction
between the polymer and liquid molecules, the constant x being specific to the
particular system considered. T h e second term, aG,/an, is obtainable directly
from the network theory (equation (2.7)). T h e ratio of swollen to unswollen
The elasticity and related properties of rubbers
795
volume being given by l/v,, we have, for an isotropic swelling,
A, = A, = A, = 1/v?.
(5.5)
Substituting into equation (2.7), and introducing (2.16), we thus obtain for the
work corresponding to the network expansion
For a constant pressure process 6G = 6A+pSV (from equation (5.2)). T h e term
pSV being negligible in the present case, we may therefore write
a ~ , a ~ , aw - -a w a v , = pR*y,v21j3
--
an,
an,
an,
av, an,
M,
(5.7)
where V, is the molar volume of the swelling liquid. (This derivation assumes
additivity of volumes, so that 1 n, V, = v2-'.) T h e total free energy of dilution,
taking into account both terms in equation (5.3), is therefore (from (5.4) and ( 5 . 7 ) )
+
T h e equilibrium swelling is determined by the value of v, for which the condition
i?G/an, = 0 (equation (5.1)) is satisfied, that is,
In(l-v,)+v,+Xc2~
Y
= -fl_laii3,
MC
(5.9)
This equation, which was first derived by Flory and Rehner (1943b), has
provided the fundamental basis for most of the subsequent studies of swelling in
cross-linked rubberlike polymers. It contains two physical parameters, x and M,.
Of these, the first relates solely to the un-cross-linked polymer. Its value for a number
of polymer-liquid systems may be determined from measurements of vapour
pressure or of any of the thermodynamically related properties (osmotic pressure,
Figure 21. Relation between equilibrium degree of swelling and molecular weight of network
chains for rubber in: 0 CCld; x CS,; A COHO;
corresponding curves from equation
(5.9). From Gee (1964b).
L R G Treloar
796
etc) of the swollen polymer (or polymer solution). T h e second parameter, Mc, is
of course a function of the degree of cross-linking, and may be derived from the
value of the elastic modulus. (cf $2.) An example of the applicability of equation
(5.9) is shown in figure 21, taken from the early work of Gee (194613) in which the
values of x were obtained from vapour pressure measurements and the values of Mc
from modulus measurements.
T h e physical significance of the Flory-Huggins relation is that the process of
swelling is governed primarily by the increase in entropy associated with the diffusion
of the liquid molecules into the polymer, this entropy change being essentially the
same for all swelling liquids and all rubbers. It is only in respect of the parameter x,
which is related to the specific interactions between the polymer and liquid
molecules, that the differences between different systems are exhibited.
I n the following sections we shall be considering two aspects of swelling
phenomena; first the effect of swelling on the mechanical properties of the rubber,
and secondly the dependence of the degree of swelling on the applied stresses or
strains.
5.2. Stress-strain relations foy swollen rubber
It is a simple matter to extend the elementary network theory, as outlined in $ 2,
to the case of a swollen rubber. T h e resulting equation for the work of deformation
per unit volume of the swollen rubber in a pure homogeneous strain is (Treloar 1958)
w =~NkTv,l'3(X,2+h,2+X32-33)
(5.10)
in which N is the number of chains per unit volume of the unswollen rubber and
A,, A, and A, are the principal extension ratios referred to the unstrained (ie stressfree) swollen dimensions. T h e corresponding relations between the principal
stresses (referred to the final strained swollen area) and the corresponding extension
o.9b
io
40
io
s'o
Id0
!O
Extension C%l
Figure 22. Variation of
x
with strain for different swelling ratios l / v 2 for rubber in toluene.
From Gee (1946a).
The elasticity and related properties of rubbers
797
ratios are of the form
(5.11)
These equations reduce to those for an unswollen rubber (equations (2.7) and
= 1. T h e effect of the swelling is thus to reduce the modulus
in the ratio vli3 without affecting the form of the stress-strain relations,
These conclusions are not entirely borne out by experiment. For the case of
simple extension equation (5.11) may be converted to the form
( 2 . 1 5 ~ )on
) putting U ,
f‘ = G~,-’is(h- l / h 2 )
(5.12)
in which f’ is the force per unit unstrained unswollen area. Figure 22 shows a
typical plot of the quantity x = f’/Tv,-l’s( X - 1/A2) against X (Gee 1946a), which
according to equation (5.12) should yield a single horizontal line, independent of
the degree of swelling. T h e results show that x decreases not only with increasing
strain, but also with increasing swelling (decreasing U,). These results, which have
been amply confirmed in later studies, indicate that with increasing swelling the
deviations from the theoretical form of force-extension curve diminish and
ultimately vanish.
I t has already been indicated in $ 2 that these deviations from the statistical
theory may be represented by the Mooney equation
(5.13)
2(X- 1/h2)(C,+ CJX)
containing two arbitrary constants C, and C,. T o take account of the effect of
f
=
b
2
Volume fraction of rubber,
Figure 23. ‘Mooney’ plot of 4 (equation
(5.14)) for various degrees of swelling. From Gumbrell et al (1953).
Figure 24. Dependence of constant C, in
equation (5.14) on degree of swelling:
0 natural rubber; 0 butadiene-styrene;
x butadiene-acrylonitrile. From Gumbrell et a1 (1953).
L R G TreZoar
798
swelling Gumbrell et aZ (1953) introduced a factor V,-’/S on the right hand side of
equation (5.13) to correspond to that in (5.12), so that
f fU21/3
+ = 2(h-
1/X2)
=
c,+C,/X.
(5.14)
+
On this basis a plot of against 1jX should yield a straight line of slope C,. Their
experimental data (figures 23 and 24) conformed remarkably accurately with this
equation, and showed that the C, term diminished with increasing swelling and
ultimately vanished at a value of o, of about 0.25, while C, remained approximately
constant.
It is seen from figure 24 that the general nature of these deviations from the
statistical theory is independent of the type of rubber and the particular swelling
liquid employed. Though no generally accepted explanation has been put forward,
it is tempting to identify the constant 2C, with G or N k T in the statistical theory and
to interpret it as a ‘genuine’ network parameter, while treating C, as an independent
parameter not directly affected by the network (Mullins 1956). I n the absence of a
more complete understanding, any such interpretation must, however, be treated
with caution, (For further discussion see 0 6.)
5.3. Dependence of swelling equilibrium on strain
T h e stress-strain relations given above define the stresses acting on a rubber at
any particular degree of swelling, as defined by the parameter l / u z , regardless of
whether this degree of swelling represents the equilibrium degree of swelling under
the conditions considered. T h e further question of defining the equilibrium degree
of swelling when the rubber is in contact with the liquid whilst under stress will now
be examined.
This problem was originally treated by Flory and Rehner (1944) and by Gee
(1946b), who considered the case of simple extension, and by Treloar (1950a), who
dealt with the more general case of a pure homogeneous strain. T h e condition for
equilibrium with respect to liquid content, in the presence of a stress, is that the
change in free energy for a small change in liquid content shall be equal to the work
done by the applied forces. For the general case the result is expressed by equations
of the form
(5.15)
where t , is the principal (true) stress in the 1-direction, and I, is the corresponding
principal extension ratio referred to the unstrained unswollen dimensions. If, for
example, I, and t , are fixed, the equilibrium swelling is determined by the value of
a , which satisfies equation (5.15). Similar expressions apply for the other two
directions.
For the case of simple extension 1, = Z3 = l/Z1vz and t, = t , = 0. This gives the
equation
(5.16)
The elasticity and related properties of rubbers
799
which may be solved for v2. For a uniform two-dimensional extension (1, = 13) the
corresponding condition for equilibrium is
l n ( 1 - ~ , ) + v ~ + x PK
~ ~ ~ + -=v 0.
~Z~~
(5.17)
Mc
Experimentally, these equations have been found to give a very satisfactory
representation of the changes in swelling for vulcanized rubbers in simple extension
(Flory and Rehner 1944, Gee 1946b) and in two-dimensional extension and
(uniaxial) compression (Treloar 1950b). An example is shown in figure 25. I n
simple extension or two-dimensional extension the swelling is increased by the
strain, while in compression it is reduced.
L2 (referred to unswollen state)
Figure 25. Dependence of equilibrium degree of swelling on strain for two-dimensional
extension. Curves from equation (5.17). From Treloar (1950b).
5.4. Torsion of cylinder
T h e calculation of the change in equilibrium swelling of a cylindrical rod when
subjected to torsion is a more complex problem than any of those discussed above,
owing to the fact that the state of stress and strain is inhomogeneous, the local stress,
and hence the local degree of swelling, being a function of the radial position of the
element considered. I n a theoretical treatment of this problem by the writer
(Treloar 1972) an approximate solution was obtained by making use of the equations developed by Rivlin (1949a) for the distribution of stress in an isotropic
cylinder subjected to combined axial extension and torsion about the axis (0 6).
From these equations the change in u2 in a radial shell dr at a distance r from the
axis was obtained by the application of the swelling equilibrium equations discussed
above. Integration of the change of swollen volume with respect to Y then yielded
the total change of swollen volume, AV, for the whole cylinder. T h e result was
expressed by the formula
(5.18)
L R G Treloar
800
in which V is the swollen volume at an axial extension ratio /I3 (referred to the
unswollen unstrained axial length), AV is the change in swollen volume produced
by a torsion (in radians per unit strained length), a, is the unstrained unswollen
radius, and v2 is the volume fraction of rubber corresponding to the swollen volume
V , Since (for a positive tensile stress) the denominator of (5.18) is a negative
quantity, this formula predicts a reduction of swelling on twisting, the amount of
this reduction being proportional to the square of the torsion.
T h e above formula is applicable only for small values of twist, that is, for small
changes of swollen volume. For larger values of twist an explicit algebraical solution
is not obtainable. For this case, however, it is still possible to obtain a solution,
+
4
Figure 26. Effect of torsion on swelling of cylinder: A, approximate theory, equation (5.18)
and B, exact theory (both Treloar 1972); C, experimental (Loke et al 1972).
using numerical computational methods. A comparison of the more exact solution
with the approximate formula (5.18) shows the latter to be sufficiently accurate for
practical purposes for values of +a, up to about 0.2 in a typical case (figure 26).
Experiments by Loke et a1 (1972) on the swelling of a twisted rubber cylinder
substantiated the general form of dependence of AV/V on torsion, though there was
a significant quantitative discrepancy from the theory (figure 26), amounting to
between 12% and 23% of the expected volume change.
T h e origin of the reduction of swelling due to torsion is to be found in the
existence of a radial compressive stress which, as Rivlin has shown ($6), also varies
as the square of the torsional strain.
T h e problem is of interest also in providing an analogy to the problem of the
change in volume of a dry rubber cylinder when subjected to torsion. T h e application of Flory's modified gaussian network theory (discussed in 5 4), which takes
account of the finite compressibility of the rubber, to the problem of torsion yields a
resultant reduction of volume which is similarly proportional to the square of the
torsion (Treloar 1969a). This second-order volume change has exactly the same
origin as the reduction of swollen volume in the swollen rubber cylinder, though the
The elasticity and related properties of rubbers
80 1
detailed processes are of course different. This is an example of the close formal
similarity between the changes in swollen volume of a swollen rubber, which behaves
as if it were highly compressible, and the changes in volume of a slightly compressible dry rubber under similar conditions of stress.
6. Phenomenological theories of rubber elasticity
Attention has been drawn in previous sections to certain deviations between the
observed forms of stress-strain relations for rubbers and the corresponding forms
predicted by the statistical theory, which are particularly apparent in the case of
simple extension. Some reference has also been made to certain other formulations
of an empirical or semi-empirical nature, for example, the Mooney equation, which
are designed to provide a more realistic representation of the actual properties, T h e
need for such theories arises particularly in the application of rubbers to problems
of the engineering type involving the design of components of predictable
mechanical properties.
These so-called ‘phenomenological’ theories have certain disadvantages, as well
as advantages, compared with the statistical theory. T h e latter, being based on a
specific molecular or structural model, enables us to understand why rubbers
possess certain particular properties, and is therefore relevant to the problem of
producing rubbers of any desired type from materials of different chemical constitutions. It is also not limited in its application to the purely mechanical properties, but can be applied, as we have seen, to many other problems, such as
swelling in solvents, thermodynamic or thermoelastic effects, photoelasticity, etc,
in all of which fields it has provided a profound physical insight into the molecular
mechanisms involved in the phenomena examined. Thus, despite its lack of complete precision, the statistical theory establishes a solid basic framework which is
essential for physical understanding. T h e phenomenological theories, on the other
hand, when not merely descriptive, are inspired by purely mathematical considerations, and it remains a matter for experiment to determine to what extent these
correspond to physical reality. While such theories offer the possibility of a greater
degree of refinement than the statistical theory, they are also capable of misuse, and
if not treated with proper caution, for example if extrapolated beyond the range of
observation, may lead to errors of a more serious and more fundamental character
than are likely with the statistical theory.
I n this section we shall first consider the particular type of general mathematical
formulation in terms of strain invariants developed by Rivlin, and its application to
( a ) problems of the engineering type, and ( b ) the general representation of the
mechanical properties of rubbers. This will be followed by a discussion of the form
of the observed deviations from the statistical theory and their possible molecular
interpretation. Finally, other types of mathematical approach to the formulation
of the mechanical properties of rubbers will be briefly reviewed.
6.1, Rivlin’s formulation of large-deformation theory
T h e most general homogeneous deformation of an elastic body may be resolved
into a ‘ pure’ strain, corresponding to three principal extensions (or compressions)
in three mutually perpendicular directions, together with a rotation of the body as a
whole. T h e rotational component involves no net work by the external forces, and
L R G Treloar
802
hence is independent of the properties of the material, which are therefore completely definable in terms of a pure homogeneous strain (figure 3).
For a complete specification of the elastic properties of the material it is sufficient
to know the form of the function W defining the work of deformation or elastically stored energy per unit volume in terms of the three principal extension ratios
A,, A, and A,. I t is pointed out by Rivlin (1948)that the form of W cannot be chosen
completely arbitrarily, but is subject to certain mathematical limitations. Thus, if
the material is isotropic in the unstrained state the function W must be symmetrical
in A,, A, and A,. Rivlin also argued that it must be an even-powered function of these
three variables. T h e three simplest even-powered functions are the so-called
strain inaariants
I, = A,, + A,, + A,,
I,
=
A,, A,,
+ A,2 A,, + A,2
A,2
I3 = A12 A,, A,.
1
(6.1)
For a constant volume deformation, corresponding to an incompressible
material, with which the present discussion is primarily concerned, I, = 1. T h e
two remaining equations may then be written
+ A,, + A,2
Il
=
I,
= l/A,Z+
A,2
1/A,2+ 1/A,2.
i
On this basis the stored energy function W must be expressible in terms of these
two strain invariants only. T h e most general form can thus be represented by the
series
W = clm(I1
- 3)'((I2- 3)"
(6.3)
in which I, - 3 and I, - 3 automatically vanish in the unstrained state (I, = I, = 3).
T h e two simplest expressions derivable from (6.3) are
w = (?,(Il- 3 )
(6.4a)
W = C 2 ( 4- 3 )
(6.4b)
in which C, and C, are constants. We have already seen that (6.4a), with C, = BNkT,
is the form derived from the gaussian network theory (equation ( 2 . 7 ) ) .
T h e most general first-order expression in I, and I, is obtained by combination
) (6.4b) to give
of ( 6 . 4 ~and
w = C1(I,- 3 ) + C2(12- 3).
(6.5)
This form of stored energy function was originally derived by Mooney (1940) on
the basis that the material obeys Hooke's law in simple shear, or in a simple
shear superimposed in a plane at right angles to a preceding uniaxial extension or
compression.
6.2. General stress-strain reIations
T h e relations between the principal stresses t,, t , and t, (figure 3 ) and the
principal extension ratios A,, A, and A, in a pure homogeneous strain of the most
general type involve the partial derivatives of W with respect to the strain invariants
The elasticity and related properties of rubbers
803
I, and I, (Rivlin 1948), that is,
E+
t , - t , = 2(A,2 - A,,)
A 3 2 3
with similar expressions for t, - t,, etc. As with the statistical theory, the assumption
of incompressibility ( I , 1 ) implies that only dijerences of principal stresses are
determinable. For the particular case when one of the principal stresses, say t,, is
zero, the other two are given by
For the Mooney type of stored-energy function equations (6.6) and (6.7) become
t, - t, = 2( A,2 )-: A
for the general stress system, and
(C,
+
t, = 2(A,, - A,) (C, + A,, C,)
t , = 2(A,, - A,) (C, + A,, C,)
C,)
X2,
i
t, = 0
(6.6a)
(6.7a)
for the case when t, = 0. For the special case C, = 0 (statistical theory) equations
(6.6a) and (6.7a) reduce further to
t , - t, = 2C,(A12 - A,)
(6.6b)
and
t,
=
2C,( A,, - A,)
t, = 2C1(A,2-
t, = 0
(6.7b)
A,2)
respectively.
6.3. Particular stress-strain relations
6.3.1. Simple extension (or uniaxial compression). For a simple extension we put
A, = A; A, = A, = A-'l2. Equations (6.7) then yield
For the Mooney form of W this becomes
t = 2 ( P - l / A ) (C, + C,/A)
(6.8a)
which transforms to equation (5.13) on putting t = Af.
6.3.2. Simple shear. Putting A, = A; A, = 1; A,
1,-3
1jX we have
= ~2+~-2-=
2 y2
= 1,-3
=
(6.9)
where y is the shear strain (cf equation (2.12)). T h e shear stress t,, (or t,,) may be
shown to be (Rivlin 1948)
I$+::(
t,, = t,,
=
2 .-
(6.10)
a04
L R G Treloar
which for the Mooney form of W gives
t12 = t2l = 2(CI + C,) Y
(6.10~)
implying Hooke’s law in shear (in accordance with the basic assumption of the
Mooney theory).
6.4. Stress components: normal stresses in shear
T h e analysis of Rivlin brings out a number of respects in which the effects of
large deformations are different not only in magnitude but also in kind from the
corresponding small deformation effects. An illustration is provided by the case of
simple shear (figure 27). I n addition to the tangential stresses tI2, tZ1,given by
A’
B
9‘
C
(6.10) there exist normal stresses tll, t2, and t3, on planes perpendicular to the
x, y and x axes. On account of the volume incompressibility, only dzfferences between
these normal stresses are obtainable. These are given by
(6.11)
It is seen that while the tangential components of stress (6.10) are identical in form
to those derived on the basis of the classical (small-strain) theory of elasticity, that is,
proportional to the shear strain, the normal components (6.11) are proportional to
the square of the shear strain. These normal components of stress are peculiar to
large elastic deformations, and have no analogue on the classical theory. Their
existence is quite general, and does not depend on the particular form of stored
energy function chosen.
T h e meaning of this result is that a large shear strain cannot be produced by a
shear stress acting alone; normal stresses must also be applied.
T h e origin of these normal stresses can be partly explained in terms of the
geometry of a large shear strain (figure 28). I n a small shear strain the principal axes
The elasticity and related properties of rubbers
805
of the strain ellipsoid lie at an angle of 45" to the direction of shearing, and the
principal stresses (acting in the directions of the principal axes) are equal and
opposite ( t , = -t,). I n a large shear strain the major axis of the strain ellipsoid is
inclined at an angle x to the direction of shearing (less than 45") given by
cot x
=
(6.12)
A.,
I n addition, from equations of the type (6.7), it follows that (for the case when
t , = 0) the numerical value of the tensile stress t , increases more rapidly than that of
Figure 28. Inclination of principal axes in simple shear.
the compressive stress t, as the strain is increased. On resolving the stresses t, and
t , parallel to the 1-direction the resultant therefore no longer vanishes, as it does
when t , = - t , and x = 45".
6.5. Torsion of cylinder
T h e above distinction between two types of stress components, one of the
classical form and the other having no classical analogue, is found in other more
complex problems involving shear strain. A typical example is provided by the
case of the torsion of a cylinder (figure 29). For this the tangential component of
I
/
I
/
/
/
/
Figure 29. Torsion of cylinder.
stress (teJ on a plane normal to the axis (z), for a Mooney-type material is given
(Kivlin 1948) by
to, = W r ( C , + C,)
(6.13)
where
#
is the angular twist per unit axial length and
Y
is the radial coordinate.
L R G Treloar
806
T h e corresponding normal component t, in the axial direction is
tG2
=
- #2{( C,- 2C,) (a2 - r2)+ 2a2C,}
(6.14)
where a is the radius of the cylinder. T h e resultant forces required to maintain the
state of torsion consist of a couple M about the axis together with a total normal
force N acting on the end surface given by
M = 4 a 4 (C, + C,)
N = - I ,np a4( c, 2C2).
+
(6.15)
(6.16)
T h e first of these equations is identical to the classical solution for a material of
shear modulus 2(C,+ C,). T h e second, which being negative represents a compressive force or thrust, is proportional to the square of the torsion, and has no
analogue on the classical theory. If this force is not applied the cylinder will
elongate on twisting.
Cb)
( a)
Figure 30. Distribution of end pressure on twisted cylinder: (a) statistical theory; (b) Mooney
equation.
T h e pressure distribution corresponding to equation (6.14) is parabolic, and is
illustrated in figure 30 for the case of the Mooney form of W, and for the gaussian
theory (C, = 0).
3 (rad m-'1
Figure 31. Relation between torsional couple and amount of torsion for cylinder of 1 inch
diameter. From Rivlin and Saunders (1951).
The elasticity and related properties of rubbers
807
T h e results represented by equations (6.15) and (6.16) have been verified by
Rivlin and Saunders (1951), whose data are reproduced in figures 31 and 32. From
a comparison of the values of M and N they obtained the ratio C,/C, = 118. This
compares with a value C,/C, = 1/7 obtained by Rivlin (1947) from the distribution of pressure across the end surface of a twisted cylinder, as represented by
equation (6.17).
yb2 (rad' rr?)
Figure 32. Relation between normal thrust and square of torsion. From Rivlin and Saunders
(1951).
Figure 33. Pure homogeneous strain. Difference of principal stresses in plane of sheet tl - t 2
plotted against AI2- At2. From Treloar (1948).
More complex problems which have been treated by Rivlin and his associates
include the combined torsion and extension of a cylinder and of a cylindrical tube
(Rivlin 1949a) the two-dimensional extension of a sheet containing a circular hole
(Rivlin and Thomas 1951), the eversion (turning inside out) of a tube (Gent and
Rivlin 1952) and the flexure or bending of a thick sheet (Rivlin 1949a,b).
808
L R G Treloar
6.6. Experimental determination of f o r m of stored energy function
I n $ 2 experimental data for rubber in extension, compression and shear were
presented and compared with the statistical theory, These simple types of strain,
however, are not adequate in themselves to determine unambiguously the properties of a rubber in all possible types of strain. For this purpose it is necessary to
examine the behaviour under a pure homogeneous strain of the most general type,
corresponding to any desired range of the independent strain invariants Il and 12.
Figure 34. Pure homogeneous strain. Principal stress tl (or t z ) plotted against XI2-hg2 (or
AB2- As2) for values of fi (or fi) shown. From Treloar (1948).
Figure 35. Pure homogeneous strain. Principal stresses plotted according to equations (6.7a),
C2/C1= 0.05.
This can be achieved by subjecting a sheet of rubber to two different stresses t, and
t, in two perpendicular directions, and measuring the corresponding principal
extension ratios A, and h2 in the plane of the sheet. (From the incompressibility
condition (2.3) the corresponding A, in the thickness direction is then determined.)
From the first experiment of this kind, carried out by the writer (Treloar 1948), a
The elasticity and related properties of rubbers
809
plot of t, - t, against XI2 - X Z 2 (figure 33) yielded fair agreement with the statistical
theory (equation (6.6b)). Closer examination, however, showed this result to be to
some extent illusory, since in this type of experiment, in which both A, and A,
exceed unity,
is in general very small, so that (6.6b) differs in practice only
slightly from (6.6a), with the result that a C, term (or, more generally, a term in
+x 11=5
d
1.4
+
IO
e
++
+ +
0I1=7
+11=9
0I1=11
26
18
'2
x 11=5
0I1=7
+11=9
0 Il = 11
x I,=5
0 J2 = 10
+ I2= 20
0.1
-
++ $+ +
I
I
I
I
I
1
0 I, = 30
. ' .e
1;
Figure 36. Pure homogeneous strain. Dependence of aW/aI, on Il and I,. From Rivlin and
Saunders (195 1).
aWja1, in equation ( 6 . 6 ) ) , if present, would probably escape detection. A more
critical test is provided by a plot of t , or t, separately against h12- A,, or hZ2respectively, on the basis of equations ( 6 . 7 ~ ~ I)n. such a plot (figure 34) the divergence from the statistical theory is immediately apparent. To test the applicability
of the Mooney equation the same data may be plotted alternatively against
A,,) (1 AI2 C2/C,) (equations ( 6 . 7 ~ ) ) By
. choosing
( 1 h,2 C,/C,) or
(X12a quite small value of C,/C,, namely 0-05, the independent arrays of points were
brought on to a single continuous curve (figure35). However, this curve was
+
+
L R G Treloar
810
definitely nonlinear, indicating that the Mooney equation, while giving a greatly
improved fit to the data, was still not entirely adequate.
Rivlin and Saunders repeated the above experiment under conditions such that
Il was held constant while I2was varied, and vice versa. For each value of Il and I,
they calculated both aW/aI, and aW/aI,, by insertion of the corresponding values of
t, and t , into equations (6.7). From their results, shown in figure 36, they drew
the conclusion that the stored energy function could be expressed in the form
W = Ci(I1- 3) +f(I2-3)
(6.17)
in which the function f diminishes progressively with increasing 12.This is equivalent to the statement that W contains two terms, of which the first corresponds to
o'
I
I
I
I
I
4
5
6
7
8
12
Figure 37. Pure homogeneous strain. Dependence of aW/aI, and aW/a12on I, for various
values of Il. From Obata et al (1970).
the statistical theory, while the second (correction) term decreases with increasing
strain. I n numerical terms the ratio of the second term to the first fell from about
1jS to about 1/30 over the range of strain covered by their experiments.
It is rather doubtful whether the data are sufficiently accurate to justify the
conclusion drawn by Rivlin and Saunders, as represented by equation (6.17), and
a more detailed investigation by Obata et a1 (1970), using a more elaborate automatic biaxial stretching device, indicates that neither aW/aI, nor aW/aI, can be
regarded as constant, and that each is a function of both I, and 1,(figure 37). Their
data do not suggest that either of these functions is reducible to any simple mathematical form.
Other possible forms of representation of the properties of rubber will be
considered later.
6.7. Significance of deviations fyom statistical theory
I n discussing the deviations from the statistical theory, which are usually
represented in the case of simple extension by the Mooney equation, as for example
in figure 23, it is important to bear in mind that in this form of representation the
The elasticity and related properties of rubbers
811
constants C, and C, are to be regarded as purely empirical parameters, and cannot
be automatically identified with the derivatives a WjaI, and a WjaI, in the general
stress-strain relations (6.6). T h e distinction, which is all too often overlooked,
arises from the consideration that in simple extension I, and I2 are not mathematically independent variables, since each is a function of the single strain parameter A. Thus a linear plot of f / ( h - 1/X2)against l / h , though consistent with the
I/
Figure 38. Contributions of aW/aI, and aW/aI, terms to total force (per unit unstrained area)
in simple extension: 0 directly measured; 0 calculated from 2W/811and aW/aI,. From
Obata et al (1970).
assumption aW/aI, = C,,aW/aI, = C, (equations (6.8) and (6.8a)) could equally
well arise from a linear variation of aWjaI, with l / h together with aWjaI, = 0, or
from some intermediate form of variation of both a W/aI, and a WjX, with strain.
This is explicitly shown by the work of Obata et al referred to above. By extrapolation of their pure homogeneous strain data these authors were able to estimate
the independent contributions of 2W/211and aW/GI, to the total stress in simple
extension, as given by equation (6.8). Their result (figure 38) shows that the term
in aWja12contributes only a small fraction (between 0 and 10%) to the total stress.
T h e variation off/(X- l/X2)with strain is therefore to be associated primarily with a
large i3W/i?I1term coupled with a much smaller aWji31, term, both of which are
strain dependent.
A still more fundamental objection to the use of the Mooney equation to
represent the deviations from the statistical theory in simple extension is that this
equation is based on the assumption of a linear stress-strain relation in simple
shear. But the observed deviations from linearity in shear (figure 6 ) are of a similar
kind to the deviations in simple extension. T h e basic assumption of the Mooney
equation is therefore invalid.
Notwithstanding these objections, it is remarkable that formal agreement with
the Mooney equation in simple extension appears to be almost universal, having
been obtained not only for natural rubber, but also for butadiene-styrene and
L R G Treloar
812
butadiene-acrylonitrile rubbers (Gumbrell et a1 1953), butyl and silicone rubbers
(Ciferri and Flory 1959), hydrofluorocarbon (Viton) rubbers (Roe and Krigbaum
1963), and cross-linked polythenes (Gent and Vickroy 1967), for example. Moreover, the values of the ratio C,/C, are usually rather large, typically between about
0.60 and 1-00 (Gumbrell et a1 1953). Suggested explanations of these deviations,
however, are scarce and rather unconvincing. T h e observation by Gumbrell et a1
(1953) that the value of C, diminishes with increasing swelling in a manner which
is independent of both the rubber and the swelling liquid (figure 24) suggests that
any explanation must be of a very general character.
m
';"E
z
1
'"'-
I
I
Extension
0
0
P
"
9
*I*
-lX
t
k1h-
4
-
'Dm O"0~
Compression
I
'
'
0.6
I/h
I
I
I
I
10
.
3
5
1
7
I/A
I
I
9
I1
Figure 39. Mooney plot of data for natural rubber in extension (A > 1) and in uniaxial compression (A< 1). (Note change of scale at h = 1.) From Rivlin and Saunders (1951).
Ciferri and Flory (1959), from a comparison of the C, values for different
rubbers showing varying degrees of hysteresis, attributed the deviations from the
statistical theory solely to the failure to obtain true equilibrium between stress and
strain, due to the presence of irreversible processes such as the breakdown of
entanglements between chains. This hypothesis receives some support from the
interesting observation of Kraus and Moczvgemba (1964) that carboxy-terminated
polybutadiene networks, which contain no 'loose ends' such as occur with conventional methods of vulcanization, show almost perfect agreement with the
statistical theory, that is, C, = 0. A comparable result has been obtained by
Price et a1 (1970) with natural rubbers cross-linked in solution, this having the effect
of reducing entanglements between chains. Krigbaum and Roe (1965) in a careful
review of the evidence on this subject point out, however, that certain rubbers may
show significant C, values even when precautions are taken to avoid all irreversible
effects. Moreover, on the hypothesis of Ciferri and Flory comparable deviations
would be expected in all types of strain. Yet, as already noted, the deviations from
the statistical theory observed under a uniaxial compression are practically negligible
(cf figure 5 ) . This point is more explicitly brought out by a 'Mooney' plot of the
data for extension and compression for the same rubber, as in figure 39. For the
example shown it is found that whereas in simple extension C,/C, N 1.0, in uniaxial
compression C, N 0.
Other possible explanations of the discrepancies from the statistical theory are
concerned with inter-chain packing effects, or with energy barriers to internal
rotation, etc. Theories taking into account these effects which have been presented
by Di Marzio (1962) and by Krigbaum and Koneko (1962) respectively yield results
showing some similarity to the experimental deviations. Thomas (1955), by
including an arbitrary additional term into the expression for the free energy of the
The elasticity and related properties of rubbers
813
single chain, was also able to reproduce the main features of the general strain data
of Rivlin and Saunders. Much work remains to be done, however, before the origin
of the deviations from the gaussian network theory, as exhibited in the most
general type of strain, can be regarded as established with any degree of confidence.
6.8. Alternative forms of representation
From the purely phenomenological standpoint many attempts have been made to
obtain a more realistic mathematical formulation of the elastic properties of rubbers
than that provided either by the statistical theory or by the two-constant Mooney
form of stored energy function. It is not practicable to discuss these in full detail,
but an attempt will be made to indicate the nature and scope of some of the main
lines along which these developments have proceeded.
First of all, attention is drawn to the more general theory of Mooney (1940).
This is based on an arbitrary nonlinear stress-strain relation in simple shear, which
replaces the linear relation used in the more restricted form of the theory considered
in the preceding pages. For the more general case the stored energy function is
shown to be expressible in the form of the even-powered series
a
W=
C {LIZn(
n=l
+
+
X3-2n
- 3 ) + Bzn(
+
+ X32n - 3)).
(6.18)
This form of development is analogous to, but rather more restrictive than, the
Rivlin formulation (6.3).
Other authors have developed more specific forms of stored energy function,
a number of which have been referred to in a paper by Alexander (1968). Gent and
Thomas (1958) proposed the two-constant expression
W = C,(Il-3)+kln(12/3)
(6.19)
which they showed to give results substantially similar to those deduced from the
Thomas (1953) theory, referred to above. Other types of formulation have included
additional terms associated with the limited extensibility of the network. Thus
Isihara et aZ(1951), using the non-gaussian statistical theory as a basis, added a term
in ( I , - 3)2 to the Mooney formula to give
w = CIO(I1- 3 ) + CO&
-3 )
+
- 3)2
C20(12
(6.20)
while Biderman (1958) included also a cubic term, that is,
W = Cio(I1-3) + Col(I2- 3 ) + C2o(I1- 3)'+ C3o(I1- 3)3.
(6.21)
Clearly this type of formulation can be carried to any desired degree of refinement
if it is intended to represent the stress-strain data for a particular type of rubber
in a purely empirical manner. An example of this kind of elaboration is provided
by the work of Tschoegl (1970), who showed that the complete force-extension
curve for a carbon-reinforced natural rubber vulcanizate could be represented by
the formula
w = Cl0(I1- 3 ) COl(4- 3 ) C& - 3 ) (I,- 3 )
(6.22a)
while for a 'pure-gum' butadiene-styrene rubber the best fit was obtained with the
formula
W = c10(11-3)+
co1(I,-3)+C ~ 2 ( I 1 - 3 ) ~ ( I 2 - 3 ) ' *
(6.22b)
Such formulations are open to the objection already referred to in connection with
+
+
8 14
L R G Treloar
the Mooney equation. T h e elaboration is misleading, since a polynomial expression
in terms of the single variable h would contain all the information which can
justifiably be represented by these more complex formulae, without carrying the
unwarranted implication of general validity for the three-dimensional strain case.
Some authors have proposed formulations which include the ‘non-gaussian’
effects more directly, and also take into account different types of strain. Thus HartSmith (1966) finds that the expressions
aw
- - G exp {k,(I, - 3),}
a11
(6.23)
give a good fit to the writer’s data on compression (or equibiaxial extension) as well
as to Rivlin and Saunders’ pure homogeneous strain data. By a further elaboration
of this theme Alexander (1968) arrived at the more complicated five-parameter
equation
I
W = C, exp @(I,- 3)2) dI, + C, In ((I,-:)
+
7)
+ c3(r2
- 3)
(6.24)
which was found to give good agreement with his simple extension and equibiaxial
extension data on polychloroprene rubber.
Other authors have questioned the necessity of the Rivlin formulation involving
strain invariants which are even-powered functions of the extension ratios. This
point of view has been adopted, for example, by Varga, who develops a series
expression for the difference of principal stresses (in pure homogeneous strain), of
which the first- and second-order terms are (Varga 1966 p115)
+
+ + a3e3>
t, - t , = al(e, - e,) (1 a2(e1 e,)
(6.25)
the e$ being strains (hi- 1). T h e first-order term al(e,-e,) alone is found to give
good agreement with experiment up to strains (e$) of about 1.0 but is seriously
inadequate for higher strains. T h e inclusion of the term in curly brackets in
formula (6.25) gives a fair representation of the writer’s data for different types of
strain.
A promising line of attack has been suggested by Valanis and Landel (1967), who
have proposed as a hypothesis that the stored energy function may be capable of
representation as the sum of separable functions of the three independent extension
ratios, that is,
(6.26)
= w( A,)
w( A,) w( h3).
w
+
+
Symmetry considerations require that these separate functions shall be identical.
T h e problem is thus reduced to the determination of the form of the function w(h)
corresponding to a single strain variable. This formulation is not a mathematical
necessity, and its justification must therefore be based on physical or experimental
rather than mathematical considerations. It is pointed out that both the gaussian
network theory and the Mooney equation satisfy this hypothesis, as does also the
non-gaussian network theory in one of its simplest formulations ; the hypothesis
therefore has an a priori plausibility. Certain forms of W, for example those
including products of I, and I, in equation (6.3) are, however, excluded.
The elasticity and related properties of rubbers
815
2.5,
2.0-
1.51.0-
.
-a
N
h
-
0.5-
I
-
0-
-0.5 -
-‘%O
kL-----0.5
0
( A , w’ ( A , )
0.5
-
1.0
1.5
2.0
5
A, w’ (A,) 1 / 2 p
Figure 40. Representation of various stress-strain data in terms of equation (6.27): A Becker
biaxial;
Becker uniaxial; 0 Rivlin and Saunders; 0Treloar equi-biaxial. From
Valanis and Landel (1967).
+
in the case of an incompressible rubber (I, = l), to pure shear. For this case we
have also (since A, = 1/A, and t, = 0),
t, - t, = - t, = Al-lw’(Al-l) - constant.
(6.27b)
Thus if t, and t, are measured, as functions of A,, equations(6.27~)and (6.276)
are sufficient to enable the function Aw’(A) to be evaluated for values of X either
greater or less than unity, subject only to an undetermined constant c, given by
c = (Aw’(A)),,, = w’(1).
(6.28)
T h e results may then be applied to the derivation of the principal stress differences
in any other type of strain, for on substituting back into equation (6.27) the constant c disappears. According to this equation, for all types of strain a plot of t , -,!i
against A, w’( A,) - A, w’( A,) should yield a single straight line of unit slope, the
w’(A) being derived from a pure shear experiment on the same rubber. This
expectation has been tested by Valanis and Landel, using various authors’ data for
simple extension, uniaxial compression and biaxial extension. As a basis for
deriving w’(A)the data of Becker (1967) for pure shear were used and a scale factor
or modulus (2p) was introduced to render the different rubbers comparable. T h e
results, as shown in figure 40, give strong support to the proposed theory.
L R G Treloar
816
Valanis and Landel also proposed that over the range of A from about 0.35 to
2.5 the function w’(A) approximates to the logarithmic form
w’(A) = 2 p In A.
(6.29)
This, however, was based on the incorrect assumption that c in equation (6.28) may
be put equal to zero. T h e correct interpretation of their observations, on the basis
of equations ( 6 . 2 7 ~ and
) (6.28)) would give (for A, > 1)
(6.29~)
with, presumably, a comparable correction for the case A < 1.t
0
Figure 41. Plot oft, - t z against A2 at various A, for uniaxial strain (e),and biaxial strain (other
symbols). From Obata et al (1970).
Further support for the Valanis-Landel hypothesis is provided by the work of
Obata et a1 (1970) referred to above. These authors plotted their uniaxial and
biaxial strain data in the form shown in figure 41. According to equation (6.27) or
( 6 . 2 7 ~ the
) differences between t,-t, for two different values of A,, at constant A,,
are independent of A,; the curves relating t,-t, to A, for different values of the
parameter A, are therefore parallel. T h e accuracy with which this prediction is
fulfilled is shown in figure 42 in which the curves for different values of A, have been
displaced vertically so as to be brought into coincidence. T h e curve shown represents the pure shear data, displaced in a similar way.
Reference may here be made to an earlier paper, by Carmichael and Holdaway
(1961)) which bears a close relation to Valanis and Landel’s formulation. These
authors use as a basis the generalixed Mooney equation, in which the stored energy
function is represented by equation (6.18). By regrouping of terms this can be
expressed in the form
= <D(A,2)+<D(h,2)f(D(h,2)-(D(l)
(6.30)
w
in which (on Mooney’s theory) only even powers of A are allowed. Carmichael and
Holdaway point out that this restriction is physically without foundation, and
The curve referred to as 2p In A in figure 2 of Valanis and Landel (1967) is incorrectly
drawn. (Landel, private communication.)
The elasticity and related properties of rubbers
817
proceed to apply equation (6.30) without any restriction on the form of a. I n this
sense it becomes equivalent to (6.26). T h e authors show that it is possible to
represent the data of Treloar (1944) for simple extension, shear and equibiaxial
extension very accurately on this basis, using formulae containing three adjustable
parameters. T h e stress-strain relation for simple shear is given in the form
t12
=
A sinh ,By
2(1 +y2/4)”a
(6.3 1)
in which y is the shear strain and A and p are adjustable parameters. For uniaxial
extension or compression an additional parameter B is required, that is,
t, = &A[exp{ j ( h- A-,)}
- exp { --p(X1‘2 - h-’/2))] - B(X2+ hF2- h - X-l).
(6.32)
Unfortunately the form of @ in (6.30) is not explicitly derived, so that a direct
comparison of these conclusions with those of other workers is not possible.
Figure 42. Superposition of data shown in figure 41 on curve derived from pure shear experiment. Symbols for points as in figure 41. From Obata et al (1970).
An important recent development is that of Ogden (1972a). This also diverges
from the Rivlin type of formulation in discarding the principle (for which the
arguments are debatable) that the stored energy should be expressible in terms of
even-powered functions I, and I2 of the extension ratios. For an incompressible
rubber Ogden expresses W in the form of the series
(6.33)
in which the p n are constants and the an are not necessarily integers, and may be
either positive or negative. This formulation (which incidentally conforms to the
Valanis-Landel hypothesis) includes the statistical theory (n, = 2) and the Mooney
equation (n, = 2, n2 = -2) as special cases. T h e principal stresses are given by
(6.34)
L R G Treloar
818
It is shown that the inclusion of two terms only in the series is sufficient to describe
the writer’s data for simple extension and pure shear, but is inadequate to account
for the equibiaxial extension data. T o represent all three types of strain a threeterm expression is required. This contains the six adjustable parameters
al = 1.3
p1 = 6.3 kgcm-2
a2 = 5.0
p 2 = 0.012 kgcm-2
013
= -2.0
(6.35)
p 3 = - 0.1 kg cm-2.
T h e degree of agreement with experiment is shown in figure 43.
h
Figure 43. Representation of data for simple extension (0)’
equibiaxial extension (e),and
pure shear (+) by Treloar (1944) on basis of equation (6.34). (f is the force per unit
unstrained area.) From Ogden (1972a).
Ogden’s formulation has the merit of mathematical simplicity, which arises
from the fact that all the terms in (6.33) and (6.34) are of identical form. This
advantage is apparent in the application of the theory to such problems as the combined extension and torsion of cylindrical rods or tubes, which have been fully
treated by Ogden and Chadwick (1972)’ and combined axial and torsional shear
(Ogden et aZ1973). I n the first of the above papers it is shown that the three-term
expression, with the values (6.35) of the constants adjusted only by a common
scaling factor, accounts very well for the torsional data of Rivlin and Saunders
(1951).
The elasticity and related properties of rubbers
819
6.9. Compressible rubbers
I n the foregoing discussion it has been assumed throughout that all deformations take place at constant volume, so that I, in equation (6.1) is equal to unity.
In reality, rubbers have a finite compressibility, which though small (relative to
the shear compliance) is not negligible in all contexts, for example, in relation to
changes of internal energy during deformation ($4).A suitable adaptation of the
gaussian network theory which is applicable so long as the volume changes are small
has been given by Flory (1961) and in a modified form by Treloar (1969a). I n this
adaptation a separate term, representing the free energy associated purely with the
change of volume, is added to the free energy of network deformation. This treatment is equivalent to the assumption that in respect of volume changes the rubber
has the properties of an ordinary liquid, with a constant value of compressibility.
I n all other respects the large-deformation properties are essentially unaffected by
this modification.
I n the more general case when the volume changes are large the situation is more
complex. This case can arise in such materials as foam rubbers, where the bulk
modulus (ie reciprocal of compressibility) is of the same order of magnitude as the
shear modulus, or in normal rubbers under conditions of restraint involving very
high hydrostatic stress components. General theoretical treatments of compressible
rubbers have been put forward by Blatz and KO (1962) and Blatz (1963) and also
by Ogden (1972b).
Baltz and KOintroduce a modified set of strain invariants J,, J, and J,, such that
(6.36)
They then write for the stored energy
W = C Cl,,( J1 - 3)'( 5 2 - 3)"( 5, - 3),.
(6.37)
On expansion of this expression in terms of strains ei ( = Ai- 1) and differentiation
with respect to J,, 5, and 5, one obtains (putting e, + e, + e, = 0)
aWjaJ, = A + B ( e , + e , + e , ) +
aWjaJ,
=
C+D(e,+e,+e,)+
aWja5, = E + F ( e , + e , + e , ) +
...= A + B 0 + ...
...= C+DB+ ...
...= E + F 0 + ... .
1
(6.38)
I n the large-strain theory the principal stresses are readily derived in terms of the
derivatives aWja1,. I n the classical theory of elasticity, on the other hand, the
stresses are directly related to the strains ei. Hence if equations (6.38) are limited to
small strains (ie by neglecting squares and products of ei etc) the two theories must
be equivalent, and by comparing the coefficients of corresponding terms certain
identities may be established.
T h e principal stresses fi (referred to the unstrained cross-sectional areas) are
given by equations of the type
(6.39)
32
L R G Treloar
820
which, on introducing (6.38) and retaining only first-order terms in e, becomes
fi= (1 - ei) {2(1+ 2 4 ( A +BO)- 2( 1- 2ei) (C+ DO)}+(1 + 0 ) ( E + Fe).
(6.40)
T h e corresponding relations from the classical theory of elasticity are
fi= 2pei + (K- 2p/3) O
(6.41)
in which p and K are the shear and bulk moduli, respectively. By comparing
coefficients in (6.40) and (6.41) Blatz and KO derive the results
A+C=p/2
(6.42)
2A-2C = - E
2B-20+ E + F = K - 2 ~ 1 3 .
(6.43)
(6.44)
They now introduce a subsidiary parameter f such that
(6.45)
A =PfP
c = P(1 -f
)/2
(6.46)
p(1-2f).
(6.47)
which gives, from (6.43),
E
=
T h e further development is restricted to the case where both aW/i3Jl and
aW/aJ2are constants, so that B = D = 0. Equation (6.44) then gives
F = K-2p/3-E = K-p(5/3-2f).
(6.48)
Introducing the above equations for E and F into the third of equations (6.38) they
obtain an expression for aW/aJ3. T h e complete set of strain invariants then becomes
2W/8Jl = pf12
aw/aJ,= (P/2) (1-f)
aWjaJ3 = p(l-2f)+{K-p(5/3
(6.49)
-2f)}(J3-1)
and the principal stresses take the form (from (6.39) )
fi
Ai = p{fAi2 - (1 - f ) / A i 2 }
+ 5 3 aW/aJ3.
(6.50)
It is seen that the first two of equations (6.49) correspond to the Mooney equation
(Hooke's law in simple shear), the parameters f and (1 -f) being the fractional
contributions of the C, and C, terms in that equation (cf equation (6.13a)). I t is
only in the J3 term that the finite compressibility appears. It is to be noted also that
J3 - 1 represents the relative increase in volume under the applied stress. This term
is eliminated when only differences of principal stresses are considered, as in the
usual experimental tests, that is,
This equation is formally equivalent to the Mooney equation for pure homogeneous strain ( 6 . 6 ~ for
) an incompressible rubber.
Blatz and KO have applied the foregoing theory to their experimental data for
polyurethane rubbers (continuum) and polyurethane foam rubbers, using uniaxial
extension, strip-biaxial (A2 = 1) and equibiaxial (A, = A,) tests. For the case of
uniaxial extension the data for the continuum rubber were accurately represented by
821
The elasticity and related properties of rubbers
i3W/2Il = constant, aWja5, = 0, while for a foam rubber containing 47% rubber by
volume in either uniaxial or biaxial extension aW/aJ, was constant and i3W/i3Jl
was very small or zero. T h e latter results are summarized in table 4 below.
Table 4. Values of parameters for polyurethane foam rubbers?
Type of strain
2 p (N mm-*)
f
0.26,
0.13
0.07
Simple extension
Strip-biaxial extension
Equibiaxial extension
0.20,
0.18,
-049
V
0.25
0.25
0.25
From Blatz and KO (1962).
T h e effects of the finite compressibility are only brought in by introducing a
further assumption, which enables .I3to be determined. For this purpose Blatz and
KOpropose a relation between longitudinal and transverse extension ratios in simple
extension of the form
A, = A,-”
(6.52)
in which v is equivalent to Poisson’s ratio in the case of small strains. (For an
incompressible rubber A, = hl-’/2.) This gives
- A11-2v.
(6.53)
J3 -
Introduction of this relation into the constitutive equation (6.39) corresponding to
the lateral dimension in simple extension (i = 2, f, = 0) yields aW/aI3,that is,
(6.54)
T h e resulting value of aWja5, may then be reintroduced into (6.39) and applied
to any other type of strain, with J3 treated as a measured variable.
From measurements of lateral dimensions Blatz and KOwere able to substantiate
the relation (6.52) for simple extension, the value of v being 0.25. They showed also
that consistent results were obtainable from their biaxial strain data, interpreted on
the same basis (table 4).
6.10. Compressibility of ‘solid’ rubbers
Stress-strain data for ordinary ‘solid’ or continuum rubbers in the usual tests
do not provide a sufficiently accurate basis for the examination of the aWja5, term,
owing to the smallness of the volume changes ( J 3 - 1) involved. T o obtain sufficiently reliable data it is necessary to apply very high hydrostatic pressures. Under
these conditions the volume changes are associated primarily with the intermolecular
potential field. T o treat this problem Blatz and KO adapted a formula due to
Murnaghan ( 1 9 5 9 namely,
p
K
k
= -(J3-k-
1)
(6.55)
where p is the pressure and k is related to the intermolecular repulsive term. This
was found to fit the data of Bridgman (1945) for a butyl rubber tread stock up to
50 kbar, with the value K = 13-3. They show further that the resulting aWjaJ,
term may be put into a form which is consistent with their general formulation
L R G Treloar
822
based on (6.53) provided that
k=
5+2v
6( 1 - 2v) ’
(6.56)
For k = 13.3 this yields the result U = 0.463.
T h e significance of this figure, which is inconsistent with the value of Poisson’s
ratio for small strains, for which Blatz and KO quote the value 0,49997, is not
entirely clear.
6.10.1. General comment. Further work will be required before the practical value
of the Blatz and KO theory can be properly assessed. At first sight it seems to the
writer that a rather elaborate structure has been built on a somewhat limited
foundation, that is, constancy of CWjaJ, and aWjaJ,, which we have already seen
to be far from justified in the case of natural rubber in the normal solid form.
Whether this assumption is more generally applicable to compressible (eg foam)
rubbers remains to be seen. Finally attention may be drawn to the assumption that
i?W/CJ3 is independent of J, and J2) which is involved in the application of the
formula (6.54) to the general strain. This assumption is shown to fit the biaxial
strain data for the foam rubber, but before it can be regarded as generally valid it
would be desirable to have further data covering in particular compressive types of
strain, for which J3 < 1.
6.1 1. Ogden’s theory
Ogden’s theory (Ogden 1972b) for compressible rubbers is a natural extension
of the same author’s theory for incompressible rubbers (Ogden 1972a), discussed
above. For the compressible case, as in the theory of Blatz and KO,a term associated
with the volume (A, A, A3) is introduced into the stored energy function, previously
given by (6.33). T h e modified form of W thus becomes
~
w= p(Al.n+A2.n+A3.11n
3) + F ( J 3 )
(6.57)
an
where J3 = A, A, A3 and F is an unspecified function. As in the incompressible case
the values of an are not necessarily integers. T h e corresponding principal stresses ti
(referred to the deformed areas) are
(6.58)
where F’( J3) = 8WjaJ,. (In the corresponding incompressible case the term
involving J3 is replaced by an arbitrary hydrostatic stress p (equation (6.34).)
Equations (6.57) and (6.58) are transformed by the introduction of two new
functions f(J3) and g( J3) defined by
(6.59)
f(J3) = F( 5 3 ) + (2pn) In J3
(6.60)
to give
W = pn{(Ala* + A Z +~Aga,~ -~ 3)!01, - In J3}+ A g ( J3)
J3tZ= p,,(Ai.~-1)+AJ3g’(J3)
i = 1,2,3
(6.61)
(6.62)
The elasticity and related properties of rubbers
823
the suffix n implying summation over n. Equation (6.62) reduces to the classical
theory in the case of small strains, the equivalent Lam4 constants ( p and h) being
p = ’ pLnan
(summed over n)
h = A.
(6.63)
T h e application of equation (6.63) may be illustrated by the case of a simple
extension in the ratio A,, with A, = A, = Ji’2 h1-l!2. Since f z = 0 we have
fl = f 1 - f 2
J 3a d 2> *
= Pn(A1an - h-ad2
1
(6.64)
For a normal (ie slightly compressible) rubber J3 differs from 1 by about
and
hence (6.64) is indistinguishable from the form derived for the incompressible
case, that is,
(6.64~)
This result, which is easily generalized, means that the introduction of a small
compressibility has no significant effect on the stress-strain relations, so long as very
high stresses are not involved. It follows that the degree of agreement with experimental data in simple extension, pure shear and equibiaxial extension obtained on
the basis of Ogden’s original theory, referred to earlier, is retained in the modified
form of the theory.
T h e corollary of this is that in order to investigate the form of the compressibility
term it is necessary to work at high values of hydrostatic stress. For this purpose,
Ogden, like Blatz and KO, makes use of the data of Bridgman (1945) and the
corresponding Murnaghan relation (6.55). This he modifies slightly to give
(6.65)
where /3 is a constant. For high values of p this may be approximated by
p
= ~lp-1
J?(P+1).
(6.65 a )
This equation describes the experimental pressure-volume data of Bridgman for
two rubbers, and also those of Adams and Gibson (1930) very satisfactorily, with
values of p between 9 and 11.
It is clear that in relation to the compressibility term the practical consequences
of Ogden’s theory are very similar to those of the Blatz and KO theory. T h e conclusion that for slightly compressible rubbers the form of the conventional stressstrain relations is not significantly dependent on the inclusion of the compressibility
term implies that the theory has little bearing on the properties of these materials,
except under the rather special condition of high hydrostatic pressure. But under
this condition, conversely, the pressure-volume relationship is substantially independent of the specifically rubberlike properties (ie the large distortional or shear
strain behaviour), and is a function primarily of the intermolecular forces, which
are comparable to those in a liquid. Thus while Ogden’s formulation has a greater
generality than that of Blatz and KOin that it avoids arbitrary assumptions about the
form of the stored energy function, its practical assessment must await its application
to highly compressible or foamed rubbers, in which the respective contributions of
the dilatational and distortional terms to the stored energy are comparable in
magnitude.
824
L R G Treloar
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