Treloar theory of network elasticity

The elasticity and related properties of rubbers

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1973 Rep. Prog. Phys. 36 755

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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, UK

Abstract In Section 1 the relation of rubbers to other classes of polymers and the

molecular basis of rubber elasticity are briefly examined. In 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 stress- strain 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. The 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 Page

1. Historical survey . . 758 1.1. Early developments . . 758

1.3. Statistical development . , 760 1.4. The glassy state . . 762 1.5, Crystallization . . 763

2. Network theory . . 763 2.1. Entropy of single chain . . 763 2.2. Definition of the network . . 763 2.3. Basic assumptions. . 764 2.4. Calculation of network entropy . . 764 2.5. Particular stress-strain relations . . 765 2.6. General stress-strain relations, , 766 2.7. Experimental verification . . 767 2.8. Limited extensibility of chains : non-gaussian statistics . . 770

3. Photoelasticity . . 775

3.2. Network theory . . 776 3.3. Significance of theoretical conclusions . . 777 3.4. Experimental examination . . 778 3.5. Effect of swelling on the stress-optical coefficient . , 780 3.6. The equivalent random link . . 780 3.7. Temperature coefficient of optical anisotropy . . 782

4. Thermodynamics of rubber elasticity . . 783 4.1. Elementary theory . . 783 4.2. Later thermodynamic analyses . . 786 4.3. Experimental determination of f,/f . , 788 4.4. Thermoelastic data for torsion . 791

1.2. Meyer’s kinetic theory . , 759

2.9. The equivalent random link . . 774

3.1. Basic concepts . , 775

4.5. Conclusion . , 793

5. Swelling and mechanical properties . . 794 5.1. Mechanism of swelling . . 794

5.4. Torsion of cylinder . . 799

5.2. Stress-strain relations for swollen rubber . . 796 5.3. Dependence of swelling equilibrium on strain . . 798

6. Phenomenological theories of rubber elasticity . . 801 6.1. Rivlin’s formulation of large-deformation theory . , 801 6.2. General stress-strain relations. . 802 6.3, Particular stress-strain relations . . 803 6.4. Stress components: normal stresses in shear . . 804 6.5. Torsion of cylinder . . 805 6.6. Experimental determination of form of stored energy function 808 .

The elasticity and related properties of rubbers 757

6.7. Significance of deviations from statistical theory . . 810 6.8. Alternative forms of representation . , 813

6.10. Compressibility of ‘solid’ rubbers . . 821 6.9. Compressible rubbers . . 819

6.11. Ogden’s theory . . 822

References . , 824

75 8 L R G Treloar

1. Historical survey

The 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.

The 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. The 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&pe or 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.


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. The 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. The 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.

The 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). The acceptance of the concept of a high polymer con- stituted 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. He 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. He 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. The 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. In thermodynamic terms this implies that the elasticity is associated with a reduction of configurational entropy on extension. The 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. The Gough-Joule effects follow immediately from the initial assumptions of the kinetic theory. If dU, d W and dQ 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

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

Since for an increase in length d Wis 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.

The 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).

d U = dQ+dW. (1.1)

dQ = -dW. ( 1 -2)

1 .3 . Statistical development The original conception of the kinetic theory was quickly followed up by

detailed mathematical treatments, first of the chain molecule considered as an iso- lated 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 dxdydx was shown to be defined by the gaussian probability function

p(x, y, 2) dx dy dz = (b3/.rr3’z) exp { - b2(x2 +y2 + 2))) dx dy dx (1 .3 )


where b2 = 3/2n12. (1.3a)

This function may be resolved into three independent component probabilities of the type

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.

p ( x) dx = ( b / d 2 ) exp ( - b2 x2) dx (1.4)

r ( b )

Figure 2. Distribution of (a) end-to-end distance Y, and (b) x component of Y , for long-chain molecule.

The 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) r2 exp ( - b2 r2 ) 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 . The root mean square end-to-end distance for the free chain, obtained from equation (1.5)) is

(p)"2 = (3/2b2)'/2 = lnlh ( 1 4 and is proportional to the square root of the number of links in the chain.

7 62 L R G Treloar

The 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. The more accurate (non-gaussian’ theory, in which this approximation is avoided, will be considered in 5 2.

1.3.1. Application to real molecules. The treatment of the molecule in terms of an idealized randomly jointed chain of equal links is adopted for mathematical con- venience only. In an actual molecule the successive bonds are connected at a definite angle-the valence angle. The simplest of such structures is the paraffin- type 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 RMS length of 2.01n1’2 A, where n is the total number nf bonds in the chain.

1.4. The glassy state K’ot all polymers are rubberlike, and even in rubber itself the rubbery pro-

perties 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. In 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).

The rubbers are distinguished from the glassy polymers primarily by the strength of their intermolecular forces. In a typical glassy polymer such as poly- methyl methacrylate (Perspex), the presence of polar groups attached to the main chain increases the intermolecular forces, so raising the glass transition temperature


to + 100 "C. It is only when heated above this temperature that such a polymer begins to develop rubberlike properties.

1.5. Crystallization The failure to exhibit rubberlike properties may arise alternatively from

crystallization. Many synthetic polymers, for example, nylon, polyethylene, poly- propylene, 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. In 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. The 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

The application of the statistical theory to the problem of a network of long- chain 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,

where K is Boltzmann's constant. Application to equation (1.3) thus gives, for the chain entropy s,

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).

S = k l n P (2.1)

s = c - kb'(x2 + y 2 + 9) = c - kb2 r2 (2.2)

2.2. Dejinition of the network The 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. The 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). The 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


oriented we have 2 = = z"i = +G, where 3 is the mean square chain vector length in the unstrained state. Putting this equal to 2 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)

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

( U 1 (61

Figure 3. Pure homogeneous strain: (a ) unstrained state; (h ) strained state.

(AU = 0). The 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)

W = iNkT(A1' + A,* + - 3) . (2.7)

Hence from (2.5) the work of deformation per unit volume is

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. The condition for constancy of volume (equation (2.3)) requires that

A, = A A, = A, = A-li2. (2.8)

W = $NkT(A2+2/A-3). (2.9)

Insertion in equation (2.7) yields the work of deformation

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 cross- sectional area, we have then

(2.10)

766 L R G Treloar

2.5.2. Uniaxial compression. Simple extension corresponds to the case h > 1. The 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 - l ih . (2.12)

The work of deformation (2.7) thus becomes

W = &NkT(X2 - 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

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(hI2 - A,,) etc. (2.15a)

I n a practical problem one of the principal stresses is usually known, and the remaining stresses are thus determinable. In a simple extension, for example, the lateral stresses ( t , and t3) are zero and hence by introducing (2.8) into (2 .15~) we obtain

t , = N k T ( h 2 - l jh) . (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.

I t 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’

ti = NkTXi2+p i = 1,2,3 (2.15)

The elasticity and related properties of Tubbers 767

M, of the chains (ie segments of molecules between successive points of cross- linkage), 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. The 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) The 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) The 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. The 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

theoretical. Figure 4. Force-extension curve for uniaxial extension : curve A, experimental; curve R,

The 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. The 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. The ‘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,. The equivalent shear stress is readily obtained in terms of the corresponding shear strain y. The 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. The 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


general pure homogeneous strain, considered in $ 6 . The deviations, however, are by no means negligible. These appear most clearly in the curve for simple extension. The 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. The 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, = NkT, 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 /h2) 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 force- extension 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. The 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. In 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.

The 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%. The 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 cross- linking 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 The 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. In 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. In reality, the probability must fall to zero at Y = nl, corresponding to the maximum extensibility of the chain. The 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 approxi- mations 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. The form of this function (figure 8) is such that f + 00 as rln1-t 1 ; its series expansion is

= 3 ( $ ) + J 9 (a) Y 3 +T7J(;;1)3+-isj 297 r 1539 (2) r 7 + ... . (2.19a)

The 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.

The 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,

772 L R G Treloar

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 non- gaussian force-extension relation for the network

f = h~vk Tnl/z(g-1( hn-l/~) - A-% g-1( 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

r l n l 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. The 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.

&lore refined 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 .19~) 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 .19~) 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, The 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 force- extension 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. In 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 para-

meters N , the number of chains per unit volume, and n, the number of random links per chain, were treated as independently adjustable parameters. In 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.

The 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).

The 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). The 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.) In 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(cis- isoprene) 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. The 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. The results, shown in figure 11, indicate a very satisfactory agreement with


this theory. The 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.

I 1

I I I I I 1 O.do.2 0 4 0 6 0.8

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, Mc = 5505; curve C, Mc = 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. The 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, On 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 long- chain 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.

776 L R G Treloar

The 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 yz are the mean polarizabilities for directions of electric field respectively parallel and perpendicular

r l n l Figure 12. Relative optical anisotropy y1-y2 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

3 r 2 36 r 4 1 0 8 ~ 6 y1-y2 = n ( a 1 - a 2 ) J -J +- - +=- - +... . [ ( ) 175 (n l ) 815 (d) ( 3 ' 1 a )

For small values of rlnl the first term only of this series provides an adequate approximation, that is,

( 3 . l b )

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


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)

i 3 1 2 - 5 p l -p - -N("l-"2) 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 47T -= -p. n2+2 3

The birefringence so obtained is given by

(3-3)

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 cross- sectional area measured in the strained state, as given by the statistical theory (equation (2.16)), is

The ratio of birefringence to stress, or stress-optical coefficient (C), is thus t = NkT(A2- l / A ) . (3.5)

The 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 The 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 cross- linking (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 . The 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 .15~) 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. The 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-')

extension. Figure 13. Relation between birefringence n l - n 2 and stress for natural rubber in simple

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 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). In 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?

M , C m2 N-l) Cure Chain molecular weight Stress-optical coefficient

Peroxide 2150 4210 5270 7400

Radiation 7334 13360 15290

0.183 0.195 0.198 0.200

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 cross-linking (as measured by the chain molecular weight iV,) the variation of

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 stress- optical 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 non- crystalline, the suggested explanation of Gent and Vickroy would not in this case apply.

3.5. EfSect of swelling on the stress-optical coe8cient The 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 cis- and 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 -a2 ranging, in the case of poly(cis- isoprene), for example, from 38.5 to 65-5 x cm3, compared with 48.0 for the unswollen polymer, The 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 polariza- tion or internal field effect. Comparable results have also been obtained for polyethylene (Gent and Vickroy 1967).

3.6. The 'equivalent random link' In the statistical theory the polymer chain is represented in terms of an idealized

model of randomly jointed links which are optically anistropic. The actual molecule


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. In 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.

The 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 value of b, and hence also a specific value of mean square length 2 in the gaussian 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

n, 1, = R,

n, 1,” = r i giving

The link of length 1, in the randomly jointed chain, as defined by (3.9), may be called the ‘equivalent random link’.

The 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 - a2) 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

782 L R G Treloar

mean square length (equations (3.9)), for the same total chain length. The 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)

Number of Number of monomers chain C-C

bonds per R d{ln - %)I d ( l i T ) 81 - Pt a1-az random random

Polymer cm3) cm3) link, q link (cal mol-l)

Per

Natural rubber 27.1, 47.0, 1.73 5.19 - 270 (C,H,), (cis)

(C5H8), (trans)

(C,H,), (96% cis)

(CH,),

Gutta percha? 28.3, 96.2, 3.39 10.17 + 40

Polybutadiene 29.2, 77.7, 2.66 7.94 + 85

Polyethylene? 7.2, 133.4 18.5 18.5 + 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 m and 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 stress- optical 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). The 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. The 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 In 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. Th‘ 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 al-az 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 con- figuration 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.


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

I I I I I f

1.9 2.0 2.1 2.2 2.3 2.4 2

I I r ( KII

that in natural

5 x IO3

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 (1 968).

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. The same general conclusions are obtained from thermoelastic studies ($ 4), though the energy difference between trans and gauche configurations for polythene from thermoelastic data ( N 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).

4. Thermodynamics of rubber elasticity 4.1. Elementary theory

I n this section we return to the consideration of the thermodynamics of rubber elasticity. In $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. The 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

In 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. The 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.

where c is a constant, that is, the

We have then

f = c ~ J 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 (NkT 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 = 3NkTE (4.4)

(4.5) E = € - - 0 +%)

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 * 785

ob lo i o 30 40 50 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.

The 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,

f (%Ip,* = T (4.7)

the subscript p indicating constancy of the surrounding (atmospheric) pressure. The result (4.7) is in harmony with the statistical theory, as represented by equation (2.10).

In 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. The 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


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. The 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)

The 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 low- molecular-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.

In Flory’s derivation the force-extension relation (2.10) given by the elementary gaussian network theory is replaced by the formula

f =+---J vkT 1 li Y O

(4.14)

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. The 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 3 of an identical set of free chains.

- -

31

788 L R G Treloar

The significance of the factor e/z will be apparent on recalling that in the 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 2 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),

from which we obtain

(4.15)

(4.16)

(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. The 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. The stress was determined from transducer measurements of the deflection of a stiff spring. In

t In 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).


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 /aT) , , 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. The 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. In 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

Method Polymer Reference f ,if Allen et a1 (1963) Allen et a1 (1971) Roe and Krigbaum (1962) I Ciferri (1961) I Shen (1969)

Natural rubber

\ Boyce‘and Treloar (1970) Allen et a1 (1968) Ciferri et a1 (1961) Butyl rubber {

Silicone rubber Price et a1 (1969b) Polyethylene Ciferri et a1 (1961)

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

- 0.42 k 0.05 0.25 k 0.01

Constant V Constant V Constant p

Constant p Constant p Torsion Constant V Constant p Constant V Constant p

I t is important to note that these derivations of fer 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. The 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).

The value of fer is, 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 fer corresponds to a positive temperature coefficient of G, which implies that the more

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). The 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).

The 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

e t 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,


and is more likely to be associated with the extreme sensitivity of the calculated value of fer to 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 fer from 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 fer at 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 fer on 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 fer 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 terms of fer. I n all cases f e i f was found to be independent not only of the strain, 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.

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.

In 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). The 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). In 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)

The difference between these two coefficients is therefore

Mep - MeV = - MTP. (4.23)

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. The 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 stress- temperature 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. The 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.

These conclusions are of considerable interest.


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 dependence of chain dimensions (2) on temperature. Despite this success, however, 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

J I 1

20 40 60 Temperature ('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

794 L R G Treloar

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

The 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. The 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 inter- relations 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. The condition for equilibrium is that the free energy change for further absorption of liquid shall be zero, that is,

- 0 2G _ - 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 - T S + p V = A + p V ( 5 4 where A is the Helmholtz free energy. The 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. Thus

aG aG, aG, an, an, an,

+-. _ - -- (5.3)

The 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} an, (5.4)

in which v2 is the volume fraction of polymer in the mixture and R the gas constant per mole. This expression contains a semi-empirical term xv22 which 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. The second term, aG,/an, is obtainable directly from the network theory (equation (2.7)). The ratio of swollen to unswollen


volume being given by l/v,, we have, for an isotropic swelling,

Substituting into equation (2.7), and introducing (2.16), we thus obtain for the work corresponding to the network expansion

A, = A, = A, = 1/v?. ( 5 . 5 )

For a constant pressure process 6G = 6A+pSV (from equation (5.2)). The term pSV being negligible in the present case, we may therefore write

a ~ , a ~ , aw a w a v , (5.7) -- - -- = pR*y,v21j3

an, an, an, av, an, M, where V, is the molar volume of the swelling liquid. (This derivation assumes additivity of volumes, so that 1 + n, V, = v2-'.) The total free energy of dilution, taking into account both terms in equation (5.3), is therefore (from (5.4) and (5 .7) )

The equilibrium swelling is determined by the value of v, for which the condition i?G/an, = 0 (equation (5.1)) is satisfied, that is,

(5.9) Y In( l -v , )+v,+Xc2~ = -fl_laii3,

MC 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).

796 L R G Treloar

etc) of the swollen polymer (or polymer solution). The 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.

The 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. The 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 stress- free) swollen dimensions. The corresponding relations between the principal stresses (referred to the final strained swollen area) and the corresponding extension

o.9b io 40 i o s'o Id0 Extension C%l

!O

Figure 22. Variation of x with strain for different swelling ratios l / v2 for rubber in toluene. From Gee (1946a).


ratios are of the form

(5.11)

These equations reduce to those for an unswollen rubber (equations (2.7) and ( 2 . 1 5 ~ ) ) on putting U, = 1. The 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

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 / A 2 ) against X (Gee 1946a), which according to equation (5.12) should yield a single horizontal line, independent of the degree of swelling. The 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) containing two arbitrary constants C, and C,. T o take account of the effect of

f = 2(X- 1/h2) (C,+ CJX)

Figure 23. ‘Mooney’ plot of 4 (equation (5.14)) for various degrees of swelling. From Gumbrell et al (1953).

b

2 Volume fraction of rubber,

Figure 24. Dependence of constant C, in equation (5.14) on degree of swelling: 0 natural rubber; 0 butadiene-styrene; x butadiene-acrylonitrile. From Gum- brell et a1 (1953).

798 L R G TreZoar

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

= c,+ C,/X. + = 2(h- 1/X2) (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 The 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. The 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)


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 ~ ~ ~ + - v ~ Z ~ ~ PK = 0. (5.17)

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.

Mc

L2 (referred to unswollen state)

extension. Curves from equation (5.17). From Treloar (1950b). Figure 25. Dependence of equilibrium degree of swelling on strain for two-dimensional

5.4. Torsion of cylinder The 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. The result was expressed by the formula

(5.18)

800 L R G Treloar

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.

The 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.

The 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.

The 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. The 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. The 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 con- stitutions. 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. The 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. The rotational component involves no net work by the external forces, and

802 L R G Treloar

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 elastic- ally 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. The three simplest even-powered functions are the so-called strain inaariants

(6.1) 1 I, = A,, + A,, + A,,

I, = A,, A,, + A,2 A,, + A,2 A,2

I3 = A12 A,, A,,. For a constant volume deformation, corresponding to an incompressible

material, with which the present discussion is primarily concerned, I, = 1. The two remaining equations may then be written

i Il = A,2 + A,, + A,2

I, = l/A,Z+ 1 /A,2+ 1/A,2.

On this basis the stored energy function W must be expressible in terms of these two strain invariants only. The 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) . The 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 ) ) .

The most general first-order expression in I, and I, is obtained by combination of ( 6 . 4 ~ ) and (6.4b) to give

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


I, and I, (Rivlin 1948), that is,

t , - t , = 2( A,2 - A,,) E+ 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:) (C, + X,2 C,) (6.6a) for the general stress system, and

t, = 0 i t , = 2( A,, - A,,) (C, + A,, C,) t , = 2( A,, - A,,) (C, + A,, C,)

(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

(6.6b) and

t , - t , = 2C,(A12 - A,,)

t, = 0 t , = 2C,( A,, - A,,) t , = 2C1(A,2- A,2)

respectively.

(6.7b)

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, = 1jX we have

1,-3 = 1,-3 = ~ 2 + ~ - 2 - 2 = y 2 (6.9) where y is the shear strain (cf equation (2.12)). The shear stress t,, (or t,,) may be shown to be (Rivlin 1948)

t,, = t,, = 2 .- (::+$I (6.10)

a04 L R G Treloar

A’ B

which for the Mooney form of W gives

t12 = t2l = 2(CI + C,) Y ( 6 . 1 0 ~ ) implying Hooke’s law in shear (in accordance with the basic assumption of the Mooney theory).

9‘

6.4. Stress components: normal stresses in shear The 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). In addition to the tangential stresses tI2, tZ1, given by

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


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 = A,. (6.12)

In 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 The 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.

806 L R G Treloar

The corresponding normal component t , in the axial direction is

tG2 = - #2{( C, - 2C,) (a2 - r2) + 2a2 C,} (6.14) where a is the radius of the cylinder. The 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,) (6.15)

N = - I ,np a4( c, + 2C2). (6.16)

The first of these equations is identical to the classical solution for a material of shear modulus 2(C,+ C,). The 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.

( a ) Cb)

Figure 30. Distribution of end pressure on twisted cylinder: (a) statistical theory; (b) Mooney equation.

The 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).


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 t l - t2 plotted against A I 2 - 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 rm 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 tz) 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


0.1

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

e.'. - ++ $+ +

I I I I I 1

+- x 11=5 0 I 1 = 7 + 1 1 = 9 d + e

+ + ++ 0 I 1 = 1 1 1.4 IO 18 26

'2

x 11=5 0 I 1 = 7 + 1 1 = 9 0 Il = 11

x I , = 5 0 J2 = 10 + I2 = 20 0 I, = 30

1; Figure 36. Pure homogeneous strain. Dependence of aW/aI, on Il and I,. From Rivlin and

Saunders (1 95 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 hZ2- respectively, on the basis of equations ( 6 . 7 ~ ~ ) . In such a plot (figure 34) the diver- gence from the statistical theory is immediately apparent. To test the applicability of the Mooney equation the same data may be plotted alternatively against (X12- A,,) (1 + AI2 C2/C,) (equations ( 6 . 7 ~ ) ) . By choosing a 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

( 1 + h,2 C,/C,) or

810 L R G Treloar

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 I2 was 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'zo7

I I I I I 4 5 6 7 8

1 2

Figure 37. Pure homogeneous strain. Dependence of aW/aI, and aW/a12 on I, for various

the statistical theory, while the second (correction) term decreases with increasing strain. In 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 auto- matic 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.

values of Il. From Obata e t al (1970).

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


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). The 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/811 and 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 extra- polation of their pure homogeneous strain data these authors were able to estimate the independent contributions of 2W/211 and aW/GI, to the total stress in simple extension, as given by equation (6.8). Their result (figure 38) shows that the term in aWja12 contributes only a small fraction (between 0 and 10%) to the total stress. The variation off/(X- l/X2) with strain is therefore to be associated primarily with a large i3W/i?I1 term 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. The 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

812 L R G Treloar

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. More- over, 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. The 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.

1

I I m

0 0

Compression

E z P

';" '"'- Extension

" 9 4

*I* -lX -

I t k1h- I I I 1 I I 'Dm O"0~ ' ' I 0.6 1.0 3 5 7 9 I1

I / h I / A

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


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 { L I Z n ( + + X3-2n - 3 ) + Bzn( + + X32n - 3)). (6.18)

n=l

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 ) + C20(12 - 3)2 (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( I1 -3 )~ ( I2 -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. The 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 Hart- Smith (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

W = C, exp @(I, - 3)2) dI, + C, In ((I,-:) + 7 ) + c3(r2 - 3) I (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)

t, - t , = al(e, - e,) (1 + a2(e1 + e,) + a3 e3> (6.25)

the e$ being strains (hi- 1). The 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. The 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,

w = w( A,) + w( A,) + w( h3). (6.26)

Symmetry considerations require that these separate functions shall be identical. The 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.


2.5,

2.0-

1.5-

1.0- a -- 0.5-

0-

N . h

I - - -0.5 -

-‘%O kL----- -0.5 0 0.5 1.0 1.5 2.0 5

( A , w’ ( A , ) - 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; 0 Treloar 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) The 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. The results, as shown in figure 40, give strong support to the proposed theory.

816 L R G Treloar

Valanis and Landel also proposed that over the range of A from about 0.35 to

w’(A) = 2p In A. (6.29) This, however, was based on the incorrect assumption that c in equation (6.28) may be put equal to zero. The correct interpretation of their observations, on the basis of equations (6 .27~) and (6.28)) would give (for A, > 1)

2.5 the function w’(A) approximates to the logarithmic form

with, presumably, a comparable correction for the case A < 1.t

0

(6 .29~)

Figure 41. Plot oft, - tz 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 .27~) 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. The 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. The 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

w = <D(A,2)+<D(h,2)f(D(h,2)-(D(l) (6.30)

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.)


proceed to apply equation (6.30) without any restriction on the form of a. I n this sense it becomes equivalent to (6.26). The 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. The stress-strain relation for simple shear is given in the form

A sinh ,By t12 = 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 pn 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. The principal stresses are given by

(6.34)

818 L R G Treloar

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 three- term expression is required. This contains the six adjustable parameters

al = 1.3

a2 = 5.0

013 = -2.0

p1 = 6.3 kgcm-2

p 2 = 0.012 kgcm-2

p 3 = - 0.1 kg cm-2.

The degree of agreement with experiment is shown in figure 43.

(6.35)

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).


6.9. Compressible rubbers In the foregoing discussion it has been assumed throughout that all deforma-

tions 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 KO introduce a modified set of strain invariants J,, J, and J,, such that

They then write for the stored energy

W = C Cl,,( J1 - 3)'( 5 2 - 3)"( 5, - 3),.

(6.36)

(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)

(6.38) 1 aWjaJ, = A + B ( e , + e , + e , ) + ...= A+B0+ ... aWjaJ, = C + D ( e , + e , + e , ) + ...= C+DB+ ... aWja5, = E + F ( e , + e , + e , ) + ...= E + F 0 + ... .

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.

The principal stresses fi (referred to the unstrained cross-sectional areas) are given by equations of the type

(6.39)

32

820 L R G Treloar

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) The 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 (6.43)

(6.44) 2B-20+ E+ F = K - 2 ~ 1 3 .

They now introduce a subsidiary parameter f such that (6.45)

c = P(1 - f )/2 (6.46)

E = p(1-2f). (6.47)

The further development is restricted to the case where both aW/i3Jl and

F = K-2p /3 -E = K - p ( 5 / 3 - 2 f ) . (6.48)

Introducing the above equations for E and F into the third of equations (6.38) they obtain an expression for aW/aJ3. The complete set of strain invariants then becomes

A = P f P

which gives, from (6.43),

aW/aJ2 are constants, so that B = D = 0. Equation (6.44) then gives

(6.49)

(6.50)

2W/8Jl = pf12

aw/aJ, = (P/2) (1 -f) aWjaJ3 = p(l-2f)+{K-p(5/3 - 2 f ) } ( J 3 - 1 )

and the principal stresses take the form (from (6.39) ) fi Ai = p{fAi2 - (1 - f ) / A i 2 } + 53 aW/a J3.

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 ( A 2 = 1) and equibiaxial (A, = A,) tests. For the case of uniaxial extension the data for the continuum rubber were accurately represented by


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. The 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 V

Simple extension 0.26, 0.13 0.25

Equibiaxial extension 0.18, - 0 4 9 0.25 Strip-biaxial extension 0.20, 0.07 0.25

From Blatz and KO (1962).

The effects of the finite compressibility are only brought in by introducing a further assumption, which enables .I3 to be determined. For this purpose Blatz and KO propose 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

J 3 - - A 1 1-2v. (6.53)

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)

The 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 KO were 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,

K k p = - (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

822 L R G Treloar

based on (6.53) provided that 5+2v

6( 1 - 2v) ’ k = (6.56)

For k = 13.3 this yields the result U = 0.463. The 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). The modified form of W thus becomes

w= p(A l .n+A2.n+A3.11- 3) + F( J3) (6.57)

where J3 = A, A, A3 and F is an unspecified function. As in the incompressible case the values of an are not necessarily integers. The corresponding principal stresses ti (referred to the deformed areas) are

~

an n

(6.58)

where F’( J3) = 8 WjaJ,. involving J3 is replaced by an arbitrary hydrostatic stress p (equation (6.34).)

functions f( J3) and g( J3) defined by

(In the corresponding incompressible case the term

Equations (6.57) and (6.58) are transformed by the introduction of two new

(6.59) f( J3) = F( 5 3 ) + (2 pn) In J3

(6.60)

to give W = pn{( Ala* + A Z ~ ~ ~ + Aga, - 3)!01, - In J3} + Ag( J3) (6.61)

J3tZ = p,,(Ai.~-1)+AJ3g’(J3) i = 1,2 ,3 (6.62)


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 = ’ pLn an (summed over n)

h = A. (6.63)

The application of equation (6.63) may be illustrated by the case of a simple

(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,

extension in the ratio A,, with A, = A, = Ji’2 h1-l!2. Since f z = 0 we have

fl = f 1 - f 2 = Pn(A1 an - h-ad2 1 J 3 a d 2 > *

( 6 . 6 4 ~ )

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.

The 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. The conclusion that for slightly compressible rubbers the form of the conventional stress- strain 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 KO in 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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Treloar theory of network elasticity

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