Mechanical Properties of Solid Polymers

By Ian M. Ward and John Sweeney

Providing an updated and comprehensive account of the properties of solid polymers, the book covers all aspects of mechanical behaviour.  This includes finite elastic behavior, linear viscoelasticity and mechanical relaxations, mechanical anisotropy, non-linear viscoelasicity, yield behavior and fracture. New to this edition is coverage of polymer nanocomposites, and molecular interpretations of yield, e.g. Bowden, Young, and Argon.

The book begins by focusing on the structure of polymers, including their chemical composition and physical structure.  It goes on to discuss the mechanical properties and behaviour of polymers, the statistical molecular theories of the rubber-like state and describes aspects of linear viscoelastic behaviour, its measurement, and experimental studies.

Later chapters cover composites and experimental behaviour, relaxation transitions, stress and yielding. The book concludes with a discussion of breaking phenomena.

• Language: English
• Publisher: Wiley
• Release date: Oct 22, 2012
• ISBN: 9781118413197

Book preview

Preface

This book is the third edition of Mechanical Properties of Solid Polymers and follows the format of the first two editions in writing the chapters as separate units. Therefore, each chapter can be regarded as a self-contained introduction and review of progress in the different aspects of the mechanical behaviour.

Since the publication of the second edition in 1983, the subject has advanced considerably in many respects, especially with regard to non-linear viscoelasticity, yield and fracture. We have altered some chapters very little, notably those dealing with viscoelastic behaviour and the earlier research on anisotropic mechanical behaviour and rubber elasticity, only adding sections to deal with the latest developments.

On the other hand, it has been necessary to change substantially the chapters on non-linear viscoelasticity, yield and fracture and in some cases incorporate material from the second edition of An Introduction to the Mechanical Properties of Solid Polymers. A separate chapter is also added on polymer composites.

In all cases, the approach of the previous textbooks has been followed. This is to obtain a formal description of the behaviour using the mathematical techniques of solid mechanics, followed by an attempt to seek understanding in terms of the molecular structure and morphology.

Finally, we wish to thank Margaret Ward for undertaking a substantial part of the initial typing of the new text and Glenys Bowles for providing secretarial assistance.

I. M. Ward

J. Sweeney

1

Structure of Polymers

The mechanical properties that form the subject of this book are a consequence of the chemical composition of the polymer and also of its structure at the molecular and supermolecular levels. We shall therefore introduce a few elementary ideas concerning these aspects.

1.1 Chemical Composition

1.1.1 Polymerisation

Linear polymers consist of long molecular chains of covalently bonded atoms, each chain being a repetition of much smaller chemical units. One of the simplest polymers is polyethylene, which is an addition polymer made by polymerising the monomer ethylene, CH2=CH2, to form the polymer.

Note that the double bond is removed during the polymerisation (Figure 1.1). The well-known vinyl polymers are made by polymerising compounds of the form.

where X represents a chemical group; examples are as follows:

Polypropylene

FIGURE 1.1 (a) The polyethylene chain (CH2)n in schematic form (larger spheres, carbon; smaller spheres, hydrogen) and (b) sketch of a molecular model of a polyethylene chain.

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Polystyrene

and

poly(vinyl chloride)

Natural rubber, polyisoprene, is a diene, and its repeat unit

contains a double bond.

A condensation reaction is one in which two or more molecules combine into a larger molecule with or without the loss of a small molecule (such as water). One example is the formation of polyethylene terephthalate (the polyester used for Terylene and Dacron fibres and transparent films and bottles) from ethylene glycol and terephthalic acid:

Another common condensation polymer is nylon 6,6.

1.1.2 Cross-Linking and Chain-Branching

Linear polymers can be joined by other chains at points along their length to make a cross-linked structure (Figure 1.2). Chemical cross-linking produces a thermosetting polymer, so called because the cross-linking agent is normally activated by heating, after which the material does not soften and melt when heated further, for example Bakelite and epoxy resins. A small amount of cross-linking through sulfur bonds is needed to give natural rubber its characteristic feature of rapid recovery from a large extension.

FIGURE 1.2 Schematic diagram of a cross-linked polymer.

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Very long molecules in linear polymers can entangle to form temporary physical cross-links, and we shall show later that a number of the characteristic properties of solid polymers are explicable in terms of the behaviour of a deformed network.

A less extreme complication is chain branching, where a secondary chain initiates from a point on the main chain, as is illustrated for polyethylene in Figure 1.3. Low-density polyethylene, as distinct from the high-density linear polyethylene shown in Figure 1.1, possesses on average one long branch per molecule and a larger number of small branches, mainly ethyl (—CH2—CH3) or butyl (—(CH2)3—CH3) side groups. The presence of these branch points leads to considerable differences in mechanical behaviour compared with linear polyethylene.

FIGURE 1.3 A chain branch in polyethylene.

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1.1.3 Average Molecular Mass and Molecular Mass Distribution

Each sample of a polymer contains molecular chains of varying lengths, that is of varying molecular mass (Figure 1.4). The mass (length) distribution is of importance in determining the properties of the polymer, but until the advent of gel permeation chromatography [1,2] it could be determined only by tedious fractionation procedures. Most investigations therefore quoted different types of average molecular mass, the commonest being the number average and the weight average , defined as

Unnumbered Display Equation

where Ni is the number of molecules of molecular mass Mi, and Σ denotes summation over all i molecular masses.

FIGURE 1.4 The gel permeation chromatograph trace gives a direct indication of the molecular distribution. (Results obtained in Marlex 6009 by Dr. T. Williams.)

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The weight average molecular mass is always higher than the number average, as the former is strongly influenced by the relatively small number of very long (massive) molecules. The ratio of the two averages gives a general idea of the width of the molecular mass distribution.

Fundamental measurements of average molecular mass must be performed on solutions so dilute that intermolecular interactions can be ignored or compensated for. The commonest techniques are osmotic pressure for the number average and light scattering for the weight average. Both methods are rather lengthy, so in practice an average molecular mass was often deduced from viscosity measurements of either a dilute solution of the polymer (which relates to Mn) or a polymer melt (which relates to Mw). Each method yielded a different average value, which made it difficult to correlate specimens characterised by different groups of workers.

The molecular mass distribution is important in determining flow properties, and so may affect the mechanical properties of a solid polymer indirectly by influencing the final physical state. Direct correlations of molecular mass to viscoelastic behaviour and brittle strength have also been obtained.

1.1.4 Chemical and Steric Isomerism and Stereoregularity

A further complication of the chemical structure of polymers lies in the possibility of different chemical isomeric forms within a repeat unit or between a series of repeat units. Natural rubber and gutta percha are chemically both polyisoprene, but the former is the cis form and the latter is the trans form (see Figure 1.5). The characteristic properties of rubber are a consequence of the loose packing of molecules (i.e. large free volume) that arises from its structure.

FIGURE 1.5 cis-1,4-Polyisoprene and trans-1,4-polyisoprene.

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Vinyl monomer units

can be added to a growing chain either head-to-tail:

or head-to-head:

Head-to-tail substitution is usual, and only a small proportion of head-to-head linkages can produce a reduction in the tensile strength because of the loss of regularity.

Stereoregularity provides a more complex situation, which we will examine in terms of the simplest type of vinyl polymer (Figure 1.6) and for which we shall suppose that the polymer chain is a planar zigzag. Two very simple regular polymers can be constructed. In the first (Figure 1.6(a)) the substituent groups are all added in an identical manner to give an isotactic polymer. In the second regular polymer (Figure 1.6(b)) there is an inversion of the manner of substitution between consecutive units, giving a syndiotactic polymer for which the substituent groups alternate regularly on opposite sides of the chain. The regular sequence of units is called stereoregularity, and stereoregular polymers are crystalline and can possess high melting temperatures. The working range of a polymer is thereby extended compared with the amorphous atactic form, whose range is limited by the lower softening point. The final alternative structure is formed when the orientation of successive substituents takes place randomly (Figure 1.5(c)) to give an irregular atactic polymer that is incapable of crystallising. Polypropylene (—CH2CHCH3—)n was for many years obtainable only as an atactic polymer, and its widespread use began only when stereospecific catalysts were developed to produce the isotactic form. Even so, some faulty substitution occurs and atactic chains can be separated from the rest of the polymer by solvent extraction.

FIGURE 1.6 A substituted α-olefin can take three stereosubstituted forms.

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1.1.5 Liquid Crystalline Polymers

Liquid crystals (or plastic crystals as they are sometimes called) are materials that show molecular alignment in one direction but not three-dimensional crystalline order. During the last 20 years, liquid crystalline polymers have been developed where the polymer chains are so straight and rigid that small regions of almost uniform orientation (domains) separated by distinct boundaries are produced. In the case where these domains occur in solution, polymers are termed lyotropic. Where the domains occur in the melt, the polymers are termed thermotropic.

An important class of lyotropic liquid crystal polymers are the aramid polymers such as polyparabenzamide

and polyparaphenylene terephthalamide

better known as Kevlar, which is a commercially produced high stiffness and high strength fibre. It is important to emphasise that although Kevlar fibres are prepared by spinning a lyotropic liquid crystalline phase, the final fibre shows clear evidence of three- dimensional order.

Important examples of thermotropic liquid crystalline polymers are copolyesters produced by condensation of hydroxybenzoic acid (HBA)

and 2,6-hydroxynaphthoic acid (HNA)

most usually in the proportions HBA:HNA = 73:27.

In addition to these main-chain liquid crystalline polymers, there are also side-chain liquid crystalline polymers, where the liquid crystalline nature arises from the presence of rigid straight side-chain units (called the mesogens) chemically linked to an existing polymer backbone either directly or via flexible spacer units.

The review by Noël and Navard [3] gives further information on liquid crystalline polymers, including methods of preparation.

1.1.6 Blends, Grafts and Copolymers

A blend is a physical mixture of two or more polymers. A graft is formed when long side chains of a second polymer are chemically attached to the base polymer. A copolymer is formed when chemical combination exists in the main chain between two or more polymers, [A]n, [B]n, and so on. The two principal forms are block copolymers ([AAAA…] [BBB…]) and random copolymers, the latter having no long sequences of A or B units.

All these processes are commonly used to enhance the ductility and toughness of brittle homopolymers or increase the stiffness of rubbery polymers. An example of a blend is acrylonitrile–butadiene–styrene copolymer (ABS), where the separate rubber phase gives much improved impact resistance.

The basic properties of polymers may be enhanced by physical as well as chemical means. An important example is the use of finely divided carbon black as a filler in rubber compounds. Polymers may be combined with stiffer filaments, such as glass and carbon fibres, to form a composite. We shall show later that some semi-crystalline polymers may be treated as composites at a molecular level.

It must not be forgotten that all useful polymers contain small quantities of additives to aid processing and increase the resistance to degradation. The physical properties of the base polymer may be modified by the presence of such additives.

1.2 Physical Structure

The physical properties of a polymer of a given chemical composition are dependent on two distinct aspects of the arrangement of the molecular chains in space.

1. The arrangement of a single chain without regard to its neighbour: rotational isomerism.

2. The arrangement of chains with respect to each other: orientation and crystallinity.

1.2.1 Rotational Isomerism

Rotational isomerism arises because of the alternative conformations of a molecule that can result from the possibility of hindered rotation about the many single bonds in the structure. Spectroscopic techniques [4] developed in small molecules have been extended to polymers, and as an example we illustrate (Figure 1.7) the alternative trans and gauche conformations in the glycol residue of polyethylene terephthalate [5]: the former is a crystalline conformation, but the latter is present in amorphous regions.

FIGURE 1.7 Polyethylene terephthalate in the crystalline trans conformation (a) and in the gauche conformation present in ‘amorphous’ regions (b). (Adapted from Grime, D. and Ward, I.M. (1958) The assignment of infra-red absorptions and rotational isomerism in polyethylene terephthalate and related compounds. Trans. Faraday Soc., 54, 959. Copyright (1958) Royal Society of Chemistry.)

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To pass from one rotational isomeric form to another requires that an energy barrier be surmounted (Figure 1.8), so that the possibility of the chain molecules changing their conformations depends on the relative magnitude of the energy barrier compared with thermal energies and the perturbing effects of applied stress. Hence, arises the possibility of linking molecular flexibility to deformation mechanisms, a theme to which we will return on several occasions.

FIGURE 1.8 Potential energy for rotation (a) around the CC bond in ethane and (b) around the central CC bond in n-butane. (Reprinted from McCrum, N.G., Read, B.E., Williams, G. (1991) Anelastic and Dielectric Effects in Polymeric Solids, Dover Publications, New York. Copyright (1991) Dover Publications.)

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1.2.2 Orientation and Crystallinity

When we consider the arrangement of molecular chains with respect to each other, there are again two largely separate aspects, those of molecular orientation and crystallinity. In semi-crystalline polymers, this distinction may at times be an artificial one.

When cooled from the melt, many polymers form a disordered structure called the amorphous state. Some of these materials, such as polymethyl methacrylate, polystyrene and rapidly cooled (melt-quenched) polyethylene terephthalate, have a comparatively high modulus at room temperature, but others, such as natural rubber and atactic polypropylene, have a low modulus. These two types of polymer are often termed glassy and rubber-like, respectively, and we shall see that the form of behaviour exhibited depends on the temperature relative to a glass–rubber transition temperature (Tg) that is dependent on the material and the test method employed. Although an amorphous polymer may be modelled as a random tangle of molecules (Figure 1.9(a)), features such as the comparatively high density [6] show that the packing cannot be completely random. X-ray diffraction techniques indicate no distinct structure, rather a broad diffuse maximum (the amorphous halo) that indicates a preferred distance of separation between the molecular chains.

FIGURE 1.9 Schematic diagrams of (a) unoriented amorphous polymer and (b) oriented amorphous polymer.

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When an amorphous polymer is stretched, the molecules may be preferentially aligned along the stretch direction. In polymethyl methacrylate and polystyrene, such molecular orientation may be detected by optical methods, which measure the small difference between the refractive index in the stretch direction and that in the perpendicular direction. X-ray diffraction methods for relaxed amorphous polymers still reveal no evidence of three-dimensional order, so the structure may be regarded as a somewhat oriented tangled skein (Figure 1.9(b)) that is oriented amorphous but not crystalline.

In polyethylene terephthalate, however, stretching produces both molecular orientation and small regions of three-dimensional order, termed crystallites, because the orientation processes have brought the molecules into adequate juxtaposition for regions of three-dimensional order to form.

Many polymers, including polyethylene terephthalate, also crystallise if they are cooled slowly from the melt. In this case, we may say that they are crystalline but unoriented. Although such specimens are unoriented in the macroscopic sense, that is, they possess isotropic bulk mechanical properties, they are not homogeneous in the microscopic sense and often show a spherulitic structure under a polarising microscope.

In summary, it may be said that for a polymer to crystallise the molecule must have a regular structure, the temperature must be below the crystal melting point and sufficient time must be available for the long molecules to become ordered in the solid state.

The structure of the crystalline regions of polymers can be deduced from wide-angle X-ray diffraction patterns of highly stretched specimens. When the stretching is uniaxial, the patterns are related to those obtained from fully oriented single crystals. The crystal structure of polyethylene was determined by Bunn [7] as long ago as 1939 (Figure 1.10).

FIGURE 1.10 Arrangement of molecules in polyethylene crystallites. (Reprinted with permission from Hill, R. (ed.) (1953) Fibres from Synthetic Polymers, Elsevier, Amsterdam. Copyright (1953) Elsevier Ltd.)

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In addition to the discrete reflections from the crystallites, the diffraction pattern of a polymer shows diffuse scattering attributed to amorphous regions. Such polymers are said to be semi-crystalline, with the crystalline fraction being controlled by molecular regularity. By comparing the relative amounts of crystalline and amorphous scattering of X-rays, the crystallinity has been found to vary from more than 90 per cent for linear polyethylene to about 30 per cent for oriented polyethylene terephthalate.

The first model to describe the structure of a semi-crystalline polymer was the so-called fringed micelle model (Figure 1.11), which is a natural development of the imagined situation in an amorphous polymer. The molecular chains alternate between regions of order (the crystallites) and disorder (the amorphous regions).

FIGURE 1.11 The fringed micelle representation of crystalline polymers. (Reprinted with permission from Hill, R. (ed.) (1953) Fibres from Synthetic Polymers, Elsevier, Amsterdam. Copyright (1953) Elsevier Ltd.)

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The fringed micelle model was called into question by the discovery of polymer single crystals grown from solution [8]. Linear polyethylene, for example, forms single crystal lamellae with lateral dimensions of the order of 10–20 μm and thickness of 10nm. Electron diffraction shows that the molecular chains are oriented approximately normal to the lamellar surface. As the molecules are typically about 1 μm in length, it follows that they must be folded back and forth within the crystals. The simplest geometric arrangement is that the folds are sharp and regular producing the adjacent re-entry model shown in Figure 1.12(a). This model provoked controversy and an alternative switchboard model shown schematically in Figure 1.12(b) was proposed [9].

FIGURE 1.12 Crystallites with folded lamellar crystals of thickness ζ in the direction of the c axis for (a) regular folding and (b) irregular folding of the chain molecules.

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The crystallisation of polymers from the melt has proved even more controversial, as a single molecule is unlikely to be laid down on a crystalline substrate without interference from its neighbours, and it might be expected that the highly entangled topology of the chains that exists in the melt would be substantially retained in the crystalline state. These issues were explored to great effect by neutron scattering measurements, notably by Fischer [10,11] and also other researchers [12]. The neutron scattering measurements showed that the radii of gyration in the melt and in the semi-crystalline state for polyethylene quenched from the melt, for polyethylene oxide crystallised by slow cooling and for isotactic polypropylene, isothermally crystallised, were almost identical. On the basis of these results, Fischer proposed the solidification model shown in Figure 1.13 [10], where straight sequences of the original melt are incorporated into the growing lamellae without long-range diffusional motion.

FIGURE 1.13 Chain conformation (a) in the melt and (b) in the crystal according to the solidification model. (Adapted from Stamm, M., Fischer, E.W., Dettenmaier, M. and Convert, P. (1979) Chain conformation in the crystalline state by means of neutron scattering methods. Discuss. Chem. Soc. (London), 68, 263. Copyright (1979) RSC.)

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Although it is accepted that kinetic factors determine the growth rate of crystallisation and the morphology, there is still debate in this area also. The theory proposed by Lauritzen and Hoffman [13] led the field and predicted the growth rate as the function of the degree of supercooling, the temperature difference between the crystallisation temperature and the melting point. It was assumed that the free energy barrier associated with nucleation of the crystallisation was energetic in origin. An alternative model for chain folding in polymer crystals has been proposed by Sadler and Gilmer [14], which assumes that the free energy barrier for nucleation is predominantly entropic. For a comprehensive review of these theories and related issues, the reader is referred to an excellent review in Reference [15] and also to Reference [16].

There is, of course, much evidence to support the existence of a lamellar morphology in crystalline polymers. Typically, spherulites l–10 μm in diameter are formed, which grow outwards until they impinge upon neighbouring spherulites (Figure 1.14). The spherulitic textures are formed by the growth of dominant lamellae from a central nucleus in all directions by a twisting of these lamellae along the fibrils, the intervening spaces being filled in by subsidiary lamellae, possibly due to low molecular weight material. This is shown schematically in Figure 1.15, where, for ease of illustration, regular chain folding is sketched. For a good review of polymer morphology, see the text by Bassett [17] and also more recent work directed by the same author.

FIGURE 1.14 A photograph of typical spherulitic structure under a polarising microscope.

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FIGURE 1.15 A model of the lamellar arrangement in a polyethylene spherulite. The small diagrams of the a, b, c axes show the orientation of the unit cell at various points. (Adapted from Takayanagi, M. (1963) Viscoelastic properties of crystalline polymers. Memoirs of the Faculty of Engineering Kyushu Univ., 23, 1. Copyright (1963) Kyushu University.)

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Orientation through plastic deformation (drawing) destroys the spherulitic structure. What remains is determined to a large extent by the degree of crystallinity. Mechanical testing, described in the subsequent chapters, has helped to establish several models. At one extreme, some highly oriented, highly crystalline specimens of linear polyethylene behave as blocks or lamellae of crystalline material, connected together by tie molecules or crystalline bridges and separated by the amorphous component. Such materials in some respects can be treated as microscopic composites. At the other extreme one has materials such as polyethylene terephthalate in which the crystalline and amorphous components are so intermixed that a single-phase model appears to be more appropriate.

The current state of knowledge suggests that chain folding and the threading of molecules through the crystalline region both occur in typical polymers.

A schematic attempt to illustrate this situation, and other types of irregularity, is given in Figure 1.16.

FIGURE 1.16 Schematic composite diagram of different types of order and disorder in oriented polymers. (Reproduced from Hosemann, R. (1962) Crystallinity in high polymers, especially fibres. Polymer, 3, 393. Copyright (1962) Elsevier Ltd.)

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References

1. Vaughan, M.F. (1960) Fractionation of polystyrene by gel filtration. Nature, 188, 55.

2. Moore, J.C. (1964) Gel permeation chromatography. I. A new method for molecular weight distribution of high polymers. J. Polym. Sci. A, 2, 835.

3. Noël, C. and Navard, P. (1991) Liquid crystal polymers. Prog. Polym. Sci., 16, 55.

4. Mizushima, S. I. (1954) Structure of Molecules and Internal Rotation, Academic Press, New York.

5. Grime, D. and Ward, I.M. (1958) The assignment of infra-red absorptions and rotational isomerism in polyethylene terephthalate and related compounds. Trans. Faraday Soc., 54, 959.

6. Robertson, R.E. (1965) Polymer order and polymer density. J. Phys. Chem., 69, 1575.

7. Bunn, C.W. (1939). The crystal structure of long-chain normal paraffin hydrocarbons: the shape of the >CH2 group. Trans. Faraday Soc., 35, 482.

8. Fischer, E.W. (1957) Stufen- und spiralförmiges Kristallwachstum bei Hochpolymeren, Naturforschung, 12a, 753; Keller, A. (1957) A note on single crystals in polymers: evidence for a folded chain configuration. Philos. Mag., 2, 1171; Till, P.H. (1957) The growth of single crystals of linear polyethylene. J. Polym. Sci., 24, 301.

9. Flory, P.J. (1962) On the morphology of the crystalline state in polymers. J. Amer. Chem. Soc., 84, 2857.

10. Fischer, E.W. (1978) Studies of structure and dynamics of solid polymers by elastic and inelastic neutron scattering. Pure and Appl. Chem., 50, 1319.

11. Stamm, M., Fischer, E.W., Dettenmaier, M. and Convert, P. (1979) Chain conformation in the crystalline state by means of neutron scattering methods. Discuss. Chem. Soc. (London), 68, 263.

12. Stamm, M., Schelten, J. and Ballard, D.G.H. (1981) Determination of the chain conformation of polypropylene in the crystalline state by neutron scattering. Coll. Pol. Sci., 259, 286.

13. Lauritzen, J.I. and Hoffman, J.D. (1960) Theory of formation of polymer crystals with folded chains in dilute solution. J. Res. Nat. Bur. Std., 64A, 73.

14. Sadler, D.M. and Gilmer, G.H. (1984) A model for chain folding in polymer crystals: rough growth faces are consistent with the observed growth rates. Polymer, 25,1446.

15. Gedde, U.W. (1995) Polymer Physics, Chapman and Hall, London.

16. Strobl, G. (1997) The Physics of Polymers, 2nd edn, Springer, Berlin.

17. Bassett, D.C. (1981) Principles of Polymer Morphology, Cambridge University Press, Cambridge.

18. McCrum, N.G., Read, B.E., Williams, G. (1991) Anelastic and Dielectric Effects in Polymeric Solids, Dover Publications, New York.

19. Hill, R. (ed.) (1953) Fibres from Synthetic Polymers, Elsevier, Amsterdam.

20. Takayanagi, M. (1963) Viscoelastic properties of crystalline polymers. Mem. Fac. Eng. Kyushu Univ., 23, 1.

21. Hosemann, R. (1962) Crystallinity in high polymers, especially fibres. Polymer, 3, 393.

Further Reading

Billmeyer, F.W. (1984) Textbook of Polymer Science, 3rd edn, John Wiley & Sons, New York.

Bower, D.I. (2002) An Introduction to Polymer Physics, Cambridge University Press, Cambridge.

Cowie, J.M.G. and Arrighi, V. (2008) Polymers: Chemistry and Physics of Modern Materials, 3rd edn, Taylor & Francis Group, Boca Raton, Florida.

Hamley, I.W. (1998) The Physics of Block Copolymers, Oxford University Press, Oxford.

Mark, J., Ngai, K., Graessley, W. et al. (2004) Physical Properties of Polymers, 3rd edn, Cambridge University Press, Cambridge.

Odian, G.G. (2004) Principles of Polymerization, 4th edn, Wiley-Interscience, Hoboken, New Jersey.

Painter, P.C. and Coleman, M.M. (2008) Essentials of Polymer Science and Engineering, DEStech Publications, Lancaster Pennsylvania.

Rubinstein, M. and Colby, R.H. (2003) Polymer Physics, Oxford University Press, Oxford.

Sperling, L. (2006) Introduction to Physical Polymer Science, 4th edn, John Wiley & Sons, New York.

Tadokoro, H. (1979) Structure of Crystalline Polymers, John Wiley & Sons, Ltd, New York.

van Krevelen, D.W. and te Nijenhuis, K. (2009) Properties of Polymers, 4th edn, Elsevier, Oxford.

Wunderlich, B. Macromolecular Physics, Vols 1 1973 Vol 2 1976, Vol 3 1980, Academic Press, New York.

Young, R.J. and Lovell, P.A. (2011) Introduction to Polymers, 3rd edn, CRC Press/Taylor & Francis Group, Boca Raton, Florida.

2

The Mechanical Properties of Polymers: General Considerations

2.1 Objectives

Discussions of the mechanical properties of solid polymers often contain two inter-related objectives. The first of these is to obtain an adequate macroscopic description of the particular facet of polymer behaviour under consideration. The second objective is to seek an explanation of this behaviour in molecular terms, which may include details of the chemical composition and physical structure. In this book, we will endeavour, where possible, to separate these two objectives and, in particular, to establish a satisfactory macroscopic or phenomenological description before discussing molecular interpretations.

This should make it clear that many of the established relationships are purely descriptive, and do not necessarily have any implications with regard to an interpretation in structural terms. For engineering applications of polymers this is sufficient, because a description of the mechanical behaviour under conditions that simulate their end use is often all that is required, together with empirical information concerning their method of manufacture.

2.2 The Different Types of Mechanical Behaviour

It is difficult to classify polymers as particular types of materials such as a glassy solid or a viscous liquid, since their mechanical properties are so dependent on the conditions of testing, for example the rate of application of load, temperature and amount of strain.

A polymer can show all the features of a glassy, brittle solid or an elastic rubber or a viscous liquid depending on the temperature and time scale of measurement. Polymers are usually described as viscoelastic materials, a generic term which emphasises their intermediate position between viscous liquids and elastic solids. At low temperatures, or high frequencies of measurement, a polymer may be glass-like with a Young's modulus of 1–10 GPa and will break or flow at strains greater than 5%. At high temperatures or low frequencies, the same polymer may be rubber-like with a modulus of 1–10 MPa, withstanding large extensions (~100%) without permanent deformation. At still higher temperatures, permanent deformation occurs under load, and the polymer behaves like a highly viscous liquid.

In an intermediate temperature or frequency range, commonly called the glass transition range, the polymer is neither glassy nor rubber-like. It shows an intermediate modulus, is viscoelastic and may dissipate a considerable amount of energy on being strained. The glass transition manifests itself in several ways, for example by a change in the volume coefficient of expansion, which can be used to define a glass transition temperature Tg. The glass transition is central to a great deal of the mechanical behaviour of polymers for two reasons. First there are the attempts to link the time–temperature equivalence of viscoelastic behaviour with the glass transition temperature Tg. Secondly, glass transitions can be studied at a molecular level by such techniques as nuclear magnetic resonance and dielectric relaxation. In this way, it is possible to gain an understanding of the molecular origins of the viscoelasticity.

The different features of polymer behaviour such as creep and recovery, brittle fracture, necking and cold drawing are usually considered separately, by comparative studies of different polymers. It is customary, for example, to compare the brittle fracture of polymethyl methacrylate, polystyrene and other polymers, which show similar behaviour at room temperature. Similarly comparative studies have been made of the creep and recovery of polyethylene, polypropylene and other polyolefins. Such comparisons often obscure the very important point that the whole range of phenomena can be displayed by a single polymer as the temperature is changed. Figure 2.1 shows load–elongation curves for a polymer at four different temperatures. At temperatures well below the glass transition (curve A), where brittle fracture occurs, the load rises to the breaking point linearly with increasing elongation, and rupture occurs at low strains (−10%). At high temperatures (curve D), the polymer is rubber-like and the load rises to the breaking point with a sigmoidal relationship to the elongation, and rupture occurs at very high strains (~30–1000%).

FIGURE 2.1 Load–elongation curves for a polymer at different temperatures. Curve A, brittle fracture; curve B, ductile failure; curve C, cold drawing; curve D, rubber-like behaviour.

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In an intermediate temperature range below the glass transition (curve B), the load–deformation relationship resembles that of a ductile metal, showing a load maximum, i.e. a yield point before rupture occurs. At slightly higher temperatures (curve C), still below the glass transition, the remarkable phenomenon of necking and cold drawing is observed. Here, the conventional load–elongation curve again shows a yield point and a subsequent decrease in conventional stress. However, with a further increase in the applied strain, the load falls to a constant level at which deformations of the order of 300–1000% are accomplished. At this stage, a neck has formed and the strain in the specimen is not uniform. (This is discussed in detail in Chapter 12.) Eventually, the load begins to rise again and finally fracture occurs.

It is usual to discuss the mechanical properties in the different temperature ranges separately, because different approaches and mathematical formalisms are adopted for the different features of mechanical behaviour. This conventional treatment will be followed here, although it is recognised that it somewhat arbitrarily isolates particular facets of the mechanical properties of polymers.

2.3 The Elastic Solid and the Behaviour of Polymers

Mechanical behaviour is, in most general terms, concerned with the deformations which occur under loading. In any specific case, the deformations depend on details such as the geometrical shape of the specimen or the way in which the load is applied. Such considerations are the province of the plastics engineer, who is concerned with predicting the performance of a polymer in a specified end use. In our discussion of the mechanical properties of polymers, we will ignore such questions as these, which relate to solving particular problems of behaviour in practice. We will concern ourselves only with the generalised equations termed constitutive relations, which relate stress and strain for a particular type of material. First it will be necessary to find constitutive relations that give an adequate description of the mechanical behaviour. Secondly, where possible, we will obtain a molecular understanding of this behaviour by a molecular model that predicts the constitutive relations.

One of the simplest constitutive relations is Hooke's law, which relates the stress σ to the strain e for the uniaxial deformation of an ideal elastic isotropic solid. Thus

Unnumbered Display Equation

where E is the Young's modulus.

There are five important ways in which the mechanical behaviour of a polymer may deviate from that of an ideal elastic solid obeying Hooke's law. First, in an elastic solid the deformations induced by loading are independent of the history or rate of application of the loads, whereas in a polymer the deformations can be drastically affected by such considerations. This means that the simplest constitutive relation for a polymer should in general contain time or frequency as a variable in addition to stress and strain. Secondly, in an elastic solid all the situations pertaining to stress and strain can be reversed. Thus, if a stress is applied, a certain deformation will occur. On removal of the stress, this deformation will disappear exactly. This is not always true for polymers. Thirdly, in an elastic solid obeying Hooke's law, which in its more general implications is the basis of small-strain elasticity theory, the effects observed are linearly related to the influences applied. This is the essence of Hooke's law; stress is exactly proportional to strain. This is not generally true for polymers, but applies in many cases only as a good approximation for very small strains; in general, the constitutive relations are non-linear. It is important to note that non-linearity is not related to recoverability. In contrast to metals, polymers may recover from strains beyond the proportional limit without any permanent deformation.

Fourthly, the definitions of stress and strain in Hooke's law are only valid for small deformations. When we wish to consider larger deformations a new theory must be developed in which both stress and strain are defined more generally.

Finally, in many practical applications (such as films and synthetic fibres) polymers are used in an oriented or anisotropic form, which requires a considerable generalisation of Hooke's law.

It will be convenient to discuss these various aspects separately as follows: (1) behaviour at large strains in Chapters 3 and 4 (finite elasticity and rubber-like behaviour, respectively); (2) time-dependent behaviour in Chapters 5–7 and 10 (viscoelastic behaviour); (3) the behaviour of oriented polymers in Chapters 8 and 9 (mechanical anisotropy); (4) non-linearity in Chapter 11 (non-linear viscoelastic behaviour); (5) the non-recoverable behaviour in Chapter 12 (plasticity and yield) and (6) fracture in Chapter 13 (breaking phenomena). However, it should be recognised that we cannot hold to an exact separation and that there are many places where these aspects overlap and can be brought together by the physical mechanisms, which underlie the phenomenological description.

2.4 Stress and Strain

It is desirable at this juncture to outline very briefly the concepts of stress and strain. For a more comprehensive discussion, the reader is referred to standard textbooks on the theory of elasticity [1-6].

2.4.1 The State of Stress

The components of stress in a body can be defined by considering the forces acting on an infinitesimal cubical volume element (Figure 2.2) whose edges are parallel to the coordinate axes 1, 2 and 3. In equilibrium, the forces per unit area acting on the cube faces are:

Px on the 23 plane,

Py on the 31 plane,

Pz on the 12 plane.

FIGURE 2.2 The stress components.

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These three forces are then resolved into their nine components in the 1, 2 and 3 directions as follows:

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The first subscript refers to the direction of the normal to the plane on which the stress acts, and the second subscript to the direction of the stress. In the absence of body torques, the total torque acting on the cube must also be zero, and this implies three equalities:

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Therefore, the components of stress are defined by six independent quantities:σ11,σ22 and σ33, the normal stresses, and σ12,σ23 and σ31, the shear stresses.

These form the six independent components of the stress tensor Σ or σij:

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The state of stress at a point in a body is determined when we can specify the normal components and the shear components of stress acting on a plane drawn in any direction through the point. If we know these six components of stress at a given point, the stresses acting on any plane through this point can be calculated. (See Reference [1], Section 67; and Reference [2], Section 47.)

2.4.2 The State of Strain – Engineering Components

The displacement of any point X (see Figure 2.3) in the body may be resolved into its components u1, u2 and u3 parallel to 1, 2 and 3 (Cartesian coordinate axes chosen in the undeformed state) so that if the coordinates of the point in the undisplaced position were (X1, X2, X3), they become (X1 + u1, X2 + u2, X3 + u3) on deformation. In defining the strains, we are not interested in the displacement or rotation but in the deformation. The latter is the displacement of a point relative to adjacent points. Consider a point very close to X, which in the undisplaced position had coordinates (X1+ dX1, X2+ dX2, X3+ dX3) and let the displacement, which it has undergone, have components (u1+ du1, u2+ du2, u3+ du3). The quantities required are then du1, du2 and du3, the relative displacements.

FIGURE 2.3 The displacements produced by deformation.

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If dX1, dX2 and dX3 are sufficiently small, that is infinitesimal:

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Thus, we require to define the nine quantities:

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For convenience, these nine quantities are regrouped and denoted as follows:

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The first three quantities e11, e22 and e33 correspond to the fractional expansions or contractions along the 1, 2 and 3 axes of an infinitesimal element at X – the normal strains. The second three quantities e23, e31 and e12 correspond to the components of shear strain in the 23, 31 and 12 planes respectively. The last three quantities and do not correspond to a deformation of the element at X, but are the components of its rotation as a rigid body.

The concept of shear strain can be conveniently illustrated by a diagram showing the two-dimensional situation of shear in the 23 plane (see Figure 2.4). ABCD is an infinitesimal square that has been displaced and deformed into the rhombus A′B′C′D′, θ1 and θ2 being the angles that A′D′ and A′B′ make with the 2 and 3 axes, respectively.

FIGURE 2.4 Shear strains.

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Now,

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The shear strain in the 23 plane is given by

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2w1 =θ1-θ2 does not correspond to a deformation of ABCD but to twice the angle through which AC has been rotated.

Therefore, the deformation is defined by the first six quantities e11, e22, e33 e23, e31, e12 that are called the components of strain. It is important to note that engineering strains have been defined. In Chapter 3, we take a more general approach and examine a number of strain-related tensor quantities in Section 3.1.5. For the purposes of this chapter, which concerns small strains, we define the strain tensor i j as

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where i and j take values 1, 2 and 3, and we sum over all possible values. Then,

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in terms of the engineering components of strain.

2.5 The Generalised Hooke's Law

The most general linear relationship between stress and strain is obtained by assuming that each of the tensor components of stress is linearly related to all the tensor components of strain and vice versa. Thus

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and

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where a, b, … , a′, b′, … are constants. This is the generalised Hooke's law.

In tensor notation, we relate the second-rank tensor σij to the second-rank strain tensor ij by fourth-rank tensors Cijkl and Sijkl. Thus,

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or equivalently

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where

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and

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The fourth-rank tensors Sijkl and Cijkl contain the compliance and stiffness constants respectively, with i,j,k,l taking values l, 2 and 3.

It is customary to adopt an abbreviated nomenclature in which the generalised Hooke's law relates the six independent components of stress to the six independent components of the engineering strains.

We have

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where σp represents σ11, σ22, σ33 σ13, σ23,or σ12 and q represents σp represents e11, e22, e33 e13, e23,or e12. We form matrices cpq and spq in which p and q take the values 1, 2, … , 6. In the case of the stiffness constants, the values of p and q are obtained in terms of i,j,k,l by substituting 1 for 11, 2 for 22, 3 for 33, 4 for 23, 5 for 13 and 6 for 12. For the compliance constants, rather more complicated rules apply owing to the occurrence of the factor-2 difference between the definition of the tensor shear strain components and the definition of engineering shear strains. Thus

Sijkl = spq, when p and q are 1, 2 or 3,

2Sijkl = spq, when either p or q are 4, 5 or 6,

4Sijkl = spq, when both p and q are 4, 5 or 6.

A typical relationship between stress and strain is now written as

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The existence of a strain–energy function (see Reference [2], p. 149; Reference [3], p. 267) provides the relationships

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and reduces the number of independent constants from 36 to 21. We then have

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and similarly

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These matrices define the relationships between stress