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Behavior of Macromolecules pp 67–117Cite as

Polymer elasticity discrete and continuum models

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Part of the book series: Advances in Polymer Science ((POLYMER,volume 46))

Abstract

There is at present available in the literature on polymers and on materials science a wealth of information regarding measurements of mechanical properties. These properties are dependent upon many relevant physical parameters and most measurements take this into account. There is also available a great deal of information regarding the relations between molecular structure and macroscopic physical properties and many calculations have been made. The bridge between these two extremes (the macro and the micro) is constructed primarily by the use of models of structure.

Composites theory, which has developed from classical elasticity, combined with modelling techniques may point the way forward to a complete theory of the behaviour of polymers. However, it is clear from the literature that many experimentalists do not appreciate the niceties of the mathematical theories of elasticity and of continuum mechanics, nor, in some cases, the inaccuracies inherent in their experimental methods, while nearly all theorists have no conception of the problems encountered by the experimentalist when dealing with real materials and samples of finite size. We have therefore attempted in this review to bring theory and experiment closer together by highlighting some of the problems both of the theoretician and of the experimentalist.

After an introductory chapter we review in Chap. 2 the classical definition of stress, strain and modulus and summarize the commonly used solutions of the equations of elasticity. In Chap. 3 we show how these classical solutions are applied to various test methods and comment on the problems imposed by specimen size, shape and alignment and also by the methods by which loads are applied. In Chap. 4 we discuss non-homogeneous materials and the theories relating to them, pressing the analogies with composites and the value of the concept of the representative volume element (RVE). Chapter 5 is devoted to a discussion of the RVE for crystalline and non-crystalline polymers and scale effects in testing. In Chap. 6 we discuss the methods so far available for calculating the elastic properties of polymers and the relevance of scale effects in this context.

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8 References

  1. Love, A. E. H.: A treatise on the mathematical theory of elasticity. 4th Edit. Dover Press, New York 1944

    Google Scholar 

  2. Green, A. E., Adkins, J. E.: Large elastic deformations. Clarendon Press, Oxford 1970

    Google Scholar 

  3. Sokolnikoff, I. S.: Mathematical theory of elasticity. 2nd Edit. McGraw-Hill, New York 1956

    Google Scholar 

  4. Eringen, A. C. (ed.): Continuum Physics. Vol. 4, Academic Press, New York 1976

    Google Scholar 

  5. Saint-Venant, B.: Mémoires de l'académie des sciences des savants étrangers 14, 233 (1855)

    Google Scholar 

  6. Pochhammer, L.: Z. Reine Angew. Math. 81, 33 (1876)

    Google Scholar 

  7. Chree, C.: Trans. Camb. Phil. Soc. 14, 250 (1889)

    Google Scholar 

  8. Filon, L. N. G.: Phil. Trans. Roy. Soc. A 198, 147 (1902)

    Google Scholar 

  9. Lur'e, A. I.: Three dimensional problems of the theory of elasticity. Intersci. New York 1964

    Google Scholar 

  10. Toupin, R. A.: Arch. Rat. Mech. Anal. 18, 83 (1965)

    Article  Google Scholar 

  11. Horgan, C. O.: J. Elast. 2, 335 (1972)

    Google Scholar 

  12. Folkes, M. J., Arridge, R. G. C.: J. Phys. D: Appl. Phys. 8, 1058 (1975)

    Article  Google Scholar 

  13. Arridge, R. G. C. et al.: J. Mater. Sci. 11, 788 (1976)

    Article  Google Scholar 

  14. Fama, M. E. D.: Q. J. Mech. Appl. Math. 25, 479 (1972)

    Google Scholar 

  15. Vendhan, C. P., Archer, R. R.: Int. J. Solids Structures 14, 305 (1978)

    Article  Google Scholar 

  16. Arridge, R. G. C., Folkes, M. J.: Polymer 17, 495 (1976)

    Article  Google Scholar 

  17. Lekhnitskii, S. G.: Theory of elasticity of an anisotropic body. Holden-Day, San Francisco, 1963

    Google Scholar 

  18. Hearmon, R. F. S.: An introduction to applied anisotropic elasticity. Oxford Univ. Press, London 1961

    Google Scholar 

  19. Raumann, G.: Proc. Phys. Soc. 79, 1221 (1962)

    Article  Google Scholar 

  20. Ladizesky, N. H., Ward, I. M.: J. Macromol. Sci. B 5, 745 (1971)

    Google Scholar 

  21. Biot, M. A.: J. Appl. Phys. 10, 860 (1939)

    Article  Google Scholar 

  22. Pipkin, A. C., Rogers, T. G.: J. Appl. Mech. 38, 634 (1971)

    Google Scholar 

  23. Spencer, A. J. M.: Deformations of Fibre-reinforced materials. Clarendon Press, Oxford 1972

    Google Scholar 

  24. Thomson, W. T.: Vibration theory and applications. George Allen and Unwin, London 1965

    Google Scholar 

  25. Timoshenko, S.: Vibration problems in Engineering. 2nd Edit. Van Nostrand et al., New York 1937

    Google Scholar 

  26. Ritchie, I. G. et al.: J. Phys. D: Appl. Phys. 8, 1733 (1975)

    Article  Google Scholar 

  27. Hertz, H.: Z. f. Math. (Crelle) 92 (1881)

    Google Scholar 

  28. McCrum, N. G.: Priv. comm.

    Google Scholar 

  29. Hashin, Z.: Int. J. Engng. Sci. 7, 11 (1969)

    Article  Google Scholar 

  30. Laws, N., McLaughlin, R.: Proc. Roy. Soc. A 359, 251 (1978)

    Google Scholar 

  31. Lee, E. H.: Q. Appl. Math. 13, 183 (1955)

    Google Scholar 

  32. Takayanagi, M.: Proc. 4th Int. Conf. Rheology, Part 1. Intersci. New York 1965

    Google Scholar 

  33. Smith, J. B. et al.: J. Polym. Sci., Polym. Phys. Ed. 13, 2331 (1975)

    Google Scholar 

  34. Arridge, R. G. C., Barham, P. J.: Polymer 19, 603 (1978)

    Article  Google Scholar 

  35. Ward, I. M.: Mechanical properties of solid polymers. Wiley, New York 1971

    Google Scholar 

  36. Wedgewood, A. R., Seferis, J. C.: Polymer 22, 966 (1981)

    Article  Google Scholar 

  37. Gibson, A. G., Davies, G. R., Ward, I. M.: Polymer 19, 683 (1978)

    Article  Google Scholar 

  38. Odell, J. A., Keller, A.: Polymer Engng. Sci. 17, 544 (1977)

    Article  Google Scholar 

  39. Cooper, J. W., McCrum, N. G.: J. Mater. Sci. 7, 1221 (1972)

    Article  Google Scholar 

  40. Gilmour, I. W., Trainor, A., Haward, R. N.: J. Polym. Sci., Polym. Phys. Ed. 16, 1277 (1978)

    Google Scholar 

  41. Godovskii, Yu, K., Slonimskii, G. L., Alekseev, V. F.: Vysokomol. Soedin. 11 A, 1181 (1969)

    Google Scholar 

  42. Zener, C. M.: Elasticity and anelasticity of metals. Chicago Univ. Press, Chicago 1965

    Google Scholar 

  43. Nowacki, W.: Dynamic problems of thermoelasticity. Noordhoff Internat. Publ. Groningen 1975

    Google Scholar 

  44. Nowick, A. S., Berry, B. S.: Anelastic relaxation in crystalline solids. Academic Press, New York 1972

    Google Scholar 

  45. Barham, P. J., Arridge, R. G. C.: Polymer 20, 509 (1979)

    Article  Google Scholar 

  46. Sander, B.: Gefügekunde der Gesteine. J. Springer, Vienna 1930.

    Google Scholar 

  47. Hill, R.: J. Mech. Phys. Solids 11, 357 (1963)

    Article  Google Scholar 

  48. Takayanagi, M.: Mem. Fac. Engng. Kyushu Univ. 23, 1 and 41 (1963)

    Google Scholar 

  49. Van Fo Fy, G. K., Savin, G. N.: Mekh. Polim. 1, 151 (1965)

    Google Scholar 

  50. Pickett, G.: Trans. ASME 664, 176 (1944)

    Google Scholar 

  51. Arridge, R. G. C., Lock, M. W. B.: J. Phys. D: Appl. Phys. 9, 329 (1976)

    Article  Google Scholar 

  52. Smith, G. E., Spencer, A. J. M.: J. Mech. Phys. Solids 18, 81 (1970)

    Article  Google Scholar 

  53. Broutman, L. J., Agarwal, B. D.: Polymer Engng. Sci. 14, 581 (1974)

    Article  Google Scholar 

  54. Whitney, J. M.: J. Compos. Mat. 1, 121 (1967)

    Google Scholar 

  55. Rosen, B. W.: Proc. Roy. Soc. A 319, 79 (1970)

    Google Scholar 

  56. Hashin, Z., Rosen, B. W.: Trans. ASME, J. Appl. Mech. 31, 223 (1964)

    Google Scholar 

  57. Hashin, Z., Shtrikman, S.: J. Mech. Phys. Solids 11, 127 (1963)

    Article  Google Scholar 

  58. Walpole, L.: J. Mech. Phys. Solids 17, 235 (1969)

    Article  Google Scholar 

  59. Kröner, E.: Z. Physik 151, 504 (1958)

    Article  Google Scholar 

  60. Roscoe, R.: J. Mech. Phys. Solids 20, 91 (1972)

    Article  Google Scholar 

  61. Walpole, L.: J. Mech. Phys. Solids 14, 151 (1966)

    Article  Google Scholar 

  62. Hill, R.: J. Mech. Phys. Solids 12, 199 (1964)

    Article  Google Scholar 

  63. Eshelby, J. D.: Proc. Roy. Soc. A 241, 376 (1957)

    Google Scholar 

  64. Wunderlich, B.: Macromolecular Physics, Vol. 1. Academic Press, New York 1973

    Google Scholar 

  65. Arridge, R. G. C.: Continuum Models of Discrete Systems 4. Brulin, O., Hsieh, R. K. T. (eds.) North Holland, Amsterdam 1981

    Google Scholar 

  66. Haward, R. N., Daniels, H. E., Treloar, L. R. G.: J. Polym. Sci., Polym. Phys. Ed. 16, 1169 (1978)

    Google Scholar 

  67. Gaskell, P. H., Gibson, J. M., Howie, A. in: The structure of non-crystalline materials. Gaskell, P. H. (ed.) Taylor and Francis, London 1977, p. 181

    Google Scholar 

  68. Hendus, H., Illers, K., Ropte, E.: z. Kolloid z.z. Polym. 216, 110 (1967)

    Article  Google Scholar 

  69. Matsuo, M.: Japan Plast. 2, 6 (1968)

    Google Scholar 

  70. Dawkins, J. V. in: Block Copolymers. Allport, D. C., Janes, W. H. (eds.) Appl. Sci. Publ. London 1973

    Google Scholar 

  71. Allport, D. C., Mohajer, A. A. in: Block Copolymers, Allport, D. C., Janes, W. H. (eds.) Appl. Sci. Publ. London 1973

    Google Scholar 

  72. Arridge, R. G. C., Folkes, M. J.: J. Phys. D: Appl. Phys. 5, 344 (1970)

    Article  Google Scholar 

  73. Gray, W., McCrum, N. G.: J. Polym. Sci. A-2, 7, 1329 (1969)

    Article  Google Scholar 

  74. Maeda, M. et al.: J. Polym. Sci. A-2, 8, 1303 (1970)

    Article  Google Scholar 

  75. Buckley, C. P.: J. Mater. Sci. 9, 100 (1974)

    Article  Google Scholar 

  76. Andrews, E. H.: Pure and Appl. Chem. 39, 179 (1974)

    Google Scholar 

  77. Patel, J., Phillips, P. J.: Polym. Lett. 11, 771 (1973)

    Article  Google Scholar 

  78. Halpin, J. C., Kardos, J. C.: J. Appl. Phys. 43, 2235 (1972)

    Article  Google Scholar 

  79. Owen, A. J., Ward, I. M.: Mater. Sci. 6, 485 (1971)

    Article  Google Scholar 

  80. Raumann, G., Saunders, D. W.: Proc. Phys. Soc. 77, 1028 (1961)

    Article  Google Scholar 

  81. Ward, I. M.: Proc. Phys. Soc. 80, 1176 (1962)

    Article  Google Scholar 

  82. McCullough, R. L. et al.: Polym. Engng. and Sci. 16, 371 (1976)

    Article  Google Scholar 

  83. Seferis, J. C., McCullough, R. L., Samuels, R. J.: Polym. Engng. and Sci. 16, 334 (1976)

    Article  Google Scholar 

  84. Halpin, J. C., Kardos, J. L.: Polym. Engng. and Sci. 16, 344 (1976)

    Article  Google Scholar 

  85. Barham, P. J., Arridge, R. G. C.: J. Polym. Sci., Polym. Phys. Ed. 15, 1177 (1977)

    Google Scholar 

  86. Arridge, R. G. C., Barham, P. J.: J. Polym. Sci., Polym. Phys. Ed. 16, 1297 (1978)

    Google Scholar 

  87. Peterlin, A.: Copolymers, Polyblends and Composites: Adv. Chem. Ser. 142, Am. Chem. Soc. 1975

    Google Scholar 

  88. Cox, H. L.: Brit. J. Appl. Phys. 3, 72 (1952)

    Article  Google Scholar 

  89. Peterlin, A. in: Ultra-high modulus polymers. Ciferri, A., Ward, I. M. (eds.) Appl. Sci. Publ. London 1979, p. 279

    Google Scholar 

  90. Ciferri, A., Ward, I. M. (eds.): Ultra-high modulus polymers. Appl. Sci. Publ. London 1979

    Google Scholar 

  91. Chen, F. C., Young, K.: J. Math. Phys. 18, 1412 (1977)

    Article  Google Scholar 

  92. Capaccio, G., Ward, I. M.: Polymer 15, 233 (1974)

    Article  Google Scholar 

  93. Gibson, A. G. et al.: J. Mater. Sci. 9, 1193 (1974)

    Article  Google Scholar 

  94. Odell, J. A., Grubb, D. T., Keller, A.: Polymer 19, 617 (1978)

    Article  Google Scholar 

  95. Zwijnenburg, A., Pennings, A. J.: Colloid and Polym. Sci. 259, 868 (1978)

    Google Scholar 

  96. Barham, P. J., Keller, A. Polym. Lett. 17, 591 (1979)

    Article  Google Scholar 

  97. Smith, P. et al.: Polym. Bull. 1, 733 (1979)

    Article  Google Scholar 

  98. Smith, P., Lemstra, P. J.: J. Mater. Sci. 15, 505 (1980)

    Article  Google Scholar 

  99. Treloar, L. R. G.: Polymer 1, 95 (1960)

    Article  Google Scholar 

  100. Rasmussen, R. S.: J. Chem. Phys. 16, 712 (1948)

    Article  Google Scholar 

  101. McCullough, R. L., Eisenstein, A. J., Weikart, D. F.: J. Polym. Sci., Polym. Phys. Ed. 15, 1837 (1977)

    Google Scholar 

  102. Lifson, S., Warshel, A.: J. Chem. Phys. 49, 5116 (1968)

    Article  Google Scholar 

  103. Williams, D. E.: J. Chem. Phys. 47, 4680 (1967)

    Article  Google Scholar 

  104. Mizushima, S., Shimanouchi, T.: J. Amer. Chem. Soc. 71, 1320 (1949)

    Article  Google Scholar 

  105. Shimanouchi, T., Asahima, M., Enomoto, S.: J. Polym. Sci. 59, 93 (1962)

    Article  Google Scholar 

  106. Shauffele, K. F., Shimanouchi, T.: J. Chem. Phys. 47, 3605 (1967)

    Article  Google Scholar 

  107. Odajima, A., Maeda, T.: J. Polym. Sci. C15, 55 (1966)

    Google Scholar 

  108. Born, M., Huang, K.: The dynamical theory of crystal lattices. Clarendon Press Oxford 1954

    Google Scholar 

  109. Schachtschneider, J. H., Snyder, R. G.: Spectrochim. Acta 19, 117 (1963)

    Article  Google Scholar 

  110. Wobser, G., Blasenbrey, S.: Kolloid Z. Polym. 241, 985 (1970)

    Article  Google Scholar 

  111. Hosemann, R.: CRC Crit. Rev. Macromol. Sci. 351 (1972)

    Google Scholar 

  112. Falk, F.: Colloid and Polymer Sci. 258, 142 and 365 (1980)

    Article  Google Scholar 

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Authors and Affiliations

  1. H. H. Wills Physics Laboratory, University of Bristol, Royal Fort, Tyndall Avenue, BS 81 TL, Bristol, England

    Robert G. C. Arridge & Peter J. Barham

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  1. Robert G. C. Arridge
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  2. Peter J. Barham
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Dedicated to Professor Manfred Gordon, University of Essex, with best wishes for his 65th birthday.

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© 1982 Springer-Verlag

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Arridge, R.G.C., Barham, P.J. (1982). Polymer elasticity discrete and continuum models. In: Behavior of Macromolecules. Advances in Polymer Science, vol 46. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0023984

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  • DOI: https://doi.org/10.1007/BFb0023984

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  • Print ISBN: 978-3-540-11640-0

  • Online ISBN: 978-3-540-39453-2

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