A review is given of several current network models (constrained chain model, localization model, liquidlike model, and eight-chain model) that have been developed to explain nonideal rubbery behavior. The stress-strain relationships for both dry and swollen networks are discussed. It is shown that the constrained chain model of Flory, Erman, and Monnerie provides the best description of the considered models for the stress-strain response of both the dry state and swollen state properties of cross-linked rubber networks. The localization model of Gaylord and Douglas fits the dry-state stress-strain data very well, but the predictions of the mechanical response in the swollen state are not as good as those obtained with the constrained chain model. The liquidlike model of DiMarzio yields reasonable description of extension behavior, but it does not explain the observed modest decrease of the elastic modulus (reduced stress) with increasing compression; nor does the liquidlike model predict the decreasing modulus with increasing swelling correctly The eight-chain model of Arruda and Boyce puts forward a stress-strain relation that includes the finite extensibility of the network chains. It is shown that at moderate deformation ratios, the eight-chain model does not provide even qualitative agreement with the experimental stress-strain data for both the dry and swollen states.
Get full access to this article
View all access options for this article.
References
1.
[1] Allen, G., Kirkham, M. J., Padget, J., and Price, C.: Thermodynamics of rubber at constant volume. Trans. Faraday Soc., 67, 1278-1278 (1971).
2.
[2] Arruda, E. M. and Boyce, M. C.: Evolution of plastic anisotropy in amorphous polymers during finite straining. Int. J. Plast., 9, 697-697 (1993).
3.
[3] Arruda, E. M. and Boyce, M. C.: A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J. Mech. Phys. Solids, 41, 389-389 (1993).
4.
[4] Besseling, J. and van der Giessen, E.: Mathematical Modelling of Inelastic Deformations, Chapman Hall, London, 1994.
5.
[5] Bradley, G. L.: Determination of the ultimate capacity of elastomeric bearings under axial loading. Master's thesis, University of Maryland, College Park, 1997.
6.
[6] Deam, R. T. and Edwards, S. F.: The theory of rubber elasticity. Phil. Trans. Roy. Soc. London, A, 280, 317-317 (1978).
7.
[7] DiMarzio, E. A.: Contribution to a liquid-like theory of rubber elasticity. J. Chem. Phys., 36, 1563-1563 (1962).
8.
[8] DiMarzio, E. A.: Contribution to a liquid-like theory of rubber elasticity: 2. Existence of a(A(A,, + y + A):) term. Polymer, 35, 1819-1819 (1994).
9.
[9] Douglas, J. F. and McKenna, G. B.: The effect of swelling on the elasticity of rubber: Localization model description. Macromolecules, 26, 3282-3238 (1993).
10.
[10] Edwards, S. F.: The theory of rubber elasticity. Brit. Polym. J., 9, 140-140 (1977).
11.
[11] Erman, B. and Mark, J. E.: Interpretation of stress-strain isotherms for elastomers cross linked in solution. Macro-molecules, 20, 2892-2892 (1987).
12.
[12] Erman, B. and Mark, J. E.: Stress-strain isotherms for elastomers cross linked in solution. 2. Interpretation in terms of the constrained chain model. Macromolecules, 25, 1919-1919 (1992).
13.
[13] Erman, B. and Mark, J. E.: Structure and Properties of RubberlikeNetworks, Oxford University Press, New York, 1997.
14.
[14] Erman, B. and Monnerie, L.: Theory of elasticity of amorphous networks: Effect of constraints along chains. Macro-molecules, 22, 3342-3342 (1989).
15.
[15] Erman, B. and Monnerie, L.: Corrections. Macromolecules, 25, 4456-4456 (1992).
16.
[16] Ferry, J. D.: 14scoelastic Properties of Polymers, Wiley, New York, 1980.
17.
[17] Flory, P J.: Principles ofPolymer Chemistry, Cornell University Press, Ithaca, NY1953.
18.
[18] Flory, P J.: Theory of elasticity of polymer networks: The effect of local constraints on junctions. J. Chem. Phys., 66, 5720-5720 (1977).
19.
[19] Flory, P J. and Erman, B.: Theory of elasticity of polymer networks. Macromolecules, 15, 800-800 (1982).
20.
[20] Flory, P J. and Rehner, J.: Statistical mechanics of cross-linked polymer networks. J. Chem. Phys., 11, 521-521 (1943).
21.
[21] Flory, P J. and Tatara, Y.: The elastic free energy and the elastic equation of state: Elongation and swelling of poly-dimethylsiloxane networks. J. Polym. Sci., Polym. Phys. Ed., 13, 683-683 (1975).
22.
[22] Frenkel, J.: A theory of elasticity, viscosity and swelling in polymeric rubber-like substances. Rubber Chem. Tech., 13, 264-264 (1940).
23.
[23] Gaylord, R. J. and Douglas, J. F.: Rubber elasticity: A scaling approach. Polym. Bull., 18, 347-347 (1987).
24.
[24] Gaylord, R. J. and Douglas, J. F.: The localisation model of rubber elasticity. Polym. Bull., 23, 529-529 (1990).
25.
[25] Gottlieb, M. and Gaylord, R. J.: Experimental tests of entanglement models of rubber elasticity: 1. Uniaxial extension-compression. Polymer, 24, 1644-1644 (1983).
26.
[26] Gottlieb, M. and Gaylord, R. J.: Experimental tests of entanglement models of rubber elasticity: 2. Swelling. Macro-molecules, 17, 2024-2024 (1984).
27.
[27] Graessley, W W.: Entangled linear, branched and network polymer systems: Molecular theories. Adv Polym. Sci., 47, 67-67 (1982).
28.
[28] Haward, R. N.: Strain hardening of thermoplastics. Macromolecules, 26, 5860-5860 (1993).
29.
[29] Hermans, J. J.: Deformation and swelling of polymer networks containing comparatively long chains. Trans. Fara-day Soc., 43, 591-591 (1947).
30.
[30] Higgs, P G. and Ball, R. C.: Trapped entanglements in rubbers. A unification of models. Europhys. Lett., 8, 357-357 (1989).
31.
[31] Higgs, P G. and Gaylord, R. J.: Slip-links, hoops and tubes: Tests of entanglement models of rubber elasticity. Polymer; 31, 70-70 (1990).
32.
[32] James, H. M. and Guth, E.: Theory of elastic properties of rubber. J. Chem. Phys., 11, 455-455 (1943).
33.
[33] James, H. M. and Guth, E.: Statistical thermodynamics of rubber elasticity. J. Chem. Phys., 15, 669-669 (1947).
34.
[34] Mark, J. E. and Erman, B.: Rubberlike Elasticity: A Molecular Primer, Wiley-Interscience, New York, 1988.
35.
[35] McKenna, G. B. and Hinkley, J. A.: Mechanical and swelling behavior of well characterized polybutadiene networks. Polymer; 27, 1318-1318 (1986).
36.
[36] Mooney, M.: A theory of large elastic deformation. J. Appl. Phys., 11, 582-582 (1940).
37.
[37] Mullins, L.: Determination of degree of crosslinking in natural rubber vulcanizates: Part IV Stress-strain behavior at large extensions. J. Appl. Polym. Sci., 2, 257-257 (1959).
38.
[38] Pak, H. and Flory, P J.: Relationship of stress to uniaxial strain in crosslinked poly(dimethylsiloxane) over the full range from large compression to high elongations. J. Polym. Sci., Polym. Phys. Ed., 17, 1845-1845 (1979).
39.
[39] Rivlin, R. S.: Elastic deformations of isotropic materials. Phil. Trans. Roy. Soc. London, A, 240, 459-459 (1948).
40.
[40] Rivlin, R. S. and Saunders, D. W.: Large elastic deformations of isotropic materials: VII. Experiments on the deformation of rubber. Phil. Trans. Roy Soc. London, A, 243, 251-251 (1951).
41.
[41] Ronca, G. and Allegra, G.: An approach to rubber elasticity with internal constraints. J. Chem. Phys., 63, 4990-4990 (1975).
42.
[42] Rubin, M. B.: Plasticity theory formulated in terms of physically based microstructural variables. Int. J. Solids Struct., 31, 532-532 (1994).
43.
[43] Tervoort, T A.: Constitutive modelling of polymer glasses: Finite, nonlinear viscoelastic behaviour of polycarbonate. Doctoral thesis, Technische Univeriteit Eindhoven, CIP-G Gegevens Koninklijke Bibliotheek, den Haag, the Netherlands, 1996.
44.
[44] Treloar, L.R.G.: The photoelastic properties of short-chain molecularnetworks. Trans. Faraday Soc., 50, 881-881 (1954).
45.
[45] Van der Hoff, B.M.E.: The stress strain relation of swollen rubbers. Polymer, 6, 397-397 (1965).
46.
[46] Varga, O.H.: Stress-Strain Behavior of Elastic Materials: Selected Problems of Large Deformations, Interscience, New York, 1966.
47.
[47] Wall, F T.: Statistical thermodynamics of rubber. J. Chem. Phys., 11, 527-527 (1943).
48.
[48] Wang, M. C. and Guth, E.: Statistical theory of networks of non-Gaussian flexible chains. J. Chem. Phys., 20, 1144-1144 (1952).
49.
[49] Wu, P D. and van der Giessen, E.: On improved network models for rubber elasticity and their applications to orientation in glassy polymers. J. Mech. Phys. Solids, 41, 427-427 (1993).
