Previous formulations of rubber elasticity theory have been based on the assumption that either the deformation, or force field associated with crosslink deformations is the same as in a continuum. There is no apriori justification for such an assumption. In the present theory, consideration is given to a completely arbitrary distribution of crosslink deformations. Generalized equations for the six components of the macroscopic stress tensor and for the stored strain energy function are formulated. The unknown deformation functions of the radius vectors between crosslinks are then found by postulating that the deformation functions are such as to minimize the free energy of the network, subject to the constraints imposed by the macroscopic stress components applied to the body. The problem is solved using the calculus of variations. At vanishing strains, the minimum free energy deformation system for the crosslink network is found to be affine. However, at finite strains the free energy criterion of equilibrium shows the deformation system to be non-affine and a functional of the particular function chosen to describe the macromolecule’s conformational behavior. Non-Gaussian and Gaussian chain statistics are examined.
KeywordsWork Function Uniaxial Tension Rest State Deformation Function Rensselaer Polytechnic Institute
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- 1.W. Kuhn, Kolloid-Z. 68, 2 (1934); 76, 258 (1936).Google Scholar
- 3.H. James and E. Guth, J. Chem. Phys. 11, 455 (1943); 15, 669 (1947).Google Scholar
- 6.L. R. G. Treloar, Trans. Faraday Soc., 42, 77, 83, (1946); 50, 881 (1954); The Physics of Rubber Elasticity, 2nd Edition, Oxford Press (1958).Google Scholar
- 7.S. S. Sternstein, Unpublished work at Rensselaer Polytechnic Institute, 1961–1965.Google Scholar
- 8.S. S. Sternstein, “The Micromechanics of Fiber Networks”, in Cellulose and Cellulose Derivatives, Vol. 5, Bikales and Segal, Eds., J. Wiley, New York (1971).Google Scholar
- 9.S. S. Sternstein and A. H. Nissan, Trans. of the Oxford Symposium on Formation and Structure of Paper, p. 319, B.P. & B.M.A., London (1961).Google Scholar
- 10.R. Weinstock, Calculus of Variations, McGraw-Hill, New York (1952).Google Scholar
- 11.L. Elsgolc, Calculus of Variations, Pergammon Press, London (1962).Google Scholar
- 12.P. Thirion, Institut Francais du Caoutchouc, Paris, Personal Communication.Google Scholar
- 13.G. M. Lederle, “A Stress-Strain Law for Rubberlike Materials From Minimum Free Energy Considerations,” Ph.D. Dissertation, Rensselaer Polytechnic Institute (1968).Google Scholar