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  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 80, Issue 1-2
  • January 1978, pp. 99-137

A nonlinear singularly perturbed Volterra integrodifferential equation occurring in polymer rheology

  • A. S. Lodge (a1), J. B. McLeod (a1) and J. A. Nohel (a1)
  • DOI:
  • Published online: 14 November 2011

We study the initial value problem for the nonlinear Volterra integrodifferential equation

where μ > 0 is a small parameter, a is a given real kernel, and F, g are given real functions; (+) models the elongation ratio of a homogeneous filament of a certain polyethylene which is stretched on the time interval (— ∞ 0], then released and allowed to undergo elastic recovery for t > 0. Under assumptions which include physically interesting cases of the given functions a, F, g, we discuss qualitative properties of the solution of (+) and of the corresponding reduced problem when μ = 0, and the relation between them as μ → 0+, both for t near zero (where a boundary layer occurs) and for large t. In particular, we show that in general the filament does not recover its original length, and that the Newtonian term —μy′ in (+) has little effect on the ultimate recovery but significant effect during the early part of the recovery.

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1B. Bernstein , E. A. Kearsley and L. J. Zapas A Study of Stress Relation with Finite Strain. Trans. Soc. Rheol. 7 (1963), 391410.

6V. Lakshmikantham and S. Leela Differential and Integral Inequalities (London: Academic Press, 1969).

10A. S. Lodge Constitutive Equations from Molecular Network Theories for Polymer Solutions. Rheol. Acta 7 (1968), 379392.

11A. S. Lodge The Compatibility Conditions for Large Strains. Quart. J. Mech. Appl Math. 4 (1951), 8593.

12A. S. Lodge and J. Meissner On the Use of Instantaneous Strains Superposed on Shear and Elongational Flows of Polymeric Liquids, to Test the Gaussian Network Hypothesis and to Estimate the Segment Concentration and its Variation during Flow. Rheol. Acta 11 (1972), 351352.

13A. S. Lodge and J. Meissner Comparison of Network Theory Predictions with Stress/Time Data in Shear and Elongation for a Low-density Polyethylene Melt. Rheol. Acta 12 (1973), 4147.

17J. A. Nohel Some Problems in Nonlinear Volterra Integral Equations. Bull. Amer. Math. Soc. 68 (1962), 323329.

18J. A. Nohel and D. F. Shea Frequency Domain Methods for Volterra Equations. Advances in Math. 22 (1976), 278304.

19W. Walter Differential and Integral Inequalities (Berlin: Springer, 1970).

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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