Proceedings of the Royal Society of Edinburgh Section A Mathematics · 1978 · 62 citations · 9 references
Natural SciencesMaterials ScienceMathematicsDifferential EquationsFluid DynamicsRheologyPolymer PhysicsPolymer RheologyRheological ModelingSingular Perturbation TheoryViscoelastic BehaviorViscoplastic BehaviorElastic RecoveryInitial Value FormulationInitial Value Problem FormulationPolyethylenePolymer Melt RheologyRheology Of Polymer SolutionsViscoelastic Properties Of PolymersVolterra Equations
Synopsis 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 —μ y ′ in (+) has little effect on the ultimate recovery but significant effect during the early part of the recovery.
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