a1 Mathematics Research Center, University of Wisconsin, Madison, U.S.A.
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.
(Received March 29 1977)
1 sponsored by The U.S. Army under Contract No. DAAG29-75-C-0024
2 sponsored by The National Science Foundation under Grant No. ENG 75-18397
3 sponsored by The U.S. Army under Grant No. DAHC04-74-G-0012
4 sponsored by The National Science Foundation under Grant No. MCS 75-21868.