Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



A model for one-dimensional, nonlinear viscoelasticity

Author: R. C. MacCamy
Journal: Quart. Appl. Math. 35 (1977), 21-33
MSC: Primary 73.45; Secondary 45K05
DOI: https://doi.org/10.1090/qam/478939
MathSciNet review: 478939
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Abstract: The problem

$\displaystyle {u_{tt}} = a\left( 0 \right)\sigma {\left( {{u_x}} \right)_x} + \... ...u_o}\left( x \right), \qquad {u_t}\left( {x, 0} \right) = {u_1}\left( x \right)$

is considered. The essential hypotheses are that

$\displaystyle a\left( t \right) = {a_\infty } + A\left( t \right), {a_\infty } ... ... 1, 2, \sigma \left( 0 \right) = 0, \sigma '\left( \xi \right) \ge \epsilon > 0$

. It is shown that the problem has a unique classical solution for all $ t$ if the data are sufficiently small and, if $ f$ is suitably restricted, this solution tends to zero as $ t$ tends to infinity. It is shown that the problem provides a special model for elastic materials which exhibit a memory effect.

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Additional Information

DOI: https://doi.org/10.1090/qam/478939
Article copyright: © Copyright 1977 American Mathematical Society