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 | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] D. R. Bland, The theory of linear viscoelasticity, Pergamon Press, New York, 1960 MR 0110314
  • [2] B. D. Coleman and M. E. Gurtin, Waves in materials with memory. II. On the growth and decay of one-dimensional acceleration waves, Arch. Rat. Mech. 19, 239-265 (1965) MR 0195336
  • [3] C. Corduneanu, Integral equations and stability of feedback systems, Academic Press, New York, 1973 MR 0358245
  • [4] C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Diff. Eq. 7, 554-569 (1970) MR 0259670
  • [5] James Glimm and P. D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Mem. Amer. Math. Soc. 101 (1970) MR 0265767
  • [6] J. M. Greenberg, A-priori estimates for flows in dissipative materials (preprint) MR 0450796
  • [7] J. M. Greenberg, R. C. MacCamy and V. J. Mizel, On the existence, uniqueness and stability of solutions of the equation $ \rho - {\chi _{tt}} = E\left( {{\chi _x}} \right){\chi _{xx}} + \lambda {\chi _{xxt}}$, J. Math. Mech. 17, 707-728 (1968)
  • [8] P. D. Lax, Development of singularities of solutions of non-linear hyperbolic differential equations, J. Math. Phys. 5, 611-613 (1964) MR 0165243
  • [9] Stig-Olaf London, An existence result on a Volterra equation in a Banach space (preprint)
  • [10] R. C. MacCamy, Existence uniqueness and stability of $ {u_{tt}} = \frac{\partial }{{\partial x}}\left( {\sigma \left( {{u_x}} \right) + \lambda \left( {{u_x}} \right){u_{xt}}} \right)$, Indiana Univ. Math. J. 20, 231-338 (1970) MR 0265790
  • [11] R. C. MacCamy, Nonlinear Volterra equations on a Hilbert space, J. Diff. Eq. 16, 373-393 (1974) MR 0377605
  • [12] R. C. MacCamy, Remarks on frequency domain methods for Volterra integral equations, J. Math. Anal. Appl.
  • [13] R. C. MacCamy, An integro-differential equation with applications in heat flow, Q. Appl. Math. (this issue)
  • [14] R. C. MacCamy and J. S. W. Wong, Stability theorems for some functional equations, Trans. Amer. Math. Soc. 164, 1-37 (1972) MR 0293355
  • [15] R. R. Nachlinger and L. T. Wheeler, Wave propagation and uniqueness in one-dimensional simple materials, J. Math. Anal. Appl. 48, 294-300 (1974) MR 0351231
  • [16] T. Nashida, Global smooth solutions for the second order quasilinear equation with first-order dissipation (preprint)
  • [17] J. A. Nohel and D. F. Shea, Frequency domain methods for Volterra equations (preprint) MR 0500024

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Additional Information

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

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