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 \[ {u_{tt}} = a\left ( 0 \right )\sigma {\left ( {{u_x}} \right )_x} + \int _0^t {\dot a} \left ( {t - \tau } \right )\sigma {\left ( {{u_x}} \right )_x}d\tau + f, \qquad 0 < x < 1, \qquad t > 0, \\ u\left ( {0, t} \right ) \equiv u\left ( {1, t} \right ) \equiv 0, \\ u\left ( {x, 0} \right ) = {u_o}\left ( x \right ), \qquad {u_t}\left ( {x, 0} \right ) = {u_1}\left ( x \right )\] is considered. The essential hypotheses are that \[ a\left ( t \right ) = {a_\infty } + A\left ( t \right ), {a_\infty } > 0, A \in {L^1}\left ( {0, \infty } \right ), \\ {\left ( { - 1} \right )^k}{a^{\left ( k \right )}}\left ( t \right ) \ge 0, k = 0, 1, 2, \sigma \left ( 0 \right ) = 0, \sigma ’\left ( \xi \right ) \ge \epsilon > 0\]. 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.
- D. R. Bland, The theory of linear viscoelasticity, International Series of Monographs on Pure and Applied Mathematics, Vol. 10, Pergamon Press, New York-London-Oxford-Paris, 1960. MR 0110314
- Bernard D. Coleman and Morton E. Gurtin, Waves in materials with memory. II. On the growth and decay of one-dimensional acceleration waves, Arch. Rational Mech. Anal. 19 (1965), 239–265. MR 195336, DOI https://doi.org/10.1007/BF00250213
- Constantin Corduneanu, Integral equations and stability of feedback systems, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Mathematics in Science and Engineering, Vol. 104. MR 0358245
- Constantine M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations 7 (1970), 554–569. MR 259670, DOI https://doi.org/10.1016/0022-0396%2870%2990101-4
- James Glimm and Peter D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Memoirs of the American Mathematical Society, No. 101, American Mathematical Society, Providence, R.I., 1970. MR 0265767
- J. M. Greenberg, A priori estimates for flows in dissipative materials, J. Math. Anal. Appl. 60 (1977), no. 3, 617–630. MR 450796, DOI https://doi.org/10.1016/0022-247X%2877%2990005-1
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)
- Peter D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Mathematical Phys. 5 (1964), 611–613. MR 165243, DOI https://doi.org/10.1063/1.1704154
Stig-Olaf London, An existence result on a Volterra equation in a Banach space (preprint)
- R. C. MacCamy, Existence uniqueness and stability of solutions of the equation $u_{tt}=(\partial /\partial x)(\sigma (u_{x})+\lambda (u_{x})u_{tt})$, Indiana Univ. Math. J. 20 (1970/71), 231–238. MR 265790, DOI https://doi.org/10.1512/iumj.1970.20.20021
- Richard C. MacCamy, Nonlinear Volterra equations on a Hilbert space, J. Differential Equations 16 (1974), 373–393. MR 377605, DOI https://doi.org/10.1016/0022-0396%2874%2990021-7
R. C. MacCamy, Remarks on frequency domain methods for Volterra integral equations, J. Math. Anal. Appl.
R. C. MacCamy, An integro-differential equation with applications in heat flow, Q. Appl. Math. (this issue)
- R. C. MacCamy and J. S. W. Wong, Stability theorems for some functional equations, Trans. Amer. Math. Soc. 164 (1972), 1–37. MR 293355, DOI https://doi.org/10.1090/S0002-9947-1972-0293355-X
- R. Ray Nachlinger and Lewis T. Wheeler, Wave propagation and uniqueness in one-dimensional simple materials, J. Math. Anal. Appl. 48 (1974), 294–300. MR 351231, DOI https://doi.org/10.1016/0022-247X%2874%2990235-2
T. Nashida, Global smooth solutions for the second order quasilinear equation with first-order dissipation (preprint)
- J. A. Nohel and D. F. Shea, Frequency domain methods for Volterra equations, Advances in Math. 22 (1976), no. 3, 278–304. MR 500024, DOI https://doi.org/10.1016/0001-8708%2876%2990096-7
D. R. Bland, The theory of linear viscoelasticity, Pergamon Press, New York, 1960
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)
C. Corduneanu, Integral equations and stability of feedback systems, Academic Press, New York, 1973
C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Diff. Eq. 7, 554–569 (1970)
James Glimm and P. D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Mem. Amer. Math. Soc. 101 (1970)
J. M. Greenberg, A-priori estimates for flows in dissipative materials (preprint)
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)
P. D. Lax, Development of singularities of solutions of non-linear hyperbolic differential equations, J. Math. Phys. 5, 611–613 (1964)
Stig-Olaf London, An existence result on a Volterra equation in a Banach space (preprint)
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)
R. C. MacCamy, Nonlinear Volterra equations on a Hilbert space, J. Diff. Eq. 16, 373–393 (1974)
R. C. MacCamy, Remarks on frequency domain methods for Volterra integral equations, J. Math. Anal. Appl.
R. C. MacCamy, An integro-differential equation with applications in heat flow, Q. Appl. Math. (this issue)
R. C. MacCamy and J. S. W. Wong, Stability theorems for some functional equations, Trans. Amer. Math. Soc. 164, 1–37 (1972)
R. R. Nachlinger and L. T. Wheeler, Wave propagation and uniqueness in one-dimensional simple materials, J. Math. Anal. Appl. 48, 294–300 (1974)
T. Nashida, Global smooth solutions for the second order quasilinear equation with first-order dissipation (preprint)
J. A. Nohel and D. F. Shea, Frequency domain methods for Volterra equations (preprint)
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© Copyright 1977
American Mathematical Society