On energies for nonlinear viscoelastic materials of single-integral type
Authors:
Morton E. Gurtin and William J. Hrusa
Journal:
Quart. Appl. Math. 46 (1988), 381-392
MSC:
Primary 73F99; Secondary 73B30, 73G10
DOI:
https://doi.org/10.1090/qam/950610
MathSciNet review:
950610
Full-text PDF Free Access
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Additional Information
- Shlomo Breuer and E. Turan Onat, On the determination of free energy in linear viscoelastic solids, Z. Angew. Math. Phys. 15 (1964), 184–191 (English, with German summary). MR 178645, DOI https://doi.org/10.1007/BF01602660
- Theodore Allen Burton, Volterra integral and differential equations, Mathematics in Science and Engineering, vol. 167, Academic Press, Inc., Orlando, FL, 1983. MR 715428
- Bernard D. Coleman, Thermodynamics of materials with memory, Arch. Rational Mech. Anal. 17 (1964), 1–46. MR 171419, DOI https://doi.org/10.1007/BF00283864
- Bernard D. Coleman and Victor J. Mizel, On the stability of solutions of functional-differential equations, Arch. Rational Mech. Anal. 30 (1968), 173–196. MR 229933, DOI https://doi.org/10.1007/BF00253873
- Bernard D. Coleman and David R. Owen, On the thermodynamics of materials with memory, Arch. Rational Mech. Anal. 36 (1970), 245–269. MR 269165, DOI https://doi.org/10.1007/BF00249514
- 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
- W. A. Day, A theory of thermodynamics for materials with memory, Arch. Rational Mech. Anal. 34 (1969), 85–96. MR 249038, DOI https://doi.org/10.1007/BF00247460
- W. A. Day, Restrictions on relaxation functions in linear viscoelasticity, Quart. J. Mech. Appl. Math. 24 (1971), 487–497. MR 317631, DOI https://doi.org/10.1093/qjmam/24.4.487
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- M. E. Gurtin and I. Herrera, On dissipation inequalities and linear viscoelasticity, Quart. Appl. Math. 23 (1965), 235–245. MR 189346, DOI https://doi.org/10.1090/S0033-569X-1965-0189346-9
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- J. J. Levin, A nonlinear Volterra equation not of convolution type, J. Differential Equations 4 (1968), 176–186. MR 225117, DOI https://doi.org/10.1016/0022-0396%2868%2990034-X
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- Victor J. Mizel and Victor Trutzer, Stochastic hereditary equations: existence and asymptotic stability, J. Integral Equations 7 (1984), no. 1, 1–72. MR 747535
J. W. Nunziato, E. K. Walsh, K. W. Schuler, and L. M. Barker, Wave propagation in nonlinear viscoelastic solids, in S. Flugge (ed.), Handbuch der Physik VI a/4, Springer, 1–108 (1974)
- Michael Renardy, William J. Hrusa, and John A. Nohel, Mathematical problems in viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 35, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. MR 919738
V. Volterra, Sur la théorie mathématique des phénomènes héréditaires, J. Math. Pures Appl. 7, 249–298 (1928)
- Morton E. Gurtin and William J. Hrusa, On the thermodynamics of viscoelastic materials of single-integral type, Quart. Appl. Math. 49 (1991), no. 1, 67–85. MR 1096233, DOI https://doi.org/10.1090/qam/1096233
S. Breuer and E. T. Onat, On the determination of the free energy in linear viscoelastic solids, Z. Angew. Math. Phys. 15, 184–191 (1964)
T. A. Burton, Volterra Integral and Differential Equations, Academic Press, 1983
B. D. Coleman, Thermodynamics of materials with memory, Arch. Rat. Mech. Anal. 17, 1–46 (1964)
B. D. Coleman and V. J. Mizel, On the stability of solutions of functional-differential equations, Arch. Rat. Mech. Anal. 30, 173–196 (1968)
B. D. Coleman and D. R. Owen, On the thermodynamics of materials with memory, Arch. Rat. Mech. Anal. 36, 245–269 (1970)
C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations 7, 554–569 (1970)
W. A. Day, A theory of thermodynamics for materials with memory, Arch. Rat. Mech. Anal. 34, 85–96 (1969)
W. A. Day, Restrictions on relaxation functions in linear viscoelasticity, Quart. J. Mech. Appl. Math. 24, 487–497 (1971)
W. A. Day, The Thermodynamics of Simple Materials with Fading Memory, Springer, 1972
M. E. Gurtin and I. Herrera, On dissipation inequalities and linear viscoelasticity, Quart. Appl. Math. 23, 235–245 (1965)
J. K. Hale, Sufficient conditions for stability and instability of autonomous functional-differential equations, J. Differential Equations 1, 452–482 (1965)
J. K. Hale, Theory of Functional Differential Equations, Springer, 1977
W. J. Hrusa and M. Renardy, On a class of quasilinear partial integrodifferential equations with singular kernels, J. Differential Equations 64, 195–220 (1986)
J.J. Levin, A nonlinear Volterra equation not of convolution type, J. Differential Equations 4, 176–186 (1968)
J. J. Levin and J. A. Nohel, Perturbations of a nonlinear Volterra equation, Michigan Math. J. 12, 431–447 (1965)
V. J. Mizel and V. Trutzer, Stochastic hereditary equations: existence and asymptotic stability, J. Integral Equations 7, 1–72 (1984)
J. W. Nunziato, E. K. Walsh, K. W. Schuler, and L. M. Barker, Wave propagation in nonlinear viscoelastic solids, in S. Flugge (ed.), Handbuch der Physik VI a/4, Springer, 1–108 (1974)
M. Renardy, W. J. Hrusa, and J. A. Nohel, Mathematical Problems in Viscoelasticity, Longman Scientific and Technical, Essex, England and John Wiley, New York, 1987
V. Volterra, Sur la théorie mathématique des phénomènes héréditaires, J. Math. Pures Appl. 7, 249–298 (1928)
M. E. Gurtin and W. J. Hrusa, On the thermodynamics of viscoelastic materials of single-integral type, Forthcoming
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Article copyright:
© Copyright 1988
American Mathematical Society