On the stability for linear viscoelastic solids
Authors:
C. Giorgi and B. Lazzari
Journal:
Quart. Appl. Math. 55 (1997), 659-675
MSC:
Primary 73F15; Secondary 45K05, 73H10
DOI:
https://doi.org/10.1090/qam/1486541
MathSciNet review:
MR1486541
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Abstract: The stability and asymptotic behaviour of solutions to the dynamical initial past-history problem within the theory of linear viscoelasticity is investigated. First, a family of free energies is introduced as a Liapunov functional via the energy method. Then, emphasizing the role of thermodynamic inequalities, the asymptotic stability property is derived by means of some original a priori integral estimates of the solution.
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B. Lazzari and E. Vuk, Un teorema di esistenza e unicità per un problema dinamico in viscoelasticità lineare, Atti Sem. Mat. Fis. Univ. Modena 35, 309–317 (1987)
- Mauro Fabrizio, An existence and uniqueness theorem in quasi-static viscoelasticity, Quart. Appl. Math. 47 (1989), no. 1, 1–8. MR 987890, DOI https://doi.org/10.1090/S0033-569X-1989-0987890-3
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M. Fabrizio, Sull’invertibilità dell’equazione costitutiva della viscoelasticità lineare, Rend. Mat. Acc. Naz. Lincei, (9) 3, 141–148 (1992)
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C. Giorgi, Alcune conseguenze delle restrizioni termodinamiche per mezzi viscoelastici lineari, Quad. Dip. Mat. Univ. Cattolica Brescia, 6/89 (internal report)
B. D. Coleman and W. Noll, Foundations of linear viscoelasticity, Rev. Mod. Phys. 33, 239–249 (1961)
M. J. Leitman and G. M. C. Fisher, The linear theory of viscoelasticity, in Encyclopedia of Physics, Vol. VIa/3, C. Truesdell ed., Springer-Verlag, Berlin, 1973
W. A. Day, The Thermodynamics of Simple Materials with Fading Memory, Springer, Berlin, 1972
M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM, Philadelphia, 1992
W. A. Day, Time-reversal and the symmetry of the relaxation function of a linear viscoelastic material, Arch. Rational Mech. Anal. 40, 155–159 (1971)
M. Fabrizio and A. Morro, Viscoelastic relaxation function compatible with thermodynamics, J. Elasticity 19, 63–75 (1988)
M. Fabrizio, C. Giorgi, and A. Morro, Free energies and dissipation properties for systems with memory, Arch. Rational Mech. Anal. 125, 341–373 (1994)
M. Fabrizio, Proprietà di stabilità per sistemi con memoria, Atti Acc. Peloritana dei Pericolanti, Classe I - Sc. Fis. Mat. Nat., 68, Suppl.n.1 (1990)
G. Duvault and J. L. Lions, Inequalities in Mechanics and Physics, Springer, Berlin, 1976
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37, 297–308 (1970)
M. Fabrizio and B. Lazzari, On the existence and the asymptotic stability of solutions for linearly viscoelastic solids, Arch. Rational Mech. Anal. 116, 139–152 (1991)
C. Baiocchi, Soluzioni ordinarie e generalizzate del problema di Cauchy per equazioni differenziali astratte non lineari del secondo ordine in spazi di Hilbert, Ricerche Mat. 16, 27–95 (1967)
B. Lazzari and E. Vuk, Un teorema di esistenza e unicità per un problema dinamico in viscoelasticità lineare, Atti Sem. Mat. Fis. Univ. Modena 35, 309–317 (1987)
M. Fabrizio, An existence and uniqueness theorem in quasi-static viscoelasticity, Quart. Appl. Math. 47, 1–8 (1989)
C. Giorgi and B. Lazzari, Uniqueness and stability in linear viscoelasticity: some counterexamples, in Proceedings of the Vth International Meeting on Waves and Stability in Continuous Media, Sorrento, Italy, 9–14 October, 1989, ed. S. Rionero, World Scientific Publishing, River Edge, NJ, 1991, pp. 146-153
M. E. Gurtin and I. Herrera, On dissipation inequalities and linear viscoelasticity, Quart. Appl. Math. 23, 235–245 (1965)
M. Fabrizio and A. Morro, On uniqueness in linear viscoelasticity: a family of counterexamples, Quart. Appl. Math. 45, 263–268 (1987)
M. Fabrizio, Sull’invertibilità dell’equazione costitutiva della viscoelasticità lineare, Rend. Mat. Acc. Naz. Lincei, (9) 3, 141–148 (1992)
G. Gentili and A. Bozza, Inversion and quasi-static problems in linear viscoelasticity, Meccanica (to appear)
P. Benvenuti, Sulla teoria ereditaria delle deformazioni lineari dinamiche, Rend. Mat. 24, 255–290 (1965)
C. Giorgi, Alcune conseguenze delle restrizioni termodinamiche per mezzi viscoelastici lineari, Quad. Dip. Mat. Univ. Cattolica Brescia, 6/89 (internal report)
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© Copyright 1997
American Mathematical Society