On uniqueness in linear viscoelasticity: a family of counterexamples
Authors:
A. Morro and M. Fabrizio
Journal:
Quart. Appl. Math. 45 (1987), 321-325
MSC:
Primary 73F05
DOI:
https://doi.org/10.1090/qam/895102
MathSciNet review:
895102
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Abstract: Within the framework of linear viscoelasticity a two-parameter family of relaxation functions are exhibited which allow nonuniqueness of the infinitesimal strain, for a given stress, in the space of sinusoidal-in-time strain histories. These relaxation functions are shown to be compatible with thermodynamics and to have a positive equilibrium elastic modulus.
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G. Capriz and M. E. Gurtin, letter to G. Fichera referred to in [4]
G. Fichera, Sul principio della memoria evanescente, Rend. Sem. Mat. Univ. Padova 68, 245–259 (1982)
M. Fabrizio and A. Morro, Thermodynamic restrictions on the relaxation functions in linear viscoelasticity, Mech. Res. Comm. 12, 101–105 (1985)
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R. M. Christensen, Theory of viscoelasticity, Academic Press, 1971
A. C. Pipkin, Lectures on viscoelasticity theory, Springer, 1972
D. Graffi, Sui problemi della ereditarietà lineare, Nuovo Cimento 5, 53–71 (1928)
M. J. Leitman and G. M. C. Fisher, The linear theory of viscoelasticity, Encyclopedia of physics VIa/3. Springer, 1973
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E. Fermi, Thermodynamics, Dover, 1956; §11
M. W. Zemanski, Heat and thermodynamics, McGraw-Hill, 1957; §10.9
- 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
G. Fichera, Avere una memoria tenace crea gravi problemi, Arch. Rat. Mech. Anal. 70, 101–112 (1979)
D. Graffi, On the fading memory, Applicable Analysis 15, 295–311 (1983)
G. Capriz and M. E. Gurtin, letter to G. Fichera referred to in [4]
G. Fichera, Sul principio della memoria evanescente, Rend. Sem. Mat. Univ. Padova 68, 245–259 (1982)
M. Fabrizio and A. Morro, Thermodynamic restrictions on the relaxation functions in linear viscoelasticity, Mech. Res. Comm. 12, 101–105 (1985)
M. Fabrizio and A. Morro, Viscoelastic relaxation functions compatible with thermodynamics, J. Elasticity, to appear
G. Fichera, On linear viscoelasticity, Mech. Res. Comm. 12, 241–242 (1985)
R. M. Christensen, Theory of viscoelasticity, Academic Press, 1971
A. C. Pipkin, Lectures on viscoelasticity theory, Springer, 1972
D. Graffi, Sui problemi della ereditarietà lineare, Nuovo Cimento 5, 53–71 (1928)
M. J. Leitman and G. M. C. Fisher, The linear theory of viscoelasticity, Encyclopedia of physics VIa/3. Springer, 1973
B. D. Coleman and D. R. Owen, A mathematical foundation for thermodynamics, Arch. Rat. Mech. Anal. 54, 1–104 (1974)
B. D. Coleman, On thermodynamics, strain impulses and viscoelasticity, Arch. Rat. Mech. Anal. 17, 230–254 (1964)
R. M. Bowen and P. J. Chen, Thermodynamic restriction on the initial slope of the stress-relaxation function, Arch. Rat. Mech. Anal. 51, 278–284 (1973)
E. Fermi, Thermodynamics, Dover, 1956; §11
M. W. Zemanski, Heat and thermodynamics, McGraw-Hill, 1957; §10.9
M. Fabrizio, An existence and uniqueness theorem in quasi-static viscoelasticity, to appear
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Article copyright:
© Copyright 1987
American Mathematical Society