Some quasi-static problems with elastic and viscous boundary conditions in linear viscoelasticity
Authors:
Carlo Alberto Bosello and Giorgio Gentili
Journal:
Quart. Appl. Math. 54 (1996), 687-696
MSC:
Primary 73F15; Secondary 35Q72, 49J45
DOI:
https://doi.org/10.1090/qam/1417232
MathSciNet review:
MR1417232
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Abstract: We study the quasi-static behaviour of a linearly viscoelastic body which is subject to boundary forces respectively of elastic type and of viscous type. The ensuing problems exhibit dynamic boundary conditions. We impose on the memory kernel only those restrictions deriving from thermodynamics and, making use of the Fourier transform method, we show existence and uniqueness of the solution to each problem.
W. A. Day, The thermodynamics of simple materials with fading memory, Springer Tracts in Natural Philosophy, vol. 22, Springer-Verlag, New York, 1972
M. Fabrizio, An existence and uniqueness theorem in quasi-static viscoelasticity, Quart. Appl. Math. 47, 1–8 (1989)
M. Fabrizio and A. Morro, Viscoelastic relaxation function compatible with thermodynamics, J. Elasticity 19, 63–75 (1988)
M. Fabrizio and A. Morro, On uniqueness in linear viscoelasticity: a family of counterexamples, Quart. Appl. Math. 45, 321–325 (1987)
M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM Studies in Applied Mathematics, vol. 12, Philadelphia, 1992
G. Gentili, Alcune proprietà per la funzione di rilassamento in viscoelasticità lineare, Riv. Mat. Univ. Parma (4) 14, 121–133 (1988)
D. Graffi, Sui problemi della ereditarietà lineare, Nuovo Cimento 5, 53–71 (1928)
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985
J. L. Lions, Equations Differentielles Operationnelles et Problèmes aux Limites, Springer-Verlag, Berlin-Gottingen-Heidelberg, 1961
A. C. Pipkin, Lectures on Viscoelasticity Theory, Springer, Berlin, 1972
R. E. Showalter, Hilbert Space Methods for Partial Differential Equations, Pitman, London and San Francisco, 1977
F. Treves, Basic Linear Partial Differential Equations, Academic Press, New York, 1975
G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, Plenum Press, New York, London, 1987
W. A. Day, The thermodynamics of simple materials with fading memory, Springer Tracts in Natural Philosophy, vol. 22, Springer-Verlag, New York, 1972
M. Fabrizio, An existence and uniqueness theorem in quasi-static viscoelasticity, Quart. Appl. Math. 47, 1–8 (1989)
M. Fabrizio and A. Morro, Viscoelastic relaxation function compatible with thermodynamics, J. Elasticity 19, 63–75 (1988)
M. Fabrizio and A. Morro, On uniqueness in linear viscoelasticity: a family of counterexamples, Quart. Appl. Math. 45, 321–325 (1987)
M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM Studies in Applied Mathematics, vol. 12, Philadelphia, 1992
G. Gentili, Alcune proprietà per la funzione di rilassamento in viscoelasticità lineare, Riv. Mat. Univ. Parma (4) 14, 121–133 (1988)
D. Graffi, Sui problemi della ereditarietà lineare, Nuovo Cimento 5, 53–71 (1928)
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985
J. L. Lions, Equations Differentielles Operationnelles et Problèmes aux Limites, Springer-Verlag, Berlin-Gottingen-Heidelberg, 1961
A. C. Pipkin, Lectures on Viscoelasticity Theory, Springer, Berlin, 1972
R. E. Showalter, Hilbert Space Methods for Partial Differential Equations, Pitman, London and San Francisco, 1977
F. Treves, Basic Linear Partial Differential Equations, Academic Press, New York, 1975
G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, Plenum Press, New York, London, 1987
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Article copyright:
© Copyright 1996
American Mathematical Society
