Monotonic, completely monotonic, and exponential relaxation functions in linear viscoelasticity
Authors:
Gianpietro Del Piero and Luca Deseri
Journal:
Quart. Appl. Math. 53 (1995), 273-300
MSC:
Primary 73F15; Secondary 73F05
DOI:
https://doi.org/10.1090/qam/1330653
MathSciNet review:
MR1330653
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Abstract: A priori restrictions on the relaxation function of linear viscoelasticity are studied under regularity assumptions weaker than those usually made in the literature. The new set of assumptions is sufficient to define, by a limit procedure, the work done in deformation processes in which some parts are subject either to extreme retardations or to extreme accelerations. The use of such processes results in a considerable simplification of the proofs of some classical results. Under the same assumptions, we give a characterization of the monotonicity of the relaxation function in terms of work. We also extend an earlier one-dimensional characterization of complete monotonicity due to Day, and prove that the work done in every closed path in stress-strain space is nonnegative if and only if the relaxation function is of exponential type.
B. D. Coleman, Thermodynamics of materials with memory, Arch. Rat. Mech. Anal. 17, 1–45 (1964)
B. D. Coleman, On thermodynamics, strain impulses and viscoelasticity, Arch. Rat. Mech. Anal. 17, 230–254 (1964)
W. A. Day, On monolonicity of the relaxation functions of viscoelastic materials, Proc. Cambridge Phil. Soc. 67, 503–508 (1970)
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-Verlag, 1972
G. Gentili, Alcune proprietà per la funzione di rilassamento in viscoelasticità lineare, Riv. Mat. Univ. Parma 14, 121–133 (1988)
M. E. Gurtin and I. Herrera, On dissipation inequalities and linear viscoelasticity, Quart. Appl. Math. 23, 235–245 (1965)
P. R. Halmos, Finite-Dimensional Vector Spaces, Van Nostrand, 1958
M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, 1974
R. L. Jeffery, The Theory of Functions of a Real Variable, The University of Toronto Press, 1951
A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Prentice Hall, 1970
H. König and J. Meixner, Lineare Systeme und lineare Transformationen, Math. Nachr. 19, 256–322 (1958)
M. J. Leitman and G. M. C. Fischer, The Linear Theory of Viscoelasticity, Handbuch der Physik, Vol. VIa/3, Springer-Verlag, 1973
D. V. Widder, The Laplace Transform, Princeton University Press, 1941
B. D. Coleman, Thermodynamics of materials with memory, Arch. Rat. Mech. Anal. 17, 1–45 (1964)
B. D. Coleman, On thermodynamics, strain impulses and viscoelasticity, Arch. Rat. Mech. Anal. 17, 230–254 (1964)
W. A. Day, On monolonicity of the relaxation functions of viscoelastic materials, Proc. Cambridge Phil. Soc. 67, 503–508 (1970)
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-Verlag, 1972
G. Gentili, Alcune proprietà per la funzione di rilassamento in viscoelasticità lineare, Riv. Mat. Univ. Parma 14, 121–133 (1988)
M. E. Gurtin and I. Herrera, On dissipation inequalities and linear viscoelasticity, Quart. Appl. Math. 23, 235–245 (1965)
P. R. Halmos, Finite-Dimensional Vector Spaces, Van Nostrand, 1958
M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, 1974
R. L. Jeffery, The Theory of Functions of a Real Variable, The University of Toronto Press, 1951
A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Prentice Hall, 1970
H. König and J. Meixner, Lineare Systeme und lineare Transformationen, Math. Nachr. 19, 256–322 (1958)
M. J. Leitman and G. M. C. Fischer, The Linear Theory of Viscoelasticity, Handbuch der Physik, Vol. VIa/3, Springer-Verlag, 1973
D. V. Widder, The Laplace Transform, Princeton University Press, 1941
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© Copyright 1995
American Mathematical Society
