Free energies in the frequency domain: the scalar case

Golden, J. M.

doi:10.1090/qam/1739041

Author: J. M. Golden
Journal: Quart. Appl. Math. 58 (2000), 127-150
MSC: Primary 74D05; Secondary 30E20, 74A20
DOI: https://doi.org/10.1090/qam/1739041
MathSciNet review: MR1739041
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A general closed expression is given for the isothermal minimum free energy of a linear viscoelastic material in terms of Fourier-transformed quantities. A one-parameter family of free energies is constructed, ranging continuously from the maximum to the minimum free energies.

A. J. Staverman and F. Schwarzl, Thermodynamics of viscoelastic behaviour, Proc. Konink. Nederl. Akad. Wettensch. B55, 474-485 (1952)

D. R. Bland, The theory of linear viscoelasticity, International Series of Monographs on Pure and Applied Mathematics, Vol. 10, Pergamon Press, New York-London-Oxford-Paris, 1960. MR 0110314
S. C. Hunter, Tentative equations for the propagation of stress, strain and temperature fields in viscoelastic solids, J. Mech. Phys. Solids 9 (1961), 39–51. MR 165780, DOI https://doi.org/10.1016/0022-5096%2861%2990037-0
Shlomo Breuer and E. Turan Onat, On recoverable work in linear viscoelasticity, Z. Angew. Math. Phys. 15 (1964), 12–21 (English, with German summary). MR 178644, DOI https://doi.org/10.1007/BF01602660

S. Breuer and E. T. Onat, On the determination of free energy in viscoelastic solids, Z. Angew. Math. Phys. 15, 185-191 (1964)

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, A general theory of dissipation in materials with memory, Arch. Rational Mech. Anal. 27 (1967), 255–274. MR 220474, DOI https://doi.org/10.1007/BF00281714
Bernard D. Coleman and David R. Owen, A mathematical foundation for thermodynamics, Arch. Rational Mech. Anal. 54 (1974), 1–104. MR 395502, DOI https://doi.org/10.1007/BF00251256
Bernard D. Coleman and David R. Owen, On thermodynamics and elastic-plastic materials, Arch. Rational Mech. Anal. 59 (1975), no. 1, 25–51. MR 381526, DOI https://doi.org/10.1007/BF00281515
W. A. Day, Thermodynamics based on a work axiom, Arch. Rational Mech. Anal. 31 (1968), no. 1, 1–34. MR 1553518, DOI https://doi.org/10.1007/BF00251512
W. A. Day, Some results on the least work needed to produce a given strain in a given time in a viscoelastic material and a uniqueness theorem for dynamic viscoelasticity, Quart. J. Mech. Appl. Math. 23 (1970), 469–479. MR 273881, DOI https://doi.org/10.1093/qjmam/23.4.469

W. A. Day, The thermodynamics of materials with memory, in Materials with Memory, ed. D. Graffi, Liguori, Naples, 1979

M. Fabrizio and A. Morro, Thermodynamic restrictions on 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 19 (1988), no. 1, 63–75. MR 928727, DOI https://doi.org/10.1007/BF00041695
Dario Graffi and Mauro Fabrizio, On the notion of state for viscoelastic materials of “rate” type, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 83 (1989), 201–208 (1990) (Italian, with English summary). MR 1142459

D. Graffi and M. Fabrizio, Non unicità dell’energia libera per materiali viscoelastici, Atti Accad. Naz. Lincei 83, 209-214 (1990)

A. Morro and M. Vianello, Minimal and maximal free energy for materials with memory, Boll. Un. Mat. Ital. A (7) 4 (1990), no. 1, 45–55 (English, with Italian summary). MR 1047512
Mauro Fabrizio and Angelo Morro, Mathematical problems in linear viscoelasticity, SIAM Studies in Applied Mathematics, vol. 12, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1153021
Mauro Fabrizio, Claudio Giorgi, and Angelo Morro, Free energies and dissipation properties for systems with memory, Arch. Rational Mech. Anal. 125 (1994), no. 4, 341–373. MR 1253168, DOI https://doi.org/10.1007/BF00375062
M. Fabrizio, Existence and uniqueness results for viscoelastic materials, Crack and contact problems for viscoelastic bodies (Udine, 1994) CISM Courses and Lect., vol. 356, Springer, Vienna, 1995, pp. 53–102. MR 1411685
M. Fabrizio, C. Giorgi, and A. Morro, Internal dissipation, relaxation property, and free energy in materials with fading memory, J. Elasticity 40 (1995), no. 2, 107–122. MR 1364749, DOI https://doi.org/10.1007/BF00042457
A. Morro, Wave solutions in linear viscoelastic materials, Crack and contact problems for viscoelastic bodies (Udine, 1994) CISM Courses and Lect., vol. 356, Springer, Vienna, 1995, pp. 227–258. MR 1411687
Gianpietro Del Piero and Luca Deseri, On the concepts of state and free energy in linear viscoelasticity, Arch. Rational Mech. Anal. 138 (1997), no. 1, 1–35. MR 1463802, DOI https://doi.org/10.1007/s002050050035
Gianpietro Del Piero and Luca Deseri, On the analytic expression of the free energy in linear viscoelasticity, J. Elasticity 43 (1996), no. 3, 247–278. MR 1415545, DOI https://doi.org/10.1007/BF00042503
Morton E. Gurtin and William J. Hrusa, On energies for nonlinear viscoelastic materials of single-integral type, Quart. Appl. Math. 46 (1988), no. 2, 381–392. MR 950610, DOI https://doi.org/10.1090/S0033-569X-1988-0950610-9
J. M. Golden and G. A. C. Graham, Boundary value problems in linear viscoelasticity, Springer-Verlag, Berlin, 1988. MR 958684
N. I. Muskhelishvili, Singular integral equations, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR 0355494
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

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon, Oxford, 1937

I. N. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York, 1972

Philip M. Morse and Herman Feshbach, Methods of theoretical physics. 2 volumes, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. MR 0059774