Thermomechanically consistent formulations of the standard linear solid using fractional derivatives

• Autorzy: Lion A.
• Języki publikacji: EN

Abstrakty

EN

We study the thermomechanical properties of a frequently used fractional generalisation of the standard linear solid. Its mathematical structure arises from an ordinary linear differential equation between stress and strain when replacing the first order time rates by fractional derivatives of the order .... If the parameters ... and ... are not further restricted, the model leads to an unphysical behaviour. In the case of harmonic deformations the dissipation modulus can become negative. This corresponds to a negative entropy production and violates the second law of thermodynamics. Then we propose two generalisations of the standard linear solid which are based on a so-called thermodynamically consistent fractional rheological element. It possesses a non-negative free energy and rate of dissipation for arbitrary deformation processes and is compatible with the second law of thermodynamics. The differential equations between stress and strain of the proposed generalisations contain also fractional derivatives of different orders but both the dynamic moduli and the relaxation spectra are non-negative functions of their arguments. No restrictions on the material parameters are required.

Słowa kluczowe

EN

thermomechanical properties; linear solid; second law of thermodynamics; thermodynamically consistent fractional rheological element

PL

własności termomechaniczne; liniowe ciała stałe; drugie prawo termodynamiki

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Archives of Mechanics
• Rocznik: 2001
• Tom: Vol. 53, nr 3
• Strony: 253--273
• Opis fizyczny: Bibliogr. 21 poz.

Twórcy

autor

Lion A.

• Institute of Mechanics, University of Kassel, D-34109 Kassel, Mönchebergstraße 7

Bibliografia

• 1. R. L. BAGLEY, P. J. TORVIK, Fractional calculus- a different approach to the analysis of viscoelastically damped structures, J. AIAA, 21, 741-748, 1983.
• 2. R. L. BAGLEY, P. J. TORVIK, On the fractional calculus of viscoelastic behaviour, J. of Rheology, 30, 133-155, 1986.
• 3. M. CAPUTO, F. MAINARDI, Linear models in anelastic solids, Riv. II Nuovo Cimento (Serie. II), 1, 161-198, 1971.
• 4. C. FRIEDRICH, Relaxation functions of rheological constitutive equations with fractional derivatives: thermodynamical constraints, Rheological Modelling: Thermodynamical and Statistical Approaches, 321-330, Springer Verlag, Berlin, Heidelberg 1991.
• 5. B. GROSS, Mathematical structure of the theories of linear viscoelasticity, Hermann Ltd., Paris 1968.
• 6. S. HARTMANN, A. LION, P. HAUPT, Zur Modellierung von Kriecheigensehaften von Beton, Zeitschr. Angew. Math. Mech., 78, 457-458, 1997.
• 7. P. HAUPT, A. LION, E. BACKHAUS, On the dynamic behaviour of polymers during finite strains: constitutive modelling and identification of parameters, Int. J. Solids Struct., 37, 3633-3646, 2000.
• 8. P. HAUPT, Continuum mechanics and theory of materials, Springer Ltd., 2000.
• 9. N. HEYMANS, J. C. BAUWENS, Fractal rheological models and fractional differential equations for viscoelastic behaviour, Rheologica Acta, 33, 210-219, 1994.
• 10. R. C. KOELLER, Applications of fractional calculus to the theory of viscoelasticity, J. of Applied Mechanics, 51, 299-307, 1984.
• 11. A. LION, On the thermodynamics of fractional damping elements, Continuum Mech. Thermodyn., 9, 83-96, 1997.
• 12. A. LION, Thixotropic behaviour of rubber under dynamic loading histories: experiments and theory, J. Mech. Phys. Solids., 46, 895-930, 1998.
• 13. A. LION, Strain-dependent dynamic properties of filled rubber: a nonlinear viscoelastic approach based on structural wariables, Rubber Chem. Technol., 72, 410-429, 1999.
• 14. A. LION, Thermomechanik von Elastomeren: Experimente und Materialtheorie, Habilitation Thesis, University of Kassel, Department of Mechanical Engineering, Institute of Mechanics, Germany 2000.
• 15. F. MAINARDI, E. BONETTI, The application of real order derivatives in linear viscoelasticity, Rheologica Acta, 26, 64-67, 1988.
• 16. R. METZELER, W. SCHICK, H. G. KILIAN, T. F. NONNEMACHER, Relaxation in filled polymers: a fractional calculus approach, J. Chem. Phys., 103, 7180-7186, 1995.
• 17. T. F. NONNENMACHER, Fractional relaxation equations for viscoelasticity and related phenomena, Rheological Modelling: Thermodynamical and Statistical Approaches, 309-320, Springer Verlag, 1991.
• 18. K. B. OLDHAM, J. SPANIER, The fractional calculus: theory and applications of differentiation and integration to arbitrary order, Academic Press London 1974.
• 19. A. SCHMIDT, S. OEXL, L. GAUL, Modellierung des viskoelastischen Werkstoffverhaltens von Kunststoffen mit fraktionalen Zeitableitungen, Proceedings of the 18th CAD-FEM Users Meeting, International Congress on FEM Technology, 2000.
• 20. N. W. TSCHOEGL, The phenomenological theory of linear viscoelastic behaviour, Springer Ltd., 1989.
• 21. J. D. FERRY, Viscoelastic properties of polymers, John Wiley and Sons.