Abstract
Viscoelastic constitutive equations are constructed by assuming that the stress is a nonlinear function of the current strain and of a set of internal variables satisfying relaxation equations of fractional order. The dependence of the relaxation equations on the strain can also be nonlinear. The resulting constitutive equations are examined as mapping between appropriate Sobolev spaces. The proposed formulation is easier to implement numerically than history-based formulations.
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References
Christensen R.M. (1971). Theory of Viscoelasticity: An introduction. Academic Press, New York
Simo J.C., Hughes T.J.R. (1998). Computational Inelasticity. Springer, New York
Coleman B.D., Gurtin M. (1967). Thermodynamics with internal state variables. J. Chem. Phys. 47: 597–613
Saut J.C., Joseph D.D. (1983). Fading memory. Arch Rat. Mech. Anal. 81: 53–95
Kelbert M.Y., Chaban I.Y. (1986). Relaxation and propagation of pulses in fluids (Izv Ak. Nauk, ser). Mech. Fluids Gases 5: 153–160
Gripenberg G., Londen S.O., Staffans O.J. (1990). Volterra Integral and Functional Equations. Cambridge University Press, Cambridge
Kohlrausch, F.: Über die elastische Nachwirkung bei der Torsion, Poggendorfer Annalen (Annalen der Physik und Chemie Lpzg) 119, 337–568 (1863)
Friedrich C. (1991). Relaxation and retardation function of the Maxwell model with fractional derivatives. Rheol. Acta 30: 151–158
Bagley R.L., Torvik P.J. (1986). On the fractional calculus model of viscoelastic behavior. J. Rheol. 30: 133–155
Renardy M. (1982). Some remarks on the propagation and non-propagation of discontinuities in linearly viscoelastic liquids. Rheol. Acta 21: 251–254
Hanyga A., Seredyńska M. (1999). Asymptotic ray theory in poro- and viscoelastic media. Wave Motion 30: 175–195
Hanyga A., Seredyńska M. (2002). Asymptotic wavefront expansions in hereditary media with singular memory kernels. Quart. Appl. Math. LX: 213–244
Hanyga A. (2003). Well-posedness and regularity for a class of linear thermoviscoelastic materials. Proc. R. Soc. Lond. A 459: 2281–2296
Gripenberg G. (2001). Non-smoothing in a single conservation law with memory. Elect. J. Diff. Equ. 2001(08): 1–8
Gripenberg G., Londen S.-O. (1995). Fractional derivatives and smoothing in nonlinear conservation laws. Diff. Integr. Equ. 8: 1961–1976
Cockburn B., Gripenberg G., Londen S.-O. (1996). On convergence to entropy solutions of a single conservation law. J. Diff. Eqn. 128: 206–251
Chen, P.J.: Growth and Decay of Waves in Solids, vol. VIa/3 of Handbuch der Physik, pp 303–402. Springer, Berlin (1973)
Hanyga, A.: Regularity of solutions of nonlinear problems with singular memory. In: 2nd Canadian Conference on Nonlinear Solid Mechanics, Simon Frazer University, Vancouver (2002)
Enelund M., Mähler L., Runesson K., Josefson B.L. (1999). Formulation and integration of the standard linear viscoelastic solid with fractional order rate laws. Int. J. Solids Struct. 36: 2417–2442
Hanyga A. (2003). An anisotropic Cole-Cole viscoelastic model of seismic attenuation. J. Comput. Acoustics 11: 75–90
Miller K.S., Ross B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York
Podlubny I. (1998). Fractional Differential Equations. Academic Press, San Diego
Soula, M., Chevalier, Y.: La dérivee fractionnaire en rhéologie des polymères—Application aux comportements élastiques et viscoélastiques linéaires et non-linéaires des élastomères. In: ESAIM: Proceedings Fractional Differential Systems: Models, Methods and Applications, vol. 5, pp 193–204 (1998). URL http://www.emath.fr/proc/Vol5
Hanyga A. (2001). Wave propagation in media with singular memory. Math. Comput. Mech. 34: 1399–1422
Atanackovic T.M. (2001). A modified Zener model of viscoelastic body. Contin. Mech. Thermodyn. 14: 137–148
Adolfsson K., Enelund M. (2003). Fractional derivative viscoelasticity at large deformations. Nonlinear Dyn. 33: 301–321
Palade L.I., Attané P., Huilgol R.R., Mena B. (1999). Anomalous stability behavior of a properly invariant constitutive equations which generalises fractional derivative models. Int. J. Eng. Sci. 37: 315–329
Freed A., Diethelm K., Luchko Y. (2002). Fractional-order viscoelasticity (FOV): constitutive development using the fractional calculus: first annual report. Tech. Rep. NASA/TM-2002-211914, NASA
White J.L., Metzner A.B. (1963). Development of constitutive equations for polymeric melts and solutions. J. Appl. Polym. Sci. 7: 1867–1889
Wagner M.H., Laun H.M. (1978). Nonlinear shear creep and constrained elastic recovery of an LDPE melt. Rheol. Acta 17: 138–148
Larson R.G. (1988). Constitutive Laws for Polymer Melts and Solutions. Birkhäuser, Boston
Schapery R.A. (1966). An engineering theory of nonlinear viscoelasticity with applications. Int. J. Solids Struct. 2: 407–425
Schapery R.A. (1969). On the characterization of nonlinear viscoelastic materials. Polym. Eng. Sci. 9: 295–310
Valanis K.C., Landel R.F. (1967). The strain-energy function of a hyperelastic material in terms of the extension ratios. J. Appl. Phys. 38: 2997–3002
Green A.E., Rivlin R.S. (1957). The mechanics of nonlinear materials with memory. 1. Arch. Rat. Mech. Anal. 1: 1–21
Hanyga, A., Seredyńska, M.: Multiple-integral viscoelastic constitutive equations. J. Nonlinear Mech. (2006). doi:10.1016/j.ijnonlinmec.2007.02.2003
Hanyga, A.: An anisotropic Cole-Cole model of seismic attenuation. In: Shang, E.-C., Li, Q., Gao T.F. (Eds.), Theoretical and Computational Acoustics 2001, pp 319–334, World-Scientific, Singapore (2002). In: Proceedings of the 5th International Conference on Computational and Theoretical Acoustics, Beijing, 21-25 May 2001
Carcione J.M., Cavallini F. (1994). A rheological model for anelastic anisotropic media with applications to seismic wave propagation. Geophys. J. Int. 119: 338–348
Day S.M., Minster J.B. (1984). Numerical simulation of wavefields using a Padé approximant method. Geophys. J. R. Astr. Soc. 78: 105–118
Carcione J.M., Kosloff D., Kosloff R. (1998). Wave propagation simulation in a linear viscoacoustic medium. Geophys. J. R. Astr. Soc. 93: 393–407
Cole K.S., Cole R.H. (1941). Dispersion and absorption in dielectrics, I: Alternating current characteristics. J. Chem. Phys. 9: 341–351
Torvik P.J., Bagley R.L. (1983). On the appearance of the fractional derivative in the behavior of real material. J. Appl. Mech. 51: 294–298
Batzle M., Hofmann R., Han D.-H., Castagna J. (2001). Fluids and frequency dependent seismic velocity of rocks. Lead. Edge 20: 168–171
Soula M., Vinh T., Chevalier Y. (1997). Transient responses of polymers and elastomers deduced from harmonic responses. J. Sound Vibration 205: 185–203
Friedrich C., Braun H. (1992). Generalized Cole-Cole behavior and its rheological relevance. Rheol. Acta 31: 309–322
Bagley R.L., Torvik P.J. (1983). Fractional calculus–A different approach to the analysis of viscoelastically damped structures. AIAA J. 21: 741–748
Widder D.V. (1946). The Laplace Transform. Princeton University Press, Princeton
Doetsch G. (1958). Einführung in Theorie und Anwendung der Laplace Transformation. Birkhäuser Verlag, Basel
Eldred B.L., Baker W.P., Palazzotto A.N. (1995). Kelvin-Voigt versus fractional derivative model as constitutive relation for viscoelastic materials. AIAA J. 33: 547–550
Samko S.G., Kilbas A.A., Marichev O.I. (1993). Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, Amsterdam
Gorenflo, R.: Fractional Calculus: Some Numerical methods, Springer, Wien, 1997, CISM Courses and Lectures, vol. 38
Lubich C. (1986). Discretized fractional calculus. SIAM J. Math. Anal. 17: 704–719
Lubich C. (1988). Convolution quadrature and discretized fractional calculus, I. Numer. Math. 52: 129–145
Lubich C. (1988). Convolution quadrature and discretized fractional calculus, II. Numer. Math. 52: 413–425
Hanyga A., Lu J.-F. (2005). Wave field simulation for heterogeneous transversely isotropic porous media with the JKD dynamic permeability. Comput. Mech. 36: 196–208. doi:10.1007/s00466-004-0652-3
Zeidler E. (1985). Nonlinear Functional Analysis and its Applications, vol. II/B. Springer, New York
Corduneanu C. (1991). Integral Equations and Applications. Cambridge University Press, Cambridge
Diethelm K., Ford N.J. (2003). Analysis of fractional differential equations. J. Math. Anal. Appl. 265: 229–248
Diekmann O., Lunel S.M.V., Walther H.-O., Gils S.A. (1995). Delay Equations, Functional-, Complex-, and Nonlinear Analysis. Springer, New York
Bernstein B., Kearsley E.A., Zapas L.J. (1963). A study of stress relaxation with finite strains. Trans. Soc. Rheol. 7: 391–410
Laun H.M. (1978). Description of the non-linear shear behavior of a low-density polyethylene melt by means of an exerimentally determined strain dependent memory function. Rheol. Acta 17: 1–15
Staverman A.J., Schwarzl F. (1952). Thermodynamics of viscoelastic behavior (model theory). Proc. Konink. Nederlands Akad. van Wetenschapen B55: 474–485
Breuer S., Onat E.T. (1964). On recoverable work in linear viscoelasticity. ZAMP 15: 13–21
Seredyńska M., Hanyga A. (2000). Nonlinear Hamiltonian equations with fractional damping. J. Math. Phys. 41: 2135–2156
Desch W., Grimmer R. (2001). On the well-posedness of constitutive laws involving dissipation potentials. Trans. Am. Math. Soc. 353: 5095–5120
Fabrizio M., Morro A. (1992). Mathematical Problems in Linear Viscoelasticity. SIAM, Philadelphia
Diethelm, K., Freed, A.D.: The FracPECE subroutine for the numerical solution of differential equations of fractional order. pp 57–71 Gesellschaft für wissenschaftliche Dataverarbeitung, Göttingen (1999)
Diethelm, K., Freed, A.D., Ford, N.: A predictor-corrector approach to the numerical solution of fractional differential equations. Nonlinear Dyn. 22, 3–22 (2002)
Enelund M., Olsson P. (1998). Damping described by fading memory – Analysis and application to fractional derivative models. Int. J. Solids Struct. 36: 939–970
Enelund M., Lesieutre G.A. (1999). Time-domain modeling of damping using anelastic displacement fields and fractional calculus. Int. J. Solids Struct. 36: 4447–4472
Diethelm K. (1997). An algorithm for the numerical solution of differential equations of fractional order. Elect. Trans. Numer. Anal. 5: 1–6
Newmark N.M. (1959). A method of computation for structural dynamics. ASCE J. Eng. Mech. Div. 5: 67–94
Cook R.D., Malkhus D.S., Plesha M.E., Witt R.J. (2002). Concepts and Applications of Finite Element Analysis. 4th edn. Wiley, New York
Day W.A. (1970). Restrictions on the relaxation functions in linear viscoelasticity. Quart. Jl Mech. Appl. Math. 24: 487–497
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Communicated by J.-J. Marigo
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Hanyga, A. Fractional-order relaxation laws in non-linear viscoelasticity. Continuum Mech. Thermodyn. 19, 25–36 (2007). https://doi.org/10.1007/s00161-007-0042-0
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DOI: https://doi.org/10.1007/s00161-007-0042-0