Summary
A new class of constitutive models is derived for viscoelastic media with finite strains. The models employ the so-called fractional derivatives of tensor functions.
We introduce fractional derivatives for an objective tensor which satisfies some natural assumptions. Afterwards, we construct fractional differential analogs of the Kelvin-Voigt, Maxwell, and Maxwell-Weichert constitutive models. The models are verified by comparison with experimental data for viscoelastic solids and fluids. We consider uniaxial tension of a bar and radial oscillations of a thick-walled spherical shell made of the fractional Kelvin-Voigt incompressible material. Explicit solutions to these problems are derived and compared with experimental data for styrene butadiene rubber and synthetic rubber. It is shown that the fractional Kelvin-Voigt model provides excellent prediction of experimental data. For uniaxial tension of a bar and simple shear of an infinite layer made of the fractional Maxwell compressible material, we develop explicit solutions and compare them with experimental data for polyisobutylene specimens. It is shown that the fractional Maxwell model ensures fair agreement between experimental data and results of numerical simulation. This model allows the number of adjustable parameters to be reduced significantly compared with other models which ensure the same level of accuracy in the prediction of experimental data.
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References
Bagley, R. L., Torvik, P. J.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol.27, 201–210 (1983).
Bagley, R. L., Torvik, P. J.: On the fractional calculus model of viscoelastic behavior. J. Rheol.30, 133–155 (1986).
Friedrich, C.: Relaxation and retardation functions of the Maxwell model with fractional derivatives. Rheol. Acta30, 151–158 (1991).
Glockle, W. G., Nonnenmacher, T. F.: Fractional relaxation and the time-temperature superposition principle. Rheol. Acta33, 337–343 (1994).
Heymans, N., Bauwens, J.-C.: Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheol. Acta33, 210–219 (1994).
Koeller, R. C.: Applications of fractional calculus to the theory of viscoelasticity. J. Appl. Mech.51, 299–307 (1984).
Nonnenmacher, T. F., Glockle, W. G.: A fractional model for mechanical stress relaxation. Phil. Mag. Lett.64, 89–93 (1991).
Oldham, K. B., Spanier, J.: The fractional calculus. New York: Academic Press 1974.
Srivastava, H. M., Menocha, H. L.: A treatise of generalizing functions. New York: Wiley 1984.
Miller, K. S., Ross, B.: An introduction to the fractional calculus and fractional differential equations. New York: Wiley 1994.
Glockle, W. G., Nonnenmacher, T. F.: Fractional integral operators and Fox functions in the theory of viscoelasticity. Macromolecules24, 6426–6435 (1991).
Ngai, K. L.: Evidences of universal behavior of condensed matter at low frequences/long times. In: Non-Debay relaxation in condensed matter (Ramakrishnan, T. V., Raj Lakshmi, M., eds.) pp. 78–94. Singapore: World Scientific 1987.
Oldroyd, J. G.: On the formulation of rheological equations of state. Proc. R. Soc. London Ser.A200, 523–541 (1950).
Engler, H.: Global regular solution to the dynamic antiplane shear problem in nonlinear viscoelasticity. Math. Z.202, 251–259 (1989).
Renardy, M., Hrusa, W. J., Nohel, J. A.: Mathematical problems in viscoelasticity. New York: Longman 1987.
Leonov, A. I.: Nonequilibrium thermodynamics and rheology of viscoelastic polymeric media. Rheol. Acta15, 85–98 (1976).
White, J. L., Metzner, A. B.: Development of constitutive equations for polymeric melts and solutions. J. Appl. Polym. Sci.7, 1867–1889 (1963).
Pearson, G., Middleman, S.: Elongation flow behavior of viscoelastic liquids: modelling bubble dynamics with viscoelastic constitutive relations. Rheol. Acta17, 500–510 (1978).
Janssen, L. P. B. M., Janssen-van Rosmalen, R: An analysis of flow induced formation of long fibers. Rheol. Acta17, 578–588 (1978).
La Mantia, F. P.: Non linear viscoelasticity of polymeric liquids interpreted by means of a stress dependence of free volume. Rheol. Acta16, 302–308 (1977).
Giacomin, A. J., Jeyaseelan, R. S.: A constitutive theory for polyolefins in large amplitude oscillatory shear. Polym. Engng Sci.35, 768–777 (1995).
Bloch, R., Chang, W. V., Tschoegl, N. W.: The behavior of rubberlike materials in moderately large deformations. J. Rheol.22, 1–32 (1978).
Derman, D., Zaphir, Z., Bodner, S. R.: Nonlinear anelastic behavior of a synthetic rubber at finite strains. J. Rheol.22, 239–258 (1978).
Burton, T. A.: Stability and periodic solutions of ordinary and functional differential equations. Orlando: Academic Press 1983.
Glucklich, J., Landel, R. F.: Strain energy function of styrene butadiene rubber and the effect of temperature. J. Polym. Sci. Polym. Phys. Ed.15, 2185–2199 (1977).
Titomanlio, G., Spadaro, G., La Mantia, F. P.: Stress relaxation of a polyisobutylene under large strains. Rheol. Acta19, 477–481 (1980).
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Drozdov, A.D. Fractional differential models in finite viscoelasticity. Acta Mechanica 124, 155–180 (1997). https://doi.org/10.1007/BF01213023
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DOI: https://doi.org/10.1007/BF01213023