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Determining the relaxation tensor in linear viscoelasticity of integral type

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An Erratum to this article was published on 01 February 1996

Abstract

The dynamic equation of linear viscoelasticity of integral type is considered. This equation contains a fourth-order relaxation tensor which only depends on time and accounts for the viscoelastic effects. We show that this tensor is identified by suitable boundary stresses, provided that the displacement vector field solves a given Cauchy-Dirichlet problem for the viscoelastodynamics equation.

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References

  1. C. Baiocchi, Sulle equazioni differenziali astratte lineari del primo e del secondo ordine negli spazi di Hilbert. Ann. Mat. Pura Appl.,76 (1967), 233–304.

    Article  MATH  MathSciNet  Google Scholar 

  2. H. T. Banks, R. H. Fabiano, and Y. Wang, Inverse problem techniques for beams with tip body and time hysteresis damping. Mat. Apl. Comput.,8 (1989), 101–118.

    MATH  MathSciNet  Google Scholar 

  3. A. Ben-Menahem and S. J. Singh, Seismic Waves and Sources. Springer-Verlag, New York, 1981.

    MATH  Google Scholar 

  4. D. W. Brewer and R. K. Powers, Parameter estimation for a Volterra integro-differential equation. Appl. Numer. Math.,9 (1992), 307–320.

    Article  MATH  MathSciNet  Google Scholar 

  5. C. Cavaterra and M. Grasselli, An inverse problem for the linear viscoelastic Kirchhoff plate. Quart. Appl. Math., to appear.

  6. C. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity. J. Differential Equation,7 (1970), 554–569.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity. SIAM Stud. Appl. Math., vol.12, SIAM, Philadelphia, 1992.

    MATH  Google Scholar 

  8. W. N. Findley, J. S. Lai, and K. Onaran, Creep and Relaxation of Nonlinear Viscoelastic Materials. Series in Appl. Math and Mech., vol.18, North-Holland, Amsterdam, 1976.

    MATH  Google Scholar 

  9. J. M. Golden and G. A. C. Graham, Boundary Value Problems in Linear Viscoelasticity. Springer-Verlag, Berlin-Heidelberg, 1988.

    MATH  Google Scholar 

  10. M. Grasselli, S. I. Kabanikhin, and A. Lorenzi, An inverse hyperbolic integrodifferential problem arising in Geophysics I. Siberian Math. J.33 (1992), 415–426.

    Article  MathSciNet  Google Scholar 

  11. M. Grasselli, S. I. Kabanikhin, and A. Lorenzi, An inverse hyperbolic integrodifferential problem arising in Geophysics II. Nonlinear Anal. TMA,15 (1990), 283–298.

    Article  MATH  MathSciNet  Google Scholar 

  12. M. Grasselli, An identification problem for an abstract linear hyperbolic integrodifferential equation with applications. J. Math. Anal. Appl.,171 (1992), 27–60.

    Article  MATH  MathSciNet  Google Scholar 

  13. M. Grasselli, An inverse problem in three-dimensional linear thermoviscoelasticity of Boltzmann type. Ill-posed Problems in Natural Sciences (ed. A. N. Tikhonov), VSP, Zeist, 1992, 284–299.

    Google Scholar 

  14. M. Grasselli, On an inverse problem for a linear hyperbolic integrodifferential equation. Forum Math., in press.

  15. M. L. Heard, A class of hyperbolic Volterra integrodifferential equations. Nonlinear Anal. TMA,8 (1984), 79–93.

    Article  MATH  MathSciNet  Google Scholar 

  16. W. J. Hrusa, J. A. Nohel, and M. Renardy, Mathematical Problems in Viscoelasticity. Pitman Monographs and Surveys in Pure and Applied Math., vol.35, Longman, Harlow, 1987.

    MATH  Google Scholar 

  17. T. J. R. Hughes and J. Marsden, Mathematical Foundations of Elasticity. Prentice-Hall, Englewood Cliffs, 1983.

    MATH  Google Scholar 

  18. J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, I. Springer-Verlag, Berlin, 1972.

    Google Scholar 

  19. J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1. RMA8, Masson, Paris, 1988.

    MATH  Google Scholar 

  20. A. Lorenzi, An identification problem related to a nonlinear hyperbolic integrodifferential equation. Nonlinear Anal. TMA, to appear.

  21. A. E. Osokin and Iu. V. Suvorova, Nonlinear governing equation of a hereditary medium and methodology of determining its parameters. J. Appl. Math. Mech.,42 (1978), 1214–1222.

    Article  MATH  MathSciNet  Google Scholar 

  22. A. C. Pipkin, Lectures on Viscoelasticity. Appl. Math. Sci., vol.7, Springer-Verlag, New York, 1972.

    MATH  Google Scholar 

  23. Yu. N. Rabotnov, Elements of Hereditary Solid Mechanics. Mir Publishers, Moscow, 1980.

    MATH  Google Scholar 

  24. J. L. Sackman, Prediction and identification in viscoelastic wave propagation. Wave Propagation in Viscoelastic Media (ed. F. Mainardi), Pitman Res. Notes in Math., vol.52, Longman, Harlow, 1987, 218–234.

    Google Scholar 

  25. T. A. Tobias and Yu. K. Engelbrecht, Inverse problems for evolution equations of the integrodifferential type. J. Appl. Math. Mech.,49 (1985), 519–524.

    Article  MATH  MathSciNet  Google Scholar 

  26. N. W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behavior. Springer-Verlag, Heidelberg, 1989.

    MATH  Google Scholar 

  27. L. v. Wolfersdorf and L. Rost, On mathematical modelling of absorption and dispersion of seismic waves for low frequencies. Math. Methods Appl. Sci.,14 (1991), 563–572.

    Article  MATH  MathSciNet  Google Scholar 

  28. L. v. Wolfersdorf, On an inverse problem for acoustic and seismic waves. Appl. Anal.,45 (1992), 309–319.

    Article  MATH  MathSciNet  Google Scholar 

  29. L. v. Wolfersdorf, On identification of memory kernels in linear viscoelasticity. Math. Nachr.,161 (1993), 203–217.

    Article  MATH  MathSciNet  Google Scholar 

  30. J. Janno, On an inverse problem for a hyperbolic equation. Uchen. Zap. Tartu Gos. Univ.,672 (1984), 40–46 (in Russian).

    MathSciNet  Google Scholar 

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An erratum to this article is available at http://dx.doi.org/10.1007/BF03167305.

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Grasselli, M. Determining the relaxation tensor in linear viscoelasticity of integral type. Japan J. Indust. Appl. Math. 11, 131–153 (1994). https://doi.org/10.1007/BF03167218

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  • DOI: https://doi.org/10.1007/BF03167218

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