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An Approach to Experimental Computation of an Anisotropic Viscoelastic Plate Stiffnesses

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Abstract

The viscoelastic behavior of anisotropic composite is studied in this paper. Constitutive relations and equilibrium equations are derived for a Kirchhoff plate using general linear viscoelasticity constitutive relations for the anisotropic case. The derived model parameters—plate stiffnesses—are experimental functions. An approach to these parameters identification is given for certain cases of material properties.

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REFERENCES

  1. V. I. Gorbachev and T. M. Mel’nik, ‘‘Formulation of problems in the Bernoulli-Euler theory of anisotropic inhomogeneous beams,’’ Mosc. Univ. Mech. Bull. 73, 18–26 (2018).

    Article  Google Scholar 

  2. V. I. Gorbachev and O. B. Moskalenko, ‘‘Stability of a straight bar of variable rigidity,’’ Mech. Solids 46, 645–655 (2011).

    Article  Google Scholar 

  3. V. I. Gorbachev and L. L. Firsov, ‘‘New statement of the elasticity problem for a layer,’’ Mech. Solids 46, 89–95 (2011).

    Article  Google Scholar 

  4. A. F. Arkhangelskii and V. I. Gorbachev, ‘‘Effective characteristics of corrugated plates,’’ Mech. Solids 42, 447–462 (2007).

    Article  Google Scholar 

  5. V. I. Gorbachev and L. A. Kabanova, ‘‘Formulation of problems in the general Kirchhoff–Love theory of inhomogeneous anisotropic plates,’’ Mosc. Univ. Mech. Bull. 73 (3), 60–66 (2018).

    Article  Google Scholar 

  6. V. I. Gorbachev, ‘‘The engineering theory of the deforming of the nonuniform plates from composite materials,’’ Mekh. Kompos. Mater. Konstruk. 22, 585–601 (2016).

    Google Scholar 

  7. I. N. Vekua, General Methods of Constructing Different Versions of Shell Theory (Nauka, Moscow, 1982) [in Russian].

    Google Scholar 

  8. N. A. Kil’chevskii, Fundamentals of the Analytical Mechanics of Shells (Akad. Nauk UkrSSR, Kiev, 1963) [in Russian].

  9. L. A. Agolovyari, Asymptotic Theory of Anisotropic Plates and Shells (Nauka, Moscow, 1997) [in Russian].

    Google Scholar 

  10. I. I. Vorovich, ‘‘Some results and problems of asymptotic theory of plates and shells,’’ in Proceedings of the 1st All-Union School on Theory and Numerical Methods for Shells and Plates, Gegechkori, Georgia, October 1–10, 1974 (Tbilisi Univ., Tbilisi, 1975), pp. 51–149.

  11. A. L. Goldenveiser, Theory of Elastic Thin Shells (Nauka, Moscow, 1976; Pergamon, Oxford, 1961).

  12. S. A. Ambartsumyan, General Theory of Anisotropic Shells (Nauka, Moscow, 1974) [in Russian].

    Google Scholar 

  13. X. He et al., ‘‘Application of the Kirchhoff hypothesis to bending thin plates with different moduli in tension and compression,’’ J. Mech. Mater. Struct. 5, 755–769 (2010).

    Article  Google Scholar 

  14. V. I. Andreev, B. M. Yazyev, and A. S. Chepurnenko, ‘‘On the bending of a thin plate at nonlinear creep,’’ Adv. Mater. Res. 900, 707–710 (2014).

    Article  Google Scholar 

  15. A. Constantinescu, ‘‘On the identification of elastic moduli in plates,’’ in Inverse Problems in Engineering Mechanics, Ed. by M. Tanaka and G. S. Dulikravich (Elsevier Science, Amsterdam, 1998), pp. 205–214.

    Google Scholar 

  16. H. Altenbach and V. Eremeyev, ‘‘Analysis of the viscoelastic behavior of plates made of functionally graded materials,’’ Z. Angew. Math. Mech. 88, 332–341 (2008).

    Article  MathSciNet  Google Scholar 

  17. S. W. White, S. K. Kim, A. K. Bajaj, et al., ‘‘Experimental techniques and identification of nonlinear and viscoelastic properties of flexible polyurethane foam,’’ Nonlin. Dyn. 22, 281–313 (2000).

    Article  Google Scholar 

  18. M. Sellier, C. E. Hann, and N. Siedow, ‘‘Identification of relaxation functions in glass by mean of a simple experiment,’’ J. Am. Ceram. Soc. 90, 2980–2983 (2007).

    Article  Google Scholar 

  19. R. Lakes, Viscoelastic Materials (Cambridge Univ. Press, Cambridge, 2009)

    Book  Google Scholar 

  20. R. A. Kayumov, S. A. Lukankin, V. N. Paimushin, and S. A. Kholmogorov, ‘‘Identification of mechanical properties of fiber-reinforced composites,’’ Uch. Zap. Kazan. Univ., Ser. Fiz.-Mat. Nauki 157 (4), 112–132 (2015).

    Google Scholar 

  21. A. A. Il’yushin and B. E. Pobedrya, Fundamentals of the Mathematical Theory of Thermal Viscoelasticity (Nauka, Moscow, 1970) [in Russian].

    Google Scholar 

  22. V. V. Alexandrov, V. G. Boltianskii, S. S. Lemak, N. A. Parusnikov, and V. M. Tikhomirov, Optimal Motion Control (Fizmatlit, Moscow, 2005) [in Russian].

    Google Scholar 

  23. I. Markovsky and S. Van Huffel, ‘‘Overview of total least squares methods,’’ Signal Process. 87, 2283–2302 (2007).

    Article  Google Scholar 

  24. C. L. Lawson and R. J. Hanson, Solving Least Squares Problems, Classics in Applied Mathematics (Soc. Ind. Appl. Math., Philadelphia, 1995).

    Book  Google Scholar 

  25. G. H. Golub and Ch. F. Van Loan, Matrix Computations (John Hopkins Univ. Press, Baltimore, 1996).

    MATH  Google Scholar 

  26. M. E. Wall, A. Rechtsteiner, and L. M. Rocha, ‘‘Singular value decomposition and principal component analysis,’’ in A Practical Approach to Microarray Data Analysis, Ed. by D. P. Berrar, W. Dubitzky, and M. Granzow (Springer, Boston, MA, 2003).

    Google Scholar 

  27. P. S. Shcherbakov, ‘‘Use of prior data in updating the parameter estimates,’’ Autom. Remote Control 49, 613–620 (1988).

    Google Scholar 

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Correspondence to L. A. Kabanova.

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(Submitted by A. V. Lapin)

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Kabanova, L.A. An Approach to Experimental Computation of an Anisotropic Viscoelastic Plate Stiffnesses. Lobachevskii J Math 41, 2010–2017 (2020). https://doi.org/10.1134/S1995080220100091

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  • DOI: https://doi.org/10.1134/S1995080220100091

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