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On Constraints for Displacement Gradients in Elastic Materials

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The classical, quite abstract constraint |u k,i | < 1 for elastic materials and a number of possible mathematical and physical constraints for displacement gradients are discussed Keywords: elastic material, nonlinear elasticity, constraints, displacement gradient

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References

  1. V. L. Bogdanov, A. N. Guz, and V. M. Nazarenko, “Spatial problems of the fracture of materials loaded along cracks (review),” Int. Appl. Mech., 51, No. 5, 489–560 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. V. H. Carneiro, J. Meireles, and H. Puga, “Auxetic materials—A review,” Materials Science-Poland, 31, No. 4, 561–571 (2013).

    Article  ADS  Google Scholar 

  3. C. Cattani and J. J. Rushchitsky, Wavelet and Wave Analysis as applied to Materials with Micro or Nanostructures, World Scientific, Singapore–London (2007).

    MATH  Google Scholar 

  4. K. K. Dudek, D. Attard, R. Caruana-Gauci, K. W. Wojciechowski, and J. N. Grima, “Unimode metamaterials exhibiting negative linear compressibility and negative thermal expansion,” Smart Mater. Struct., 25, No. 2, 025009 (2016).

  5. Y. B. Fu and R. W. Ogden (eds.), Nonlinear Elasticity: Theory and Applications, London Mathematical Society Lecture Note Series, 283, Cambridge University Press, Cambridge (2001).

    Google Scholar 

  6. P. Germain, Cours de Mechanique des Milieux Continua, Vol. 1. Theorie Generale, Masson et Cie Editeurs, Paris (1973).

  7. Z. A. Goldberg, “On interaction of plane longitudinal and transverse waves,” Akust. Zh., 6, No. 2, 307–310 (1960).

    Google Scholar 

  8. I. I. Goldenblatt, Nonlinear Problems of the Theory of Elasticity [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  9. A. E. Green and J. E. Adkins, Large Elastic Deformations and Nonlinear Continuum Mechanics, Oxford University Press, Clarendon Press, London (1960).

    MATH  Google Scholar 

  10. A. N. Guz, Elastic Waves in Bodies with Initial Stresses, Vol. 1–2, Naukova Dumka, Kyiv (1986).

    Google Scholar 

  11. A. N. Guz, Fundamentals of the Three-Dimensional Theory of Stability of Deformable Bodies, Ser.: Foundations of Engineering Mechanics, Springer, Berlin (1999).

  12. A. N. Guz, “Ultrasonic nondestructive method for stress analysis of structural members and near-surface layers of materials: Focus on Ukrainian research (review),” Int. Appl. Mech., 50, No. 3, 231–252 (2014).

    Article  Google Scholar 

  13. A. N. Guz, “Recognition of the achievements of the S. P. Timoshenko Institute of Mechanics by the world’s scientific community,” Int. Appl. Mech., 51, No. 1, 1–11 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  14. A. N. Guz, F. G. Makhort, and O. I. Gushcha, Introduction to Electroelasticity [in Russian], Naukova Dumka, Kyiv (1977).

    Google Scholar 

  15. A. N. Guz and J. J. Rushchitsky, “Establishing foundations of the mechanics of nanocomposites,” Int. Appl. Mech., 47, No. 1, 2–44 (2011).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. A. N. Guz and J. J. Rushchitsky, Short Introduction to Mechanics of Nanocomposites, Scientific and Academic Publishing, Rosemead, CA (2013).

    Google Scholar 

  17. A. N. Guz and J. J. Rushchitsky, “Some fundamental aspects of mechanics of nanocomposite materials,” J. Nanotechnol., 24, special issue “Nanocomposites 2013,” 1–15 (2013).

  18. A. N. Guz and J. J. Rushchitsky, “On features of continuum description of nanocomposite material,” J. Research in Nanotechnol., 1, No. 1, 50–60 (2014).

    Google Scholar 

  19. I. A. Guz and J. J. Rushchitsky, “Comparison of mechanical properties and effects in micro- and nanocomposites with carbon fillers (carbon microfibers, graphite microwhiskers, and carbon nanotubes),” Mech. Comp. Mater., 40, No. 3, 179–190 (2004).

    Article  Google Scholar 

  20. A. Hanyga, Mathematical Theory of Nonlinear Elasticity, Ellis Horwood, California (1983).

    MATH  Google Scholar 

  21. R. B. Hetnarski and J. Ignaczak, The Mathematical Theory of Elasticity, CRC Press, Taylor and Francis Group, Boca Raton (2010).

  22. G. A. Holzapfel, Nonlinear Solid Mechanics. A Continuum Approach for Engineering, Wiley, Chichester (2006).

    Google Scholar 

  23. J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge (1978).

    MATH  Google Scholar 

  24. T. C. Lim, Auxetic Materials and Structures, Springer, Singapore (2015).

    Book  Google Scholar 

  25. A. I. Lurie, Nonlinear Theory of Elasticity, North-Holland, Amsterdam (1990).

    MATH  Google Scholar 

  26. A. I. Lurie, Theory of Elasticity, Ser.: Foundations of Engineering Mechanics, Springer, Berlin (2005).

  27. F. D. Murnaghan, Finite Deformation in an Elastic Solid, John Wiley, New York 1951 (1967).

  28. V. V. Novozhilov, Foundations of the Nonlinear Theory of Elasticity, Graylock Press, New York (1953); Dover (2011).

    Google Scholar 

  29. R. W. Ogden, Nonlinear Elastic Deformations, Dover, New York (1997).

    MATH  Google Scholar 

  30. J. J. Rushchitsky, Nonlinear Elastic Waves in Materials, Ser.: Foundations of Engineering Mechanics, Springer, Heidelberg (2014).

  31. J. J. Rushchitsky, “Auxetic linearly elastic isotropic materials: restrictions on elastic moduli,” Arch. Appl. Mech., 72, No. 1, 72–76 (2015).

    Google Scholar 

  32. V. Hauk (ed.), Structural and Residual Stress Analysis, Elsevier Science B.V., Amsterdam (1997) (e-variant 2006).

  33. C. Truesdell, A First Course in Rational Continuum Mechanics, The John Hopkins University, Baltimore (1972); Academic Press, New York (1991).

    Google Scholar 

  34. C. Truesdell and W. Noll, The Nonlinear Field Theories of Mechanics, Flügge Handbuch der Physik, Band III/3, Springer Verlag, Berlin (1965); (2004).

    Google Scholar 

  35. R. Zhu, X. N. Liu, and G. L. Huang, “Study of anomalous wave propagation and reflection in semi-infinite elastic metamaterials,” Wave Motion, 55, 73–83 (2015).

    Article  MathSciNet  Google Scholar 

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Correspondence to J. J. Rushchitsky.

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Translated from Prikladnaya Mekhanika, Vol. 52, No. 2, pp. 20–35, March–April, 2016.

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Rushchitsky, J.J. On Constraints for Displacement Gradients in Elastic Materials. Int Appl Mech 52, 119–132 (2016). https://doi.org/10.1007/s10778-016-0739-5

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