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  4. On three facts of reticences in the classical mathematical modeling of elastic materials

On three facts of reticences in the classical mathematical modeling of elastic materials

MMC.
2014;
Volume 1, Number 2
: pp. 245-255
https://doi.org/10.23939/mmc2014.02.245
Received: August 31, 2014

Math. Model. Comput. Vol. 1, No. 2, pp. 245-255 (2014)

Authors:
  1. J. Rushchitsky
1
S. P. Timoshenko Institute of Mechanics

Three facts of reticences (passing over in silence, an absence of comments) in the procedures of mathematical modeling of elastic materials are described and commented. The first fact consists in a reticence of one of the first steps in the mentioned above procedure – an assumption that the kinematics of deformation is described by the linear approximation of motion of material continuum, namely by gradients of deformation. In the paper, a novel nonlinear approach to this procedure is offered. The second and third facts are associated with constitutive relations. The second fact consists in the absence of necessary comments relative to determination of smallness of strains and gradients of displacements (absence of comments relative to a criterion of applicability of the linear model) because the criterion $ |u_{i,k}|\ll 1$ is sufficiently abstract. It is shown that there exists a based on the nonlinear Cauchy relations approximate procedure of determination of threshold values of strains and gradients of deformations starting with which a nonlinearity of process appears. The third fact consists in the absence of comments relative to essential differences between the nonlinear constitutive equations, which are written for the ordered pairs "Lagrange stress tensor – Cauchy-Green strain tensor" and "Kirchhoff stress tensor – gradients of displacements". It is shown on an example of the shear stress and the Murnaghan model of nonlinear elastic deformation that deviation from the corresponding straight lines of linear deformation for different pairs differs in many times in the range of small strains and small gradients of displacements. The general estimate of facts of reticences looks positive, because for one part of scientists-mechanicians the reticences form the comfort feeling of monolithic character of the classical theory of elasticity, whereas for another part the reticences form a space for development of the theory of elasticity.

mathematical modeling of elastic materials
reticences in classical nonlinear theory of elasticity
smallness of strains and gradients of displacements
nonlinear kinematic parameters
  1. Eringen A. C. Nonlinear Theory of Continuous Media. McGraw Hill, New York (1962).
  2. Fu Y. B. Nonlinear Elasticity: Theory and Applications. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2001).
  3. Goldenblatt I. I. Nonlinear Problems of the Theory of Elasticity. Nauka, Moscow (1969), (In Russian)
  4. Green A. E., Adkins J. E. Large Elastic Deformations and Nonlinear Continuum Mechanics. Oxford University Press, Clarendon Press, London (1960).
  5. Guz A. N. Fundamentals of the Three-Dimensional Theory of Stability of Deformable Bodies. Springer, Berlin (1999).
  6. Hetnarski R. B., Ignaczak J. The Mathematical Theory of Elasticity. CRC Press, Taylor & Francis Group, Boca Raton (2011).
  7. Holzapfel G. A. Nonlinear Solid Mechanics. A Continuum Approach for Engineering. Wiley, Chichester (2006).
  8. Lur’e A. I. Nonlinear Theory of Elasticity. North-Holland, Amsterdam (1990).
  9. Lur’e A. I. Theory of Elasticity. Springer, Berlin (1999).
  10. Novozhilov V. V. Foundations of the nonlinear theory of elasticity. Graylock Press, Rochester, New York (1953).
  11. Ogden R. W.The Nonlinear Elastic Deformations. Dover, New York (1997).
  12. Truesdell C. A first course in rational continuum mechanics.The John Hopkins University, Baltimore (1972).
  13. Truesdell C., Noll W. The Nonlinear Field Theories of Mechanics. Flugge Handbuch der Physik, Band III / 3. Springer Verlag, Berlin (1965).
  14. Truesdell C., Toupin R. The Classical Field Theories, Flugge Handbuch der Physik, Band III / 1. Springer Verlag, Berlin (1960).
  15. Eringen A. C. Mechanics of Continua. John Wiley, New York (1967).
  16. Eringen A. C., Suhubi E. S. Nonlinear theory of simple microelastic solids. Int. J. Engng. Sci. 2, n.2, 189–203 (1964).
  17. Rushchitsky J. J. Nonlinear Elastic Waves in Materials. Springer, Heidelberg (2014).
  18. Murnaghan F. D. Finite Deformation in an Elastic Solid. John Wiley, New York (1951, 1967).
  19. Guz A. N., Makhort F. G., Gushcha O. I. Introduction to Electroelasticity. Naukova Dumka, Kiev (1977), (In Russian).
  20. Guz A. N. Elastic Waves in Bodies with Initial Stresses, 1, General Problems, 2, Regularities of Propagation. Naukova Dumka, Kiev (1986), (In Russian).
  21. Goldberg Z. A. On interaction of plane longitudinal and transverse waves. Akusticheskii Zhurnal, 6, n.2, 307-310 (1960), (In Russian and English).
  22. Cattani C., Rushchitsky J. J. Wavelet and wave analysis as applied to materials with micro or nanostructures. Singapore-London: World Scientific Publishing Co. Pte. Ltd. (2007).