Continuum Mechanics and Thermodynamics

, Volume 29, Issue 3, pp 715–729 | Cite as

Mechanics of deformations in terms of scalar variables

  • Valeriy A. RyabovEmail author
Original Article


Theory of particle and continuous mechanics is developed which allows a treatment of pure deformation in terms of the set of variables “coordinate-momentum–force” instead of the standard treatment in terms of tensor-valued variables “strain–stress.” This approach is quite natural for a microscopic description of atomic system, according to which only pointwise forces caused by the stress act to atoms making a body deform. The new concept starts from affine transformation of spatial to material coordinates in terms of the stretch tensor or its analogs. Thus, three principal stretches and three angles related to their orientation form a set of six scalar variables to describe deformation. Instead of volume-dependent potential used in the standard theory, which requires conditions of equilibrium for surface and body forces acting to a volume element, a potential dependent on scalar variables is introduced. A consistent introduction of generalized force associated with this potential becomes possible if a deformed body is considered to be confined on the surface of torus having six genuine dimensions. Strain, constitutive equations and other fundamental laws of the continuum and particle mechanics may be neatly rewritten in terms of scalar variables. Giving a new presentation for finite deformation new approach provides a full treatment of hyperelasticity including anisotropic case. Derived equations of motion generate a new kind of thermodynamical ensemble in terms of constant tension forces. In this ensemble, six internal deformation forces proportional to the components of Irving–Kirkwood stress are controlled by applied external forces. In thermodynamical limit, instead of the pressure and volume as state variables, this ensemble employs deformation force measured in kelvin unit and stretch ratio.


Mechanics on torus Finite deformation as scalar Constitutive equations Hyperelastisity Isotension ensemble 


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  1. 1.
    Landau, L.D., Lifshits, E.M.: Theory of Elasticity, vol. 7. Pergamon Press, Oxford (1970)Google Scholar
  2. 2.
    Andersen, H.C.: Molecular dynamics simulations at constant pressure and/or temperature. J. Chem. Phys. 72, 2384 (1980)ADSCrossRefGoogle Scholar
  3. 3.
    Parrinello, M., Rahman, A.: Polymorphic transitions in single crystals: a new molecular dynamics method. J. Appl. Phys. 52, 7182 (1981)ADSCrossRefGoogle Scholar
  4. 4.
    Ryabov, V.A.: Dynamics on a torus. Phys. Rev. E 71, 016111 (2005)ADSCrossRefGoogle Scholar
  5. 5.
    Ryabov, V.A.: Constant pressure–temperature molecular dynamics on a torus. Phys. Lett. A 359, 61 (2006)ADSCrossRefGoogle Scholar
  6. 6.
    Hilbert, D., Cohn-Vossen, S.: Anschauliche Geometrie. Springer, Berlin (1932)CrossRefGoogle Scholar
  7. 7.
    Lurie, A.I.: Theory of Elasticity. Springer, Berlin (2005)CrossRefGoogle Scholar
  8. 8.
    Bower, Allan F.: Applied Mechanics of Solids. CRC Press, Taylor & Francis Group, Boca Raton (2009)CrossRefGoogle Scholar
  9. 9.
    Born, M., Huang, K.: Dynamical Theory of Crystal Lattices. Clarendon Press, Oxford (1954)zbMATHGoogle Scholar
  10. 10.
    Ryabov, V.A.: Quantum volume. Int. J. Mod. Phys. B 29, 1550166 (2015)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Martyna, G.J., Tuckerman, M.E., Tobias, D.J., Klein, M.L.: Explicit reversible integrators for extended systems dynamics. Mol. Phys. 87, 1117 (1996)ADSCrossRefGoogle Scholar
  12. 12.
    Thurston, R.N.: Physical Acoustics, Principles and Methods. Academic, New York (1964)Google Scholar
  13. 13.
    Ray, J.R., Rahman, A.: Statistical ensembles and molecular dynamics studies of anisotropic solids. J. Chem. Phys. 80, 4423 (1984)ADSCrossRefGoogle Scholar
  14. 14.
    Lill, J.V., Broughton, J.Q.: Atomistic simulations incorporating nonlinear elasticity: slow-stress relaxation and symmetry breaking. Phys. Rev. B 47, 11619 (1994)ADSCrossRefGoogle Scholar
  15. 15.
    Miller, R.E., et al.: Molecular dynamics at constant Cauchy stress. J. Chem. Phys. 144, 184107 (2016)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.National Research Centre “Kurchatov Institute”MoscowRussia

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