Journal of Mathematical Sciences

, Volume 186, Issue 1, pp 130–138 | Cite as

On the choice of boundary conditions in problems of the local gradient approach in thermomechanics

  • Т. S. Nahirnyj
  • K. А. Chervinka
  • Z. V. Boiko

Within the framework of the local gradient approach in thermomechanics, we formulate key systems of equations of a mathematical model that describes the behavior of elastic bodies with regard for effects of a local heterogeneity. In this case, we choose the displacement vector and vector of local mass displacement or the stress tensor and the vector of local mass displacement as key functions. Based on this, we obtain and analyze solutions of the problems of the equilibrium state of a stretched layer. The problem of choice of boundary conditions in the problems of the local gradient approach is discussed. It is shown that the size effects, including the ultimate strength, depend substantially on the boundary conditions and that the assignment of the density or the divergence of the vector of local mass displacement is physically justified.


Ultimate Strength Force Load Nonlocal Elasticity Gradient Elasticity Nonlocal Theory 
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  1. 1.
    Ya. Yo. Burak and T. S. Nagirnyi, “Mathematical modeling of local gradient processes in inertial thermomechanical systems,” Prikl. Mekh., 28, No. 12, 3–23 (1992); English translation: Int. Appl. Mech., 28, No. 12, 775–793 (1992).MATHGoogle Scholar
  2. 2.
    Ya. Yo. Burak, H. I. Moroz, and Z. V. Boiko, “Mathematical model of thermomechanics with regard for dissipative processes in the formation of near-surface phenomena,” Dop. Nats. Akad. Nauk. Ukr., No. 9, 65–71 (2008).Google Scholar
  3. 3.
    Ya. Yo. Burak, H. I. Moroz, and Z. V. Boiko, “On the energy approach and thermodynamic foundations of the variational formulation of boundary-value problems of thermomechanics with regard for near-surface phenomena,” Mat. Met. Fiz.-Mekh. Polya, 52, No. 2, 55–65 (2009); English translation: J. Math. Sci., 170, No. 5, 629–641 (2010).MATHGoogle Scholar
  4. 4.
    Ya. Yo. Burak, T. S. Nahirnyj and O. R. Hrytsyna, “On an approach to taking into account a near-surface heterogeneity in thermomechanics of solid solutions,” Dop. Akad. Nauk. Ukr., No. 11, 47–51 (1991).Google Scholar
  5. 5.
    Ya. Yo. Burak, E. Ya. Chaplya, V. F. Kondrat, and O. R. Hrytsyna, “Mathematical modeling of thermomechanical processes in elastic bodies with regard for the local mass displacement,” Dop. Nats. Akad. Nauk. Ukr., No. 6, 45–49 (2007).Google Scholar
  6. 6.
    Ya. Burak, E. Chaplya, T. Nahirnyj, V. Chekurin, V. Kondrat, O. Chernukha, H. Moroz, and K. Chervinka, Physicomathematical Modeling of Complex Systems [in Ukrainian], SPOLOM, Lviv (2004).Google Scholar
  7. 7.
    GOST 2789-73. Surface Roughness. Parameter and Characteristics [in Russian], Izd. Standartov (1973).Google Scholar
  8. 8.
    O. Hrytsyna, T. Nahirnyj, and K. Chervinka, “Local gradient approach in thermomechanics,” Fiz.-Mat. Model. Inform. Tekhnol., No. 3, 72–83 (2006).Google Scholar
  9. 9.
    T. S. Nahirnyj, “On the problem of choice of influence functions in rheological kinetic equations of continuum mechanics,” Dop. Akad. Nauk. Ukr., No. 2, 49–53 (1992).Google Scholar
  10. 10.
    T. S. Nahirnyj, “Near-surface stresses in a layer. Surface tension and size effects,” Mat. Met. Fiz.-Mekh. Polya, 42, No. 4, 111–115 (1999).Google Scholar
  11. 11.
    T. S. Nahirnyj, Thermodynamic Models and Methods in Local Gradient Thermomechanics with Regard for Surface Phenomena [in Ukrainian], Author’s Abstract of Doctoral Thesis (Physicomathematical Sciences), Lviv (1998).Google Scholar
  12. 12.
    V. V. Panasyuk (editor), Fracture Mechanics and Strength of Materials, Vol. 1: V. V. Panasyuk, A. E. Andreikiv, and V. S. Parton, Foundations of Fracture Mechanics of Materials [in Russian], Naukova Dumka, Kiev (1988).Google Scholar
  13. 13.
    E. P. De Garmo, J. T. Black, and R. A. Kohser, Materials and Processes in Manufacturing, Wiley, New York (2003).Google Scholar
  14. 14.
    A. C. Eringen, Nonlocal Continuum Field Theories, Springer, New York (2002).MATHGoogle Scholar
  15. 15.
    A. C. Eringen, “Nonlocal continuum mechanics based on distributions,” Int. J. Eng. Sci., 44, No. 3–4, 141–147 (2006).MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    A. C. Eringen, Polar and Nonlocal Theories of Continua, Boğaziçi University, Istanbul (1974).Google Scholar
  17. 17.
    A. C. Eringen and D. G. B. Edelen, “On nonlocal elasticity,” Int. J. Eng. Sci., 10, No. 3, 233–248 (1972).MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    M. Lazar and G. A. Maugin, “A note on line forces in gradient elasticity,” Mech. Res. Commun., 33, No. 5, 674–680 (2006).MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    M. Lazar and G. A. Maugin, “Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity,” Int. J. Eng. Sci., 43, No. 13–14, 1157–1184 (2005).MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    G. A. Maugin, “Nonlocal theories or gradient-type theories: A matter of convenience?,” Arch. Mech., 31, 15–26 (1979).MathSciNetMATHGoogle Scholar
  21. 21.
    R. D. Mindlin, “Elasticity, piezoelectricity and crystal lattice dynamics,” J. Elasticity, 2, No. 4, 217–282 (1979).CrossRefGoogle Scholar
  22. 22.
    T. Nahirnyj and K. Tchervinka, “Interface phenomena and interaction energy at the surface of electroconductive solids,” Comput. Meth. Sci. Technol., 14(2), 105–110 (2008).Google Scholar
  23. 23.
    K. Santaoja, “Gradient theory from the thermomechanics point of view,” Eng. Fract. Mech., 71, No. 4–6, 557–566 (2004).CrossRefGoogle Scholar
  24. 24.
    Z. Tang, S. Shen, and S. N. Atluri, “Analysis of materials with strain-gradient effects: A meshless local Petrov–Galerkin (MLPG) approach, with nodal displacements only,” Comput. Model. Eng. Sci., 4, No. 1, 177–196 (2003).MATHGoogle Scholar
  25. 25.
    R. A. Toupin, “Elastic materials with couple-stresses,” Arch. Ration. Mech. Anal., 11, 385–414 (1962).MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    C. Truesdell and R. A. Toupin, “The classical field theory,” in: S. Flügge (editor), Handbuch der Physik, Vol. III/1, Springer, Berlin (1960), pp. 226–793.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  • Т. S. Nahirnyj
    • 1
    • 2
  • K. А. Chervinka
    • 3
  • Z. V. Boiko
    • 1
  1. 1.Center of Mathematical Modeling, Pidstryhach Institute for Applied Problems of Mechanics and MathematicsUkrainian National Academy of SciencesLvivUkraine
  2. 2.University of Zielona GóraZielona GóraPoland
  3. 3.Franko National University of LvivLvivUkraine

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