Mechanics of Solids

, 46:387 | Cite as

Derivation of equations of state for ideal crystals of simple structure

Article

Abstract

We consider an approach to the derivation of thermodynamic equations of state by averaging the dynamic equations of particles of the crystal lattice. Microscopic analogs of macroscopic variables such as pressure, volume, and thermal energy are introduced. An analysis of the introduced variables together with the equations of motion permits obtaining the equation of state. Earlier, this approach was used to obtain the equation of state in the Mie-Grüneisen form for a one-dimensional lattice. The aim of this paper is to develop and generalize this approach to the three-dimensional case. As a result, we obtain the dependence of the Grüneisen function on the volume, which is compared with the computations performed according to well-known models with experimental data taken into account. It is proved that the Grüneisen coefficient substantially depends on the form of the strain state. Moreover, we refine the equation of state; namely, we show that the Grüneisen coefficient depends on the thermal energy, but this dependence in the three-dimensional case is much weaker than in the one-dimensional case. A refined equation of state containing a nonlinear dependence on the thermal energy is obtained

Keywords

equation of state Mie-Grüneisen equation crystal particle dynamics method 

References

  1. 1.
    V. A. Palmov, Vibrations of Elasto-Plastic Bodies (Nauka, Moscow, 1976; Springer, Berlin, 1998).Google Scholar
  2. 2.
    P. A. Zhilin, “Mathematical Theory of Inelastic Bodies,” Uspekhi Mekh. 2(4), 3–36 (2003).Google Scholar
  3. 3.
    S. B. Segletes, “Thermodynamic Stability of the Mie-Grüneisen Equation of State and Its Relevance to Hydrocode Computations,” J. Appl. Phys. 70(5), 2489–2499 (1991).ADSCrossRefGoogle Scholar
  4. 4.
    V. N. Zharkov and V. A. Kalinin, Equations of State for Solids at High Pressures and Temperatures (Nauka, Moscow, 1968; Consultants Bureau, New York, 1971).Google Scholar
  5. 5.
    A. I. Melker and A. V. Ivanov, “Dilatons of Two Types,” Fiz. Tverd. Tela 28(11), 3396–3402 (1986) [Sov. Phys. Solid State (Engl. Transl.) 28 (11), 1912–1914 (1986)].Google Scholar
  6. 6.
    J. C. Salter, Introduction to Chemical Physics (McGraw Hill, New York, 1939).Google Scholar
  7. 7.
    J. S. Dugdale and D. K. C. MacDonald, “The Thermal Expansion of Solids,” Phys. Rev. 89(4), 832–834 (1953).ADSCrossRefGoogle Scholar
  8. 8.
    V. Ya. Vashchenko and V. N. Zubarev, “Concerning the Grüneisen constant,” Fiz. Tverd. Tela 5(3), 886–890 (1963) [Sov. Phys. Solid State (Engl. Transl.) 5 (3), 653–655 (1963)].Google Scholar
  9. 9.
    I. S. Grigoriev and E. Z. Melikhov (Editors), Handbook of PhysicalQuantities (Energoatomizdat, Moscow, 1991; CRC Press, Boca Raton, 1997).Google Scholar
  10. 10.
    S. B. Segletes, “A Frequency-Based Equation of State for Metals,” Int. J. Impact Engng 21(9), 747–760 (1998).CrossRefGoogle Scholar
  11. 11.
    L. V. Altshuller, “Use of Shock Waves in High-Pressure Physics,” Uspekhi Fiz. Nauk 85(2), 197–258 (1965) [Sov. Phys. Uspekhi (Engl. Transl.) 8 (1), 52–91 (1965)].Google Scholar
  12. 12.
    E. I. Kraus, “Small-Parameter Equation of State of a Solid,” Vestnik NGU. Ser. Fizika 2(2), 65–73 (2007).Google Scholar
  13. 13.
    R. V. Goldstein and A. V. Chentsov, “Discrete-Continuum Model of a Nanotube,” Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 4, 57–74 (2005) [Mech. Solids (Engl. Transl.) 40 (4), 45–59 (2005)].Google Scholar
  14. 14.
    O. S. Loboda and A. M. Krivtsov, “The Influence of the Scale Factor on the Elastic Moduli of a 3D Nanocrystal,” Izv. Akad. Nauk.Mekh. Tverd. Tela, No. 4, 27–41 (2005) [Mech. Solids (Engl. Transl.) 40 (4), 20–32 (2005)].Google Scholar
  15. 15.
    I. E. Berinskii, E. A. Ivanova, A. M. Krivtsov, and N. F. Morozov, “Application of Moment Interaction to the Construction of a StableModel ofGraphite Crystal Lattice,” Izv. Akad. Nauk.Mekh. Tverd. Tela, No. 5, 6–16 (2007) [Mech. Solids (Engl. Transl.) 42 (5), 663–671 (2007)].Google Scholar
  16. 16.
    E. A. Ivanova, A. M. Krivtsov, and N. F. Morozov, “Derivation of Macroscopic Relations of the Elasticity of Complex Crystal Lattices Taking into Account the Moment Interactions at the Microlevel,” Prikl. Mat. Mekh. 71(4), 595–615 (2007) [J. Appl.Math. Mech. (Engl. Transl.) 71 (4), 543–561 (2007)].MathSciNetMATHGoogle Scholar
  17. 17.
    A. M. Krivtsov, “Thermoelasticity of One-Dimensional Chain of Interacting Particles,” Izv. Vyssh. Uchebn. Zaved. Sev.-Kavkaz. Region. Estestv. Nauki, Special Issue. Nonlinear Problems of Continuum Mechanics, 231–243 (2003).Google Scholar
  18. 18.
    A. M. Krivtsov, “From Nonlinear Oscillations to Equation of State in Simple Discrete Systems,” Chaos, Solitons, and Fractals 17(1), 79–87 (2003).ADSMATHCrossRefGoogle Scholar
  19. 19.
    A. M. Krivtsov, Deformation and Failure of Solids with Microstructure (Fizmatlit, Moscow, 2007) [in Russian].Google Scholar
  20. 20.
    M. Born and H. Kun, Dynamical Theory of Crystal Lattices (Clarendon Press, Oxford, 1954; Izd-vo Inostr. Liter., Moscow, 1958).MATHGoogle Scholar
  21. 21.
    M. Zhou, “A New Look at the Atomic Level Virial Stress: On Continuum-Molecular System Equivalence,” Proc. Roy. Soc. London. Ser. A 459(2037), 2347–2392 (2003).ADSMATHCrossRefGoogle Scholar
  22. 22.
    V. Ph. Zhuravlev, Foundations of Theoretical Mechanics (Fizmatlit, Moscow, 2008) [in Russian].Google Scholar
  23. 23.
    B. L. Glushak, V. F. Kuropatenko, and S. A. Novikov, Studies of Material Strength under Dynamical Loads (Nauka, Novosibirsk, 1992) [in Russian].Google Scholar
  24. 24.
    A. I. Lurie, Nonlinear Theory of Elasticity (Nauka, Moscow, 1980) [in Russian].Google Scholar

Copyright information

© Allerton Press, Inc. 2011

Authors and Affiliations

  1. 1.Institute for Problems inMechanical EngineeringRussian Academy of SciencesSt. PetersburgRussia

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