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IssuesArchive of Issues2011-3pp.387-399

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A.M. Krivtsov and V.A. Kuz'kin, "Derivation of Equations of State for Ideal Crystals of Simple Structure," Mech. Solids. 46 (3), 387-399 (2011)
Year 2011 Volume 46 Number 3 Pages 387-399
DOI 10.3103/S002565441103006X
Title Derivation of Equations of State for Ideal Crystals of Simple Structure
Author(s) A.M. Krivtsov (Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Bol'shoy pr-t 61, St.†Petersburg, 199178 Russia, akrivtsov@bk.ru)
V.A. Kuz'kin (Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Bol'shoy pr-t 61, St.†Petersburg, 199178 Russia, kuzkinva@gmail.ru)
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.  V. A. Palmov, Vibrations of Elasto-Plastic Bodies (Nauka, Moscow, 1976; Springer, Berlin, 1998).
2.  P. A. Zhilin, "Mathematical Theory of Inelastic Bodies," Uspekhi Mekh. 2 (4), 3-36 (2003).
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).
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).
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)].
6.  J. C. Salter, Introduction to Chemical Physics (McGraw Hill, New York, 1939).
7.  J. S. Dugdale and D. K. C. MacDonald, "The Thermal Expansion of Solids," Phys. Rev. 89 (4), 832-834 (1953).
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)].
9.  I. S. Grigoriev and E. Z. Melikhov (Editors), Handbook of Physical Quantities (Energoatomizdat, Moscow, 1991; CRC Press, Boca Raton, 1997).
10.  S. B. Segletes, "A Frequency-Based Equation of State for Metals," Int. J. Impact Engng 21 (9), 747-760 (1998).
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)].
12.  E. I. Kraus, "Small-Parameter Equation of State of a Solid," Vestnik NGU. Ser. Fizika 2 (2), 65-73 (2007).
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)].
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)].
15.  I. E. Berinskii, E. A. Ivanova, A. M. Krivtsov, and N. F. Morozov, "Application of Moment Interaction to the Construction of a Stable Model of Graphite Crystal Lattice," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 5, 6-16 (2007) [Mech. Solids (Engl. Transl.) 42 (5), 663-671 (2007)].
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)].
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).
18.  A. M. Krivtsov, "From Nonlinear Oscillations to Equation of State in Simple Discrete Systems," Chaos, Solitons, and Fractals 17 (1), 79-87 (2003).
19.  A. M. Krivtsov, Deformation and Failure of Solids with Microstructure (Fizmatlit, Moscow, 2007) [in Russian].
20.  M. Born and H. Kun, Dynamical Theory of Crystal Lattices (Clarendon Press, Oxford, 1954; Izd-vo Inostr. Liter., Moscow, 1958).
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).
22.  V. Ph. Zhuravlev, Foundations of Theoretical Mechanics (Fizmatlit, Moscow, 2008) [in Russian].
23.  B. L. Glushak, V. F. Kuropatenko, and S. A. Novikov, Studies of Material Strength under Dynamical Loads (Nauka, Novosibirsk, 1992) [in Russian].
24.  A. I. Lurie, Nonlinear Theory of Elasticity (Nauka, Moscow, 1980) [in Russian].
Received 19 December 2008
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