Grüneisen Parameter for a Solid under Finite Strain

J. J. Gilvarry

doi:10.1103/PhysRev.102.331

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Grüneisen Parameter for a Solid under Finite Strain

J. J. Gilvarry

The Rand Corporation, Santa Monica, California

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Phys. Rev. 102, 331 – Published 15 April, 1956

DOI: https://doi.org/10.1103/PhysRev.102.331

Abstract

An evaluation of the Grünesisen parameter (or constant) from the equation of state of a solid has been obtained by Druyvesteyn and Meyering on the basis of the theory of finite strain. The result differs (by $- \frac{1}{3}$ ) from the corresponding evaluation on the Debye theory, as given by Lorentz and by Slater. The value of Druyvesteyn and Meyering is derived here without use of the formal theory of finite strain, and shown to correspond physically to a model of independent pairs of nearest neighbor atoms, rather than to the Debye model of coupled atomic vibrations. This fact resolves a paradox raised by Dugdale and MacDonald in connection with an ideal harmonic solid, and ascribed by them to neglect of finite strain. The presence of a state of finite hydrostatic pressure, upon which elastic waves or pressure changes of infinitesimal amplitude are impressed, is taken into account explicitly by means of Murnaghan's theory of finite strain, to obtain the Grüneisen parameter, as evaluated from the equation of state, on the Debye model and for a Druyvesteyn-Meyering solid. The results are identical in the two cases with the corresponding values obtained without use of the formal theory of finite strain. Hence, no basis exists for the modification at finite pressure in the Grüneisen parameter from the Debye theory, as proposed by Dugdale and MacDonald. A comparison of average values over a relatively large number of elements, of Grüneisen constants as evaluated from Grüneisen's law and from the equation of state on the Debye model, shows excellent agreement at normal and at melting temperature.

References (39)

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J. J. Gilvarry, this issue [Phys. Rev. 102, 308 (1956)]
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Omitted endnote
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[c1] H. A. Lorentz, Proc. Roy. Acad. Amsterdam 19, 1324 (1916)

[c2] J. C. Slater, Phys. Rev. 57, 744 (1940)

[c3] J. C. Slater, Introduction to Chemical Physics (McGraw-Hill Book Company, Inc., New York, 1939), pp. 238, 394, 451

[c4] J. J. Gilvarry, this issue [Phys. Rev. 102, 308 (1956)]

[c5] J. J. Gilvarry, this issue [Phys. Rev. 102, 317 (1956)]

[c6] J. J. Gilvarry, preceding paper [Phys. Rev. 102, 325 (1956)]

[c7] P. Duhem, Ann. École Norm. 23, 169 (1906)

[c8] M. J. Druyvesteyn and J. L. Meyering, Physica 8, 851 (1941)

[c9] M. J. Druyvesteyn, Physica 8, 862 (1941)

[c10] F. D. Murnaghan, Am. J. Math. 59, 235 (1937)

[c11] F. D. Murnaghan, in Applied Mechanics, Theodore von Kármán Anniversary Volume (California Institute of Technology, Pasadena, 1941), p. 121

[c12] M. J. Druyvesteyn, Philips Research Rept. 1, 77 (1946)

[c13] J. S. Dugdale and D. K. C. MacDonald, Phys. Rev. 89, 832 (1953)

[c14] L. Brillouin, Ann. phys. 3, 267, 328 (1925)

[c15] L. Brillouin, Les Tenseurs en Mécanique et en Élasticité (Masson et Cie., Paris, 1949), Chaps. 10-12

[c16] J. C. Slater (private communication)

[c17] F. D. Murnaghan, Finite Deformation of an Elastic Solid (John Wiley and Sons, Inc., New York, 1951), Chap. 4

[c18] C. Truesdell, J. Rational Mech. and Anal. 1, 173 (1952)

[c19] M. A. Biot, J. Appl. Phys. 11, 522 (1940)

[c20] E. Grüneisen, in Handbuch der Physik (Verlag Julius Springer, Berlin, 1926), pp. 1-59

[c21] M. Born and E. Brody, Z. Physik 6, 132 (1921)

[c22] D. J. Hooton, Phil. Mag. 46, 422, 433 (1955)

[c23] J. E. Mayer and M. G. Mayer, Statistical Mechanics (John Wiley and Sons, Inc., New York, 1940), pp. 243, 251

[c24] A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Dover Publications, New York, 1944), fourth edition, pp. 94, 99, 104

[c25] H. Jeffreys, Proc. Cambridge Phil. Soc. 26, 101 (1930)

[c26] J. J. Gilvarry, Phys. Rev. 96, 934 (1954)

[c27] E. Madelung, Physik. Z. 11, 898 (1910)

[c28] A. Einstein, Ann. Physik 34, 170, 590 (1911)

[c29] [C. Zwikker, Physical Properties of Solid Materials (Interscience Publishers, Inc., New York, 1954), p. 90]

[c30] L. S. Ornstein and F. Zernike, Proc. Roy. Acad. Amsterdam 19, 1289, 1304 (1916)

[c31] L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics (McGraw-Hill Book Company, Inc., New York, 1935), p. 160

[c32] H. C. Corben and P. M. Stehle, Classical Mechanics (John Wiley and Sons, Inc., New York, 1950), p. 202

[c33] R. E. Peierls, Quantum Theory of Solids (Oxford University Press, London, 1955), p. 31

[c34] Omitted endnote

[c35] Omitted endnote

[c36] D. S. Hughes and J. L. Kelly, Phys. Rev. 92, 1145 (1953)

[c37] T. H. K. Barron, Phil Mag. 46, 720 (1955)

[c38] C. Zener, Elasticity and Anelasticity of Metals (University of Chicago Press, Chicago, 1948), p. 30

[c39] J. J. Gilvarry, J. Chem. Phys. 23, 1925 (1955)

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Grüneisen Parameter for a Solid under Finite Strain

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