DOI: 10.1029/jb093ib05p04688. Temperature and pressure derivatives of elastic constants with application to the mantle

Temperature and pressure derivatives of elastic constants with application to the mantle

10.1029/jb093ib05p04688

Crossref journal-article

American Geophysical Union (AGU)

Journal of Geophysical Research: Solid Earth (13)

Abstract

The temperature and pressure derivatives of the elastic moduli M of solids can be cast into the form of dimensionless logarithmic anharmonic {DLA} parameters, ∂ ln M/∂ In p = {M} at constant temperature, pressure, or entropy (T, P, S), where p is the density. These parameters show little variation from material to material and are expected to show little variation with temperature at high temperature. Most of the available derivative data for ionic solids has been renormalized and analyzed for dependency on ion type, crystal structure, and other parameters. The {DLA} parameters exhibit little variation and little correlation with crystal structure for most close‐packed halides and oxides. There are small systematic variations with ionic radius, Gruneisen's γ, and the bulk modulus‐rigidity ratio (K/G). Temperature and pressure derivatives are correlated because of the importance of the volume‐dependent, or extrinsic, terms. The intrinsic terms {K}v and {G}v are also highly correlated, even for open‐packed structures, where ∂ ln M/α∂T = {M}v. These correlations make it possible to estimate the derivatives of high‐pressure phases. The spinel forms of olivine are predicted to have “normal” derivatives, and therefore the magnitude of the modulus or velocity jump associated with the olivine‐spinel transition near 400 km should be similar to that measured in the laboratory. The actual size of the 400‐km discontinuity is much less, indicating the presence of substantial quantities of minerals other than olivine in the upper mantle or transition region. Recent calculations in apparent support of a homogeneous olivine‐rich (>60%) mantle are based on choices for the derivatives of β‐ and γ‐Mg2SiO4, which are unlike other ionic crystals. There is no evidence that these phases should be anomalous in their physical properties. The temperature and pressure derivatives of ionic crystals depend on the nature of the ions and their coordination.

Bibliography

Anderson, D. L. (1988). Temperature and pressure derivatives of elastic constants with application to the mantle. Journal of Geophysical Research: Solid Earth, 93(B5), 4688â4700. Portico.

Authors 1

Don L. Anderson (first)

References 30 Referenced 85

10.1126/science.157.3793.1165
10.1111/j.1365-246X.1967.tb02144.x
{'key': 'e_1_2_1_4_1', 'first-page': '85', 'article-title': 'Petrology of the mantle', 'volume': '3', 'author': 'Anderson D. L.', 'year': '1970', 'journal-title': 'Mineral. Soc. Am. Spec. Pap.'} / Mineral. Soc. Am. Spec. Pap. / Petrology of the mantle by Anderson D. L. (1970)
10.1016/0031-9201(87)90039-2
10.1029/JB092iB13p13968
10.1029/JB075i017p03494
10.1038/320321a0
10.1029/JZ072i014p03661
10.1029/JB088iB04p03549
10.1029/GL011i003p00229
10.1038/324449a0
10.1029/JB092iB06p04853
10.1785/BSSA0290030463 / Bull. Seismol. Soc. Am. / The variation of seismic velocities within a simplified Earth model in accordance with the theory of finite strain by Birch F. (1939)
10.1029/JB073i002p00817
10.1029/JB083iB03p01257
{'key': 'e_1_2_1_17_1', 'volume-title': 'Tensors in Mechanics and Elasticity', 'author': 'Brillouin L.', 'year': '1964'} / Tensors in Mechanics and Elasticity by Brillouin L. (1964)
10.1029/JB080i008p01070
10.4294/jpe1952.30.245
10.1098/rspa.1944.0023
10.1063/1.1722628
10.1007/978-3-642-99531-6_1
10.1029/GM039p0231
10.1029/JB088iB10p08145
10.1029/JB075i023p04478
{'key': 'e_1_2_1_26_1', 'first-page': '39', 'volume-title': 'CRC Handbook of Physical Properties of Rocks', 'author': 'Sumino Y.', 'year': '1984'} / CRC Handbook of Physical Properties of Rocks by Sumino Y. (1984)
10.4294/jpe1952.31.125
10.1119/1.1987046
10.1029/GL012i007p00417
10.1007/978-1-4612-4928-3_7
10.1016/0031-9201(83)90045-6

Dates

Type	When
Created	17 years, 6 months ago (Feb. 6, 2008, 3:28 p.m.)
Deposited	1 year, 11 months ago (Sept. 22, 2023, 10:07 p.m.)
Indexed	3 months, 4 weeks ago (May 2, 2025, 1:04 p.m.)
Issued	37 years, 3 months ago (May 10, 1988)
Published	37 years, 3 months ago (May 10, 1988)
Published Online	12 years, 11 months ago (Sept. 20, 2012)
Published Print	37 years, 3 months ago (May 10, 1988)

Funders 0

None

BibTeX

@article{Anderson_1988, title={Temperature and pressure derivatives of elastic constants with application to the mantle}, volume={93}, ISSN={0148-0227}, url={http://dx.doi.org/10.1029/jb093ib05p04688}, DOI={10.1029/jb093ib05p04688}, number={B5}, journal={Journal of Geophysical Research: Solid Earth}, publisher={American Geophysical Union (AGU)}, author={Anderson, Don L.}, year={1988}, month=may, pages={4688–4700} }

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