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Volume 70, 1940-41
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– 137 –

Elastic Constants of the Earth's Mantle.

[Read before Auckland Institute, April 17, 1940; received by the Editor, April 12, 1940; issued separately, September, 1940.]

The solid portion of the Earth outside the central core is commonly referred to in seismological work as the Earth's “mantle.” It extends to a depth below the surface which Jeffreys (1939) has now fixed as 2900 km. within an error of not more than a few kilometres. The purpose of the present paper is to obtain figures giving the usual elastic constants for the materials of the Earth's mantle, apart from the crustal layers. Values for the crustal layers have been obtained from near earthquake studies and are liable to vary somewhat from region to region. Studies of distant earthquakes show that, below the crustal layers, the Earth conforms closely to spherical symmetry, and it is now possible to give fairly accurate values of the elastic constants at various depths below these layers.

The determination to fair accuracy becomes possible on account of the recent accurate determination by Jeffreys (1939A) of the velocities α and β, respectively, of P and S seismic waves throughout the Earth's mantle, and also the work of the writer on the density distribution. In the following calculations, values of the density ρ will be those obtained in the most recent publication (Bullen, 1940) of the writer on the Earth's density variation, and use will be made of the assessment there given of the limits of error in the density figures.

The Lamé elastic constants λ and μ may thus be found directly from the standard equations:—

α2=(λ+2μ)/ρ,

β2=μ/ρ

The bulk modulus or incompressibility k and Poisson's ratio σ are obtained using:—

k=λ+⅔μ

σ=½λ/(λ+μ)

In Table I the results are expressed in terms of the depth d km. below the Earth's surface, the thickness of the crustal layers being taken as 33 km. The units of λ, μ and k are 10−12 dynes/cm. in each case; σ of course is a pure ratio.

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For theoretical purposes, it is sometimes convenient to have various results in terms of values of r/R, where R is the radius of the Earth's sub-crustal material, and r the distance of a point from the Earth's centre. For this purpose Table II has been constructed. Values of the density ρ, in gm./cm.3, as found during the preparation of the density paper (Bullen, 1940), are also given in Table II to three decimal places. The third place has been included merely to facilitate theoretical work and has no significance in practice.

The density figures used were shown (1940) to be probably accurate within rather less than 1 per cent. at practically all depths, and it is likely that, except between the depths of 413 and 984 km. (corresponding to values of r/R of 0·94 and 0·85), the values of the elastic constants given are accurate within an order of 1 per cent. Between the particular depths just mentioned, the uncertainty may be a little greater on account of a slightly increased uncertainty in the velocity figures in the region here involved.

Values of the constants μ, k and σ to two significant figures are also given by Gutenberg and Richter (1939), using a completely independent set of velocity figures, and earlier density figures of the present writer. Within the errors of the two determinations, the agreement is very satisfactory.

As with the density and velocities, there are sudden changes in the gradients of the elastic constants when r/R is 0·94, i.e. at a depth. of about 413 km.

Table I.
d λ μ k σ
33 0.74 0.63 1.16 0.269
100 0.80 0.67 1.24 0.272
200 0.90 0.74 1.38 0.275
300 1.01 0.81 1.54 0.277
400 1.12 0.89 1.71 0.279
413 1.14 0.90 1.73 0.280
500 1.42 1.10 2.15 0.283
600 1.69 1.32 2.57 0.282
700 1.90 1.52 2.91 0.278
800 2.06 1.69 3.19 0.275
900 2.20 1.81 3.40 0.275
1000 2.33 1.89 3.59 0.276
1200 2.54 2.03 3.89 0.278
1400 2.76 2.15 4.20 0.281
1600 3.00 2.27 4.52 0.284
1800 3.27 2.39 4.87 0.288
2000 3.55 2.51 5.23 0.292
2200 3.81 2.63 5.57 0.295
2400 4.06 2.76 5.90 0.298
2600 4.32 2.88 6.23 0.300
2800 4.47 3.01 6.47 0.299
2900 4.49 3.03 6.51 0.299
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Table II.
d r/R ρ λ μ k σ
33 1.00 3.320 0.74 0.63 1.16 0.269
96 0.99 3.379 0.80 0.67 1.24 0.272
160 0.98 3.436 0.86 0.71 1.33 0.274
223 0.97 3.491 0.92 0.75 1.42 0.276
286 0.96 3.543 0.99 0.80 1.52 0.277
350 0.95 3.594 1.06 0.85 1.62 0.278
413 0.94 3.643 1.14 0.90 1.73 0.280
476 0.93 3.824 1.35 1.05 2.04 0.283
540 0.92 3.988 1.54 1.19 2.33 0.283
603 0.91 4.136 1.70 1.33 2.58 0.282
666 0.90 4.267 1.84 1.46 2.81 0.279
731 0.89 4.383 1.95 1.58 3.00 0.277
794 0.88 4.480 2.05 1.68 3.17 0.275
921 0.86 4.624 2.23 1.83 3.45 0.275
1047 0.84 4.708 2.38 1.92 3.66 0.277
1174 0.82 4.785 2.51 2.01 3.85 0.278
1301 0.80 4.859 2.64 2.09 4.04 0.279
1427 0.78 4.931 2.79 2.17 4.24 0.281
1554 0.76 5.002 2.94 2.25 4.44 0.283
1681 0.74 5.070 3.10 2.32 4.66 0.285
1807 0.72 5.137 3.28 2.40 4.88 0.288
1934 0.70 5.204 3.46 2.47 5.11 0.291
2061 0.68 5.269 3.63 2.55 5.33 0.293
2188 0.66 5.333 3.79 2.62 5.54 0.295
2315 0.64 5.396 3.95 2.70 5.75 0.297
2442 0.62 5.459 4.12 2.78 5.97 0.299
2568 0.60 5.521 4.28 2.86 6.18 0.300
2695 0.58 5.583 4.44 2.94 6.39 0.301
2822 0.56 5.645 4.48 3.02 6.49 0.299
2885 0.55 5.676 4.49 3.03 6.51 0.299

References.

Bullen, K. E., 1940. The Problem of the Earth's Density Variation, Bulletin of Seismological Assocn. of America, vol. 30, no. 3.

Gutenberg, B., and Richter, C. F., 1939. On Seismic Waves (Fourth Paper), Gerlands Beiträge zur Geophysik, vol. 54, pp. 94–136.

Jeffreys, H., 1939. The Times of PcP and ScS, M.N.R.A.S. Geophysical Supplement, vol. 4, pp. 537–547.

— 1939A. The Times of P, S and SKS, and the Velocities of P and S, M.N.R.A.S. Geophys. Suppl., vol. 4, pp. 498–533.