It is obvious that if the earth shrinks, its crust must become too large for the reduced spheroid, and must be compressed or distorted to fit the new form. The amount of distortion required for any given shrinkage is easily computed from the ratio of the radius to the circumference of a sphere, which is approximately 1 : 6.28. If, for example, the radius shortens 5 miles, each great circle must on the average be compressed, wrinkled, or otherwise distorted to the extent of about 31 miles, or, in reversed application, if the mountain foldings on any great circle together show a shortening of 100 miles, the appropriate radial shortening is 16 miles. The ratio of 1 : 6+ furnishes a convenient check on hypotheses that assign specific thrusts to specific sinkings of adjacent segments. A segment 3000 miles across, for example, such as the bottom of the North Atlantic basin, sinking three miles, about the full depth of the basin, would give a lateral thrust of about 2.2 miles, a little over a mile on each side, a trivial amount compared with the foldings on the adjacent continental borders.
The influence of the domed form of the surface.—Because of the spheroidal form of the earth, each portion of the crust is ideally an arch or dome. When broad areas like the continents are considered, it is the dome rather than the arch that is involved, and in this the thrust is ideally toward all parts of the periphery. It is probably for this reason that mountain ranges so often follow curved or angulated lines, or outline rude triangles or polygons. The sigmoidal courses of the ranges of southern Europe, the looped chains of the eastern border of Asia, and the curved ranges of the Antillean region, are notable examples. The border ranges of the Americas, of the Thibetan plateau, and of other great segments, illustrate the polygonal tendency. The general distribution of the great ranges is such that a nearly equal portion of crustal crumpling is thrown across each great circle, as theory demands. The common generalization that mountain ranges run chiefly in oblique directions, as northeast-southwest, northwest-southeast, is but a partial view of the more general fact that the lines of distortion must lie in all directions to accommodate the old crust to the new geoid, if there be equable contraction in all parts.
Theoretical strength of domes of earth-dimensions.—As the domed form of the crust has played an important part in theories of deformation, it is important to form quantitative conceptions of the strength of ideal domes having the figure and dimensions of segments of the earth’s crust. According to Hoskins,[271] a dome corresponding perfectly to the sphericity of the earth, formed of firm crystalline rock of the high crushing strength of 25,000 pounds to the square inch, and having a weight of 180 pounds to the cubic foot, would, if unsupported below, sustain only 1⁄525 of its own weight.[272] This result is essentially independent of the extent of the dome, and also of its thickness, provided the former is continental and the latter does not exceed a small fraction of the earth’s radius. If this ideal case be modified by supposing the central part of the spherical dome to rise above the average surface, the supporting power will not be materially changed unless the central elevation is a considerable fraction of the radius of the dome. Assuming a central elevation of two miles—to represent the protrusion of the continental segments—the results for domes of different horizontal extent are as follows:[273]
THEORETICAL STRENGTH OF IDEAL DOMES ARCHED TWO MILES ABOVE THE AVERAGE SURFACE OF THE SPHERE.
| Diameter of given dome arched 2 miles above sphere. | Multiplier of 1/525 i.e. the supporting proportion of a spherical dome. | Proportion of its own weight sustained by given dome arched 2 miles above sphere. | |
|---|---|---|---|
3,000 | miles | 1.006 | 1/522 |
400 | ” | 1.396 | 1/376 |
240 | ” | 2.11 | 1/249 |
160 | ” | 3.49 | 1/150 |
80 | ” | 10.97 | 1/48 |
From this table it will be seen that for domes of continental dimensions the supporting strength equals only a very small fraction of the dome’s own weight. Increasing the thickness of the shell increases its actual supporting power, but the proportion is somewhat less when the whole sphere is concerned. The problem has not been worked out for domes of limited extent. For rough estimates, where the dimensions of the dome are of continental magnitude, each mile of thickness may be taken as supporting a layer of about 10 feet of its own material. If the hypothetical level of no stress be placed at 8 miles depth, the shell above this, by reason of its domed shape, could relieve its own pressure on that below to an amount equal only to the weight of about 80 feet of rock over its surface, even if its form and structure were ideal. If the shell were thick enough (817 miles) to embrace one-half the volume of the earth, its supporting power would be a little more than the weight of one and one-half miles of rock. As the radius of the earth is less than 4000 miles, the extreme supporting power reckoned on this basis would be only about 8 miles of rock-depth. It is interesting, if not significant, to observe that this depth barely reaches the minimum shrinkage that will serve, according to current estimates, to account for the crustal shortening of the great mountain-making periods. It is as if the shrinkage stresses accumulated to the full extent of the stress-resisting power of the whole sphere, and then collapsed. It is not safe, however, to give much weight to this coincidence, for higher densities and probably higher resistances to distortion come into play in the deeper horizons. If these resistances are proportional to the higher densities of the interior, the deductions would remain the same. If the effective rigidity of the earth as a whole is that of steel, as deduced by Kelvin and Darwin from tidal and other observations, or twice that of steel, as inferred by Milne from the transmission of seismic vibrations, the supporting power of the body of the earth dependent on its sphericity would be appreciably higher.
It would seem clear from the foregoing considerations that something more than the mere crust of the earth has been involved in the great deformations. Indeed it is not clear that the fullest resources of stress-accumulation which the spheroidal form of the earth affords are sufficient to meet the demands of the problem, unless the rigidity of the earth be taken at a much higher value than that of surface-rock, and this is perhaps an additional argument for the high rigidities inferred from tides and seismic waves.
In view of the doubtful competency of even the thickest segments to accumulate the requisite stresses, there is need to consider modes of differential stress-accumulation other than those dependent on sphericity.
Stress-accumulation independent of sphericity.—The principle of the dome is brought into play whenever an interior shell shrinks away, or tends to shrink away, from an outer one which does not shrink. In this case, there is a free outer surface and a more or less unsupported under surface toward which motion is possible. The dome may, therefore, yield by crushing or by contortion. The computations given above are for cases of this kind. But where the thickness becomes great and the dome involves a large part or even all of a sector of the earth, freedom of motion beneath is small, and to readjust the matter to a new form, strains must be developed widely throughout the sector, and must involve regions where the pressure is extremely great on all sides, and crushing in the usual sense impossible. Assuming the correctness of the modern doctrine that such pressure increases rigidity, instead of the older doctrine that it gives plasticity, it becomes reasonable to assume that stress-differences would be distributed throughout the mass, and bring into play a large portion of its stress-accumulating competency. When the mass yielded, it would not be by crushing, but by “flowage,” which would be more or less general throughout the mass. It might, however, be partially concentrated, as, for example, on the borders of sectors of different specific gravity.
Stress-differences may arise from physical changes within the rock itself. Whenever there is a re-aggregation of matter, or a change of any kind which involves change of volume, a change of stress is liable to be involved. It may be of the nature of relief or of intensification. In an earth built up by the haphazard infall of matter, a very heterogeneous mass must result, and the subsequent changes may be supposed to be intimately distributed through the mass, being slight at any point, but present at innumerable points. An immeasurable number of small stress-differences may, therefore, be developed throughout the mass. Until these overmatch the effective strength of the mass, they may continue to accumulate. These are not necessarily connected with stresses that arise from sphericity, and may work more or less independently of them. It is not improbable that the great stress-accumulating power of the globe finds an essential part of its explanation in supplemental considerations of this kind, and not wholly in its spheroidal form.