Assuming that the volume of the shell is increased by approximately 10% by its fusion, it follows that the thickness of the molten layer should be approximately 31 km in order to increase the pressure on the roof of the molten layer by the critical value of 31.7 (over the internal pressure already existing at this depth).
Let us now determine the approximate scale of the tensile deformations of the exosphere having the same thickness (100km), e.g., by assuming that the area of the magmatic chamber underneath is approximately 3.14 In order to simplify the calculations we shall assume a cylindrical magmatic chamber with an internal radius of 100km and a height (thickness) of 31 km. For infinitely thick cylinder walls the stresses of radial compression and tangential tension are equal to the product of the excess of the internal over the external pressure and the square of the inner radius of the cylinder divided by the squared radius of the surface inside the cylinder wall for which the stress is being determined.
Near the inner surface of the cylinder wall the maximum compressive radial stress (equal to the difference between the internal and the external pressures) reaches 31.7 At a distance of approximately 17-18 km from this surface the radial compressive stress and the tangential tensile stress are approximately equal to the strength of the substance at the respective depth. At a distance of 177 km the magnitudes of both stresses amount to only one-tenth of the strength of the shell, i. e., they may be considered to be negligible. The tensile stresses in the roof of the magmatic chamber can be approximated with the aid of the formulas for the determination of stresses in uniformly loaded plates fixed along the entire edge.
According to calculations, the radial tensile stress near the contour reaches a magnitude of about i.e., only 1/25-1/40 of the tensile strength of the roof, while the tangential tensile strength in the roof is approximately 5.9 (1 / 100 —1 / 160 the tensile strength of the roof). The maximum sag of the roof is approximately 36.1 cm.
Thus, the tensile stresses in the roof of the magmatic chamber of the above dimensions as well as its sag are insignificant in comparison with its strength, and are capable of generating only elastic deformations of the roof. The tensile stresses in the roof will equal the roof's tensile strength only when the radius of the magmatic chamber is approximately 480-650km. In this case the maximum sag of the roof will be 190-640m. Such dimen
sions of magmatic chambers may be regarded as critical, their further increase would facilitate the generation of irreversible deformations of the roof.
In the case of the solidification of the molten matter in a magmatic chamber of critical dimensions the inward sag of the roof will be too small to generate substantial compressive crustal deformations. Obviously, compressive crustal deformations in large areas entail physicochemical changes of matter in very extensive magmatic chambers in the shell, and possibly even throughout the entire geosphere. It is quite probable that such alterations in the state of matter of an entire shell layer (e. g., its fusion) may have occurred in the recent past history of the Earth. Investi gations into the Earth's thermal history made by several scientists have revealed that the decay of radioactive elements in the bowels of the Earth following its formation caused its progressive heating; the fusion of a thick layer within the shell took place at a certain stage of this heating at a particular depth below its surface.
In the USSR painstaking investigations of the Earth's thermal history and temperature were conducted by Lyubimova over many years (1955, 1956, 1957, and 1958). In her various studies she inevitably concluded that the shell substance fused at some depth between 100 and 700 km. According to Lyubimova's calculations the fusion of a layer 600 km thick within the shell would have increased the Earth's radius by approximately 50 km.
Obviously, fusion proceeded slowly, in parallel with the accumulation of the radiogenic heat inside the Earth. Initially fusion must have taken place in a thin layer at a depth exceeding 100km, with the thickness of the molten layer subsequently increasing with the increasing radiogenic heat in both directions — toward depths exceeding 100km, with the thickness of the layers. Fusion in the outer layers must have ceased upon reaching a certain maximum thickness corresponding to the maximum concentration of radiogenic heat in the bowels of the Earth. This was followed by the cooling of the outer layers of the Earth's shell as a result of the loss of heat through continuous radiation into space. In the deeper layers the fusion may have continued for some time, gradually decreasing with the depletion of the radioactive elements and the corresponding decrease in internal heat. This cooling, which started in the outer layers, must have spread progressively downward.