Publication Rights Reserved
Accurate information on the in-situ state of stress in the near surface rocks of the earth's crust is essential to the solution of many problems in rock mechanics. Most of the theoretical estimates that have been made are based on questionable assumptions and none can accurately predict stresses in localized regions. It is frequently suggested that the state of stress should be considered a hydrostatic pressure of intensity equal to that produced by the weight of overlying rocks. This opinion, often called Helm's hypothesis, is proposed because it is considered that the rocks will, in geological time flow, so as to alleviate completely any stress-differences. However, proof that stress-differences do exist and are maintained is, according to Jeffreys, afforded by the presence on the earth's surface of high Mountain ranges and deep oceans. After having analyzed a variety of models for the earth, Jeffreys concludes that stress-differences of the order of 20,000 psi must exist in the upper 50 kilometres of the crust.
Lame obtained a solution for the stresses developed in the earth assuming that it could be considered as a homogeneous isotropic elastic sphere which instantaneously acquired a gravitational field. It is interesting to note that this model predicts that the maximum tangential stress, of the order of 1 million psi for typical rocks, occurs at the surface where the radial stress is zero. Salustowicz has modified Lame's solution considering the earth as a thin homogeneous, elastic and isotropic spherical shell filled with an incompressible material. This assumption yields the prediction that except for the unusual condition of Poisson's ratio equal to 0.5, the tangential or lateral stresses [ ] are less than the radial or vertical stress [ ].
[Tensile stresses are considered positive throughout this paper]
where: pg is the average weight density of the rocks overlying the point [P] at which the stresses act, d is the depth below the surface of the point P, is Poisson's ratio.
Both parts of Eq. 1 are identical to those derived by Mindlin for the vertical and horizontal stresses at a depth 'd' below the horizontal surface of a semi-infinite isotropic elastic mass which was prevented from expanding laterally upon the "supposed application" of a gravitational field. A major criticism of such analyses concerns the assumption that the gravitational field did not exist until the earth acquired its present state. In reality, gravity has been active throughout the formation of the earth and has exerted a cumulative effect in developing stresses within the earth. In particular, gravity was acting on deposited material even before it could be considered elastic.
Brown and Goodman have analyzed a situation more representative of rock deposition and formation processes. They consider the stresses and deformations generated during the buildup of a body in a gravitational field by the continuous deposition of a sequence of incremental layers. Each new layer is presumed to be free to move laterally over the pre-existing surface before becoming elastic and forming an integral part of the body.* The analysis is applied to the case of a hollow sphere built up by this process of accretion.