Discontinuum Mechanics of linear rock masses

Discontinuum Mechanics is based on two foundations: the idealization of the internal structure of substances, and the principle of the arithmetic mean value. For a linear discontinuum, the methods of Crystallography are used to describe their structure, in which nodes stands for the position of the homologous points such as molecules, grains, or any other motif. The principle of the mean value states that the value of any discrete function at a node is equal to the arithmetic average of the values of this function at the neighboring nodes. This principle leads to an elliptical linear differential equation of the second order, with coefficients given by the statistical second moments of the Cartesian components of the metric vector. As long as those coefficients are components of a symmetric second-order tensor, there are three mutually perpendicular directions, in which the variances reach their extreme values and the covariances are zero, showing that the mean value principle transforms the discontinuum into a statistically equivalent orthotropic continuum. The application of Discontinuum Mechanics to linear rock masses is simple and direct. It is sufficient to define the motif as the plane of geological discontinuity, and the metric vector as the normal vector to this plane, whose magnitude is the spacing between two consecutive planes belonging to the same set of discontinuities. Furthermore, heterogeneities of rock masses can be also included in this description by substituting the plane of discontinuity by a zone of weakness. In this way, the magnitude of the normal vector results from the product of the spacing and the RQD or the RMR. In going further in this theory, at least three conclusions show up. First, in the evaluation of rock masses, it is necessary to determine the orientation, spacing, and RMR for each set of discontinuities. Second, the conventional method of finding the principal families from the contours of equal concentration of poles on the stereographic projection excluding the spacing does not have any physical meaning. Third, any problem of rock mechanics can be solved in a rigorous and reliable way by integrating the differential equation of the mean value.