Closed-form/least-squares failure envelopes for rock strength

Using test data for three rock types from a published ASTM interlaboratory study of the precision of rock-property test methods, closed-form solutions for rock-strength failure envelopes are presented for pairs of Mohr stress circles in a loading series. Linear, parabolic, and hyperbolic envelopes are used to evaluate the coordinates of the envelopes' points of tangency with the stress circles. Four types of least-squares and three types of cubic-spline curves are then fit to the points of tangency for a series of three or more circles. The least-squares results based on closed-form linear solutions are quite similar to many of the least-squares results based on closed-form parabolic and hyperbolic solutions. Currently available software makes it quite feasible to explore the capabilities of a wide variety of combinations of the options offered in this paper. (C) 2000 Elsevier Science Ltd. All rights reserved.