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...