The Coulomb Failure Criterion governs failure of INTACT rock: \(\tau = c + \mu_i\,\sigma_n\), where \(c\) is the rock's cohesive strength (a genuine, material-dependent intercept, \(c>0\)) and \(\mu_i\) is the internal (static) coefficient of friction. It defines a straight-line failure ENVELOPE on the Mohr diagram, offset above the origin by the cohesion, that a Mohr circle must touch before new fracturing occurs.
Byerlee's Law governs FRICTIONAL SLIDING on a PRE-EXISTING surface (a joint, fault or other plane of weakness) that already has essentially zero cohesion across it: \(\tau = 0.85\,\sigma_n\) for \(\sigma_n<200\ \text{MPa}\), and \(\tau = 50\ \text{MPa} + 0.6\,\sigma_n\) for \(\sigma_n>200\ \text{MPa}\). Remarkably, this frictional-sliding line is nearly the SAME for almost all rock types (Byerlee 1978), because it depends only on frictional contact between rock surfaces, not on the rock's own cohesive strength.
Fig. E1 — Coulomb envelope for intact-rock failure (positive cohesion intercept c) vs. Byerlee's Law for frictional sliding on a pre-existing, cohesionless surface (through the origin).
Because Byerlee's line passes through (or very near) the origin while the Coulomb envelope is offset upward by \(c\), a pre-existing weakness typically requires LESS shear stress to reactivate than intact rock needs to fracture afresh at the same normal stress — UNLESS the pre-existing plane is badly oriented relative to the principal stresses, which is exactly the situation explored in E(2).
E(2) When is a new fracture more favourable than reactivating an existing plane?
Yes. Whether slip occurs on a pre-existing plane or a fresh fracture forms depends on the plane's ORIENTATION relative to the principal stresses, not just on which failure line (Coulomb vs. Byerlee) sits lower. A plane oriented at the optimal reactivation angle \(2\theta \approx \arctan(1/\mu)\) from \(\sigma_1\) resolves the maximum possible shear-to-normal stress ratio and slips easily, well below the Coulomb envelope's threshold. But a plane oriented far from that optimum — nearly parallel or nearly perpendicular to \(\sigma_1\) — resolves very little shear stress relative to the normal stress clamped across it, and plots on the Mohr diagram BELOW the Byerlee line: it is effectively "locked" and will not slip no matter how much the Mohr circle grows. As the circle continues to enlarge (increasing differential stress), it will eventually touch the (lower-threshold, but non-zero-cohesion) Coulomb envelope at some OTHER, favourably-oriented angle FIRST — producing a brand-new fracture through intact rock before the badly-oriented old weakness ever reaches its own frictional-sliding condition.
Fig. E2 — a misoriented pre-existing plane (small 2θ) plots below the Byerlee line and stays locked, while the growing Mohr circle already reaches the Coulomb envelope at a favourably-oriented new-fracture angle.
E(3) Pi-diagram: fold axis, Pi circle and axial plane
Given. Eleven poles to bedding plotted on a lower-hemisphere equal-area net, forming a girdle: four in the NW quadrant (trend ≈310–320°, plunge 20–40°), two steep NW poles (trend ≈330–340°, plunge 60–70°), two steep SW poles (trend ≈210–220°, plunge 60–70°), three SE poles (trend ≈130–140°, plunge 20–40°); and one axial-trace measurement plotted on the primitive circle at azimuth 270° (horizontal).
Find. The Pi circle, the Pi axis (fold axis), the axial-plane orientation, and a complete fold description.
Fig. E3 — poles to bedding (○) define the Pi circle (blue great circle); its pole is the Pi axis (red); the axial-trace measurement (□) together with the Pi axis fixes the axial plane.
Approach. For a cylindrical fold, all poles to bedding lie on ONE great circle (the Pi circle); the POLE to that great circle is the Pi axis (= fold axis). The axial plane is the unique great circle containing both the Pi axis and the independently-measured axial-trace lineation.
Fit the Pi circle. Converting each pole to a unit vector and finding the eigenvector of the smallest eigenvalue of the summed outer-product (orientation) matrix gives the pole to the best-fit great circle through all 11 points: \(\boxed{\text{Pi axis} \approx 049^{\circ}/07^{\circ}}\) (trend/plunge).
Pi circle orientation. Converting that pole to its own plane (strike = trend−90°, dip = 90°−plunge) gives the Pi circle itself: \(\boxed{319^{\circ}/84^{\circ}\text{NE}}\) (strike/dip).
Axial plane. The plane containing both the Pi axis (049/07) and the axial-trace measurement (270/00) is found from their cross product: pole at trend 180°, plunge 80°, giving \(\boxed{\text{axial plane} \approx 090^{\circ}/10^{\circ}\text{S}}\) (strike/dip).
(d) Fold description. With the fold axis plunging only 7° and the axial plane dipping only 10°, this is a non-plunging (sub-horizontal), recumbent fold (Fleuty's classification: axial-plane dip <10° = recumbent). The two limb-pole clusters span a wide range of plunges (20–40° on the shallow limb vs. 60–70° on the steep limb), so the limbs dip by clearly different amounts — the fold is asymmetric. Because the plotted poles never bunch into a single tight point (which would indicate an isoclinal, near-zero interlimb angle) the fold is best described as moderately close rather than tight or isoclinal. Combining at least three standard terms: a non-plunging, recumbent, asymmetric, moderately close, cylindrical fold.
Check: pole positions were taken from the printed figure's approximate cluster description (quadrant + trend/plunge range per cluster), not from a point-by-point reading of each individual symbol — the eigenvector fit's residuals (mean angular deviation from the fitted great circle ≈ 8°, consistent with typical hand-plotted field data scatter) confirm the fit is stable, but a reader working from the original plot should re-fit against their own point-by-point reading if better precision is required.
