Fig. 5: Criteria for bulk failure.

a The Mohr-Coloumb-Griffith failure curve (broken curve) shows the combinations of effective normal and shear stress that permit bulk failure^73,74,75. The effective normal stress is the difference between normal stress and pore pressure; normal stresses are positive in compression and negative in tension. The axes have been normalised by dividing the shear and effective normal stresses by the tensile strength. The Mohr circles (continuous semi-circles) describe states of stress between the maximum and minimum effective normal stresses (corresponding to where the circles meet the normal-stress axis on the right and left). Failure occurs where the failure envelope meets a Mohr circle. It is tensile when the curves meet along the normal-stress axis. The minimum effective normal stress then equals the crust’s tensile strength, so that the [Effective Normal Stress]/[Tensile Strength] is -1. The diameter of a Mohr circle gives the ratio S_d/σ_T. The largest circle that can meet the failure envelope at tensile failure^73,74,75 has a value of S_d/σ_T = 4. Conditions for failure when S_d/σ_T = 2 are shown for comparison. b Internal rupture (white ellipse) and relaxation (A–B–C) explains stress drops without measurable changes in surface and corrected uplift or in conditions in the pressure source (purple). The trend for simple elastic relaxation (A–B–D) would produce a measurable decrease in surface and corrected uplift and require a reduction in deforming pressure. Campi Flegrei’s behaviour is consistent with internal rupture (A–B–C).