Extension Strain and Rock Strength Limits for Deep Tunnels, Cliffs, Mountain Walls and the Highest Mountains

Brittle rock can fail in tension even when all principal stresses are compressive. The culprit is Poisson’s ratio, but marked stress anisotropy due to a neighbouring free surface, and due to a raised principal tangential stress is also necessary. Extension strain-induced failure causes fracture initiation in tension. Propagation in unstable shear may occur if the tunnels or mine openings are deep enough, and if they are located in hard, brittle, sparsely jointed rock. Both in laboratory uniaxial compression test samples with strength σ_c and in deep tunnels, extension fracturing and acoustic emission begin when the principal applied or induced stress reaches the magnitude of tensile strength divided by Poisson’s ratio σ_t/ν. The traditionally expected fracture initiation when the principal or maximum tangential stress σ₁ or σ_θ = 0.4 ± 0.1 × σ_c can actually be explained with arithmetic. Using related logic, cliffs and the near-vertical mountain walls frequented by rock climbers, may have erosional or glacial origin, but extension strain limits their height, including vertical walls of sheeting joints and long continuous fractures. Shear failure seems to be reserved for occasional major rock avalanches. Equations with soil mechanics origin involving Coulomb parameters c and φ and density that may apply to vertical cuts in soil, give greatly exaggerated heights for rock cliffs and mountain walls since rock is brittle and favours failure in tension. Tensile strength, Poisson’s ratio and density are suggested for estimating the maximum heights of rock cliffs and mountain walls, not compression strength and density. However, overall mountain heights are limited by critical state maximum shear strength, or by the slightly lower brittle–ductile transition strength.