Mechanics of curved surfaces, with application to surface-parallel cracks

Abstract

[1] The surfaces of many bodies are weakened by shallow enigmatic cracks that parallel the surface. A re-formulation of the static equilibrium equations in a curvilinear reference frame shows that a tension perpendicular to a traction-free surface can arise at shallow depths even under the influence of gravity. This condition occurs if σ11k1 + σ22k2 > ρg cosβ, where k1 and k2 are the principal curvatures (negative if convex) at the surface, σ11 and σ22 are tensile (positive) or compressive (negative) stresses parallel to the respective principal curvature arcs, ρ is material density, g is gravitational acceleration, and β is the surface slope. The curvature terms do not appear in equilibrium equations in a Cartesian reference frame. Compression parallel to a convex surface thus can cause subsurface cracks to open. A quantitative test of the relationship above accounts for where sheeting joints (prominent shallow surface-parallel fractures in rock) are abundant and for where they are scarce or absent in the varied topography of Yosemite National Park, resolving key aspects of a classic problem in geology: the formation of sheeting joints. Moreover, since the equilibrium equations are independent of rheology, the relationship above can be applied to delamination or spalling caused by surface-parallel cracks in many materials.