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

Abstract

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 σ 11 k 1 + σ 22 k 2 pg cosβ, where k 1 and k 2 are the principal curvatures (negative if convex) at the surface, σ 11 and σ 22 are tensile (positive) or compressive (negative) stresses parallel to the respective principal curvature arcs, p 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. © 2011 by the American Geophysical Union.