Effect of surface curvature on the normal stress normal to the topographic surface, with application to sheeting joints

The differential equations of equilibrium reveal that the rate at which the normal stress perpendicular to the topographic surface changes with depth (i.e., the "tensile stress partial derivative") depends on the two principal curvatures at a point on the surface (k2 and k3) and the associated surface-parallel compressive stresses (P22 and P33), neither of which need be principal stresses. For static equilibrium at any point in or on a body, the divergence of the stress tensor and the body force vector must sum to zero. This relationship can be specialized to the case of a traction-free surface, as the topographic surface of the Earth is commonly idealized. The specialized solution shows that the tensile stress partial derivative at the surface equals P22k2 + P33k3 - ρgcosβ, where ρ is the material density, g is gravitational acceleration, and β is the slope. This general solution is independent of material rheology, does not require the solution of a boundary value problem, and applies to bodies of arbitrary shape. The tensile stress normal to the surface is zero at a traction-free surface, so where the tensile stress derivative is positive, the tensile stress normal to the surface must become positive in the subsurface. This condition can be met if the compressive stresses are sufficiently high (negative) and the surface is convex (k<0) in at least one direction. This provides a fundamental explanation for the opening of near-surface fractures parallel to the surface of the Earth (i.e., sheeting joints), a phenomenon that has been an enigma for more than two centuries. Numerous observations and measurements reveal that sheeting joints characteristically develop beneath convex surfaces in massive rocks where high compressive stresses parallel the surface, consistent with the prediction here. The proposed mechanism provides a fundamental explanation for sheeting joints; erosion of overburden, by itself, does not. Sheeting joints develop well in rocks with high uniaxial compressive strengths because they can sustain the high compressive stresses necessary to generate surface-normal tensile stresses given typical curvatures in landscapes. The effect described here also provides a fundamental explanation for the common observation that the stress normal to the surface of the Earth is not just a function of material density and depth in the shallow subsurface.