Effects of small-amplitude periodic topography on combined stresses due to gravity and tectonics

Topographic perturbations of gravitational body forces and horizontal tectonic stresses can be substantial, non-intuitive, and important in terms of subsurface engineering and rock fracture near the surface of the Earth. For (co)sinusoidal topography where the amplitude (A) is small relative to the wavelength (L), adjustments to published plane strain (two-dimensional) approximate elastic solutions for stresses in uniform, isotropic rock allow effects of gravity and a uniform regional horizontal stress (T) to be distinguished. These first-order solutions contain a characteristic stress and three geometric terms, one that varies linearly with elevation, one that decays exponentially with depth, and a (co)sinusoidal term; elastic moduli do not enter the solutions. The first-order solutions are useful approximations for A/L < 0.04. Both gravity and regional compression yield a compression parallel to the surface at ridge crests. Gravity, by itself, causes a localized horizontal tension below valley bottoms. Regional horizontal compression, by itself, contributes a localized vertical tension beneath ridge crests. If T is about an order of magnitude less compressive than ρgA, where ρ is rock density, and g is gravitational acceleration, then effects of gravity dominate effects of the regional compression near the topographic surface. These conditions promote opening of vertical fractures at valley bottoms. Conversely, if T is about an order of magnitude more compressive than ρgA, then effects of regional compression dominate the effects of gravity near the topographic surface. These conditions promote the opening of sheeting joints, macroscopic fractures that open near to and essentially parallel to the topographic surface.