An instability mechanism for the sliding motion of finite depth of bulk granular materials

Summary Field observations and experimental records indicate that the primary mode of motion of many large landslides is that ofsliding rather thanflowing. Most of shear during sliding is concentrated at the base of slides, with little or no mixing taking place away from the base. This sliding motion may generate strong pressure waves at the interface between the quasi-static deforming granular mass and the grain-inertia dominated rapid granular flow, thus inducing a Kelvin-Helmholtz type instability mechanism for large landslides. The existence of a transitional zone in granular flow is essential for the generation of this type of instability waves. A model using a finite depth of elastic sliding bulk granular materials riding on a basal granular shear flow layer is estabilished to represent the sliding motion of these large volume of bulk granular materials. The balance and the stability of this sliding system are investigated under the perturbation of internal pressure waves. The generated instability waves will force favorable phase shifts between the overburden pressure and the sliding velocity, leading to a net reduction in the total power loss due to friction. The depth of sliding mass will affect the generation of this type of instability waves. Both analytical and numerical results show that smaller depth slides can induce stronger instability waves than larger depth slides do.Notation a perturbation wave amplitude - C nondimensional instability wave speed - C _i growth rate, the imaginary part ofC - C _r wave phase speed, the real part ofC - c _p compressional wave speed in elastic medium - c _s shear wave speed in elastic medium - D nondimensional depth of sliding mass - d depth of sliding mass - G shear modulus of elastic medium - H nondimensional basal depth - h depth of basal shear zone - i - K Coulomb friction coefficient - P _xx, P_yy lateral and normal pressures in granular material, respectively - P _xy shear stress in granular material - p ₀ amplitude of perturbation pressure - p _yy perturbation pressure - r nondimensional complex wave number of instability wave - S nondimensional wave number of shear wave - t time scale - U uniform sliding velocity of a landslide inx direction - u, v velocities inx direction andy direction, respectively - u ₀ granular flow velocity in the basal shear zone - V, V _c nondimensional sliding velocity and its critical velocity, respectively - W power loss to friction - internal friction angle - , Lame's potentials, and are time-independent amplitudes of and , respectively - perturbation wave surface profile - wave number of perturbation wave, _r and _i are the real and imaginary parts of - Poisson's ratio of elastic medium - wave frequency of perturbation wave - , _g density of granular material - stress component in elastic medium - Rankine's earth pressure coefficient - -K ² - Re{}, Im{} the real and imaginary parts of complex quantity inside {}, respectively - , the divergence and the curl of perturbation wave velocities, respectively - Laplacian operator - _ij Kronecker delta; _ij =1 fori=j, _ij =0 forij - ()i, ()j, ()ij tensor - ()1, ()e in sliding mass - ()2, ()b in ground