Landslides modeled as bifurcations of creeping slopes with nonlinear friction law

Abstract

This paper investigates landslide as a consequence of the unstable slide of an initially stationary or creeping slope triggered by a small perturbation, such as the effect due to rainfall. Motivated by Skempton's (1985) observation on clay and siltstone containing low clay fraction, the one state variable friction law proposed by Ruina (1983), in which shear resistance along the slip surface (τ) depends on both sliding history (or the state) and the creeping velocity of the slope (V), is extended to model both fluid-saturated rock and soil joints. Presuming that the slope is shallow and infinitely long, a system of three coupled nonlinear first-order differential equations, relating τ, V and u (displacement of the slip surface), is formulated. Linear stability analysis suggests that an equilibrium state (or a critical point) of a slope can be classified as either an asymptotically stable spiral point, an asymptotically stable improper point, or an asymptotically unstable saddle point in the dimensionless shear stress-velocity (s-v) phase plane, depending on the state parameters on the slip surface. For the special case that steady state τ is insensitive to V and the gravitational pull equals the threshold shear stress (τ0), the critical point becomes a neutrally stable equilibrium line in the s-v phase plane; trajectories in the s-v phase plane are obtained analytically for this case. Periodic solution (or Hopf bifurcation) for the system is ruled out based on physical grounds. A fully nonlinear numerical analysis is done for all possible scenarios when a finite perturbation is imposed to the equilibrium state. As the slip continues and erosion due to rainfall occurs, nonlinear parameters of the slip surface may evolve such that a previously stable slope may become unstable (i.e. bifurcation occurs) when a small perturbation is imposed. Thus, the present analysis offers a plausible explanation to why slope failure occurs at a particular rainfall, which is not the largest in the history of the slope.