A non-smooth cohesive zone model for rock fracture and contact

The effect of climate-change driven increases in temperature in high mountain areas is known to enhance the rockfall risk. One of the driving mechanisms is the fracture of rock masses that previously consisted of permafrost, but that are now subject to freeze-thaw cycles. Cohesive zone models are a high-fidelity way of modelling fracture propagation, and in particular extrinsic cohesive zone models are particularly suitable to the task of modelling rock fracture behaviour, as they can capture the full range of fracturing behaviour, from quasi-static to dynamic. As cracks in the field can progress very slowly before reaching a critical point from which they accelerate rapidly, being able to model the full range of crack speeds is essential to accurately capture the physics of rockfall initiation. Here, we propose a non-smooth cohesive zone model that allows us to combine fracture mechanics with contact mechanics, meaning that it is suitable both to model the formation of cracks and the subsequent contact of surfaces as newly formed blocks fall in a unified manner. Further, writing our problem in this way allows us to include frictional behaviour within a monolithic linear complementarity problem, which enables very efficient numerical resolution. We can prove mathematically that the discrete-in-time-and-space problem is well posed for a small enough time-step, meaning that the solution is unique and will not suffer from "solution jumps" (as can happen in quasi-statics). As such, the evolution of the fracture in the rock is continuous, matching the reality, and the shape of the newly-formed rock mass can be accurately captured. Our formulation is also well-adapted for extension to fully-coupled systems that include thermal effects, so as to accurately capture freeze-thaw cycles and properly integrate permafrost behaviour, and thus have a complete model of the system under climate-change-driven loading.