FLEXURE WITH DAMAGE

Ductile behavior in rocks is often associated with plasticity due to dislocation motion or diffusion under high pressures and temperatures. Ductile behavior can also occur in brittle materials. Ismat and Mitra (2001) describe folding under elasto-frictional conditions by cataclastic flow at shallow crustal levels. Power law increases in acoustic emissions and strain acceleration toward failure in experiments also suggest brittle creep. Although the rheology of the rock is viscoelastic, the underlying mechanism is brittle deformation. Damage has been used to describe inelastic behavior of solids in engineering, and covers a wide range of phenomena from microfracture in brittle materials to dislocation creep in the mantle. We apply continuum damage to describe the inelastic behavior of brittle materials and the temporal and spatial changes in rheology. We use this empirical method to simulate folding through the problem of a plate under flexure.

We use a numerical method to obtain quasi-static solutions to the Navier equation. We use the program GeoFEST v 4.5 (Geophysical Finite Element Simulation Tool), developed by NASA Jet Propulsion Laboratory, to generate solutions for each time step. Where the von Mises stresses exceed the critical stress, we apply damage to the elements and reduce the shear modulus of the element. Damage is calculated for each time step by a power law relationship of the ratio of the critical stress to the von Mises stress and the critical strain to the von Mises strain, accounting for relaxation of the material due to increasing damage. To test our method, we apply damage rheology to an semi-infinite plate deforming under its own weight. Where stresses exceed the critical stress, we simulate the formation of damage and observe the time-dependent relaxation of the stress and strain to levels below the yield strength. We simulate a wide range of behavior from slow relaxation to instantaneous failure, over timescales that span six orders of magnitude. Using this method, stress relaxation produces perfectly-plastic behavior in cases where failure does not occur. For cases of failure, we observe a rapid increase in damage, analogous to the acceleration of microcrack formation and acoustic emissions prior to failure. Thus continuum damage mechanics can be used to simulate irreversible deformation and viscoelastic rheology in brittle materials.