Damage formation associated with bending under a constant moment

Deformation within the Earth's lithosphere is largely controlled by the rheology of the rock. Fracture and faulting are characterized by elastic rheologies with brittle mechanisms, while folding and flow are characterized by plastic and/or viscous rheologies due to ductile mechanisms. However, it has been recognized that deformation that resembles ductile behavior can be produced within the confines of the brittle lithosphere. Specific examples are folds that form in the shallow crust, steep hinges at subduction zones that are accompanied by seismicity, and large-scale deformation at plate boundaries. In these cases, the brittle lithosphere behaves elastically with fracture and faulting yet produces ductile behavior. In this paper, we attempt to simulate such ductile behavior in elastic materials using continuum damage mechanics. Engineers utilize damage mechanics to model the continuum deformation of brittle materials. We utilize a modified form of damage mechanics that represents a reduction in frictional strength of preexisting fractures and faults. We use this empirical approach to simulate the bending of the lithosphere under the application of a constant moment.We use numerical simulations to obtain elastostatic solutions for plate bending and where the longitudinal stress at a particular node exceeds a yield stress, we apply damage to reduce Young's modulus at the node. Damage is calculated at each time step by a power-law relationship of the ratio of the yield stress to the longitudinal stress and the yield strain to the longitudinal strain. This results in the relaxation of the material due to increasing damage. To test our method, we apply our damage rheology to an infinite plate deforming under a constant bending moment. We simulate a wide range of behaviors from slow relaxation to instantaneous failure, over timescales that span six orders of magnitude. Using this method, stress relaxation produces elastic-perfectly plastic behavior in cases where failure does not occur. For cases of failure, we observe a rapid increase in damage leading to failure, analogous to the acceleration of microcrack formation and acoustic emissions prior to failure. The changes in the rate of damage accumulation in failure cases are similar to the changes in b-values of acoustic emissions observed in triaxial compression tests of fractured rock and b-value changes prior to some large earthquakes. Thus continuum damage mechanics can simulate the phenomenon of ductile behavior due to brittle mechanisms as well as observations of laboratory experiments and seismicity.