Investigations in structural geology and geophysics often apply continuum mechanics to solve problems relating to the deformation of the crust and lithosphere, such as the bending of strata due to a laccolith, bending of the lithosphere under island chains, bending of the lithosphere at an ocean trench. These solutions are largely based on linear elasticity. However, many processes involve stresses that exceed the elastic limit. Thus plastic, as well as elastic, deformation occurs. The temporal and spatial changes in material rheology and the nonlinear nature of the process preclude the development of analytical solutions in these complex cases. To account for the evolving rheology, we apply continuum damage mechanics as an empirical method to solve the problem of a bending beam. 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 a 2-D simple beam deforming under its own weight. This problem can be considered an analog for folding. 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 plastic limit. Thus damage can be used as a proxy for irreversible deformation in the fold hinge area, representing brittle fracture and microcracking where extensional fiber stresses dominate, and material dissolution where compressional fiber stresses dominate.

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