On the solvability of incompressible Stokes with viscoplastic rheologies in geodynamics

Abstract Plasticity/failure is an essential ingredient in geodynamics models as earth materials cannot sustain unbounded stresses. However, many questions remain as to appropriate models of plasticity as well as effective solvers for these strongly nonlinear systems. Here we present some simplified model problems designed to elucidate many of the issues involved for the description and solution of viscoplastic problems as currently used in geodynamic modeling. We consider compression and extension of a viscoplastic layer overlying an isoviscous layer and introduce a single plastic yield criterion which includes the most commonly used viscoplasticity models: von Mises, depth‐dependent von Mises, and Drucker‐Prager. We show that for all rheologies considered, successive substitution schemes (aka Picard iteration) often stall at large values of the nonlinear residual, producing spurious solutions. However, combined Picard‐Newton schemes can be effective for rheologies that are independent of the dynamic pressure. Difficulties arise when solving incompressible Stokes problems for rheologies that depend on the dynamic pressure such as Drucker‐Prager viscoplasticity. Analysis suggests that incompressible Stokes can become ill‐posed when the dependence of the deviatoric stress tensor on dynamic pressure (i.e., | ∂ τ / ∂ p ′ | ) becomes large. We demonstrate empirically that, in these cases, Newton solvers can fail by introducing spurious shear bands and discuss the consequence of interpreting the results of nonconverged computations. Even for problems where solvers converge, Drucker‐Prager viscoplasticity can produce dynamic pressures that deviate significantly from lithostatic and both the velocity and pressure fields should be evaluated to determine whether solutions are geologically reasonable.