An Eulerian projection method for quasi-static elastoplasticity

Awell-established numerical approach to solve the Navier–Stokes equations for incompressible žuids is Chorin’s projection method [1], whereby the žuid velocity is explicitly updated, and then an elliptic problem for the pressure is solved, which is used to orthogonally project the velocity eld to maintain the incompressibility constraint. In this paper, we develop a mathematical correspondence between Newtonian žuids in the incompressible limit and hypo-elastoplastic solids in the slow, quasi-static limit. Using this correspondence, we formulate a new xed-grid, Eulerian numerical method for simulating quasi-static hypo-elastoplastic solids, whereby the stress is explicitly updated, and then an elliptic problem for the velocity is solved, which is used to orthogonally project the stress to maintain the quasi-staticity constraint. We develop a nite-dišerence implementation of the method and apply it to an elasto-viscoplastic model of a bulk metallic glass based on the shear transformation zone theory. We show that in a two-dimensional plane strain simple shear simulation, the method is in quantitative agreement with an explicit method. Like the žuid projection method, it is e›cient and numerically robust, making it practical for a wide variety of applications. We also demonstrate that the method can be extended to simulate objects with evolving boundaries. We highlight a number of correspondences between incompressible žuid mechanics and quasi-static elastoplasticity, creating possibilities for translating other numerical methods between the two classes of physical problems.