### A Quasi-static Projection Method for Three-dimensional Hypoelastoplasticity

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Nicholas Boffi, Harvard University
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A variety of materials such as metals, rocks and bulk metallic glasses can be represented using hypoelastoplasticity (HEP), where the deformation gradient tensor is additively decomposed into elastic and plastic parts. Furthermore, many processes of practical relevance are in the long-timescale limit, where the material stresses remain in quasi-static force balance. Recent work has demonstrated in two dimensions that the underlying equations for quasi-static HEP can be efficiently solved by a tensorial generalization of Chorin's projection method for the incompressible Navier-Stokes equations.

The quasi-static projection algorithm is composed of two steps: an intermediate "advection step" that consists of partially integrating the full HEP equations, and a "projection step" that enforces the quasi-staticity constraint and solves for the updated velocities. After solving for the updated velocities, the advection step is completed by considering the remaining terms from the full HEP equations. We present the first results of this algorithm in three dimensions. To do this, we develop an MPI-based parallel three-dimensional multigrid solver using C++ templates capable of solving an arbitrary linear system for an arbitrary data type (e.g. scalars, vectors, complex numbers) at each point in space to compute the projection step efficiently. Simulation results and applications to shear banding in bulk metallic glasses are presented and discussed.

**Abstract Author(s): **Nicholas M. Boffi, Chris Rycroft