An Eulerian projection method for quasi-static elastoplasticity

Hypoelastic-plastic models, which invoke an additive decomposition of the total rate of deformation into elastic and plastic parts, are quite common in various branches of solid mechanics. In the framework of these models, the Eulerian velocity field is usually regarded as the basic field of interest. As the timescale characterizing plastic deformation is typically small compared to elastic wave travel times, and when the loading rates are not very high, quasi-static stress equilibrium is essentially maintained throughout the deformation process. This means that the inertial term involving the time derivative of the velocity field (acceleration) can be omitted from the linear momentum balance equation, which in turn implies that there is no way to explicitly update the velocity field in time. Retaining the inertial term and employing an explicit simulation method would make it prohibitively computationally expensive to consider realistic strain rates, which is physically relevant in many problems.

In the paper attached below (by Chris Rycroft, Yi Sui and Eran Bouchbinder) we address this problem by building a mathematical correspondence between Newtonian fluids in the incompressible limit and hypoelastic-plastic solids in the quasi-static limit. In the incompressible fluids problem, it is the pressure field (rather than the velocity field) that cannot be updated explicitly in time. A well-established numerical approach to solve this problem is Chorin's projection method, whereby the fluid velocity is explicitly updated, and then an elliptic problem for the pressure is solved, which is used to orthogonally project the velocity field to maintain the incompressibility constraint.

Using this correspondence between these two different classes of physical problems, we formulate a new fixed-grid, Eulerian projection method  -- analogous to Chorin's projection method for incompressible fluids -- for simulating quasi-static hypoelastic-plastic 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 finite-difference implementation of the method and apply it to a specific elasto-viscoplastic model. We demonstrate via a few examples that the method is in quantitative agreement with an explicit method. We also demonstrate that the method can be extended to simulate objects with evolving boundaries.

We find that the numerical method developed here offers a useful practical approach for dealing with hypo-elastoplastic materials in the quasi-static limit. One of the main advantages of the fluid projection method is that it maintains the incompressibility condition through a single algebraic problem for the pressure, which is generally well-conditioned and can be carried out efficiently, and we find that many of the same benefits remain valid for the elasto-plasticity method we develop. Throughout the paper, we find a surprising number of correspondences between the two methods, such as analogous considerations for boundary conditions or the uniqueness of solutions. The mathematical connection opens up interesting possibilities for translating numerical methods for incompressible fluid mechanics over to quasi-static elastoplasticity and vice versa.