Numerical methods and improvements for simulating quasi-static elastoplastic materials

TL;DR

This work addresses the computational challenge of simulating quasi-static elastoplastic materials, such as bulk metallic glasses, by leveraging a mathematical correspondence with incompressible fluid dynamics. It introduces a second-order in time projection method, a finite element projection step yielding SPD systems, and an adaptive global timestepping strategy, validated on a two-dimensional STZ-based BMG model. The approach achieves true second-order temporal accuracy across stress, plasticity, and velocity fields, while enabling substantial reductions in timestep counts without sacrificing accuracy. The methods are general to elastoplastic materials and pave the way for higher-order spatial discretizations and timestepping schemes in future work.

Abstract

Hypo-elastoplasticity is a framework suitable for modeling the mechanics of many hard materials that have small elastic deformation and large plastic deformation. In most laboratory tests for these materials the Cauchy stress is in quasi-static equilibrium. Rycroft et al. discovered a mathematical correspondence between this physical system and the incompressible Navier-Stokes equations, and developed a projection method similar to Chorin's projection method (1968) for incompressible Newtonian fluids. Here, we improve the original projection method to simulate quasi-static hypo-elastoplasticity, by making three improvements. First, drawing inspiration from the second-order projection method for incompressible Newtonian fluids, we formulate a second-order in time numerical scheme for quasi-static hypo-elastoplasticity. Second, we implement a finite element method for solving the elliptic equations in the projection step, which provides both numerical benefits and flexibility. Third, we develop an adaptive global time-stepping scheme, which can compute accurate solutions in fewer timesteps. Our numerical tests use an example physical model of a bulk metallic glass based on the shear transformation zone theory, but the numerical methods can be applied to any elastoplastic material.

Figures (7)

• Figure 1: Schematic illustrations of (a) the projection method for incompressible Navier--Stokes equations; (b) the projection method for QS elastoplasticity.
• Figure 2: Snapshots of the effective temperature $\chi$, the scaled non-dimensionalize pressure $p/s_Y$ and the scaled non-dimensionalize magnitude of the deviatoric stress $\Bar{s}/s_Y$ in a shearing simulation. The simulation was obtained by using an $N=360$ resolution grid, with constant timestep size $\Delta t=0.5 \cdot 10^4 \cdot h\cdot t_s/L$. The FEM procedure described in Sec. \ref{['sec: FEM projection']} is used for the projection step.
• Figure 3: The accuracy plot for the reduced model and the full model. In the plot, the errors for the three fields, $E_{\mathbf{v}}$, $E_{\boldsymbol\sigma}$ and $E_{\chi}$, are non-dimensionalized by the scaling $E_{\mathbf{v}}/U$, $E_{\boldsymbol\sigma}/s_Y$ and $E_{\chi}/\chi_{\infty}$, respectively. The slopes of the log--log plot are calculated using linear regression of the log--log solution accuracy values from grids of sizes $N=280, 360, 504$. The slope values are labeled in the plot, and we see that all fields reach full second-order accuracy in their solution fields for both the reduced and the full models.
• Figure 4: (a) Comparison of the evolution of $Q^{*}$ versus time $t/t_s$, of the constant timestepping scheme $\Delta t=0.5 \cdot 10^4 \cdot h\cdot t_s/L$, and the adaptive timestepping scheme using $Q^{*}_{\text{tol}}=0.001 h^2/L^2$; For the adaptive timestepping scheme, $Q^{*}$ exceeded $Q^{*}_{\text{tol}}$ only rarely, at the onset of plastic deformation around time $t=11.5\cdot 10^4 t_s$. The scheme then quickly adjusted $\Delta t$ and $Q^{*}$ remained within $Q^{*}_{\text{tol}}$ for the rest of the simulation. (b) Comparison of the $\Delta t$ step size over time for the two schemes.
• Figure 5: (a) Comparison of accuracy of solution fields. Both the constant and the adaptive timestepping schemes reached the same level of solution accuracy; (b) Comparison of the number of timesteps taken for the two schemes to reach the same level of solution accuracy. The adaptive timestepping scheme took significantly fewer timesteps.