Simulations of a stretching bar using a plasticity model from the shear transformation zone theory

Abstract

An Eulerian simulation is developed to study an elastoplastic model of amorphous materials that is based upon the shear transformation zone theory developed by Langer and coworkers [1]. In this theory, plastic deformation is controlled by an effective temperature that measures the amount of configurational disorder in the material. The simulation is used to model ductile fracture in a stretching bar that initially contains a small notch, and the effects of many of the model parameters are examined. The simulation tracks the shape of the bar using the level set method. Within the bar, a finite difference discretization is employed that makes use of the essentially nonoscillatory (ENO) scheme. The system of equations is moderately stiff due to the presence of large elastic constants, and one of the key numerical challenges is to accurately track the level set and construct extrapolated field values for use in boundary conditions. A new approach to field extrapolation is discussed that is second order accurate and requires a constant amount of work per gridpoint.

Reference map technique for finite-strain elasticity and fluid–solid interaction

“…The fluid-solid interface is described by a level set function φ held at cell corners, with solid grid points corresponding to φ ≤ 0 and fluid grid points corresponding to φ > 0. The level set update is carried out using the hybrid approach described by Rycroft [31] that makes use of the method described by Chopp [5] to update points straddling the interface, coupled to a second-order fast marching method to determine points that are further away.…”

“…The level set field can also be used to carry out linear field extrapolation using the equations described by Aslam [1] and the algorithms described by Rycroft [31]; the routines can be applied in both directions, so that fluid fields can be extrapolated to solid grid points and vice versa. In addition, a routine has been written to allow linear extrapolation of the cell-centered stress fields using the corner-centered level set function.…”

“…10 giving an Eulerian numerical scheme where the stress tensor and the velocity vector fields are stored and updated. Hypoelasticity is simple to use and effective in certain simulations (as in [36,31]), however it carries physical drawbacks that make it inappropriate for our work here: In view of future intentions with simulating non-equilibrium or thermalized materials, hypoelastic relations are problematic as they lack connection to a thermodynamic potential. Moreover, as a consequence, certain processes can cause hypoelastic laws to give pathological results, such as a non-zero stress at zero deformation after cyclic straining [17].…”

“…The calculations presented in this section can be applied by considering the fields available at a single grid point. It is assumed that by making use of existing extrapolation methods [1,31] that linear extrapolations of all fields are available at the grid pointthese are labeled with an "ex" superscript. In this appendix, the prodecures are described to make use of these fields to construct ghost field values, labeled with a "g" superscript, which are then employed in the finite-difference update.…”

Fracture Toughness of Metallic Glasses: Annealing-Induced Embrittlement

“…The STZ model of amorphous plasticity [14][15][16][17] has been recently shown to emerge within a systematic formulation of non-equilibrium thermodynamics [18,19] and to capture a wide range of glassy deformation phenomena [17,[20][21][22][23][24]. Its main advantage in our context is that it offers a way to quantify the degree of structural relaxation and the deformation-driven evolution of structural disorder.…”