Attaining the rate-independent limit of a rate-dependent strain gradient plasticity theory

Abstract

The existence of characteristic strain rates in rate-dependent material models, corresponding to rate-independent model behavior, is studied within a back stress based rate-dependent higher order strain gradient crystal plasticity model. Such characteristic rates have recently been observed for steady-state processes, and the present study aims to demonstrate that the observations in fact unearth a more widespread phenomenon. In this work, two newly proposed back stress formulations are adopted to account for the strain gradient effects in the single slip simple shear case, and characteristic rates for a selected quantity are identified through numerical analysis. Evidently, the concept of a characteristic rate, within the rate-dependent material models, may help unlock an otherwise inaccessible parameter space.

A steady-state modeling framework incorporating the Kuroda–Tvergaard model: demonstrated on single crystal crack growth

“…The existence of a characteristic velocity is in accordance with the findings of e.g. Nielsen and Niordson [17], Juul et al [12], El-Naaman et al [5]. -The behavior of either monotonic increase or decrease leads to an identification of a characteristic velocity at which the macroscopic toughness becomes independent of the rate sensitivity i.e.…”

“…Generally, when a back stress of approximately 60% of the critically resolved shear stress was reached, the solution started to diverge rendering solution impossible within the present framework. Similar problems are also experienced with this type of model for traditional Lagrangian frameworks (El-Naaman et al [6,5]) and the steady-state framework inherits similar stability issues. [1100]…”

“…The intersection of curves (defining the characteristic velocity) are marked by arrows, and it establishes a point at which the crack tip shielding is independent of the rate-sensitivity, m, despite the fact that the plastic zone can be different and that it depends on rate-sensitivity at this specific velocity. What is intriguing is the fact that the Kuroda-Tvergaard model is not formulated in a rate-independent version, but the results presented may still be interpreted as the rate-independent limit, although the numerical calculation for m → 0 is unstable and essentially out of reach (see also discussion in El-Naaman et al [5]). Thus, the existence of the characteristic velocity establishes a possibility of future studies of rate-independent results by the Kuroda and Tvergaard model.…”

“…As discussed by many authors (e.g. in Gurtin [9] Evers et al [7], Bayley et al [2], El-Naaman et al [6,5], Husser and Bargmann [10]), the back stress can be interpreted as a long range internal stress field that evolves with spatial gradients of the geometrically necessary dislocation (GND) density. However, by increasing the length parameter, or diminishing size, the back stress plays an increasing role.…”