Practical Time Averaging of Nonlinear Dynamics with Application to Plasticity from Discrete Dislocation Dynamics

Abstract

A novel approach to meso/macroscale plasticity is proposed that does not involve postulating constitutive assumptions beyond those embodied in Discrete Dislocation Dynamics (DD) methodology and macroscopic elastic response. It involves forefront ideas in the mechanics of solids (e.g. continuum mechanics of defects) and applied mathematics, e.g. Young measure theory of averaging for nonlinear Ordinary Differential Equation (ode) systems and their approximation by numerical techniques. This is achieved by a carefully designed coupling of an exact, non-closed Partial Differential Equation (pde) based theory (Mesoscale Field Dislocation Mechanics, MFDM) representing the evolution of averaged dislocation dynamics with DD simulations, MFDM utilizes inputs obtained from space-time averaged response of fast, local DD simulations. The rationale behind using this coupled pde-ode based approach instead of a completely DD based approach is the vast separation in time scales between plasticity applications that operate at quasi-static loading rate (ranging between 10-6 s-1 to 1 s-1), and the fundamental time scale of dislocation motion as embodied in DD on the order of nanoseconds. Thus, it is impractical to use conventional DD to reach appreciable strain rates using realistic loading rates. We discuss the method behind generating the required constitutive inputs for MFDM from DD simulations using a rigorous mathematical theory of averaging ode response, and its essential adaptation for practical implementation that we have called Practical Time Averaging (PTA). In the final part, we derive statements for the evolution of coarse variables that represent an averaged behavior of microscopic dislocation dynamics. We show that the exact averaged evolution equations are extremely cumbersome and non-closed and, more often than not, an infinite hierarchy, and therefore it is more reasonable to close it at a low level using inputs obtained from averaged, stress-coupled, interaction dynamics of dislocations. This acts as a further justification for our coupled MFMD-DD approach.