Some Quantitative Analogies Between Large-Strain Plasticity and Rectilinear Fluid Flow

Abstract

Abstract The mechanics of chip formation involves a complex interplay between surface plastic flow, material microstructure and tool-chip interface dynamics. At the continuum scale, this process has often been approximated as being one of pure shear along a very narrow zone — the shear plane. The primary assumption behind this approximation is that of steady or time-independent plastic flow. This approximation addresses the exception rather than the rule, since most metal cutting processes involve unsteady time-dependent plastic flows. In this work, we attempt to develop parallels with an analogous problem in fluid mechanics — flow past a rigid fixed body — to quantify the unsteady nature of plastic flow. Here the role of the body is played by the tool, and the flow field corresponds to the plastically deforming material. The prerequisite for fully exploiting this kinematic analogy is full-field material flow measurements. This is provided by digital image correlation (DIC), a family of techniques that use in situ imaging of the cutting process to obtain instantaneous full-field material displacements. Given this kinematic flow information, we show how one can describe the cutting process more accurately using the fluid flow analogy, and without having to resort to specific preconceived models. For steady flows, the analogy is more than merely cosmetic — the pathlines, streamlines and streaklines, all determined kinematically, coincide exactly. The analogy also allows consideration of time-dependent unsteady plastic flows that are usually beyond the purview of theoretical analyses. One can, for instance, ask the question of whether the transition from steady to unsteady plastic flow observed in metal cutting can be described by a dimensionless parameter, analogous to the Reynolds number for fluids. We show that in the case of unsteady flows, the actual deformation field is far removed from that described by conventional shear plane or slip line field models. Now the streaklines develop undulations or folds for the same boundary conditions as for the steady case (cutting velocity, depth of cut). Of the three flow lines, we attempt to extract information from streaklines, since they contain information about both spatial and temporal gradient. We present analyses of these streakline curvature features using simple geometric techniques that reflect both the spatial and temporal flow evolution. Our results shed light on the importance of considering unsteady flows in chip formation and machining. Borrowing ideas from fluids to describe these flows appears to hold significant promise for quantifying unsteady flows and their consequences for practical machining operations.