Non-proportional plastic deformation at the micron scale: Single crystal Cu cantilever beams subjected to orthogonal bending

Abstract

Experiments involving abrupt non-collinear changes in the direction of loading in the plastic range have been performed on micron-scale, single crystal Cu cantilever beams to provide the first data of its kind on non-proportional loading. The data is used to assess whether existing strain gradient plasticity (SGP) theories are capable of reproducing complex deformation histories representative of micron-scale metal forming processes, for which non-proportional loading is common. The data is also used to explore an issue that has arisen in efforts to develop SGP that is sufficiently accurate for engineering applications and yet not overly complex. Specifically, using a combination of experimentation and computation, the paper examines the differences in predictions made by two classes of theories presently in the mainstream, termed “incremental” and “non-incremental”, when non-proportional plastic loading occurs at the micron scale. Orthogonal bend experiments are performed on Cu single crystal cantilever beams with square cross-sections that are symmetrically oriented with respect to the vertical and horizonal bending axes. In Stage 1, the force applied to the end of the cantilever is vertical, producing bending in the vertical plane. Abruptly, in Stage 2, a horizontal force is applied with either the vertical force held constant (force control) or the vertical end-displacement of the beam held constant (displacement control). Three cantilever sizes, with widths of the square cross-section of 2, 5 and 20 microns, have been tested. The strength elevation for cantilever widths decreasing from 20 to 2 microns is about a factor of three as compared to what would be expected based on conventional plasticity theory. The incremental and non-incremental SGP theories both capture the full non-proportional loading history, including the size effect. However, they differ in their predictions of behavior in the early portion of Stage 2, due to the abrupt change in the loading path. This difference will be assessed with the aid of experimental test data.