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Non-proportional plastic deformation at the micron scale: Single crystal Cu cantilever beams subjected to orthogonal bending
Department of Mechanical and Industrial Engineering, Louisiana State University, Baton Rouge, LA 70803, USA.
KTH, School of Engineering Sciences (SCI), Engineering Mechanics, Material and Structural Mechanics.ORCID iD: 0000-0002-9509-2811
KTH, School of Engineering Sciences (SCI), Engineering Mechanics. Institute of Materials Physics and Technology, Hamburg University of Technology, Hamburg, Germany.ORCID iD: 0000-0002-3327-6711
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA.
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2026 (English)In: Journal of the mechanics and physics of solids, ISSN 0022-5096, E-ISSN 1873-4782, Vol. 206, article id 106375Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Elsevier BV , 2026. Vol. 206, article id 106375
Keywords [en]
Incremental and non-incremental SGP formulation, Sequential cantilever orthogonal bending, Strain gradient plasticity
National Category
Applied Mechanics
Identifiers
URN: urn:nbn:se:kth:diva-371980DOI: 10.1016/j.jmps.2025.106375ISI: 001587404000001Scopus ID: 2-s2.0-105017225274OAI: oai:DiVA.org:kth-371980DiVA, id: diva2:2009471
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QC 20251028

Available from: 2025-10-28 Created: 2025-10-28 Last updated: 2025-10-28Bibliographically approved

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Dahlberg, Carl F. O.Fischer, Tim

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