Chapter 9 Constitutive Equations for Severe Plastic Deformation Processes Robert Goldstein, Sergei Alexandrov, and Marko Vilotic Abstract Processes of severe plastic deformation significantly change material properties. Therefore, it is reasonable to expect that conventional constitutive equations of plasticity theory are not applicable for severe plastic deformation processes. The main assumption made in the present paper is that the relative spin (the difference between the velocity spin and the spin of the principal axes of stress) should be included in constitutive equations to adequately describe severe plastic deformation processes. An experimental procedure to verify this assumption is proposed and partly carried out. Keywords Severe plastic deformation • Ideal flow • Material spin • Upsetting • Constitutive equations 9.1 Introduction Severe plastic deformation is a widely used method to improve material properties. Reviews of this method are provided, for example, in [1, 2]. During such deformation, intensive microstructural rearrangements occur in the metal. Therefore, it is reasonable to assume that constitutive equations for describing material flow in severe plastic deformation processes should be different from conventional constitutive equations used for analysis and design of traditional metal forming processes. Analysis of the mode of deformation in typical severe plastic deformation processes shows that an essential difference between severe plastic deformation processes and traditional metal forming processes is that the difference between the velocity spin and the spin of the principal axes of stress is more pronounced in the former processes. Therefore, it is reasonable to hypothesize that a measure of this relative spin should be included in constitutive equations. It is interesting to note that the relative spin referred to above may be very large in the vicinity of frictional interfaces in traditional metal forming processes. For example, in the case of rigid perfectly plastic material the theoretical relative spin approaches infinity in the vicinity of maximum friction surfaces [3]. However, a narrow layer of intensive plastic deformation often appears in the vicinity of frictional surfaces in real metal forming processes [4–7], which confirms the hypothesis made above. Note that constitutive equations that include various relative spins are widely adopted in the mechanics of granular materials [8, 9]. An attempt to use such models for describing material flow in severe plastic deformation processes has been made in [10, 11]. The present paper further develops the approach introduced in [10, 11]. In particular, the process of multi-stage upsetting of a long bar between V-shape dies is considered. It is shown that the mode of deformation of infinitesimal line segments on the axes of symmetry of the process possesses some qualitative properties inherent in severe plastic deformation processes and some qualitative properties inherent in ideal flow processes. A review of the ideal flow theory is provided in [12]. A property of ideal flows is that the velocity spin is equal to the spin of the principal axes of stress. Therefore, in a sense, ideal flow processes are opposite to severe plastic deformation processes. An experimental procedure to reveal a possible effect of the relative spin on material response is proposed. In particular, the procedure includes the process of the multistage upsetting of a long bar between V-shape dies. The distribution of the equivalent strain in this process is determined experimentally. This distribution can be used to select an appropriate ideal flow process and a appropriate severe deformation process, which are also included in the general experimental procedure. R. Goldstein (*) • S. Alexandrov Laboratory on Mechanics of Strength and Fracture of Materials and Structures, Institute for Problems in Mechanics of the Russian Academy of Sciences, Prospect Vernadskogo, 101-1, 119526 Moscow, Russia e-mail: goldst@ipmnet.ru M. Vilotic Faculty of Technical Sciences, University of Novi Sad, Obradovich Street, 6, 21000 Novi Sad, Serbia #The Society for Experimental Mechanics, Inc. 2017 W.C. Ralph et al. (eds.), Mechanics of Composite and Multi-functional Materials, Volume 7, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-41766-0_9 73
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