Processing Maps for the Hot Forming of Polycrystalline Metallic Materials Using the Garofalo Equation
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e than 30 years several models have been developed for the determination of the most stable areas, or zones of stability, in the process of hot working of metallic materials. There has also been interest in analyzing the zones of greater thermodynamic efficiency of these processes, something necessary for the optimization and research on new materials. Consequently, these models have been used to calculate the values of the efficiency of the process of plastic flow of hot working of polycrystalline metallic materials. These calculations of efficiency and stability of the mentioned processes can be represented in maps as the temperature versus true strain rate (plane T; e_ ) of the processes. This gives rise to efficiency and stability maps that, joined, constitute the so-called processing maps. The theoretical frame of reference for the study of stability of dynamical systems starts with the theoretical works of Lyapunov in 1892, first through his PhD
IGNACIO RIEIRO is with the Department of Mathematics, Universidad de Castilla-La Mancha, Av. Carlos III s/n, 45071 Toledo, Spain. OSCAR A. RUANO is with the Department of Physical Metallurgy, CENIM-CSIC, Av. Gregorio del Amo 8, 28040 Madrid, Spain. Contact e-mail: [email protected]. Manuscript submitted November 21, 2019.
METALLURGICAL AND MATERIALS TRANSACTIONS A
thesis[1] and later through various works,[2,3] and in the theoretical work of Ziegler.[4] This last work helped to approximate the theoretical framework to a more realistic referential frame than that of the general works of Lyapunov that were more theoretical. On the other hand, Ziegler uses the concepts of thermodynamics of irreversible processes and also thermomechanics and mechanics of continuous media. The criteria of Lyapunov and those of Ziegler predict different conditions of temperature and strain rate for the stability of materials but both are valid to study the hot-working behavior of materials. In the present work, the Lyapunov criteria are not considered since this was done in a previous publication by Rieiro et al.[5] In the present work, only the stability criterion of Ziegler[4] is analyzed, which is related to plastic flow at large strains. Ziegler uses the mechanics of the continuum for the case of large plastic flow and especially the thermodynamics of irreversible processes, considering extreme principles. Later, and using the previously mentioned frameworks, the dynamic materials model, DMM, was introduced, attributed to Prasad.[6] This model, over the years, has suffered criticism, modifications, extensions and reviews by very different researchers, but it resists as a model to evaluate the stability (among other objectives of the model). Because of their importance in clarifying DMM, the works of Gegel, Malas et al., should be mentioned.[7,8]
The DMM is based on the previously referenced theoretical frameworks,[6–8] and a line of stability criteria applied to specific metallic materials under hot-forming conditions is set. The DMM is the synthesis and confluence of the dynamic theory of material
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