Application of Invariant Set Theory to
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I.
INTRODUCTION
H O T - w o r k i n g operations, such as rolling, forging, and extrusion, often encounter material instabilities, such as dynamic recrystallization and shear localization, that can complicate control of these operations. It is therefore important to predict the onset of material instabilities so that they can be either controlled or avoided. In some instances, the instability is desired, for example, when dynamic recrystallization is exploited to refine grain size. Whether the instability is desirable or undesirable, however, control of the hot working process is improved if the instability can be predicted. This article considers the phenomenon of dynamic recrystallization within a state variable, nonlinear system dynamics framework. We define dynamic recrystallization as a recrystallization process that occurs as a material experiences a nonzero strain rate. The macroscopic effect of dynamic recrystallization on rate-dependent constitutive behavior is to introduce a maximum in the stress/strain response, with possible additional oscillations in the flow stress resulting from repeated cycles of recrystallization. Figure 1 presents a representative oscillating stress/strain curve for high-purity lead at relatively high homologous temperature.t1} The phenomenon is normally associated with low stacking fault energy materials that have a low rate of static and dynamic recovery, thereby permitting high dislocation densities and, consequently, high stored elastic internal energy densities to drive recrystallization. Excellent reviews of dynamic recrystallization include those by Sakai and Jonas I2} a n d C a h n . {3]
Although criteria have been proposed to predict the onset of dynamic recrystallization, most, if not all, of
STUART BROWN, Richard P. Simmons Associate Professor of Materials Manufacturing, and JOHN WLASSICH, Research Assistant, Department of Materials Science and Engineering, are with the Massachusetts Institute of Technology, Cambridge, MA 02139. Manuscript submitted September 23, 1991. METALLURGICAL TRANSACTIONS A
these criteria are applicable under a limited set of operating conditions. Luton and Sellars t4} and Sakai and Jonas t2] define the onset of dynamic recrystallization v i a critical strain criteria. The fluctuations in macroscopic stress/strain behavior are correlated with different threshold strains that represent the onset or completion of cycles of recrystallization. These critical strain measures are valid only under isothermal, constant strainrate conditions--conditions that are not frequently met during typical hot-working processes, such as rolling, forging, or extrusion. The shortcoming of a criterion based on strain derives from the inability of strain to represent the state of a hot-worked metal under nonsteady conditions. Strain is not a state variable at elevated temperatures, since the microstructure continues to evolve through thermally activated processes that continue in the absence of deformation. Other variables are therefore necessary to represent the t
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