Developing the Processing Maps Using the Hyperbolic Sine Constitutive Equation
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INTRODUCTION
THE prediction of flow stress and load needed for the deformation of an alloy is an important step for the design of an industrial hot-working operation. The flow stress is closely related to the both intrinsic and extrinsic workabilities of the material being deformed. The intrinsic workability is governed by the mechanisms of microstructural changes at a given deformation condition. However, the processing variables, i.e., strain, strain rate, and temperature actually control the microstructural evolutions and thereby the intrinsic and extrinsic workabilities of a material. Different constitutive equations have been used to interconnect the mentioned workability factors and derive an applicable formula for measuring the flow stress as a function of processing variables.[1–3] Even though the constitutive equations are widely used to estimate the deformation loads, the behavior of the material in terms of stable or unstable flow should be also analyzed. One of the welldeveloped practical methods to avoid the occurrence of
AMIR MOMENI, Assistant Professor, is with the Department of Materials Science and Engineering, Hamedan University of Technology, Hamedan, Iran. Contact e-mail: [email protected] KAMRAN DEHGHANI, Associate Professor, is with the Department of Mining and Metallurgy, Amirkabir University of Technology, Tehran, Iran. GOLAM REZA EBRAHIMI, Associate Professor, is with the Department of Materials and Polymer Engineering, Hakim Sabzevari University, Sabzevar, Iran. SHAHAB KAZEMI, Assistant Professor, is with the D. M. S. E., Bu Ali Sina University, Hamedan, Iran. Manuscript submitted October 14, 2012. METALLURGICAL AND MATERIALS TRANSACTIONS A
an unstable flow and premature fracture is plotting a processing map. The approach of processing map, originally proposed by Raj.[4] is defined as a representation of microstructural changes and restoration mechanisms of a given material. However, some questions still arise about the reliability of this method.[5] A processing map illustrates a superimposition of power dissipation map (PDM) and instability regions which are depicted on the basis of dynamic materials model (DMM).[6] DMM is a continuum model in which an instability criterion based on the principles of irreversible thermodynamic, as applied to large plastic flow, is utilized to mark flow instability regimes.[7] The power dissipation and the instability maps are plotted in terms of deformation temperature and logarithm of strain rate to correlate different regions with the processing variables. The principles of this approach and its applications to the hot deformation of a wide range of materials were described by Prasad and other researchers.[8–11] In the DMM model, the constitutive equation used to relate the flow stress to strain rate and temperature is of power-law type, as follows: r ¼ K_em
½1
where K is a temperature-dependent constant and m stands for the strain rate sensitivity parameter. This equation gives a good measurement of the behavior of the workpiece when it consumes the energ
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