A Simplified Approach for Developing Constitutive Equations for Modeling and Prediction of Hot Deformation Flow Stress
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SINCE hot working plays an important role in the industry for the production of materials with required shape, microstructure, and mechanical properties, the understanding of the microstructural evolution during elevated temperature deformation and characterizing the constitutive behavior describing material flow are among the main requirements.[1–3] The modeling of hot flow stress and the prediction of flow curves are important in metal-forming processes from the mechanical and metallurgical standpoints because any feasible mathematical simulation needs accurate flow description. As a result, considerable research has been carried out to model the flow stress of metals and alloys.[4–9] A constitutive relationship is a mathematical representation of flow stress as a function of temperature, strain rate, strain, and occasionally other factors such as chemical composition.[9,10] Usually, the effect of temperature and strain rate is incorporated in the Zener– Hollomon parameter (Z), which is a temperaturecompensated strain rate parameter. The simplest and most widely applied method in the literature for constitutive analysis is the modeling of flow stress using the expressions which relate Z to the flow stress as shown in Eq. [1], in which Q is the hot deformation activation energy, e_ is the strain rate, T is the absolute temperature, and finally A¢, A¢¢, A, n¢, b, n, a are the material’s parameters.[11–15] The stress multiplier a is an adjustable constant which brings ar into the correct
HAMED MIRZADEH, Assistant Professor, is with the School of Metallurgy and Materials Engineering, College of Engineering, University of Tehran, P.O. Box 11155-4563 Tehran, Iran. Contact e-mail: [email protected] Manuscript submitted January 28, 2015. METALLURGICAL AND MATERIALS TRANSACTIONS A
range that gives linear and parallel lines in ln_e vs lnfsinhðarÞg plots and it can be estimated by a b/n¢. Z ¼ e_ exp
Q RT
8 0 < A0 rn ¼ fðrÞ ¼ A00 expðbrÞ : A½sinhðarÞn
½1
The Zener–Hollomon parameter can be related to flow stress in different ways as indicated by fðrÞ in Eq. [1]. It is well known that the power law description of stress is preferred for relatively low stresses. Conversely, the exponential law is suitable for high stresses. Finally, the hyperbolic sine law can be used for a wide range of temperatures and strain rates. However, the description of flow stress by the expressions of Eq. [1] is incomplete, because no strain for determination of flow stress is specified. Therefore, characteristic stresses that represent the same deformation or softening mechanisms for all flow curves, such as steady state or peak stress, should be used for this purpose. In general, the peak stress is a widely accepted one in order to find the hot working constants as demonstrated for magnesium alloys,[9,10] steels,[16] intermetallic compounds,[17] aluminum alloys,[18] and copper alloys.[19] To make it possible to model the whole flow curve, the conventional approach is to express the constants of the hyperbolic sine equation as functions of strain using
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