The Expanded Lever Rule on the Phase Transformation Rate Curve for Phase Volume Fraction Calculation in Thermal Simulati
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THERMAL simulation is a common experimental method used to study a range of materials.[1–3] Recorded data in thermal simulations, including time, temperature, and dilatation, can imply phase transformation information, such as phase start and finish temperatures, phase kinetics, phase transformation rate, and phase volume fractions. Especially, phase volume fractions, which are important parameters for microstructure characterization, can provide primary references for analyzing material properties. Metallography analysis[4,5] and the lever rule[6–8] are widely used to count phase volume fractions, but both have limitations. The former results in large deviations when two phases are similar in morphology and the latter cannot work when two phases overlap. Therefore, neither method can calculate reliable volume fractions for two overlapped phases that are similar in morphology. To address this problem, the expanded lever rule has been put forward in our previous study.[8] The expanded lever rule has an advantage as it
XUAN-WEI LEI, RONG-BO YANG, LIN-LIN XIAO, JIA-JUN DING, XUE-HUI ZHANG, CHAO-BIN LAI, and TONG-XIANG LIANG are with the Faculty of Materials, Metallurgy and Chemistry, Jiangxi University of Science and Technology, Ganzhou 341000, China. Contact e-mails: [email protected], [email protected] Manuscript submitted January 21, 2020.
METALLURGICAL AND MATERIALS TRANSACTIONS A
can calculate phase volume fractions when two phases are only partly overlapped. Principles of the expanded lever rule on the differential dilatation curve and a simple principle (approximate symmetry principle) in applying the expanded lever rule were introduced. Reliability of the expanded lever rule has been exhibited and verified by metallography analysis but the steps for the expanded lever rule on the differential dilatation curve are complicated, which affects its calculation accuracy and widespread application. In this study, the expanded lever rule on the phase transformation rate curve with simplified steps is introduced. The approximate symmetry principle in applying the expanded lever rule is optimized and the measurement uncertainty is discussed. Experiments are conducted to illustrate the application of the expanded lever rule. The revealed advantages can promote its application.
II. THE EXPANDED LEVER RULE ON THE PHASE TRANSFORMATION RATE CURVE A. Principles of the Expanded Lever Rule on the Phase Transformation Rate Curve Metal with length L0 at initial temperature T0 will expand to LT at temperature T if no phase transforms. Thermal dilatation is approximated as[9,10]
DðTÞ ¼ LT L0 ¼ aðT T0 ÞL0 ;
½1
where a is the expansion coefficient. If a phase transformation occurs during a cooling process, the dilatation should be expressed as DðTÞ ¼
8 < D1 ðTÞ ¼ a1 T T00 L0 ; T>Ts Dc ðTÞ ¼ a1 T T00 L0 ½1 fðTÞ þ a2 T T000 L0 fðTÞ; Tf T Ts ; : 00 D2 ðTÞ ¼ a2 T T0 L0 ; T
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