Superplastic Constitutive Equation Including Percentage of High-Angle Grain Boundaries as a Microstructural Parameter
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SUPERPLASTICITY refers to the ability of materials to exhibit high uniform elongation when pulled in tension while maintaining a stable microstructure.[1] This phenomenon has considerable industrial potential for the manufacture of complex sheet structures. It has been widely accepted that the constitutive relationship for superplasticity can be expressed in the following generalized form:[2]
K. WANG, formerly Research Fellow with the Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, P.R. China, is now Associate Professor with College of Materials Science and Engineering, Chongqing University, 174 Shapinba Main Street, Chongqing 400030, P.R. China. F.C. LIU, formerly Postgraduate with the Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, is now Research Fellow with the Department of Mechanical Engineering, Brigham Young University, Provo, UT 84602. P. XUE, Assistant Professor, D. WANG, Associate Professor, and B.L. XIAO and Z.Y. MA, Professors, are with the Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences. Contact email: [email protected] K. Wang and F.C. Liu have contributed equally to this work and share first authorship. Manuscript submitted February 28, 2015. Article published online November 16, 2015 546—VOLUME 47A, JANUARY 2016
p D0 Eb Q b r r0 n exp ; e_ ¼ A kT RT d E
½1
where e_ is the strain rate, A is a dimensionless value, D0 is the pre-exponential constant for diffusivity, E is Young’s modulus, b is Burger’s vector, k is Boltzmann’s constant, T is the absolute temperature, Q is the activation energy dependent on the rate-controlling process, R is the gas constant, d is the inverse grain size, p is the grain size exponent, r is the applied stress, r0 is the threshold stress, and n is the stress exponent. The superplastic data analysis for a large number of powder metallurgy-processed Al-based alloys demonstrated that the activation energies were close to that for the grain boundary diffusion of Al (84 kJ/mol), the inverse grain size dependence p and the stress exponent n were equal to 2 in Eq. [1], and the dimensionless value, A, was previously considered to be less than 50.[3–5] Equation [1] shows that for the classical constitutive equation, the grain size is the sole microstructural determining parameter of the superplastic properties, and therefore, grain refinement is considered to be the sole method for increasing the superplastic deformation rate or reducing deformation temperature. Therefore, many processing methods,[6,7] such as equal channel angular pressing (ECAP),[8] high-pressure torsion (HPT),[9] cold rolling,[10] and friction stir processing (FSP),[11] were used to refine the grains of various alloys METALLURGICAL AND MATERIALS TRANSACTIONS A
in order to shift the optimum superplastic deformation to a lower temperature and/or higher strain rate.[12–14] Generally, the
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