Influence of grain size on the constitutive response and substructure evolution of MONEL 400
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I. INTRODUCTION
THE strength of polycrystalline metals and alloys is well documented to increase with decreasing grain size. This strengthening effect has been empirically shown to follow the linear relation s 5 M(s0 1 ky ? d2n)
[1]
where s is either the yield stress or flow stress at a fixed strain, s0 is referred to as a friction stress, d is the grain diameter, ky is the “unpinning constant” at the yield point (similarly, kf is used when referring to the unpinning value at some finite plastic strain), and M is the Taylor orientation factor. The terms s0, ky , kf , M, and n are material-dependent constants. The parameter ky was originally considered to be a measure of the stress required at the tip of a dislocation pileup to unlock pinned dislocations[1] or to create new dislocations in the adjacent grain;[2] later analysis,[2] as described subsequently, suggests that it reflects the stress necessary to cause a dislocation emission from a boundary or source. Physically, this reflects the resistance of grain or phase boundaries to the spread of slip bands.[1,3–8] This strengthening relationship, with n 5 1/2, was formulated to account for the grain-size dependency of the yield strength of mild steels by Hall[9] and Petch.[10] Over the past 40 years, this equation, referred to as the “Hall–Petch equation,” has been used to correlate a wide range of grain-size (or microstructural-unit size, i.e., cell-size, subgrain, etc.) vs yield or flow-stress data. Hall–Petch relationships have been shown, in several reviews, to correlate with a number of additional mechanical GEORGE T. GRAY III, Team Leader, Dynamic Properties, MST-8, and SHUH RONG CHEN, Technical Staff Member, are with the Los Alamos National Laboratory, Los Alamos, NM 87545. KENNETH S. VECCHIO, Professor, is with the Department of AMES, University of California-San Diego, La Jolla, CA 92093-0411. Manuscript submitted January 26, 1998. METALLURGICAL AND MATERIALS TRANSACTIONS A
properties, including indentation hardness, tensile ductileto-brittle transition temperature, fatigue-limit strength, fatigue-crack growth resistance, and polycrystalline creep rate.[1–3,11–13] In many of these cases, the n 5 1/2 power relation has been verified over a wide range of grain sizes, and the dependency of yield strength on grain size can be significant for many materials, while, in other materials, grain-size effects on yielding are quite modest. Typical values for the material parameters s0 and ky , for a large number of metals and alloys, including some ordered intermetallics, have been summarized by Hall.[1] The range in effects of grain size on yield or flow stress at small strains is given in Table I for several materials. Although alternative relationships to n 5 1/2 (i.e., n values ranging from 1/3 to 1) have been offered by several authors, including Baldwin,[14] Kocks,[15] Conrad,[16] Anderson et al.,[17] and Mecking,[18] the n 5 1/2 dependency has been found to be widely applicable to steels, as well as to a large number of other metals and alloys.[1] Several in-
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