Influence of Grain Size on the Constitutive Response and Substructure Evolution of MONEL 400 GEORGE T. GRAY III, SHUH RONG CHEN, and KENNETH S. VECCHIO The influence of grain size on the constitutive behavior (strain-rate and temperature dependence of the yield stress and strain hardening) and substructure evolution of MONEL 400 was investigated. Increasing the grain size from 9.5 to 202 m was seen to reduce the quasi-static yield strength from 290 to 115 MPa, while having a minimal effect on the work-hardening response. Increasing the strain rate from quasi-static to dynamic strain rates (3000 s -1 ) was seen to increase the yield and overall flow-stress levels, but has no effect on the strong grain-size dependency exhibited by this alloy. The persistent influence of grain size to large strains is inconsistent with previous d -1/2 pileup grain-size modeling in the literature, which predicts convergence at large strains. Substructure evolution differ- ences between the grain interiors and adjacent to grain boundaries supports differential defect storage processes which are consistent with previously published work-hardening d -1 modeling arguments for grain size–dependent strengthening in polycrystals. The integration of grain-size dependency into constitutive modeling using the mechanical threshold stress (MTS) model is discussed. The MTS model is shown to provide a robust constitutive description capturing yielding, large-strain work hardening, and grain-size effects simultaneously. The MTS model is, additionally, shown to satisfacto- rily address the experimentally observed transients due to strain-rate or temperature-path dependency. I. INTRODUCTION properties, including indentation hardness, tensile ductile- to-brittle transition temperature, fatigue-limit strength, THE strength of polycrystalline metals and alloys is well fatigue-crack growth resistance, and polycrystalline creep documented to increase with decreasing grain size. This rate. [1–3,11–13] In many of these cases, the n = 1/2 power strengthening effect has been empirically shown to follow relation has been verified over a wide range of grain sizes, the linear relation and the dependency of yield strength on grain size can be  = M( 0 + k y  d -n ) [1] significant for many materials, while, in other materials, grain-size effects on yielding are quite modest. Typical val- where  is either the yield stress or flow stress at a fixed ues for the material parameters  0 and k y , for a large number strain,  0 is referred to as a friction stress, d is the grain of metals and alloys, including some ordered intermetallics, diameter, k y is the “unpinning constant” at the yield point have been summarized by Hall. [1] The range in effects of (similarly, k f is used when referring to the unpinning value grain size on yield or flow stress at small strains is given at some finite plastic strain), and M is the Taylor orientation in Table I for several materials. Although alternative relation- factor. The terms  0 , k y , k f , M, and n are material-dependent ships to n = 1/2 (i.e., n values ranging from 1/3 to 1) constants. The parameter k y was originally considered to be have been offered by several authors, including Baldwin, [14] a measure of the stress required at the tip of a dislocation Kocks, [15] Conrad, [16] Anderson et al., [17] and Mecking, [18] pileup to unlock pinned dislocations [1] or to create new dislo- the n = 1/2 dependency has been found to be widely applica- cations in the adjacent grain; [2] later analysis, [2] as described ble to steels, as well as to a large number of other metals subsequently, suggests that it reflects the stress necessary to and alloys. [1] cause a dislocation emission from a boundary or source. Several in-depth review articles have summarized the Physically, this reflects the resistance of grain or phase effects of externally imposed constraints and microstructural boundaries to the spread of slip bands. [1,3–8] This strengthen- variables, such as stress state, temperature, strain rate, etc., ing relationship, with n = 1/2, was formulated to account on the parameters in the Hall–Petch relation. [1,2,3] Changes for the grain-size dependency of the yield strength of mild in the propensity for a material to cross-slip, via either struc- steels by Hall [9] and Petch. [10] Over the past 40 years, this tural ordering or decreases in stacking-fault energy, have equation, referred to as the “Hall–Petch equation,” has been been shown to increase the Hall–Petch slope due to a reduc- used to correlate a wide range of grain-size (or micro- tion of the number of active slip systems. [1,19] In mild steels, structural-unit size, i.e., cell-size, subgrain, etc.) vs yield or k y is relatively composition insensitive, [1,3] while, in copper- flow-stress data. based alloys, [20] k y is seen to increase substantially with Hall–Petch relationships have been shown, in several increased alloying. Variations in strain rate from 10 -4 to reviews, to correlate with a number of additional mechanical 2350 s -1 were found to have no effect on the k y term for the lower yield stress of pure iron tested at room tempera- GEORGE T. GRAY III, Team Leader, Dynamic Properties, MST-8, and ture. [21] The k y term for copper has been found to be indepen- SHUH RONG CHEN, Technical Staff Member, are with the Los Alamos dent of strain rate, over the range from 0.001 to 100 s -1 , as National Laboratory, Los Alamos, NM 87545. KENNETH S. VECCHIO, well as to strains of 0.20. [22] Professor, is with the Department of AMES, University of California-San While a number of models have been proposed to explain Diego, La Jolla, CA 92093-0411. Manuscript submitted January 26, 1998. the dependence of plastic flow on grain size, two general METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 30A, MAY 1999—1235