The law of mixtures applied to the plastic deformation of two- phase alloys of coarse microstructures
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I.
INTRODUCTION
A
number of technically important alloys consist of mixtures of soft and hard phases or constituents, of coarse microstructures (i.e., with mean-intercept-length of each phase of the order of 1/xm or larger) and compositions such that each constituent is present in relatively large amounts (typically more than 10 pct by volume). Examples of this type of composite alloy are a//3 brasses, dual-phase steels, and cemented carbides. As a group, these alloys offer useful combinations of high strength and good ductility. In plastically deforming composite alloys the softer phase always deforms plastically, but the harder phase may deform either elastically only, or plastically, depending on phase properties, microstructure, and load. In many cases, the plastic deformation behavior of two-phase alloys is characterized by large differences of strain and/or stress between the phases, and the mechanical properties reflect a synergistic interaction of the properties of the constituent phases. The plastic incompatibility between the phases causes both a strong contribution to strain hardening and high local stress concentrations which may lead to premature fracture by void initiation and propagation, unless relieved by local plastic flow. The interaction strain hardening in several two-phase alloys of specific compositions has been successfully estimated tL21 by applying the dislocation concepts and continuum mechanics of dispersion hardening theory/3J In the present work, we are explicitly concerned with identifying the contribution of each of the phases to the strength and deformation of a two-phase alloy, in general. We will follow a suggestion of Embury and Duncan I4~that the plastic deformation behavior of two-phase alloys can be understood comprehensively by combining the law of mixtures with the dislocation-continuum strain hardening theory. It is an objective of this paper to demonstrate that the law of mixtures for stress and strain, properly modified, provides a valid and useful framework for the analysis of the partition of stress and of strain between the phases during the plastic deformation of two-phase alloys. The analysis will be applied to two representative alloys, namely spheroidized steels (FeJFe3C) and duplex stainless steels K. CHO, Research Assooate, and J GURLAND, Professor Emeritus, are with the Division of Engineering, Brown University, Providence, RI 02912. Manuscript submitted August 17, 1987. METALLURGICAL TRANSACTIONS A
(F%/Fe~), the former with elastically deforming hard particles (Fe3C), the latter with a plastically deforming hard phase (Fee). II. THE MECHANICAL THEORY OF PHASE MIXTURES
A. Law of Mixtures Applied to Plastic Deformation The mechanical theory of phase mixtures is well developed for linear properties. For instance, for elastic modulus, Figure 1 shows the high and low limits proposed by Paul, 151which correspond, respectively, to parallel and series arrangements of the constituents. More closely spaced upper and lower bounds were calculated by Hashin and Shtrikm
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