Theoretical calculation of the stress-strain behavior of dual-phase metals with randomly oriented spheroidal inclusions
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INTRODUCTION
DUAL-PHASE metals constitute an important class of engineering materials. Unlike most ordinary metal-matrix composites, a dual-phase metal usually consists of two ductile phases, both capable of undergoing plastic flow. The nonlinear stress-strain behavior of both phases can be markedly different, and therefore, the overall behavior of the system will depend on the volume fraction of both phases. In addition, the morphology of the dual-phase system is also an important factor influencing its overall response. While the microstructures of some systems are interconnected, most dual-phase metals exhibit the inclusion/matrix microgeometry. In the latter case, the shape of inclusions and whether the softer or the harder phase (in the sense of flow stress at a given total strain) serves as the surrounding matrix will also affect the final elastoplastic response. This article is concerned with the development of a theoretical principle for the determination of the overall stressstrain behavior of a dual-phase metal which exhibits such an inclusion/matrix microstructure. The inclusions are taken to be spheroidal in shape, with a common aspect ratio (the length-to-diameter ratio), and are randomly oriented and homogeneously dispersed in the matrix. This allows one to examine the effect of inclusion shape--from very thin discs (a --> O) to oblate inclusions (c~ < 1), from spheres (a = 1) to prolate inclusions (~ > 1), and all the way to very long needles (a --+ oo)--on the overall yield strength of the isotropic system. In this theoretical development, no restriction will be placed on whether the inclu-
A. BHATTACHARYYA, Research Scientist, is with the Department of Aerospace Engineering, Texas A & M University, College Station, TX 77843-3141. G.J. WENG, Professor, is with the Department of Mechanical and Aerospace Engineering, Rutgers University, New Bnmswick, NJ 08903. Manuscript submitted April 26, 1995. METALLURGICAL AND MATERIALSTRANSACTIONS A
sions are harder or softer than the matrix. Our focus will be on the influence of these factors on the overall response of the two-phase system, with an application to dual-phase steels at the end. A schematic diagram of such a system is shown in Figure 1(a). While the mechanical behavior of a two-phase isotropic material containing randomly oriented spheroidal inclusions has been a subject of considerable interest in the past, the research has been primarily on the elastic properties (e.g., Wu [1] and Tandon and Wengt2]), and it was only very recently that the problem involving randomly oriented spheroidal inclusions in a ductile matrix was considered by Bhattacharyya and WengJ 3] Their study was directed toward the ordinary metal-matrix composite, where inclusions are elastic and share a common elastic property. In the present problem, however, the plastic state of the inclusions is orientation dependent, and therefore, the inclusions do not share the same secant moduli at a given stage of external loading. Consequently, the theory developed by Bhattacharyya
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