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I.

INTRODUCTION

T H E fracture toughness of materials which fail by the nucleation, growth, and coalescence of voids nucleated at second-phase particles will be considered. While many materials contain more than one type of second-phase particle, it will be assumed that one can distinguish between two particle types, primary and secondary. Voids nucleate first at the primary particles, and such voids are termed primary voids. In steels, the primary particles are the oxides and sulfides formed in the liquid steel and during solidification. The secondary particles are more strongly bonded to the matrix, and voids are nucleated at these particles late in the fracture process (if at all); such voids are termed secondary voids. In the simplest picture of ductile fracture, voids nucleated at the primary particles can grow to impingement or coalesce by the formation of void sheets due to the nucleation of voids at the secondary particles. There is a considerable body of experimental evidence which suggests the fracture toughness of materials failing by microvoid coalescence is strongly dependent on the spacing of the primary particles. A theoretical treatment of the role of particle spacing on fracture toughness was presented by Rice and Johnson. m Rice and Johnson considered a single void of initial radius, R0, centered a distance X0 from the initially sharp crack tip. Using the Rice and Tracey ~21void growth equations, their own resuits for the strain distribution and stresses ahead of the blunting crack tip (in the absence of a void), and assure-

J.W. BRAY, formerly Graduate Student, Carnegie Mellon University, is Supervisor, Mechanical Metallurgy, Reynolds Metal Company, Richmond, VA 23219. K.J. HANDERHAN, formerly Graduate Student, Carnegie Mellon University, is President, Ellwood City Forge, Ellwood City, PA 16117. W.M. GARRISON, Jr., and A.W. THOMPSON, Professors, are with the Department of Metallurgical Engineering and Materials Science, Carnegie Mellon University, Pittsburgh, PA 15213-3890. Manuscript submitted July 17, 1989. METALLURGICAL TRANSACTIONS A

hag fracture to occur when the vertical void radius equaled the ligament between the blunting crack tip and the growing void, Rice and Johnson obtained a numerical relationship between the critical crack tip opening displacement at fracture (61c) and the quantities X0 and R0. In applying their results, they suggested that X0 be considered the average three-dimensional nearest-neighbor distance between particles and R0, the average particle radius; this measure of spacing seems reasonable, as particle spacings on the fracture surface appear to correspond more closely to this spacing than others, such as the average nearest-neighbor distance on a plane. This spacing can be approximated by X0 = 0.89Rof-l/3. [211 Thus, their numerical results suggest

t3lc = XoF(f)

[1]

where F is a function which increases slowly with decreasing primary particle volume fraction, f. While the Rice and Johnson calculation suggests a strong influence of particle spacing on toughne