On the flow behavior of constrained ductile phases
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I.
INTRODUCTION
N U M E R O U S studies have shown that improved fracture toughness can be achieved by the incorporation o f a ductile second phase into a brittle matrix. Examples o f current o r potential technological significance are tungsten carbide toughened with cobalt network, tl,:~ zirconia toughened with zirconium network, t3~ alumina toughened with dispersed molybdenum, tal magnesia toughened with cobalt and nickel particles o r fibers, ]5] and glass-enamels toughened with dispersed aluminum and nickel particles.[6] Successful toughening has also been observed in titanium aluminide ]7~ and molybdenum disilicide L8,9] reinforced with n i o b i u m pancake o r filament. The primary toughening mechanism o f ductile reinforcement has been attributed to the bridging o f ductile ligaments. I~°-~4] When the size o f the bridging zone in the wake o f the c r a c k tip is small relative to the c r a c k length and the specimen dimension, the contribution to fracture toughness from bridging can be estimated by extending the cohesive force m o d e l]~5] to the ligament bridging ma2a and can be written as AG =
or(u) du
[1]
where o-(u) is the n o m i n a l stress carried by the constrained ductile reinforcement for a given c r a c k opening u, Vf is volume fraction o f the reinforcement, u* is the c r a c k opening at the point when the ductile reinforcement fails, and the definite integral, designated as ~ in the text, is the work o f rupture o f the constrained ductile ligament. Thus, the key to predict the increased fracture toughness is to calculate o-(u) as a function o f c r a c k opening. Recognizing that o-(u) is different from that L. XIAO, Research Assistant, and R. ABBASCHIAN, Chairman and Professor, are with the Department of Materials Science and Engineering, University of Florida, Gainesvitle, FL 32611. Manuscript submitted January 3 1 , 1992. METALLURGICAL TRANSACTIONS A
measured in a simple tensile test, several investigators have attempted to relate o-(u) to the uniaxial stress-strain properties o f the ductile phase. The methods used included a slip-line field analysis, m,12] finite element methods, ~1~'13] spring models, 11°'~4J and geometric models. [11,13,~6] Experimental determination o f the ~ ( u ) has also been conducted with a tensile test on a single constrained ductile phase. ]16,17381 Ashby e t a l . t16] found that the work o f rupture was enhanced by limited decohesion at the interface. Deve e t a l . , ~81 however, revealed a more complicated situation; i . e . , whether or not a limited decohesion was beneficial to a high work o f rupture depended on the work-hardening capability o f the ductile phase. Furthermore, Ashby e t al.tl6] also found that the work o f rupture normalized with both the yield strength and the size o f the ductile phase (lead metal in their study) was independent on the size o f the lead in the matrix. Thus, the work o f rupture measured from one size o f ductile phase can be extended to other sizes o f ductile phase. These results indicate that more experi
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