A dislocation shielding model for the fracture of semibrittle polycrystals
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I.
INTRODUCTION
GRAIN size effects on fracture toughness, the ductileto-brittle transition temperature (DBTT), and cleavage, in general, have been thoroughly investigated for more than three decadesJ t-t31 Most o f the early theoretical interpretations were treated in terms o f dislocation mechanics, lj-6j Later, explanations based upon continuum fracture mechanics t7-J31 became popular since one could analytically 19,~°1 o r numerically m,~21 calculate the local stress field and relate it to an "independently" determined macroscopic cleavage fracture stress,t7~ This "independence" begs the issue somewhat, since the local "macroscopic" stress determined from a blunt notch bent beam is compared to the local "macroscopic" stress in front o f a c r a c k tip. Thus, one continuum field is being compared to another. While this approach has enjoyed considerable success, particularly where brittle secondphase particles nucleate cleavage, it is dissatisfying in that it does not speak to the theoretical stress which must locally exist to separate cleavage planes. In a contemporary development, but with minimal experimental confirmation, dislocation mechanics has resurfaced starting with the Rice-Thomson model, It41 which addresses dislocation shielding. A series o f articles by Lin and Thomson, I~'~6j Weertman and co-workers, 117n8] D a i and Li, Iw'2°l addressed equilibrium issues, mostly concerned with dislocation emission and the transition between emission and fracture. There has been some advance in dealing with silicon where St. Johnt2jl and Hirsch et alJ22J have utilized the kinetics o f dislocation/cracktip interactions to explain the DBTT. In large part, however, the experimental data have addressed the crack-tip emission o f dislocations, since these could be studied by thin-film electron microscopy, t23,241 birefringence, t23,26~ o r etch pits o f alkali halides, i27] T o our knowledge, there R . R . KELLER, Postdoctoral Fellow, is with the Institut fiir Werkstoffwissenschaften, University of Erlangen-Niirnberg, D-8520 Erlangen, Federal Republic of Germany. Y. KATZ, Visiting Professor, on leave from the Nuclear Research Center, Beer-Sheva, Israel, H. HUANG, Graduate Student, and W . W . GERBERICH, Professor, are with the Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455. Manuscript submitted March 3 0 , 1992. METALLURGICAL TRANSACTIONS A
has been only one experimentally based study attempting to link these dislocation shielding models to polycrystalline fracture toughness,t28~ In that, a link-up was attempted from transmission electron microscope (TEM) observations in single and polycrystalline Fe-Si to polycrystalline fracture toughness. This used Li's t2°~ original analysis, which gave the equilibrium n u m b e r o f dislocations emitted from a mode II c r a c k as blocked by a grain boundary. An asymptotical solution tzS1 gives 4(1-/,) ~dd[ ~ d ] - - K - 7y N - - - /zb
[11
where p and p. are Poisson's ratio and shear modulus, respectively, b is the Bu
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