Longitudinal Relaxation of a Thermally Stressed Fiber by Prismatic Dislocation Punching
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LONGITUDINAL RELAXATION OF A THERMALLY STRESSED FIBER BY PRISMATIC DISLOCATION PUNCHING DAVID C. DUNAND AND ANDREAS MORTENSEN Department of Materials Science and Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139 ABSTRACT A model predicting the number of prismatic loops dislocation punched at the ends of a cylindrical fiber by thermal mismatch stresses is presented and compared to another based on a mismatching ellipsoid. The longitudinal stress in the fiber and the interfacial shear stress are derived by adapting a shear-lag model to the plastic portion of the interface. In certain cases, the central part of the fiber is strained by plastic and elastic interfacial shear until it exhibits no mismatch with the matrix. This leads to a critical fiber length above which the number of punched loops is constant. INTRODUCTION When a composite is subjected to a temperature change, internal stresses are generated if matrix and reinforcement have different coefficients of thermal expansion (CTE). In the case of metal matrix composites (MMCs), plastic relaxation occurs in the ductile matrix when these stresses are high enough to generate dislocations. Prismatic loop punching due to thermal stresses has been observed in metals with submicroscopic particles of spherical [1] and irregular [2,3] shape, as well as MMCs reinforced with large particles [4] and whiskers [5,6]. We have recently used silver chloride (a transparent salt deforming by slip in a manner very similar to metals) to image in the bulk such prismatic loops punched by microspheres, particles and fibers [7,8]. Fig. 1 shows a row of decorated loops punched by a mismatching glass fiber embedded in silver chloride. Taya and Mori [6] have modelled prismatic punching by a mismatching ellipsoid. They use Eshelby's equivalent inclusion model [9] and assume a continuous distribution of dislocations to predict the punching distance. In the model presented here, we use the shear-lag approximation by Cox [10] to derive, in the case of a mismatching cylindrical fiber, the interfacial shear stress and the fiber stress when the interface is plastically deformed. We also give expressions for the number of prismatic loops punched by the fiber and perform a parametric study to illustrate the model and compare it to that of Taya and Mori [6]. A more detailed version of this model will be published elsewhere [11].
Figure I: Row of decorated prismatic dislocation loops punched by a glass fiber embedded in silver chloride, quenched from 673 K to room temperature. Experimental procedures are given in Reference [7]. Mat. Res. Soc. Symp. Proc. Vol. 209. ©1991 Materials Research Society
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THEORY
Consider a fiber embedded in an infinite matrix, both phases being initially stress-free at high temperature. If the matrix CTE is larger than that of the fiber (as is the case in most MMC systems), the matrix shrinks more than the fiber upon cooling, resulting in a stressed interface with the fiber in compression and the matrix in tension. At f
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