Relaxed configuration of a row of punched prismatic dislocation loops

Scripta METALLURGICA Vol. 25, pp. 607-612, 1991 Pergamon Press plc et MATERIALIA Printed in the U.S.A. All rights reserved RELAXED CONFIGURATION OF A ROW OF PUNCHED PRISMATIC DISIX)CATION LOOPS David C. Dunand and Andreas Mortensen Department of Materials Science and Engineering Massachusetts Institute of Technology, Cambridge, MA 02139, USA. (Received December 19, 1990) Introduction Punched rows of coaxial, circular prismatic dislocation loops can be produced in a crystal by several mechanisms: growth of a second phase in a ductile matrix (1), growth of gas bubbles in irradiated metals (2), mechanical indentation (3), elastic constant mismatch strains induced by isostatic pressurization (4) or by shear (5) and strain mismatch due to a difference of coefficient of thermal expansion and temperature change (6,7). The loops within such a row are nucleated one by one at their source, from which they are repelled by following loops until an equilibrium configuration is reached where nucleation stops and all loops are at rest, unable to overcome lattice friction. The resulting loop spacings in the row are of interest since they govern the local dislocation density in, as welt as the size of, the resulting plastic zone. Unlike the related problem of a pile-up of straight dislocations of infinite length, there exists no closed-form expression for loop spacing within such a row. From expressions for the elastic stress field surrounding a single loop in an isotropic crystal, Bullough and Newman (8) computed the equilibrium configuration of such a row after making the simplifying assumption that only fwst and second nearest neighboring loops interact. They considered a row of ten loops for three different values of a dimensionless parameter v which combines the lattice properties and the loop diameter. They noted that their neglect of the interaction of loops separated by more than two neighbors is invalid when loops arc closely spaced, i.e., for a large number of loops or a high value of the parameter v. In this paper, we first explore the simpler case where all but nearest neighbors interactions are negligible. This yields a closed-form solution for the problem. We then use this solution as a tool to compute equilibrium configurations for loop numbers and values of the parameter v beyond those explored by BuUough and Newman (8). From the results, we deduce a "best-fit" relationship between the dimensionless length of the row and its total number of loops, which can be used to apply results of our calculations easily. The smallest loop spacing in a row is given to define the range of validity of the calculations. The largest loop spacing in a row is also calculated, to allow measurement of lattice friction stress by microstmctural observation of such loop rows. We then calculate the equilibrium configuration of two rows of prismatic loops of opposite Burgers vector placed on opposite sides of their source. This situation can arise when a two-phase material is sheared (9) and is mathematically equivalent to a single row attracted to a free surface. Finally, we give a simple solution for the average backstress exerted on an obstacle by the relaxed row of loops. Theory Equilibrium confimn'ati0n 0f loop_ rows Consider a circular prismatic dislocation of diameter d and Burgers vector b lying in the plane z=0 with its center at the origin of a cylindrical coordinate system p, ~ and z. The shear stress on the glide cylinder r.n(~') induced by this loop in the infinite, isolropic crystal is given by Kroupa (10) and Bullough and Newman (8) as : Tr~O = ~(~ "~'' K[(~'2 + 1)'1/2] " (~.. ~.-1). E[(~.2 + 1) "1/2] • (1) 2 ~d(l-1)) (~.2+ i:/2! where