Forces on Dislocations in Multilayer Structures
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FORCES ON DISLOCATIONS IN MULTILAYER STRUCTURES SOMIR V. KAMAT*, J.P. HIRTH* AND B. CARNAHAN** Ohio State University, Dept. of Metallurgical Engineering, Wlumbus, Ohio, 43210 Dept. of Chemical Engineering, Ann University of Michigan, Arbor, Michigan, 48109 ABSTRACT In multilayer structures, dislocation forces that arise from coherency strains are well understood in both the isotropic and Image forces produced by elastic anisotropic elastic cases. inhomogeneity have also been developed for single layers surrounded by extended phases. Image forces for the multilayers are shown to reduce to the simpler single embedded layer case to a good approximation. Image forces are shown to lead to a " stand-off " position of interface dislocations from the interface, with several implications for the properties of multilayer structures. INTRODUCTION Multilayer structures are of current interest as an example of nanometer-scale, artificially constructed microstructures (1). For strained multilayer structures in particular, the mechanical properties are important in connection with the stability of the structures. In the strained multilayers, forces arising from coherency strains are well understood in both the isotropic and anisotropic elastic cases (2-5) and have been included in analyses of the vulnerability of the multilayers to damage by dislocation or crack injection (5). In all of these treatments, effects arising from the elastic inhomogeneity of the component layers have been neglected. In a recent analysis (6), the image forces associated with elastic inhomogeneity have been derived for screw dislocations in multilayer structures. The results showed that multiple image terms were important within a given layer, but that, to a good approximation (within five percent) the multilayer situation could be represented by a single layer bounded by two semiinfinite layers. In this present work, we use this three-layer approximation with its simple analytical form, to determine the equilibrium " stand-off " position of a dislocation from the interface in a strained multilayer structure. We treat the case of an infinite straight screw dislocation with its axis parallel to the interfaces of a given layer. The stand-off distance is determined by a balance of the forces caused by image forces and coherency forces. FORMULATION The geometry of the multilayer system is as shown in Fig. 1. The layer thickness is h, the dislocation is at position X in the material with the smaller shear modulus, and the reduced distance is defined by the parameter a = (X / h). The image force in the three-layer approximation for the isotropic case (6) is Mat. Res. Soc. Symp. Proc. Vol. 103. '1988 Materials Research Society
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B
A
A
B
A
X
a---
hB
;
A
B
B
x
-hA-Ia--
hB-
0,.
Fig. 1 a. Multilayer structure and its three layer approximation b. Unstrained A and B layers and strained configuration (dashed lines). FI/L
Tyz bz
a-(x: )
where
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=
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:
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[I] [2] [3]
57
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0 = ( P1 - MA ) /
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