Equilibrium of Linear Dislocation Arrays in Heterogeneous Materials

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Abstract
The analysis of linear dislocation arrays, piled up in the soft phase of a two-phase material, is extended. The equilibrium positions of dislocations are determined by numerical analysis. A result of the computation is the confirmation of the linear approximation L/n = (A/σ) (α+βK), 0<K≤1. Here, L is the length of the array, n is the number of dislocations, σ is the applied stress, α and β are constants for a fixed n, and A is the usual material constant for the dislocation field. The heterogeneity parameter K is defined as a function of the elastic constants of the neighboring phases, ranging from 0 to 1 for the present study. It is shown that for large arrays of either screw dislocations against a welded boundary or edge dislocations against a slipping boundary, α and β are practically constants for all large n's and α =̇ 2 and β =̇ 1.1. For edge arrays against a welded boundary, only a simple case is examined, i.e., when Poisson's ratio for the two neighboring phases is identical. The linear relation is good in certain ranges. By using the above relation, an inverse square root expression for the applied stress and the length of the array (the Hall-Petch type) is derived for heterogeneous materials.