The Question of Statistical Superposition

In the previous subsection, we considered the emergence of the parameter F in eqn (11.54) on the basis of an elastic treatment of the interaction between a single dislocation and a single obstacle. In order to complete the analysis associated with eqn (11.54) we must now determine the parameter Leg which characterizes the distribution of obstacles. To do so, we will invoke statistical arguments of varying 

A more sophisticated consideration of the forces that are exerted upon a dislocation as it glides through a medium populated by a distribution of obstacles is depicted in fig. 11.26. In this case, the fundamental idea that is being conveyed is to divide the character of various obstacles along the lines of whether or not they are localized or diffuse and whether or not they are strong or weak . These different scenarios result in different scaling laws for the critical stress to induce dislocation motion. 

11 Points, Lines and Walls Defect Interactions and Material Response 

If we take the result given above and exploit it in the context of eqn (11.54) we obtain one of the crucial scaling relations for obstacle limited motion of dislocations, namely. 

This result is one which Kocks (1977) goes so far as to note already contains the most important result of dislocation theory namely, that the flow stress is proportional to the square root of concentration . Indeed, to the extent that we are interested in understanding the observed macroscopic plastic properties of materials, scaling relations involving the concentration of obstacles are pivotal. 

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