Does anyone know a rationale for this characteristic of all metals?
You will find it in most texts on the mechanical behavior of materials. I imagine Astronuc will be able to recommend good ones.
You need to be somewhat familiar with at least the basics of dislocation theory. But simplistically put, at low to moderate temperatures, grain boundaries act as barriers to dislocation movement. For a given material with a specific dislocation density at zero stress, the total number of dislocations that pile up against grain boundaries at a given applied stress is conserved. If the "number of grain boundaries" per unit volume is small (i.e., large grain size), then the number of dislocations built up at the edge of a slip plane is large. The stress build up, at this point on the grain boundary is proportional to the number of dislocations there, and hence, is proportionately large. When this stress exceeds some critical value, the dislocations get to cross the grain boundary and propagate into the neighboring grain. So, if the grain size is large, it takes a relatively small applied stress to make the number of dislocations piled up against the GB sufficient to achieve this critical stress. In other words, a large grain size makes for a low yield strength through easier dislocation movement.
The actual details of what makes the inverse square root behavior is unknown to me, beyond the knowledge that there is some tricky positional dependence on Frank-Reed multiplication - this possibly plays a role.
Sorry, but I was jointly refering the Hall-Petch Effect and the inverse Hall-Petch Effect. Most of the properties of nitinol follow this same type of curve.
Until I read this post, I wasn't aware of such a thing as the Inverse Hall-Petch effect.
I just Googled the term and it seems, at first glance, that this is expected in nanocystalline microstructures (hence, the absence of the term in older texts). I don't know anything about it yet, nor specifically, of it's relevance to the mechanical behavior of nitinol SMAs.
When I first read the term in your post, I thought it might be related to the high-temperature regime where the regular HP behavior is violated.
From what I have seen of it, it represents a characteristic curve where something like transformation temperature begins to increase with increased annealing time/temperature, but at a certain point, for nitinol it is around 300 C, the transformation temperature begins to decrease. I was going nuts about trying to understand it since the trend also applies to hardness and young's modulus.
The best explaination I have been able to guess for this is that disorder and participates build up prior to that temperature. At 300 degrees I guess the grains could migrate to relieve internal stress. This point is very well defined because I have seen with slight changes in temperature when annealing, the flexibility will dramatically change. When I was heating a wire at around 300 C, when I blew on it, it became stiff. When I stopped blowing and it warmed the wire dropped down and 'relaxed'. I am thinking of using an explaination like this to justify the relationship. I know that since I have no formal background this could be completely wrong, but any comments are greatly appreicated.
Got to refresh my memory, but I've thought the inverse Hall-Petch effect results when dislocation based plasticity 'fails' (isn't the primary mechanism of plasticity any longer) - happening when grains are of the order of 10 nm or so (really approximate figure) - when they are "simply" absent due to the size scale (dislocations essentially "are" the grain boundaries or get thrown there). Deformation then takes place via grain boundary sliding, time-dependent processes (diffusion at least) .... and when "traditional" dislocation plasticity ceases to be active is probably something relatively complex as well. Some publications have presented some "pretty consistent" numerical results predicting the behavior, many of the experimental results I think are somewhat debatable.
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