How to estimate the strength contribution from low angle grain boundaries
low angle grain boundaries form the subgrain structure in a metal. It is usually assumed that subgrains contribute to the material's strength and hardness similar to the grains. Thus, it is assumed that a Hall-Petch type relation exists between the subgrain structure and the strength of a material. This type of correlation usually includes both the 'subgrain size' and its 'orientation (angle)'.
For further discussion please find the attached document. The source for the attached document is:
FJ Humphreys, M. Hatherly. Recrystallization and related annealing phenomena. 2nd Ed, Elsevier, 2004.
Visualise the LAGB as a stack of paralle edge dislocations. Minimum stress to pass the wall with another dislocation of same type will be if it goes midway between wall dislocations. A reasonable if a bit rough approximation would be a bit more than twice the peak stress between two parallel dislocations on slip planes of half the wall dislocation spacing.
I add on to Prof. Steve Roberts answer:
1) At high temperature creep, in class II alloys, the dislocations can arrange themselves as cells or sub-grains. The driving force for their formation is the reduction in elastic energy and critical dislocation density is required (see D. L. Holt, J Applied Physics, 1970, 41, 3197). The sub-grain size remains constant in steady-state creep and the cell/sub-grain size depends strongly on the applied stress, and is inversely related to stress: this is well known (please see #1 R.K. Bhargava, PhD Thesis, "The influence of substructure on the elevated temperature deformation of an austenitic alloy", Univ. Cincinnati, 1975. # 2. O.D. Sherby etal., Metall Trans., 1977, 8A,p.843. # 3. A. Orlova, Mater. Sci. Eng, 1996, A220, p.117. # 4. S. Takeuchi, A.S. Argon, J Mater. Sci, 1976, 11, p.1542).
2) The sub-grain strengthening may not be correct, and a different view point is presented that the strengthening is related to the free dislocation density which is more important: see O. Ajaja, A.J. Ardell, Scripta Metall., 1977, 11, p.1089.
3) Coming to your question: The development of heterogeneous dislocation structure, i.e., cells or sub-grains formation (leads to soft and hard zones) has been addressed by many researchers, but the composite model by Nix and Ilschner might be useful: "Mechanisms Controlling Creep of Single Phase Metals and Alloys" W.D. Nix, B. Ilschner, "Strength of Metals and Alloys - ICSMA", Proceedings of the 5th International Conference, Aachen, Federal Republic of Germany, August 27–31, 1979, Vol 3, p. 1503.
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