Editing Plasticity (physics) (section)

==== Grain boundary constraint in polycrystals ====
The GB constraint for polycrystals can be explained by considering a grain boundary in the xz plane between two single crystals A and B of identical composition, structure, and slip systems, but misoriented with respect to each other. To ensure that voids do not form between individually deforming grains, the GB constraint for the bicrystal is as follows:
ε<sub>xx</sub><sup>A</sup> = ε<sub>xx</sub><sup>B</sup> (the x-axial strain at the GB must be equivalent for A and B), ε<sub>zz</sub><sup>A</sup> = ε<sub>zz</sub><sup>B</sup> (the z-axial strain at the GB must be equivalent for A and B), and ε<sub>xz</sub><sup>A</sup> = ε<sub>xz</sub><sup>B</sup> (the xz shear strain along the xz-GB plane must be equivalent for A and B). In addition, this GB constraint requires that five independent slip systems be activated per crystallite constituting the GB. Notably, because independent slip systems are defined as slip planes on which dislocation migrations cannot be reproduced by any combination of dislocation migrations along other slip system’s planes, the number of geometrical slip systems for a given crystal system - which by definition can be constructed by slip system combinations - is typically greater than that of independent slip systems. Significantly, there is a maximum of five independent slip systems for each of the seven crystal systems, however, not all seven crystal systems acquire this upper limit. In fact, even within a given crystal system, the composition and Bravais lattice diversifies the number of independent slip systems (see the table below). In cases for which crystallites of a polycrystal do not obtain five independent slip systems, the GB condition cannot be met, and thus the time-independent deformation of individual crystallites results in cracks and voids at the GBs of the polycrystal, and soon fracture is realized. Hence, for a given composition and structure, a single crystal with less than five independent slip systems is stronger (exhibiting a greater extent of plasticity) than its polycrystalline form.
{| class="wikitable"
|+ The number of independent slip systems for a given composition (primary material class) and structure (Bravais lattice).<ref>{{cite book |last1=Partridge |first1=Peter |title=Deformation and Fatigue of Hexagonal Close Packed Metals |date=1969 |location=University of Surrey}}</ref><ref>{{cite journal |last1=Groves |first1=Geoffrey W. |last2=Kelly |first2=Anthony |title=Independent Slip Systems in Crystals |journal=Philosophical Magazine |date=1963 |volume=8 |issue=89 |pages=877–887 |doi=10.1080/14786436308213843|bibcode=1963PMag....8..877G }}</ref>
|-
! Bravais lattice !! Primary material class: # Independent slip systems
|-
| Face centered cubic || Metal: 5, ceramic (covalent): 5, ceramic (ionic): 2
|-
| Body centered cubic || Metal: 5
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| Simple cubic || Ceramic (ionic): 3
|-
| Hexagonal || Metal: 2, ceramic (mixed): 2
|}