Editing Plasticity (physics) (section)

== Time-independent yielding and plastic flow in crystalline materials ==
<ref>{{cite book |last1=Courtney |first1=Thomas |title=Mechanical Behavior of Materials |date=2005 |publisher=Waveland Press, Inc |location=Long Grove, Illinois |isbn=978-1-57766-425-3 |edition=Second}}</ref>

Time-independent plastic flow in both single crystals and polycrystals is defined by a critical/maximum resolved [[shear stress]] (''τ''<sub>CRSS</sub>), initiating dislocation migration along parallel slip planes of a single slip system, thereby defining the transition from elastic to plastic deformation behavior in crystalline materials.

=== Time-independent yielding and plastic flow in single crystals === 
The critical resolved shear stress for single crystals is defined by Schmid’s law ''τ''<sub>CRSS</sub>=σ<sub>y</sub>/m, where σ<sub>y</sub> is the yield strength of the single crystal and ''m'' is the Schmid factor. The Schmid factor comprises two variables λ and φ, defining the angle between the slip plane direction and the tensile force applied, and the angle between the slip plane normal and the tensile force applied, respectively. Notably, because ''m'' > 1, ''σ''<sub>''y''</sub> > ''τ''<sub>CRSS</sub>.

==== Critical resolved shear stress dependence on temperature, strain rate, and point defects ====
[[File:Critical Resolved Shear Stress Versus Temperature.png|class=skin-invert-image|thumb|The three characteristic regions of the critical resolved shear stress as a function of temperature]]There are three characteristic regions of the critical resolved shear stress as a function of temperature. In the low temperature region 1 (''T'' ≤ 0.25''T''<sub>m</sub>), the [[strain rate]] must be high to achieve high ''τ''<sub>CRSS</sub> which is required to initiate dislocation glide and equivalently plastic flow. In region 1, the critical resolved shear stress has two components: athermal (''τ''<sub>''a''</sub>) and thermal (''τ''*) shear stresses, arising from the stress required to move dislocations in the presence of other dislocations, and the resistance of point defect obstacles to dislocation migration, respectively. At ''T'' =&nbsp;''T''*, the moderate temperature region 2 (0.25''T''<sub>m</sub>&nbsp;<&nbsp;''T''&nbsp;<&nbsp;0.7''T''<sub>m</sub>) is defined, where the thermal shear stress component ''τ''*&nbsp;→&nbsp;0, representing the elimination of point defect impedance to dislocation migration. Thus the temperature-independent critical resolved shear stress τ<sub>CRSS</sub> = τ<sub>a</sub> remains so until region 3 is defined. Notably, in region 2 moderate temperature time-dependent plastic deformation (creep) mechanisms such as solute-drag should be considered. Furthermore, in the high temperature region 3 (''T''&nbsp;≥&nbsp;0.7''T''<sub>m</sub>) έ can be low, contributing to low τ<sub>CRSS</sub>, however plastic flow will still occur due to thermally activated high temperature time-dependent plastic deformation mechanisms such as Nabarro–Herring (NH) and Coble diffusional flow through the lattice and along the single crystal surfaces, respectively, as well as dislocation climb-glide creep.

==== Stages of time-independent plastic flow, post yielding ====
[[File:Plastic Stress Versus Strain.png|class=skin-invert-image|thumb|The three stages of time-independent plastic deformation of single crystals]]During the easy glide stage 1, the work hardening rate, defined by the change in shear stress with respect to shear strain (''dτ''/''dγ'') is low, representative of a small amount of applied shear stress necessary to induce a large amount of shear strain. Facile dislocation glide and corresponding flow is attributed to dislocation migration along parallel slip planes only (i.e. one slip system). Moderate impedance to dislocation migration along parallel slip planes is exhibited according to the weak stress field interactions between these dislocations, which heightens with smaller interplanar spacing. Overall, these migrating dislocations within a single slip system act as weak obstacles to flow, and a modest rise in stress is observed in comparison to the yield stress. During the linear hardening stage 2 of flow, the work hardening rate becomes high as considerable stress is required to overcome the stress field interactions of dislocations migrating on non-parallel slip planes (i.e. multiple slip systems), acting as strong obstacles to flow. Much stress is required to drive continual dislocation migration for small strains. The shear flow stress is directly proportional to the square root of the dislocation density (τ<sub>flow</sub> ~''ρ''<sup>½</sup>), irrespective of the evolution of dislocation configurations, displaying the reliance of hardening on the number of dislocations present. Regarding this evolution of dislocation configurations, at small strains the dislocation arrangement is a random 3D array of intersecting lines. Moderate strains correspond to cellular dislocation structures of heterogeneous dislocation distribution with large dislocation density at the cell boundaries, and small dislocation density within the cell interior. At even larger strains the cellular dislocation structure reduces in size until a minimum size is achieved. Finally, the work hardening rate becomes low again in the exhaustion/saturation of hardening stage 3 of plastic flow, as small shear stresses produce large shear strains. Notably, instances when multiple slip systems are oriented favorably with respect to the applied stress, the τ<sub>CRSS</sub> for these systems may be similar and yielding may occur according to dislocation migration along multiple slip systems with non-parallel slip planes, displaying a stage 1 work-hardening rate typically characteristic of stage 2. Lastly, distinction between time-independent plastic deformation in body-centered cubic transition metals and face centered cubic metals is summarized below.
{| class="wikitable"
|+ Comparison between the time-independent plastic deformation of body centered cubic transition metals and face centered cubic metals, highlighting the critical resolved shear stress, work hardening rate, and necking strain during tensile testing.
|-
! Body-centered cubic transition metals !! Face-centered cubic metals
|-
| Critical resolved shear stress = high (relatively) & strongly temperature-dependent || Critical resolved shear stress = low (relatively) & weakly temperature-dependent
|-
| Work hardening rate = temperature-independent || Work hardening rate = temperature-dependent
|-
| Necking strain increases with temperature || Necking strain decreases with temperature
|}

=== Time-independent yielding and plastic flow in polycrystals ===
Plasticity in polycrystals differs substantially from that in single crystals due to the presence of grain boundary (GB) planar defects, which act as very strong obstacles to plastic flow by impeding dislocation migration along the entire length of the activated slip plane(s). Hence, dislocations cannot pass from one grain to another across the grain boundary. The following sections explore specific GB requirements for extensive plastic deformation of polycrystals prior to fracture, as well as the influence of microscopic yielding within individual crystallites on macroscopic yielding of the polycrystal. The critical resolved shear stress for polycrystals is defined by Schmid’s law as well (τ<sub>CRSS</sub>=σ<sub>y</sub>/ṁ), where σ<sub>y</sub> is the yield strength of the polycrystal and ''ṁ'' is the weighted Schmid factor. The weighted Schmid factor reflects the least favorably oriented slip system among the most favorably oriented slip systems of the grains constituting the GB. 

==== 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
|-
| Simple cubic || Ceramic (ionic): 3
|-
| Hexagonal || Metal: 2, ceramic (mixed): 2
|}

==== Implications of the grain boundary constraint in polycrystals ====
Although the two crystallites A and B discussed in the above section have identical slip systems, they are misoriented with respect to each other, and therefore misoriented with respect to the applied force. Thus, microscopic yielding within a crystallite interior may occur according to the rules governing single crystal time-independent yielding. Eventually, the activated slip planes within the grain interiors will permit dislocation migration to the GB where many dislocations then pile up as geometrically necessary dislocations. This pile up corresponds to strain gradients across individual grains as the dislocation density near the GB is greater than that in the grain interior, imposing a stress on the adjacent grain in contact. When considering the AB bicrystal as a whole, the most favorably oriented slip system in A will not be the that in B, and hence τ<sup>A</sup><sub>CRSS</sub> ≠ τ<sup>B</sup><sub>CRSS</sub>. Paramount is the fact that macroscopic yielding of the bicrystal is prolonged until the higher value of τ<sub>CRSS</sub> between grains A and B is achieved, according to the GB constraint. Thus, for a given composition and structure, a polycrystal with five independent slip systems is stronger (greater extent of plasticity) than its single crystalline form. Correspondingly, the work hardening rate will be higher for the polycrystal than the single crystal, as more stress is required in the polycrystal to produce strains. Importantly, just as with single crystal flow stress, τ<sub>flow</sub> ~ρ<sup>½</sup>, but is also inversely proportional to the square root of average grain diameter (τ<sub>flow</sub> ~d<sup>-½</sup> ). Therefore, the flow stress of a polycrystal, and hence the polycrystal’s strength, increases with small grain size. The reason for this is that smaller grains have a relatively smaller number of slip planes to be activated, corresponding to a fewer number of dislocations migrating to the GBs, and therefore less stress induced on adjacent grains due to dislocation pile up. In addition, for a given volume of polycrystal, smaller grains present more strong obstacle grain boundaries. These two factors provide an understanding as to why the onset of macroscopic flow in fine-grained polycrystals occurs at larger applied stresses than in coarse-grained polycrystals.