Editing Plasticity (physics) (section)

=== 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
|}