Deformation of a single crystal
Figure 9: The stress-strain curve of a single crystal. The gradient of the linear region in stage ll is G/200.
When a single crystal is plastically deformed, different behaviors are observed in its stress-strain curve depending on the strain. These behaviors are divided into three stages (Figure 9). Here, only stage l and ll will be discussed.
This is a stage of low linear hardening which may be absent, or account for as much as 40 percent shear strain depending on the testing conditions (mainly how much lattice rotation is required to start a second slip system deforming, see Slip in Single Crystals). Since only one slip system is operative, the dislocations can glide easily without being impeded by dislocations from other slip systems. Hence, this stage is often referred to as ‘easy glide’.
In stage l, dislocations of a single slip system which are parallel to each other all glide in one direction. Experimental data has shown that there is a small but finite work hardening rate associated with this stage. This is due to the accumulation of dislocation debris in the form of dipoles.
The extent of stage 1 depends on the purity of the crystal. If the impurities form a dispersion of second phases (e.g. silicon and iron phases in aluminium), stage 1 hardening is reduced or even eliminated. This is because the small inclusions encourage localized slip on other than the primary slip plane, so other slip systems are activated. On the other hand, impurities which form a solid solution with the crystal tend to enhance the extent of stage 1.
As the crystal is deformed, the tensile axis rotates towards the slip direction. Stage 2 is initiated when the tensile axis has rotated to a position where two slip systems share the largest Schmid factor. At this stage, two slip systems are activated, forming a secondary slip system in addition to the primary system which exists in stage 1. The dislocations that move on both slip systems can interact with each other to form jogs, locks and pile ups. Consequently, the crystal becomes work-hardened in this stage.
Stage 2 work hardening is characterized by an approximately linear stress-strain curve whose slope (the hardening rate) is about G/200 (where G is the shear modulus), which has a mild dependence on temperature or strain rate (i.e. the hardening rate is constant).
Using dimensional analysis, it can be shown that the fundamental relationship for the flow stress \( \tau \) in stage 2 is:\[ \tau = \alpha \;b\;\;G\sqrt \rho \]
This is known as the Taylor equation. Here α is a dimensionless number, G is the shear modulus, b is the Burgers vector and ρ is the forest dislocation density of the system.
The term α represents an average interaction strength between dislocations and its value depends on the inherent complexity of detailed dislocation theory. The interaction can vary from entirely elastic between dislocations with perpendicular Burgers vectors, to energy storing when intersection leads to formation of a jog. The magnitude is typically in the range 0.5 – 1.0.