# Strain Rate of Metal Deformation in the Machining Process from a Fluid Flow Perspective
## 2 Materials and Methods  
### Microscopic Foundation of Metal Cutting Research from a Fluid Flow Perspective  
The plastic deformation control mechanism has been studied for a long time. From a microscopic point of view, the plastic deformation of a metal material mainly involves the movement and proliferation of dislocations, including slip and twinning. However, the plastic deformation mechanism of metals is complex, as metal cutting is a process of dislocation accretion and annihilation. An increase in the density of movable dislocations will lead to work hardening. At the same time, dislocation movement leads to dislocation interaction, which makes the study of the plastic deformation mechanism more complicated. Slip is the main deformation form in the cutting process, so here we only study the plastic deformation mechanism from a dislocation slip aspect. The accepted mechanisms include heating activation, dislocation damping and the relativistic effect of the dislocations [18].  
Dislocations can be multiplied through several mechanisms. According to the dislocation mechanism of the Frank-Read source (Figure 2), the dislocation line AB is pinned by dislocation nodes at both ends. The time needed to remove the node constraint is \(t_{t}\), and the time spent by moving dislocation between the adjacent dislocation node is \(t_{w}\). When the dislocation moves to the obstacle under applied stress, the dislocation will be successfully transformed to the other side of the obstacle by a certain frequency due to the influence of heat. The transition frequency is  
\[v=v_{0}\exp[-\frac{(E_{c}-\tau v^{*})}{kT}] \tag{1}\]  
where \(v_{0}\) is the vibration frequency of a dislocation heated near an obstacle, \(E_{c}\) is the energy needed to overcome the short-range barrier, \(\tau\) is the applied stress, \(v^{*}\) is the thermal active volume, \(k\) is the Boltzmann constant and \(T\) is the temperature. Thus, the time spent by moving dislocation between the adjacent dislocation node is  
\[t_{t}=v^{-1} \tag{2}\]  
The moving dislocation will slip after removing the node pinning. The viscous damping B of the dislocation should be considered when it slips in the barrier (slip speed \(v_{s}=\tau b/B\), b is the burgers vector; \(B\) is the viscous damping). Assuming that the distance between the adjacent dislocation node is \(L\), then the time required for the dislocation slip \(L\) distance is  
\[t_{w}=\frac{L}{v_{s}} \tag{3}\]  
Figure 1: Apparatus used by [PERSON] (1952) [8]. A—Baffle chamber; B–Cutting tool; C–Baffle; D—Camera.  
The speed of the dislocation that moves in distance \(L\) is  
\[v_{d}=\frac{L}{t_{t}+t_{w}} \tag{4}\]  
According to the Orowan [19] equation \(\dot{\gamma}=\rho bv_{d}\), the relationship between the stress and strain rate can be acquired:  
\[\dot{\gamma}=\rho bL/(v_{0}^{-1}\exp((E_{c}-\tau v^{*})/kT)+BL/\tau b) \tag{5}\]  
At the low strain-rate stages, usually \(t_{t}>>t_{w}\), and the thermal activation mechanism is dominant in the plastic deformation process of the materials. The relationship between the stress and strain rate can be sampled as  
\[\dot{\gamma}=\rho bLkTv_{0}/\exp((E_{c}-\tau v^{*}) \tag{6}\]  
The strain rate increases with increasing cutting speed, and the shear yield stress increases due to the strain rate effect. \(t_{t},t_{w}\) will decrease; \(t_{w}\) is inversely proportional to the yield stress; that is, \(t_{t}\) follows an exponential decay with increasing yield stress and decays faster than \(t_{w}\). A critical value exists to make the two parameters equal; consequently, \(t_{t}<<t_{w}\). At this time, the damping mechanism of the dislocation motion is dominant, and the relationship between the stress and strain rate can be sampled as  
\[\tau=\frac{B}{\rho b^{2}}\dot{\gamma}=\eta\dot{\gamma} \tag{7}\]  
This equation represents the properties of fluid. The sensitivity of the material yield stress-to-strain rate will increase rapidly according to their linear relationship. Hence, research of the cutting process can start with fluids.