According to this classical relation, the force components in machining polycrystalline copper should increase with the decrease of grain size. Indeed, it is the case when the grain size decreases from 16.88 to 14.75 nm. The tangential force increases by 4.6%, and the thrust force increases by 31.6%. However, the Hall–Petch relation is apparently not applicable for polycrystalline machining with grain sizes of 5.32 to 14.75 nm RG-7388 cost (i.e., cases C2 to C6), in which the cutting forces decrease with the decrease of grain size. In recent years, it has been discovered that when the grain size of nano-structured materials is MK5108 clinical trial smaller than a critical value, the Hall–Petch relation could

be inversed [37–39]. In other words, as the fraction of grain boundary atoms increases to a significant level, work softening will become dominant. The inverse

Hall–Petch relation indicates that a smaller grain size increases the volume fraction of grain boundary, which facilitates the activation of other deformation mechanisms such as grain boundary sliding and thereby lowers material strength. The inverse Hall–Petch relation indeed matches up with our observation of nano-scale polycrystalline machining in the particular grain size range. Apparently, the decrease in cutting forces with the decrease of grain size is the result of yield strength reduction. The Givinostat supplier decrease in cutting force can also be further explained as strengthening due to dislocation activity below a critical grain size is PAK6 ceased, and the kick-in of other mechanisms leads to work softening and thus lowers the force required by the tool to remove the material. In particular, Mohammadabadi and Dehghani developed a modified

Hall–Petch equation, which incorporates the negative slope observed between grain size and yield stress [40]. It is in the following form: (7) where σ in is internal stress along the grain boundary that depends on parameters such as grain boundary thickness, lattice distortions, and grain size, and f gb is the volume fraction of the grain boundary. Figure 16 shows the yield stress of polycrystalline copper as a function of grain size under both the conventional Hall–Petch relation and the modified Hall–Petch relation. It can be seen that if the conventional Hall–Petch relation is followed, the yield stress should increase exponentially with grain size reduction. However, the modified Hall–Petch relation indicates that with the decrease of grain size, the yield stress grows at a slower pace to its peak position when the grain size is around 14 nm, and then it starts to drop if the grain size is below this critical value. Note that there are also other literature reporting that for some metals, the critical grain size for the inverse Hall–Petch to take over is about 10 to 15 nm [38, 41–43]. Figure 16 Predicted yield stress for nano-structured copper as a function of grain size.