# Ultra-Precision Cutting Mechanism of KDP Crystal in Microplastic Region via Heating Assistance
## 2 Theory of Ductile/Brittle Transition  
KDP surfaces include a plastic and a brittle zone. The plastic surface layer becomes brittle when the processing depth increases to a certain depth, known as the critical depth of ductile/brittle transition. When the processing depth is less than the critical depth, insignificant cracks or brittle collapses occur on crystal surfaces due to the plastic slip in this region. When the process depth increases beyond the critical depth, the number and severity of pits and cracks increase due to the brittle fractures in the zone. Thus, the surface quality drops [22; 23; 24; 25; 26]. By increasing the surface temperatures of the KDP crystals, the ductile zone can be expanded. Cutting in the ductile zone can reduce the surface cracks and subsurface damage in the material. The processing principle is shown in Figure 1. When the cutting depth of the tool is constant, thermal-assisted cutting causes the ductile area of the material to expand. As the temperature increases, the range of the plastic zone expands, whereas the range of the brittle zone continues to decrease. The plastic processing of the materials in the plastic area can reduce surface pits and cracks, further improving surface quality. As a typical brittle material, the processing depth of KDP crystals occurs in the brittle region. The ultraprecision single-point diamond cutting of KDP crystals can be described as the elastic deformation changes to plastic deformation at the yield point. During the plastic deformation process, with increases in the shear stress and slip, the fracture strength is reached, and the chip is separated from the matrix. However, due to the low fracture toughness of the KDP crystal (0.74-0.8 MPam\({}^{1/2}\)), the elastic and strength limits of the material are close. Therefore, fracture failure occurs when the load on the KDP crystal exceeds its elastic limit. During the cutting process, defects such as pits, scratches, and cracks appear on the processed crystal surface. The overall fracture causes the sample to be scrapped. If the cutting depth is controlled within the critical depth of the ductile/brittle transition, the material deforms in the plastic zone via slip, improving the surface machining quality. The thermal assisted machining process method proposed in this paper provides a useful reference for the efficient and high-precision machining of similar brittle materials. Nanoindentation experiments were used to measure the mechanical parameters of crystalline materials [27; 28; 29; 30; 31]. A diamond indenter was used in the nanoindentation experiment. The indenter acted on the sample during the indentation process, and the loading force and depth of the test sample were obtained. The Knoop hardness, critical fracture toughness, and elastic modulus of the material were calculated. Nanoindentation experiments can simulate the material removal behavior during the cutting process. The deformation of materials on both sides of the nanoscratch groove reveals effective information on material removal [32; 33; 34; 35]. The relationship between the critical depth of the ductile/brittle transition, \(H\); Knoop hardness, \(HK\); material's elastic modulus, \(E\); and critical fracture toughness, \(K_{IC}\), was measured by [PERSON] et al. [36] via the microindentation method under static conditions. They can be expressed as  
\[H=n\left(\frac{E}{HK}\right)\left(\frac{K_{IC}}{HK}\right)^{2} \tag{1}\]  
The proportional coefficient \(n\) is determined by the indenter geometry (0.15 for a diamond glass indenter). The crack was less than 10% in their experiment. The critical depth of the ductile/brittle transition of KDP crystals was calculated using Formula (1) [37; 38; 39]. To determine the value of the ductile/brittle transition depth of KDP crystals, we conducted nanoindentation experiments on the surfaces of KDP crystal. The experimental  
Figure 1: High-temperature cutting principle of KDP crystal. sample was a large KDP 80 mm \(\times\) 40 mm \(\times\) 5 mm crystal.