## 3 Model
### Strategy for Obtaining Better Smoothness of Cut Walls
#### 3.2.12. Modification Mechanism Preferable for Smooth Separation
If we assume for the estimations in these formulas that \(\gamma_{sw}\approx\gamma_{tw}=1/e=0.368\), the condition of separation  
Figure 6: (**a**) Obtaining, or (**b**) not obtaining a smooth separation, depending on the dominance of one or another material modification mechanism: (**a**) The dominance of the thermal mechanism of formation of the impact zone, i.e., \(R_{T}>R_{TD}\), where \(R_{T}\) is the radius of the cylindrical impact zone around the filament axis due to the thermal mechanism, and \(R_{TD}\) is the similar radius of the impact zone due to the thermal deformation mechanism, with the radius of the filament being \(r_{f}<R_{T}\), \(R_{TD}\). \(\pi\) is the indicated in Figure 3A plane of the cross-section of impact zones of the filament, which are sequentially generated by moving the laser beam relative to the material, \(u\) is the speed and direction of relative movement of the laser beam and material. \(T=T_{strain}\) and \(T=T_{c}\) are the positions of the corresponding isotherms in the material that determine the boundaries of the modification zones for these mechanisms. 1 is a straight and smooth line of material separation during its subsequent mechanical splitting in this case. (**b**) The dominance of the thermal deformation mechanism when \(R_{TD}>R_{T}\). As can be seen, at radii \(r\) such that \(R_{T}<r\leq R_{TD}\), there is a characteristic area of damage and cracking of the material as a result of irradiation. 2 is a zigzag-like line (surface) of separation, randomly changing direction, obtained in this case by mechanical splitting after irradiation since it passes along random microcracks and boundaries between pieces of material formed as a result of cracking in the indicated damage area of the material. Such a surface 2 corresponds to a rough, non-smooth separation of the material. with a better smoothness (82) is satisfied when simple relations between the corresponding thresholds of these 3 mechanisms are fulfilled:  
\[w_{mod}^{T}<\ w_{mod}^{TD}\ \text{and}\ w_{mod}^{T}<\ w_{mod}^{SW}. \tag{83}\]  
Given the expressions (66) or (67) for \(w_{mod}^{T}\) and (66) for \(w_{mod}^{TD}\), the first inequality in (81) corresponds to the following condition:  
\[T_{strain}<T_{c}, \tag{84}\]  
for amorphous materials, and:  
\[T_{mdt}<T_{c}. \tag{85}\]  
for crystalline materials. These conditions indicate the simple physical fact that for obtaining a separation with a better smoothness, it is important that the point of intense volumetric expansion, estimated as \(T_{exp}=T_{strain}\) for amorphous materials (or, respectively, as \(T_{exp}=T_{mdt}\) for crystalline materials), is reached earlier (i.e., at a lower VED \(w\)) than the critical for material cracking point \(T_{c}\) in the mechanism of the thermally-induced deformation. By substituting expression (26) for \(T_{c}\) into (84) and (85), one then obtains a dimensionless criterion for obtaining a separation with a better smoothness for amorphous materials  
\[\frac{\alpha_{T}Y(T_{strain}-\ T_{i})}{\sigma_{c}}<\ 1, \tag{86}\]  
and, correspondingly, a similar criterion for crystalline materials:  
\[\frac{\alpha_{T}Y(T_{melt}-\ T_{i})}{\sigma_{c}}<\ 1. \tag{87}\]  
Besides, from the second inequality in (83), \(w_{mod}^{T}<\ w_{mod}^{SW}\), similarly, when substituting the expression (66) or (67) for \(w_{mod}^{T}\) and the expression (46) for \(w_{mod}^{SW}\), we find the following dimensionless criterion for obtaining a separation with a better smoothness:  
\[\frac{C_{2}\ \rho_{2}(T_{strain}-T_{i})}{(1+\eta)\sigma_{c}}<1 \tag{88}\]  
for amorphous materials, and:  
\[\frac{C_{2}\ \rho_{2}(T_{melt}-T_{i})}{(1+\eta)\sigma_{c}}<1 \tag{89}\]  
for crystalline materials.