## 3 Model
### Strategy for Obtaining Better Smoothness of Cut Walls
#### 3.2.10. Step of Filaments following and PRR in the Approximation of \"Long Time\" of Energy Release  
In the approximation of a \"long time\" of energy release \(t_{E}\), when the condition (11) opposite to (10) and (33) is fulfilled:  
\[t_{E}>R^{2}/(4\chi), \tag{69}\]  
the spatio-temporal dependence of the temperature around the filament has the form (30). But for calculation of the maximal in time temperature at a radius \(r\), it is sufficient to consider times \(t\leq t_{E}\), described by (29). For a fixed \(r\), this function monotonically increases with \(t\), so the maximum is reached at \(t=t_{E}\):  
\[T_{max}(r)-T_{i}=T(r,\ t=t_{E})-T_{i}=\frac{q_{0}}{4\pi C\rho\chi H}\ B\bigg{(} \frac{r^{2}}{4\chi t_{E}}\bigg{)}. \tag{70}\]We recall that since \(r\leq R\), in this expression according to (69) the parameter \(z=r^{2}\)\(/(4\chi t_{E})\)\(\leq 1\), or \(\approx 1\). To find the radius of modification \(R\) from the expression (70), we use the method as in Ref. [59], when for any special function (in the given case for \(B(z)\)), the values of which are known only in a tabulated form, its approximation is used in the form of a relatively simple analytical exponential function \(B_{1}(z)\). In this case, as can be easily shown by a numerical computer interpolation of tabulated values, \(B(z)\approx B_{1}(z)\), where the approximation is \(B_{1}(z)=B_{0}\)\(\exp(-Az^{1/2})\) with \(B_{0}=5.47\) and \(A=3.3\). Calculations of \(B_{1}(z)\) using this simple analytical formula in comparison with tabulated values of \(B(z)\) show that this approximation is valid in the range \(0.01\leq z\leq 2.6\) (covering the interesting for us range \(0.01\leq z\leq 1\)) with a relative accuracy of at least 12%. With the use of this approximation, the ratio (70) takes the following simple form:  
\[T_{max}(r)-T_{i}=\frac{q_{0}}{4\pi C\rho\chi H}B_{0}\exp\biggl{(}-A\frac{r}{2 \sqrt{\chi t_{E}}}\biggr{)}. \tag{71}\]  
Substituting here \(r=R\), \(T_{max}=T_{mod}\) and \(q_{0}=Q_{tw}/t_{E}\), we now can easily solve this equation with respect to \(R\):  
\[R_{T,TD}=\frac{1}{A}2\sqrt{\chi t_{E}}\ \ln(Q_{tw}/Q_{th}),\,\text{where}\ Q_{ th}=4\pi\chi t_{E}HC\rho(T_{mod}-T_{i})/B_{0}. \tag{72}\]  
By substituting here the general expression (39) for \(Q_{tw}\) and taking also into account the relations (24) and (25), we obtain the dependence of \(R,s_{1 max}=2R\) and \(s_{1}\) on the filament radius \(r_{f}\) and the VED \(w\). For the thermal mechanism, when \(R=R_{T}\), we obtain:  
\[s_{1}\leq s_{1 max}=2R_{T},\,\,\text{where}\ R_{T}=\frac{1}{A}2\sqrt{\chi t_{E} }\ \ln\biggl{[}\frac{K_{tw}A_{P-H}B_{0}(w-w_{dis})r_{f}^{2}}{4\chi t_{E}w_{mod}^{ T}}\biggr{]}, \tag{73}\]  
\[w_{mod}^{T}=C_{2}\rho_{2}(T_{strain}-T_{i})\,\,\text{for amorphous materials}, \tag{74}\]  
\[w_{mod}^{T}=C_{2}\rho_{2}(T_{melt}-T_{i})\,\,\text{for crystalline materials}. \tag{75}\]  
For the thermally-induced deformation when \(R=R_{TD}\), respectively:  
\[s_{1}\leq s_{1 max}=2R_{TD},\,\,\text{where}\ R_{TD}=\frac{1}{A}2\sqrt{\chi t_ {E}}\ \ln\biggl{[}\frac{K_{TW}A_{P-H}B_{0}(w-w_{dis})r_{f}^{2}}{4\chi t_{E}w_{mod}^{ TD}}\biggr{]}, \tag{76}\]  
\[w_{mod}^{TD}=C_{2}\rho_{2}(T_{c}-T_{i})\,\,=\frac{C_{2}\rho_{2}}{\alpha_{T}Y} \sigma_{c}. \tag{77}\]  
The condition for the applicability of Equations (72), (73) and (76) is the obvious relation \(R_{T}\), \(R_{TD}>0\), i.e., the fulfilment of the condition \(Q_{tw}/Q_{th}\geq 1\), as well as the fulfilment of the inequality (69), rewritten as \(R_{T}{}^{2}\), \(R_{TD}{}^{2}<4\chi t_{E}\). From these limitations, we find the range of acceptable \(Q_{tw}\):  
\[Q_{tw}=(1\div\exp A)\ Q_{th}=(1\div 27)\ Q_{th}. \tag{78}\]  
Similar to the above calculations for the frequency limits \(f_{1}\)\(w_{2}\), we find the repetition frequencies in the considered mode (69), which allow minimising the mentioned above influence of the temperature gradient and heating from the previous filament at the location and at the time of occurrence of the next, new filament.