# Prediction of Cutting Parameters of HVOF-Sprayed Stellite 6
## 2 The Methodology of the Work
### Theoretical Background
#### 2.1.3 Derivation of the Equations for Cutting Parameters Prediction
The material plasticity constant \(K_{plmat}\) (\(\mu\)m) that characterizes the capacity of the material to deform for material characterized by Young's modulus of elasticity \(E_{mat}\) (MPa), derived from the topographic elements of the cut walls (2):  
\[K_{plmat}=\frac{Ra\cdot a_{p}}{Y_{ret}};K_{plmat}=\frac{10^{12}}{E_{mat}^{2}} \tag{2}\]  
The equations of equilibrium on the cut are (3):  
\[\frac{Ra\cdot a_{p}}{Y_{ret}}=\frac{10^{12}}{E_{mat}^{2}};\frac{Ra\cdot a_{p}} {Y_{ret}}-\frac{10^{12}}{E_{mat}^{2}}=0 \tag{3}\]  
According to (3), consequently we obtain Equations (4)-(6):  
\[E_{mat}=\sqrt{\frac{10^{12}}{K_{plmat}}} \tag{4}\]  
\[E_{mat}=\sqrt{\frac{10^{12}\cdot Y_{ret}}{Ra\cdot a_{p}}} \tag{5}\]  
\[E_{mat}=\sqrt{\frac{10^{12}\cdot Y_{ret}}{K_{pl}}} \tag{6}\]  
where the strain surface area \(K_{pl}\) (m\({}^{2}\)) is given by Equation (7):  
\[K_{pl}=Ra\cdot a_{p} \tag{7}\]  
The main topographic elements are calculated according to Equations (8)-(12):  
\[Ra=\frac{10^{12}\cdot Y_{ret}}{E_{mat}^{2}\cdot a_{p}} \tag{8}\]  
\[Y_{ret}=\frac{10^{-12}\cdot Ra\cdot a_{p}}{E_{mat}^{2}};Y_{ret}=\frac{K_{pl}} {K_{plmat}} \tag{9}\]\[v_{c}=\frac{10^{6}\cdot\sqrt{10^{-3}\cdot Ra}}{\sqrt{E_{mat}}} \tag{10}\]  
\[a_{p}=\frac{10^{12}\cdot Y_{ret}}{\left(E_{mat}^{2}\cdot Ra\right)} \tag{11}\]  
\[\frac{Ra\cdot a_{p}}{Y_{ret}}=\frac{10^{12}}{E_{mat}^{2}};\frac{Ra\cdot a_{p}} {Y_{ret}}-\frac{10^{12}}{E_{mat}^{2}}=0 \tag{3}\]0  
For modular stress, we derived relations from surface topography (13) to (14):  
\[\frac{Ra\cdot a_{p}}{Y_{ret}}=\frac{10^{12}}{E_{mat}^{2}};\frac{Ra\cdot a_{p}} {Y_{ret}}-\frac{10^{12}}{E_{mat}^{2}}=0 \tag{3}\]1  
\[\frac{Ra\cdot a_{p}}{Y_{ret}}=\frac{10^{12}}{E_{mat}^{2}};\frac{Ra\cdot a_{p}} {Y_{ret}}-\frac{10^{12}}{E_{mat}^{2}}=0 \tag{3}\]2  
where \(E_{rz}\) is the component of \(E_{mat}\) for compression and \(E_{ret}\) is component of \(E_{mat}\) for tension, for strain stress real \(\sigma_{rztrue}\) (15), or contractual \(\sigma_{rzng}\) (16):  
\[\frac{Ra\cdot a_{p}}{Y_{ret}}=\frac{10^{12}}{E_{mat}^{2}};\frac{Ra\cdot a_{p}} {Y_{ret}}-\frac{10^{12}}{E_{mat}^{2}}=0 \tag{3}\]3  
\[\frac{Ra\cdot a_{p}}{Y_{ret}}=\frac{10^{12}}{E_{mat}^{2}};\frac{Ra\cdot a_{p}} {Y_{ret}}-\frac{10^{12}}{E_{mat}^{2}}=0 \tag{3}\]4  
The more complex relation for roughness \(Ra_{q}\) already accurately accentuates the position of \(a_{p0}\) and the yield strength \(Re\) at the depth of cut \(a_{p}\) according to (17):  
\[\frac{Ra\cdot a_{p}}{Y_{ret}}=\frac{10^{12}}{E_{mat}^{2}};\frac{Ra\cdot a_{p}} {Y_{ret}}-\frac{10^{12}}{E_{mat}^{2}}=0 \tag{3}\]5  
According to \(Ra_{q}\), the strain stress \(\sigma_{rzq}\) is derived (18):  
\[\frac{Ra\cdot a_{p}}{Y_{ret}}=\frac{10^{12}}{E_{mat}^{2}};\frac{Ra\cdot a_{p}} {Y_{ret}}-\frac{10^{12}}{E_{mat}^{2}}=0 \tag{3}\]6  
For true residual stress \(\sigma_{rztue}\) (MPa) and for engineering residual stress \(\sigma_{rzng}\) (MPa) the following Equations (19) and (20) apply:  
\[\frac{Ra\cdot a_{p}}{Y_{ret}}=\frac{10^{12}}{E_{mat}^{2}};\frac{Ra\cdot a_{p}} {Y_{ret}}-\frac{10^{12}}{E_{mat}^{2}}=0 \tag{3}\]7  
\[\frac{Ra\cdot a_{p}}{Y_{ret}}=\frac{10^{12}}{E_{mat}^{2}};\frac{Ra\cdot a_{p}} {Y_{ret}}-\frac{10^{12}}{E_{mat}^{2}}=0 \tag{3}\]8  
The specific elongation \(\varepsilon\) is derived as f (\(Y_{ret}\), \(E_{mat}\)) in the form (21):  
\[\frac{Ra\cdot a_{p}}{Y_{ret}}=\frac{10^{12}}{E_{mat}^{2}};\frac{Ra\cdot a_{p}} {Y_{ret}}-\frac{10^{12}}{E_{mat}^{2}}=0 \tag{3}\]9  
[PERSON]'s elongation (22):  
\[E_{mat}=\sqrt{\frac{10^{12}}{K_{plmat}}} \tag{4}\]0  
Equation (23) applies to the cutting force \(F_{c}\) and Equation (24) applies to calculate the area \(S_{c}\) on which the cutting tool acts:  
\[E_{mat}=\sqrt{\frac{10^{12}}{K_{plmat}}} \tag{4}\]1  
\[E_{mat}=\sqrt{\frac{10^{12}}{K_{plmat}}} \tag{4}\]2In addition to the continuous functions described above, knowledge of the values of these functions at the level of the mechanically neutral plane in the cut, i.e., at the \(a_{p0}\) (25) level, is necessary for our further analytical work:  
\[E_{mat}=\sqrt{\frac{10^{12}}{K_{plmat}}} \tag{4}\]3  
where \(v_{c0}\) = speed at the neutral plane (m-min\({}^{-1}\)), \(f_{0}\) = feed at the neutral plane (mm), \(R_{mp0}\) = ratio of machining parameters at the neutral plane (-), \(Ra_{0}\) = surface roughness at the neutral plane (\(\mu\)m), \(Y_{ret0}\) = deviation of the cut trace from the vertical plane at the neutral plane (mm), \(\delta_{0}\) = angle of deviation from the vertical plane at the neutral plane (\({}^{\circ}\)), \(D_{gr0}\) = diameter of structural grain at the neutral plane (\(\mu\)m), \(\sigma_{rzeng0}\) = technical deformation stress at the neutral plane (MPa), \(\sigma_{rztru0}\) = actual deformation stress at the neutral plane (MPa), \(\sigma_{re20}\) = residual stress at the neutral plane (MPa), \(\sigma_{ret0}\) = modular tensile stress component at the neutral plane (MPa), \(t_{cut0}\) = continuous time at the neutral plane (s), \(S_{c0}\) = cutting area at the neutral plane (mm\({}^{2}\)), \(F_{c0}\) = cutting force at the neutral plane (N) and \(a_{p0}\) = depth of cut at the neutral plane (mm).