6.7: The Blade Tip Problem

Last updated
Save as PDF

Page ID: 33853

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

During the physical modeling of the cutting process it has always been assumed that the blade tip is sharp. In other words, that in the numerical calculation, from the blade tip, a hydrostatic pressure can be introduced as the boundary condition along the free sand surface behind the blade. In practice this is never valid, because of the following reasons:

The blade tip always has a certain rounding, so that the blade tip can never be considered really sharp.
Through wear of the blade a flat section develops behind the blade tip, which runs against the sand surface (clearance angle \(\ \leq\) zero)
If there is also dilatancy in the sand underneath the blade tip it is possible that the sand runs against the flank after the blade has passed.
There will be a certain under-pressure behind the blade as a result of the blade speed and the cutting process.

A combination of these factors determines the distribution of the water under-pressures, especially around the blade tip. The first three factors can be accounted for in the numerical calculation as an extra boundary condition behind the blade tip. Along the free sand surface behind the blade tip an impenetrable line element is put in, in the calculation. The length of this line element is varied with 0.0·h_i, 0.1·h_i and 0.2·h_i. It showed from these calculations that especially the water under-pressures on the blade are strongly determined by the choice of this boundary condition as indicated in Figure 6-18 and Figure 6-19.

Screen Shot 2020-08-17 at 11.04.04 PM.png — Figure 6-18: The water pore pressures on the blade as function of the length of the wear section w.

Screen Shot 2020-08-17 at 11.05.10 PM.png — Figure 6-19: The water pore pressure in the shear zone as function of the length of the wear section w.

It is hard to estimate to what degree the influence of the under-pressure behind the blade on the water under- pressures around the blade tip can be taken into account with this extra boundary condition. Since there is no clear formulation for the under-pressure behind the blade available, it will be assumed that the extra boundary condition at the blade tip describes this influence. If there is no cavitation the water pressures forces W₁ and W₂ can be written as:

\[\ \mathrm{W}_{1}=\frac{\mathrm{p}_{1 \mathrm{m}} \cdot \rho_{\mathrm{w}} \cdot \mathrm{g} \cdot \mathrm{v}_{\mathrm{c}} \cdot \varepsilon \cdot \mathrm{h}_{\mathrm{i}}^{2} \cdot \mathrm{w}}{\mathrm{k}_{\mathrm{m a x}} \cdot \mathrm{s i n}(\beta)}\tag{6-30}\]

And

\[\ \mathrm{W}_{2}=\frac{\mathrm{p}_{2 \mathrm{m}} \cdot \rho_{\mathrm{w}} \cdot \mathrm{g} \cdot \mathrm{v}_{\mathrm{c}} \cdot \varepsilon \cdot \mathrm{h}_{\mathrm{i}} \cdot \mathrm{h}_{\mathrm{b}} \cdot \mathrm{w}}{\mathrm{k}_{\max } \cdot \sin (\alpha)}\tag{6-31}\]

In case of cavitation W₁ and W₂ become:

\[\ \mathrm{W}_{1}=\frac{\rho_{\mathrm{w}} \cdot \mathrm{g} \cdot(\mathrm{z}+\mathrm{1 0}) \cdot \mathrm{h}_{\mathrm{i}} \cdot \mathrm{w}}{\sin (\beta)}\tag{6-32}\]

And

\[\ \mathrm{W}_{2}=\frac{\rho_{\mathrm{w}} \cdot \mathrm{g} \cdot(\mathrm{z}+\mathrm{1 0}) \cdot \mathrm{h}_{\mathrm{b}} \cdot \mathrm{w}}{\sin (\alpha)}\tag{6-33}\]