6.11: Determination of the Coefficients c1, c2, d1 and d2

If only the influence of the water under-pressures on the forces that occur with the cutting of saturated packed sand under water is taken in to account, equations (6-14) and (6-15) can be applied. It will be assumed that the non- cavitating process switches to the cavitating process for that cutting velocity v_c, for which the force in the direction of the cutting velocity F_h is equal for both processes. In reality, however, there is a transition region between both processes, where locally cavitation starts in the shear zone. Although this transition region starts at about 65% of the cutting velocity at which, theoretically, full cavitation takes place, it shows from the results of the cutting tests that for the determination of the cutting forces the existence of a transition region can be neglected. In the simplified equations the coefficients c₁ and d₁ represent the dimensionless horizontal force (or the force in the direction of the cutting velocity) in the non-cavitating and the cavitating cutting process. The coefficients c₂ and d₂ represent the dimensionless vertical force or the force perpendicular to the direction of the cutting velocity in the non- cavitating and the cavitating cutting process. For the non-cavitating cutting process:

\[\ \mathrm{F_{\mathrm{ci}}=\frac{c_{\mathrm{i}} \cdot \rho_{\mathrm{w}} \cdot \mathrm{g} \cdot \mathrm{v}_{\mathrm{c}} \cdot \mathrm{h}_{\mathrm{i}}^{2} \cdot \varepsilon \cdot \mathrm{w}}{\mathrm{k}_{\mathrm{m}}}}\tag{6-71}\]

In which:

\[\ \mathrm{c_1}=\left(\begin{array}{left}\frac{\left(\mathrm{p}_{1 \mathrm{m}} \cdot \frac{\mathrm{s i n}(\phi)}{\sin (\beta)}+\mathrm{p}_{\mathrm{2 m}} \cdot \frac{\mathrm{h}_{\mathrm{b}}}{\mathrm{h}_{\mathrm{i}}} \cdot \frac{\sin (\alpha+\beta+\phi)}{\sin (\alpha)}\right) \cdot \sin (\alpha+\delta)}{\sin (\alpha+\beta+\delta+\phi)}\\
-\mathrm{p}_{2 \mathrm{m}} \cdot \frac{\mathrm{h}_{\mathrm{b}}}{\mathrm{h}_{\mathrm{i}}} \cdot \frac{\sin (\alpha)}{\sin (\alpha)}\end{array}\right)\cdot \mathrm{\frac{(a_1 \cdot k_i +a_2 \cdot k_{max})}{k_{max}}}\tag{6-72}\]

And:

\[\ \mathrm{c_2}=\left(\begin{array}{left}\frac{\left(\mathrm{p}_{1 \mathrm{m}} \cdot \frac{\mathrm{s i n}(\phi)}{\sin (\beta)}+\mathrm{p}_{\mathrm{2 m}} \cdot \frac{\mathrm{h}_{\mathrm{b}}}{\mathrm{h}_{\mathrm{i}}} \cdot \frac{\sin (\alpha+\beta+\phi)}{\sin (\alpha)}\right) \cdot \cos (\alpha+\delta)}{\sin (\alpha+\beta+\delta+\phi)}\\
-\mathrm{p}_{2 \mathrm{m}} \cdot \frac{\mathrm{h}_{\mathrm{b}}}{\mathrm{h}_{\mathrm{i}}} \cdot \frac{\cos (\alpha)}{\sin (\alpha)}\end{array}\right)\cdot \mathrm{\frac{(a_1 \cdot k_i +a_2 \cdot k_{max})}{k_{max}}}\tag{6-73}\]

And for the cavitating cutting process:

\[\ \mathrm{F}_{\mathrm{c i}}=\mathrm{d _ { i }} \cdot \rho_{\mathrm{w}} \cdot \mathrm{g} \cdot(\mathrm{z}+\mathrm{1 0}) \cdot \mathrm{h}_{\mathrm{i}} \cdot \mathrm{w}\tag{6-74}\]

In which:

\[\ \mathrm{d}_{1}=\frac{\left(\frac{\sin (\phi)}{\sin (\beta)}+\frac{\mathrm{h}_{\mathrm{b}}}{\mathrm{h}_{\mathrm{i}}} \cdot \frac{\sin (\alpha+\beta+\phi)}{\sin (\alpha)}\right) \cdot \sin (\alpha+\delta)}{\sin (\alpha+\beta+\delta+\phi)}-\frac{\mathrm{h}_{\mathrm{b}}}{\mathrm{h}_{\mathrm{i}}} \cdot \frac{\sin (\alpha)}{\sin (\alpha)}\tag{6-75}\]

And:

\[\ \mathrm{\mathrm{d}_{2}=\frac{\left(\frac{\sin (\phi)}{\sin (\beta)}+\frac{h_{\mathrm{b}}}{h_{\mathrm{i}}} \cdot \frac{\sin (\alpha+\beta+\phi)}{\sin (\alpha)}\right) \cdot \cos (\alpha+\delta)}{\sin (\alpha+\beta+\delta+\phi)}-\frac{h_{\mathrm{b}}}{h_{\mathrm{i}}} \cdot \frac{\cos (\alpha)}{\sin (\alpha)}}\tag{6-76}\]

The values of the 4 coefficients are determined by minimizing the cutting work that is at that shear angle β where the derivative of the horizontal force to the shear angle is zero. The coefficients c₁, c₂, d₁ and d₂ are given in Miedema (1987 September) and in 0 and 0 for the non-cavitating cutting process and 0 and 0 for the cavitating cutting process as functions of α, δ, φ and the ratio h_b/h_i.

6.11.1. Approximations

Assuming δ=2/3·φ the coefficients can be approximated by:

\[\ \begin{array}{left}\alpha=30^{\circ}\quad\text{ and }\quad\mathrm{h_{b} / h_{i}=1:}\\
\mathrm{c_{1}=0.0427 \cdot e^{0.0509 \cdot \varphi} \quad\text{ and }\quad c_{2}=0.0343 \cdot e^{0.0341 \cdot \varphi}}\\
\mathrm{d_{1}=0.3027 \cdot e^{0.0516 \cdot \varphi} \quad\text{ and }\quad d_{2}=-0.3732+0.0219 \cdot \varphi}\end{array}\tag{6-77}\]

\[\ \begin{array}{ll}\mathrm{\alpha=30^{\circ} \quad\text { and }\quad h_{b} / h_{i}=2 \text { : }} \\ \mathrm{c_{1}=0.0455 \cdot 0^{0.0511 \cdot \varphi} \quad\text { and }\quad c_{2}=0.0304 \cdot e^{0.0356 \cdot \varphi}} \\ \mathrm{d_{1}=0.4795 \cdot e^{0.0490 \cdot \varphi}  \quad\text { and }\quad  d_{2}=-0.5380+0.0159 \cdot \varphi}\end{array}\tag{6-78}\]

\[\ \begin{array}{ll}\mathrm{\alpha=30^{\circ} \quad\text { and }\quad h_{b} / h_{i}=3:} \\ \mathrm{c_{1}=0.0457 \cdot e^{0.0512 \cdot \varphi} \quad\text { and }\quad c_{2}=0.0312 \cdot e^{0.0348 \cdot \varphi} }\\ \mathrm{d_{1}=0.6418 \cdot e^{0.0478 \cdot \varphi}  \quad\text { and }\quad  d_{2}=-0.7332+0.0094 \cdot \varphi}\end{array}\tag{6-79}\]

\[\ \begin{array}{ll}\mathrm{\alpha=45^{\circ} \quad\text { and }\quad h_{b} / h_{i}=1:}  \\ \mathrm{c_{1}=0.0485 \cdot e^{0.0577 \cdot \varphi} \quad \text { and } \quad c_{2}=0.0341 \cdot e^{0.0255 \cdot \varphi}} \\ \mathrm{d_{1}=0.2618 \cdot e^{0.0603 \cdot \varphi} \quad \text { and } \quad d_{2}=-0.0287+0.0081 \cdot \varphi}\end{array}\tag{6-80}\]

\[\ \begin{array}{ll}\mathrm{\alpha=45^{\circ} \quad\text { and }\quad h_{b} / h_{i}=2: } \\ \mathrm{c_{1}=0.0545 \cdot e^{0.0580 \cdot \varphi} \quad\text { and }\quad c_{2}=0.0281 \cdot e^{0.0238 \cdot \varphi} }\\ \mathrm{d_{1}=0.3764 \cdot e^{0.0577 \cdot \varphi}  \quad\text { and } \quad d_{2}=-0.0192-0.0017 \cdot \varphi}\end{array}\tag{6-81}\]

\[\ \begin{array}{ll}\mathrm{\alpha=45^{\circ} \quad\text { and }\quad h_{b} / h_{i}=3:}  \\ \mathrm{c_{1}=0.0551 \cdot e^{0.0589 \cdot \varphi} \quad\text { and }\quad c_{2}=0.0286 \cdot e^{0.0199 \cdot \varphi}} \\ \mathrm{d_{1}=0.4814 \cdot e^{0.0563 \cdot \varphi}  \quad\text { and } \quad d_{2}=-0.0295-0.0116 \cdot \varphi}\end{array}\tag{6-82}\]

\[\ \begin{array}{left}\mathrm{\alpha=60^{\circ}\quad\text{ and }\quad h_{b} / h_{i}=1:}\\
\mathrm{c_{1}=0.0474 \cdot e^{0.0688 \cdot \varphi} \quad\text{ and }\quad c_{2}=-0.2902+0.0203 \cdot \varphi-0.000334 \cdot \varphi^{2}}\\
 \mathrm{d_{1}=0.2342 \cdot e^{0.0722 \cdot \varphi} \quad\text{ and }\quad d_{2}=+1.0548-0.0343 \cdot \varphi}\end{array}\tag{6-83}\]

\[\ \begin{array}{left}\mathrm{\alpha=60^{\circ}\quad\text{ and }\quad h_{b} / h_{i}=2:}\\
\mathrm{c_{1}=0.0562 \cdot e^{0.0686 \cdot \varphi} \quad\text{ and }\quad c_{2}=-0.3550+0.0235 \cdot \varphi-0.000403 \cdot \varphi^{2}}\\
 \mathrm{d_{1}=0.3148 \cdot e^{0.0695 \cdot \varphi} \quad\text{ and }\quad d_{2}=+1.2737-0.0516 \cdot \varphi}\end{array}\tag{6-84}\]

\[\ \begin{array}{left}\mathrm{\alpha=60^{\circ}\quad\text{ and }\quad h_{b} / h_{i}=3:}\\
\mathrm{c_{1}=0.0593 \cdot e^{0.0692 \cdot \varphi} \quad\text{ and }\quad c_{2}=-0.3785+0.0250 \cdot \varphi-0.000445 \cdot \varphi^{2}}\\
 \mathrm{d_{1}=0.3889 \cdot e^{0.0680 \cdot \varphi} \quad\text{ and }\quad d_{2}=+1.4708-0.0685 \cdot \varphi}\end{array}\tag{6-85}\]

The shear angle β can be approximated by, for the non-cavitating case:

\[\ \mathrm{\beta=\frac{\pi-\alpha-\varphi-\delta}{3}-0.0037 \cdot \frac{h_{b}}{h_{i}}}\tag{6-86}\]

The shear angle β can be approximated by, for the cavitating case:

\[\ \mathrm{\beta=1-\frac{1}{6} \cdot \alpha-\frac{2}{7} \cdot(\varphi+\delta)-0.057 \cdot \frac{h_{b}}{h_{i}}}\tag{6-87}\]