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

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    33959
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    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 vc, for which the force in the direction of the cutting velocity Fh 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 αδφ and the ratio hb/hi.

    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:

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


    6.11: Determination of the Coefficients c1, c2, d1 and d2 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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