4.1: Cutting Dry Sand

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In dry sand the cutting processes are governed by gravity and by inertial forces. Pore pressure forces, cohesion and adhesion are not present or can be neglected. Internal and external friction are present. The cutting process is of the Shear Type with discrete shear planes, but this can be modeled as the Flow Type, according to Merchant (1944). This approach will give an estimate of the maximum cutting forces. The average cutting forces may be 30%-50% of the maximum cutting forces.

Dry sand cutting is dominated by gravitational and inertial forces and by the internal and external friction angles. The cutting mechanism is the Shear Type. This is covered in Chapter 5: Dry Sand Cutting.

Screen Shot 2020-08-16 at 11.29.01 PM.png — Figure 4-1: The Shear Type in dry sand cutting.

The forces K₁ and K₂ on the blade, chisel or pick point are now:

\[\ \begin{aligned} \mathrm{K}_{1}&= \frac{\mathrm{W}_{2} \cdot \sin (\delta)+\mathrm{W}_{1} \cdot \sin (\alpha+\beta+\delta)+\mathrm{G} \cdot \sin (\alpha+\delta)}{\sin (\alpha+\beta+\delta+\varphi)} \\ &+\frac{-\mathrm{I} \cdot \cos (\alpha+\beta+\delta)-\mathrm{C} \cdot \cos (\alpha+\beta+\delta)+\mathrm{A} \cdot \cos (\delta)}{\sin (\alpha+\beta+\delta+\varphi)} \end{aligned}\tag{4-1}\]

\[\ \begin{aligned} \mathrm{K}_{2}&= \frac{\mathrm{W}_{2} \cdot \sin (\alpha+\beta+\varphi)+\mathrm{W}_{1} \cdot \sin (\varphi)+\mathrm{G} \cdot \sin (\beta+\varphi)}{\sin (\alpha+\beta+\delta+\varphi)} \\ &+\frac{+\mathrm{I} \cdot \cos (\varphi)+\mathrm{C} \cdot \cos (\varphi)-\mathrm{A} \cdot \cos (\alpha+\beta+\varphi)}{\sin (\alpha+\beta+\delta+\varphi)} \end{aligned}\tag{4-2}\]

The normal forces N₁ on the shear plane and N₂ on the blade are:

\[\ \mathrm{N}_{1}=\mathrm{K}_{1} \cdot \cos (\varphi) \quad\text{ and }\quad \mathrm{N}_{2}=\mathrm{K}_{2} \cdot \cos (\delta)\tag{4-3}\]

The horizontal and vertical forces on the blade, chisel or pick point are:

\[\ \mathrm{F}_{\mathrm{h}}=-\mathrm{W}_{2} \cdot \sin (\alpha)+\mathrm{K}_{2} \cdot \sin (\alpha+\delta)+\mathrm{A} \cdot \cos (\alpha)\tag{4-4}\]

\[\ \mathrm{F}_{\mathrm{v}}=-\mathrm{W}_{2} \cdot \cos (\alpha)+\mathrm{K}_{2} \cdot \cos (\alpha+\delta)-\mathrm{A} \cdot \sin (\alpha)\tag{4-5}\]

The equilibrium of moments around the blade tip is:

\[\ \mathrm{\left(N_{1}-W_{1}\right) \cdot R_{1}-G \cdot R_{3}=\left(N_{2}-W_{2}\right) \cdot R_{2}}\tag{4-6}\]

Analyzing these equations results in the following conclusions:

Since the argument in the cosine of the inertial term in the force K₁ is always greater than 90 degrees, the cosine is negative and the term as a whole is positive. This results in positive forces on the blade, chisel or pick point and also positive normal forces.
There are no forces directly proportional to the (mobilized) blade height or the length of the shear plane, so the equilibrium of moments does not play a role. The Curling Type and the Tear Type will not occur. The acting points of the forces R₁ and R₂ will be adjusted by nature to form an equilibrium of moments.
When the argument of the sine in the denominator gets close to 180 degrees, the forces become very large. If the argument is greater than 180 degrees, the forces would become negative. Since both conditions will not happen in nature, nature will find another cutting mechanism, the wedge mechanism.