The Delft Sand, Clay & Rock Cutting Model, 2019a

12.6. The Cavitating Wedge
The Delft Sand, Clay & Rock Cutting Model.
Also for the cavitating process, a case will be discussed. The calculations are carried out for blade angles α of 65,
70, 75, 80, 85, 90, 95, 100, 105, 110, 115 and 120, while the smallest angle is around 60º depending on
the possible solutions. Also the cutting forces are determined with and without a wedge, so it’s possible to carry
out
step 6.
The case concerns a sand with an internal friction angle of 30, a soil interface friction angle of 20 fully
mobilized, a friction angle between the soil cut and the wedge equal to the internal friction angle, an initial
permeability ki of 6.2*10 -5 m/s and a residual permeability kmax of 17*10 -5 m/s. The blade dimensions are a width
of 0.25 m and a height of 0.2 m, while a layer of sand of 0.05 m is cut with a cutting velocity of 0.3 m/s at a water
depth of 0.6 m, matching the laboratory conditions. The values for the acting points of the forces, are e2=0.35,
e3=0.55 and e4=0.32, based on the finite element calculations carried out by Ma (2001).
Figure 12-24 and Figure 12-25 show the results of the calculations. Figure 12-24 shows the wedge angle θ, the
shear angle β, the mobilized internal friction angle λ and the mobilized external friction angle δe as a function of
the blade angle α. Figure 12-25 shows the horizontal and vertical cutting forces, with and without a wedge.
With the cavitating cutting process, the wedge angle θ always results in an angle of 90-, which matches the
theory of Hettiaratchi and Reece (1975).The reason of this is that in the full cavitation situation, the pore pressures
are equal on each side of the wedge and form equilibrium in itself. So the pore pressures do not influence the ratio
between the grain stresses on the different sides of the wedge. From Figure 12-25 it can be concluded that the
transition point between the conventional cutting process and the wedge process occurs at a blade angle of about
77 degrees.
In the non-cavitating cases this angle is about 70 degrees. A smaller angle of internal friction results in a higher
transition angle, but in the cavitating case this influence is bigger. In the cavitating case, the horizontal force is a
constant as long as the external friction angle is changing from a positive maximum to the negative minimum.
Once this minimum is reached, the horizontal force increases a bit. At the transition angle where the horizontal
forces with and without the wedge are equal, the vertical forces are not equal, resulting in a jump of the vertical
force, when the wedge starts to occur.
12.7. Limits.
Instead of carrying out the calculations for each different case, the limits of the occurrence of the wedge can be
summarized in a few graphs. Figure 12-26 shows the upper and lower limit of the wedge for the non-cavitating
case as a function of the angle of internal friction φ. It can be concluded that the upper and lower limits are not
symmetrical around 90º, but a bit lower than that. An increasing angle of internal friction results in a larger
bandwidth for the occurrence of the wedge. For blade angles above the upper limit most probably subduction will
occur, although there is no scientific evidence for this. The theory developed should not be used for blade angles
above the upper limit yet. Further research is required. The lower limit is not necessarily the start of the occurrence
of the wedge. This depends on whether the cutting forces with the wedge are smaller than the cutting forces without
the wedge. Figure 12-28 shows the blade angle where the wedge will start to occur, based on the minimum of the
horizontal cutting forces with and without the wedge. It can be concluded that the blade angle where the wedge
starts to occur is larger than the minimum where the wedge can exist, which makes sense. For high angles of
internal friction, the starting blade angle is about equal to the lower limit.
For the cavitating case the upper and lower limit are shown in Figure 12-27. In this case the limits are symmetrical
around 90º and with an external friction angle of 2/3 of the internal friction angle it can be concluded that these
limits are 90º+δ and 90º-δ. The blade angle where the wedge will start to occur is again shown in Figure 12-28.
The methodology applied gives satisfactory results to determine the cutting forces at large cutting angles. The
results shown in this paper are valid for the non-cavitating and the cavitating cutting process and for the soils and
geometry as used in this paper. The wedge angles found are, in general, a bit smaller then 90- for the noncavitating
case and exactly 90- for the cavitating case, so as a first approach this can be used.
The mobilized external friction angle δ e varies from plus the maximum for small blade angles to minus the
maximum for large blade angles, depending on the blade angle.
The cutting forces with the wedge do not increase much in the non-cavitating case and not at all in the cavitating
case, when the cutting angle increases from 60 to 120.
If the ratio between the thickness of the layer cut and the blade height changes, also the values of the acting points
e2, e3 and e4 will change slightly.
It is not possible to find an explicit analytical solution for the wedge problem and it’s even difficult to automate
the calculation method, since the solution depends strongly on the values of the acting points.
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