The Cutting of Water Saturated Sand at Large Cutting Angles

Dr.ir. S.A. Miedema[1]

Ma Yasheng[2]

Abstract: In the last decennia extensive research has been carried out into the cutting of water saturated sand. In the cutting of water-saturated sand, the phenomenon of dilatation plays an important role. In fact the effects of gravity, inertia, cohesion and adhesion can be neglected at cutting speeds in the range of 0.5 10 m/s. In the cutting equations, as published by Miedema, there is a division by the sine of the sum of the blade angle, the shear angle, the angle of internal friction and the soil/interface friction angle. When the sum of these angle approaches 180, a division by zero is the result, resulting in infinite cutting forces. This may occur for example for a=80, b=30, f=40 and d=30. When this sum is greater then 180 degrees, the cutting forces become negative. It is obvious that this cannot be the case in reality and that nature will look for another cutting mechanism. Hettiaratchi and Reece, 1975 [7] found a mechanism which they called boundary wedges for dry soil. At large cutting angles a triangular wedge will exist in front of the blade, not moving relative to the blade. This wedge acts as a blade with a smaller blade angle. In fact, this reduces the sum of the 4 angles mentioned before to a value much smaller than 180. The existence of a dead zone (wedge) in front of the blade when cutting at large cutting angles will affect the value and distribution of vacuum water pressure on the interface. He, 1998 [8], proved experimentally that also in water saturated sand at large cutting angles a wedge will occur. Although the number of experiments published is limited, his research is valuable as a starting point to predict the shape of the wedge. At small cutting angles the cutting forces are determined by the horizontal and vertical force equilibrium equations of the sand cut in front of the blade. These equations contain 3 unknowns, so a third equation/condition had to be found. The principle of minimum energy is used as a third condition to solve the 3 unknowns. This has proved to give very satisfactory results finding the shear angle and the horizontal and vertical cutting forces at small cutting angles. At large cutting angles, a 4^th unknown exists, the wedge angle or virtual blade angle. This means that a 4^th equation/condition must be found in order to determine the wedge angle. There are 3 possible conditions that can be used: The principle of minimum energy, The circle of Mohr, The equilibrium of moments of the wedge. These methods are discussed and the choice made is explained.

1 INTRODUCTION

In dredging and tunneling there is a strong interaction between the material to be handled and the equipment handling the material. To be able to make an optimal design of the equipment, the physical processes and mechanisms involved in the interaction should be known. In dredging and tunneling processes, the excavation is the first and primary step.

In the water saturated sand cutting process, the dilatancy phenomenon plays an important role in determining the cutting force. As a result of shear in the sand package the pore volume changes, shown in fig.1. The flowing water which fills the increased volume experiences a certain resistance, causing pore water under pressure. By this reason, grain force increases and so the required cutting forces. If the volume strain rate is high enough, there is a chance that the pore pressure reaches the saturated water vapor pressure and cavitation occurs.

Fig.1 The cutting process modeled as a continuous process.

Since seventies and eighties the cutting process in saturated sand is extensively researched at Delft Hydraulics in Delft[13], at the Delft University of Technology and at the Mineral Technology Institute[19]. In pore water pressure calculations, near a underwater slope [14], a forward moving breach face is supposed just in front of the blade, keeping in contact with the blade tip.

Conventional analyses generally assume that all the soil within the rupture surface, isolated by the outer bounding stress (and velocity) discontinuity, is deforming. This is not always so and dead zones of soil, with no movement relative to the interface, can develop within the rupture surface. These zones, which can be very large, are capable of drastically altering the effective interface geometry. In two-dimensional soil failure problems these zones resemble regular prisms of soil, so can be simply described as wedges. As these boundary wedges are fixed to the interface there is no necessity for the soil interface friction to be fully mobilized(d<d_f).

A series of tests with rake angles 90, 105 and 120 degrees under fully saturated and densely compacted sand condition was performed by Jisong He [8] at dredging technology section of Delft University of Technology. The experimental results showed that the failure pattern with large rake angles is quite different from that with small rake angles. For large rake angles a dead zone is formed in front of the blade but not for small rake angles. In the tests he carried out, both a video camera and film camera were used to capture the failure pattern. The video camera was fixed on the frame which is mounted on the main carriage, translates with the same velocity as the testing cutting blade. Shown in the static slide of the video record, as in fig.2, the boundary wedges exist during the test cutting.

Fig.2 Diagram of the failure pattern with Rake angle 120.

2 Cutting theory, literature survey.

If the cutting process is assumed to be stationary, the water flow through the pores of the sand can be described in a blade motions related coordinate system. The determination of the water vacuum pressures in the sand around the blade is then limited to a mixed boundary conditions problem. The potential theory can be used to solve this problem. For the determination of the water vacuum pressures it is necessary to have a proper formulation of the boundary condition in the shear zone. Miedema (1984, [20]) derived the basic equation for this boundary condition. In 1985 [21, 22] and 1986 [24] a more extensive derivation is published. If it is assumed that no deformations take place outside the deformation zone, then:

(1)

applies for the sand package around the blade.

The boundary condition is in fact a specific flow rate that can be determined with the following hypothesis. As shown in Fig.3, for a sand element in the deformation zone, the increase in the pore volume per unit of blade width, is:

in m^3/m	(2)
in which:

For the residual pore percentage is chosen for n_max on the basis of the ability to explain the water under-pressures, measured in the laboratory tests. The per unit width volume flow rate flowing to the sand element, is equal to:

in m^3/m

(3)

With the aid of Darcy's law the next differential equation can be derived for the specific flow rate. Perpendicular to the deformation zone:

(4)

The partial derivative is the derivative of the water under-pressure perpendicular on the boundary of the area, in which the water under-pressures are calculated (in this case the deformation zone). The boundary conditions on the other boundaries are indicated in fig.3.

Fig.3 The volume balance over the shear zone. The meaning of parameters:

wedge angle; shear angle

A hydrostatic pressure distribution is assumed on the boundaries between sand and water. This pressure distribution equals zero in the calculation of the water under-pressures, if the height difference over the blade is neglected. The boundaries that form the edges in the sand package are assumed to be impermeable.

Making equation (4) dimensionless is similar to that of the breach equation of Meijer and Van Os[14]. In the breach problem the length dimensions are normalized by dividing them by the breach height, while in the cutting of sand they are normalized by dividing them by the cut layer thickness. Equation (4) in normalized format:

	(5)
with:

This equation is made dimensionless with:

(6)

The accent indicates that a certain variable or partial derivative is dimensionless. The next dimensionless equation is now valid as a boundary condition in the deformation zone:

(7)

The storage equation also has to be made dimensionless, which results in the next equation :

(8)

The water under-pressures distribution in the sand package can now be determined using the storage equation and the boundary conditions. Because the calculation of the water under-pressures is dimensionless the next transformation has to be performed to determine the real water under-pressures. The real water under-pressures can be determined by integrating the derivative of the water under-pressures in the direction of a flow line, along a flow line, so:

over the shear zone

(9)

This is illustrated in fig.4.

Fig.4: the flow of the pore water towards the shear zone.

Over shear zone P calculated is , over interface between wedge and cut sand , over interface between wedge and original sand , and over the blade .

3 Numerical water pore pressure calculations.

The water vacuum pressures in the sand package on and around the blade are numerically determined using the finite element method. A standard program package is used (Segal 2001, [29]). Within this package, available "subroutines" a program is written, with which water vacuum pressures can be calculated and be output graphically and numerically. As shown in fig.5, SEPRAN model is made of three parts, the original sand layer, the cut sand layer, and the wedge. The solution of such a calculation is however not only dependent on the physical model of the problem, but also on the next points:

1. The size of the area in which the calculation takes place.
2. The size and distribution of the elements
3. The boundary conditions

The choices for these three points have to be evaluated with the problem that has to be solved in mind. These calculations are about the values and distribution of the water under-pressures in the shear zone and on the blade, on the interface between wedge and cut sand, between wedge and the original sand layer. A variation of the values for point 1 and 2 may therefore not influence this part of the solution. This is achieved by on the one hand increasing the area in which the calculations take place in steps and on the other hand by decreasing the element size until the variation in the solution was less than 1%. The distribution of the elements is chosen such that a finer mesh is present around the blade tip, the shear zone and on the blade, also because of the blade tip problem. A number of boundary conditions follow from the physical model of the cutting process, these are:

vacuum pressures and the hydrostatic pressure it is valid to take a zero pressure as the boundary condition.

The boundary conditions along the boundaries of the area where the calculation takes place that are located in the sand package are not determined by the physical process. For this boundary condition there is a choice among:

1. A hydrostatic pressure along the boundary.
2. A boundary as an impermeable wall.
3. A combination of a known pressure and a known specific flow rate.

None of these choices complies with the real process. Water from outside the calculation area will flow through the boundary. This also implies, however, that the pressure along this boundary is not hydrostatic. If, however, the boundary is chosen with enough distance from the real cutting process the boundary condition may not have an influence on the solution. The impermeable wall is chosen although this choice is arbitrary. Fig. 6 and fig. 7 give an impression of the equi-potential line in the model area.

Fig. 5: The boundaries of SEPRAN model

Fig. 6 Contour of equi-pressure line.

Fig. 7 color contour of equi-pressure line.

4 Calculation of the cutting forces.

The forces that act on the blade during the cutting of soil, are transmitted on the blade through grain stresses and water pressures from wedge and cut soil.

The forces on the cut layer are shown in fig. 8. These forces are:

1. Normal stress force N₂ between the wedge and cut layer.

2. Shear stress force S₂ as a result of the internal friction of the sand between the wedge and cut layer.

3. Cohesion force of the sand C₂ between wedge and cut layer.

4. Water pressure difference force W₂ between the wedge and cut layer resulting from p₂.

5. Cohesion force of the sand C₁ between cut layer and ground soil.

6. Normal stress force N₁ between the cut layer and ground floor.

7. Shear stress force S₁ as a result of the internal friction of the sand between the cut layer and ground floor.

8. Water pressure difference force W₁ between the cut layer and ground floor resulting from p₁.

9. The force as a result of the acceleration of the sand T.

10. Weight of the sand wedge G₂.

11. The force W₆ as a result of the water resistance.

The normal force N₁ and the shear force S₁ are related according:

with

(10)

The normal force N₂ and the shear force S₂ are related as follows:

with

(11)

For the horizontal force equilibrium can now be found:

(12)

And for the vertical force equilibrium can be found:

(13)

Fig. 8 the forces on the cut layer

(14)

For the determination of the forces on the blade only the force K₂ is of importance. For this force can now be derived as (K₂ = K₂₁ + K₂₂):

	(15)
	(16)

The force K₂₁ is the water under-pressures part, while force K₂₂ is the part of the gravity, the inertia forces, the cohesion, the adhesion and the water resistance in the force K₂. The forces on the wedge layer are shown in fig. 9. These forces are:

1. The earlier mentioned forces N₂, S₂, C₂ and W₂.

2. A, shear stress as a result of the adhesion between the soil and the blade.

3. Water under-pressures on the blade p₄ , resulting in the force W₄.

4. Normal stress, resulting in the force N₄.

5. Shear stress as a result of the soil/steel friction, S₄.

6. Normal stress force N₃ between the wedge and ground floor.

7. Shear stress force S₃ as a result of the internal friction of the sand between the wedge and ground floor.

8. Water pressure difference force W₃ between the wedge and ground floor resulting from p₃.

9. Cohesion force of the sand C₃ between wedge and ground soil.

10. Weight of the sand wedge G₂.

Fig. 9 the forces on the wedge.

The normal force N₃ and the shear force S₃ are related as follows:

with

(17)

The normal force N₄ and the shear force S₄ are related as follows:

with

(18)

Horizontal force equilibrium

(19)

Vertical force equilibrium

(20)

So

(21)

This force can now be derived into two parts (K₄ = K₄₁ + K₄₂):

	(22)
	(23)

The K₄₁ part is pore water part, which of important effect on the blade force. The forces that act on the blade during the cutting of soil, are transmitted on the blade through grain stresses and water pressures. These forces are indicated in fig.10.

Fig. 10 the forces on blade

The forces are:

1. The earlier mentioned forces N₄, S₄, A and W₄.

2. water force W₅ behind the blade

The resulting water force on the blade W₄ can be determined theoretically. The resulting water force W₅ behind the blade can be determined by the angle of internal friction from measurements. Since the grain force K₄ is known, forces on the blade can be calculated now.

The following forces acting on the blade per unit width can be calculated as:

The horizontal force F_h.

(24)

The vertical force F_v.

(25)

If there is no cavitation the water pressures forces W₁, W₂, W₃ and W₄ can be written as:

,	(26), (27)
,	(28), (29)

Under cavitation situation the pore pressure reaches water pressure with z as water depth.

So pore pressure forces become:

,	(30), (31)
,	(32), (33)

5 The equilibrium of moments.

To solve equation 24 and 25, the wedge angle a has to be known. Since this angle is the 4^th unknown parameter, K₁, K₂ and b are the first 3, a 4^th equation or condition must be found. Using the condition of minimum energy as used by Zhao & Miedema, 2000/2001 [32, 33], does not give satisfactory results. For the full-cavitational situation a minimum exists, but the wedge angles found do not match the observations of He, 1998 [8]. For the non-cavitational situation the minimum does not exists, so another equation/condition had to be found.

Using the Mohr circle encounters some complications. In principle the Mohr circle can be used for an infinitesimal small element, where an equilibrium of moments exists, based on constant stresses on the boundaries of the element. The points of action of the resulting forces are on the center of the boundaries. The water sub-pressures should be distributed equally on all boundaries, resulting in an equilibrium of the forces resulting from the grain stresses. This last condition is correct for the full cavitational situation, but not for the non-cavitational situation. In the latter case, the Mohr circle for the soil stresses can be used. From the calculation and measurements of He, 1998 [8] and Zhao & Miedema, 2000/2001 [32, 33] it appears that the points of action of the forces on the wedge are not in the centers of the 3 boundaries of the wedge. Still the Mohr circle approach gives wedge angles that match the observations of He.

Using the equilibrium equation of moments on the wedge around the blade tip is of course the best approach, if the points of action of the forces are known. Ma & Miedema, 2001 [34] derived this equilibrium equation and estimated values for the points of action of the forces on the wedge, equation 34 shows this equilibrium.

(34)

The parameters e₂, e₃ and e₄ represent the points of action on the 3 boundaries as a fraction of the boundary length. The values estimated are: e₂ = 0.35, e₃ = 0.56 and e₄ = 0.40, based on finite element calculations. Fig. 11 shows the resulting moment as a function of the shear angle b and the wedge angle a. From this graph the conclusion can be drawn that the wedge angle should be about 55.

Fig. 11: Moment versus wedge angle by using polynomial regression for:
θ=90⁰; α=15⁰,20⁰,25⁰,30⁰;δ=28⁰;φ=42⁰; H_i=1;H_b=3;k_i/k_max=0.25

6 CONCLUSIONS

When cutting water saturated san at large cutting angles (>60), a wedge may occur in front of the blade, reducing the cutting forces. The correct way of finding the geometry of this wedge is to use the horizontal, vertical and moment equilibrium equations, combined with the principle of minimum energy. This results in an estimate of the wedge geometry and cutting forces that match the observations of He, 1998. Further research has to be carried out to improve the model and make it more reliable.

REFERENCES

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LIST OF SYMBOLS USED

A	Surface	m
b	width of the blade of blade element	m
e	volume strain	%
Fci	cutting force (general)	kN
g	gravitation acceleration	m/s
hi	initial layer thickness	m
k	permeability	m/s
ki	initial permeability	m/s
kmax	maximum permeability	m/s
km	effective permeability	m/s
l	length of the shear zone	m
n	normal on an edge	m
ni	initial pore percentage	%
nmax	maximum pore percentage	%
p	pressure (pore pressure)	kPa
patm	atmospheric pressure	kPa
pcalc	calculated dimensionless pressure (pore pressure)	-
pdamp	saturated vapor pressure (12 cm water column)	kPa
preal	real acting pressure (pore pressure)	kPa
p1m	average pore pressure in the shear zone	-
p2m	average pore pressure on the interface between wedge and cut sand	-
p3m	average pore pressure on the interface between the wedge and the original sand	-
p4m	average pore pressure on the blade	-
q, q1 ,q2	specific flow rate	m/s
Q	flow rate per unit blade width	m/s
s	length of a flow line	m
s	measure for the layer thickness	m
t	time	s
Dt	time interval	s
vc	cutting velocity perpendicular on the blade edge	m/s
V	volume increase per unit of blade width	m
x	coordinate	m
y	coordinate	m
z	coordinate	m
z	water depth	m
a	blade angle (without wedge) wedge angle(for wedge)	rad
b	shear angle	rad
q	Blade angle(for wedge)	rad
j	angle of internal friction	rad
d	soil/steel angle of friction	rad
rw	water density	ton/m

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[1] Associate Professor, Delft University of Technology, Mechanical Engineering, Dredging Engineering, Mekelweg 2, 2628 CD Delft, The Netherlands, Tel: +31-15-2788359, Fax: +31-15-2781397, s.a.miedema@wbmt.tudelft.nl, http://www.ocp.tudelft.nl/dredging/home.htm.

2 MSc student, Delft University of Technology, Mechanical Engineering, Dredging Engineering, Mekelweg 2, 2628 CD Delft, The Netherlands, Tel: +31-15-2786780, y.ma@student.tudelft.nl.