Finite Element Calculations to Determine the Pore Pressures
when Cutting Water Saturated Sand at Large Cutting Angles
Y. Zhao[1]
Dr.ir. S.A. Miedema[2]
Abstract: In the cutting of water-saturated sand, the
phenomenon of dilatation plays an important role. The existence of dead zone in
front of the blade when cutting at large cutting angles will affect the value
and distribution of vacuum water pressure on the interface. This paper attempts
to develop a finite element method for predicting the occurrence of boundary
soil wedges in water saturated soil cutting and to evaluate the interaction
between these two factors. First the authors consider the non-cavitation
situation with finite element approach, focusing on the dimensionless pore
vacuum pressure and plot of its distribution on the interfaces. Then, by using
an electrical analogon for the pore vacuum pressures, an analytical method is
applied. The results of the analytical method is very good for a first estimate
of the pore vacuum pressures. By the end, another important condition, full
cavitation situation, is analysed and calculated.
Keywords: Water saturated sand, dilatancy, pore water pressure, finite element
calculation, cavitation
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/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 certain resistance, cause 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<df).
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.
This paper attempts to develop a systematic method, following a finite
element approach to estimate the conditions for the occurrence of sand wedge
and evaluate the influence of the sand wedge on the cutting forces
theoretically.
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 which: |
(2) |
For the residual pore percentage is chosen for nmax 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
|
|
(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-pressures 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 of this area 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:
|
with: |
(5) |
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:
|
|
(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 The blade tip problem
The blade tip problem is a very important problem but very easy to get
neglected. During the physical modeling of the cutting process it has always
been assumed that the blade tip is sharp. In other words, that in the numerical
calculation, from the blade tip, a hydrostatic pressure can be introduced as
the boundary condition along the free sand surface behind the blade. In
practice this is never valid, because of the reasons of rounding, wearness,
dilatancy and sub-pressure behind the blade. A combination of these factors
determines the distribution of the water under-pressures, especially around the
blade tip. The first three factors can be accounted for in the numerical calculation as an extra
boundary condition behind the blade tip. Along the free sand surface behind the
blade tip an impenetrable line element is put in, in the calculation. The
length of this line element is varied with 0.0hi , 0.1hi
and 0.2hi. It showed from these calculations that especially the
water under-pressures on the blade are strongly determined by the choice of
this boundary condition as indicated in fig.5.
Fig.5: The water sub-pressures distribution on
the blade and shear zone
as function of the length of the flat wear
section w.
It is hard to estimate to what degree the influence of the
under-pressure behind the blade on the water under-pressures around the blade
tip can be taken into account with these extra boundary condition. Since there
is no clear formulation for the under-pressure behind the blade available, it
will be assumed that the extra boundary condition at the blade tip describes
this influence. The model is built up with the length of the shear zone behind the blade as 0.2hi. In the SEPRAN mesh produced for the finite
element calculation, as shown in fig.6, We can see the existence of blade tip.
Behind the curve 13, which stand for blade, curve 9 is in the same straight
line with curve 8 and curve 10, standing for an impermeable line element as
blade tip, with the length of 0.2hi.
4 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.6,
SEPRAN model is made of three parts, the original sand layer, the cut sand
layer, and the wedge.
fig.6: The boundaries of SEPRAN model
The solution of such a calculation is however not only dependent on the
physical model of the problem, but also on the next points:
- The size of the area in which the
calculation takes place.
- The size and distribution of the elements
- 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% (see fig.7
and fig.8).
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:
Fig.7 coarse mesh
Fig.8 fine mesh
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:
- A hydrostatic pressure along the boundary.
- A boundary as an impermeable wall.
- 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.9 and Fig.10 give an
impression of the equi-potential line in the model area, while Fig.11 to Fig.14
show the two-dimensional distribution of the water vacuum pressures on shear
zone, interface between wedge and cut sand, interface between wedge and
original sand, and on the blade respectively.
Fig.9 Contour of equi-pressure line
Fig.10 color contour of equi-pressure line
|
|
|
|
|
|
Fig.11 pressure
distribution on shear zone(upleft), on the interface between wedge and cut
sand(upright), on the interface between the wedge and the original
sand(downleft) and on blade(downright).
5 CALCULATION OF 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.12. 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:
|
|
(10) |
The normal force N2
and the shear force S2 are related as follows:
|
|
(11) |
For the horizontal
force equilibrium can now be found:
|
|
(12) |
And for the vertical
force equilibrium can be found:
|
|
(13) |
Fig.12 the forces on the cut layer
We can get
|
|
(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. K22 part is neglected in the
Visual Basic program calculation.
The forces on the
wedge layer are shown in fig.13.
Fig.13 the forces on wedge
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.
The normal force N3
and the shear force S3 are related as follows:
|
|
(17) |
The normal force N4
and the shear force S4 are related as follows:
|
|
(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 K42 part is neglected in the Visual Basic
programme calculation.
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.14.
Fig.14 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 Fh.
|
|
(24) |
The vertical force Fv.
|
|
(25) |
If there is no
cavitation the water pressures forces W1, W2, W3
and W4 can be written as:
|
|
(26) |
|
|
(27) |
|
|
(28) |
|
|
(29) |
6 ANALYTICAL WATER PORE PRESSURE CALCULATIONS
As is shown in fig.4, the water can flow from 4 directions to the shear
zone where the dilatancy takes place. Two of those directions go through the
sand which has not yet been deformed and thus have a permeability of ki
, while the other two directions go through the deformed sand and thus have a
permeability of kmax. Fig.4 shows that the flow lines in 3 of the 4
directions have a more or less circular shape, while the flow lines above the
blade have the character of a straight line. If a point on the shear zone is
considered, then the water will flow to that point along the 4 flow lines as
mentioned above. Along each flow line, the water will encounter a certain
resistance. One can reason that this resistance is proportional to the length
of the flow line and reversibly proportional to the permeability of the sand,
the flow line passes. Fig.15 shows a point on the shear zone and it shows the 4
flow lines. The length of the flow lines can be determined with the equations
30, 31, 32 and 33. The variable "Lmax" in these equations
is the length of the shear zone, which is equal to hi/sin(b), while the variable "L" starts at
the free surface with a value zero and ends at the wedge tip with a value
"Lmax".
Fig.15: The flow lines used
in the analytical method.
|
with |
(30) |
|
with |
(31) |
|
with |
(32) |
|
with |
(33) |
The total resistance on the flow lines can be determined by dividing the
length of a flow line by the permeability of the flow line. The equations 34,
35, 36 and 37 give the resistance of each flow line.
|
|
(34) |
|
|
(35) |
|
|
(36) |
|
|
(37) |
Since the 4 flow lines can be considered as 4 parallel resistors, the
total resulting resistance can be determined according to the rules for parallel
resistors. Equation 38 shows this rule.
|
|
(38) |
The resistance Rt in fact replaces the hi/kmax
part of the equations 5 and 6, resulting in equation 39 for the determination of
the pore vacuum pressure of the point on the shear zone.
|
|
(39) |
The average pore vacuum pressure on the shear zone can be determined by
summation or integration of the pore vacuum pressure of each point on the shear
zone. Equation 40 gives the average pore vacuum pressure by summation.
|
|
(40) |
The determination of the average pore vacuum pressure on the blade and other
interfaces cannot be carried out by integration or summation, because the
calculation only gives the pore vacuum pressure at the tip (edge) of the
interfaces. It is known that the pore vacuum pressure at the top of the
interface between the wedge and cut sand equals zero, because the sand at that
point is in direct contact with the surrounding water. If the pore vacuum
pressure distribution on the interface is considered linear, then the average
pore vacuum pressure equals 50% of the pore vacuum pressure at the blade edge.
|
|
(41) |
However fig.11 shows that this distribution is not linear. Going from
the tip (edge) to the top of the interface, first the pore vacuum pressure
increases until it reaches a maximum and then it decreases (non-linear) until
it reaches zero at the top. In this graph, the top is left and the tip is
right. Thus the pore vacuum pressure equals zero at the free water surface
(most right point of the graph).
Because the distribution of the pore vacuum pressure is non-linear, a
shape factor has to be used. From the FEM calculations of Miedema [24] and Zhao
[32] it is known, that the shape of the pore vacuum pressure distribution on
the interface depends strongly on the ratio of the length of the shear zone and
the length of the blade, and on the length of the flat wear zone (as shown in
figure 10). A high ratio should result in a shape factor higher then 2, while a
low ratio should result in a factor smaller than 0.5. Equation 42 gives the
ratio in a modified form, which is empirical function get from comparison with
the FEM plot. The value of the power has been determined by trial and error.
|
|
(42) |
In the past decades many research has been carried out into the
different cutting processes. The more fundamental the research, the less the
theories can be applied in practice. The analytical method as described in this
paper, gives a method to use the basics of the sand cutting theory in a very
practical and pragmatic way. One has to consider that usually the accuracy of
the output of a complex calculation is determined by the accuracy of the input
of the calculation, in this case the soil mechanical parameters. Usually the
accuracy of these parameters is not very accurate and in many cases not
available at all. The accuracy of less then 10% of the analytical method
described in this paper is small with regard to the accuracy of the input. This
does not mean however that the accuracy is not important, but this method can
be applied for a quick first estimate.
By introducing some shape factors to the shape of the streamlines, the
accuracy of the analytical model can be improved.
7 FULL CAVITATION SITUATION
Under cavitation
situation the pore pressure reaches water pressure 
with z as water depth.
So pore pressure
forces become:
|
|
(45) |
|
|
(46) |
|
|
(47) |
|
|
(48) |
for simplification, to
reduce the forces into dimensionless we neglect the other parameters because
they are constant. With this simplification the relationship we pursued wont
be affected.
So
|
|
(49) |
|
|
(50) |
|
|
(51) |
|
|
(52) |
So the dimensionless
grain force is
|
|
(53) |
|
|
(54) |
When 
equal to zero,
|
|
(55) |
the value of 
reach its minimum,
so is 
.
This equation requires
iterative method to solve.
From this relationship we can determine the angle of shear zone when the
grain force K2 is minimal
under cavitation condition. From the former equations we know, that in the
components of Fh and Fv, K2 is the only part
that is relevant to 
. So we can say when K2 reaches its minimum, the
horizontal and vertical forces on blade reach their minimum also. The natural
tendency is when the minimum force (energy) is acquired, so is the stable
situation. So the geometry property is got.
With blade angle equal to 90 to 40, under two
different steel-sand friction angle d, the relationship between 
and
is calculated. In
=90 ,
situation the vertical force on the blade is zero, which is
easy to understand because on the vertical blade the friction between sand
wedge and the blade surface is the only source of vertical force. With 
increased, both the vertical and horizontal force are
increased.
The three angles 
,
,
form a stable relationship with each other, no matter what
is the value of 
.
For perfectly smooth blade, under full-cavitation situation, the forces
for non-wedge are always smaller compared to relevant forces for wedge. So
there will be no wedge exist.
For
blade angle 90 and 80, the wedge will exist to attain the smaller cutting forces. But
between blade angle 80and 70, the situation changes. Somehow if f is below 25, then no matter what the blade angle is, the wedge
will exist. As shown in Fig.16. The two planes which stand for the cutting
forces of wedge and non-wedge respectively intersect with each other. Between
the intersect line, the change of the pattern happens. Is there a wedge or not
will depend on which one give the smaller cutting force.
From the force balance
on wedge zone(equation 19,20), taking account of grain forces and pore water pressure
forces only, we get the equation:
|
|
(56) |
Input the value of
water pressure forces under full cavitation situation (equation 46,47 and 48)
we get:
|
|
(57) |
so under cavitation situation
Fig.16 3-D plot of minimal horizontal forces
under different internal friction angle and blade angle,
For wedge and non-wedge cutting respectively
|
|
(58) |
From the force balance
on wedge zone(equation 19,20), we can also get the formula presentation of
grain forces between wedge and original soil K3.
|
|
(59) |
For sand, there is no
tension force. So K3 must be a positive value. That means:
|
|
(60) |
When take 
then 
. The value of 
can never be as big as 
.
8 CONCLUSIONS
A model for the cutting forces in saturated sand at large cutting angles
has been derived mathematically, by application of numerical pore pressure
calculations and existing cutting theory and taking into account the effect of
a dead zone wedge. For full cavitation situation the results of calculation
show clearly a transition phase between wedge cutting and non-wedge cutting.
The happen of transition is depend on the blade angle and soil internal
friction angle. For non-cavitation situation both the numerical and analytical
model give realistic simulation of practical situation. But from the cutting
forces we calculated we didnt find the solution. We cant decide the wedge
geometry from wedge cutting force, so it is impossible to compare the wedge and
non-wedge situation.The reason for that is the subject of furter research work.
REFERENCES
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 |