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

The Delft Sand, Clay & Rock Cutting Model.
The boundary condition is in fact a specific flow rate (Figure 6-8) that can be determined with the following
hypothesis. For a sand element in the deformation zone, the increase in the pore volume per unit of blade length
is:
V A x hi
x l sin( )
(6-19)
nmax
ni
1
n
(6-20)
max
It should be noted that in this book the symbol ε is used for the dilatation, while in previous publications the symbol
e is often used. This is to avoid confusion with the symbol e for the void ratio.
For the residual pore percentage nmax is chosen on the basis of the ability to explain the water under-pressures,
measured in laboratory tests. The volume flow rate flowing to the sand element is equal to:
V
x
Q l sin( ) vc
l sin( )
t
t
(6-21)
With the aid of Darcy's law the next differential equation can be derived for the specific flow rate perpendicular
to the deformation zone:
Q k p k p
q q q v sin( )
l g n g n
i
max
1 2 c
w 1 w 2
(6-22)
The partial derivative p/n 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 Figure 6-8. 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 impenetrable. Making equation (6-22)
dimensionless is similar to that of the breach equation of Meijer and van Os (1976). 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 (6-22) in normalized format:
k p p g v h sin( )
n
k k h
i
w c i
'
with: n
' '
max n
1 n
2
max i
(6-23)
This equation is made dimensionless with:
p
'
p
'
n
n g v h /k
w c i max
(6-24)
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:
max 1 2
'
ki
p p
sin( )
k n n
'
(6-25)
Page 130 of 454 TOC Copyright © Dr.ir. S.A. Miedema