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

The Delft Sand, Clay & Rock Cutting Model.
N1
N1
sin
h w
i
(7-65)
Substituting equation (7-64) in equation (7-65) gives for the normal stress on the shear plane:
s c hi w s a hb
w
cos( )
sin sin sin
N1
h w sin( )
i
c
s
sin
cos( ) r sin
sin( )
(7-66)
Assuming a fixed strain rate factor λs for cohesion and tensile strength, the normal stress minus the shear strength
(cohesion) has to be bigger than the tensile strength, where the tensile strength is negative (compressive stresses
are positive).
N1 s c
s T
(7-67)
Substituting equation (7-66) into equation (7-67) gives the condition for ductile failure:
sin
cos( ) r sin
c c
sin( )
s s s T
(7-68)
The transition from the Flow Type mechanism to the Tear Type mechanism is at the moment where the equal
sign is used in the above equation, resulting in a critical ratio between the tensile strength and the shear strength,
still also depending on the ac ratio r according to:
sin
r cos( ) sin( )
sin
c
sin( )
T
(7-69)
Figure 7-27 shows the critical ratio curves of the ratio of the tensile strength to the shear strength (cohesion) of the
transition of the Flow Type mechanism to the Tear Type mechanism. Since the tensile strength is considered to
be negative, the more negative this ratio, the higher the relative tensile strength. Below a curve the Flow Type
may be expected, above a curve the Tear Type. Only negative ratios should be considered, since the tensile
strength cannot be positive. The figure shows that for r=1 (high adhesive forces) the curve just touches a ratio of
zero, but never becomes negative, meaning the Tear Type will never occur. For smaller r values the curves are
more negative for a decreasing r value. The minimum for r is zero (no adhesion). The figure also shows that all
curves (except the r=0 curve) start with a positive value, then decrease with an increasing blade angle to a minimum
value and with a further increasing blade angle increase again to positive values. For blade angles larger than 90
degrees tensile failure will never occur. Because of the choice of the parameter hb, the blade height, at constant
blade height the length of the blade is increasing with a decreasing blade angle. This means that the adhesive force
on the blade increases with a decreasing blade angle, resulting in increasing normal stresses on the shear plane.
Higher normal stresses suppress tensile failure. On the other hand, an increasing blade angle will increase the
normal stress on the shear plane because of the force equilibrium. So we have two effects, the normal stresses on
the shear plane will decrease with an increasing blade angle because of the decrease of the adhesive force and the
normal stresses will increase with an increases blade angle because of the force equilibrium. The result is a curve
with a minimum.
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