30
to its frictional shear strength and can be expressed as F f = RA. In general, F f , and as a result, R, are depth- dependent, due to
the dependence of shear strength on geostatic stress. However, it is convenient to approximate R as a constant, which results
in Eq. (6.2).
While Eq. (6.1) can be used to describe penetration resistance in a variety of materials, the choice of a single C and R does
not always capture penetration-velocity characteristics in granular media under transonic impact velocities. In this chapter, a
least-square fitting error analysis is used to demonstrate the existence of two distinct regimes of penetration resistance and
their importance in accurately describing rapid penetration in sand. Conventionally, C and R are taken as the values that
produce the least-square best fit to the projectile velocity-depth data. In this analysis, different sets of C and R are assumed
for the high-velocity and low-velocity data, and the least-square fit residual is used to identify the transition zone from one
set of Poncelet parameters to the other. It is demonstrated that the resulting two-step Poncelet model can more accurately
capture velocity-penetration data for spherical projectiles penetrating sand at transonic velocities.
6.2 Experiments and Analysis Methods
The analysis utilizes data produced by launching a 14 mm diameter stainless steel sphere into dry Ottawa sand targets prepared under dense and loose packing conditions [6]. Impact velocities of approximately 300 m/s were achieved using an
electro-pneumatic launcher. The bulk compaction wave speed in dry sand has been reported to be about 80 m/s [4]; thus these
impacts produce transonic projectile speeds. Photonic Doppler velocimetry (PDV) was used to obtain projectile velocity
time-histories. Single integration was then used to produce penetration depth data. PDV measurements yielded high-resolution data for P(v), from the point of impact to approximately 0.1v 0 . The final stage of penetration was not resolved due to the
collapse of the cavity and the loss of the PDV signal. As a result, the terminal penetration depth was unknown.
The velocity-time data obtained from the PDV signal was smoothed using a 41-point fourth-order Savitsky-Golay algorithm, which operates as a piecewise polynomial fit to the data [8]. Projectile penetration depth data was produced by integrating the raw PDV velocity data over time. Least- square fits were produced using Eq. (6.2) and using either a single C and
R or two sets of C and R separated by a velocity transition zone. In the latter case, the least-square fit error was plotted for
the two Poncelet parameter sets as a function of the transition velocity, in order to identify the transition velocity resulting in
the best fit. The procedure also provides a means to extrapolate to the terminal penetration depth of the projectile where v = 0.
The choice of the velocity-independent bearing stress term, R, in Eq. (6.2) was informed by quasi-static penetration tests
of long rods having the same diameter as the projectile [8, 9]. Best fit values for R were restricted to positive values, with
zero being reported in place of negative best fit values.
Values for the transition velocity ranges were obtained by least- square fitting error analysis performed on the data sets
described in Table 6.1. In order to identify the transition zone, two separate sets of C and R were fitted to high and low velocities. In order to assess the high-velocity range, the Poncelet equation was fitted to the data from the impact velocity to a lower
velocity limit that was incrementally decreased in successive analyses. Best fit values for C and R, as well as Chi squared
error were recorded for each transition velocity. The process was repeated for the low velocity fit by beginning with the data
from the lowest recorded velocity to an incrementally increased upper velocity limit. In all cases, the Chi squared error was
recorded. Trends in the Chi squared error for the two separate fits were used to identify the optimal transition velocity. The
Chi squared error was computed as the sum of the square of the difference between the data points (y i ) and fitted data (y)
points divided by the standard deviation (σ i ):
Chi
2
2
=
-
æ
è
ç
ö
ø
÷
å
i
i
i
y y
s
(6.3)
Table 6.1 Test parameters for impact of spheres into dry Ottawa sand
Test designation [6]
Density (kg/m
3 )
Impact velocity (m/s)
Dense Dry OS (1420)
1817
298
Dense Dry OS (1426)
1817
269
Loose Dry OS (1427)
1587
304
Loose Dry OS (1428)
1587
296
B. Kenneally et al.
to its frictional shear strength and can be expressed as F f = RA. In general, F f , and as a result, R, are depth- dependent, due to
the dependence of shear strength on geostatic stress. However, it is convenient to approximate R as a constant, which results
in Eq. (6.2).
While Eq. (6.1) can be used to describe penetration resistance in a variety of materials, the choice of a single C and R does
not always capture penetration-velocity characteristics in granular media under transonic impact velocities. In this chapter, a
least-square fitting error analysis is used to demonstrate the existence of two distinct regimes of penetration resistance and
their importance in accurately describing rapid penetration in sand. Conventionally, C and R are taken as the values that
produce the least-square best fit to the projectile velocity-depth data. In this analysis, different sets of C and R are assumed
for the high-velocity and low-velocity data, and the least-square fit residual is used to identify the transition zone from one
set of Poncelet parameters to the other. It is demonstrated that the resulting two-step Poncelet model can more accurately
capture velocity-penetration data for spherical projectiles penetrating sand at transonic velocities.
6.2 Experiments and Analysis Methods
The analysis utilizes data produced by launching a 14 mm diameter stainless steel sphere into dry Ottawa sand targets prepared under dense and loose packing conditions [6]. Impact velocities of approximately 300 m/s were achieved using an
electro-pneumatic launcher. The bulk compaction wave speed in dry sand has been reported to be about 80 m/s [4]; thus these
impacts produce transonic projectile speeds. Photonic Doppler velocimetry (PDV) was used to obtain projectile velocity
time-histories. Single integration was then used to produce penetration depth data. PDV measurements yielded high-resolution data for P(v), from the point of impact to approximately 0.1v 0 . The final stage of penetration was not resolved due to the
collapse of the cavity and the loss of the PDV signal. As a result, the terminal penetration depth was unknown.
The velocity-time data obtained from the PDV signal was smoothed using a 41-point fourth-order Savitsky-Golay algorithm, which operates as a piecewise polynomial fit to the data [8]. Projectile penetration depth data was produced by integrating the raw PDV velocity data over time. Least- square fits were produced using Eq. (6.2) and using either a single C and
R or two sets of C and R separated by a velocity transition zone. In the latter case, the least-square fit error was plotted for
the two Poncelet parameter sets as a function of the transition velocity, in order to identify the transition velocity resulting in
the best fit. The procedure also provides a means to extrapolate to the terminal penetration depth of the projectile where v = 0.
The choice of the velocity-independent bearing stress term, R, in Eq. (6.2) was informed by quasi-static penetration tests
of long rods having the same diameter as the projectile [8, 9]. Best fit values for R were restricted to positive values, with
zero being reported in place of negative best fit values.
Values for the transition velocity ranges were obtained by least- square fitting error analysis performed on the data sets
described in Table 6.1. In order to identify the transition zone, two separate sets of C and R were fitted to high and low velocities. In order to assess the high-velocity range, the Poncelet equation was fitted to the data from the impact velocity to a lower
velocity limit that was incrementally decreased in successive analyses. Best fit values for C and R, as well as Chi squared
error were recorded for each transition velocity. The process was repeated for the low velocity fit by beginning with the data
from the lowest recorded velocity to an incrementally increased upper velocity limit. In all cases, the Chi squared error was
recorded. Trends in the Chi squared error for the two separate fits were used to identify the optimal transition velocity. The
Chi squared error was computed as the sum of the square of the difference between the data points (y i ) and fitted data (y)
points divided by the standard deviation (σ i ):
Chi
2
2
=
-
æ
è
ç
ö
ø
÷
å
i
i
i
y y
s
(6.3)
Table 6.1 Test parameters for impact of spheres into dry Ottawa sand
Test designation [6]
Density (kg/m
3 )
Impact velocity (m/s)
Dense Dry OS (1420)
1817
298
Dense Dry OS (1426)
1817
269
Loose Dry OS (1427)
1587
304
Loose Dry OS (1428)
1587
296
B. Kenneally et al.