29
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
L. Lamberson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society
for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-59947-8_6
Chapter 6
Observations of Velocity-Dependent Drag and Bearing Stress
in Sand Penetration
B. Kenneally, M. Omidvar, S. Bless, and M. Iskander
Abstract Data from rapid penetration tests into granular media were analyzed using a two-step Poncelet equation to determine values for the Poncelet drag coefficient on a projectile. The data were from a spherical projectile launched into both
densely and loosely packed Ottawa sand at transonic impact velocities of approximately 300 m/s. Velocity data were obtained
and integrated over time to produce data for penetration depth. Three approaches were investigated: (1) Fitting the data with
a constant drag coefficient, (2) fitting the data with a constant drag coefficient along with a constant velocity- independent
term computed as the average of the quasi-static penetration resistance into the same target, and (3) by separating the data
into two segments about a transition velocity, and using best fit values for the inertial drag and bearing stress terms. The latter
approach produced superior fits compared to the other two procedures investigated. The result indicates a velocity dependence of both the inertial drag coefficient and bearing stress on a projectile during rapid penetration into granular media.
Keywords Projectile · Sand · Poncelet · Drag · Penetration Depth
6.1 Introduction
The phenomenology of rapid penetration into granular media as well as the applications of Poncelet’s equation have been
extensively investigated [1–7]. In particular, Omidvar et al. [7] identified the existence of distinct regimes for penetration
resistance in sand. The Poncelet equation considers the force on a projectile as due to two terms:
-
=
+
m
dv
dt
C Av RA
r
2
(6.1)
For a projectile penetrating a target at starting velocity v 0 , the penetration depth of the projectile at instantaneous penetration velocity, v, is given by
P
m
C A
v
R
C
v
R
C
=
+
+
æ
è
ç
ç
ç
ç
ö
ø
÷
÷
÷
÷
2
0
2
2
r
r
r
ln
(6.2)
where, m is the projectile mass, A is the projected projectile area, ρ is the target density, and C and R are the inertial drag
coefficients and the velocity-independent bearing resistance, respectively. The drag coefficient, C, is adopted to describe the
inertial force needed to accelerate sand particles ahead of the projectile, which results in a velocity squared-dependent resistance to penetration, F i = CρAv
2
. The bearing stress, R, describes velocity-independent resistance of sand, F f , which is related
B. Kenneally · M. Omidvar
Department of Civil and Environmental Engineering, Manhattan College, Riverdale, NY, USA
e-mail: bkenneally01@manhattan.edu; momidvar01@manhattan.edu
S. Bless (*) · M. Iskander
Civil and Urban Engineering Department, Tandon School of Engineering, New York University, Brooklyn, NY, USA
e-mail: sbless@nyu.edu
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
L. Lamberson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society
for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-59947-8_6
Chapter 6
Observations of Velocity-Dependent Drag and Bearing Stress
in Sand Penetration
B. Kenneally, M. Omidvar, S. Bless, and M. Iskander
Abstract Data from rapid penetration tests into granular media were analyzed using a two-step Poncelet equation to determine values for the Poncelet drag coefficient on a projectile. The data were from a spherical projectile launched into both
densely and loosely packed Ottawa sand at transonic impact velocities of approximately 300 m/s. Velocity data were obtained
and integrated over time to produce data for penetration depth. Three approaches were investigated: (1) Fitting the data with
a constant drag coefficient, (2) fitting the data with a constant drag coefficient along with a constant velocity- independent
term computed as the average of the quasi-static penetration resistance into the same target, and (3) by separating the data
into two segments about a transition velocity, and using best fit values for the inertial drag and bearing stress terms. The latter
approach produced superior fits compared to the other two procedures investigated. The result indicates a velocity dependence of both the inertial drag coefficient and bearing stress on a projectile during rapid penetration into granular media.
Keywords Projectile · Sand · Poncelet · Drag · Penetration Depth
6.1 Introduction
The phenomenology of rapid penetration into granular media as well as the applications of Poncelet’s equation have been
extensively investigated [1–7]. In particular, Omidvar et al. [7] identified the existence of distinct regimes for penetration
resistance in sand. The Poncelet equation considers the force on a projectile as due to two terms:
-
=
+
m
dv
dt
C Av RA
r
2
(6.1)
For a projectile penetrating a target at starting velocity v 0 , the penetration depth of the projectile at instantaneous penetration velocity, v, is given by
P
m
C A
v
R
C
v
R
C
=
+
+
æ
è
ç
ç
ç
ç
ö
ø
÷
÷
÷
÷
2
0
2
2
r
r
r
ln
(6.2)
where, m is the projectile mass, A is the projected projectile area, ρ is the target density, and C and R are the inertial drag
coefficients and the velocity-independent bearing resistance, respectively. The drag coefficient, C, is adopted to describe the
inertial force needed to accelerate sand particles ahead of the projectile, which results in a velocity squared-dependent resistance to penetration, F i = CρAv
2
. The bearing stress, R, describes velocity-independent resistance of sand, F f , which is related
B. Kenneally · M. Omidvar
Department of Civil and Environmental Engineering, Manhattan College, Riverdale, NY, USA
e-mail: bkenneally01@manhattan.edu; momidvar01@manhattan.edu
S. Bless (*) · M. Iskander
Civil and Urban Engineering Department, Tandon School of Engineering, New York University, Brooklyn, NY, USA
e-mail: sbless@nyu.edu