Models Penetration

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Models Penetration
Volume 108 , October 2017 , Pages 3-26
Author links open overlay panel Charles E. Anderson Jr.
The mechanics of penetration of projectiles into targets
Int J Eng Sci , 16 ( 1 ) ( 1978 ) , pp. 1 - 100
Mechanics of penetration: analysis and experiment
Int J Eng Sci , 16 ( 11 ) ( 1978 ) , pp. 879 - 904
Int J Eng Sci , 16 ( 11 ) ( 1978 ) , pp. 793 - 920
Ballistic impact: the status of analytical and numerical modeling
Int J Impact Eng , 7 ( 1 ) ( 1988 ) , pp. 9 - 35
Non-ideal projectile impact on targets
Int J Impact Eng , 22 ( 2-3 ) ( 1999 ) , pp. 95 - 395
Penetration experiments with 6061-T6511 aluminum targets and spherical-nose steel projectiles at striking velocities between 0.5 and 3 km/s
Int J Impact Eng , 24 ( 2000 ) , pp. 57 - 67
J.A. Zukas (Ed.) , High velocity impact dynamics , John Wiley & Sons , NY, NY ( 1990 )
Penetration data for a medium caliber tungsten sinter alloy penetrator into aluminum alloy 7020 in the velocity regime from 250 m/s to 1900 m/s
F. Gálvez , V. Sánchez-Gálvez (Eds.) , Proc. 23rd Int. symp. ballistics , 2 , S.L., Madrid, Spain , Gráficas Couche ( 2007 ) , pp. 1437 - 1452
J Appl Phys , 19 ( 1948 ) , pp. 563 - 582
Experimental test of the theory of penetration by metallic jets
J Appl Phys , 27 ( 1 ) ( 1956 ) , pp. 63 - 68
Penetration of a rod into a semi-infinite target
J Franklin Inst , 272 ( 1961 ) , pp. 275 - 284
Analysis of high-velocity projectile penetration mechanics
J Appl Phys , 37 ( 4 ) ( 1966 ) , pp. 1579 - 1587
J Appl Mech , 30 ( 1963 ) , pp. 384 - 390
Long-rod penetration, target resistance, and hypervelocity impact
Int J Impact Eng , 14 ( 1993 ) , pp. 1 - 14
Hypervelocity penetration modeling: momentum vs. energy and energy transfer mechanisms
Int J Impact Eng , 26 ( 2001 ) , pp. 809 - 822
The influence of projectile hardness on ballistic performance
Int J Impact Eng , 22 ( 6 ) ( 1999 ) , pp. 619 - 632
John Wiley & Sons, Inc , NY, NY ( 1982 )
A theory for the deceleration of long rods after impact
J Mech Phys Solids , 15 ( 1967 ) , pp. 387 - 399
Penetration of a rod into a target at high velocity
Fizika Goreniya I Vzryva , 2 ( 2 ) ( 1966 ) , pp. 99 - 106
translated Combustion, Explosion, and Shock Waves , pp. 63-66, July 1965
Further results in the theory of long rod penetration
J Mech Phys Solids , 17 ( 1969 ) , pp. 141 - 150
Penetration of 6061-T651 aluminum targets with rigid long rods
J Appl Mech , 55 ( 1988 ) , pp. 755 - 760
Penetration of strain-hardening targets with rigid spherical-nose projectiles
J Appl Mech , 58 ( 1991 ) , pp. 7 - 10
Penetration of 6061-T6511 aluminum targets by ogive-nose steel projectiles with striking velocities between 0.5 and 3.0 km/s
Int J Impact Eng , 23 ( 1999 ) , pp. 723 - 734
Penetration experiments with 6061-T6511 aluminum targets and spherical-nose steel projectiles at striking velocities between 0.5 and 3.0 km/s
Int J Impact Eng , 24 ( 1 ) ( 2000 ) , pp. 57 - 67
Long rod penetration models—Part I. A flow field model for high speed long rod penetration
Int J Mech Sci , 28 ( 8 ) ( 1986 ) , pp. 535 - 548
Long rod penetration models—Part II. Extensions to the hydrodynamic theory of penetration
Int J Mech Sci , 28 ( 9 ) ( 1986 ) , pp. 599 - 612
The theory of indentation and hardness
Proc R Soc , 57 ( 3 ) ( 1945 ) , pp. 147 - 159
An examination of long-rod penetration
Int J Impact Eng , 11 ( 4 ) ( 1991 ) , pp. 481 - 501
Dynamic expansion of spherical cavities in metals
I. Sneddon , R. Hill (Eds.) , Progress in solid mechanics , 1 , North-Holland , NY ( 1960 ) , pp. 85 - 164
Dynamic spherical cavity expansion of strain-hardening materials
J Appl Mech , 58 ( 1991 ) , pp. 1 - 6
Effects of strain hardening and strain-rate sensitivity on the penetration of aluminum targets with spherical-nosed rods
Int J Solids Struct , 35 ( 28-29 ) ( 1998 ) , pp. 3737 - 3753
Cavitation and the influence of headshape in attack of thick targets by non-deforming projectiles
J Mech Phys Solids , 28 ( 1980 ) , pp. 249 - 263
A theoretical estimate of temperature effects during rod penetration
Proc. 9th Int. symp. ballistics , 2 , Shriverham, UK ( 1986 ) , pp. 307 - 314
Calculations of elastic-plastic flow
B. Adler , S. Fernback , M. Rotenberg (Eds.) , Methods of computational physics , 3 , Academic Press , NY, NY ( 1964 )
Adaptive Lagrangian modeling of ballistic penetration of metallic targets
Comput Methods Appl Mech Eng , 142 ( 1997 ) , pp. 269 - 301
The effect of target inertia on the penetration of aluminum targets by rigid ogive-nosed long rods
Int J Impact Eng , 91 ( 2016 ) , pp. 6 - 13
“A comment on ‘The effect of target inertia on the penetration of aluminum targets by rigid ogive-nosed long rods’ by T. L. Warren,” Letter to the Editor
Int J Impact Eng , 93 ( 2016 ) , pp. 231 - 233
Essential physics of target inertia in penetration problems missed by cavity expansion models
Int J Impact Eng , 98 ( 2016 ) , pp. 97 - 104
Response to: ‘A comment on ‘The effect of target inertia on the penetration of aluminum targets by rigid ogive-nosed long rods by T. L. Warren’ by Z. Rosenberg and E. Dekel,” Letter to the Editor
Int J Impact Eng , 93 ( 2016 ) , pp. 234 - 235
Dynamic perforation of viscoplastic plates by rigid projectiles
Int J Eng Sci , 21 ( 6 ) ( 1983 ) , pp. 577 - 591
Int J Eng Sci , 25 ( 4 ) ( 1987 ) , pp. 473 - 482
A time-dependent model for long-rod penetration
Int J Impact Eng , 16 ( 1 ) ( 1995 ) , pp. 19 - 48
Long-rod penetration: cylindrical vs. spherical cavity expansion for the extent of plastic flow
C. Van Niekerk (Ed.) , Proc 17th symp. ballistics , 3 , Morleta Park, South African , South African Ballistics Organisation ( 1998 ) , pp. 319 - 326
Analysis of the terminal phase of penetration
Int J Impact Eng , 29 ( 2003 ) , pp. 69 - 80
Modification of the Walker-Anderson penetration model to include exit failure modes and fragmentation
C. Van Niekerk (Ed.) , Proc. 17th Int. symp. ballistics , 3 , Morleta Park, South African , South African Ballistics Organisation ( 1998 ) , pp. 267 - 274
An analytic velocity field for back surface bulging
W.G. Reinecke (Ed.) , Proc. 18th Int. symp. ballistics , 2 , Technomic Publishing Company , Lancaster, PA ( 1999 ) , pp. 1239 - 1246
A consistent flow approach to model penetration and failure of finite-thickness metallic targets
W.G. Reinecke (Ed.) , Proc. 18th Int. symp. ballistics , 2 , Technomic Publishing Company , Lancaster, PA ( 1999 ) , pp. 761 - 768
C.E. Anderson
Jr. , B.L. Morris , D.L. Littlefield (Eds.) , SwRI Report 3593/001 , Southwest Research Institute , San Antonio TX ( 1992 )
Time-resolved penetration of long rods into steel targets
Int J Impact Eng , 16 ( 1 ) ( 1995 ) , pp. 1 - 18
Target effective flow stress calibrated using the Walker-Anderson penetration model
Proc. 28th Int. symp. ballistics , 2 , DESTech Publications, Inc. ( 2014 ) , pp. 1242 - 1253
A penetration model based on experimental data
Int J Impact Eng , 80 ( 2015 ) , pp. 24 - 35
A constitutive model and data for metals subjected to large strain, high strain rates, and high temperatures
Proc. 7th Int. symp. ballistics , The Hague, The Netherlands ( 1983 ) , pp. 541 - 548
Survivability modeling in DARPA's adaptive vehicle make (AVM) program
Proc. 27th Int. symp. ballistics , 2 , Lancaster, PA , DEStech Publications ( 2013 ) , pp. 967 - 979
Penetration efficiency as a function of target obliquity and projectile pitch
Tungsten into steel penetration including velocity, L / D , and impact inclination effects
I.R. Crewther (Ed.) , Proc. 19th Int. ballistics symp , 3 , Interlaken, Switzerland ( 2001 ) , pp. 1133 - 1140
The erosion transition of tungsten-alloy long rods into aluminum targets
Int J Impact Eng , 44 ( 2007 ) , pp. 2168 - 2191
© 2017 Elsevier Ltd. All rights reserved.
Analytical penetration models, including assumptions and results, are reviewed.
Applicability and limitations of penetration models are highlighted.
A review of analytical penetration models has been conducted and summarized. Models include the Poncelet equation, hydrodynamic theory, modified hydrodynamic theory, Recht-Ipson, Tate-Alekseevskii, cavity expansion, Ravid-Bodner, Walker-Anderson, and similitude modeling. These models describe, depending upon assumptions, rigid-body penetration, eroding penetration, steady-state and transient penetration, and perforation. Model assumptions are highlighted, and examples are provided of model predictions against experimental data. The manuscript has 30 figures, many of which compare model results to experimental data; 59 reference citations are included.
vector potential, Eqs. (50) and (59)
projectile radius ( Fig. 14 ), Eq. (39)
maximum hardness at bottom of crater, Eq. (16)
rate of deformation tensor, Eq. (62)
unit vectors in r, θ , and ϕ directions
Force, normal force, tangential force
distance between projectile nose to target rear (back) surface
mass of L / D 1 projectile, Eq. (16)
penetration depth of a L / D = 1 projectile, Eq. (16)
penetration channel (crater) radius
extent of plastic zone in projectile
velocity of eroded projectile debris, Eq. (37)
axial and radial extent of plastic flow field (Ravid-Bodner model)
extent of plastic flow field (Walker-Anderson model)
target shear stress, Eqs. (48) and (49)
Several good articles exist in the literature that review penetration, perforation, and other aspects of terminal ballistics. Backman and Goldsmith [1] present an extensive survey (278 references) of the interaction of projectiles and targets. They looked at semi-infinite targets, thin plate penetration and perforation, and intermediate and thick targets, covering the entire velocity range. Jonas and Zukas [2] reviewed available analytical methods for the study of projectile-armor interactions, with an emphasis on numerical simulations (167 references), focused on the velocity range of 0.5 to 2.0 km/s. A special issue of the International Journal of Engineering Science [3] was dedicated to penetration mechanics, covering experimental investigations, analytical modeling, and numerical modeling. Anderson and Bodner [4] reviewed analytical and numerical modeling of ballistic impact (88 references). Goldsmith [5] provides an extensive review of non-ideal projectile impact on targets (367 references). The above articles concentrate, in varying detail, on experimental procedures/results, data trends, analytical modeling, numerical modeling, and the mechanics of projectile-target interactions.
There has been considerable progress in analytical modeling since the publication of Refs. [1] , [2] , [3] , [4] . This article provides a review of analytical models that have been developed to describe the process of penetration into metallic targets, subject to several constraints. Firstly, the model has to provide significant insight into the mechanics of penetration. This constraint eliminates empirical models that require experimental data to fit an assumed functional form. It is noted that empirical models can be very useful, but their range of validity is limited by the experimental data used to determine the model parameters. Secondly, the model provided a major advance, and was not a slight modification of an existing model. Any errors of omission are those solely by the author, either because the model did not meet the selection criteria, or simply because we were unaware of the model.
In today's world of large-scale numerical simulations, it can be asked: “Why is there a need for analytical models?” There are several answers to this question. If an analytic model can be developed that captures the essence of the phenomenology, then understanding has been demonstrated, and the essential and relevant physics/mechanics have been applied/incorporated. This fundamental understanding leads to improved designs. Further, analytical models are typically hundreds to thousands times faster in execution than numerical simulations. Thus, analytical models provide tools for predictions, rapid design studies, first-order optimization, and risk assessments. For example, a typical risk assessment might run thousands (maybe millions) of Monte Carlo scenarios, which would be prohibitively time-consuming for numerical simulations, but which are well suited for analytical models.
With the above as background, the arrangement of the paper is largely chronological by model development. The focus is on metallic targets.
Jean-Victor Poncelet (1788–1867) was a French engineer and mathematician. He developed an ordinary differential equation to describe penetration of rigid projectiles. Newton's second law gives the projectile deceleration as a result of a resisting force:
(1) M d v d t = − F = − ( A + B v 2 )
The A is a static target resistance term, and the B v 2 term states that the resisting force is proportional to the square of the velocity. In traditional fluid mechanics, this v 2 term is usually called a drag term. Since the projectile is non-deforming, the cross-sectional area remains constant, and Eq. (1) can be written as:
(2) ρ p L d v d t = − ( a + b v 2 )
( L is the effective length of the projectile if the cross-section is not a constant.) Eq. (2) can be integrated to find the total depth of penetration ( P ) by using the chain rule of differentiation, d v/ dt = ( d v/ dz )( dz / dt ) = v( d v/ dz ):
(3) 1 ρ p L ∫ 0 P d z = − ∫ V 0 v d v a + b v 2
Thus,
(4) P L = ρ p 2 b ln ( 1 + b V 2 a )
(A more general case, with a resisting force proportional to the velocity as well as the velocity squared, in addition to the constant term, can be written. Subsequent integration of the equation results in an arctangent term in addition to the logarithmic term. This form of the resisting force is generally not used.)
Forrestal and Piekutowski [6] measured the depth of penetration for maraging steel projectiles with an ogival nose penetrating 6061-T6 aluminum as a function of impact velocity. The depths of penetration, normalized by the length of the projectile, are plotted in Fig. 1 . The Rockwell C hardness of the projectiles varied between Rc38 and Rc53, depending upon the specific alloying and heat treatment. A least-squares regression fit through the experimental data gives the values of the parameters a and b , given by the solid line in Fig. 1 (several points, above 1500 m/s, that fall below the general trend were ignored in the regression analysis; they will be discussed later).
Fig. 1. . Normalized penetration for very hard steel projectiles into 6061-T6 aluminum. Data from Ref. [6] .
For small values of bV 2 / a , the natural logarithm can be expanded to give:
(5) P L = ρ p V 2 2 a
which is shown as the dashed line in Fig. 1 . Thus, the depth of penetration is proportional to the square of the impact velocity. The resistive force, Eq. (2) , is approximately a constant, independent of the impact velocity, since the parameter b does not enter into Eq. (5) .
The normalized penetration for blunt-nose tungsten-alloy projectiles into a 7000 series aluminum target is shown in Fig. 2 . The experimental data are from two data sets generated by researchers at the Ernst-Mach-Institut [ 7 , 8 ]. A least-squares regression was conducted on all data below 0.7 km/s; the result is shown with the solid line. However, if four data points that are circled are deleted from the analysis, the resulting fit is shown with the dashed line. The fitted standard error is 45% less for the dashed curve than for the solid curve. Because we know that at sufficiently high impact velocities that the projectile will begin to deform, this Poncelet analysis is suggestive that the operative mechanics (that is, the assumption of rigid-body penetration) is not true for these four omitted data points. Note that without experimental evidence, it can only be said that this analysis is suggestive. Although not very satisfying, an alternative explanation is that the Poncelet parameters change during penetration. However, a similar analysis of Poncelet fits to the data for a completely different set of experiments, described in Section 6.3 , also show that a better fit to the experimental data is achieved if some of the higher velocity data points are not included. In these experiments, recovered projectiles of the deleted data points show considerable deformation.
Fig. 2. . Normalized penetration for tungsten-alloy projectiles into a structural grade aluminum. Date are from Refs. [7 , 8] .
One last comment is worth mentioning before proceeding. The small argument expansion, given by Eq. (5) , is only valid to about 0.3 km/s in Fig. 2 , given the values of a and b (shown in the figure). Thus, for this projectile-target combination, the assumption of a constant resisting force is only valid to about 0.3 km/s; at higher impact velocities, a “drag” resistive force must also be included to reproduce the experimental data.
We will see that the Poncelet equation will manifest itself several times in the development of penetration models.
The hydrodynamic theory of penetration emerged during WWII and was applied to shaped-charge jet penetration. The theory was developed by Birkhoff, MacDougall, Pugh, and Taylor [9] ; the authors acknowledged in a footnote that independent work of Hill, Mott and Pack in England led to the same results. The operative equation is derived from conservation of momentum, assuming that the strengths and viscosity of target and penetrator materials can be neglected, that is, the problem can be treated by hydrodynamics. Thus, the momentum equation is:
(6) ∂ v ∂ t + 1 2 ∇ ( v 2 ) − v × ( ∇ × v ) = − 1 ρ ∇ P r
where ∇ is the vector operator. With the assumption that the jet and target materials are incompressible, the term on the right-hand side can be written as ∇( P r / ρ ). A further simplification is done by using a coordinate transformation, as shown in Fig. 3 . It is assumed that the jet moves at a constant velocity v, and that the penetration velocity, u , is also a constant; thus the projectile-target interface does not move in Fig. 3 b. The momentum equation, Eq. (6) , then becomes
(7) [ v × ( ∇ × v ) ] = ∇ ( 1 2 v 2 + P r ρ )
with the assumptions of hydrodynamic and incompressible materials under steady-state conditions. The dot product of both sides of the equation by the vector velocity gives:
(8) v • ∇ ( 1 2 v 2 + P r ρ ) = 0
since [ v × (∇ × v )] is perpendicular to v . Thus, we have that the gradient of the term in the parenthesis is perpendicular to v . The term within the parenthesis is usually called Bernoulli's equation. It is important to note that while the units of the terms within the parenthesis in Eq. (8) have units of specific energy, Eq. (8) was derived from the momentum equation. 1
Fig. 3. . Schematic of an incompressible, hydrodynamic jet penetrating an incompressible hydrodynamic target.
Along the projectile-target centerline, Eq. (8) becomes:
(9) ∂ ∂ z ( 1 2 v 2 + P r ρ ) = ∂ ∂ z ( 1 2 ρ v 2 + P r ) = 0

Eq. (9) is integrated from the back of the projectile to the projectile-target interface, and then from the interface into the target. The pressure at the back of the projectile is zero, and the target is assumed to be semi-infinite (the pressure at infinity is zero). Using the nomenclature from Fig. 3 b, so that the velocity at the projectile-target interface is zero, and the target velocity at infinity is – u , integration of Eq. (9) gives:
(10) 1 2 ρ p ( v − u ) 2 = 1 2 ρ t u 2
This is the hydrodynamic, steady-state, incompressible model for penetration. Solving for t
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