An examination of long-rod penetration

doi:10.1016/0734-743X(91)90015-8

International Journal of Impact Engineering

Volume 11, Issue 4, 1991, Pages 481-501

https://doi.org/10.1016/0734-743X(91)90015-8 Get rights and content

Abstract

The one-dimensional modified Bernoulli theory of Tate [J. Mech. Phys. Solids15, 287–399 (1967)] is often used to examine long-rod penetration into semi-infinite targets. The theory is summarized and the origins of the target resistance term examined. Numerical simulations were performed of a tungsten-alloy, long-rod projectile into a semi-infinite hardened steel target at three impact velocities sufficiently high to result in projectile erosion. The constitutive responses of the target and projectile were varied parametrically to assess the effects of strain hardening, strain-rate hardening, and thermal softening on penetration response. The results of one of the numerical simulations were selected to compare and contrast in detail with the predictions of the Tate model.

References (35)

A. Tate
Further results in the theory of long rod penetration
J. Mech. Phys. Solids
(1969)
A. Tate
Long rod penetration models—Part II. Extensions to the hydrodynamic theory of penetration
Int. J. Engng Sci.
(1986)
J. Awerbuch et al.
Analysis of the mechanics of perforation of projectiles in metallic plates
Int. J. Solids Struct.
(1974)
A. Tate
A comment on a paper by Awerbuch and Bodner concerning the mechanics of plate perforation by a projectile
Int. J. Engng Sci.
(1979)
S.R. Bodner
Reply to the comments of Yuan Wenzue, Zhou Lanting, and M. Xiaoqing on the ballistic penetration theory of Awerbuch and Bodner
Int. J. Solids Struct.
(1986)
R.C. Batra et al.
Steady state penetration of rigid perfectly plastic targets
Int. J. Engng Sci.
(1986)
Z. Rosenberg et al.
On the hydrodynamic theory of long-rod penetration
Int. J. Impact Engng
(1990)
P.H. Pidsley
A numerical study of long rod impact onto a large target
J. Mech. Phys. Solids
(1984)
R.C. Batra et al.
Steady state axisymmetric deformations of a thermoviscoplastic rod penetrating a thick thermoviscoplastic target
Int J. Impact Engng
(1991)
C.E. Anderson et al.
Ballistic impact: the status of analytical and numerical modeling
Int. J. Impact Engng
(1988)

V. Hohler et al.

Hypervelocity impact of rod projectiles with L/D from 1 to 32

Int. J. Impact Engng

(1987)

W.W. Predebon et al.

Inclusion of evolutionary damage measures in Eulerian wavecodes

Comput. Mech.

(1991)

J. Lankford et al.

Microstructural dependence of high strain rate deformation and damage development in tungsten heavy alloys

L.L. Wilson et al.

Experimental rod impact results

Int. J. Impact Engng

(1989)

A. Tate

A theory for the deceleration of long rods after impact

J. Mech. Phys. Solids

(1967)

V.P. Alekseevskii

Penetration of a rod into a target at high velocity

J. Liss et al.

Constraint to side flow in plates

J. Appl. Mech.

(1983)

Cited by (104)

A simplified and modified model for long rod penetration based on ovoids of Rankine
2021, International Journal of Impact Engineering
This paper proposes an alternative to the Alekseevskii-Tate model, denoted by AT, for erosion of a long rod penetrator. Specifically, the model uses simplified equations of the model for long rod penetration based on ovoids of Rankine developed in (Roisman et al., 2001). The resulting modified ovoid of Rankine model, denoted by MOR, proposes differential equations for the velocity of the projectile’s tail, the penetration velocity into the target, the erosion of the projectile’s mass and the penetration depth, as well as an algebraic equation to determine the point of separation of the target material from the projectile’s surface. The MOR model uses shock conditions to determine the initial value of the penetration velocity and it predicts the initial transient decrease of this velocity to its value for quasi-steady-state projectile erosion. This model also includes an expression for the crater diameter which predicts values close to experimental data. Relative to the AT model, the MOR model introduces a single empirical parameter in a modified evolution for mass erosion which significantly influences the penetration depth without affecting the crater diameter. This parameter models the interface between the deforming projectile’s nose and its rigid tail where the projectile’s tail is rapidly decelerating due to mushrooming and mass erosion. Its value is determined by comparing model predictions of the penetration depth with experimental data. Future research is needed to replace this empirical parameter with an analytical model and to better model the termination process of penetration.
Influence of the mushroomed projectile's head geometry on long-rod penetration
2021, International Journal of Impact Engineering
Self-sharpening has been observed in experiments and simulations of long-rod penetration. Penetration performances are influenced by different nose shapes occurred in the penetration of long rods made by different materials. In this paper, to model this nose shape effect, the resistance of the target in the standard Alekseevskii-Tate model is replaced by the resistance determined by the spherical cavity expansion approximation for general nose shapes. The function of nose shape is constructed to establish a modified model of long-rod penetration, in which the nose geometry during the long rod penetration is represented by two main parameters, i.e., the diameter ratio of rod nose and shank $η$ and the nose shape factor $N^{*}$ . The influences of $η$ and $N^{*}$ on long-rod penetration and their mechanisms are analyzed. The penetration performance decreases with an increasing diameter ratio $η$ , whose physical basis is the increased radial flow of rod material around the nose, which in turn, leads to the loss of kinetic energy due to larger cavity formed in the targets. The nose shape factor $N^{*}$ reflects the sharpness of nose shapes, and a smaller value of $N^{*}$ corresponds to a sharper nose. $N^{*}$ obviously affects the ballistic performance of long-rod penetration, however, depending on the initial impact velocity. In general, the influence of $N^{*}$ is relatively smaller than that of $η$ .
A unified model for dwell and penetration during long rod impact on thick ceramic targets
2019, International Journal of Impact Engineering
Citation Excerpt :
Consequently, the value of Rt obtained from these models cannot be considered to be a unique material constant. Improvements to the hydrodynamic models [41–43] were made by modifying the Bernoulli's equation approach from a momentum balance consideration along the centerline of penetration [27]. Using this approach, Walker and Anderson [44] proposed a time-dependent model for long-rod penetration into a semi-infinite target.
In this manuscript, we present an improved model for long rod penetration into thick ceramic targets by using a modified version of the Walker–Anderson model along with our dynamic expanding cavity model. This model is simpler to use and captures additional physics such as the dwell-penetration transition phenomena in ceramics as compared to the Walker–Anderson model. The target response is assumed to occur in a hemispherical region containing nested comminuted, cracked and elastic regions of deformation. A dynamic expanding cavity model, recently developed by the authors, along with an exponential pressure-shear response of brittle materials in the form of the extended Mohr-Coulomb model, is used to capture the stress fields in these regions. Incorporating this constitutive model reduces the requirement on (often expensive) experimental data as it uses a single set of parameters to predict the response of many brittle ceramics, which is especially beneficial in understanding and exploring the behavior of new materials. The predictions of the model are in good agreement with experimental results and is also used to quantify the effect of amorphization on boron carbide.
A three-stage model for the perforation of finite metallic plates by long rods at high velocities
2019, Defence Technology
Citation Excerpt :
Moreover, the A-T model with a constant Rt cannot describe well the long rod penetration over a wide range of impact velocities, especially for metallic target. Anderson and Walker [16] pointed out that there is pressure gradient in the region where the pressure is higher than the target strength. Later on, Anderson and Walker [17] investigated the target resistance under various impact velocities by curve fitting the A-T model predictions to the experimental data for tungsten alloy long rod penetration into semi-infinite armor steel targets.
A three-stage theoretical model is presented herein to predict the perforation of a thick metallic plate struck normally by a long rod at high velocities. The model is suggested on the basis of the assumption that the perforation of a thick metallic plate by a long rod can be divided into three stages: (1) initial penetration; (2) plug formation and (3) plug slipping and separation. Various analytical equations are derived which can be employed to predict the ballistic limit, residual velocity and residual length of the long rod. It is demonstrated that the present model predictions are in good agreement with available experimental results for the perforation of finite steel targets struck normally by steel as well as tungsten alloy long rods at high velocities. It is also demonstrated that the dynamic maximum shear stress of a plate material has strong effect on plug formation and plug thickness which, in turn, exerts considerable influence on the residual velocities and lengths of a long rod at impact velocities just above the ballistic limit.
An improved dynamic expanding cavity model for high-pressure and high-strain rate response of ceramics
2017, International Journal of Solids and Structures
Citation Excerpt :
Ceramic materials such as aluminum oxide (Al2O3), aluminum nitride (AlN), silicon carbide (SiC) and boron carbide (B4C) are promising candidates for structural and armor applications due to a combination of their properties, such as high elastic moduli, hardness, compressive strength, penetration resistance and low density. In order to understand their behavior under impact loading, numerous approaches such as hydrodynamic theory (Alekseevskii, 1966; Anderson and Walker, 1991; Birkhoff et al., 1948; Tate, 1986, 1969, 1967), micromechanical models (Curran et al., 1993), principles of momentum balance (Walker, 2003, 2002; Walker and Anderson, 1998a,b, 1995), dimensional analysis (Clayton, 2016, 2015) and spherical (Forrestal, 1986; Forrestal and Longcope, 1990; Forrestal and Tzou, 1997; Galanov et al., 2008; Gao et al., 2006; Hopkins, 1960; Kartuzov et al., 1999; Luk and Forrestal, 1987; Satapathy, 2001, 1997; Satapathy and Bless, 2000, 1996) and cylindrical (Guo et al., 2013; Kartuzov et al., 2002) cavity expansion analyses have been employed. The basic objective of all these approaches is to relate ballistic parameters such as depth of penetration, size of the cavity, deceleration and erosion of the projectile, etc. to material properties and geometrical factors.
An extended Mohr–Coulomb (M–C) model, capable of capturing the high pressure-dependent shear deformation of ceramics, is incorporated into a spherical dynamic expanding cavity model (d-ECM) to capture the dynamic response of semi-infinite brittle ceramics under steady-state penetration. It is shown that this improved d-ECM can better predict the target resistance of structural ceramics reported in penetration experiments. The brittle ceramic is assumed to undergo cracking when the hoop stress reaches its tensile strength and is subsequently comminuted when the radial stress reaches its compressive strength. The constitutive behavior of the comminuted ceramic is predicted using a modified form of the extended M–C model. The extended M–C model captures the strain rate-independent exponential pressure-shear response of intact ceramics in a normalized universal strength model. A single set of three parameters can be used to describe the response of all intact ceramics. This constitutive model has been suitably modified into a two-parameter exponential model, applicable to comminuted ceramics. These two parameters have been calibrated using experimental penetration data along with analytical estimations, and it has been shown that a single set of universal parameters can be used to describe the response of most comminuted ceramics. By incorporating these models, the improved d-ECM can be used to estimate the target resistance of a ceramic when relevant experimental data is not readily available. The results of the analysis further reveal that, in the case of steady-state penetration into semi-infinite monolithic targets, the properties of the comminuted material have a greater influence on the target resistance of ceramics than those of its intact state. The model proposed is applicable for thick monolithic ceramic targets under steady-state penetration conditions, and not for layered targets.
Analytical models for penetration mechanics: A Review
2017, International Journal of Impact Engineering
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.

View all citing articles on Scopus

View full text

An examination of long-rod penetration

Abstract

J. Mech. Phys. Solids

Int. J. Engng Sci.

Int. J. Solids Struct.

Int. J. Engng Sci.

Int. J. Solids Struct.

Int. J. Engng Sci.

Int. J. Impact Engng

J. Mech. Phys. Solids

Int J. Impact Engng

Int. J. Impact Engng

Int. J. Impact Engng

Comput. Mech.

Int. J. Impact Engng

A theory for the deceleration of long rods after impact

J. Mech. Phys. Solids

Penetration of a rod into a target at high velocity

Constraint to side flow in plates

J. Appl. Mech.