The fundamental assumptions of cavity expansion models for penetration - revisited

doi:10.1016/j.ijimpeng.2020.103723

International Journal of Impact Engineering

Volume 146, December 2020, 103723

https://doi.org/10.1016/j.ijimpeng.2020.103723 Get rights and content

Highlights

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The CCE and SCE models predict an incorrect dependence on the penetration velocity.
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Errors in the predictions of the CCE and SCE models are shown to be significant.
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The CCE and SCE models may not accurately predict general pene- tration processes.
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Penetration models should be validated relative to numerical and ex- perimental results.

Abstract

The fundamental assumptions motivating the use of cylindrical (CCE) and spherical (SCE) cavity expansion solutions for the prediction of penetration mechanics are revisited. Predictions of an ovoid of Rankine model (OR), which has been validated relative to axisymmetric numerical simulations, are used as a reference solution for normal penetration into a metallic target. In previous papers it has been observed that the average axial resistance stress is independent of the penetration velocity until a critical value which denotes the onset of separation of the target material from the projectile’s surface. The main conclusion of this paper is that the velocity fields in the CCE and SCE models do not accurately model a realistic flow field for penetration of a projectile. The results in this paper quantify the errors caused by low estimates of the static resistance stress and compensation due to incorrect dependence of the resistance stress on the penetration velocity predicted by the CCE and SCE models.

Introduction

Penetration mechanics has been of interest for many decades and a review up until 1978 can be found in [2]. Results from a large number of experiments can be found in [1] and a modern comprehensive presentation of ballistics is contained in [17].

Here, attention is limited to normal penetration of a long rigid projectile into a thick metal target. Fig. 1 shows a rigid axisymmetric projectile which has a nose shape characterized by $r = \hat{r} (ξ),$ where r is the radial coordinate, ξ is an axial coordinate relative to a point moving with the projectile and $\hat{r} (ξ)$ defines the shape of the projectile’s surface, which transitions from zero radius at its nose to the tail radius R.

Following Hill [10], who suggested that the target material melts at the projectile-target interface, the shear stress applied to the projectile is negligible. Consequently, the resistance to penetration can be expressed as an average (relative to the area of the projectile’s tail) axial resistance stress Σ given by integrating the contact pressure P (i.e., the negative of the normal component of the traction vector) over the projectile’s surface $Σ = \frac{2}{R^{2}} \int_{0}^{r_{s}} P r d r .$ In this equation, r_s is the point of separation of the target material from the projectile’s surface, which may occur near the projectile’s tail or nose. For quasi-steady-state penetration of a rigid projectile into a metal target, Hill [10] determined that the resistance stress Σ can be expressed in the form $Σ = Σ_{Y} Y + Σ_{V} ρ V^{2},$ where Σ_Y controls the static resistance stress, Y is the constant yield strength in uniaxial tension, ρ is the constant target density and Σ_V controls the dynamic resistance stress which depends on the current penetration velocity V in the negative e_z direction (see Fig. 1). For a rigid projectile, the penetration velocity V and penetration depth D are determined by integrating the equations and initial conditions $\dot{V} = - \frac{π R^{2} Σ}{M}, V (0) = V_{0}, \dot{D} = V, D (0) = 0,$ where M is the projectile’s mass, V₀ is the impact velocity and a dot $\dot{()}$ denotes differentiation with respect to time t.

The desire for physical understanding fuels the development of simplified engineering models which attempt to provide analytical equations that can be used to describe the penetration process. Static and dynamic solutions for expansion of cylindrical and spherical cavities of current radius a(t) into a target material have been developed to determine the pressure P applied to the cavity’s surface. Solutions for a number of different materials can be found in [e.g., [3], [4], [5], [7], [8], [11], [13], [20]].

For decades researchers have followed the work of Goodier [9] to determine Cylindrical Cavity Expansion (CCE) and Spherical Cavity Expansion (SCE) models for penetration mechanics. These CCE and SCE models are based on the following two fundamental assumptions:

(A1) The contact pressure P applied by the target on the projectile in (1) can be approximated by the pressure P required to expand the cavity in the CCE or SCE solutions.
(A2) The rate of expansion $\dot{a}$ can be determined by the shape of the projectile and its current axial penetration velocity V.

At present, the most accurate and general analysis of penetration problems is based on the use of computational methods to solve the nonlinear field equations in the target and projectile regions along with the boundary conditions at the target-projectile interface and the other surfaces. Analysis of numerical results focused on obtaining insight into the main physical mechanisms which are important in different penetration applications can be found in [e.g. [16], [17]].

The objective of this paper is to revisit the validity of the assumptions (A1) and (A2). Specifically, use is made of the analytical Ovoid of Rankine (OR) model developed in [15], [18], [21] which determines the motion of a rigid projectile with the shape of an ovoid of Rankine penetrating an incompressible elastic-perfectly plastic target. It was shown in [19] that predictions of axisymmetric numerical simulations of an ovoid of Rankine projectile penetrating a compressible aluminum target with the yield strength specified by the Johnson-Cook model [12] can be accurately modeled using the OR model when the yield strength is adjusted to a constant effective value Y_OR.

It is not suggested that the OR model replace other models for penetration, especially because the OR model is limited to an incompressible elastic-perfectly plastic material and a ovoid of Rankine nose shape. However, the CCE and SCE formulations for an ovoid of Rankine nose shape can be easily developed. Specifically, the detailed predictions of the CCE, SCE and OR models presented here are used to quantify errors caused by the unrealistic flow fields in the CCE and SCE models for penetration of a projectile.

Section snippets

Basic equations for the OR model

The analytical approach developed in [15], [18], [21] considers a rigid ovoid of Rankine shaped projectile penetrating an incompressible elastic-perfectly-plastic target (see Fig. 1). Specifically, this approach uses an assumed realistic flow field (see Fig. 2) to obtain an expression for Σ in (2). This section summarizes some basic equations of this solution.

Using cylindrical polar base vectors e_r, e_θ and e_z, a moving ovoid of Rankine shaped projectile with tail radius R is given by $\begin{matrix} ξ = z - x (t) - \frac{R}{2}, \end{matrix}$

Cavity expansion models

Neglecting acceleration terms, the pressure P required for quasi-steady state expansion of cavities in the CCE and SCE models is a function of time only, which can be expressed in the form [e.g., [20]] $P = P_{Y} Y + P_{V} ρ {\dot{a}}^{2} .$ Using the assumption (A1), (13) is substituted into (1) and (2), the target is assumed to remain in contact with the projectile’s surface $(r_{s} = R),$ and the static solutions in [3] are used to deduce that $Σ_{Y} = P_{Y} = \frac{1}{\sqrt{3}} [1 + \ln (\frac{\sqrt{3} G}{Y})],$ for the CCE model and $Σ_{Y} = P_{Y} = \frac{2}{3} [1 + \ln (\frac{2 G}{Y})],$ for the SCE model.

Comparison of the predictions of the models

To validate the OR model, the axisymmetric numerical simulations in [19] used a projectile modeled as a truncated ovoid of Rankine with the geometry and mass specified by $R = 3.55 [mm], L = 74.7 [mm], M = 23.3 [gm] .$ In these simulations, the projectile was forced to move at constant velocity V and the value of the average resistance stress Σ was determined after the target material attained a steady-state velocity field. The range of applied velocities was $V = 0.5 - 3.0$ [km/s]. These simulations used a

Conclusions

The fundamental assumptions (A1) and (A2) of the use of cavity expansion solutions for the CCE and SCE penetration models have been revisited in this paper. The problem of penetration of a rigid projectile with the shape of an ovoid of Rankine into a compressible aluminum target has been used as a reference solution. Although the ovoid of Rankine model was developed for an incompressible, rigid-plastic target material, it was shown in [19] that the effects of compressibility, strain hardening,

Author Statement

I am the sole author of this paper and therefore am responsible for all of the content.

Declaration of Competing Interest

The author declares no conflict of interest.

Acknowledgments

The author would like to acknowledge helpful discussions with Dr. Z. Rosenberg.

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The fundamental assumptions of cavity expansion models for penetration - revisited

Highlights

Abstract

Introduction

Section snippets

Basic equations for the OR model

Cavity expansion models

Comparison of the predictions of the models

Conclusions

Author Statement

Declaration of Competing Interest

Acknowledgments

Int. J. Eng. Sci.

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