Cavitation phenomenon in penetration of rigid projectiles into elastic-plastic targets

Highlights

• Cavitation during penetration of rigid projectiles into elastic-plastic targets is studied.

• An analytical criterion for projectile-target separation is developed.

• Formulae for the cavitation critical velocity V_CR and contact zone size x* are obtained.

• Penetration of ogive and Rankine ovoid nose projectiles into aluminum targets are investigated.

• The effect of projectile and medium properties on V_CR and on x* are studied.

Abstract

This paper presents an analytical study on the cavitation phenomenon during penetration of a rigid projectile into an elastic-plastic target. The developed analytical approach is based on the general solution of the quasi-static and non-stationary dynamic cylindrical/spherical cavity expansion problem with assigned kinematic boundary condition at the cavity boundary (variable, time dependent radial velocity). An analytical projectile-target separation criterion is developed for a projectile having an arbitrary shape and for a target described by a simplified material model with a locked equation of state and a linear shear failure relationship. This relatively simple model may represent the behavior of different materials reasonably well yet allow an analytical solution of the problem. The paper derives the analytical expression of the contact zone size using both the cylindrical and spherical cavity expansion approaches and presents the related normal contact stresses acting on the projectile nose surface that maintains contact with the target. Examples of the developed formulation is presented for ogive nose and Rankine ovoid projectiles hitting aluminum targets, with a constant shear failure envelope, at different velocities and comparisons with known analytical and numerical solutions are shown. The effect of projectile and medium properties on the size of the contact zone are studied.

Introduction

The term "cavitation" may have several meanings in the scientific literature:

• In fluid mechanics, cavitation describes a phenomenon where rapid pressure changes in a liquid lead to formation of small vapor-filled cavities in places where the pressure is relatively low [1].

• In studies on the dynamic or quasi-static expansion of a spherical and cylindrical cavity in an elastic-plastic media, the term "cavitation" is frequently used in relation to the pressure acting at the cavity wall denoted as the “cavitation stress/pressure” [2], [3], [4], the “cavitation fields” [5], [6], [7], [8], [9] that are the stress/pressure/velocity distributions along the radial coordinate and “cavitation instability” that is defined as a critical state in the material response at which constant applied load induces spontaneous expansion of a cavity [10].

• When an underwater vehicle is moving at a sufficiently high speed, its surface pressure is lower than the saturated vapor pressure of local water, thus forming a cavity that can surround a major part of its body. This phenomenon is called “natural super cavitation” [11]. When super cavitation occurs, only a small part of the body front is in contact with the water, while the rest of the body is in contact with the vapor in the formed cavity.

Hill [12] borrowed the term “cavitation” from hydrodynamics and applied it to problems of penetration into solid media to describe the case of partial contact between the projectile and the target. Hill's analysis for the penetration of ogive-nosed projectiles resulted in determination of the threshold velocity (V_CR) beyond which cavitation occurs, and some energy is invested in expanding the projectile channel's diameter. The present analysis follows the definition given by Hill and aims at contributing to enhance our understanding on this important, complex and not fully understood topic.

Since Hill's pioneering study, a very small number of articles have been published on the cavitation phenomenon in problems of projectile penetration into elastic-plastic targets [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27].

Evaluation of the rather small number of papers on this topic shows that the topic deserves more elaboration to clarify several fundamental aspects and to compare the different approaches. Unfortunately, there is hardly any supporting experimental data on cavitation. This is apparently due the difficulties involved, such as the relatively high velocity at which cavitation occurs and the corresponding high stresses in the projectile body, that may exceed the projectile elastic stress level cause deformation and even failure, thus affecting the cavitation development.

Other difficulties arise from the very short duration of cavitation occurrence, the fact that cavitation cannot be inspected directly and the high resolution required to determine the cavitation region. Therefore, cavitation is mostly studied theoretically. Such studies indicate that the importance of cavitation for common engineering practice is rather limited. Nevertheless, this is one of the more complex issues in penetration mechanics, that may become more important and influential when extremely high strength projectiles are considered or when applications with other specific target materials are implemented. There is no doubt that regardless of its practical importance, its academic importance is high and deserves attention. Such analysis requires an advanced and delicate formulation of the dynamic projectile-target interaction problem.

When penetration into a thick target is discussed, the mechanical interaction between the projectile and the target should be carefully simulated to allow calculation of the contact stresses (pressures). This analysis capability is especially important for cavitation analysis. Cavitation occurs at a certain distance along the projectile nose when the conditions are such that the "projectile - target" contact pressure drops to zero and separation occurs between the target and the projectile nose surface. It may be noted that in fact, on the transition of numerical calculations from a certain time step to the following, different algorithms for contact stress calculation may calculate a negative contact pressure (i.e. tensile stresses). This is of course non-physical and the contact pressure must be set then to zero. In the spherical/cylindrical cavity expansion models analyses, this instant of zero stress activates a new boundary condition at the cavity boundary replacing the kinematic boundary condition (when the contact is always maintained, and the contact stresses are always positive). Beyond the point of separation there is a part of the projectile nose surface which is disconnected from the target. Obviously, the total resistance force on the projectile nose is only due to the contact stresses and the cavitation zone is not affecting the overall resistance. Therefore, contact stresses should always be checked, to control the zero contact stresses in the cavitation zone. The above contact stresses check is performed in [[13], [14], [15], [21], [22], [23], [24], [25], 28, 29].

It should be noted that there are many penetration models which cannot consider the cavitation and projectile-target separation phenomenon. For example, in the simplest versions of the different spherical and cylindrical cavity expansion models (denoted as SCE and CCE correspondingly) full contact along the nose is pre-assumed and the contact stresses are always positive (i.e. compression). These contact stresses are expressed, for example, in [30, 31] as:

$${\sigma }_n={\sigma }_{ST}+{\rho }_0{V}_n^2$$

where ${\sigma }_{ST}$is the radial quasi-static constant stress at the cavity boundary and the normal projectile velocity $V_n$ is the radial velocity at the cavity boundary.

The vast majority of publications on the cavitation phenomenon deals with the penetration of ogive and spherical-nosed projectiles [19, 20] as well as ovoid shaped projectiles [21], [22], [23], [24], [25] and concentrate on their penetration into metallic targets. Rubin's publications [21], [22], [23], [24], [25] conclude that for a steady-state projectile motion at a constant penetration velocity, the drag force acting on the projectile in the tunneling penetration stage in a metallic elastic-perfectly-plastic target is related to the velocity as follows: when the velocity is below a certain critical velocity, the projectile axial resistant force is constant, whereas when the velocity is higher, the projectile axial resistant force is not constant and it increases at increasing velocity.

Except for single studies on cavitation in penetration into soils [14] and concrete [27], cavitation has not been studied with regard to penetration into geological materials or concrete despite their importance as common barriers against projectiles penetration.

This paper presents an analytical study on the cavitation phenomenon using a general solution of the problem of the quasi-static [32] and non-stationary dynamic [33] cylindrical/spherical cavity expansion with assigned kinematic boundary condition at the boundary (variable, time dependent radial velocity). An analytical projectile-target separation criterion is developed for a projectile having an arbitrary shape and for a target described by a simplified material model with a locked equation of state and a linear shear failure relationship. This material model of the target material may reasonably well represent the behavior of different materials yet allow an analytical solution of the problem.

This paper derives the analytical expression of the contact zone size between the projectile nose and the target, using both the cylindrical and the spherical cavity expansion approaches and presents the related normal contact stresses acting on the projectile nose surface.

The developed solution may be applied to different materials such as metal, concrete, soil and rock as well as to materials with simpler behavior (e.g. constant shear failure envelope) like aluminum.

As examples of the application of the developed formulae, penetration of ogive and Rankine ovoid nose projectiles with different velocities into aluminum targets described by a constant shear failure envelope is investigated.

The obtained solution is based on the CCE and SCE approaches and is compared with the hybrid DISC method [28], with numerical 2D analysis in metallic targets that were reported in [19] and with other analytical and numerical solutions [23, 24].

The effect of projectile and medium properties on the size of the contact zone are studied.