Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Analytical and Energy-Based Methods for Penetration Mechanics

  • F. Teixeira-DiasEmail author
  • N. Smith
  • E. Galiounas
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_207-1

Synonyms

Definitions

The terms perforation and penetration are used frequently throughout this chapter. It should however be noted that there is a significant difference between the two terms. Penetration refers to the ballistic impact of a target which results in the projectile becoming embedded or ricocheting off the target. Perforation describes the projectile impacting the target and passing completely through it. Plugging refers to the mechanical process of the formation of a plug of target material once it has been impacted by a projectile.

Introduction

A comprehensive understanding of ballistic impact has been essential ever since the invention of rudimentary weapons such as the bow and arrow. There has been since then a need to develop effective means of protection such as the shield. The rapid advancement of weapon technology has led to the progress of protective structure technology in equal measure (Krauthammer, 2008). Scientists and engineers have for many years striven to design protective structures that provide an ideal balance of resistance to ballistic impact and financial and time constraints, which accompany all research in modern society.

Although experimental validation of theories, new solutions, and developments is important to the advancement of the field, it is not always time and cost-effective. The same can be said, albeit in a different scale, for numerical modeling approaches using the finite element (FE) method or equivalent tools. These have gained notoriety for the accuracy of its results, without the need to address the health and safety and procurement logistics that experimental approaches require. However, the time input into the FE modeling is large, as is the computational expense required to give high accuracy results.

The above suggests that there is a need in the research of ballistic impact and protective structure technology to obtain sufficiently accurate results, quickly and cheaply, prior to progressing to higher-order testing and analysis. This can be achieved most effectively using analytical methods. There are a number of currently established analytical models, each adopting different approaches, and some of these will be discussed in the following sections, focusing on the use of an energy-based principles to derive analytical models.

Ballistic impact dynamics can be broken up into three distinct areas of study, namely, (i) interior ballistics, (ii) exterior ballistics, and (iii) terminal ballistics. Interior ballistics is the examination of the motion and forces acting on a projectile, while it is in the launching device (e.g., the gun barrel). Exterior ballistics concerns the motion and forces of the projectile during free flight. Finally, terminal ballistics describes the interaction between the projectile and the target during impact (Borvik, 2000), which is the main focus of this chapter.

Impact can be defined in a simple way as two objects striking each other, and it is governed by their motion, reaction forces, energy, and material properties. In the scope of this chapter, impact is thought of as a mass Mp – termed the projectile – traveling with velocity v and striking a stationary second mass Mt, termed the target.

The impact of a projectile on a target can result in penetration or perforation. The former is defined as a projectile’s entrance into a target without fully completing its passage through the body (Mackman and Goldsmith, 1978). In practice, this can mean that either the striker is embedded into the target material or it rebounds. The latter describes a ballistic impact which completely pierces the target (Zukas, 1980).

A major characteristic of protective structures and their design is the ballistic limit velocity (BLV) (Borvik, 2000), which is often referred to as the minimum perforation velocity – i.e., the minimum projectile velocity that ensures perforation (Zhang and Stronge, 1996). This velocity is governed by a number of parameters, such as the projectile and target material properties and projectile mass and target configuration (thickness, orientation, etc.). The residual velocity is the projectile velocity after it has perforated the target. The definition of the BLV implies that if v0 = v, then vr = 0, where v0 is the projectile velocity just before impact, v is the BLV of the target, and vr is the residual velocity of the projectile. The residual velocity is zero if the target is struck by a projectile at its BLV (Velmurugan et al., 2014).

The scope of this chapter is limited to the analysis of impacts on ductile targets, leaving the analysis of impact on ceramics and composite materials to other chapters. For the purpose of the analyses in the following sections, targets can be classified as thin, intermediate, thick, or semi-infinite, as described in Table 1 (Borvik, 2000; Mackman and Goldsmith, 1978; Greszczuk et al., 1982). Additionally, different target configurations can be considered, namely, (i) monolithic, (ii) in-contact multilayered, and (iii) spaced multilayered. These are schematically shown in Fig. 1.
Fig. 1

Schematic representation of different target configurations: (a) monolithic, (b) in-contact multilayered, and (c) spaced multilayered

Table 1

Classification of targets in the scope of terminal ballistics

Classification

Description

Thin

Negligible stress and strain gradients throughout the thickness of the target

Intermediate

Considerable influence of rear surface (distal boundary) on the deformation process during penetration

Thick

Influence of the distal boundary only felt after substantial travel of the projectile in the target

Semi-infinite

Negligible influence of the distal boundary on the penetration process

A significant amount of research has been done with the aim of understanding the phenomena occurring in targets with different configurations when impacted by projectiles and understanding the effect of the air gaps within spaced multilayered shields. It is observed that the BLV decreased with increasing number of layers and that bigger air gaps provided increased target effectiveness (Zukas, 1980; Cao et al., 2012).

Ballistic Impact and Perforation Mechanics

In the scope of ballistic impact, it is widely accepted that there are six main distinct perforation mechanisms (Jia et al., 2014). Of these, the most common in ductile (e.g., metallic) targets are (i) ductile hole growth, (ii) plugging, and (iii) petalling, shown schematically in Fig. 2. Ductile hole growth (Fig. 2a) usually occurs due to penetration by a conical projectile and is the consequence of the projectile pushing the target material radially outward with respect to the forward motion of the projectile. This can occur in ductile metals, polymers, and some ductile composite materials.
Fig. 2

Schematic representation of perforation mechanisms occurring in ductile targets: (a) ductile hole growth, (b) plugging, and (c) petalling

Plugging (see Fig. 2b) is highly sensitive to the impact angle and the shape of the nose of the projectile. It occurs most frequently in finite thickness targets impacted normally by blunt or hemispherical shaped projectiles traveling close to the target’s BLV. One consequence of this mechanism is the formation of a plug of material, dislodged from the target, of approximately the same diameter as the projectile. The plug is formed by adiabatic shear as it is generally accepted that the initiating mechanism leading to plugging is the occurrence of plastic shear instability at a location of high-stress concentration, in an otherwise uniformly straining solid. Plugging failure is thus initiated by plastic (permanent) deformation due to high stress in a small area, the primary zone of the target, while the rest of the target remains unaffected – the secondary zone. In plugging, plastic shear deformation energy is converted almost entirely into heat. This increases the temperature in the primary zone. The temperature rise occurs due to high strain rates stopping heat from propagating through the target material into the secondary zone. This increase in temperature results in additional local plastic flow, further localizing plastic strain. This process continues until either the material fractures or the plug exits the target (Krauthammer, 2008; Borvik et al., 2001).

Petalling (see Fig. 2c) is produced by high radial and circumferential tensile stresses originating from the passage of the initial stress wave. This phenomenon is observed most frequently on ductile thin plates. Ogival or conical nose projectiles traveling at low velocities induce high-stress fields at their tip which, due to their forward motion, enforce significant bending on the target. This creates a star-shaped failure pattern on the target. Eventually, the tensile strength of the material is reached, the star-shaped deformation increases, initiating the formation of radial cracks, and petals appear. Petalling is accompanied by large plastic deformations and flow and permanent flexure (bending) (Horne, 1979).

There are numerous different approaches to analytically describe these perforation mechanisms, and this chapter will focus solely on analytical energy-based methods for the analysis of plugging, with a brief introduction to analytical methods for petalling at the end.

Perforation by Plugging

Numerous analytical models and approaches have been proposed to model ballistic perforations with plugging failure. A brief description of the advantages and disadvantages of three of the highest-profile models is presented in the following paragraphs.

Structural Models

The structural approach considers the stresses and strains that lead to the formation of the plug, in addition to the resistance of the structure as a whole. This method is based on the principles of the conservation of linear and angular momentum, over two stages of penetration. The first stage begins with the condition that the projectile and plug move with a common velocity. As the plug moves through the target, the contact length with the target decreases, causing shearing. This shear force induces bending of the surrounding target material, resulting in an increase in hole radius. The second stage is dominated by the enlargement of the secondary target zone. During this second phase, perforation occurs when the strain experienced by the target reaches the material’s limit due to stretching and bending moments (Woodward and de Morton, 1987; Jaramaz et al., 2005).

The three-stage penetration model, built upon the above two-stage model, is based on the forward motion of a variable mass body, expressed by considering the forces involved (Mackman and Goldsmith, 1978; Awerbuch and Bodner, 1973) as in
$$\displaystyle \begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}(mv) = -\left( F_{\mathrm{i}} + F_{\mathrm{c}} + F_{\mathrm{s}} \right) {} \end{aligned} $$
(1)
where Fi is the inertial force, Fc is the compressive force, and Fs is the force due to shear. This simple theoretical equation is applied to each stage of the process, deriving expressions for effective mass velocity using experimental approximations of average hole diameter and plug length. Below is a description of each stage of the process, with reference to Fig. 3, which shows an illustration of each process:
Stage I
Shearing forces are not yet relevant, but the projectile is experiencing inertial and compressive loading.
Fig. 3

The three-stage model showing the formation of the plug and shear stresses: (a) Stage I, (b) Stage II, (c) end of Stage II, and (d) Stage III. Due to compression d1 < d2 < d3 < d4

Stage II

Shear force begins to act on the projectile due to its motion relative to the target plate. Similar to Stage I, the effective mass of the projectile is changing with the formation of the plug. This stage ends when the projectile and plug are completely joined, traveling as one body, and x = h is satisfied.

Stage III

The third and final stage starts with the projectile and plug moving as a rigid body. In this stage, the effective mass does not vary any more, but its motion is resisted by the shearing force acting along the circumference of the plug. Ejection occurs when the shear strain reaches its maximum value, with respect to the target material. As in the structural model, there is the possibility that the projectile may become embedded within the target.

It should be noted that a few assumptions have to be made for this model, namely, that it relies on empirical geometrical parameters and the particle velocity, due to elastic stress wave propagation in the target, is neglected. This is relevant because a change in the material may alter the elastic stress wave propagation speed. It is also assumed that plastic stress waves are negligible within the process, assuming that their velocity is significantly smaller than that of the projectile. This may in some cases be an unrealistic assumption because the model is derived for high impact velocities, excluding the possibility of the projectile embedding within the target. This model is however useful as it can provide a means of quickly determining perforation velocities, contact times, and force-time histories (Mackman and Goldsmith, 1978).

Energy-Based Models

This section provides a brief description of the principles and derivations involved in analyzing ballistic perforations using energy-based approaches. The basic principle behind all derivations is the conservation of energy, which helps predict the ballistic performance of both monolithic and multilayered targets. This is, in general terms,
$$\displaystyle \begin{aligned} \sum E_{\mathrm{in}} = \sum E_{\mathrm{out}} {} \end{aligned} $$
(2)
where Ein and Eout are the input (initial) and output (final) energies of the system. This states that energy cannot be destroyed or created but only transformed/transferred from one form to another. In the scope of ballistic impact, this means that the total energy of the system before and after a projectile impacts a target is unchanged. This basic principle is used to determine velocities.

Similar to the structural models, energy-based models also rely on a number of assumptions, which will be made clear from the outset to avoid confusion. It is assumed that only the contact area of the projectile is relevant, thermal effects are neglected, target surfaces are planar, both projectile and target are initially stress free, and the friction between the frontal face of the projectile and the plug is not relevant.

To derive a generalized model, two basic principles must be introduced: (i) the relationship between elastic strength and impact velocity and (ii) the conservation of momentum. The elastic-wave velocities of the projectile and target, cp and ct, respectively, are calculated as a function of the elastic strength and impact velocity. The elastic-wave speed in the projectile is
$$\displaystyle \begin{aligned} c_{\mathrm{p}} = \sqrt{\frac{E_{\mathrm{p}}}{\rho_{\mathrm{p}}}} {} \end{aligned} $$
(3)
where Ep and ρp are the projectile’s Young’s modulus and density, respectively. The elastic-wave velocity of the target can be expressed similarly as
$$\displaystyle \begin{aligned} c_{\mathrm{t}} = \sqrt{\frac{\lambda_{\mathrm{t}}+2G_{\mathrm{t}}}{\rho_{\mathrm{t}}}} {} \end{aligned} $$
(4)
where λt and Gt are Lame’s constant and shear strength for the material of the target, defined as
$$\displaystyle \begin{aligned} \lambda_{\mathrm{t}} = \frac{v_{\mathrm{t}} E_{\mathrm{t}}}{(1+v_{\mathrm{t}})(1-2 v_{\mathrm{t}})} \quad \mathrm{and}\quad G_{\mathrm{t}} = \frac{E_{\mathrm{t}}}{2(1+v_{\mathrm{t}})} {} \end{aligned} $$
(5)
Here, vt is the target’s Poisson ratio. With the above equations, it is possible to define an expression for the contact compressive stress σc, by relating the relative velocity V and material properties of the projectile and target (Velmurugan et al., 2014):
$$\displaystyle \begin{aligned} \sigma_{\mathrm{c}} = \varphi V = \left(\frac{\rho_{\mathrm{t}} c_{\mathrm{t}}\rho_{\mathrm{p}} c_{\mathrm{p}}}{\rho_{\mathrm{t}} c_{\mathrm{t}}+\rho_{\mathrm{p}} c_{\mathrm{p}}}\right) V {} \end{aligned} $$
(6)
Additionally, the conservation of momentum applied to the whole model yields
$$\displaystyle \begin{aligned} v_{\mathrm{f}} = \left(\frac{M_{\mathrm{p}}}{M_{\mathrm{p}}+M_{\mathrm{g}}}\right)v_0 {} \end{aligned} $$
(7)
where vf is the free impact final velocity, Mp is the mass of the projectile, Mg is the mass of the plug (the material from the target), and v0 is the initial projectile velocity. A generalized model can now be discussed with the understanding of these concepts.

Monolithic Targets

The energy balance equation that describes the energy transfer of a projectile impacting a target can be defined as (Recht and Ipson, 1963)
$$\displaystyle \begin{aligned} E_{\mathrm{k} 0} = E_{{\mathrm{f}}\mathrm{n}} + W_{\mathrm{n}} + E_{\mathrm{k}\mathrm{p}} + E_{\mathrm{k}\mathrm{g}} {} \end{aligned} $$
(8)
where Ek0 is the total kinetic energy of the projectile and target of the total energy of the system – before impact, Efn is the energy lost to deformation and heat during free impact, Wn is the additional work lost due to the peripheral shear area, Ekp is the kinetic energy of the projectile after impact, and Ekg is the kinetic energy of the plug after impact. Expanding this expression by substituting in the general expression of kinetic energy gives
$$\displaystyle \begin{aligned} \frac{1}{2}M_{\mathrm{p}} v_0^2 = E_{{\mathrm{f}}\mathrm{n}} + W_{\mathrm{n}} + \frac{1}{2}M_{\mathrm{p}} v_{\mathrm{r}}^2 + \frac{1}{2}M_{\mathrm{g}}v_{\mathrm{r}}^2 {} \end{aligned} $$
(9)
where vr is the residual velocity of the projectile, assumed to be the same for the plug. During perforation, all kinetic energy is lost due to the presence of the peripheral shear layer at the plug boundary. Considering a purely inelastic collision – sometimes referred to as free impact – it first appears that only kinetic energy is lost. There is a pressure increase, however, due to the shear resistance at the collision interface (i.e., between the projectile and the target). This infers that the strain energy will change during the process. Hence, the collision cannot be considered to only yield kinetic energy losses.
Understanding this, it is possible to derive an expression for the energy lost to deformation and heat during free impact, Efn. By energy conservation, this must equal the difference between initial and final kinetic energies, expressed as
$$\displaystyle \begin{aligned} E_{{\mathrm{f}}\mathrm{n}} = \frac{1}{2}\left( \frac{M_{\mathrm{g}}}{M_{\mathrm{p}}+M_{\mathrm{g}}} \right)M_{\mathrm{p}} v_0^2 {} \end{aligned} $$
(10)
It is also possible to define the additional energy lost due to the peripheral shear area Wn. For simplicity, the expression of Wn is derived for an initial velocity equal to the ballistic limit velocity (BLV) of the target, that is, v0 = v. As stated previously, approaching the derivation in this manner allows the residual velocity to be ignored as it is zero. The derivation yields
$$\displaystyle \begin{aligned} W_{\mathrm{n}}^\star = \frac{1}{2}\left( \frac{M_{\mathrm{p}}}{M_{\mathrm{p}}+M_{\mathrm{g}}} \right)M_{\mathrm{p}} \left(v^\star\right)^2 {} \end{aligned} $$
(11)
In fact, the BLV is used here merely to give the reader a physical, easy to understand property. It is widely accepted that Wn is insensitive to change in velocity, therefore remaining constant (Velmurugan et al., 2014). It is, however, important to note that this only holds true as long as the dynamic shear stress of the target material remains constant and for speeds up to or above adiabatic slip.

In-Contact Multilayered Targets

In the case of an in-contact multilayered target such as the one shown in Fig. 4, the BLV is expressed by reconsidering Eq. (10) for a residual velocity equal to zero (vr = 0). This is to say, the projectile has just enough energy to complete full perforation. To achieve this, Efn and Wn must be expressed in a general form for each target layer, i.e., for \(E_{{\mathrm {f}}\mathrm {n}}^i\) and \(W_{\mathrm {n}}^i\). These energy terms will be derived in the following paragraphs. It should be noted that in what follows, i denotes the current target layer.
Fig. 4

Plugging in an in-contact multilayered target: (a) initial penetration stage, (b) formation of the first plug, and (c) formation of the second plug

As the projectile travels through the target, it is subjected to compression contact stresses that increase incrementally, induced by the addition of multiple plugs from the multilayered target. The compressive stress has been defined previously in Eq. 6 and can be used to define the relative velocity of the projectile. With both of these considerations, \(E_{{\mathrm {f}}\mathrm {n}}^i\) can be expressed as
$$\displaystyle \begin{aligned} E_{{\mathrm{f}}\mathrm{n}}^i = \frac{1}{2}\left(\frac{M_{\mathrm{g}}^i\varOmega^j}{M_{\mathrm{g}}^i+\varOmega^j}\right)\left(\frac{\sigma_{{\mathrm{c}}}^i}{\varphi^i}\right)^2 {} \end{aligned} $$
(12)
with
$$\displaystyle \begin{aligned} \varOmega^j = {M_{\mathrm{p}}^{i-1}+\sum_{j=1}^{i-1}M_{\mathrm{g}}^j} {} \end{aligned} $$
(13)
This, however, does not include the aforementioned incremental stress
$$\displaystyle \begin{aligned} \sigma_\tau=\frac{4h^i\tau^i}{d} \end{aligned} $$
(14)
where τi and hi are the dynamic shear strength and the thickness of the i-th target layer. The expression of the incremental contact stress is determined by considering the mechanical impedance resistance caused by the peripheral shear area, which is assumed to be the area of the frontal face of the projectile. This can be added to Eq. 12 to give
$$\displaystyle \begin{aligned} E_{{\mathrm{f}}\mathrm{n}}^i = \frac{1}{2}\left(\frac{M_{\mathrm{g}}^i\varOmega^j}{M_{\mathrm{g}}^i+\varOmega^j}\right) \frac{\left(\sigma_{{\mathrm{c}}}^i\right)^2+\sigma_\tau^2}{\left(\varphi^i\right)^2} {} \end{aligned} $$
(15)
The additional energy dissipated into the peripheral shear area \(W_{\mathrm {n}}^i\) can be defined by considering the average work done by the projectile in order to displace the plug of the i-th layer
$$\displaystyle \begin{aligned} W_{\mathrm{n}}^i = \frac{1}{2}\pi d\left(h^i\right)^2\tau^i {} \end{aligned} $$
(16)
Recalling the assumption that the residual velocity is zero (vr = 0), substituting Eqs. (15) and (16) into Eq. (9), rearranging for the ballistic limit velocity v, and simplifying yields
$$\displaystyle \begin{aligned} v^{\star i} = \frac{4h^i\tau^i\varphi^i M_{\mathrm{g}}^i}{d\varOmega^j}\left[ 1+\sqrt{\frac{\varOmega^j}{M_{\mathrm{g}}^i}\left( 1+\frac{\pi d^3}{16\tau^i\left(\varphi^i\right)^2 M_{\mathrm{g}}^i} \right)} \right] {} \end{aligned} $$
(17)
A generalized expression for the residual velocity can now be derived by rewriting Eq. (9) for multilayered targets as
$$\displaystyle \begin{aligned} \frac{1}{2}M_{\mathrm{p}}^{i-1}\left(v_{\mathrm{r}}^{i-1}\right)^2 &= E_{{\mathrm{f}}\mathrm{n}}^i + W_{\mathrm{n}}^i + \frac{1}{2}M_{\mathrm{p}}^{i-1}\left(v_{\mathrm{r}}^i\right)^2\\ &\quad + \frac{1}{2}M_{\mathrm{g}}^i\left(v_{\mathrm{r}}^i\right)^2 {} \end{aligned} $$
(18)
where
$$\displaystyle \begin{aligned} M_{\mathrm{p}}^{i-1} = M_{\mathrm{p}} + \sum_{j=1}^{i-1}M_{\mathrm{g}}^j {} \end{aligned} $$
(19)
$$\displaystyle \begin{aligned} E_{{\mathrm{f}}\mathrm{n}}^i = \frac{1}{2}\left( \frac{M_{\mathrm{g}}^i}{M_{\mathrm{p}}^{i-1} + M_{\mathrm{g}}^i} \right)M_{\mathrm{p}}^{i-1}\left(v_{\mathrm{r}}^{i-1}\right)^2 {} \end{aligned} $$
(20)
$$\displaystyle \begin{aligned} W_{\mathrm{n}}^i = \frac{1}{2}\left( \frac{M_{\mathrm{p}}^{i-1}}{M_{\mathrm{p}}^{i-1} + M_{\mathrm{g}}^i} \right) M_{\mathrm{p}}^{i-1}\left(v^{\star i}\right)^2 {} \end{aligned} $$
(21)
Substituting these expressions into Eq. (18) and rearranging for the residual velocity of the i-th target layer \(v_{\mathrm {r}}^i\) give
$$\displaystyle \begin{aligned} v_{\mathrm{r}}^i = \frac{M_{\mathrm{p}}^{i-1}}{M_{\mathrm{p}}^{i-1} + M_{\mathrm{g}}^i}\sqrt{\left(v_{\mathrm{r}}^{i-1}\right)^2 - \left(v^{\star i}\right)^2} {} \end{aligned} $$
(22)

Case Study: An In-Contact Metallic Three-Layered Target

The equations of the ballistic limit velocity and residual velocity for a three-layered target configuration hit by a cylindrical projectile (see Fig. 5) are presented here. Although this case study looks at an in-contact layout, the model described in the previous section can be applied to both spaced and in-contact configurations, simply by considering the gaps as additional layers and using the corresponding material properties (e.g., for air). Expressions for the BLV of the first layer and residual velocity are
$$\displaystyle \begin{aligned}v^{\star 1} = \left(\frac{{4h^1\tau^1\varphi^1}M_{\mathrm{g}}^1}{dM_{\mathrm{p}} }\right) \left[ 1+\sqrt{\frac{M_{\mathrm{p}}+M_{\mathrm{g}}^1}{M_{\mathrm{g}}^1}\left( 1+\frac{\pi d^3}{16\tau^1\left(\varphi^1\right)^2 M_{\mathrm{g}}^1} \right)} \right] {} \end{aligned} $$
(23)
$$\displaystyle \begin{aligned} v_{\mathrm{r}}^1 = \frac{M_{\mathrm{p}}}{M_{\mathrm{p}}+M_{\mathrm{g}}^1}\sqrt{v_0^2 - \left(v^{\star 1}\right)^2} {} \end{aligned} $$
(24)
Fig. 5

Plugging in a generic in-contact three-layered target hit by a cylindrical projectile: (a) target configuration and (b) plug formation

Identically, the BLV and residual velocity for the second layer of the target can be expressed as
$$\displaystyle \begin{aligned} v^{\star 2} = \frac{{4h^2\tau^2\varphi^2}M_{\mathrm{g}}^2}{d\left(M_{\mathrm{p}} + M_{\mathrm{g}}^1\right)} \left[ 1+\sqrt{\frac{M_{\mathrm{p}}+M_{\mathrm{g}}^1+M_{\mathrm{g}}^2}{M_{\mathrm{g}}^2}\left( 1+\frac{\pi d^3}{16\tau^2\left(\varphi^2\right)^2 M_{\mathrm{g}}^2} \right)} \right] {} \end{aligned} $$
(25)
$$\displaystyle \begin{aligned} v_{\mathrm{r}}^2 = \frac{M_{\mathrm{p}}+M_{\mathrm{g}}^1}{M_{\mathrm{p}}+M_{\mathrm{g}}^1 + M_{\mathrm{g}}^2}\sqrt{\left(v_{\mathrm{r}}^1\right)^2 - \left(v^{\star 2}\right)^2} {} \end{aligned} $$
(26)
The most notable change in the above equations is the summation of plug masses. The mass ratio in Eq. 25 now includes the mass of the projectile, the mass of the plug from the first layer, and the mass of the plug from the second layer. It should also be noted that the velocity term has also changed. What was the initial projectile velocity in Eq. 23 has become the velocity at the exit of the first layer or, in other words, the residual velocity of the first layer. Similarly, the BLV of the second layer has been used instead of the BLV from the first layer. Finally, considering the third layer of the target, the BLV and residual velocity become
$$\displaystyle \begin{aligned} v^{\star 3} = \frac{{4h^3\tau^3\varphi^3}M_{\mathrm{g}}^3}{d\left(M_{\mathrm{p}} + \sum_{j=1}^2 M_{\mathrm{g}}^j\right)} \left[ 1+\sqrt{\frac{M_{\mathrm{p}} + \sum_{j=1}^3 M_{\mathrm{g}}^j}{M_{\mathrm{g}}^3}\left( 1+\frac{\pi d^3}{16\tau^3\left(\varphi^3\right)^2 M_{\mathrm{g}}^3} \right)} \right] {} \end{aligned} $$
(27)
$$\displaystyle \begin{aligned} v_{\mathrm{r}}^3 = \frac{M_{\mathrm{p}} + \sum_{j=1}^2 M_{\mathrm{g}}^j}{M_{\mathrm{p}}+\sum_{j=1}^3 M_{\mathrm{g}}^j}\sqrt{\left(v_{\mathrm{r}}^2\right)^2 - \left(v^{\star 3}\right)^2} {} \end{aligned} $$
(28)
Similar to the second layer, another plug mass, \(M_{\mathrm {g}}^3\), is added to the conservation of momentum expression, and the velocities used are incremented by 1.
The above equations for the BLV and residual velocities can now be applied to a specific ballistic impact case. Consider a steel-aluminum-steel 10 − 5 − 8 [mm] three-layered target impacted by a cylindrical hard-steel projectile at 680 m/s, schematically shown in Fig. 6a. For the configuration and material properties listed in Table 2, the final residual velocity – the postimpact velocity of the projectile and plugs – is 89 m/s.
Fig. 6

Steel-aluminum-steel three-layered target impacted by a hard-steel blunt cylindrical projectile: (a) projectile and target layout (all dimensions in millimeters) and (b) through-target velocity profile

Table 2

Case study data and results: steel-aluminum-steel three-layered target impacted by a hard-steel blunt cylindrical projectile

Property

Symbol

Units

Projectile

Layer 1

Layer 2

Layer 3

Thickness

hi

mm

10

5

8

Diameter

d

mm

10

Length

l

mm

40

Density

ρi

kg/m3

7850

7850

2710

7850

Poisson ratio

vi

0.3

0.3

0.3

0.3

Young’s modulus

Ei

GPa

210

190

70

190

Shear stress

τi

MPa

350

250

350

Wave speed

cp,t

m/s

5172.2

5708.1

5896.7

5708.1

Velocities

Symbol

Units

Projectile

Layer 1

Layer 2

Layer 3

Initial velocity

v0

m/s

680.0

BLV

vi

m/s

255.4

481.1

103.5

Residual velocity

\(v_{\mathrm {r}}^i\)

m/s

504.2

145.9

89.1

Target BLV

\(v_{\mathrm {t}}^\star \)

m/s

666.9

Solving the equations iteratively, it is also possible to find the ballistic limit velocity of the composite target – the maximum impact velocity that leads to a zero final residual velocity, which in the present case is 666.9 m/s. A summary of these results is shown in Fig. 6b.

Perforation by Petalling

Perforations occurring under either impulsive or static loading can lead to failure through the formation of petals protruded from the surface of the target, bending away from the surface of the penetrator. The mechanisms of deformation involved are known to be synergistic, with their interaction being complex and dependent on a multitude of physical variables. These involve material properties, the impact regime, and geometric characteristics. Petalling is normally caused by impacting thin flat plates with a conical projectile. The analytical approach described in the following paragraphs is based on an energy balance. The main features of interest in perforation-led petalling are the overall energy dissipation due to the formation of the petals and the geometry of the deformed zone. Deformation mechanisms are considered separately into the overall energy balance.

In what follows, the target is assumed to conform to a rigid-perfectly plastic material model with approximate applicability to large ductile deformation. The energy balance allows the location of crack arrest and, in principle, should also lead to the determination of the petal size. A complete formulation for the deformation mode of petalling, however, has not been developed yet. Analytical models can capture the entire perforation, providing complete formulations that consider crack initiation, which defines the number of petals, crack extension with simultaneous petal curling, and final petal rotation.

Petalling is commonly observed in thin ductile targets subjected to localized impulsive loading (Goldsmith, 2001). This failure mechanism is associated with propagation of radial cracks away from the point of impact, leading to the formation of symmetric petals that will eventually curl due to bending. Petalling is most likely to occur in targets struck by conical or ogive-nosed projectiles at relatively low impact velocity or by blunt projectiles at speeds near the ballistic limit (Mackman and Goldsmith, 1978). In addition to the localized deformation of the target, global deformation – known as dishing – may also occur, partly deforming the plane of the target.

A number of analytical methods have been proposed to evaluate the energy dissipated due to petalling, with a few also examining the geometry of the deformed pattern. Closely related to petalling is the mechanism of ductile hole growth (see Fig. 2a). By considering deformations in the quasi-static regime and by introducing an incremental relationship between principal stresses and strains to an energy balance, the plastic work required for enlargement of a hole on a sheet is
$$\displaystyle \begin{aligned} W_{{\mathrm{h}}} = \frac{\pi d^2h\sigma_{\mathrm{y}}}{3} {} \end{aligned} $$
(29)
where d is the diameter of the projectile shank, h is the sheet thickness, and σy is the yield strength of the target (Taylor, 1948). The above equation is derived for a rigid-perfectly plastic material where σy is the von Mises stress. Interestingly, Eq. 29 is independent of the nose geometry and is velocity insensitive in the absence of inertial effects. These effects can nonetheless be considered by redefining the work as
$$\displaystyle \begin{aligned} W_{\mathrm{h}} = \pi d^2h\left[ \frac{1}{2}\sigma_{\mathrm{y}} + \rho \left(v\tan\alpha\right)^2 \right] {} \end{aligned} $$
(30)
where α is the nose semi-cone angle, ρ is the density of the target material, and v is the projectile velocity (Thomson, 1955). These models are, however, developed for ductile hole growth, where the propagation of radial cracks and the formation of petals are not relevant.
Crack initiation in targets is a topic that has not been widely studied, with only one analytical model available in literature (Atkins et al., 1998). It is observed that as the deformation progresses, it is possible for the increment of plastic work required for radial displacement of material to become greater than the combination of cracking and plastic work. By introducing an energy balance based on these modes, the following expression can be derived for the number of cracks, n, resulting from an impact by a conical projectile
$$\displaystyle \begin{aligned} n = \frac{\pi \sigma_{\mathrm{y}} h\varepsilon_{\mathrm{f}}\tan\alpha}{G_c} {} \end{aligned} $$
(31)
where εf is the elongation at fracture and Gc is the fracture toughness of the target material. It should be noted, however, that in current terminology, Gc is known as the strain energy release rate at fracture and is a measure of the energy consumed per unit area of crack growth.
For the case of a rigid-work-hardening material, where necking precedes fracture, the plastic work required for the formation of necked regions has to be added to the energy balance. The shape of the neck is described by means of a new parameter dh∕dr, which corresponds to the variation of the plate thickness h in the necked regions with respect to the radial position r. This parameter can only be obtained experimentally, and it has been suggested that it could be a material property (Atkins et al., 1998). For a rigid-work-hardening material, the number of cracks is
$$\displaystyle \begin{aligned} n = &\frac{\pi \sigma_{\mathrm{y}} h}{G_c}\left[ \eta\tan\alpha \right.\\ &\left.+ \frac{\mathrm{d}r}{\mathrm{d}h}\left(e^\eta-\sin\alpha\right)\left(e^{-2\eta} - e^{-2\varepsilon_{f}}\right) \right] {} \end{aligned} $$
(32)
where η is the hardening exponent. Equation 32 suggests that the number of cracks increases with the increase of the cone half-angle.

Petalling has been described as a two-stage process, initiated by cracking and formation of n equally sized triangular petals, followed by rigid body rotation of the petals around their root (Landkof and Goldsmith, 1985). Cracking is assumed to occur instantaneously, while the petals are treated as triangular beams of fixed length. Plastic hinges then propagate along the petals up to the point of crack arrest, with the associated energy dissipation. Thereafter, rigid body rotation of the petals follows under the influence of the fully plastic bending moment. Rotation occurs with the petal tips in continuous contact with the projectile, while the process terminates as soon as this rotation allows the projectile to go through.

Cross-References

References

  1. Atkins AG, Afzal Khan M, Liu JH (1998) Necking and radial cracking around perforations in thin sheets at normal incidence. Int J Impact Eng 21:521–539CrossRefGoogle Scholar
  2. Awerbuch J, Bodner SR (1973) Experimental investigation of normal perforation of projectiles in metallic plates. Technical report, National Technical Information service, SpringfieldGoogle Scholar
  3. Borvik T (2000) Ballistic penetration and perforation of steel plates. Technical report, Norwegian University of Science and Technology – NUSTGoogle Scholar
  4. Borvik T, Leinum JR, Solberg JK, Hopperstad OS, Langseth M (2001) Observations on shear plug formation in Weldox 460e steel plates impacted by blunt-nosed projectiles. Int J Impact Eng 25:553–572CrossRefGoogle Scholar
  5. Cao S, Zhang W, Deng T, Wei G (2012) Experimental investigation on the ballistic performance of monolithic and layered metal plates subjected to impact by blunt rigid projectiles. Int J Impact Eng 49:115–129CrossRefGoogle Scholar
  6. Goldsmith W (2001) Impact: the theory and physical behaviour of colliding solids. Dover Publications Inc., New YorkzbMATHGoogle Scholar
  7. Greszczuk LB, Zukas JA, Nicholas T, Swift HF, Curran DR (1982) Impact dynamics. Wiley, New YorkGoogle Scholar
  8. Horne MR (1979) Plastic theory of structures, Pergamon Press, OxfordGoogle Scholar
  9. Jaramaz S, Elek P, Mickovic D (2005) Modeling of perforation of plates and multilayered metal targets. Int J Solids Struct 42:1209–1224CrossRefGoogle Scholar
  10. Jia X, Huang Z, Zu X, Gu X, Xiao Q (2014) Theoretical analysis of the disturbance of shaped charge jet penetrating a woven fabric rubber composite armor. Int J Impact Eng 65:69–78CrossRefGoogle Scholar
  11. Krauthammer T (2008) Modern protective structures. CRC Press, Boca RatonCrossRefGoogle Scholar
  12. Landkof B, Goldsmith W (1985) Petalling of thin, metallic plates during penetration by cylindro-conical projectiles. Int J Solids Struct 21:245–266CrossRefGoogle Scholar
  13. Mackman ME, Goldsmith W (1978) The mechanics of penetration of projectiles into targets. Int J Sci 16:1–99Google Scholar
  14. Recht RF, Ipson TW (1963) Ballistic perforation dynamics. J Appl Mech 30:384–390CrossRefGoogle Scholar
  15. Taylor G (1948) The formation and enlargement of a circular hole in a thin plastic sheet. Q J Mech Appl Math 1:103–124MathSciNetCrossRefGoogle Scholar
  16. Thomson WT (1955) An approximate theory of armor penetration. J Appl Phys 26:80–82CrossRefGoogle Scholar
  17. Velmurugan R, Sikarwar RS, Gupta NK (2014) Influence of fibre orientation and thickness on the response of glass/epoxy composites subjected to impact loading. Compos Part B 60:627–636CrossRefGoogle Scholar
  18. Woodward RL, de Morton ME (1987) A structural model for thin plate perforation by normal impact of blunt projectiles. Int J Impact Eng 6:129–140CrossRefGoogle Scholar
  19. Zhang TG, Stronge WJ (1996) Theory for ballistic limit of thin ductile tubes hit by blunt missiles. Int J Impact Eng 18:735–752CrossRefGoogle Scholar
  20. Zukas JA (1980) Impact dynamics: theory and experiments. Technical report, US Army Armament Research and Development Command, MarylandGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for Infrastructure and Environment (IIE), School of EngineeringThe University of EdinburghEdinburghUK
  2. 2.Jaguar Land Rover LimitedWhitleyUK
  3. 3.Rolls-Royce PLCFiltonUK

Section editors and affiliations

  • Filipe Teixeira-Dias
    • 1
  1. 1.The University of Edinburgh - School of EngineeringEdinburghUK