Analytical and Energy-Based Methods for Penetration Mechanics
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 M_{p} – termed the projectile – traveling with velocity v and striking a stationary second mass M_{t}, 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 v_{0} = v^{⋆}, then v_{r} = 0, where v_{0} is the projectile velocity just before impact, v^{⋆} is the BLV of the target, and v_{r} 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).
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
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).
- Stage I
- Shearing forces are not yet relevant, but the projectile is experiencing inertial and compressive loading.
- 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
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.
Monolithic Targets
In-Contact Multilayered Targets
Case Study: An In-Contact Metallic Three-Layered Target
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 | h^{i} | mm | – | 10 | 5 | 8 |
Diameter | d | mm | 10 | – | – | – |
Length | l | mm | 40 | – | – | – |
Density | ρ^{i} | kg/m^{3} | 7850 | 7850 | 2710 | 7850 |
Poisson ratio | v^{i} | – | 0.3 | 0.3 | 0.3 | 0.3 |
Young’s modulus | E^{i} | GPa | 210 | 190 | 70 | 190 |
Shear stress | τ^{i} | MPa | – | 350 | 250 | 350 |
Wave speed | c_{p,t} | m/s | 5172.2 | 5708.1 | 5896.7 | 5708.1 |
Velocities | Symbol | Units | Projectile | Layer 1 | Layer 2 | Layer 3 |
Initial velocity | v_{0} | m/s | 680.0 | – | – | – |
BLV | v^{⋆i} | 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.
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
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