Espacios naturales protegidos (incluida Red Natura)

In recent years, there have been enormous advances in the use of computa- tional approaches to understand the penetration behaviour of projectiles into targets and to simulate the blast response of structures. For the scientist and the engineer, the computational approach will complement the experiments very well – not least because the computer can provide a qualitative and quantitative account for every step in the penetration and failure processes.

More importantly, in recent years, they have become very good predictive tools, and it would be fair to say that most, if not all, government defence research labs in the world use computational methods in a predictive capacity.

4.5.1 Types

Computational approaches require constitutive models that describe the stress and deformation behaviour of the material under dynamic load- ing conditions. These models are commonly used within a computational code called a ‘hydrocode’. This is a dynamic computational code where the

Monolithic penetrator

Segmented penetrator

Seg-tel penetrator

FIGURE 4.21

Monolithic, segmented and seg-tel penetrators. The seg-tel penetrator here shows the presence of low-density spacers.

conservation of mass, momentum and (sometimes) energy equations are solved with the constitutive equations for the materials. The name ‘hydro- code’ arose simply because the early dynamic computations did not consider strength as the (very fast) collisions studied did not warrant it.

Initially, the problem is discretised into the appropriate geometry with a number of nodes and elements making up the complete geometry of the problem. Initial conditions such as boundary constraints and velocities are specified to the respective nodes. Then, using integration, which involves a calculated discrete time step of integration, and conservation equations, a solution for a single incremental cycle can be found. Integrating the veloci- ties, we can calculate the displacements; having these, we can then work out the strain rates, strains, stresses, pressures, nodal forces and so on. This pro- cess is then repeated over many cycles to produce the final solution. More details on the theory of hydrocodes can be found in Anderson, Jr. (1987) and Zukas (2004).

Codes are formulated usually to use what is called the finite element method and are typically explicit as outlined above. Sometimes, codes will employ the finite difference method, and this differs from the finite element method by the way in which the mathematics is handled (Zukas 1990b, 2004).

There are two fundamental descriptions for the way the geometry is dis- cretised and the way in which the conservation equations are described mathematically and solved numerically. An Eulerian description uses fixed nodes and cells and allows mass, momentum and energy to flow across cell boundaries. Analysis of the material that flows in and out of the cell enables us to calculate the change in mass, pressure, temperature and so on. Another approach that is often used in hydrocodes is the Lagrangian description. Here, the problem is spatially discretised with the complete geometry of the problem being defined by a mesh. As each time step progresses, cells can be stretched and compressed; the deformation of each individual cell is ulti- mately controlled by the nodal forces. Sadly, using a Lagrangian scheme, it is very difficult to simulate dynamic impact phenomena without experienc- ing highly compressed cells, and normally, there is a requirement to discard highly deformed cells from the calculation in a process called erosion. This can lead to inaccuracies in the solution. It is also found that shock fronts can- not be accommodated in modern-day computational codes in the same way they appear physically in nature, that is, as a finite but very thin discontinu- ity. Consequently, shock fronts are required to be smeared over a number of cells by the application of an artificial viscosity function. The use of artificial viscosity and erosion is not ideal; however, they do enable simulations of very fast events to be run.

Discrete element codes and smoothed particle hydrodynamics codes are particularly useful for modelling the failure of brittle materials as these are able to track material separation, whereas Lagrangian and Eulerian codes tend to have limited material separation characteristics.

4.6 Summary

In this chapter, some very simple approaches to understand the penetration mechanisms of a projectile into a target have been covered. It is worth not- ing that students can do some sensible work using some well-thought-out analytical models that are available in the literature; some of the more per- tinent ones have been summarised here. In recent years, we have seen that computational codes have become extremely adept at predicting penetration into targets and modelling the structure failure response to blast. This has been due to the advances in material property measurements including mea- suring strain visually using digital image correlation, improvement in high- speed camera technologies and fidelity improvements in data acquisition systems. Code developments also continue to improve. Of course, it is also notable that computer technology continues to move forward in leaps and bounds, and with them the complexity of problems that can be simulated.

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5

Stress Waves

5.1 Introduction

Projectile impacts and explosions result in the formation of waves of stress that can propagate deep into a target. Stress waves (and indeed shock waves) are important as they travel at very high velocities, and consequently, failure of the target linked to these waves can occur a long distance ahead of the penetrating projectile. This is very important when the spall failure of mate- rials is considered. Understanding wave propagation mechanisms is also important when the design of armour is considered – especially in the case of brittle-based systems where small tensile waves can cause catastrophic failure. In this chapter, the physics behind stress waves and the special case where a shock wave is formed within the target material are examined.

Any contact between a moving object and stationary object will produce a wave that will emanate from the point of impact and move into the projec- tile and the target simultaneously. For very-low-velocity collisions in strong materials, the wave is most likely to be elastic in nature. Increasing the veloc- ity of impact will result in an inelastic (plastic) wave being formed. Elastic wave velocities can be easily measured using ultrasonic techniques, as dis- cussed in Chapter 2. Transducers comprised of a piezo-electric crystal that can be forced to oscillate when a voltage is applied; the oscillation results in an ultrasonic waveform with a frequency that is typically within the range of 0.02–20 MHz.

It is possible to design a multi-layered armour such that the stresses that are transmitted and reflected between each individual layer are optimised to minimise the degree of damage caused by the impacting projectile. First, the velocity of the material that is picked up by the wave and carried along with it will be established. This velocity is called the ‘particle velocity’.