Shock-wave and high strain-rate phenomena in matter: modeling and applications

The understanding of the material response in case of high strain-rate, impact or shock loading is fundamental in several applications, such as e.g. ballistic, nuclear and military fields. The objective of the investigation of wave propagation in solids is the development of reliable methods for the prediction of dynamic events, such as high velocity or high energy impacts and detonation of explosives. In these events, usually, both high compression and expansion are involved and it is necessary to know physical, mechanical and thermodynamic properties of involved materials in wide ranges of densities, temperatures and deformations. The first chapter describes the problems correlated to shock regime and tries to collect the tools necessary to cope with the description of high dynamic regime. The concept of shock-wave phenomena is introduced and then the description of planar wave propagation in solid is reported. The analysis starts from the propagation of elastic wave and then extends to shock-wave propagation in fluids and solid. For each case presented, the corresponding numerical model is developed, aiming to highlight the main features of the investigated phenomenon. At the end of the first chapter, the propagation of cylindrical waves is also treated and the results are compared with the planar case. In the usual continuum mechanics treatment, the complete stress tensor, which describes the material condition state, is divided into two components: the deviatoric and the hydrostatic tensors. In high strain-rate and shock loading conditions, the choice of the material constitutive equations, which includes both strength model (deviatoric component) and equation of state (hydrodynamic component), is of fundamental importance: in chapter 2 the attention is focused on the hydrodynamic response, while in chapter 3 on the material strength. In impact, shock or generally high strain-rate regime, the material covers a wide range of different possible states. For example, some parts of the material can be subjected to high pressure and temperature and this implies that the thermodynamic response prevails on the mechanical one. On the other hand, other parts of material can remain in a low pressure condition, in which it is not possible to neglect the mechanical strength, which becomes the dominant part of the material response. In chapter 2, the concept of Equation Of State (EOS) is introduced, starting from the thermodynamic theory. The main characteristics of the EOS are described, with particular reference to their implementation in most of the commercial FE and hydrodynamic codes. In the last part of chapter 2, an overview of some EOS is reported, focusing the attention on their formulation and range of applicability: for each EOS presented, the trend of the pressure is reported on the basis of the data available in literature for different materials. For the visco-thermo-plastic behaviour description, the definition of constitutive relations is needed, in which the flow stress is defined as a function of all the variables of interest. Usually, in plasticity, the independent variables are: deformation (both plastic and volumetric), strain-rate, temperature and pressure. This aspect is treated in chapter 3, in which the concepts at the basis of the definition of strength material models are described, with particular reference to the high strain-rate and shock-wave regime. In this perspective, an overview of all the variables of interest in such kind of problems is examined. Then, the most common strength models, usually implemented in commercial FE codes, are examined, paying attention to the meaning of each model parameter and the availability of data for different materials, for which the plastic behaviour is analyzed and compared. After this, also some failure models, which should be defined in a numerical model for the complete description of the material behaviour, are presented and in the final part of the chapter 3, a procedure for the material model identification, based on a numerical inverse method, is presented. In chapters 5 to 7 the shock-wave propagation in matter, due to the interaction of high energy particle beams with solids is analyzed. The main objective of this study is to build safe and reliable numerical models able to estimate the damage on targets due to the impact of high energy proton beam in CERN Large Hadron Collider (LHC). To do this, all the concepts introduced in the previous chapter are used. In chapter 5 the problem is introduced on its generality: after a brief introduction to the LHC, the interaction between intense beam and solid targets is investigated from a qualitative and phenomenological point of view. This allows the comprehension of the involved phenomena, which is necessary for the construction of the numerical model described in chapters 6 and 7. In chapter 6, the numerical simulation of the high energy deposition on cylindrical bars, hit at the centre of one face by 8 proton bunches of the LHC at 7 TeV is performed. Two cases are analyzed, varying the material of the target (copper and tungsten). For each case a Lagrangian 2D axisymmetric simulation is performed, starting from the energy deposition map. In chapter 7, the impact of high energy proton beams against 3D structures is described: the FE solution is obtained in case of Lagrangian and pure structural analysis solved with an explicit time integration method, on 3D solid elements. Two different cases are reported. In the first one, the impact is simulated on the simplified model of a Tertiary collimator and the impact condition implies that the target is impacted perpendicularly to the free surface by 8 protons bunches at 5 TeV. The description of this case is of particular interest for the evaluation of the consequences of the impact near a free surface. For the second case, the objective is the description of a numerical procedure, for a soft coupling between the FLUKA and the LS-DYNA codes, developed in collaboration with the FLUKA Team at CERN. The main objective is evaluating the influence of the change in density on the deposition phase. In order to achieve this, a great number of bunches (60) are supposed to impact against a tungsten parallelepiped. The studies presented in chapter 5 to 7 are developed within the European project EuCARD (European Coordination for Accelerator Research & Development), which is co-funded by the European Commission within the Framework Programme 7 Capacities Specific Programme. In more details, the Work Package (WP) 8 is involved. The WP8, named ColMat, Collimations & Materials for higher beam power beam, has the main objectives the development of material and machine components related to collimation system and intercepting devices. The last chapter is dedicated to the description of an experiment (first-of-its-kind), in which the responses of different materials are measured under the controlled impact of a high energy proton beam with the aim to validate numerical models and extract material data in terms of strength and EOS models. The experiment was performed in the HiRadMat facility at CERN in October 2012 and some preliminary results are shown. The description of the experiment is also reported, paying particular attention to the most relevant aspects of the design phase, including the choice of materials and impact conditions.