Speaker
Description
The understanding of the material response in the case of high strain-rate, impact or shock loading is fundamental in several applications, including also high particle beam impacts. In order to be able to predict dynamic events it is necessary to adopt accurate and reliable numerical methods. To do this, first of all it was necessary to understand which are the involved variables. To reach this goal, a general study of the wave and shock-wave propagation in solids should be performed. In general, in case of shock loading conditions, producing the propagation of waves, the material is allowed to deform only in the impact direction, while in the orthogonal direction the deformation is prevented (or limited) by the inertia. This implies that a uniaxial strain state is generated to which also a great level of pressure is associated. The results obtained from simple numerical models, in case of purely hydrodynamic material, demonstrated that no signals move ahead the shock front: the shock front is supersonic relative to the undisturbed material. Otherwise, in case of shocked material, the sound speed is higher than the signal speed: the shock front is subsonic with respect to the shocked material. This means that any disturbance can catch the shock front from behind and explains why a shock front is spontaneously generated starting from a quite smooth pressure signal. Another important point is that, after the end of the planar shock, the shocked material returns in the undisturbed condition. This is not true in case of cylindrical wave propagation, for which, due to the axisymmetric constraint, the shock is always followed by a negative pressure wave. After that, the case of solid material was considered, for which also the material strength had to be taken into account. For low pressure wave that material remains elastic and a single elastic front is generated, which propagates at the solid sound speed. For medium intensity pressure wave, the material enters the plastic domain: two different pressure fronts are generated (elastic and plastic), which moves at different velocity (the plastic wave moves with a lower velocity). Depending on the intensity of the shock, the material could yield also during the unloading phase. The last case implies that a very strong shock is generated with a single plastic supersonic front.
The results of the studies of the wave propagation regimes in solids were very useful in the prediction of the evolution in high energy deposition scenarios: by looking the results in terms of path on the EOS surface allowed to observe that in the part close to the maximum energy absorption, during the deposition phase, there is a sudden increase in the internal energy with a negligible variation in the material density (isochoric transformation), but when the rarefaction process starts, the material could be expanded (reaching lower values of pressure and density). For the part of the target, in which low values of energy are deposited, the deposition phase leaves this quite undisturbed: the consequent growth in pressure is limited and no significant changes in density occur. On the other hand, when the shock-wave reaches these elements, they are strongly compressed, reaching high values both in density and pressure.
An improvement in the numerical models, is a soft coupling between FLUKA code and FE codes. The method was applied to simulate the beam impact against a 3D tungsten target. The FLUKA model was built using the voxel geometry and the FE one using 3D solid elements. Comparing the results obtained for coupled and non-coupled simulations it was possible to make some important considerations. The material, in which a great amount of energy is deposited, is subjected to a significant density reduction during the shock-wave propagation. This implies that the material becomes more transparent to the next proton bunches, and consequently the probability of interaction decreases. This provokes the so called tunnelling effect. The consequences of this are that the proton beam penetrates more in depth in the material in the beam axis direction and the energy is more diluted over the target. The density distribution emphasizes the tunnelling: the density modification involves much higher longitudinal coordinates increasing the number of bunches. The results in terms of pressure showed that the maximum of pressure remains more or less in the same longitudinal position with respect of the first bunch, but the pressure wave starts to travel in the transversal directions. The fact that the energy deposited by the following bunches is lower and more widespread implies that the pressure increment, consequent to the next bunches, is reduced in the zone in which the first bunch deposited a greater amount of energy. On the other hand, it should increase in the part of the target, where there is an increment in density. The results obtained showed also that until 8-10 bunches (~200 ns), the differences between the two simulations are negligible. Thanks to this, it was possible to conclude that the results obtained from the uncoupled analysis could be considered reliable if few bunches are impacted. By increasing the number of bunches, however, the change in density becomes relevant, because the shock-waves has the time to travel away from the impacted zone, producing a great rarefaction in the middle. The consequence is the coupled and uncoupled simulations give more and more different results increasing the number of bunches. In particular two situations could be developed, depending on the position on the target. For the target part situated in the neighbourhood of that in which the maximum energy was deposited by the first bunch, the uncoupled simulation overestimates the pressure. This could be easily explained since those elements are subjected to the greatest density variation. Consequently, they become more transparent to the proton beam and the successive bunches deposit on them lower energy, which implies lower growth in pressure. Otherwise, elements that are situated along the beam direction at higher longitudinal coordinates (far from the maximum energy region), are subjected to higher energy deposition during the successive bunches due to the tunnelling effect with respect to the first bunch. Obviously, this provokes an underestimation of the pressure level in case of uncoupled analysis.