## 3 Results and Discussions
When the incident shock wave reflects off the surface of the foam, a compaction wave is generated in the foam. The compaction wave, on reaching the end of the foam, reflects off the end-wall, generating pressure/stress on the end-wall flange. A strong shock wave generated in water has been shown to transmit to air [10], and similarly from granular medium to air [8; 9]. Figure 6 presents the variation of pressure in the air cavity in the end-wall of the shock tube, as measured by sensor-2, without and with the foams of various densities, fully supported against the end-wall. For the \"no foam\" condition (baseline value) as well as in the presence of foams of densities less than 0.46 g/cc, there is significant rise in pressure. In addition, the pressure rises sharply to a peak value indicating compaction wave transmission, drops to an intermediate value and then remains steady till the end of the measurement  
Figure 4: Aluminum foam samples of increasing foam density after shock exposure. Figure 5: The SEM and 3D CT images of the foam of density 0.31 g/cc. (**a**) SEM SE image of the front surface before shock exposure, (**b**) SEM SE image of the front surface after shock exposure, (**c**) 3D CT view before shock exposure, and (**d**) 3D CT view after shock exposure. (1.6 ms). Table 1 gives the numerical values of the post-peak \"steady\" levels of the pressures behind the foam. Compared to the empty shock tube condition, the pressure in the cavity is almost the same in the presence of 0.28 g/cc foam [4], but reduces to 75% of the \"baseline value\" for 0.31-0.36 g/cc foams. When foams of density greater than 0.46 g/cc are present, the cavity pressure does not rise sharply, and the rise drops to 5% of the \"baseline value\" for 0.46 to 0.49 g/cc foams (shock mitigation). Thus, the present work shows that it is possible to attenuate shocks in small air-cavities by using closed cell aluminum foams of adequate density. The compaction wave transmission stated above may require an extended explanation. A protrusion of the foam surface extending into the cavity was observed at the back of the shock exposed foam, as shown in Figure 5d, which is the means of shock transmission into air in the cavity. When the compaction wave reaches the back end of the foam, an acoustic impedance mismatch is realized at the cavity, as the cavity contains air. The mismatch of impedance at the foam-air interface reflects an unloading wave back into the foam in that region, as a result of which the foam extrudes into the air cavity. For the lower density foams, the sudden rise in air pressure in the cavity suggests that the extrusion must have been rapid, generating a piston effect, launching a shock wave. As this shock wave is generated by the foam by virtue of the compaction front in it, the wave is termed as a foam transmitted shock. The pressure rise and the rate at which it rises in the cavity depends on whether (and how quickly) the foam extrudes into the cavity which, in turn, depends on whether the reflected pressure exceeds the compressive strength of the foam. Since the compressive strength increases with foam density [1], while the reflected pressure changes only marginally, there is a critical density (between 0.36 to 0.46 g/cc) above which there is no extrusion and no sudden rise in cavity pressure. The effect of foam density on the total compression can be explained by comparing the plastic strength of the foams with the incident and the reflected pressures. The plastic strength of foams under quasi-static compression is known to vary from about 2 MPa (20 bar) for 0.28 g/cc density to 5 MPa (50 bar) for 0.49 g/cc density [20]. Studies have variously shown that the plastic strength either remains unchanged [21; 22; 23; 24] or increases with strain rate in the range 200-800 s\({}^{-1}\)[25]. The pressure behind the reflected shock (from the foam front face) was of the order of 50 bar, which caused (an instantaneous) deformation and extrusion in lower density foams, which reduced with increasing foam density.