Blast mitigation using monolithic closed cell aluminium foam

Blast mitigation using monolithic closed cell aluminium foam | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Blast mitigation using monolithic closed cell aluminium foam Chitralekha Dey, Amol Anant Gokhale This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4018253/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Blast protection using cellular materials is being actively pursued at research and technology levels. The present work uniquely demonstrates generation of stress waves, strain waves and mass velocities in monolithic closed cell aluminium foams of different densities and lengths, subjected to simulated blast loads, and their combined effect on blast attenuation. The foams were assumed to be resting against a rigid end wall. If the numerically calculated stress at the back face was found less than the applied stress at the front face, the interaction was termed as blast mitigation or attenuation. The results show, ‘pressure mitigation’ to occur for low-density foams whose plastic strength is less than the applied pressure, but pressure amplification for high-density foams whose plastic strength is higher than the applied pressure. The pressure amplification observed in shorter length high density foams transformed to pressure mitigation if the foams were sufficiently long. Based on these results and other stress, strain and velocity related diagnostics, the underlying mechanism behind blast wave amplification/mitigation and its relation with foam density and length is proposed. Blast foam compaction wave reflected pressure end wall stress length density Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 1. Introduction Closed-cell aluminium foams act as excellent energy-absorbing materials in the face of quasi-static and impulse loads. Under quasi-static compression, aluminium foams show three distinct regimes - at low stress the primary mechanism of elastic deformation is cell wall bending. The elastic regime is followed by the ‘plateau stress’ region, where high strains are achieved at a constant stress called the plateau stress. This region is responsible for the high energy absorption characteristics of foam. In this region, localized plastic collapse is initiated in the cells, which then propagates in bands within the foam. The third region is the ‘densification region’. When all the cells collapse and the cell walls come in contact with each other, the foam behaves like the parent solid material. Densification starts when there is a steep rise in stress with a further increase in strain [ 1 ]. Li et al. [ 2 ] have investigated foams under impact loading through numerical means and observed two distinct modes in which the foams deform- the ‘random mode’ and band front foam. The random mode occurs under low impact velocity where the shear bands of deformation are randomly oriented within the foam mass. The ‘band front mode’ occurs under high impact velocity, where the deformation starts at the impact location and then progresses in the direction of impact. The authors concluded that strength enhancement in the random mode is derived from the strain rate sensitivity of the cell wall material, whereas in band front mode, it is derived from inertia. The response of foam when subjected to impact load is available in the literature [ 3 – 7 ]. The effects of foam density on the load transfer, depth of penetration and deformation pattern under indentation impact have been studied experimentally and numerically [ 3 ]. The shape and size of the individual cells which is again a function of foam density were shown to affect the way through which the indenter penetrates inside the foam. Whereas in the low-density foams, the indenter penetrates through a higher distance and there is a thin zone of damaged cells around the indenter, for the high-density foam, this depth is less and the damaged cells accumulate below the indenter. Hanssen et al. [ 8 ] carried out full-scale blast experiments to study the response of foam panels. Charges were detonated at a fixed scaled distance from the foam panels which were connected to a pendulum. The swing of the pendulum was used as a measure of the energy and impulse transferred to the foam panels. Some experiments were also carried out with an aluminium cover plate on the foam panels. It was shown in the study that, the swing in the pendulum or the impulse transferred to the panel increased when foam was present. However, analytical solutions based on shock wave theory did not justify the experimental results. The authors concluded that the increase in impulse transfer occurred due to the convexity formed on the back face of the panel. Thus, the surface effects were identified as the cause of the amplification. Xue et al. [ 9 ] carried out a numerical analysis of monolithic solid plates and sandwich plates with tetragonal truss cores of the same mass and material against blast loads. The authors have commented that, over the range of load levels examined, the maximum deflection in sandwich plates with a relative core density of 0.08 is 50–85% of that in solid plate. Dharmasena et al. [ 10 ] performed air blasts on a square honeycomb cored sandwich panel made of super-austenitic stainless-steel alloy and solid plates of the same aerial density. The tests were carried out for three impulse levels by varying the weight of the explosive while keeping the standoff distance constant. The authors reported that the sandwich panels are much more advantageous, as the back face deflections are significantly lower than the solid plates of the same mass. Khondabi et al. [ 11 ] studied the performance of sandwich panels with aluminium face sheets and graded PU foam core with different layering arrangements against blast load. The authors tested single layer, double layer and triple layered cores. It was observed that the triple layered cores with relative densities in descending order reduced the back face sheet deflection substantially when compared with single layer panel. Jinlong et al. [ 12 ] studied the blast performance of Fiber-Metal-Laminate (FML) double layer sandwich plates. The effects of geometrical and material parameters on the dynamic response of the panel were studied. Finally, it was reported that FML double layered sandwich plate had better anti-blast performance as compared to metal sandwich plates of equal mass. Xue et al. [ 13 ] studied the blast performance of sandwich plates with three different cores – pyramidal truss, square honeycomb and folded plate. Subsequently, the performance of these sandwich plates was compared with solid plates of the same material and weight. Limited optimization studies were carried out with the parameters – core thickness, face sheet thickness, core member aspect ratio and relative density. The authors commented that well-designed sandwich plates can withstand significantly higher impulses as compared to solid plates of the same weight. Yazici et al. [ 14 ] studied the effect of foam infill on the blast resistance of corrugated steel core sandwich panels by experimental and numerical means. The authors studied the effect of face sheet thickness, corrugated core sheet thickness and boundary conditions (simply supported and encastre supported) on the blast performance of the panels. It is reported that the foam infill reduced the front face and back face deflections by more than 50% at 3 ms after the loading at a weight penalty of only 2.3%. However, if the face sheet thickness and corrugated sheet thickness are increased, the advantage of the foam filling reduces. Karagiozova et al. [ 15 ] have carried out an experimental and numerical investigation on the behaviour of peripherally supported steel face sheet-polystyrene core and steel face sheet-aluminium honeycomb core sandwich panels subjected to blast loading. They observed that the rate at which the velocity of the free face (i.e., the loading face) of the sandwich was attenuated by the core determined the transmitted pressure and, consequently, the deflections in the unsupported parts of the back face. The authors observed that the panels with honeycomb core were more efficient than the polystyrene foam core in pressure attenuation. Liu et al. [ 16 ] studied the response of aluminium foam cored sandwich panels subjected to blast load and compared it with stand-alone mild steel sheets. It was observed that the sandwich panels attenuated the peak blast pressure by approximately 65% as compared to stand-alone mild steel plates. Further, it was seen that the blast energy was primarily dissipated by the formation and growth of cracks in the honeycomb. It was felt that multiple reflections occurring at the interfaces between the foam cell walls and air in the foam cavities were responsible for energy dissipation. Karagiozova et al. [ 17 ] studied blast wave attenuation in a sandwich panel consisting of a steel face sheet and Cymat aluminium foam core through analytical and numerical means. They proposed that unloading plastic waves rather than shock waves were generated in the foam, resulting in foam compaction and energy absorption. Zhu et al. [ 18 ] reported pressure amplification under blast loading in materials such as polymeric foams and metallic foams. The authors explained that the use of such materials causes pressure amplification due to positive changes in impedance at two locations in the path of the blast wave: the air-foam interface and the foam-end wall interface. In the absence of the foam, amplification takes place only once: at the air-end wall interface. However, the authors did not comment on the contribution of energy dissipation due to densification in foams. Petel et al. [ 19 ] carried out experimental studies on open-cell polyurethane and polyethylene foams under blast and shock loading. Experiments were carried out with an impermeable membrane placed at the mid-thickness to isolate the effect of gas filtration, if any, in the creation of precursor waves in the open cell foam. It was found that the stress signal due to the precursor wave was unchanged in magnitude and time of arrival, indicating that the precursor wave was not driven by flow filtration. Nian et al. [ 20 ] have experimentally investigated the response of cellular concrete foams of two different crushing strengths to blast loading inside a shock tube. The authors have remarked that when the imposed load was greater than the crushing strength of the concrete foam, an elastic wave and a subsequent plastic wave was generated inside the foam. The authors noted that if the length of the foam panel was more than a critical value, blast mitigation occurred, as the plastic wave did not reach the wall at the end of the shock tube. For smaller lengths, the plastic waves reached the end wall, producing stress on the end wall which was higher than the plateau stress but less than the applied pressure. For lengths less than a threshold value, pressure amplification, associated with significant crushing of the foam occurred. While many authors have pointed out the advantage of using cellular cores with respect to pressure attenuation and reduced deflection at the back face, others reported stress enhancement. It is also reported in the literature that, the length of the foam plays a critical role in deciding between pressure attenuation and enhancement. However, a comprehensive study of the transmission of compaction waves through blast-loaded closed-cell aluminium foams, variation of loading face velocity and their effect on pressure attenuation/amplification is not reported. Moreover, the effect of foam length and foam density on the above responses is lacking. In this work, numerical analysis of the behaviour of closed-cell aluminium foams of various densities covering plastic strengths above and below the applied stress under blast loading arising from the detonation of 1 kg TNT at a standoff distance of 1 m from the front face of the foam is studied. It is assumed that the foam panel is resting on a fixed steel plate which acts as an end wall, and the pressure at the end wall is calculated. The variation of the end wall pressure is studied as a function of foam density in the range 270 to 900 kg/m 3 and an analysis of the compaction waves travelling through the foam is carried out. Further, the length of the foam (termed as thickness in the literature presented above) is varied for both the low and high-density foams and the corresponding changes in the transmitted stress are studied. 2. Creation of finite element model In this section, details of the finite element model comprising of the general construction of the foam panel, the material model used to create foam in the finite element environment and blast loading is presented. 2.1 Finite Element Model Cylindrical foam panel of 40 mm diameter and 25 mm length was modelled. The charge is assumed to be placed at a distance of 1 m from the centre of the loading face. The foam rests against a fixed steel plate acting as the end wall. The blast loading is incident on the front surface of the foam, herein referred to as loading face. Due to the uniform loading and the symmetry associated with the cylindrical geometry of the foam, a quarter symmetric model was prepared in Abaqus explicit Finite Element software (Fig. 1 ). General contact was defined between all surfaces. Symmetry boundary conditions on the X and Y faces of the foam were applied. All degrees of freedom on the end wall were constrained. Since there is no plastic Poisson’s ratio in foams [ 21 ], boundary conditions were not applied on the circumferential surface. 2.2 Material model ‘Crushable foam model’ in Abaqus explicit software, developed on the basis of ‘Isotropic constitutive model’ by Deshpande and Fleck [ 22 ] was used to model foam. Quasi-static compression tests were performed on foams of various densities. The experimental stress-strain curves (Fig. 2 ) were subsequently used in the material model. Foams exhibit a quasi-elastic deformation zone at very low stresses during which the deformation is largely elastic with small amount of local plastic deformation attributed to stress concentration at cell nodes, which can contribute significantly to the total deformation [ 23 ]. This is followed by the onset of plastic instability, indicated by the first peak in the stress-strain curve, which represents plastic collapse within bands of cells. The plateau region represents the dynamic balance between densification in the collapsed cells and cell collapse in fresh volumes of the foam. As more and more bands of collapsed and densified cells form, the entire volume of the foam gets occupied by such bands resulting in significant rise in stress. The material model was validated against total deformation [ 24 ] and stress at the end wall [ 25 ] under shock tube loading of the foams. 2.3 Loading definition The blast load was simulated using the CONWEP code available in Abaqus software. The code calculates the total blast pressure \(P\left(t\right)\) on a surface as a function of time by using the following pressure function (Eq. 1), where \({P}_{i}\) is the incident pressure, \({P}_{r}\) is the reflected pressure and \(\theta\) is the angle between the normal to the loading face and the direction of the blast wave propagation. The surface on which the blast pressure is incident, the mass of the explosive, and the location of the charge with respect to the structure are specified. \(P\left(t\right)={P}_{r}{cos}^{2}\theta +{P}_{i}{(1+cos}^{2}\theta -2cos\theta )\) 1 A typical \(P\) versus \(t\) blast wave is shown in Fig. 3 . On arrival of the blast wave at a certain standoff distance, the pressure rises from the ambient pressure P 0 to peak incident pressure P s . The time duration of the wave is t d , which is the time it takes for the pressure to drop from the reflected pressure P s to the ambient pressure close to the loading face. According to the literature [ 26 ], the detonation of 1 kg TNT from a standoff distance of 1 m from the centre of a loading face (assumed to be rigid) gives a peak reflected pressure of 5 MPa and a time duration of the wave equal to 0.5 ms. 3. Results and Discussion In this section diagnostics regarding the propagation of the blast wave through the foam will be presented. This involves tracking the displacements, velocities, strains and stresses at various positions along the length of the foam as a function of time. Since the wave propagation characteristics are quite distinct for the low-and the high-density foams, these are studied and presented separately. The foam of 270 kg/m 3 density having lower plastic strength as compared to applied pressure is chosen to represent the low-density category and the 600 kg/m 3 density foam having higher plastic strength than the applied pressure, is chosen to represent the high-density category. Also, the baseline foam length is taken as 25 mm. 3.1 Displacements at various locations of the foam along length with time When the foam is exposed to the blast wave, the loading face as well as the interior of the foam undergoes displacements towards the stationary face. The evolution of displacements at various distances from the loading face as a function of time is shown in Fig. 4 for two representative foam densities mentioned earlier. Several observations can be made from this figure. It is seen that for a given foam density, the displacement is the highest at the loading face, decreasing with increasing distance from the loading face. This is to be expected since the distal face of the foam is stationary. Secondly, the farther the location from the loading face, the longer it takes to initiate the displacement, which indicates the wave nature of the displacement. Thirdly, the displacements in the high-density foam are negligible compared to those in the low-density foam. Whereas the total deformation of the low-density foam is 9 mm (36%), it is only 0.4 mm for the high-density foam for which the plastic strength is higher than the applied pressure. A related observation is that there is no noticeable recovery of displacement in the low-density foam, implying that the displacements are associated with plastic deformation. Although there may be a small amount of displacement recovery in the high-density foam, most of it is permanent in nature, meaning that even at applied stresses below the plastic strength, there is a small amount of plastic deformation. The very small levels of total displacements seen in high-density foam preclude drawing any further inferences on the nature of displacements in high-density foam. 3.2 Material velocity along the foam length The evolution of material velocity of the loading face (x = 0 mm) and two representative distances from the loading face (x = 12 and x = 21 mm) for the low- and high-density foams is shown in Fig. 5 . In all the cases, the velocity rises sharply to a peak value and then drops rapidly to zero. It takes longer for the velocity to rise as the distance from the loading face increases. Also, the peak velocity attained is the highest at the loading face, decreasing towards the stationary face. Whereas very high velocity (82 m/s) is achieved by the loading face in the low-density foam, it is much lower at 11 m/s in the high-density foam. The drop in peak velocity with increasing distance from the loading face is more in low-density foam than in high-density foam. The significant reduction in velocity with increasing distance from the loading face in the low-density foam is attributed to plastic dissipation. Finally, the time taken by the velocity at each location to drop down to zero is higher in the low-density foam (lower deceleration) as compared to the high-density one (high deceleration), which is due to the largely elastic nature of the interaction between the blast wave and the latter case. 3.3 Strain evolution along the foam length The differential displacements at various locations in the foam result in permanent densification, and evolution of strain with time. The study of the evolving strain distribution along the length of the foam provides insight into the travel of the blast wave through the foam. Figure 6 shows the plastic strain distribution along the foam length in low- and high-density foams at the end of the loading cycle. It is seen from Fig. 6 that, for the low-density foam, the strains at different distances from the loading face are in the range of 40–60%, which is within the plateau region of the stress-strain curve, whereas for the high-density foam, the strains are very low and lie in the elastic region of the stress-strain curve (Fig. 2 ). Thus, under the blast load, the low-density foam deforms plastically within the plateau region and not entering the densification region, whereas the deformation in the high-density foam sees only elastic deformation plus a small amount of local plastic yielding referred earlier. Also, for the low-density foam, the strain marginally decreases from the loading face to the centre of the foam and then increases towards the end wall. The reason for this behaviour can be explained as follows: as the applied stress exceeds the plastic strength of the foam, plastic compaction waves are created inside the foam. As a result, energy dissipation takes place through plastic deformation, reducing the strains in the distal regions. When these waves reach the end wall and reflect from the rigid end wall, they get amplified where the reflected pressure is high, justifying the increase in strain levels near the end wall. However, for the high-density foam, the strain increases steadily from the loading face to the end wall. The behaviour can be explained thus: since the plastic strength of these foams exceed the applied load, the waves generated inside the foam are largely elastic. These elastic waves do not dissipate energy, and hence they do not attenuate during travel through the foam. Upon reflection from the end wall, the elastic wave amplifies, thus explaining the increase in the strain levels towards the end wall. Figure 7 shows the strain-time profiles for foams of density 270 kg/m 3 and 600 kg/m 3 . A sudden rise in strain is seen to occur at the loading face as the compaction wave is launched. The farther the location from the loading face, the longer it takes for the strain to rise since the compaction wave is travelling in the forward direction towards the stationary face of the foam. In the low-density foam, the time duration between the arrivals of the strain waves at the chosen locations is much greater than that in the high-density foam. In 270 kg/m 3 foam, the compaction wave, which is practically a plastic wave, is calculated to have a velocity of 120 m/s whereas in the high-density foam, the velocity of the comparatively faster elastic wave is calculated to be 1200 m/s. Figure 8 shows the advancement of the compaction wave inside the foam. At t = 0 ms, the incident wave is yet to reach the front face of the foam. The foam is not compacted and no strain is developed inside the foam. At t = 0.55 ms, the blast wave strikes the front surface of the foam. A part of the incident wave gets reflected back and a part gets transmitted inside foam. From this instant, the compaction process starts in foams. The strains start developing and progresses with time. At t = 0.72 ms, the wave reaches the stationary end wall. Due to mismatch in impedance, it gets reflected back towards the loading face of the foam. This phenomenon leads to further reduction in the length of foam. 3.4 Development of stress along the foam length Stress evolution in the foam of density 270 kg/m 3 at various distances from the loading face is presented in Fig. 9 (a). It can be seen from the figure that, the stress profile at the loading face represents the blast loading profile as shown in Fig. 3 . While the peak in the applied stress occurs at 5 MPa, when the blast wave impinges on the loading face, after the subsequent deformation and densification of the foam, a reduced peak stress of 4.5 MPa is present at the loading face. The movement of the front face of the foam in the loading direction upon impingement of the blast pressure causes the reduction in stress. At x = 12 mm, which is nearly at the mid-plane of the foam the stress profile is marked with a gradual rise to a lower peak of 2.8 MPa and then a fall. As the wave progresses to x = 18 mm and 21 mm, the stress profiles show three peaks. The first peak has a magnitude of 1.8 MPa which is lower than the peaks seen closer to the loading face due to further plastic dissipation-induced pressure attenuation of the forward travelling compaction wave. The formation of the second and third peaks can be explained thus - progress of compaction wave to a certain location inside foam leads to local densification. Consequently, the flow stress corresponding to the densified foam increases. The stress becomes equal to either the flow strength of foam or equal to the applied stress, whichever is lower. In the current scenario, the second rise in stress is equal to the flow strength of densified foam which is approximately equal to 2.9 MPa. The third and distinct rise in stress occurs due to the reflection of the compaction wave from the stationary end wall and the ensuing impedance mismatch between the foam and the end wall. The second peak in stress is not observed near the end wall as it is masked by the higher amplitude reflected wave travelling towards the loading face of the foam. It may be noted that the third peak due to the reflection from the end wall is not observed in locations from x = 0 to 12 mm. This is because of the following reason - whereas the total duration of the exponentially decreasing blast loading is only 0.5 ms, the wave requires approximately 0.2 ms to reach the end wall (The velocity of the compaction wave is calculated to be 120 m/s for the low-density foam). Thus, the short-duration blast loading is over before the reflected wave reaches the front end of the foam. Figure 9 (b) shows the stress contour in the low-density foam at three-time steps – 0.55 ms, 0.65 ms and 0.75 ms. At 0.55 ms, barring the region near the loading face where the stress is high, the stress in the other regions of foam is approximately in the range of 2 MPa. A 0.65 ms, the stress in the entire foam is in the range of 2 MPa. At 0.75 ms, the stress near the end wall peaks to approximately 4 MPa whereas in the other regions, it is in the range of 2–3 MPa. Stress evolution in the foam of density 600 kg/m 3 is presented in Fig. 10 (a). Foam of density 600 kg/m 3 has a plastic strength of 10 MPa which is higher than the applied pressure. Hence, the blast wave will be propagated inside the foam as an elastic wave. The stress in the loading direction is captured at various locations designated by x = 0, 12, 18, and 21 mm. It is seen that the stress profile at the loading face is like a blast loading profile and of 5 MPa peak pressure, which is the applied pressure. This is because the high-density foam, being stiff, does not deform and the input load is seen at the loading face. Only a peak of 8 MPa stress is observed at all subsequent locations, including the end wall. The reason for not seeing multiple peaks here unlike the low-density foam can be explained thus – it is seen from the strain profile that the speed of the compaction wave in high-density foam is very high (ten times that in the low-density one). Thus, due to the high wave velocity, the forward-travelling wave is masked by the backward travelling reflected wave, and only the reflected wave from the end wall is observed. Figure 10 (b) shows the stress contour at two-time steps – 0.55 ms and 0.58 ms. At 0.55 ms, the stress near the loading face of the foam is equal to the applied blast pressure, whereas, near the end wall region, the stress is very low. At 0.58 ms, an opposite scenario occurs, where the stress near the end wall region shoots up to 8 MPa and progressively reduces towards the loading face of the foam. 3.5 Estimation of the end wall stress As stated in the introduction, foams have been shown to reduce stress transmitted to the foam-clad protected structures under impact loading. If the foam was to be used as a sacrificial cladding on structures vulnerable to blast-initiated shock waves, it is important to know the stress transmitted by the foam to the protected structure. In this section, end wall stress is determined for (a) fixed length and varying density and (b) fixed density and varying length. 3.5.1 Effect of varying foam density Stress vs time variation at the end wall was determined for fixed foam length of 25 mm and varying foam densities from 270 kg/m 3 to 900 kg/m 3 . It can be observed from Figs. 11 and 12 that, as the foam density increases from 270 to 380 kg/m 3 , the peak stress reduces from 4.5 to 2.5 MPa. If the foam density is further increased, the peak stress steadily increases to 3.8 MPa for 410 kg/m 3 density and then further increases to 8 MPa for 500 kg/m 3 density. The peak stress then remains approximately constant for higher foam densities. The behaviour of the foams under blast loading can be grouped into two sets based on density – the lower density foams where the average peak end wall stress remains approximately in the range of 3 MPa and the higher density foams where the average peak end wall stress remains approximately in the range of 8 MPa. Stress amplification is defined as the ratio of the end wall stress to the applied stress. In the present scenario, it can be observed that the higher density foams starting from 500 kg/m 3 amplify the shock while attenuation occurs for densities equal to and lower than 500 kg/m 3 . The generation of end wall stress is assumed to be due to – (a) momentum transfer from the accelerating foam mass to the stationary end wall (b) impedance mismatch at the foam-end wall interface (Fig. 5 ). These two factors are considered additive in nature since they have different origins. In low density foam, material displacements are much higher and the momentum effect will be large. However, due to plastic dissipation, the incident pressure before reflection is low resulting in relatively lower reflected pressure. The net result is pressure attenuation at the end wall. In the high density foam, on the other hand, the momentum contribution will be less since the overall displacements and velocities are low. But due to lack of plastic dissipation, the incident pressure in the compaction wave remains close to the applied stress, resulting in higher contribution to the reflected stress at the end-wall. The net result is that the stress generated at the end wall is higher than the applied stress in high density foam. Mazor et al. [ 27 , 28 ] predicted that under shock tube-generated loading of elastomeric foams, the pressure enhancement at the end wall will decrease as the relative density of the foam increases. But a major difference between the cited work and the present work is that the aluminium foams deform plastically under blast load, to varying extents depending on foam density, dissipating energy causing attenuation of the stress wave till the wave reflects at the end wall. This results in a different trend in the present work. 3.5.2 Effect of varying foam length In the previous section, it was observed that foams of densities equal to and greater than 500 kg/m 3 amplify the pressure at the end wall for 25 mm length foams. An applied pressure of 5 MPa on the loading face is amplified to 8 MPa at the end wall. However, from design point of view, it is important to understand whether the pressure amplification reduces or increases further with increasing foam length. Therefore, a broad range of foam lengths were studied for the density of 600 kg/m 3 . Figure 13 shows the variation of the peak end wall stress for different foam lengths. When the foam length was reduced from the original 25 mm to 15 mm, pressure amplification increased further since there is insufficient length for stress to attenuate to any appreciable level before reflection at the end wall (Fig. 13 ). When the foam length was increased above 25 mm, amplification at the end wall reduced. Finally, attenuation was achieved for a foam length of 200 mm. With very small local plastic deformation taking place in the high density foam, it is unable to cause any attenuation in the forward traveling stress wave, a much longer foam can be expected to cause attenuation in the stress wave before reflection, reducing the overall contribution to the stress generated at the end wall. Figure 13 also shows the variation of end wall peak stress with variation in foam length for a foam density of 270 kg/m 3 . The trend in stress variation is similar to that for high density foam i.e., the peak stress reduces with increasing foam length. The lower density foam, however, can be of much shorter length to achieve pressure attenuation which is not the case in high density foam which requires a minimum of 200 mm length to achieve pressure attenuation, the difference attributed to plastic dissipation in low density foam which reduces incident stress before reflection from the end wall. Thus, the comparative diagnostic analysis of the blast-foam interaction shows that the low- and high-density foams have distinctive strain distributions along the length of the foam. In the low-density foam (270 kg/m 3 ) the strain decreases towards the mid-length of the foam sample and then increases towards the end wall, while the high-density foam (600 kg/m 3 ) shows a gradually increasing strain from the loading face to the end wall. The end wall stress under blast loading remains approximately constant as the density increases from 270 to 380 kg/m 3 . When density is further increased, the peak stress increases up to 500 kg/m 3 and then plateaus off with a further increase in foam density. The stress at the end wall is postulated to be due to two components – transfer of momentum of the foam undergoing displacement to the stationary end wall under the blast pressure and stress increase associated with reflection of the incident stress wave at the higher impedance end wall. In low density foams, the momentum effect associated with large displacements seems to dominate while in high density foams, the wave reflection at the higher impedance end wall seems to dominate in determining the stress at the end wall. Study is also carried out by varying the length of the foam with an aim to reduce the stress at the end wall. A length of 20 mm gives considerable stress attenuation in low density foams while a much larger length of 200 mm is required to give stress attenuation in high density foams. 4 Summary Stress attenuation using closed cell aluminium foam is demonstrated numerically, using a continuum based finite element model of foam, and representing the blast load using CONWEP code. Stress calculations at the back face were based on profiling the stress waves, strain waves and displacements occurring in the foam under the applied blast pressure. Under a blast-triggered incident shock, plastic compaction waves are transmitted through foams whose plastic strength is less than the applied pressure. The peak stress in the transmitted wave diminishes with travel due to energy dissipation associated with the densification of the cells. The stress generated at the end-wall is postulated to be composed of a) momentum transfer to the stationary end wall due to stagnation of large displacements in the foam associated with foam densification, which is prominent in low density foams, and b) increase in stress on reflection of the incident stress wave from the higher impedance end wall, which is prominent in high density foam due to lack of plastic dissipation in the incident wave. Lower foam density e.g., 270 kg/m 3 at 20 mm length gives substantial stress attenuation for applied peak pressure of 5 MPa with no further gain by increasing the foam length. Higher density foams give stress amplification; the foam length needs to be increased to 200 mm for any attenuation to be observed. References L. Gibson, Cellular Solids (2003), MRS Bulletin, 28(4), 270-274 https://doi.org/10.1557/mrs2003.79 L. Li, P. Xue, G. Luo, A numerical study on deformation mode and strength enhancement of metal foam under dynamic loading, Materials and Design (2016) https://doi.org/10.1016/j.matdes.2016.07.123 C. Dey, S. N. Sahu, K. Akella, A. A. Gokhale, Numerical Prediction of Quasi-Static Compression, Indentation Impact and Shock Loading Behaviour of Aluminium Foam Using Idealized Cell Geometry, Journal of dynamic behaviour of materials (2022) Volume 8, Pages 324-339 https://doi.org/10.1007/s40870-022-00336-9 X. pang, H. Du, Dynamic crushing of aluminium foams under impact crushing, Composites Part B: Engineering (2017) https://10.1016/j.compositesb.2016.12.044 M. Peroni, G. Solomos, V. Pizzinato, Impact behaviour testing of aluminium foam, International Journal of Impact Engineering (2013) https://doi.org/10.1016/j.ijimpeng.2012.07.002 M. A. Kader, M. A. Islam, P. J. Hazell, J. P. Escobedo, M. Sadatfar, A. D. Brown, G. J. Appleby-Thomas, Modelling and characterization of cell collapse in aluminium foams during dynamic loading, International Journal of Impact Engineering (2016) https://doi.org/10.1016/j.ijimpeng.2016.05.020 G. Lu, J. Shen, W. Hou, D. Ruan, L. S. Ong, Dynamic indentation and penetration of aluminium foams, International Journal of Mechanical Sciences (2008) https://doi.org/10.1016/j.ijmecsci.2007.09.006 A.G. Hanssen, L. Enstock, M. Langseth, Close-range blast loading of aluminium foam panels, International Journal of Impact Engineering (2002), Volume 27, Issue 6, Pages 593-618 https://doi.org/10.1016/S0734-743X(01)00155-5 Zhenyu Xue, John W. Hutchinson, Preliminary assessment of sandwich plates subject to blast loads, International Journal of Mechanical Sciences (2003), Volume 45, Issue 4, 687-705 https://doi.org/10.1016/S0020-7403(03)00108-5. Kumar P. Dharmasena, Haydn N.G. Wadley, Zhenyu Xue, John W. Hutchinson, Mechanical response of metallic honeycomb sandwich panel structures to high-intensity dynamic loading, International Journal of Impact Engineering (2008), Volume 35, Issue 9, 1063-1074 https://doi.org/10.1016/j.ijimpeng.2007.06.008. Khondabi, R., Khodarahmi, H., Hosseini, R. et al. Dynamic plastic response of sandwich structures with graded polyurethane foam cores and metallic face sheets exposed to uniform blast loading: experimental study and numerical simulation, J Braz. Soc. Mech. Sci. Eng.(2023) 45, 526. https://doi.org/10.1007/s40430-023-04384-7 Du, J., Su, H., Bai, J. et al. On dynamic response of double-layer rectangular sandwich plates with FML face-sheets and metal foam cores under blast loading. J Braz. Soc. Mech. Sci. Eng. 45, 31 (2023). https://doi.org/10.1007/s40430-022-03956-3 Zhenyu Xue, John W. Hutchinson, A comparative study of impulse-resistant metal sandwich plates, International Journal of Impact Engineering (2004), Volume 30, Issue 10, 1283-1305 https://doi.org/10.1016/j.ijimpeng.2003.08.007. Murat Yazici, Jefferson Wright, Damien Bertin, Arun Shukla, Experimental and numerical study of foam filled corrugated core steel sandwich structures subjected to blast loading, Composite Structures (2014), Volume 110, 98-109 https://doi.org/10.1016/j.compstruct.2013.11.016. D. Karagiozova, G.N. Nurick, G.S. Langdon, S. Chung Kim Yuen, Y. Chi, S. Bartle, Response of flexible sandwich-type panels to blast loading, Composites Science and Technology (2009), Volume 69, Issue 6, 754-763 https://doi.org/10.1016/j.compscitech.2007.12.005. H. Liu, Z.K. Cao, G.C. Yao, H.J. Luo, G.Y. Zu, Performance of aluminum foam–steel panel sandwich composites subjected to blast loading, Materials & Design (2013), Volume 47, 483-488 https://doi.org/10.1016/j.matdes.2012.12.003. D. Karagiozova, G.S. Langdon, G.N. Nurick, Blast attenuation in Cymat foam core sacrificial claddings, International Journal of Mechanical Sciences (2010), Volume 52, Issue 5, 758-776 https://doi.org/10.1016/j.ijmecsci.2010.02.002. Feng Zhu, Clifford C. Chou, King H. Yang, Shock enhancement effect of lightweight composite structures and materials, Composites Part B: Engineering (2011), Volume 42, Issue 5, 2011, 1202-1211 https://doi.org/10.1016/j.compositesb.2011.02.014. O.E. Petel, S. Ouellet, Higgins, D. L. Frost, The elastic–plastic behaviour of foam under shock loading, Shock Waves (2013), 23, 55–67 https://doi.org/10.1007/s00193-012-0414-7 Weimin Nian, Kolluru V.L. Subramaniam, Yiannis Andreopoulos, Experimental investigation on blast response of cellular concrete, International Journal of Impact Engineering (2016), Volume 96, Pages 105-115 https://doi.org/10.1016/j.ijimpeng.2016.05.021 D. Karagiozova, G.S. Langdon, G.N. Nurick, Propagation of compaction waves in metal foams exhibiting strain hardening, International Journal of Solids and Structures Volume (2012), 49, Issues 19–20, Pages 2763-2777 https://doi.org/10.1016/j.ijsolstr.2012.03.012 V.S. Deshpande, N.A. Fleck, Isotropic constitutive models for metallic foams, Journal of the Mechanics and Physics of Solids (2000), Volume 48, Issues 6–7, Pages 1253-1283 https://doi.org/10.1016/S0022-5096(99)00082-4 U Ramamurty, A Paul, Variability in mechanical properties of a metal foam, Acta Materialia (2004), Volume 52, Issue 4, Pages 869-876 https://doi.org/10.1016/j.actamat.2003.10.021 Chitralekha Dey, A. A. Gokhale, S. N. Sahu, Investigation of shock transmission and amplification/mitigation in aluminium foams, Mechanics of Advanced Materials and Structures (2023) https://doi.org/10.1080/15376494.2023.2217159 M. Thorat, V. Menezes, A. A. Gokhale, D. Chitralekha, Shock wave mediation by closed-cell aluminium foams. Journal of Performance of Constructed Facilities (2021) Volume 35 https://doi.org/10.1061/(ASCE)CF.1943-5509.0001673 J. Hetherington, P. Smith, Blast and Ballistic Loading of Structures (1994), CRC Press https://doi.org/10.1201/9781482269277 G. Mazor, G. Ben-Dor, O. Igra, S. Sorek, Shock wave interaction with cellular materials, Part I: analytical investigation and governing equations, Shock Waves 3 (1994) , 159–165 https://doi.org/10.1007/BF01414710 G. Ben-Dor, G. Mazor, O. Igra, S. Sorek, H. Onodera, Shock wave interaction with cellular materials Part II: open cell foams; experimental and numerical results, Shock Waves 3 (1994) , 167-179 https://doi.org/10.1007/BF01414711 Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4018253","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":292357625,"identity":"3339a56b-47bb-4d10-813c-d737599ce1fc","order_by":0,"name":"Chitralekha Dey","email":"","orcid":"","institution":"Indian Institute of Technology Bombay","correspondingAuthor":false,"prefix":"","firstName":"Chitralekha","middleName":"","lastName":"Dey","suffix":""},{"id":292357626,"identity":"98282244-1a9c-461d-9225-c09be4355a50","order_by":1,"name":"Amol Anant Gokhale","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAzklEQVRIie2OsQrCMBCGTwLnEnSNtJhXuCLoIl19DaWQToKjg0OnuIiuPk5EiIsP0cnJwUkcRG11cTJ1E8y3HXwf/wF4PD8JIsAE2sgAX3e1hKCDjH2ZjDJgFeQSOdAqv1A/XdabNodZDI0g+5xEe7uLFqTGmrE6gU0AQ+NI1qkWnLZlggLQAIqhO2ld6Z7iM7lVSKRQNuBkhs+kpiskxK0KQkqi4rEejZYJd6/Mdbd1nMZSrjaH/HSO23LtWjHvVyHzz365kjkVj8fj+XsezpQyzmTQ5aEAAAAASUVORK5CYII=","orcid":"https://orcid.org/0000-0003-0345-617X","institution":"Indian Institute of Technology Bombay","correspondingAuthor":true,"prefix":"","firstName":"Amol","middleName":"Anant","lastName":"Gokhale","suffix":""}],"badges":[],"createdAt":"2024-03-05 17:56:36","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4018253/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4018253/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":54993159,"identity":"9ac45b09-bfc9-4b49-95e3-e6a90674cb7e","added_by":"auto","created_at":"2024-04-19 17:30:00","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":42071,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eFinite element model of the foam showing the loading face, end wall and boundary conditions.\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-4018253/v1/79e152c1e43619e9222b55d9.png"},{"id":54993158,"identity":"0f2a85d1-80dc-4a6a-bc80-6e1ad171aa7c","added_by":"auto","created_at":"2024-04-19 17:30:00","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":71746,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eExperimental quasi-static compression stress – strain curve for foams of different densities\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-4018253/v1/7102faf47609bd6eaa8d3a45.png"},{"id":54994292,"identity":"b2511595-9e60-4b57-a95f-a4fcebcde177","added_by":"auto","created_at":"2024-04-19 17:46:00","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":15874,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eThe nature of reflected blast pressure- time profile\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-4018253/v1/60dd98d98400e76a0d8109be.png"},{"id":54993648,"identity":"70228be8-653f-4099-827c-92f082ea1f27","added_by":"auto","created_at":"2024-04-19 17:38:00","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":76323,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eEvolution of displacement with time at the loading face and at two representative distances from the loading face for foam densities 270 kg/m\u003c/strong\u003e\u003csup\u003e\u003cstrong\u003e3\u003c/strong\u003e\u003c/sup\u003e\u003cstrong\u003e (LD) and 600 kg/m\u003c/strong\u003e\u003csup\u003e\u003cstrong\u003e3 \u003c/strong\u003e\u003c/sup\u003e\u003cstrong\u003e(HD).\u0026nbsp; x = 0 mm represents the loading face, x = 12 represents the mid-face and x = 21 mm is near the stationary face.\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-4018253/v1/c26acbee973156f5f03bb033.png"},{"id":54993646,"identity":"05dc4c46-31d9-4af1-9746-a07ba2634cab","added_by":"auto","created_at":"2024-04-19 17:38:00","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":64296,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eTypical velocity- time profiles in the foams of density 270 kg/m\u003c/strong\u003e\u003csup\u003e\u003cstrong\u003e3\u003c/strong\u003e\u003c/sup\u003e\u003cstrong\u003e (LD) and 600 kg/m\u003c/strong\u003e\u003csup\u003e\u003cstrong\u003e3\u003c/strong\u003e\u003c/sup\u003e\u003cstrong\u003e(HD)\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-4018253/v1/116d076d67e827686de2ec53.png"},{"id":54993161,"identity":"8050801e-7c07-4fd7-8a8c-80deaec2f53d","added_by":"auto","created_at":"2024-04-19 17:30:00","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":96051,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eStrain distribution along the length of (a) 270 kg/m\u003c/strong\u003e\u003csup\u003e\u003cstrong\u003e3\u003c/strong\u003e\u003c/sup\u003e\u003cstrong\u003e density foam (b) 600 kg/m\u003c/strong\u003e\u003csup\u003e\u003cstrong\u003e3\u003c/strong\u003e\u003c/sup\u003e\u003cstrong\u003e density foam\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-4018253/v1/49014c3af73f763e049faf04.png"},{"id":54993168,"identity":"7f8fe935-bd17-4a6e-af6b-257aeb568e1b","added_by":"auto","created_at":"2024-04-19 17:30:00","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":67151,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eTypical strain-time profile in foam of density 270 kg/m\u003c/strong\u003e\u003csup\u003e\u003cstrong\u003e3\u003c/strong\u003e\u003c/sup\u003e\u003cstrong\u003e and 600 kg/m\u003c/strong\u003e\u003csup\u003e\u003cstrong\u003e3\u003c/strong\u003e\u003c/sup\u003e\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-4018253/v1/f3ee5a66b54a5df46de7578a.png"},{"id":54993170,"identity":"309f5920-33e8-41cf-91f6-b71def646ec4","added_by":"auto","created_at":"2024-04-19 17:30:00","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":72719,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eSchematic diagram showing the development and progress of strain inside the low density foam\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-4018253/v1/1f3746db9af1031d4d018009.png"},{"id":54993163,"identity":"4209bb28-a53c-4740-ade0-422bd09dfe96","added_by":"auto","created_at":"2024-04-19 17:30:00","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":871881,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003e(a) Stress profile as a function of time \u0026nbsp;(b) Stress Contour at different time instants inside the low-density foam\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-4018253/v1/eb93739d13940bfa405f7eee.png"},{"id":54993165,"identity":"ff401cd3-34e1-41ed-8bb2-029d27ea46cb","added_by":"auto","created_at":"2024-04-19 17:30:00","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":320879,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003e(a) Stress profile as a function of time (b) Stress Contour at different time instants inside the high-density foam\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-4018253/v1/b077c5a20db6d220b88837c9.png"},{"id":54993167,"identity":"77d689a0-f1b1-4fbb-8265-c37d7177f78c","added_by":"auto","created_at":"2024-04-19 17:30:00","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":35777,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eStress-time profile at the end wall in the presence of foams of different densities\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-4018253/v1/ea06ed11a0fd01a8d5145b5e.png"},{"id":54993160,"identity":"aa1aeafb-bcc6-446b-ae8c-be2ed8125fb6","added_by":"auto","created_at":"2024-04-19 17:30:00","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":25639,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003ePeak stress at the end wall for 5 MPa applies peak pressure for foams of different densities\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"12.png","url":"https://assets-eu.researchsquare.com/files/rs-4018253/v1/a7e1912216395b0b15c2bfbf.png"},{"id":54993169,"identity":"bd340dea-0f3a-4282-aaed-33d8fcb82f26","added_by":"auto","created_at":"2024-04-19 17:30:00","extension":"png","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":37113,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eStress-time profile at end wall for of High Density (600 kg/m\u003c/strong\u003e\u003csup\u003e\u003cstrong\u003e3\u003c/strong\u003e\u003c/sup\u003e\u003cstrong\u003e) and Low density foam (270 kg/m\u003c/strong\u003e\u003csup\u003e\u003cstrong\u003e3\u003c/strong\u003e\u003c/sup\u003e\u003cstrong\u003e) of different lengths\u003c/strong\u003e\u0026nbsp;\u003c/p\u003e","description":"","filename":"13.png","url":"https://assets-eu.researchsquare.com/files/rs-4018253/v1/7b75d5ca9511b9dcc6782d56.png"},{"id":55973087,"identity":"2be0535b-e081-4cb4-b304-8cb39ace8066","added_by":"auto","created_at":"2024-05-07 04:18:51","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1887442,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4018253/v1/051e97ea-39b5-4dea-9832-b5a901939617.pdf"}],"financialInterests":"","formattedTitle":"Blast mitigation using monolithic closed cell aluminium foam","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eClosed-cell aluminium foams act as excellent energy-absorbing materials in the face of quasi-static and impulse loads. Under quasi-static compression, aluminium foams show three distinct regimes - at low stress the primary mechanism of elastic deformation is cell wall bending. The elastic regime is followed by the \u0026lsquo;plateau stress\u0026rsquo; region, where high strains are achieved at a constant stress called the plateau stress. This region is responsible for the high energy absorption characteristics of foam. In this region, localized plastic collapse is initiated in the cells, which then propagates in bands within the foam. The third region is the \u0026lsquo;densification region\u0026rsquo;. When all the cells collapse and the cell walls come in contact with each other, the foam behaves like the parent solid material. Densification starts when there is a steep rise in stress with a further increase in strain [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. Li et al. [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e] have investigated foams under impact loading through numerical means and observed two distinct modes in which the foams deform- the \u0026lsquo;random mode\u0026rsquo; and band front foam. The random mode occurs under low impact velocity where the shear bands of deformation are randomly oriented within the foam mass. The \u0026lsquo;band front mode\u0026rsquo; occurs under high impact velocity, where the deformation starts at the impact location and then progresses in the direction of impact. The authors concluded that strength enhancement in the random mode is derived from the strain rate sensitivity of the cell wall material, whereas in band front mode, it is derived from inertia. The response of foam when subjected to impact load is available in the literature [\u003cspan additionalcitationids=\"CR4 CR5 CR6\" citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e]. The effects of foam density on the load transfer, depth of penetration and deformation pattern under indentation impact have been studied experimentally and numerically [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. The shape and size of the individual cells which is again a function of foam density were shown to affect the way through which the indenter penetrates inside the foam. Whereas in the low-density foams, the indenter penetrates through a higher distance and there is a thin zone of damaged cells around the indenter, for the high-density foam, this depth is less and the damaged cells accumulate below the indenter.\u003c/p\u003e \u003cp\u003eHanssen et al. [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e] carried out full-scale blast experiments to study the response of foam panels. Charges were detonated at a fixed scaled distance from the foam panels which were connected to a pendulum. The swing of the pendulum was used as a measure of the energy and impulse transferred to the foam panels. Some experiments were also carried out with an aluminium cover plate on the foam panels. It was shown in the study that, the swing in the pendulum or the impulse transferred to the panel increased when foam was present. However, analytical solutions based on shock wave theory did not justify the experimental results. The authors concluded that the increase in impulse transfer occurred due to the convexity formed on the back face of the panel. Thus, the surface effects were identified as the cause of the amplification. Xue et al. [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e] carried out a numerical analysis of monolithic solid plates and sandwich plates with tetragonal truss cores of the same mass and material against blast loads. The authors have commented that, over the range of load levels examined, the maximum deflection in sandwich plates with a relative core density of 0.08 is 50\u0026ndash;85% of that in solid plate. Dharmasena et al. [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e] performed air blasts on a square honeycomb cored sandwich panel made of super-austenitic stainless-steel alloy and solid plates of the same aerial density. The tests were carried out for three impulse levels by varying the weight of the explosive while keeping the standoff distance constant. The authors reported that the sandwich panels are much more advantageous, as the back face deflections are significantly lower than the solid plates of the same mass. Khondabi et al. [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e] studied the performance of sandwich panels with aluminium face sheets and graded PU foam core with different layering arrangements against blast load. The authors tested single layer, double layer and triple layered cores. It was observed that the triple layered cores with relative densities in descending order reduced the back face sheet deflection substantially when compared with single layer panel. Jinlong et al. [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e] studied the blast performance of Fiber-Metal-Laminate (FML) double layer sandwich plates. The effects of geometrical and material parameters on the dynamic response of the panel were studied. Finally, it was reported that FML double layered sandwich plate had better anti-blast performance as compared to metal sandwich plates of equal mass.\u003c/p\u003e \u003cp\u003eXue et al. [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e] studied the blast performance of sandwich plates with three different cores \u0026ndash; pyramidal truss, square honeycomb and folded plate. Subsequently, the performance of these sandwich plates was compared with solid plates of the same material and weight. Limited optimization studies were carried out with the parameters \u0026ndash; core thickness, face sheet thickness, core member aspect ratio and relative density. The authors commented that well-designed sandwich plates can withstand significantly higher impulses as compared to solid plates of the same weight. Yazici et al. [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] studied the effect of foam infill on the blast resistance of corrugated steel core sandwich panels by experimental and numerical means. The authors studied the effect of face sheet thickness, corrugated core sheet thickness and boundary conditions (simply supported and encastre supported) on the blast performance of the panels. It is reported that the foam infill reduced the front face and back face deflections by more than 50% at 3 ms after the loading at a weight penalty of only 2.3%. However, if the face sheet thickness and corrugated sheet thickness are increased, the advantage of the foam filling reduces.\u003c/p\u003e \u003cp\u003eKaragiozova et al. [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] have carried out an experimental and numerical investigation on the behaviour of peripherally supported steel face sheet-polystyrene core and steel face sheet-aluminium honeycomb core sandwich panels subjected to blast loading. They observed that the rate at which the velocity of the free face (i.e., the loading face) of the sandwich was attenuated by the core determined the transmitted pressure and, consequently, the deflections in the unsupported parts of the back face. The authors observed that the panels with honeycomb core were more efficient than the polystyrene foam core in pressure attenuation. Liu et al. [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e] studied the response of aluminium foam cored sandwich panels subjected to blast load and compared it with stand-alone mild steel sheets. It was observed that the sandwich panels attenuated the peak blast pressure by approximately 65% as compared to stand-alone mild steel plates. Further, it was seen that the blast energy was primarily dissipated by the formation and growth of cracks in the honeycomb. It was felt that multiple reflections occurring at the interfaces between the foam cell walls and air in the foam cavities were responsible for energy dissipation. Karagiozova et al. [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e] studied blast wave attenuation in a sandwich panel consisting of a steel face sheet and Cymat aluminium foam core through analytical and numerical means. They proposed that unloading plastic waves rather than shock waves were generated in the foam, resulting in foam compaction and energy absorption. Zhu et al. [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e] reported pressure amplification under blast loading in materials such as polymeric foams and metallic foams. The authors explained that the use of such materials causes pressure amplification due to positive changes in impedance at two locations in the path of the blast wave: the air-foam interface and the foam-end wall interface. In the absence of the foam, amplification takes place only once: at the air-end wall interface. However, the authors did not comment on the contribution of energy dissipation due to densification in foams. Petel et al. [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e] carried out experimental studies on open-cell polyurethane and polyethylene foams under blast and shock loading. Experiments were carried out with an impermeable membrane placed at the mid-thickness to isolate the effect of gas filtration, if any, in the creation of precursor waves in the open cell foam. It was found that the stress signal due to the precursor wave was unchanged in magnitude and time of arrival, indicating that the precursor wave was not driven by flow filtration. Nian et al. [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e] have experimentally investigated the response of cellular concrete foams of two different crushing strengths to blast loading inside a shock tube. The authors have remarked that when the imposed load was greater than the crushing strength of the concrete foam, an elastic wave and a subsequent plastic wave was generated inside the foam. The authors noted that if the length of the foam panel was more than a critical value, blast mitigation occurred, as the plastic wave did not reach the wall at the end of the shock tube. For smaller lengths, the plastic waves reached the end wall, producing stress on the end wall which was higher than the plateau stress but less than the applied pressure. For lengths less than a threshold value, pressure amplification, associated with significant crushing of the foam occurred.\u003c/p\u003e \u003cp\u003eWhile many authors have pointed out the advantage of using cellular cores with respect to pressure attenuation and reduced deflection at the back face, others reported stress enhancement. It is also reported in the literature that, the length of the foam plays a critical role in deciding between pressure attenuation and enhancement. However, a comprehensive study of the transmission of compaction waves through blast-loaded closed-cell aluminium foams, variation of loading face velocity and their effect on pressure attenuation/amplification is not reported. Moreover, the effect of foam length and foam density on the above responses is lacking.\u003c/p\u003e \u003cp\u003eIn this work, numerical analysis of the behaviour of closed-cell aluminium foams of various densities covering plastic strengths above and below the applied stress under blast loading arising from the detonation of 1 kg TNT at a standoff distance of 1 m from the front face of the foam is studied. It is assumed that the foam panel is resting on a fixed steel plate which acts as an end wall, and the pressure at the end wall is calculated. The variation of the end wall pressure is studied as a function of foam density in the range 270 to 900 kg/m\u003csup\u003e3\u003c/sup\u003e and an analysis of the compaction waves travelling through the foam is carried out. Further, the length of the foam (termed as thickness in the literature presented above) is varied for both the low and high-density foams and the corresponding changes in the transmitted stress are studied.\u003c/p\u003e"},{"header":"2. Creation of finite element model","content":"\u003cp\u003eIn this section, details of the finite element model comprising of the general construction of the foam panel, the material model used to create foam in the finite element environment and blast loading is presented.\u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Finite Element Model\u003c/h2\u003e \u003cp\u003eCylindrical foam panel of 40 mm diameter and 25 mm length was modelled. The charge is assumed to be placed at a distance of 1 m from the centre of the loading face. The foam rests against a fixed steel plate acting as the end wall. The blast loading is incident on the front surface of the foam, herein referred to as loading face. Due to the uniform loading and the symmetry associated with the cylindrical geometry of the foam, a quarter symmetric model was prepared in Abaqus explicit Finite Element software (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). General contact was defined between all surfaces. Symmetry boundary conditions on the X and Y faces of the foam were applied. All degrees of freedom on the end wall were constrained. Since there is no plastic Poisson\u0026rsquo;s ratio in foams [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e], boundary conditions were not applied on the circumferential surface.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Material model\u003c/h2\u003e \u003cp\u003e\u0026lsquo;Crushable foam model\u0026rsquo; in Abaqus explicit software, developed on the basis of \u0026lsquo;Isotropic constitutive model\u0026rsquo; by Deshpande and Fleck [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e] was used to model foam. Quasi-static compression tests were performed on foams of various densities. The experimental stress-strain curves (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) were subsequently used in the material model. Foams exhibit a quasi-elastic deformation zone at very low stresses during which the deformation is largely elastic with small amount of local plastic deformation attributed to stress concentration at cell nodes, which can contribute significantly to the total deformation [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]. This is followed by the onset of plastic instability, indicated by the first peak in the stress-strain curve, which represents plastic collapse within bands of cells. The plateau region represents the dynamic balance between densification in the collapsed cells and cell collapse in fresh volumes of the foam. As more and more bands of collapsed and densified cells form, the entire volume of the foam gets occupied by such bands resulting in significant rise in stress. The material model was validated against total deformation [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e] and stress at the end wall [\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e] under shock tube loading of the foams.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3 Loading definition\u003c/h2\u003e \u003cp\u003eThe blast load was simulated using the CONWEP code available in Abaqus software. The code calculates the total blast pressure \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(P\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e on a surface as a function of time by using the following pressure function (Eq.\u0026nbsp;1), where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({P}_{i}\\)\u003c/span\u003e\u003c/span\u003e is the incident pressure, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({P}_{r}\\)\u003c/span\u003e\u003c/span\u003e is the reflected pressure and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\theta\\)\u003c/span\u003e\u003c/span\u003e is the angle between the normal to the loading face and the direction of the blast wave propagation. The surface on which the blast pressure is incident, the mass of the explosive, and the location of the charge with respect to the structure are specified.\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(P\\left(t\\right)={P}_{r}{cos}^{2}\\theta +{P}_{i}{(1+cos}^{2}\\theta -2cos\\theta )\\)\u003c/span\u003e \u003c/span\u003e1\u003c/p\u003e \u003cp\u003eA typical \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(P\\)\u003c/span\u003e\u003c/span\u003e versus \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(t\\)\u003c/span\u003e\u003c/span\u003e blast wave is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. On arrival of the blast wave at a certain standoff distance, the pressure rises from the ambient pressure P\u003csub\u003e0\u003c/sub\u003e to peak incident pressure P\u003csub\u003es\u003c/sub\u003e. The time duration of the wave is t\u003csub\u003ed\u003c/sub\u003e, which is the time it takes for the pressure to drop from the reflected pressure P\u003csub\u003es\u003c/sub\u003e to the ambient pressure close to the loading face. According to the literature [\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e], the detonation of 1 kg TNT from a standoff distance of 1 m from the centre of a loading face (assumed to be rigid) gives a peak reflected pressure of 5 MPa and a time duration of the wave equal to 0.5 ms.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"3. Results and Discussion","content":"\u003cp\u003eIn this section diagnostics regarding the propagation of the blast wave through the foam will be presented. This involves tracking the displacements, velocities, strains and stresses at various positions along the length of the foam as a function of time. Since the wave propagation characteristics are quite distinct for the low-and the high-density foams, these are studied and presented separately. The foam of 270 kg/m\u003csup\u003e3\u003c/sup\u003e density having lower plastic strength as compared to applied pressure is chosen to represent the low-density category and the 600 kg/m\u003csup\u003e3\u003c/sup\u003e density foam having higher plastic strength than the applied pressure, is chosen to represent the high-density category. Also, the baseline foam length is taken as 25 mm.\u003c/p\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Displacements at various locations of the foam along length with time\u003c/h2\u003e \u003cp\u003eWhen the foam is exposed to the blast wave, the loading face as well as the interior of the foam undergoes displacements towards the stationary face. The evolution of displacements at various distances from the loading face as a function of time is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e for two representative foam densities mentioned earlier. Several observations can be made from this figure. It is seen that for a given foam density, the displacement is the highest at the loading face, decreasing with increasing distance from the loading face. This is to be expected since the distal face of the foam is stationary. Secondly, the farther the location from the loading face, the longer it takes to initiate the displacement, which indicates the wave nature of the displacement. Thirdly, the displacements in the high-density foam are negligible compared to those in the low-density foam. Whereas the total deformation of the low-density foam is 9 mm (36%), it is only 0.4 mm for the high-density foam for which the plastic strength is higher than the applied pressure. A related observation is that there is no noticeable recovery of displacement in the low-density foam, implying that the displacements are associated with plastic deformation. Although there may be a small amount of displacement recovery in the high-density foam, most of it is permanent in nature, meaning that even at applied stresses below the plastic strength, there is a small amount of plastic deformation. The very small levels of total displacements seen in high-density foam preclude drawing any further inferences on the nature of displacements in high-density foam.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Material velocity along the foam length\u003c/h2\u003e \u003cp\u003eThe evolution of material velocity of the loading face (x\u0026thinsp;=\u0026thinsp;0 mm) and two representative distances from the loading face (x\u0026thinsp;=\u0026thinsp;12 and x\u0026thinsp;=\u0026thinsp;21 mm) for the low- and high-density foams is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. In all the cases, the velocity rises sharply to a peak value and then drops rapidly to zero. It takes longer for the velocity to rise as the distance from the loading face increases. Also, the peak velocity attained is the highest at the loading face, decreasing towards the stationary face. Whereas very high velocity (82 m/s) is achieved by the loading face in the low-density foam, it is much lower at 11 m/s in the high-density foam. The drop in peak velocity with increasing distance from the loading face is more in low-density foam than in high-density foam. The significant reduction in velocity with increasing distance from the loading face in the low-density foam is attributed to plastic dissipation. Finally, the time taken by the velocity at each location to drop down to zero is higher in the low-density foam (lower deceleration) as compared to the high-density one (high deceleration), which is due to the largely elastic nature of the interaction between the blast wave and the latter case.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Strain evolution along the foam length\u003c/h2\u003e \u003cp\u003eThe differential displacements at various locations in the foam result in permanent densification, and evolution of strain with time. The study of the evolving strain distribution along the length of the foam provides insight into the travel of the blast wave through the foam. Figure\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e shows the plastic strain distribution along the foam length in low- and high-density foams at the end of the loading cycle. It is seen from Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e that, for the low-density foam, the strains at different distances from the loading face are in the range of 40\u0026ndash;60%, which is within the plateau region of the stress-strain curve, whereas for the high-density foam, the strains are very low and lie in the elastic region of the stress-strain curve (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). Thus, under the blast load, the low-density foam deforms plastically within the plateau region and not entering the densification region, whereas the deformation in the high-density foam sees only elastic deformation plus a small amount of local plastic yielding referred earlier. Also, for the low-density foam, the strain marginally decreases from the loading face to the centre of the foam and then increases towards the end wall. The reason for this behaviour can be explained as follows: as the applied stress exceeds the plastic strength of the foam, plastic compaction waves are created inside the foam. As a result, energy dissipation takes place through plastic deformation, reducing the strains in the distal regions. When these waves reach the end wall and reflect from the rigid end wall, they get amplified where the reflected pressure is high, justifying the increase in strain levels near the end wall. However, for the high-density foam, the strain increases steadily from the loading face to the end wall. The behaviour can be explained thus: since the plastic strength of these foams exceed the applied load, the waves generated inside the foam are largely elastic. These elastic waves do not dissipate energy, and hence they do not attenuate during travel through the foam. Upon reflection from the end wall, the elastic wave amplifies, thus explaining the increase in the strain levels towards the end wall.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e shows the strain-time profiles for foams of density 270 kg/m\u003csup\u003e3\u003c/sup\u003e and 600 kg/m\u003csup\u003e3\u003c/sup\u003e. A sudden rise in strain is seen to occur at the loading face as the compaction wave is launched. The farther the location from the loading face, the longer it takes for the strain to rise since the compaction wave is travelling in the forward direction towards the stationary face of the foam. In the low-density foam, the time duration between the arrivals of the strain waves at the chosen locations is much greater than that in the high-density foam. In 270 kg/m\u003csup\u003e3\u003c/sup\u003e foam, the compaction wave, which is practically a plastic wave, is calculated to have a velocity of 120 m/s whereas in the high-density foam, the velocity of the comparatively faster elastic wave is calculated to be 1200 m/s.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e shows the advancement of the compaction wave inside the foam. At t\u0026thinsp;=\u0026thinsp;0 ms, the incident wave is yet to reach the front face of the foam. The foam is not compacted and no strain is developed inside the foam. At t\u0026thinsp;=\u0026thinsp;0.55 ms, the blast wave strikes the front surface of the foam. A part of the incident wave gets reflected back and a part gets transmitted inside foam. From this instant, the compaction process starts in foams. The strains start developing and progresses with time. At t\u0026thinsp;=\u0026thinsp;0.72 ms, the wave reaches the stationary end wall. Due to mismatch in impedance, it gets reflected back towards the loading face of the foam. This phenomenon leads to further reduction in the length of foam.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e3.4 Development of stress along the foam length\u003c/h2\u003e \u003cp\u003eStress evolution in the foam of density 270 kg/m\u003csup\u003e3\u003c/sup\u003e at various distances from the loading face is presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e (a). It can be seen from the figure that, the stress profile at the loading face represents the blast loading profile as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. While the peak in the applied stress occurs at 5 MPa, when the blast wave impinges on the loading face, after the subsequent deformation and densification of the foam, a reduced peak stress of 4.5 MPa is present at the loading face. The movement of the front face of the foam in the loading direction upon impingement of the blast pressure causes the reduction in stress. At x\u0026thinsp;=\u0026thinsp;12 mm, which is nearly at the mid-plane of the foam the stress profile is marked with a gradual rise to a lower peak of 2.8 MPa and then a fall. As the wave progresses to x\u0026thinsp;=\u0026thinsp;18 mm and 21 mm, the stress profiles show three peaks. The first peak has a magnitude of 1.8 MPa which is lower than the peaks seen closer to the loading face due to further plastic dissipation-induced pressure attenuation of the forward travelling compaction wave. The formation of the second and third peaks can be explained thus - progress of compaction wave to a certain location inside foam leads to local densification. Consequently, the flow stress corresponding to the densified foam increases. The stress becomes equal to either the flow strength of foam or equal to the applied stress, whichever is lower. In the current scenario, the second rise in stress is equal to the flow strength of densified foam which is approximately equal to 2.9 MPa. The third and distinct rise in stress occurs due to the reflection of the compaction wave from the stationary end wall and the ensuing impedance mismatch between the foam and the end wall. The second peak in stress is not observed near the end wall as it is masked by the higher amplitude reflected wave travelling towards the loading face of the foam. It may be noted that the third peak due to the reflection from the end wall is not observed in locations from x\u0026thinsp;=\u0026thinsp;0 to 12 mm. This is because of the following reason - whereas the total duration of the exponentially decreasing blast loading is only 0.5 ms, the wave requires approximately 0.2 ms to reach the end wall (The velocity of the compaction wave is calculated to be 120 m/s for the low-density foam). Thus, the short-duration blast loading is over before the reflected wave reaches the front end of the foam. Figure\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e (b) shows the stress contour in the low-density foam at three-time steps \u0026ndash; 0.55 ms, 0.65 ms and 0.75 ms. At 0.55 ms, barring the region near the loading face where the stress is high, the stress in the other regions of foam is approximately in the range of 2 MPa. A 0.65 ms, the stress in the entire foam is in the range of 2 MPa. At 0.75 ms, the stress near the end wall peaks to approximately 4 MPa whereas in the other regions, it is in the range of 2\u0026ndash;3 MPa.\u003c/p\u003e \u003cp\u003eStress evolution in the foam of density 600 kg/m\u003csup\u003e3\u003c/sup\u003e is presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e (a). Foam of density 600 kg/m\u003csup\u003e3\u003c/sup\u003e has a plastic strength of 10 MPa which is higher than the applied pressure. Hence, the blast wave will be propagated inside the foam as an elastic wave. The stress in the loading direction is captured at various locations designated by x\u0026thinsp;=\u0026thinsp;0, 12, 18, and 21 mm. It is seen that the stress profile at the loading face is like a blast loading profile and of 5 MPa peak pressure, which is the applied pressure. This is because the high-density foam, being stiff, does not deform and the input load is seen at the loading face. Only a peak of 8 MPa stress is observed at all subsequent locations, including the end wall. The reason for not seeing multiple peaks here unlike the low-density foam can be explained thus \u0026ndash; it is seen from the strain profile that the speed of the compaction wave in high-density foam is very high (ten times that in the low-density one). Thus, due to the high wave velocity, the forward-travelling wave is masked by the backward travelling reflected wave, and only the reflected wave from the end wall is observed. Figure\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e (b) shows the stress contour at two-time steps \u0026ndash; 0.55 ms and 0.58 ms. At 0.55 ms, the stress near the loading face of the foam is equal to the applied blast pressure, whereas, near the end wall region, the stress is very low. At 0.58 ms, an opposite scenario occurs, where the stress near the end wall region shoots up to 8 MPa and progressively reduces towards the loading face of the foam.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e3.5 Estimation of the end wall stress\u003c/h2\u003e \u003cp\u003eAs stated in the introduction, foams have been shown to reduce stress transmitted to the foam-clad protected structures under impact loading. If the foam was to be used as a sacrificial cladding on structures vulnerable to blast-initiated shock waves, it is important to know the stress transmitted by the foam to the protected structure. In this section, end wall stress is determined for (a) fixed length and varying density and (b) fixed density and varying length.\u003c/p\u003e \u003cdiv id=\"Sec12\" class=\"Section3\"\u003e \u003ch2\u003e3.5.1 Effect of varying foam density\u003c/h2\u003e \u003cp\u003eStress vs time variation at the end wall was determined for fixed foam length of 25 mm and varying foam densities from 270 kg/m\u003csup\u003e3\u003c/sup\u003e to 900 kg/m\u003csup\u003e3\u003c/sup\u003e. It can be observed from Figs.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e and \u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e that, as the foam density increases from 270 to 380 kg/m\u003csup\u003e3\u003c/sup\u003e, the peak stress reduces from 4.5 to 2.5 MPa. If the foam density is further increased, the peak stress steadily increases to 3.8 MPa for 410 kg/m\u003csup\u003e3\u003c/sup\u003e density and then further increases to 8 MPa for 500 kg/m\u003csup\u003e3\u003c/sup\u003e density. The peak stress then remains approximately constant for higher foam densities. The behaviour of the foams under blast loading can be grouped into two sets based on density \u0026ndash; the lower density foams where the average peak end wall stress remains approximately in the range of 3 MPa and the higher density foams where the average peak end wall stress remains approximately in the range of 8 MPa. Stress amplification is defined as the ratio of the end wall stress to the applied stress. In the present scenario, it can be observed that the higher density foams starting from 500 kg/m\u003csup\u003e3\u003c/sup\u003e amplify the shock while attenuation occurs for densities equal to and lower than 500 kg/m\u003csup\u003e3\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eThe generation of end wall stress is assumed to be due to \u0026ndash; (a) momentum transfer from the accelerating foam mass to the stationary end wall (b) impedance mismatch at the foam-end wall interface (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e). These two factors are considered additive in nature since they have different origins. In low density foam, material displacements are much higher and the momentum effect will be large. However, due to plastic dissipation, the incident pressure before reflection is low resulting in relatively lower reflected pressure. The net result is pressure attenuation at the end wall. In the high density foam, on the other hand, the momentum contribution will be less since the overall displacements and velocities are low. But due to lack of plastic dissipation, the incident pressure in the compaction wave remains close to the applied stress, resulting in higher contribution to the reflected stress at the end-wall. The net result is that the stress generated at the end wall is higher than the applied stress in high density foam.\u003c/p\u003e \u003cp\u003eMazor et al. [\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e, \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e] predicted that under shock tube-generated loading of elastomeric foams, the pressure enhancement at the end wall will decrease as the relative density of the foam increases. But a major difference between the cited work and the present work is that the aluminium foams deform plastically under blast load, to varying extents depending on foam density, dissipating energy causing attenuation of the stress wave till the wave reflects at the end wall. This results in a different trend in the present work.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section3\"\u003e \u003ch2\u003e3.5.2 Effect of varying foam length\u003c/h2\u003e \u003cp\u003eIn the previous section, it was observed that foams of densities equal to and greater than 500 kg/m\u003csup\u003e3\u003c/sup\u003e amplify the pressure at the end wall for 25 mm length foams. An applied pressure of 5 MPa on the loading face is amplified to 8 MPa at the end wall. However, from design point of view, it is important to understand whether the pressure amplification reduces or increases further with increasing foam length. Therefore, a broad range of foam lengths were studied for the density of 600 kg/m\u003csup\u003e3\u003c/sup\u003e. Figure\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e shows the variation of the peak end wall stress for different foam lengths. When the foam length was reduced from the original 25 mm to 15 mm, pressure amplification increased further since there is insufficient length for stress to attenuate to any appreciable level before reflection at the end wall (Fig.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e). When the foam length was increased above 25 mm, amplification at the end wall reduced. Finally, attenuation was achieved for a foam length of 200 mm. With very small local plastic deformation taking place in the high density foam, it is unable to cause any attenuation in the forward traveling stress wave, a much longer foam can be expected to cause attenuation in the stress wave before reflection, reducing the overall contribution to the stress generated at the end wall.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e also shows the variation of end wall peak stress with variation in foam length for a foam density of 270 kg/m\u003csup\u003e3\u003c/sup\u003e. The trend in stress variation is similar to that for high density foam i.e., the peak stress reduces with increasing foam length. The lower density foam, however, can be of much shorter length to achieve pressure attenuation which is not the case in high density foam which requires a minimum of 200 mm length to achieve pressure attenuation, the difference attributed to plastic dissipation in low density foam which reduces incident stress before reflection from the end wall.\u003c/p\u003e \u003cp\u003eThus, the comparative diagnostic analysis of the blast-foam interaction shows that the low- and high-density foams have distinctive strain distributions along the length of the foam. In the low-density foam (270 kg/m\u003csup\u003e3\u003c/sup\u003e) the strain decreases towards the mid-length of the foam sample and then increases towards the end wall, while the high-density foam (600 kg/m\u003csup\u003e3\u003c/sup\u003e) shows a gradually increasing strain from the loading face to the end wall. The end wall stress under blast loading remains approximately constant as the density increases from 270 to 380 kg/m\u003csup\u003e3\u003c/sup\u003e. When density is further increased, the peak stress increases up to 500 kg/m\u003csup\u003e3\u003c/sup\u003e and then plateaus off with a further increase in foam density. The stress at the end wall is postulated to be due to two components \u0026ndash; transfer of momentum of the foam undergoing displacement to the stationary end wall under the blast pressure and stress increase associated with reflection of the incident stress wave at the higher impedance end wall. In low density foams, the momentum effect associated with large displacements seems to dominate while in high density foams, the wave reflection at the higher impedance end wall seems to dominate in determining the stress at the end wall.\u003c/p\u003e \u003cp\u003eStudy is also carried out by varying the length of the foam with an aim to reduce the stress at the end wall. A length of 20 mm gives considerable stress attenuation in low density foams while a much larger length of 200 mm is required to give stress attenuation in high density foams.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e"},{"header":"4 Summary","content":"\u003cp\u003eStress attenuation using closed cell aluminium foam is demonstrated numerically, using a continuum based finite element model of foam, and representing the blast load using CONWEP code. Stress calculations at the back face were based on profiling the stress waves, strain waves and displacements occurring in the foam under the applied blast pressure. Under a blast-triggered incident shock, plastic compaction waves are transmitted through foams whose plastic strength is less than the applied pressure. The peak stress in the transmitted wave diminishes with travel due to energy dissipation associated with the densification of the cells. The stress generated at the end-wall is postulated to be composed of a) momentum transfer to the stationary end wall due to stagnation of large displacements in the foam associated with foam densification, which is prominent in low density foams, and b) increase in stress on reflection of the incident stress wave from the higher impedance end wall, which is prominent in high density foam due to lack of plastic dissipation in the incident wave. Lower foam density e.g., 270 kg/m\u003csup\u003e3\u003c/sup\u003e at 20 mm length gives substantial stress attenuation for applied peak pressure of 5 MPa with no further gain by increasing the foam length. Higher density foams give stress amplification; the foam length needs to be increased to 200 mm for any attenuation to be observed.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eL. Gibson, Cellular Solids (2003), MRS Bulletin, 28(4), 270-274 https://doi.org/10.1557/mrs2003.79 \u003c/li\u003e\n\u003cli\u003eL. Li, P. Xue, G. Luo, A numerical study on deformation mode and strength enhancement of metal foam under dynamic loading, Materials and Design (2016) https://doi.org/10.1016/j.matdes.2016.07.123\u003c/li\u003e\n\u003cli\u003eC. Dey, S. N. Sahu, K. Akella, A. A. Gokhale, Numerical Prediction of Quasi-Static Compression, Indentation Impact and Shock Loading Behaviour of Aluminium Foam Using Idealized Cell Geometry, Journal of dynamic behaviour of materials (2022) Volume 8, Pages 324-339 https://doi.org/10.1007/s40870-022-00336-9 \u003c/li\u003e\n\u003cli\u003eX. pang, H. Du, Dynamic crushing of aluminium foams under impact crushing, Composites Part B: Engineering (2017) https://10.1016/j.compositesb.2016.12.044 \u003c/li\u003e\n\u003cli\u003eM. Peroni, G. Solomos, V. Pizzinato, Impact behaviour testing of aluminium foam, International Journal of Impact Engineering (2013) https://doi.org/10.1016/j.ijimpeng.2012.07.002\u003c/li\u003e\n\u003cli\u003eM. A. Kader, M. A. Islam, P. J. Hazell, J. P. Escobedo, M. Sadatfar, A. D. Brown, G. J. Appleby-Thomas, Modelling and characterization of cell collapse in aluminium foams during dynamic loading, International Journal of Impact Engineering (2016) https://doi.org/10.1016/j.ijimpeng.2016.05.020\u003c/li\u003e\n\u003cli\u003eG. Lu, J. Shen, W. Hou, D. Ruan, L. S. Ong, Dynamic indentation and penetration of aluminium foams, International Journal of Mechanical Sciences (2008) https://doi.org/10.1016/j.ijmecsci.2007.09.006\u003c/li\u003e\n\u003cli\u003eA.G. Hanssen, L. Enstock, M. Langseth, Close-range blast loading of aluminium foam panels, International Journal of Impact Engineering (2002), Volume 27, Issue 6, Pages 593-618\u003c/li\u003e\n\u003cli\u003ehttps://doi.org/10.1016/S0734-743X(01)00155-5\u003c/li\u003e\n\u003cli\u003eZhenyu Xue, John W. Hutchinson, Preliminary assessment of sandwich plates subject to blast loads, International Journal of Mechanical Sciences (2003), Volume 45, Issue 4, 687-705 \u003c/li\u003e\n\u003cli\u003ehttps://doi.org/10.1016/S0020-7403(03)00108-5. \u003c/li\u003e\n\u003cli\u003eKumar P. Dharmasena, Haydn N.G. Wadley, Zhenyu Xue, John W. Hutchinson, Mechanical response of metallic honeycomb sandwich panel structures to high-intensity dynamic loading, International Journal of Impact Engineering (2008), Volume 35, Issue 9, 1063-1074 https://doi.org/10.1016/j.ijimpeng.2007.06.008. \u003c/li\u003e\n\u003cli\u003eKhondabi, R., Khodarahmi, H., Hosseini, R. et al. Dynamic plastic response of sandwich structures with graded polyurethane foam cores and metallic face sheets exposed to uniform blast loading: experimental study and numerical simulation, J Braz. Soc. Mech. Sci. Eng.(2023) 45, 526. https://doi.org/10.1007/s40430-023-04384-7\u003c/li\u003e\n\u003cli\u003eDu, J., Su, H., Bai, J. et al. On dynamic response of double-layer rectangular sandwich plates with FML face-sheets and metal foam cores under blast loading. J Braz. Soc. Mech. Sci. Eng. 45, 31 (2023). https://doi.org/10.1007/s40430-022-03956-3\u003c/li\u003e\n\u003cli\u003eZhenyu Xue, John W. Hutchinson, A comparative study of impulse-resistant metal sandwich plates, International Journal of Impact Engineering (2004), Volume 30, Issue 10, 1283-1305 https://doi.org/10.1016/j.ijimpeng.2003.08.007. \u003c/li\u003e\n\u003cli\u003eMurat Yazici, Jefferson Wright, Damien Bertin, Arun Shukla, Experimental and numerical study of foam filled corrugated core steel sandwich structures subjected to blast loading, Composite Structures (2014), Volume 110, 98-109 https://doi.org/10.1016/j.compstruct.2013.11.016. \u003c/li\u003e\n\u003cli\u003eD. Karagiozova, G.N. Nurick, G.S. Langdon, S. Chung Kim Yuen, Y. Chi, S. Bartle, Response of flexible sandwich-type panels to blast loading, Composites Science and Technology (2009), Volume 69, Issue 6, 754-763 https://doi.org/10.1016/j.compscitech.2007.12.005. \u003c/li\u003e\n\u003cli\u003eH. Liu, Z.K. Cao, G.C. Yao, H.J. Luo, G.Y. Zu, Performance of aluminum foam\u0026ndash;steel panel sandwich composites subjected to blast loading, Materials \u0026amp; Design (2013), Volume 47, 483-488 https://doi.org/10.1016/j.matdes.2012.12.003. \u003c/li\u003e\n\u003cli\u003eD. Karagiozova, G.S. Langdon, G.N. Nurick, Blast attenuation in Cymat foam core sacrificial claddings, International Journal of Mechanical Sciences (2010), Volume 52, Issue 5, 758-776 https://doi.org/10.1016/j.ijmecsci.2010.02.002. \u003c/li\u003e\n\u003cli\u003eFeng Zhu, Clifford C. Chou, King H. Yang, Shock enhancement effect of lightweight composite structures and materials, Composites Part B: Engineering (2011), Volume 42, Issue 5, 2011, 1202-1211 https://doi.org/10.1016/j.compositesb.2011.02.014. \u003c/li\u003e\n\u003cli\u003eO.E. Petel, S. Ouellet, Higgins, D. L. Frost, The elastic\u0026ndash;plastic behaviour of foam under shock loading, Shock Waves (2013), 23,\u003cstrong\u003e \u003c/strong\u003e55\u0026ndash;67 https://doi.org/10.1007/s00193-012-0414-7 \u003c/li\u003e\n\u003cli\u003eWeimin Nian, Kolluru V.L. Subramaniam, Yiannis Andreopoulos, Experimental investigation on blast response of cellular concrete, International Journal of Impact Engineering (2016), Volume 96, Pages 105-115 https://doi.org/10.1016/j.ijimpeng.2016.05.021 \u003c/li\u003e\n\u003cli\u003eD. Karagiozova, G.S. Langdon, G.N. Nurick, Propagation of compaction waves in metal foams exhibiting strain hardening, International Journal of Solids and Structures Volume (2012), 49, Issues 19\u0026ndash;20, Pages 2763-2777\u003c/li\u003e\n\u003cli\u003ehttps://doi.org/10.1016/j.ijsolstr.2012.03.012 \u003c/li\u003e\n\u003cli\u003eV.S. Deshpande, N.A. Fleck, Isotropic constitutive models for metallic foams, Journal of the Mechanics and Physics of Solids (2000), Volume 48, Issues 6\u0026ndash;7, Pages 1253-1283 https://doi.org/10.1016/S0022-5096(99)00082-4\u003c/li\u003e\n\u003cli\u003eU Ramamurty, A Paul, Variability in mechanical properties of a metal foam, Acta Materialia (2004), Volume 52, Issue 4, Pages 869-876 https://doi.org/10.1016/j.actamat.2003.10.021 \u003c/li\u003e\n\u003cli\u003eChitralekha Dey, A. A. Gokhale, S. N. Sahu, Investigation of shock transmission and amplification/mitigation in aluminium foams, Mechanics of Advanced Materials and Structures (2023) https://doi.org/10.1080/15376494.2023.2217159 \u003c/li\u003e\n\u003cli\u003eM. Thorat, V. Menezes, A. A. Gokhale, D. Chitralekha, Shock wave mediation by closed-cell aluminium foams. Journal of Performance of Constructed Facilities (2021) Volume 35 https://doi.org/10.1061/(ASCE)CF.1943-5509.0001673 \u003c/li\u003e\n\u003cli\u003eJ. Hetherington, P. Smith, Blast and Ballistic Loading of Structures (1994), CRC Press https://doi.org/10.1201/9781482269277 \u003c/li\u003e\n\u003cli\u003eG. Mazor, G. Ben-Dor, O. Igra, S. Sorek, Shock wave interaction with cellular materials, Part I: analytical investigation and governing equations, Shock Waves 3 (1994)\u003cstrong\u003e, \u003c/strong\u003e159\u0026ndash;165 https://doi.org/10.1007/BF01414710 \u003c/li\u003e\n\u003cli\u003eG. Ben-Dor, G. Mazor, O. Igra, S. Sorek, H. Onodera, Shock wave interaction with cellular materials Part II: open cell foams; experimental and numerical results, Shock Waves 3 (1994)\u003cstrong\u003e, \u003c/strong\u003e167-179 https://doi.org/10.1007/BF01414711 \u003c/li\u003e\n\u003c/ol\u003e "}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Blast, foam, compaction wave, reflected pressure, end wall stress, length, density","lastPublishedDoi":"10.21203/rs.3.rs-4018253/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4018253/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eBlast protection using cellular materials is being actively pursued at research and technology levels. The present work uniquely demonstrates generation of stress waves, strain waves and mass velocities in monolithic closed cell aluminium foams of different densities and lengths, subjected to simulated blast loads, and their combined effect on blast attenuation. The foams were assumed to be resting against a rigid end wall. If the numerically calculated stress at the back face was found less than the applied stress at the front face, the interaction was termed as blast mitigation or attenuation. The results show, \u0026lsquo;pressure mitigation\u0026rsquo; to occur for low-density foams whose plastic strength is less than the applied pressure, but pressure amplification for high-density foams whose plastic strength is higher than the applied pressure. The pressure amplification observed in shorter length high density foams transformed to pressure mitigation if the foams were sufficiently long. Based on these results and other stress, strain and velocity related diagnostics, the underlying mechanism behind blast wave amplification/mitigation and its relation with foam density and length is proposed.\u003c/p\u003e","manuscriptTitle":"Blast mitigation using monolithic closed cell aluminium foam","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-04-19 17:29:55","doi":"10.21203/rs.3.rs-4018253/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"84ad7e9d-aeea-4fe3-924e-1e5d37d8010e","owner":[],"postedDate":"April 19th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-05-07T04:02:25+00:00","versionOfRecord":[],"versionCreatedAt":"2024-04-19 17:29:55","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4018253","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4018253","identity":"rs-4018253","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}